2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 8doi: 10.28924/ada/ma.2.8 stability of positive weak solution for generalized weighted p-fisher-kolmogoroff nonlinear stationary-state problem salah a. khafagy∗, hassan m. serag department of mathematics, faculty of science, al-azhar university, nasr city (11884), cairo, egypt salahabdelnaby.211@azhar.edu.eg, serraghm@yahoo.com ∗correspondence: salahabdelnaby.211@azhar.edu.eg abstract. in the present paper, we investigate the stability results of positive weak solution for thegeneralized fisher–kolmogoroff nonlinear stationary-state problem involving weighted p-laplacianoperator −d∆p,pu = ka(x)u[ν − υu] in ω, bu = 0 on ∂ω, where ∆p,p with p > 1 and p = p (x)is a weight function, denotes the weighted p-laplacian defined by ∆p,pu ≡ div [p (x)|∇u|p−2∇u],the continuous function a(x) : ω → r satisfies either a(x) > 0 or a(x) < 0 for all x ∈ ω, d, k, νand υ are positive parameters and ω ⊂ rn is a bounded domain with smooth boundary bu = δh(x)u + (1− δ) ∂u ∂n where δ ∈ [0, 1], h : ∂ω→ r+ with h = 1 when δ = 1. 1. introduction: in this paper we study the stability results of positive weak solution for the generalized weighted p-fisher–kolmogoroff nonlinear stationary-state problem −d∆p,pu = ka(x)f (u) = ka(x)u[ν − υu] in ω, bu = 0 on ∂ω, } (1.1) where ∆p,p with p > 1 and p = p (x) is a weight function, denotes the weighted p-laplaciandefined by ∆p,pu ≡ div [p (x)|∇u|p−2∇u] (see for details [6]), the continuous function a(x) : ω→ rsatisfies either a(x) > 0 or a(x) < 0 for all x ∈ ω, d, k, ν, ν and υ are positive parameter and ω ⊂ rn is a bounded domain with smooth boundary bu = δh(x)u + (1 − δ)∂u∂n where δ ∈ [0, 1], h : ∂ω → r+ with h = 1 when δ = 1. system (1.1) is the generalized weighted p-fisher–kolmogoroff nonlinear stationary-state problem [21], where d is the diffusion coefficient, k is theis the linear reproduction rate and u is the population density. situations where d is space-dependent are arising in more and more modelling situations of biomedical importance from diffusionof genetically engineered organisms in heterogeneous environments to the effect of white and grey received: 12 sep 2021. key words and phrases. stability; weak solution; p-laplacian.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 2matter in the growth and spread of brain tumours. problem (1.1) arises from the population biologyof one species.systems of type (1.1) have received considerable attention in the last decade (see, e.g., [18,19,24]and the references therein). it has been shown that for some certian values of ν, υ, system (1.1)has a rich mathematical structure. in [8, 23] the system (1.1) is considered under the hypothesis p (x) = (k/d) = 1, p = 2 and f (u) = u.this corresponds to the emden-fowler stationary-stateproblem of polytropic index of order one. while in [9, 19], system (1.1) is considered under thehypothesis p (x) = (k/d) = 1, p = 2 and f (u) = u − u2,where u is the population denistyof degree two.this corresponds to the logestic nonlinear stationary-state problem. due to theappearance of weighted p-laplacian operator in (1.1) and the particular cases; the extensions arechallenging and nontrivial.many authors are interested in the study of stability and instability of nonnegative solutions oflinear [2] , semilinear (see [10,26]), semiposiotne (see [3,25]), nonlinear (see [1,16]) and singular (see[17]) systems, due to the great number of applications in reaction-diffusion problems, in autocatalyticreaction, in temperature on plasma, population dynamics, etc.; see [4, 23] and references therein.also, in the recent past, many authors devoted their attention to study the weighted p-laplaciannonlinear systems (see [11,12,14,15]).tertikas in [25] have been proved the stability and instability results of positive solutions for thesemilinear system −∆u = λf (u) in ω, bu = 0 on ∂ω,under various choices of the function f . in [3], the authors have been studied the uniqueness andstability of nonnegative solutions for classes of nonlinear elliptic dirichlet problems in a ball, whenthe nonlinearity is monotone, negative at the origin, and either concave or convex. in the case p (x) = a(x) = 1, p = 2 and a function λf (u) instead of λuα +uβ, system (1.1) have been studiedby several authors (see [5, 7, 20]).khafagy in [13] have been studied the stability and instability of positive weak solution for thenonlinear system −∆p,pu + a(x)|u|p−2u = λb(x)uα in ω, bu = 0 on ∂ω. } (1.2) where 0 < α < p − 1. he proved that if 0 < α < p − 1 and b(x) > 0(< 0) for all x ∈ ω, thenevery positive weak solution u of (1.2) is linearly stable (unstable) respectively. definition 1.1. we recall that, if u be any positive weak solution of (1.1), then the linearizedequation of (1.1) about u is given by −(p − 1)div [p (x)|∇u|p−2∇φ]− (k/d)a(x)[ν − 2υu]φ = µφ, x ∈ ω, bφ = 0, x ∈ ∂ω, } (1.3) where µ is the eigenvalue corresponding to the eigenfunction φ. https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 3 definition 1.2. [3] a solution u of (1.1) is called stable solution if all eigenvalues of (1.3) arestrictly positive, which can be implied if the principal eigenvalue µ1 > 0. otherwise u unstable. 2. main results the main goal of this section is to prove the stability and instability of the positive weak solution u of (1.1). our main results are formulate in the following theorems. theorem 2.1. if α+ 1 < p < β + 1 and a(x) > 0 for all x ∈ ω, then every positive weak solution of ( 1.1) is linearly stable. proof. let u0 be any positive weak solution of (1.1), then the linearized equation bout u0 is −(p − 1)div [p (x)|∇u0|p−2∇φ]− (k/d)a(x)[ν − 2υu0]φ = µφ, x ∈ ω bφ = 0, x ∈ ∂ω. } (2.1) let µ1 be the first eigenvalue of (2.1) and let ψ(x) ≥ 0 be the corresponding eigenfunction.multiplying (1.1) by ψ and integrating over ω, we have − ∫ ω ψdiv [p (x)|∇u0|p−2∇u0]dx = (k/d) ∫ ω a(x)[νu0 − υu2 0 ]ψdx. (2.2) the first term of the l.h.s. of (2.2) may be written in the form∫ ω ψdiv [p (x)|∇u0|p−2∇u0]dx = ∫ ω ψ∇u0∇[p (x)|∇u0|p−2]dx + ∫ ω ψ[p (x)|∇u0|p−2]div(∇u0)dx. applying green’s first identity, we have∫ ω ψdiv [p (x)|∇u0|p−2∇u0]dx = ∫ ω ψ∇u0∇[p (x)|∇u0|p−2]dx − ∫ ω ∇[ψ(p (x)|∇u0|p−2)∇u0dx + ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds, = − ∫ ω ∇ψ[p (x)|∇u0|p−2]∇u0dx + ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds. (2.3) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 4from (2.3) in (2.2), we have (k/d) ∫ ω a(x)[νu0 − υu2 0 ]ψdx = ∫ ω ∇ψ[p (x)|∇u0|p−2]∇u0dx − ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds + ∫ ω a(x)ψ|u0|p−2u0]dx. (2.4) also, multiplying (2.1) by (−u0) and integrating over ω, we have −µ1 ∫ ω u0ψdx = (p − 1) ∫ ω u0div [p (x)|∇u0|p−2∇ψ]dx −(p − 1) ∫ ω u0a(x)|u0|p−2ψ +λ ∫ ω a(x)[ν − 2υu0]ψdx. (2.5) the first term of the l.h.s. of (2.5) may be written in the form∫ ω u0div [p (x)|∇u0|p−2∇ψ]dx = ∫ ω u0[p (x)|∇u0|p−2]∇ · ∇ψdx + ∫ ω u0∇ψ∇[p (x)|∇u0|p−2]dx. using green’s first identity, one have∫ ω u0div [p (x)|∇u0|p−2∇ψ]dx = − ∫ ω ∇[u0p (x)|∇u0|p−2]∇ψ + ∫ ω u0∇[p (x)|∇u0|p−2]∇ψdx + ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds, = − ∫ ω [p (x)|∇u0|p−2]∇u0∇ψ + ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds. (2.6) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 5from (2.6) in (2.5) we have −µ1 ∫ ω u0ψdx = (p − 1)[ ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds − ∫ ω [p (x)|∇u0|p−2]∇u0∇ψ] +(k/d) ∫ ω a(x)[νu0 − 2υu2 0 ]ψdx. (2.7) multiplying (2.4) by (p − 1) and adding with (2.7), we have −µ1 ∫ ω u0ψdx = (p − 1)[ ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds − ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds] +(k/d) ∫ ω a(x)[νu0 − 2υu2 0 ]ψdx −(p − 1)(k/d) ∫ ω a(x)[νu0 − υu2 0 ]ψdx. hence −µ1 ∫ ω u0ψdx = (p − 1) ∫ ∂ω [p (x)|∇u0|p−2][u0 ∂ψ ∂n − ψ ∂u0 ∂n ]ds +(k/d) ∫ ω a(x)νu0[1− (p − 1)]ψdx +(k/d) ∫ ω a(x)υu2 0 [(p − 1)− 2]ψdx. (2.8) now, when δ = 1, we have bu0 = u0 = 0 for s ∈ ∂ω and also we have ψ = 0 for s ∈ ∂ω. then∫ ∂ω [p (x)|∇u0|p−2][u0 ∂ψ ∂n − ψ ∂u0 ∂n ]ds = 0. (2.9) also, when δ 6= 1, we have ∂u0 ∂n = − δhu0 1− δ and ∂ψ ∂n = − δhψ 1− δ ,which implies again the result given by (2.9).hence − µ1 ∫ ω u0ψdx = (k/d) ∫ ω a(x)[νu0[2− p] + υu2 0 [p − 3]]ψdx. (2.10) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 6since 2 < p < 3 and a(x) > 0 for all x , then (2.10) becomes − µ1 ∫ ω u0ψdx < 0, (2.11) so µ1 > 0 and the result follows. � theorem 2.2. if 2 < p < 3 and a(x) < 0 for all x ∈ ω, then every positive weak solution of (1.1) is unstable. proof. as in the proof of theorem 1., we have − µ1 ∫ ω u0ψdx > 0, (2.12) so µ1 < 0 and the result follows. � 3. applications and related results here we introduce some examples to demonstrate the effectiveness of our results. example 3.1. consider the emden-fowler steady-state problem of polytropic index of order one [8], −∆u = λa(x)u in ω, bu = 0 on ∂ω, } (3.1) with a(x) > 0 for all x ∈ ω.here p (x) = 1, (k/d) = λ, p = 2. then according to theorem 1., every positive weak solution of (3.1) is unstable. example 3.2. consider the population denisty steady-state problem of degree two [19], −∆pu = λa(x)[u − u2] in ω, bu = 0 on ∂ω, } (3.2) with a(x) > 0 for all x ∈ ω.hence, according to theorem 1., every positive weak solution of (3.1) is stable. example 3.3. consider the chemotaxis steady-state problem of degree two [9, 19], −∆pu = λa(x)[−u + u2] in ω, bu = 0 on ∂ω, } (3.3) with a(x) > 0 for all x ∈ ω.hence, according to theorem 1., every positive weak solution of (3.1) is unstable. https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 7references [1] g. afrouzi, s. rasouli, stability properties of non-negative solutions to a non-autonomous p-laplacian equation,chaos solitons fractals. 29 (2006) 1095-1099. https://doi.org/10.1016/j.chaos.2005.08.165.[2] g. afrouzi, z. sadeeghi, stability results for a class of elliptic problems, int. j. nonlinear sci. 6 (2008) 114-117. http://www.internonlinearscience.org/upload/papers/20110307063941556.pdf.[3] i. ali, a. castro, r, shivaji, uniqueness and stability of nonnegative solutions for semipositone problems in a ball,proc. amer. math. soc. 117 (1993) 775-782. 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z. zhitao, lectures on solution set of semilinear elliptic equations (in tokyo metropolitan university), 2005.[24] s. shabani, rokn-e-vafa and t. h. tehrani, diffusive logistic equations with harvesting and heterogeneity understrong growth rate, adv. nonlinear anal. 8 (2019) 455-467. https://doi.org/10.1515/anona-2016-0208.[25] a. tertikas, stability and instability of positive solutions of semilinear problems, proc. amer. math. soc. 114 (1992)1035-1040. https://doi.org/10.1090/s0002-9939-1992-1092928-2.[26] i. voros, stability properties of nonnegative solutions of semilinear symmetric cooperative systems, electronic j. diff.eqn. 105 (2004) 1-6. https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf. https://doi.org/10.28924/ada/ma.2.8 https://doi.org/10.1090/s0002-9947-02-03005-2 https://doi.org/10.1515/anona-2016-0208 https://doi.org/10.1090/s0002-9939-1992-1092928-2 https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf 1. introduction: 2. main results 3. applications and related results references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 12doi: 10.28924/ada/ma.2.12 some investigations on a class of analytic and univalent functions involving q-differentiation ayotunde olajide lasode∗ , timothy oloyede opoola department of mathematics, faculty of physical sciences, university of ilorin, ilorin, nigeria lasode_ayo@yahoo.com, opoola.to@unilorin.edu.ng ∗correspondence: lasode_ayo@yahoo.com abstract. we use the concept of q-differentiation to define a class eq(β, δ) of analytic and univalentfunctions. the investigations thereafter includes coefficient estimates, inclusion property and someconditions for membership of some analytic functions to be in the class eq(β, δ). our results generalizesome known and new ones. 1. introduction and definitions we let ud = {z : z ∈ c, |z | < 1} represent the unit disk and a represent the class ofnormalized analytic functions of the form f (z) = z + ∞∑ m=2 amz m, z ∈ ud (1) where f (0) = 0 = f ′(0) − 1. also, let s represent a subset of a containing functions univalentin ud. a function f in s is a member of class bt (δ) of bounded turning functions of order δ if itsatisfies the geometric condition ref ′(z) > δ ∈ [0, 1), z ∈ ud. let bt (0) = bt represent the class of bounded turning functions. it is known (see [1]) that f ∈ btare univalent functions. also, a function f in s is a member of class cv(δ) of convex functions oforder δ if it satisfies the geometric condition re ( z f ′′(z) f ′(z) + 1 ) > δ ∈ [0, 1), z ∈ ud. let cv(0) = cv represent the class of convex functions.the importance of operators in geometric function theory cannot be underrated. for instancesee [2, 13,15] for some known ones.in 1908, jackson [7] (see also [3, 4, 8–11]) initiated the concept of q-calculus as follows. received: 21 jan 2022. key words and phrases. analytic functions; carathéodory functions; univalent functions; bounded turning function;coefficient bound; inclusion property and q-calculus. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.12 https://orcid.org/0000-0002-2657-7698 eur. j. math. anal. 10.28924/ada/ma.2.12 2 definition 1.1. for q ∈ (0, 1), the q-differentiation of function f ∈ a is defined by dqf (0) = f ′(0), dqf (z) = f (z)− f (qz) z(1− q) (z 6= 0) and d2qf (z) = dq(dqf (z)). (2) obviously, applying (2) in (1) gives us dqf (z) = 1 + ∞∑ m=2 [m]qamz m−1 and zd2qf (z) = ∞∑ m=2 [m − 1]q[m]qamzm−1 (3) where [m]q = 1−qm 1−q and lim q↑1 [m]q = m.for example if f (z) = zm, then by using (2), dqf (z) = dq(zm) = 1− qm 1− q z m−1 = [m]qz m−1 and observe that lim q↑1 dqf (z) = lim q↑1 ( [m]qz m−1) = mzm−1 = f ′(z) where f ′(z) is the classical differentiation.in this work, the q-differential operator was used to define a class of analytic functions andgeneralize some results. 2. relevant lemmas we represent by p the well-known class of analytic functions of the form p(z) = 1 + ∞∑ m=1 cmz m, re p(z) > 0, z ∈ ud (4) and by p(δ) ⊆ p(0) = p the class whose members are of the form pδ(z) = 1 + ∞∑ m=1 (1− δ)cmzm, re p(z) > δ ∈ [0, 1), z ∈ ud. (5) the following lemmas shall be required to proof our results. lemma 2.1 ( [14]). let g(z) = ∞∑ m=1 amz m ≺ g(z) = ∞∑ m=1 bmz m, z ∈ ud where g(z) is univalent in ud and g(ud) is a convex domain, then |am| ≤ |b1|, m ∈ n. equality holds for the function g(z) = g(τzm), |τ | = 1. the lemmas that follow are the q-analogous versions of the original ones as referenced. lemma 2.2 ( [6]). let p(z) be analytic in ud such that p(0) = 1. if re ( zdq(p(z)) p(z) + 1 ) > 3δ − 1 2δ , z ∈ ud, then for α = (δ − 1)/δ (δ ∈ [1/2, 1)), re p(z) > 2α. the constant 2α is the best possible. lemma 2.3 ( [5]). let u = u1+u2i and v = v1+v2i such that γ(u, v) : c2 −→ c is a complex-valued function such that https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 3(1) γ(u, v) is continuous in π ⊂ c2,(2) (1, 0) ∈ π and re(γ(1, 0)) > 0 and(3) re(γ(ξ + (1− ξ)u2i , v1)) ≤ ξ (0 ≤ ξ < 1)) if (ξ + (1− ξ)u2i , v1) ∈ π and v1 ≤ −12(1−ξ)(1+u 2 2) and re(γ(ξ+(1−ξ)u2i , v1)) ≥ ξ (ξ > 1) if (ξ+(1−ξ)u2i , v1) ∈ π and v1 ≥ 1 2(1− ξ)(1 + u 2 2). if p(z) ∈ p for (p(z), zdqp(z)) ∈ π and re(γ(p(z), zdqp(z))) > ξ, z ∈ ud, then rep(z) > ξ in ud. 3. main results the definition of the investigated class is as follows.a function f (z) ∈ a is a member of the class eq(β, δ) if the condition re ( dqf (z) + 1 + e iβ 2 zd2qf (z) ) > δ, δ ∈ [0, 1), β ∈ (−π, π], z ∈ ud (6) holds.when parameters in (6) are varied, the class eq(β, δ) reduces to some well-known classes ofanalytic functions that have been studied by some authors. these are cited in our corollaries andremarks.the following are the proved results. theorem 3.1. let β ∈ (−π, π] and δ ∈ [0, 1), if condition (6) holds, then eq(β, δ) ⊂ bt q(δ). bt q(δ) is the class of q-bounded turning function of order δ. proof. let p(z) = dqf (z) so that dqp(z) = d2qf (z) and for κ = (1 + e iβ)/2, then (6) can beexpressed as re(p(z) + κzdqp(z)) > δ. (7) in view of the conditions in lemma 2.3 and for p(z) in (7), we define the function γ(u, ν) = u + κν on the domain π of c2, then(i) clearly, γ(u, ν) satisfies the condition (1) in lemma 2.3,(ii) for (1, 0) ∈ π, γ(1, 0) = 1 =⇒ re(γ(1, 0)) > 0 and(iii) γ(δ + (1− δ)u2i , ν1) = δ + 1+cos δ2 ν1 + ( (1− δ)u2 + sin δ2 ν1 ) i , thus, re(γ(δ + (1− δ)u2i , ν1)) = δ + 1 + cosβ 2 ν1 ≤ δ for ν1 ≤ −12(1− δ)(1 + u22). https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 4now since γ(u, ν) satisfies all the conditions (1− 3) in lemma 2.3, then it implies that rep(z) = re(dqf (z)) > δ, z ∈ ud hence the proof is complete. � corollary 3.2 ( [1]). since class bt q(δ) is well-known to consist of univalent functions, then eq(β, δ) ⊂ bt q(δ) consists of univalent functions. corollary 3.3. lim q↑1 eq(β, δ) ⊂ bt (δ), z ∈ ud. theorem 3.4. if f ∈ a is such that re ( zdq(dqf (z) + κzd2qf (z)) dqf (z) + κzd2qf (z) ) > δ − 1 2δ , (8) then re(dqf (z) + κzd2qf (z)) > 2(δ−1)/δ, δ ∈ [1/2, 1), z ∈ ud and κ = (1 + e iβ)/2. proof. from (6), let p(z) = dqf (z) + κzd2qf (z), then by logarithmic q-differentiation we obtain zdqp(z) p(z) + 1 = zdq(dqf (z) + κzd2qf (z)) dqf (z) + κzd2qf (z) + 1. now applying lemma 2.2 gives re ( zdqp(z) p(z) + 1 ) = re ( zdq(dqf (z) + κzd2qf (z)) dqf (z) + κzd2qf (z) + 1 ) > 3δ − 1 2δ implies that re ( zdq(dqf (z) + κzd2qf (z)) dqf (z) + κzd2qf (z) ) > δ − 1 2δ and by the same lemma 2.2 the proof in complete. � corollary 3.5. if f ∈ a satisfies condition (8), then f ∈ eq(β, 2(δ−1)/δ). corollary 3.6. if f ∈ lim q↑1 eq(β, 1/2) is such that re ( z(1 + κ)f ′′(z) + κz2f ′′′(z) f ′(z) + κzf ′′(z) ) > − 1 2 , then re(f ′(z) + κzf ′′(z)) > 1/2, z ∈ ud. https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 5 corollary 3.7. if f ∈ eq(π, 1/2) is such that re ( zdq(dqf (z)) dqf (z) ) > − 1 2 , (9) then re(dqf (z)) > 1 2 . this means that if condition (9) holds, then f is a q-bounded turning function of order 1/2. now if q ↑ 1, then re ( zf ′′(z) f ′(z) ) > − 1 2 , (10) implies re(f ′(z)) > 1 2 z ∈ ud. this means that if condition (10) holds, then f is a bounded turning function of order 1/2. corollary 3.8. if f ∈ eq(0, 1/2) is such that re ( zdq(dqf (z) + zd2qf (z)) dqf (z) + zd2qf (z) ) > − 1 2 , (11) then re(dqf (z) + zd2qf (z)) > 1 2 and if q ↑ 1, re ( 2zf ′′(z) + z2f ′′′(z) f ′(z) + zf ′′(z) ) > − 1 2 implies that re(f ′(z) + zf ′′(z)) > 1/2, z ∈ ud. theorem 3.9. let β ∈ (−π, π] and δ ∈ [0, 1), then the function f (z) = z + amz m ∈ eq(β, δ), m = {2, 3, . . .} (12) if |am| ≤ 2 [m]q { |xm| − ((2 + [m − 1]q) cos θ + [m − 1]q cos(β + θ0)) } (13) where xm = 2 + [m − 1]q(1 + e iβ) |xm| = √ 2 { 2 + [m − 1]q(2 + [m − 1]q)(1 + cosβ) } ≥ 2  (14) and θ0 attains minimum at θ0 = π + arctan ( −[m − 1]q sinβ 2 + [m − 1]q(1 + cosβ) ) . (15) https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 6 proof. firstly, applying (2) in (12) gives dqf (z) = 1 + [m]qamzm−1 zd2qf (z) = [m − 1]q[m]qamzm−1 } . (16) note that it suffices to study the condition that for |z | = 1,∣∣∣∣dqf (z) + 1 + e iβ2 zd2qf (z)− 1 ∣∣∣∣ < re {dqf (z) + 1 + e iβ2 zd2qf (z) } (17) so that by putting (16) into (17) we obtain∣∣∣∣[m]qamzm−1 + 12[m − 1]q[m]q(1 + e iβ)amzm−1 ∣∣∣∣ < re { 1 + [m]qamz m−1 + 1 2 [m − 1]q[m]q(1 + e iβ)amzm−1 } . now letting |am| = r , amzm−1 = re iθ and using (14) we obtain∣∣∣∣12[m]qre iθxm ∣∣∣∣ ≤ re {1 + [m]qre iθ + 12[m − 1]q[m]q(1 + e iβ)re iθ } (18) so that 1 2 [m]qr |xm| ≤ re f (19)where f = 1 + [m]qre iθ + 1 2 [m − 1]q[m]q(1 + e iβ)re iθin (18). further simplification gives f = 1 + [m]qr cos θ + 1 2 [m − 1]q[m]qr cos θ + 1 2 [m − 1]q[m]qr cos(β + θ) + im(f) so that re f = 1 + 1 2 [m]qr{2 cos θ + [m − 1]q cos θ + [m − 1]q cos(β + θ)} = ψ. (20)now (19) becomes 1 2 [m]qr |xm| ≤ 1 + 1 2 [m]qr{(2 + [m − 1]q) cos θ + [m − 1]q cos(β + θ)} and by simplification we obtain (13).to know the values of θ where (20) attains minimum implies that ∂ψ ∂θ = − r [m]q 2 { (2 + [m − 1]q) sin θ + [m − 1]q sin(β + θ) } implies that (2 + [m − 1]q) sin θ + [m − 1]q sin(β + θ) = 0so that tan θ = −[m − 1]q sinβ 2 + [m − 1]q(1 + cosβ)which simplifies to (15). � https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 7 corollary 3.10. let f (z) = z + amzm ∈ eq(0, δ) and m = {2, 3, . . .}, then |am| ≤ 1 [m]q {√ 1 + 2[m − 1]q + [m − 1]2q + 1 + [m − 1]q } and if q ↑ 1, then |am| ≤ 1 2m2 . corollary 3.11. let f (z) = z + amzm ∈ eq(π, δ) and m = {2, 3, . . .}, then |am| 5 1 2[m]q and if q ↑ 1, then |am| ≤ 1 2m . remark 3.12. let q ↑ 1, then theorem 3.9 becomes the result in [18]. theorem 3.13 (coefficient estimates). let β ∈ (−π, π], δ ∈ [0, 1) and let g(z) = 1 + b1z + b2z 2 + · · · ∈ cv(δ). if f ∈ a belongs to eq(β, δ), then |am| ≤ 2(1− δ)|b1| [m]q|xm| , m = {2, 3, . . .} (21) where |xm| is defined in (14). proof. let f (z) ∈ eq(β, δ), therefore from (6) and using (5), dqf (z) + 1 + e iβ 2 zd2qf (z) = δ + (1− δ)p(z), z ∈ ud. (22) now putting (3) and (4) into (22) and simplifying gives 1 + ∞∑ m=2 { 1 + [m − 1]q ( 1 + e iβ 2 )} [m]qamz m−1 = 1 + ∞∑ m=2 (1− δ)cm−1zm−1 which implies that {2 + [m − 1]q(1 + e iβ)} [m]q 2 am = (1− δ)cm−1, m = {2, 3, . . .} where by applying (14) we obtain xm [m]q 2(1− δ)am = cm−1, m = {2, 3, . . .}. (23) since g(ud) is a convex domain, then from lemma 2.1, (23) becomes∣∣∣∣xm [m]q 2(1− δ)am ∣∣∣∣ = |cm−1| ≤ |b1| and simplifying further we obtain (21). � https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 8 corollary 3.14. let f (z) ∈ eq(0, δ), then |am| ≤ (1− δ)|b1|√ 1 + 2[m − 1]q + [m − 1]2q and if q ↑ 1, then |am| ≤ (1− δ)|b1| m , m = {2, 3, . . .}. corollary 3.15. let f ∈ eq(π, δ), then |am| ≤ (1− δ)|b1| [m]q and if q ↑ 1, then |am| ≤ (1− δ)|b1| m , m = {2, 3, . . .} remark 3.16. let p(z) ∈ p and φ(z) = 1 + 2 π2 ( ln 1+ √ z 1− √ z )2. if q ↑ 1,(1) β = π and g(z) = p(z), then theorem 3.13 becomes the result in [12].(2) and g(z) = p(z), then theorem 3.13 becomes the result in [16].(3) and g(z) = φ(z), then theorem 3.13 becomes the result in [18].(4) and β = 0, then theorem 3.13 becomes the result in [17]. acknowledgment. the authors would like to thank the referees for their careful reading of thismanuscript and their valuable suggestions. references [1] 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https://doi.org/10.1007/bfb0066543 https://www.jams.jp/notice/mj/50-1.html https://doi.org/10.2298/fil1614743s https://doi.org/10.1515/dema-2005-0106 https://doi.org/10.1515/dema-2005-0106 1. introduction and definitions 2. relevant lemmas 3. main results references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 5doi: 10.28924/ada/ma.3.5 updated and weaker convergence criteria of newton iterates for equations samundra regmi1, ioannis k. argyros2,∗, santhosh george3, michael i. argyros4 1department of mathematics, university of houston, houston, tx 77204, usa sregmi5@uh.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4university of oklahoma, department of computer science, norman, ok 73019, usa michael.i.argyros-1@ou.edu ∗correspondence: iargyros@cameron.edu abstract. newton iteration is often used as a solver for nonlinear equations in abstract spaces.some of the main concerns are general: criteria for convergence, error estimations on consecutiveiterates, and the location of a solution. a plethora of authors has addressed these concerns by pre-senting results based on the celebrated kantorovich theory. this article contributes in this directionby extending earlier results but without additional conditions. these extensions become possibleusing a more precise majorization than the one given in earlier articles. numerical experimentationcomplements the theoretical results involving a partial differential and an integral equation. 1. introduction nonlinear equation f (x) = 0, (1.1)plays a important role due to the fact that many applications can be brought to look like it. thecelebrated newton iteration (ni) in the following form xn+1 = xn − f ′(xn)−1f (xn), ∀ n = 0, 1, 2, . . . (1.2) is often applied to solve equation (1.1) iteratively. here, f : ω ⊂ m1 −→ m2 is differentiable perfréchet and operates between banach spaces m1 and m2, whereas set ω 6= ∅.kantorovich inaugurated the semi-local convergence of ni (slcni) analysis of ni in abstractspaces by applying the contraction mapping principle due to banach. he presented two different received: 29 apr 2022. key words and phrases. iterative processes; newton iteration; banach space; semi-local convergence.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 2proofs based on majorization and recurrent relations [12]. the newton-kantorovich theorem givesthe slcni. numerous authors applied this result, in applications and also as a theoretical tool.even a simple equation given in [1–4, 7, 10, 11] shows that convergence criteria may not besatisfied. however, ni may be convergent (see the numerical section, example 4.1). that iswhy these criteria are weakened in [2–4]. but no new conditions are added. in this study twoadditional features are presented. one involves an explicit upper bound on the smallness of initialapproximation. moreover by choosing a bit larger bound the convergence order of ni is recovered.consequently, new results can always replace corresponding ones by kantorovich [7] and others[5,8–11], since preceding results imply the one in this study but not necessarily vice versa. methodin this study uses smaller lipschitz or hölder parameters to achieve these extensions which arespecializations of earlier ones. that is no additional effort is needed. the generality of this ideaallows its application on other processes [3, 4, 11].contributions by others can be found in section 4, where comparisons take place. the majoriza-tion of ni is discussed in section 2. slcni appears in section 3. the numerical experimentationis given in section4. conclusions complete this study in section 5. 2. majorization of ni let k0, k, l0, l denote positive numbers, q ∈ (0, 1] and t stand for a positive variable. theseparametrs are connected in section 3 to initial data d = (ω, y , f, f ′, x0). define sequence {sn}by s0 = 0, s1(t) = s1 = t s2(t) = s2 = s1 + k(s1 − s0)1+q (1 + q)(1−k0sq1 ) , sn+2(t) = sn+2 = sn+1 + l(sn+1 − sn)1+q (1 + q)(1− l0sqn+1) , ∀n = 1, 2, . . . . (2.1) sequence {xn} is majorized by {sn} (see section 3). that is why convergence is studied first forsequence {sn}. lemma 2.1. suppose k0t q < 1 and l0s q n+1 < 1 ∀n = 0, 1, 2, . . . . (2.2) then, sequence {sn} is strictly increasing and converges to some limit point s∗ ∈ (0, ( 1l0 ) 1 q ]. the point s∗ is the unique least upper bound of sequence {sn}. proof. the result follows from definition of sequence {sn} and hypothesis (2.1). �let ε be a positive constant. moreover, introduce parameters by α = 1+ε, β = l (1+q)(1+ε), γ = ε (1+ε)l0 , δ = β(s2 − s1), λ = γ 1 q , h = δ1+q and u = ( 1k0 ) 1 q . furthermore, consider functions withcommon domain in t = [0, u) given as https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 3 f1(t) = ( ktq (1 + q)(1−k0tq) + t )q − γ, f2(t) klαt1+q (1 + q)2(1−k0tq) − 1 and f3(t) = (s2 + β− 1 q h 1− h )q − γ. it follows by these definitions f1(0) = −γ < 0, f2(0) = −1 < 0, f3(0) = −γ < 0 and f1(t) −→ ∞, f2(t) −→ ∞ and f3(t) −→ ∞ as t −→ u−. so, function fi , i = 1, 2, 3 have zeros in interval t by ivt (intermediate value theorem). let ηi denote the smallest such zero of functions fi ininterval t0 = (0, u), respectively.it also follows by these choices of zeros ηi k0s q 1 < 1, sq2 < γ, f1(t) < 0 at t = η1 (2.3) δ < 1, f2(t) < 0 at t = η2 (2.4) and f3(t) < 0 at t = η3. (2.5) define parameter η0 = min{ηi}. (2.6) suppose η ≤ η0. (2.7) if η0 = η1 or η0 = η2, suppose hypothesis (2.7) holds as a strict inequality.a second stronger convergence result follows. but hypotheses are easier to verify. lemma 2.2. suppose hypothesis (2.7) holds. then, sequence {sn} is strictly increasing and convergent to some s∗ ∈ (0, γ0), where γ0 = s2 + β − 1q h 1−h . moreover, for σn+2 = sn+2 − sn+1 ∀n = 0, 1, 2, . . . σn+2 ≤ βσ1+qn+1 ≤ β − 1 q δ(1+) n (2.8) and s∗ − sn+1 ≤ β− 1 q δ(1+q) n 1− δ1+q . (2.9) proof. the assertions (ij) : 0 < 1 1− l0sqj+1 ≤ α (2.10) https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 4is shown using induction. assertion (i1) is true by the choice of η1 and estimates (2.3). it followsby (i1) and sequence {sn} that 0 < s3 − s2 ≤ β(s2 − s1)1+q, or s3 < s2 + β− 1 q δ(1+q) 1 ≤ γ.so, assertion (2.8) holds for n = 1. suppose assertion (2.10) holds ∀j = 1, 2, . . . n. then, 0 < σn+1 ≤ βσ1+qnand sn+1 ≤ sn + βσ1+qn ≤ . . . ≤ s2 + β (1+q)−1 1+q−1 σ1+q2 + β (1+q)2−1 1+q−1 σ (1+q)2 2 + . . .+ β (1+q)n−1−1 1+q−1 σ (1+q)n−1 2 = s2 + β− 1 q (δ1+q + δ2(1+q) + . . .+ δ(n−1)(1+q)) (δ < 1) = s2 + β− 1 q δ1+q 1− (δ1+q)n−1 1− δ1+q < s2 + β− 1 q δ1+q 1− δ1+q = s2 + β− 1 q h 1− h = γ0. hence, estimate sqn+1 ≤ γholds if f3(t) ≤ 0 at t = η3, which is estimate (2.5). the induction for assertion (2.10) is completed.it follows that estimate (2.8) holds. notice δ = βσ2 = lα 1 + q kσ1+q1 (1 + q)(1−k0sq1 ) < 1 (2.11) also holds since it is equivalent to the second estimate in (2.4). let n = 2, 3, . . . . then, it followsin turn by assertion (2.8) sj+n − sj+1 ≤ σj+n + σj+n−1 + . . .+ σj+2 ≤ β− 1 q (δ(1+q) j+n−2 + δ(1+q) j+n−1 + . . .+ δ(1+q) j ) ≤ β− 1 q δ(1+q) n δ1+q 1− δ2n−1 1− δ1+q . (2.12) then, assertion (2.9) follows from estimate (2.12) if n −→∞. � remark 2.3. an at least as large parameter as η3 can replace it in condition (2.7) as follows. define sequences of functions ϕn on the interval t by ϕn(t) = (s2(t) + β− 1 q (δ(t)1+q + δ(t)(1+q) 2 + . . .+ δ(t)(1+q) n−1 )q − γ. (2.13) https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 5 it follows by these definition that ϕn+1(t)− ϕn(t) ≥ 0, so ϕn(t) ≤ ϕn+1(t) ∀t ∈ t. (2.14) moreover these functions have zeros in t0. these zeros are assured to exist by (ivt), since by the definitions of fucntions ϕn give ϕn(0) = −γ < 0 and ϕn(t) −→ ∞ as t −→ u−. denote the smallest such zeros of functions ϕn in t by rn, respectively. according to the proof of lemma 2.2, lim n−→∞ ϕn(t) ≤ f3(t) ∀t ∈ t. (2.15) so, this limit exists as a well defined function denoted by ψ. then, this function has zeros in t0, since ψ(0) = −γ and ψ(t) −→ ∞ as t −→ u−. denote by η4 the smallest such zero in (0, u). clearly, the proof of lemma 2.2 goes through if instead of f3(t) ≤ 0, it is shown that ψ3(t) ≤ 0 ∀t ∈ t. (2.16) define parameter η̄0 = min{η1, η2, η4}. notice that by (2.15) ψ(η3) ≤ f3(η3) = 0, so η3 ≤ η4 and consequenlty η0 ≤ η̄0. if η̄0 replaces η0 in condition (2.6), then assertion (2.16) follows. condition (2.7) becomes η ≤ η̄0. (2.17) hence, the range of initial approximations η is further extended. 3. convergence of nm the notation u(w, ρ), u[w, ρ] means the open and closed balls with radius ρ > 0 and center w ∈ x, respectively. the parameters k0, l0, k, l and t are connected with operator f as follows.consider conditions (a):suppose(a1) there exist x0 ∈ ω, t ≥ 0 such that f ′(x0)−1 ∈ l(m2,m1), ‖f ′(x0)−1f (x0)‖ ≤ t. ‖f ′(x0)−1(f ′(x1)− f ′(x0))‖ ≤ k0‖x1 − x0‖qand ‖f ′(x0)−1(f ′(x0 + τ(x1 − x0))− f ′(x0))‖ ≤ k‖τ(x1 − x0)‖q.(a2) ‖f ′(x0)−1(f ′(x)− f ′(x0))‖ ≤ l0‖x − x0‖q, ∀x ∈ ω. set b1 = u(x0, ( 1l0 ) 1 q ) ∩ω.(a3) ‖f ′(x0)−1(f ′(x + τ(y − x))− f ′(x))‖ ≤ l‖τ(y − x)‖q ∀x, y ∈ b1 and ∀τ ∈ [0, 1).(a4) conditions of lemma 2.1 or lemma 2.2 hold(a5) u[x0, t ∗] ⊂ ω. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 6notice that k0 ≤ k ≤ l0.next, conditions a are applied to show the main convergence result for ni. theorem 3.1. under conditions a sequence ni is convergent to a solution x∗ ∈ u[x0, s ∗] of equation f (x∗) = 0. moreover, upper bounds ‖x∗ − xn‖ ≤ s∗ − sn (3.1) hold ∀n = 0, 1, 2, . . . . proof. the items ‖xi+1 − xi‖ ≤ si+1 − si , (3.2)and u[xi+1, s ∗ − si+1] ⊆ u[xi , s ∗ − si ], (3.3)are shown by induction ∀i = 0, 1, 2, . . . . let u ∈ u[x1, s ∗ − s1]. it follows by condition (a1) ‖x1 − x0‖ = ‖f ′(x0)−1f (x0)‖ ≤ t = s1 − s0, ‖u − x0‖ ≤ ‖u − x1‖+ ‖x1 − x0‖ ≤ s∗ − s1 + s1 − s0 = s∗.hence, point u ∈ u[x0, s ∗−s0]. that is items (3.2) and (3.3) hold for i = 0. assume these assertionshold if i = 0, 1, . . . , n. it follows for each ξ ∈ [0, 1] ‖xi + ξ(xi+1 − xi)− x0‖ ≤ si + ξ(si+1 − si) ≤ s∗, and ‖xi+1 − xi‖ ≤ i+1∑ j=1 ‖xj − xj−1‖ ≤ i+1∑ j=1 (sj − sj−1) = si+1. it follows by induction hypotheses, lemmas and conditions (a1) and (a2) ‖f ′(x0)−1(f ′(xi+1)− f ′(x0))‖ ≤ k̄‖xi+1 − x0‖q, ≤ k̄(si+1 − s0)q ≤ k̄sqi+1 < 1. hence, the inverse of linear operator f ′(xi+1) exists. therefore, hence, f ′(v)−1 ∈ l(m2,m1) and ‖f ′(xi+1)−1f ′(x0)‖ ≤ 1 1− k̄sqi+1) , (3.4) follows as a consequence of a lemma on invertible linear operators due to banach [2, 7], where k̄ = { k0, i = 0 l0, i = 1, 2, . . . .ni gives https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 7 f (xi+1) = f (xi+1)− f (xi)− f ′(xi)(xi+1 − xi), = ∫ 1 0 (f ′(xi + ξ(xi+1 − xi))dξ − f ′(xi))(xi+1 − xi). (3.5) then, using induction hypotheses, identity (a3) and condition (??) ‖f ′(x0)−1f (xi+1)‖ ≤ l̄ ∫ 1 0 (‖xi+1 − xi‖)q (3.6) ≤ l̄ 1 + q (si+1 − si)1+q, where l̄ = { k, i = 0 l, i = 1, 2, . . . .it follows by ni, estimates (3.4), (3.6) and the definition (2.1) of sequence {sn} ‖xi+2 − xi+1‖ ≤ ‖f ′(xi+1)−1f ′(x0)‖‖f ′(x0)−1f (xi+1)‖, ≤ k̃(si+1 − si)2 2(1− l̃si+1) = si+2 − si+1, where k̃ = { k, i = 0 l, i = 1, 2, . . . . and l̃ = { k0, i = 0 l0, i = 1, 2, . . . . moreover, if v ∈ u[xi+2, s ∗ − si+2] it follows ‖v − xi+1‖ ≤ ‖v − xi+2‖+ ‖xi+2 − xi+1‖ ≤ s∗ − si+2 + si+2 − si+1 = s∗ − si+1. hence, point w ∈ u[xi+1, s ∗ − si+1] completing the induction for items (3.2) and (3.3). noticethat scalar majorizing sequence {si} is fundamental as convergent. hence, the sequence {xi} isalso convergent to some x∗ ∈ u[x0, s ∗]. furthermore, let i −→ ∞ in estimate (3.6), to conclude f (x∗) = 0. �next, the uniqueness ball for a solution is presented. notice that not all condition a are used. proposition 3.2. under center-lipschitz condition (a2) further suppose the existence of a solution p ∈ u(x0, r) ⊂ ω of equation (1.1) such that operator f ′(p) is invertible for some r > 0; a parameter r1 ≥ r given by r1 = ( 1 + q l0 − rq ) 1 q . (3.7) then, the poiny p solves uniquely equation f (x) = 0 in the domain b2 = u(x0, r1) ∩ω. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 8 proof. define linear operator q = ∫ 1 0 f ′(p̄ + ξ(p − p̄))dξ for some point p̄ ∈ b2 satisfying f (p̄) = 0. by using the definition of r1, set b2 and condition (a2) ‖f ′(x0)−1(f ′(x0)−q)‖ ≤ ∫ 1 0 l0((1− ξ)‖x0 − p‖q + ξ‖x0 − p̄‖q)dξ, < l0 1 + q (rq1 + rq) = 1, concluding that p = p̄, where the invertability of linear operator is also used together with theapproximation 0 = f (p)− f (p̄) = q(p − p̄). � remark 3.3. (1) if conditions a hold, set p = x∗ and r = s∗ in proposition 3.2. (2) lipschitz condition (a3) can be replaced by ‖f ′(x0)−1(f ′(z1 + τ(z2 − z1))− f ′(z1))‖ ≤ d‖τ(z1 − z2)‖q (3.8) for all z1 ∈ b1 and z2 = z1 − f ′(z1)−1f (z1) ∈ b1. this even smaller parameter d can replace l in the previous results. the existence of iterate z2 is assured by (a2). 4. numerical experimentation three experimenta are considered in this section. example 4.1. the parameters using example of the introduction are k0 = µ+5 3 , k = l0 = µ+11 6 . moreover,ω0 = u(1, 1 − µ) ∩ u(1, 1l0 ) = u(1, 1l0 ). set l = 2(1 + 1 3−µ) l0 < l1 and l < l1 for all µ ∈ (0, 0.5). the kantorovich criterion η ≤ 1 l1 is violated, since η > 1 l1 ∀µ ∈ (0, 0.5), where l1 is the lipschitz constant on ω. interval can be enlarged if condition of lemma 2.1 is verified. then, for µ = 0.4, we have the following; 1l0 = 0.3846, table 1. sequence (2.1) n 1 2 3 4 5 6 7 sn+1 0.2000 0.2594 0.2744 0.2755 0.2755 0.2755 0.2755 hence conditions of lemma 2.1 hold. hence condition (2.2) holds, and the interval is extended form ∅ to [0.4, o.5]. example 4.2. let us consider the two point pbvp(tpbvp) u′′ + u 3 2 = 0 u(0) = u(1) = 0. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 9 the interval [0, 1] is divided into j subintervals. set m = 1 j . denote by w0 = 0 < w1 < . . . < wj = 1 the points of subdivision with corresponding values of the function u0 = u(w0), . . . , uj = u(wj). then, the discretization of u′′ is given by u′′k ≈ uk−1 − 2uk + uk+1 m2 , ∀k = 2, 3, . . . j − 1. notice that u0 = uj = 0. it follows that the following system of equations is obtained m2u 3 2 1 − 2u1 + u2 = 0, uk−1 +m2u 3 2 k − 2uk + uk+1 = 0, ∀k = 2, 3, . . . , j − 1 uj−2 +m2u 3 2 j−1 − 2uj−1 = 0. this system can be converted into an operator equation as follows: define operator g : rj−1 −→ rj−1 whose derivative is given as g′(u) =  3 2m 2u 1 2 1 − 2 1 0 . . . 0 1 3 2m 2u 1 2 2 − 2 1 0 . . . 0 . . . . . . . . . . . . ... ... ... ... ... 0 · · · 1 0 3 2m 2u 1 2 j−1 − 2  . let z ∈ rj−1 be arbitrary. the norm is ‖z‖ = max1≤k≤j−1 ‖zk‖, where as the norm for g ∈ rj−1 × rj−1 is given as ‖g‖ = max 1≤k≤j−1 j−1∑ i=1 ‖gk,i‖. then, if u, z ∈ rj−1 for |uk | > 0, |zk | > 0, ∀k = 1, 2, . . . , j − 1 to obtain in turn ‖g′(u)− g′(z)‖ = ‖diag{ 3 2 (u 1 2 k − z 1 2 k )}‖ = 3 2 m2 [ max 1≤k≤j−1 |uk − zk | ] 1 2 = 3 2 m2‖u − z‖ 1 2 . choose as an initial guess vector 130 sinπx to obtain after four iterations u0 = [3.35740e + 01, 6.5202e + 01, 9.15664e + 01, 1.09168e + 02, 1.15363e + 02, 1.09168e + 02, 9.15664e + 01, 6.52027e + 01, 3.35740e + 01]tr ]. then, the parameters are ‖q′(u0)−1‖ ≤ 2.5582e + 01, η = 9.15311e − 05, q = 0.5, k0 = l0 = k = l = 3 200 = 0.015. then, k0ηp = 1.4351e − 04 and the following table shows that the conditions of lemma 2.1 are satisfied. example 4.3. let m1 = m2 = c[0, 1] be the set of continuous real functions on [0, 1]. the norm-max is used. set ω = u[x0, 3]. consider hammerstein nonlinear integral operator h [3,6] on https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 10table 2. sequence (2.1) n 1 2 3 4 5 6 vn+1 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 ω as h(v)(z1) = v(z1)− y(z1)− ∫ 1 0 v(z1, z2)v 3(z2)dz1 = 2, v ∈ c[0, 1], z1 ∈ [0, 1]. (4.1) where function y ∈ c[0, 1], and v is a kernel related by green’s function v(z1, z2) = { (1− z1)z2, z2 ≤ z1 z2(1− z1), z1 ≤ z2. (4.2) it follows by this definition that h′ is [h′(v)(z)](z1) = z(z1)− 3 ∫ 1 0 v(z1, z2)v 2(z2)z(z2)dz2 (4.3) z ∈ c[0, 1], z1 ∈ [0, 1]. pick x0(z1) = y(z1) = 1. it then follows from (4.1)-(4.3) that h′(x0)−1 ∈ l(m2,m1), ‖i −h′(x0)‖ < 0.375, ‖h′(x0)−1‖ ≤ 1.6, η = 0.2, l0 = 2.4, l1 = 3.6, and ω0 = u(x0, 3) ∩ u(x0, 0.4167) = u(x0, 0.4167), so l = 1.5. notice that l0 < l1 and l < l1. set k0 = k = l0. the kantorovich convergence criterion (a3) is not satisfied, since 2l1η = 1.44 > 1. therefore convergence of ni is not guaranteed. however, the new condition (2.7) is satisfied, since 2lη = 0.6 < 1. 5. conclusions an updated and weaker unified framework is presented for ni. the new analysis is finer thanbefore. convergence order 1 + q is also recovered by choosing a larger upper bound on t. newlipschitz or hölder parameters are smaller and specilizations of previous parameters. the newtheory can always replace previous ones due to weaker criterion. the strategy can be applied onother iterations [2, 3, 7, 11]. references [1] j. appell, e.d. pascale, j.v. lysenko, p.p. zabrejko, new results on newton-kantorovich approximations with appli-cations to nonlinear integral equations, numer. funct. anal. optim. 18 (1997) 1–17. https://doi.org/10.1080/ 01630569708816744.[2] i.k. argyros, unified convergence criteria for iterative banach space valued methods with applications, mathematics.9 (2021) 1942. https://doi.org/10.3390/math9161942.[3] i.k. argyros, the theory and applications of iteration methods, 2nd edition, crc press, boca raton, 2022. https: //doi.org/10.1201/9781003128915. https://doi.org/10.28924/ada/ma.3.5 https://doi.org/10.1080/01630569708816744 https://doi.org/10.1080/01630569708816744 https://doi.org/10.3390/math9161942 https://doi.org/10.1201/9781003128915 https://doi.org/10.1201/9781003128915 eur. j. math. anal. 10.28924/ada/ma.3.5 11 [4] s. singh, e. martínez, p. maroju, r. behl, a study of the local convergence of a fifth order iterative method, indianj. pure appl. math. 51 (2020) 439–455. https://doi.org/10.1007/s13226-020-0409-5.[5] f. cianciaruso, e. de pascale, newton–kantorovich approximations when the derivative is hölderian: old and newresults, numer. funct. anal. optim. 24 (2003) 713–723. https://doi.org/10.1081/nfa-120026367.[6] j.a. ezquerro, m.a. hernandez, newton’s scheme: an updated approach of kantorovich’s theory, cham switzerland,2018.[7] l.v. kantorovich, g.p. akilov, functional analysis, pergamon press, oxford, 1982.[8] f.a. potra, v. pták, nondiscrete induction and iterative processes, research notes in mathematics, 103. pitman(advanced publishing program), boston, 1984.[9] p.d. proinov, new general convergence theory for iterative processes and its applications to newton–kantorovichtype theorems, j. complex. 26 (2010) 3–42. https://doi.org/10.1016/j.jco.2009.05.001.[10] r. verma, new trends in fractional programming, nova science publisher, new york, usa, 2019.[11] t. yamamoto, historical developments in convergence analysis for newton’s and newton-like methods, j. comput.appl. math. 124 (2000) 1–23. https://doi.org/10.1016/s0377-0427(00)00417-9.[12] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987) 671–684. https://doi.org/10.1080/01630568708816254. https://doi.org/10.28924/ada/ma.3.5 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1081/nfa-120026367 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/s0377-0427(00)00417-9 https://doi.org/10.1080/01630568708816254 1. introduction 2. majorization of ni 3. convergence of nm 4. numerical experimentation 5. conclusions references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 18doi: 10.28924/ada/ma.2.18 developments of newton’s method under hölder conditions samundra regmi1, ioannis k. argyros2,∗, santhosh george3, christopher i. argyros4 1learning commons, university of north texas at dallas, dallas, tx, usa samundra.regmi@untdallas.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. the semi-local convergence criteria for newton’s method are weakened without new con-ditions. moreover, tighter error distances are provided as well as a more precise information on thelocation of the solution. 1. introduction the computation of a solution x∗ of nonlinear equation f (x) = 0 (1.1)is important in computational sciences, since many applications can be written as (1.1). here f : ω ⊆ x −→ y is fréchet-differentiable operator, x, y are banach spaces and ω 6= ∅ is aconvex and open set. but this can be attained only in special cases. that explains why mostsolution methods for (1.1) are iterative. there is a plethora of methods for solving (1.1) [1–14].among them newton’s method (nm) defined by x0 ∈ ω, xn+1 = xn − f ′(xn)−1f (xn) (1.2) seems to be the most popular [2,4]. but the convergence domain is small, limiting the applicability ofnm. that is why we have developed a technique that determines a subset ω0 of ω also containingthe iterates {xn}. hence, the hölder constants are at least as tight as the ones in ω. this crucial received: 20 mar 2022. key words and phrases. banach space; hölder condition; semi-local convergence; convergence criteria.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 2modification leads to: weaker sufficient convergence criteria, the extension of the convergencedomain, tighter error estimates on ‖x∗ − xn‖, ‖xn+1 − xn‖ and a more precise information on x∗.it is worth noticing that these advantages are obtained without additional conditions, since inpractice the evolution of the old hölderian constants require that of the new conditions as specialcases. 2. convergence we introduce certain hölder conditions crucial for the semi-local convergence. let p ∈ (0, 1].suppose there exists x0 ∈ ω such that f ′(x0)−1 ∈ l(y,x). definition 2.1. operator f ′ is center hölderian on ω if there exists h0 > 0 such that ‖f ′(x0)−1f ′(w)− f ′(x0)‖ ≤ h0‖w − x0‖p (2.1) for all w ∈ ω. set ω0 = u(x0, 1 h 1 p 0 ) ∩ω. (2.2) definition 2.2. operator f ′ is center hölderian on ω0 if there exists h > 0 such that ‖f ′(x0)−1f ′(w)− f ′(u)‖ ≤ h̃‖w − u‖p, (2.3) where h̃ = { h, w = u − f ′(u)−1f (u), u ∈ d0 k, w, u ∈ ω0. . we present the results with h although k can be used too. but notice h ≤ k. definition 2.3. operator f ′ is center hölderian on ω if there exists h1 > 0 such that ‖f ′(x0)−1f ′(w)− f ′(u)‖ ≤ h1‖w − u‖p (2.4) for all w, u ∈ ω. remark 2.4. it follows from (2.2), that ω0 ⊆ ω. (2.5) then, by (2.1)-(2.5) the following items hold h0 ≤ h1 (2.6) and h ≤ h1. (2.7) we shall assume that h0 ≤ h. (2.8) https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 3 otherwise the results that follow hold with h0 replacing h. notice that h0 = h0(x0,ω), h1 = h1(x0,ω), h = h(x0,ω0) and h0 h1 can be small (arbitrarily) [2–4]. in earlier studies [1, 5–14] the estimate ‖f ′(z)−1f ′(x0)‖ ≤ 1 1−h1‖z − x0‖ 1 p (2.9) for all z ∈ u(x0, 1 h 1 p 1 ) was found using (2.4). but, if we use (2.1) to obtain the weaker and more precise estimate ‖f ′(z)−1f ′(x0)‖ ≤ 1 1−h0‖z − x0‖ 1 p (2.10) for all z ∈ u(x0, 1 h 1 p 0 ). this modification in the proofs and exchanging h1 by h leads to the advantages as already mentioned in the introduction. that is why we omit the proofs in our results that follow. notice also that in practice the computation of h1 require that of h0 and h as special cases. hence, the applicability of nm is extended without additional conditions. let d ≥ 0 be such that ‖f ′(x0)−1f (x0)‖ ≤ d. (2.11) we assume that (2.1)-(2.3) hold from now on unless otherwise stated. first we extend the resultsby keller [11] for nm. similarly the results for the chord method can also be extended. we leavethe details to the motivated reader. for brevity we skip the extensions on the radii of convergenceballs, and only mention convergence criteria and error estimates. theorem 2.5. assume: hrλ < 1 + λ 2 + λ , d ≤ [ 1− 2 + λ 1 + λ hrλ ] λ and ū(x0, r) ⊂ ω. then, limn−→∞ xn = x∗ ∈ u(x0, r0) and f (x∗) = 0. furthermore, ‖x∗ − xn‖ ≤ ( µ 1 λ 2 + λ )(1+λ)p r µ 1 λ , where µ = hrλ 1−h0rλ 1 1+λ < 1. proof. see theorem 2 in [11]. � https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 4 theorem 2.6. assume: hdλ < 1 2 + λ ( λ 1 + λ )λ and ū(x0, r) ⊂ ω. then, limn−→∞ xn = x∗ ∈ u(x0, r0), f (x∗) = 0 and ‖x∗ − xn‖ ≤ ( λ 1 p 1− λ )(1+p)n d µ 1 p , where λ = hrp0 1−h0rp0 ( d r0 )p 1 1+p < 1 and r0 is the minimal positive root of scalar equation (2 + p)ht1+p − (1 + p)(t − d) = 0 provided that r ≥ r0. proof. see theorem 4 in [11]. � theorem 2.7. assume: hdp ≤ 1− ( p 1 + p )p , r ≥ 1 + p 2 + p − (1 + p)p d and ū(x0, r) ⊂ ω. then, limn−→∞ xn = x∗ ∈ u(x0, r), f (x∗) = 0 and ‖x∗ − xn‖ ≤ ( 1 1 + p )n [(1 + p)h 1 p d ](1+p) n h 1 p . proof. see theorem 5 in [11]. �next, we extend a result given in [6] which in turn extended earlier ones [1,7–14]. it is convenientto define function on the interval [0,∞) by g(t) = h 1 + p t1+p − t + d gβ(t) = βh 1 + p t1+p − t + d (β ≥ 0) h(t) = t1+p + (1 + p)t (1 + p)1+p − 1 , v(p) = max t≥0 h(t), δ(p) = min{β ≥ 1 : max h(t) ≤ β, 0 ≤ t ≤ t(β)} https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 5and scalar sequence {sn} by s0 = 0, sn = sn−1 − gd(sn−1) g′(sn−1) . then, we can show: theorem 2.8. assume: d ≤ 1 v(p) ( p 1 + p )p and u(x0, r̄) ⊆ ω, where r̄ is the minimal solution of equation gv (p) = 0, gv (t) = v(p)h 1 + p t1+p − t + d. proof. see theorem 2.2 in [6]. �next, we present the extensions of the work by rokne in [13] but for the newton-like method(nlm) xn+1 = xn − l−1n f (xn), where ln is a linear operator approximating f ′(xn). theorem 2.9. assume: ‖l(x)− l(x0)‖ ≤ m0‖x − x0‖p for all x ∈ ω. set ω0 = u(x0, 1 (γ2m0) 1 p ). ‖f ′(x)− f ′(y)‖ ≤ m̄‖x − y‖p for all x, y ∈ ω0, ‖f ′(x)− l(x)‖ ≤ γ0 + γ1‖x − x0‖p for all x ∈ ω0, and some γ0 ≥ 0, γ1 ≥ 0. l(x0) −1 ∈ l(y,x) with ‖l(x0) −1‖ ≤ γ2 and ‖l(x0) −1f (x0)‖ ≤ γ3, function q defined by q(t) = t1+p(γ2γ0 + γ2m0) + t( γ2m̄d p 1 + p + γ2γ0 − 1)− γ2m0γ3tp + γ3 has a smallest positive zero r > γ3, γ2m̄r p < 1, ρ = p 1− γ2m̄rp [ γ2m̄d p 1 + p + γ2γ0 + γ2γ1r p ] < 1, ū(x0, r) ⊂ ω. then limn−→∞ xn = x∗ and f (x∗) = 0. https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 6 proof. see theorem 1 in [13]. �many results on newton’s method were also reported in the elegant book in [9]. next, we showhow to extend one of them. the details of how to extend the result of them are left to the motivatedreader. theorem 2.10. suppose: conditions (2.1), (2.3), (2.8), and (c) h0 = hdp ∈ (0, ρ) where ρ is the only solution of equation (1 + p)p(1− t)1+p − tp = 0, p ∈ (0, 1] in (0, 12 ] and u(x0, s) ⊂ ω, where s = (1+p)(1−h0) (1+p)−(2+p)h0 hold. then, sequence {xn} converges to a solution x∗ of equation f (x) = 0. moreover, {xn}, x∗ ∈ u[x0, s] and x∗ is the only solution in ω ∩ u(x0, d h 1/p 0 ). moreover, the following error estimates hold ‖xn − x∗‖ ≤ en, where en = δ (1+p)n−1 p2 an 1−δ (1+p)n p a d, with δ = h1 h0 , a = 1 − h0, h1 = h0f1(h0) 1+pf2(h0) p, f1(t) = 1 1−t and f2(t) = t 1+p . finally, we extend the results by f. cianciaruso and e. de pascale in [6] who in turn extendedearlier ones [1, 5, 7, 11,12,14]. define scalar sequence {vn} for h = dph by v0 = 0, v1 = h 1 p , vn+1 = vn + (vn − vn−1)1+p (1 + p)(1− vpn ) . (2.12) next, we extend theorem 2.1 and theorem 2.3 in [6], respectively. theorem 2.11. let function f : [1,∞) −→ [0,∞), r : [0,∞) −→ [0,∞) be defined by f (t) = (1− 1 t ) 1 + p ((1 + p) 1 1−p + (t(t − 1)p) 1 1−p )1−p and r(t) = (1 + p) 1 p ((1 + p) 1 1−p + (t(t − 1)p) 1 1−p )1−p . suppose that h ≤ f (m), (2.13) where m is a global maximum for function f , given explicitly by m = 1+ √ 1+4(1+p)pp1−p 2 . then, the following assertion hold vn ≤ r(m)(1− 1 mn ), (2.14) vn+1 vn ≤ 1− 1 mn+1 1− 1 mn , (2.15) https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 7 vn ≤ vn+1 ≤ r(m) < 1 and limn−→∞ vn = v∗ ∈ [0, r(m)]. simply use h for h1 in [6]. � theorem 2.12. under condition (2.13) further suppose that r∗ = h− 1 p v∗ ≤ ρ and u(x0, ρ) ⊆ ω. then, sequence {xn} generated by nm is well defined in u(x0, v ∗), stays in u(x0, v ∗) and converges to the unique solution x∗ ∈ u[x0, v ∗] of equation f (x) = 0, so that ‖xn+1 − xn‖ ≤ vn+1 − vn and ‖x∗ − xn‖ ≤ v∗ − vn. proof. simply use h for h1 used in [6]. remark 2.13. (1) if k = h1 the last two results coincide with the corresponding ones in [6]. but if k < h1 then the new results constitute an improvement with benefits already stated in the introduction. notice that the majorizing sequence {wn} in [6] was defined for h1 = dph1 by w0 = 0, w1 = h 1 p 1 , wn+1 = wn + (wn − wn−1)1+p (1 + p)(1− wpn ) , (2.16) and the convergence criterion is h1 ≤ f (m). (2.17) it then follows by (2.7), (2.12), (2.13), (2.16) and (2.17) that h1 ≤ f (m)⇒ h ≤ f (m) (2.18) but not necessarily vice versa, unless if h = h1, vn ≤ wn, 0 ≤ vn+1 − vn ≤ wn+1 − wn and 0 ≤ v∗ ≤ w∗ = lim n−→∞ wn. (2) in view of (2.9) and (2.10) sequence {un} defined for each n = 0, 1, 2, . . . by https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 8 u0 = 0, u1 = h 1 p 1 , u2 = u1 + h0(u1 − u0)1+p (1 + p)(1−h0up1) , un+1 = un + h(un − un−1)1+p (1 + p)(1−h0upn) is a tighter majorizing sequence than {vn} and can replace it in theorem 2.11 and theorem 2.12. concerning the uniqueness of the solution x∗ we provide a result based only on (2.1). proposition 2.14. suppose: (1) the point x∗ ∈ u(x0, a) ⊂ ω is a simple solution of equation f (x) = 0 for some a > 0. (2) condition (2.1) holds. (3) there exist b ≥ a such that h0 ∫ 1 0 ((1− τ)a + τb)pdτ < 1. (2.19) let g = u[x0, b] ∩ω. then, the point x∗ is the only solution of equation f (x) = 0 in the set g. proof. let z∗ ∈ g with f (z∗) = 0. by (2.1) and (2.19), we obtain in turn for q = ∫ 1 0 f ′(x∗ + τ(z∗ − x∗))dτ ‖f ′(x0)−1(q− f ′(x0))‖ ≤ h0 ∫ 1 0 ‖x∗ + τ(z∗ − x∗)− x0‖pdτ ≤ h0 ∫ 1 0 [(1− τ)‖x∗ − x0‖+ τ‖z∗ − x0‖]pdτ ≤ h0 ∫ 1 0 ((1− τ)a + τb)pdτ < 1, showing z∗ = x∗ by the invertibility of q and the approximation q(x∗− z∗) = f (x∗)−f (z∗) = 0. �notice that if k = h1 the results coincide to the ones of theorem 3.4 in [9]. but, if k < h1then they constitute an extension. remark 2.15. (a) we gave the results in affine invariant form. (b)the results in this study can be extended more if we consider the set s = u(x1, 1 h1/p − d) provided that h1/pd < 1. moreover, suppose s ⊂ ω. then, s ⊂ ω0, so the hölderian constant corresponding to s is at least as small as k, and can replace it in all previous results. references [1] j. appell, e.d. pascale, j.v. lysenko, p.p. zabrejko, new results on newton-kantorovich approximations with appli-cations to nonlinear integral equations, numer. funct. anal. optim. 18 (1997) 1–17. https://doi.org/10.1080/ 01630569708816744. https://doi.org/10.28924/ada/ma.2.18 https://doi.org/10.1080/01630569708816744 https://doi.org/10.1080/01630569708816744 eur. j. math. anal. 10.28924/ada/ma.2.18 9 [2] i.k. argyros, s. hilout, inexact newton-type methods, j. complex. 26 (2010) 577–590. https://doi.org/10.1016/ j.jco.2010.08.006.[3] i.k. argyros, convergence and applications of newton-type iterations, springer new york, 2008. https://doi. org/10.1007/978-0-387-72743-1.[4] i.k. argyros, s. george, mathematical modeling for the solution of equations and systems of equations with appli-cations, volume-iv, nova publisher, ny, 2021.[5] f. cianciaruso, e. de pascale, newton–kantorovich approximations when the derivative is hölderian: old and newresults, numer. funct. anal. optim. 24 (2003) 713–723. https://doi.org/10.1081/nfa-120026367.[6] f. cianciaruso, e. de pascale, estimates of majorizing sequences in the newton–kantorovich method: a furtherimprovement, j. math. anal. appl. 322 (2006) 329–335. https://doi.org/10.1016/j.jmaa.2005.09.008.[7] n.t. demidovich, p.p. zabreiko, j.v. lysenko, some remarks on the newtonkantorovich method for nonlinear equa-tions with hölder continuous linearizations, izv. akad. nauk, beloruss, 3 (1993) 22-26 (russian).[8] e. de pascale, p.p. zabrejko, convergence of the newton-kantorovich method under vertgeim conditions: a newimprovement, z. anal. anwend. 17 (1998) 271–280. https://doi.org/10.4171/zaa/821.[9] j. a. ezquerro, m. hernandez-veron, mild differentiability conditions for newton’s method in banach spaces, fron-tiers in mathematics, birkhauser cham, switzerland, (2020), https://doi.org/10.1007/978-3-030-48702-7.[10] l.v. kantorovich, g.p. akilov, functional analysis in normed spaces, the macmillan co, new york, (1964).[11] h.b. keller, newton’s method under mild differentiability conditions, j. computer syst. sci. 4 (1970) 15–28. https: //doi.org/10.1016/s0022-0000(70)80009-5.[12] j.v. lysenko, conditions for the convergence of the newton-kantorovich method for nonlinear equations with hölderlinearization, dokl. akad. nauk. bssr, 38 (1994) 20-24. (in russian).[13] j. rokne, newton’s method under mild differentiability conditions with error analysis, numer. math. 18 (1971)401–412. https://doi.org/10.1007/bf01406677.[14] b.a. vertgeim, on some methods of the approximate solution of nonlinear functional equations in banach spaces,uspekhi mat. nauk. 12 (1957) 166-169 (in russian). engl. transl: amer. math. soc. transl. 16 (1960) 378-382. https://doi.org/10.28924/ada/ma.2.18 https://doi.org/10.1016/j.jco.2010.08.006 https://doi.org/10.1016/j.jco.2010.08.006 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1081/nfa-120026367 https://doi.org/10.1016/j.jmaa.2005.09.008 https://doi.org/10.4171/zaa/821 https://doi.org/10.1007/978-3-030-48702-7 https://doi.org/10.1016/s0022-0000(70)80009-5 https://doi.org/10.1016/s0022-0000(70)80009-5 https://doi.org/10.1007/bf01406677 1. introduction 2. convergence references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 6doi: 10.28924/ada/ma.2.6 solving equilibrium problem and fixed point problem by normal s-iteration process in hilbert space shamshad husain, mohd asad∗ department of applied mathematics, faculty of engineering and technology, aligarh muslim university, aligarh, india s_husain68@yahoo.com, masad19932015@gmail.com ∗correspondence: masad19932015@gmail.com abstract. the main purpose of this paper is to find a common element in the solution set of equilibriumproblem and fixed point problem of non-expansive mappings in the real hilbert space with the helpof normal s-iteration process. also, under some acceptable assumptions, we prove the sequencesinduced by above stated process converge weakly to a point in the solution set of above statedproblems. at the end, we give a numerical example to justify our work. the results studied in thiswork philosophize and boost some contemporary and known results in this direction. 1. introduction and auxiliary results everywhere in this paper except stated otherwise, let h be a real hilbert space equipped withinner product 〈·, ·〉 and induced norm ‖ · ‖. let c be a non-empty closed and convex subset of h. we denote strong and weak convergence of a sequence {xn} ∈ h by the symbols → and ⇀respectively.let t : c → h be a nonexpansive mapping. the so called fixed point problem for mapping t is tofind an element p ∈ c such that tp = p. (1) denote the set of solution of the problem (1) by fix(t ) = {p ∈ c : tp = p}. t is said to benonexpansive iff ‖tp − tq‖2 ≤ ‖p − q‖2, ∀ p, q ∈ c. received: 10 dec 2021. key words and phrases. equilibrium problem; normal s-iteration; fixed point problem; hilbert space; non-expansivemapping. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 2in 2011, d.r. sahu [4] studied problem (1) and proposed an iterative method known as normals-iteration process which is defied as follows: let x1 ∈ c be chosen arbitrarily, yn = (1− αn)xn + αntxn, xn+1 = tyn, ∀ n ≥ 1, (2) where {αn} ⊂ (0 , 1). under some acceptable conditions of {αn}, sahu proved that the sequence {xn} induced by the algorithm (2) converges weakly to an element of solution set of problem (1).the performance of normal s-iteration process is much better than mann and picard iterationprocess for nonexpansive mappings(see [4], [5]).elsewhere, let f : c × c → r be a bifunction such that for all p ∈ c, f (p, p) = 0. then the socalled equilibrium problem is to find p ∈ c such that f (p, q) ≥ 0, ∀ q ∈ c. (3) denote the solution set of problem (3) by ep (f ). problem (3) contains nash equilibrium prob-lems, fixed point problems, variational inequality problems, minimization problems and optimizationproblems as its special cases(see [7, 16]).in this paper, we consider a problem which is formulated as follows: find p ∈ c, such that p ∈ ω := fix(t ) ∩ ep (f ). (4) in past few years, many researchers have found a common solution of problem (4) by varioustechniques(see [4], [3], [2], [12], [1], [14], [10]). impelled and inspired by these approaches, the mainobjective of this paper is to find a common element in the solution set of problem (4) with the helpof normal s-iteration process in the framework of real hilbert space. also we prove some weakconvergence theorem under some acceptable conditions.now we define some basic auxiliary results which are very helpful throughout this work.the metric projection pc from h into c is defined as: for any p ∈ c, ‖p − pc(p)‖ ≤ ‖p − q‖, ∀ q ∈ c. it is to be noted that the metric projection is nonexpansive. further for any p ∈ h and s ∈ c, s = pc(p) ⇐⇒ 〈p − s, s − q〉 ≥ 0, ∀ q ∈ c. a mapping t is said to be monotone iff for all p, q ∈ h 〈tp − tq, p − q〉 ≥ 0. lemma 1.1. [9] let h be a hilbert space. then for all p, q ∈ h and α ∈ [0, 1] the followings hold:(i) ‖p − q‖2 = ‖p‖2 − ‖q‖2 − 2〈p − q, q〉;(ii) ‖p + q‖2 ≤ ‖p‖2 + 2〈q, p + q〉;(iii) ‖αp + (1− α)q‖2 = α‖p‖2 + (1− α)‖q‖2 − α(1− α)‖p − q‖2. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 3 assumption 1.1. [6] let f : c × c → r be a bi-function satisfying the subsequent conditions:(i) f (p, p) ≥ 0, ∀ p ∈ c;(ii) f is monotone, i.e. f (p, q) + f (q, p) ≤ 0, ∀ p, q ∈ c;(iii) f is upper semi continuous, i.e. for each p, q, s ∈ c, lim t→0 supf (λs + (1− λ)p, q) ≤ f (p, q); (5) (iv) for each fixed p ∈ c, the function q 7→ f (p, q) is convex and lower semi continuous; lemma 1.2. [7] assume that the bi-function f : c × c → r satisfy the conditions of assumption1.1. then for fixed r > 0 and p ∈ h, there exists s ∈ c such that f (q, p) + 1 r 〈q − p, p − s〉 ≥ 0, ∀ q ∈ c. (6) lemma 1.3. [12] assume that the bi-function f : c × c → r satisfy the conditions of assumption1.1. if for r > 0 and p ∈ h, defined a mapping t fr : h → c as follows: t fr (p) = { s ∈ c : f (s, q) + 1 r 〈q − s, s − p〉 ≥ 0, ∀ q ∈ c } . (7) then the followings hold:(i) t fr is non-empty and single valued.(ii) t fr is firmly non-expansive, i.e., ‖t fr (p)− t fr (q)‖2 ≤ 〈t fr (p)− t fr (q), p − q〉 ∀ p, q ∈ h. (iii) fix(t fr ) = ep(f ).(iv) ep(f ) is closed and convex. lemma 1.4. [11] let {an} be a sequence of non negetive real numbers such that an+1 ≤ (1− αn)an + αnδn + γn, ∀ n ≥ 0, where αn ∈ (0, 1) and δn ⊂ r satisfies the following conditions:(i) ∑∞n=0 αn =∞;(ii) lim n→∞ supδn ≤ 0.(iii) γn ≥ 0 (n ≥ 1), ∑ γn <∞.then lim n→∞ an = 0. lemma 1.5. [13] let c be a closed and convex subset of h and t : c → c be a non-expansivemapping. then(i) fix(t ) is a closed and convex subset of c;(ii) i − t is demiclosed at 0. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 4 lemma 1.6. [8] let f : c × c → r be a non linear bi-function satisfying the assumption 1.1 andlet t fr be defined as above in lemma 1.3. if for r > 0, let p, q ∈ h and r1, r2 > 0, then ‖t fr2 (q)− t fr1 (p)‖ ≤ ‖q − p‖+ ∣∣∣∣ r2 − r1r2 ∣∣∣∣‖t fr2 (q)− q‖. lemma 1.7. [15] let xn and yn be two bounded sequences in a banach space x and let βn be asequence in [0, 1] which satisfy the following conditions: 0 < lim n→∞ inf βn ≤ lim n→∞ supβn < 1. suppose xn+1 = (1− βn)zn + βnxn for all integers n ≥ 0, and lim n→∞ sup (‖zn+1 − zn‖ − ‖xn+1 − xn‖) ≤ 0, then lim n→∞ ‖xn − zn‖ = 0. 2. main result in this section we study and analyze normal s-iteration process for solving equilibrium problemand fixed point problem for nonexpansive mapping and its convergence analysis. theorem 2.1. let c ⊂ h be a nonempty closed and convex subsets of h. let f : c × c → r bea nonlinear bifunction satisfying assumption 1.1. let t : c → h be a nonexpansive mapping suchthat fix(t ) 6= . assume that ω := fix(t ) ∩ ep (f ) 6= . let {xn}be a sequence defined as follows:choose x1 ∈ h arbitrarily, yn = t frn (xn), zn = (1− αn)yn + αntyn, xn+1 = tzn, ∀ n ≥ 1, (8) where {αn} ⊂ [0 , 1] and {rn} ⊂ (0 ,∞) satisfying the following conditions: c1: lim n→∞ αn = 0, ∑∞ n=1 αn(1− αn) =∞, ∑∞ n=1 |αn − αn−1| <∞; c2: lim n→∞ inf rn > 0, ∑∞ n=0 |rn+1 − rn| <∞;then the sequence {xn} induced by process (8) converges weakly to an element in ω. proof. take p ∈ ω. then by process (8), we obtain ‖xn+1 − p‖ = ‖tzn − p‖ ≤ ‖zn − p‖, ≤ ‖(1− αn)yn + αntyn − p‖, ≤ (1− αn)‖yn − p‖+ αn‖tyn − p‖, ≤ (1− αn)‖yn − p‖+ αn‖yn − p‖, ≤ ‖yn − p‖ ≤ ‖t frn (xn)− p‖, ≤ ‖xn − p‖. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 5by using mathematical induction, we have ‖xn+1 − p‖ ≤ ‖xn − p‖ ≤ ‖x1 − p‖, ∀ n ≥ 1. hence the sequence {xn} is bounded and so are the sequences {yn}, {zn}, {tyn} and {tzn} arealso bounded.let m = supn≥0{‖yn − xn‖+ ‖xn − q‖2 + ‖tyn‖+ ‖tzn‖}.since yn = t frn (xn) and yn−1 = t frn−1(xn−1), then we obtain f (yn, q) + 1 rn 〈q − yn, yn − xn〉 ≥ 0, ∀ q ∈ c, (9) f (yn−1, q) + 1 rn−1 〈q − yn−1, yn−1 − xn−1〉 ≥ 0, ∀ q ∈ c. (10) replace q by yn in (10) and q by qn−1 in (9) and adding them with the assumption 1.1(ii),we obtain 〈yn − yn−1, yn−1 − xn−1 rn−1 − yn − xn rn 〉 ≥ 0, and hence 〈yn − yn−1, yn−1 − yn − xn−1 − rn−1 rn (yn − xn)〉 ≥ 0. this implies that by using lemma 1.6 ‖yn − yn−1‖2 ≤ 〈yn − yn−1, xn − xn−1 + ( 1− rn−1 rn ) (yn − xn)〉, ≤ ‖yn − yn−1‖ { ‖xn − xn−1‖+ ∣∣∣∣ rn − rn−1rn ∣∣∣∣‖yn − xn‖}, ‖yn − yn−1‖ ≤ ‖xn − xn−1‖+ ∣∣∣∣ rn − rn−1rn ∣∣∣∣‖yn − xn‖, from process (8)(c2), we have lim n→∞ inf rn > 0. therefore there exists r > 0 such that rn > r forlarge enough n ∈ n. then for n ≥ 1, ‖yn − yn−1‖ ≤ ‖xn − xn−1‖+ 1 r |rn − rn−1|m. (11) consider ‖xn+1 − xn‖ = ‖tzn − tzn−1‖ ≤ ‖zn − zn−1‖, ≤ ‖(1− αn)yn + αntyn − (1− αn−1)yn−1 − αn−1tyn−1‖, ≤ ‖(1− αn)yn − (1− αn)yn−1 + (1− αn)yn−1 − (1− αn−1)yn−1 + αntyn − αntyn−1 +−αntyn−1 − αn−1tyn−1‖, ≤ (1− αn)‖yn − yn−1‖+ 2|αn − αn−1|m + αn‖yn − yn−1‖, ≤ ‖yn − yn−1‖+ 2|αn − αn−1|m. (12) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 6using (11) and (12), we obtain ‖xn+1 − xn‖ ≤ ‖xn − xn−1‖+ 1 r |rn − rn−1|m + 2|αn − αn−1|m. (13) by applying lemma 1.4, we obtain lim n→∞ ‖xn+1 − xn‖ = 0. (14) by using process (8)(c1)(c2) along with lemma 1.7 and (13), we obtain lim n→∞ ‖xn − zn‖ = 0. (15) furthermore, for any p ∈ ω, we have from process (8) ‖yn − p‖2 = ‖t frn (xn)− p‖2, ≤ 〈t frn (xn)− t frn (p), xn − p〉, ≤ 〈yn − p, xn − p〉, ≤ 1 2 { ‖yn − p‖2 + ‖xn − p‖2 − ‖xn − yn‖2 } , ≤ ‖xn − p‖2 − ‖xn − yn‖2. (16) from convaxity of function x 7→ ‖x‖2 and (16), we obtain ‖xn+1 − p‖2 = ‖tzn − p‖2, ≤ ‖zn − p‖2, ≤ ‖(1− αn)yn + αntyn − p‖2, ≤ (1− αn)‖yn − p‖2 + αn‖tyn − p‖2, ≤ ‖yn − p‖2, ≤ ‖xn − p‖2 − ‖xn − yn‖2. and so, ‖xn − yn‖2 ≤ ‖xn − p‖2 − ‖xn+1 − p‖2, ≤ (‖xn − p‖ − ‖xn+1 − p‖)(‖xn − p‖+ ‖xn+1 − p‖), ≤ ‖xn − xn+1‖(‖xn − p‖+ ‖xn+1 − p‖). since the sequence {xn} is bounded and lim n→∞ ‖xn+1 − xn‖ = 0. we have lim n→∞ ‖xn − yn‖ = 0. (17) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 7further, ‖xn+1 − p‖2 = ‖tzn − p‖2, ≤ ‖zn − p‖2, ≤ ‖(1− αn)yn + αntyn − p‖2, ≤ (1− αn)‖yn − p‖2 + αn‖tyn − p‖2 − αn(1− αn)‖yn − tyn‖2, ≤ ‖yn − p‖2 − αn(1− αn)‖yn − tyn‖2, ≤ ‖xn − p‖2 − ‖xn − yn‖2 − αn(1− αn)‖yn − tyn‖2, and so, αn(1− αn)‖yn − tyn‖2 ≤ ‖xn − p‖2 − ‖xn+1 − p‖2 − ‖xn − yn‖2, ≤ ‖xn − xn+1‖(‖xn − p‖+ ‖xn+1 − p‖)− ‖xn − yn‖2, using process (8)(c1), (14) and (17), we obtain lim n→∞ ‖yn − tyn‖ = 0. (18) consider ‖zn − tzn‖ ≤ ‖zn − yn‖+ ‖yn − tyn‖+ ‖tyn − tzn‖, (19) ≤ ‖zn − yn‖+ ‖yn − tyn‖+ ‖yn − zn‖. (20) by using (15) and (18), we obtain lim n→∞ ‖zn − tzn‖ = 0. (21) since {xn} is bounded. there exists a subsequence {xni} ⊂ {xn} such that xn ⇀ p̂. since lim n→∞ ‖xn − yn‖ = 0 and {yn} is bounded, this implies that yni ⇀ p̂ ∈ c. now by (18) we have ‖tyni − yni‖ → 0. (22) from (22) and lemma 1.5, we conclude that p̂ ∈ fix(t ).next we prove that p̂ ∈ ep (f ). since yn = t frn (xn), we have f (yn, q) + 1 rn 〈q − yn, yn − xn〉 ≥ 0, ∀ q ∈ c.by using assumption 1.1(ii), we obtain 1 rn 〈q − yn, yn − xn〉 ≥ f (q, yn), and so, 〈q − yni , yni − xni rni 〉 ≥ f (q, yni ). (23) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 8 since ‖yni−xni ‖rni ≤ ‖yni−xni ‖r → 0 and yni ⇀ p̂, therefore by assumption 1.1(iv), we obtain lim ni→∞ inf f (q, yni ) ≤ lim ni→∞ 〈q − yni , yni − xni rni 〉 = 0. that is, f (q, p̂) ≤ 0, ∀ q ∈ c. (24) further for any λ ∈ (0 , 1) and q ∈ c, let qλ = λq + (1 − λ)p̂, then qλ ∈ c and so we have f (qλ, p̂) ≤ 0. it follows from the assumption 1.1 and (24), that 0 = f (qλ, qλ), ≤ λf (qλ, q) + (1− λ)f (qλ, p̂), ≤ λf (qλ, q). this implies that f (qλ, q) ≥ 0, ∀λ ∈ (0 , 1). letting λ → 0+ by assumption 1.1, we have f (p̂, q) ≥ 0, ∀ q ∈ c. this implies that p̂ ∈ ep (f ) and hence p̂ ∈ ω. this completes theproof. � 3. numerical example here we give numerical examples for supporting our main results. all codes are done by matlab2021a. example 3.1. set h = r. let c = [0 +∞). suppose t : c → h, is defined by t (p) = p 3 . it can be easily seen that, here fix(t ) = {0}. also, we define f (s, q) = 3q2 + 2sq − 5s2, it is easy to check that f satisfy the conditions of assumption 1.1. so, for rn = r > 0, t fr (p) is non-empty and single-valued for each p ∈ c. hence for r > 0, there exists s ∈ c such that f (s, q) + 1 r 〈q − s, s − p〉 ≥ 0 ∀ q ∈ c, which is equivalent to 3rq2 + (s − p + 2r s)q + (ps − 5r s2 − s2) ≥ 0, ∀ q ∈ c. after solving the above inequality, we get s = p 1+8r for each r > 0 i.e. t fr (p) = p 1+8r for each r > 0. it can be easily seen that here ep (f ) = {0}. this implies that ω := fix(t )∩ep (f ) = {0}. now, let us choose r = 1 8 , and {αn} = 1 (n+6) . {αn} satisfy the conditions of main result. table. for different initial value, we present a table of iterations here. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 9 no. of iterations x0 = 1 x0 = −11 1.000000 -1.0000002 0.150794 -0.1419233 0.023038 -0.0204074 0.003555 -0.0029645 0.000553 -0.0004346 0.000087 -0.0000647 0.000014 -0.0000098 0.000002 -0.0000019 0.000000 0.000000 0 2 4 6 8 10 12 14 16 18 20 number of iterations -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x n x 0 = 1 x 0 = -1 figure 1. graphical representation of sequence {xn} for different choices of initialvalue x0. references [1] a. moudafi, m. théra, proximal and dynamical approaches to equilibrium problems, in: m. 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singh, ∆-convergence for proximal point algorithm and fixed point problem in cat(0) spaces, fixedpoint theory appl. 2019 (2019), 8. https://doi.org/10.1186/s13663-019-0658-3. https://doi.org/10.28924/ada/ma.2.6 https://doi.org/10.1007/bf02614504 https://doi.org/10.1007/s11075-019-00688-9 https://doi.org/10.1155/2012/843486 https://doi.org/10.1155/2012/843486 https://doi.org/10.1007/978-3-319-48311-5 https://doi.org/10.1007/978-3-319-48311-5 https://doi.org/10.1016/j.jmaa.2004.04.059 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1016/j.mcm.2007.09.014 https://doi.org/10.1016/j.jmaa.2006.08.036 https://doi.org/10.1016/j.jmaa.2004.11.017 https://doi.org/10.1016/j.jmaa.2004.11.017 https://doi.org/10.1186/s13663-019-0658-3 1. introduction and auxiliary results 2. main result 3. numerical example references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 17doi: 10.28924/ada/ma.2.17 the fractal nature of drought: power laws and fractal complexity of arizona drought sepideh azizi1, tahmineh azizi2,∗ 1department of urban planning and design, shiraz university, iran sepid.azizi97@gmail.com 2department of mechanical engineering, florida state university, usa tazizi@fsu.edu ∗correspondence: tazizi@fsu.edu abstract. in this study, we explore the possibility that the drought monitor database belongs to classof fractal process which can be characterized using a single scaling exponent. the drought monitormap identifies areas of drought and labels them by intensity: d0 abnormally dry, d1 moderatedrought, d2 severe drought, d3 extreme drought, and d4 exceptional drought. the vibration analysisusing power spectral densities (psd) method has been carried out to discover whether some typeof power-law scaling exists for various statistical moments at different scales of this database. weperform multi-fractal analysis to estimate the multi-fractal spectrum of each group. we apply higuchialgorithm to find the fractal complexity of each group and then compare them for different timeintervals. our findings reveal that we have a wide range of exponents for d0-d4. therefore, d0-d4belong to class of multi-fractal process for which a large number of scaling exponents are required tocharacterize the scaling structure. 1. introduction drought is defined as a moisture deficit bad enough to have social, environmental or economiceffects. drought is a recurring feature of nearly every climate on the planet [1–5]. in many partsof the world, including north america, we have little ability to predict exactly when drought willhappen next. but if we look at history and climate data, we can be sure that drought will happenagain at some point. in the united states, a well-developed economy and agricultural systemgenerally protect citizens from the most critical effects of drought such as shortages of food andwater. however, drought still causes extreme hardship for farm and ranch families, and individualwells may run dry. besides affecting municipal water suppliers, drought affects businesses andenvironmental interests that are reliant on adequate and timely amounts of precipitation and wa-ter, such as habitat for fish and wildlife, outdoor recreation outfitters, and landscaping and car received: 2 mar 2022. key words and phrases. fractional geometry; power spectral densities (psd); multifractal analysis; fractal dimension;drought. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 2wash services [5–14]. the drought monitor map identifies areas of drought and labels them byintensity. d1 is the least intense level and d4 the most intense. d0 areas are not in drought,but are experiencing abnormally dry conditions that could turn into drought or are recovering fromdrought but are not yet back to normal.there are different indices which have been used to asses drought severity and impacts in differenttime-scales. the normalized difference vegetation index (ndvi) is one of the most widely uti-lized drought indices to determine different drought levels [15–17]. satellite databases have beenextensively used to record and quantify the changes may happen in vegetation coverage due tochanging climate conditions. the ndvi is estimated using visible and near-infrared (nir) bandsfrom advanced very-high-resolution radiometer (avhrr), terra moderate resolution imagingspectroradiometer (modis), and landsat sensors. in general, positive ndvi values demonstratevegetated areas, zero and negative values are associated with bare soil and water bodies [15]. thetime series of the average ndvi for arizona (arizona includes regions with moderate to exceptionaldrought d1-d4) shows the highest and lowest ndvi values during 10 years (2010–2020) (for themonth of jan-dec each year) (see figure (1)). the ndvi data selected from the google earth en-terprise open source which is derived using terra moderate resolution imaging spectroradiometer(modis) and would be useful to forecast the future changes in vegetation in arizona. each stateexperiences different set of impacts during a drought. we have also displayed the table of reportedimpacts during past droughts in arizona for each level of drought on the u.s. drought monitor intable (2) (source(s): ndmc, noaa, usda). when we study real world time series data, dependingon scale and higher order moments, we may confront with data that display nonlinear power-lawbehaviours. for these type data, we need to apply multifractal analysis. in multifractal analysis wediscover whether some type of power-law scaling exists for various statistical moments at differentscales. a process called mono-fractal, if it can be characterized using a single scaling exponent,or this process is a linear function of the moments. likewise, a process called multi-fractal, if wesee the scaling behavior follows a function which is non-linear in the moments. when we studyscale invariant time series data, or data with different scaling behavior, we are not able to use theclassical time series analysis and we need to perform fractal analysis.in this study, we use fractal geometry to classify drought severity from 2000 to 2021 in arizona.we perform multifractal analysis to discover whether some type of power-law scaling exists forvarious statistical moments at different scales of these data sets. we plot the multifractal spectrato compare the width of the scaling exponent for each spectrum. a quantitative analysis commonlyknown as the fractal dimension (fd) using higuchi algorithm. 2. materials, methods and results 2.1. data. here, data has been collected using u.s. drought monitor for each week of the selectedtime period (january 2000 to nov 2021) and location (arizona, usa), see figures (3) and (4). the https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 3 figure 1. normalized difference vegetation index (ndvi) arizona between 2010-2020; google earth enterprise open source u.s. drought monitor which started from 1999, is a partnership between the national droughtmitigation center (ndmc) at the university of nebraska-lincoln, the united states departmentof agriculture (usda), and the national oceanic and atmospheric administration (noaa). eachthursday, the u.s. drought monitor (usdm) will be updated to demonstrate the location andintensity of drought across the country. using the experts’ assessments, drought categories displayconditions related to dryness and drought such as observations of how much water is available instreams, lakes, and soils compared to usual time of year (source(s): ndmc, noaa, usda) [18]. 2.2. time-frequency analysis and continuous wavelet transform (cwt). continuous wavelettransform (cwt) provides a linear time-frequency representation of non-stationary signals calledscalogram by breaking the data into scales by preserving time shifts and time scales. therefore, thewavelet transform makes the analysis of the data in different frequency ranges easier and we can https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 4 figure 2. table of the reported impacts during past droughts in arizona for eachlevel of drought on the u.s. drought monitor; (source(s): ndmc, noaa, usda) extract useful information from the time intervals between its consecutive waves of the data [19].to compute the scalogram of data which is function of time and frequency, at first we split thetime series data into overlapping segments, then we need to compute the absolute value of thecontinuous wavelet transform coefficients of each segment and finally, plot it. we have displayedthe scalogram plots of drought monitor categories arizona database (2000 present) in figures(5)-(6). 2.3. vibration frequency analysis using power spectral densities (psd). the fast fourier trans-form (fft) has been used widely to analysis of vibration frequency in computing discrete fouriertransform (dft). however, fft only works accurately if we have a finite number of dominant fre-quency components. to overcome this problem, we use the power spectral densities (psd) which https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 5 figure 3. arizona percent area in u.s. drought monitor categories database (2000present); national drought mitigation center (ndmc), the u.s. department ofagriculture (usda), and the national oceanic and atmospheric administration(noaa) figure 4. histogram of arizona percent area in u.s. drought monitor categoriesdatabase (2000 present); national drought mitigation center (ndmc), the u.s.department of agriculture (usda), and the national oceanic and atmosphericadministration (noaa) https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 6 figure 5. time-frequency representations of drought monitor categories database(2000 present) using continuous wavelet transform (cwt) in two dimensionaltime-frequency space figure 6. time-frequency representations of drought monitor categories database(2000 present) using continuous wavelet transform (cwt) in three dimensionaltime-frequency-magnitude space is applied to characterize random vibration in time series data. to compute psd, we multiply eachfrequency bin of fft by its complex conjugate to get a real spectrum and then normalize the results https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 7to frequency bin width. here, we have applied the (psd) method for our database and then we fitthe logarithm power spectral densities to their frequencies in log format using least squares ap-proximation method. finally, we calculate the slope for each regression line captures the linearityof data. in figure (7), we can see the fitted least squares approximation to the logarithm of powerspectral density of arizona drought database.moreover, we have plotted the scaling exponent graphs for arizona drought database in figure (8). figure 7. fitted least squares approximation to the logarithm of power spectraldensity of arizona drought database (2000 present) obtained by wavelet tech-niques 2.4. multifractal analysis and discrete wavelet transform (dwt). fractal dimension is one of themost often used algorithm to describe the complexity of a fractal object by measuring the changesof coverings relative to the scaling factor [20–25]. it also specifies the space filling capacity of afractal object with respect to its scaling properties in the space [26–29]. the relationship betweenscaling and covering is often hard to be characterized. the variation in the number of coverings, n(ε), with respect to the scaling factor ε, can be written as n(ε) ∝ ε−d (1) where d is the fractal dimension. the relation (1) is called scaling law that is used to demonstratethe size distribution of many objects in nature. the box counting formula which has been widely https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 8 figure 8. scaling exponent of power spectral density for drought monitor cate-gories database (2000 present). applied to approximate the fractal dimension of an irregular object is defined as db = lim a→0 ln(n(a)) ln(1/a) (2) however, this monofractal dimension is not able to fully characterize complex scaling behaviors ofmany irregular objects in the real world. that’s why to study irregular objects we need to applythe multifractal algorithm. the multifractal analysis utilizes a spectrum of singularity exponents toprovide a detailed and local description of complex scaling behaviors. in order to quantify localdensities of the fractal set, we approximate the mass probability using the following formula pi(a) = ni(a) n (3) where ni(a) is the number of mass in the i th subset of measure a, n is the total mass of the set.when we scale the mass probability pi(a) with measure a of a multifractal set, it also demonstratesthe power law behavior: pi(a) ∝ aαi (4) where αi is the singularity exponent characterizing the local scaling in the i th subset. the mul-tifractal spectrum f (α) provides a statistical distribution of singularity exponents αi . in general, https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 9 f (α) may be estimated using the legendre transformation f (α) = q α− τ(q) α(q) = d τ(q) d qwhere q is the moment and τ(q) is the mass exponent of the qth order moment. in addition, themultifractal measures may be specified by scaling of qth moments of pi(a) as n(a)∑ i=1 p q(a) i ∝ aτ(q) = a(q−1)dq (5) where dq = τ(q) (q − 1) is the generalized fractal dimension. for q = 0 equation (2.4) becomes n(a) ∝ a−d0 which is similar to formula (1).from multifractal analysis results of arizona drought database (see figure (9)), we can easilysee that we have a wide range of exponents for d0-d4, which indicates they have multifractalstructure. the drought database needs to be indexed by different exponents as we decompose theminto different subsets. therefore, d0-d4 require much more exponents to characterize their scalingproperties. figure 9. the multi-fractal spectrum analysis of arizona drought monitor cate-gories database (2000 present) shows the occurrence of multi-fractality with abroad range of exponents in data structure of d0-d4. https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 102.5. higuchi fractal dimension algorithm. when we use box counting method, we compute thefractal dimension and or the complexity of a fractal process in two dimensional space [30]. however,when we are working with many real world time series data, it fails to recognize the sudden changeshappen in data [31]. to solve this problem, there are different methods such as higuchi algorithm,power spectrum analysis, and katz algorithm that help to analyze the complexity of irregulardata [32–34]. here we use higuchi algorithm. we start with a finite time series x1, x2, x3, . . . , xn .then, we create k new time series xkm of the form xm, xm+k , xm+2k , . . . , x[m+ak]where a = (n −m)/k . for each time interval k and the initial time m such that m = 1, 2, . . . , k ,we calculate the length of xkm using lkm = ∑[a] i=1 |xm+i k − xm+(i−1)k | k r where r = (n − 1)/[a]k is the curve length normalization factor. to compute the average of curvelength for each k , we calculate the mean of lkm for m = 1, 2, . . . , k and take the average for k = 1, . . . , kmax . next, we plot log(lkm) versus log(1/k) for different time interval k . finally, wecalculate the slope of regressed line which is obtained by the least-squares approximation as thehiguchi fractal dimension for time interval k = 500. we have estimated the fractal dimension ofthe arizona drought database and plotted their regression models for each data in figure (10). figure 10. plots of log(lkm) versus log(k) for time interval k = 500, the logarithmicscale and the corresponding slope of fitted regression line (the higuchi fractaldimension) for arizona drought monitor categories database (2000 present). https://doi.org/10.28924/ada/ma.2.17 eur. j. math. anal. 10.28924/ada/ma.2.17 113. discussion drought as a slowly progressed and hidden disaster takes place in normal cycles of climate andit affects adversely environment and economic the same as other disasters. therefore, character-izing the complexity of its nature will help to predict and recognize its different stages before itdamages our societies. in u.s., the national integrated drought information system (nidis) whichis a multi-agency partnership, works on facilitating drought recording, predicting, risk managementand planning at different national levels. because of obvious impacts of drought on agriculture,water supply, energy production, public health, and wildlife, we decided to find analytical andcomputational techniques to characterize the complexity of different levels of drought from mod-erate drought d1 to exceptional drought d4 using the arizona drought monitor databases. weapplied the time-frequency analysis using continuous wavelet transform (cwt) to visualize thenon-linearity in the structure of five different groups of drought levels in the frequency domain ofdata vibration. moreover, we carried out the vibration analysis using the power-law exponent and(psd) to discover the power-law and self-similarity behaviors in the structure of drought database.we performed the multi-fractal analysis in studding the multi-fractal properties of our time seriesdata. this analysis revealed the presence of a wide range of scaling exponent for d0-d4 andmulti-fractal structure of the drought database. we continued our study by measuring the fractalcomplexity in drought time series data using higuchi algorithm. this analysis helped to comparethe self-similarity of different drought levels. although these methods helped to characterize thecomplexity in the nature of our database, however it requires further studies to find an appropriatemathematical model (deterministic or stochastic) governing 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(2022) 4doi: 10.28924/ada/ma.2.4 ∗-k-operator frame for hom∗a(x ) mohamed rossafi1,∗, roumaissae el jazzar2 and ali kacha2 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco mohamed.rossafi@usmba.ac.ma 2laboratory of partial differential equations, spectral algebra and geometry department of mathematics, faculty of sciences, university ibn tofail, kenitra, morocco roumaissae.eljazzar@uit.ac.ma, ali.kacha@yahoo.fr ∗correspondence: rossafimohamed@gmail.com abstract. in this work, we introduce the concept of ∗-k-operator frames in hilbert pro-c∗-modules,which is a generalization of k-operator frame. we present the analysis operator, the synthesisoperator and the frame operator. we also give some properties and we study the tensor product of ∗-k-operator frame for hilbert pro-c∗-modules. 1. introduction duffin and schaeffer introduced the notion of frame in nonharmonic fourier analysis in 1952 [3].in 1986 the work of duffin and schaeffer were reintroduced and developed by grossman andmeyer [7]. the concept of frame on hilbert space has already been successfully extended to proc∗-algebras and hilbert modules. many properties of frames in hilbert c∗-modules are valid forframes of multipliers in hilbert modules over pro-c∗-algebras [9].operator frames for b(h) is a new notion of frames that li and cio introduced in [11] andgeneralized by rossafi in [16]. in this work we introduce the notion of ∗-k-operator frame for thespace hom∗a(x ) of all adjointable operators on a hilbert pro-c∗-module for x .this paper is divided into three sections. in section 2 we recall some fundamental definitionsand notations of hilbert pro-c∗-modules. in section 3 we introduce the ∗-k-operator frame andwe give some of its properties. lastly we investigate tensor product of hilbert pro-c∗-modules, weshow that tensor product of ∗-k-operator frames for hilbert pro-c∗-modules x and y , present an ∗-k-operator frames for x ⊗ y , and tensor product of their frame operators is the frame operatorof their tensor product of ∗-k-operator frames. received: 18 nov 2021. key words and phrases. frame; ∗-k-operator frame; k-operator frame pro-c∗-algebra; hilbert pro-c∗-modules; tensorproduct. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 22. preliminaries the basic information about pro-c∗-algebras can be found in the works [4–6,8, 12,14,15]. c∗-algebra whose topology is induced by a family of continuous c∗-seminorms instead of a c∗-norm is called pro-c∗-algebra. hilbert pro-c∗-modules are generalizations of hilbert spacesby allowing the inner product to take values in a pro-c∗-algebra rather than in the field of complexnumbers.pro-c∗-algebra is defined as a complete hausdorff complex topological ∗-algebra a whosetopology is determined by its continuous c∗-seminorms in the sens that a net {aα} converges to 0if and only if p(aα) converges to 0 for all continuous c∗-seminorm p on a [8,10,15], and we have:1) p(ab) ≤ p(a)p(b)2) p(a∗a) = p(a)2for all a, b ∈ aif the topology of pro-c∗-algebra is determined by only countably many c∗-seminorms, then it iscalled a σ-c∗-algebra.we denote by sp(a) the spectrum of a such that: sp(a) = {λ ∈ c : λ1a − a is not invertible } forall a ∈ a. where a is unital pro-c∗-algebra with unite 1a.the set of all continuous c∗-seminorms on a is denoted by s(a). if a+ denotes the set of allpositive elements of a, then a+ is a closed convex c∗-seminorms on a. example 2.1. every c∗-algebra is a pro-c∗-algebra. proposition 2.2. [8] let a be a unital pro-c∗-algebra with an identity 1a. then for any p ∈ s(a), we have:(1) p(a) = p(a∗) for all a ∈ a(2) p (1a) = 1(3) if a, b ∈ a+ and a ≤ b, then p(a) ≤ p(b)(4) if 1a ≤ b, then b is invertible and b−1 ≤ 1a(5) if a, b ∈ a+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1(6) if a, b, c ∈ a and a ≤ b then c∗ac ≤ c∗bc(7) if a, b ∈ a+ and a2 ≤ b2, then 0 ≤ a ≤ b definition 2.3. [15] a pre-hilbert module over pro-c∗-algebra a, is a complex vector space ewhich is also a left a-module compatible with the complex algebra structure, equipped with an a-valued inner product 〈., .〉 e×e → a which is c-and a-linear in its first variable and satisfiesthe following conditions:1) 〈ξ, η〉∗ = 〈η, ξ〉 for every ξ, η ∈ e2) 〈ξ, ξ〉 ≥ 0 for every ξ ∈ e https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 33) 〈ξ, ξ〉 = 0 if and only if ξ = 0for every ξ, η ∈ e. we say e is a hilbert a-module (or hilbert pro-c∗-module over a ). if e iscomplete with respect to the topology determined by the family of seminorms p̄e(ξ) = √ p(〈ξ, ξ〉) ξ ∈ e, p ∈ s(a) let a be a pro-c∗-algebra and let x and y be hilbert a-modules and assume that i and j becountable index sets. a bounded a-module map from x to y is called an operators from x to y .we denote the set of all operator from x to y by homa(x ,y). definition 2.4. [1] an a-module map t : x −→ y is adjointable if there is a map t ∗ : y −→ xsuch that 〈tξ, η〉 = 〈ξ, t ∗η〉 for all ξ ∈ x , η ∈ y , and is called bounded if for all p ∈ s(a), thereis mp > 0 such that p̄y(tξ) ≤ mpp̄x (ξ) for all ξ ∈ x .we denote by hom∗a(x ,y), the set of all adjointable operator from x to y and hom∗a(x ) = hom∗a(x ,x ) definition 2.5. [1] let a be a pro-c∗-algebra and x ,y be two hilbert a-modules. the operator t : x → y is called uniformly bounded below, if there exists c > 0 such that for each p ∈ s(a), p̄y(tξ) 6 cp̄x (ξ), for all ξ ∈ x and is called uniformly bounded above if there exists c′ > 0 such that for each p ∈ s(a), p̄y(tξ) > c′p̄x (ξ), for all ξ ∈ x ‖t‖∞ = inf{m : m is an upper bound for t} p̂y(t ) = sup {p̄y(t (x)) : ξ ∈ x , p̄x (ξ) 6 1}it’s clear to see that, p̂(t ) 6 ‖t‖∞ for all p ∈ s(a). proposition 2.6. [2]. let x be a hilbert module over pro-c∗-algebra a and t be an invertible element in hom∗a(x ) such that both are uniformly bounded. then for each ξ ∈ x ,∥∥t−1∥∥−2∞ 〈ξ, ξ〉 ≤ 〈tξ, tξ〉 ≤ ‖t‖2∞〈ξ, ξ〉. 3. ∗-k-operator frame for hom∗a(x ) we begin this section with the definition of a k-operator frame. definition 3.1. let {ti}i∈i be a family of adjointable operators on a hilbert a-module x over aunital pro-c∗-algebra, and let k ∈ hom∗a(x ). {ti}i∈i is called a k-operator frame for hom∗a(x ),if there exist two positive constants a,b > 0 such that a〈k∗ξ,k∗ξ〉 ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉,∀ξ ∈ x . (3.1) https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 4the numbers a and b are called lower and upper bound of the k-operator frame, respectively. if a〈k∗ξ,k∗ξ〉 = ∑ i∈i 〈tiξ, tiξ〉, the k-operator frame is an a-tight. if a = 1, it is called a normalized tight k-operator frame or aparseval k-operator frame. we will now move to define the ∗-k-operator frame for hom∗a(x ). definition 3.2. let {ti}i∈i be a family of adjointable operators on a hilbert a-module x overa unital pro-c∗-algebra, and let k ∈ hom∗a(x ). {ti}i∈i is called a ∗-k-operator frame for hom∗a(h), if there exists two nonzero elements a and b in a such that a〈k∗ξ,k∗ξ〉a∗ ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉b∗,∀ξ ∈ x . (3.2) the elements a and b are called lower and upper bounds of the ∗-k-operator frame, respectively.if a〈k∗ξ,k∗ξ〉∗ = ∑ i∈i 〈tiξ, tiξ〉, the ∗-k-operator frame is an a-tight. if a = 1, it is called a normalized tight ∗-k-operator frameor a parseval ∗-k-operator frame. example 3.3. let l∞ be the set of all bounded complex-valued sequences. for any u = {uj}j∈n, v = {vj}j∈n ∈ l∞, we define uv = {ujvj}j∈n, u∗ = {ūj}j∈n, ‖u‖ = sup j∈n |uj |. then a = {l∞, ‖.‖} is a c∗-algebra. then a is pro-c∗-algebra.let x = c0 be the set of all null sequences. for any u, v ∈ x we define 〈u, v〉 = uv∗ = {uj ūj}j∈n. therefore x is a hilbert a-module.define fj = {f ji }i∈n∗ by f ji = 1 2 + 1 i if i = j and f ji = 0 if i 6= j ∀j ∈ n∗.now define the adjointable operator tj : x → x , tj{(ξi)i} = (ξi f j i )i .then for every x ∈ x we have∑ j∈n 〈tjξ, tjξ〉 = { 1 2 + 1 i }i∈n∗〈ξ, ξ〉{ 1 2 + 1 i }i∈n∗ . so {tj}j is a {12 + 1 i }i∈n∗-tight ∗-operator frame.let k : h → h defined by kξ = { ξii }i∈n∗ .then for every ξ ∈ x we have 〈k∗ξ,k∗ξ〉 ≤ ∑ j∈n 〈tjξ, tjξ〉 = { 1 2 + 1 i }i∈n∗〈ξ, ξ〉{ 1 2 + 1 i }i∈n∗ . https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 5this shows that {tj}j∈n is an ∗-k-operator frame with bounds 1, {12 + 1 i }i∈n∗ . remark 3.4. (1) every ∗-operator frame for hom∗a(x ) is an ∗-k-operator frame, for any k ∈ hom∗a(x ): k 6= 0.(2) if k ∈ hom∗a(x ) is a surjective operator, then every ∗-k-operator frame for hom∗a(x ) isan ∗-operator frame. example 3.5. let x be a finitely or countably generated hilbert a-module. hom∗a(x ). let k ∈ hom∗a(x ) an invertible element such that both are uniformly bounded and k 6= 0. let {ti}i∈i be an ∗-operator frame for x with bounds a and b, respectively. we have a〈ξ, ξ〉a∗ ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉b∗,∀ξ ∈ x . or 〈k∗ξ,k∗ξ〉 ≤ ‖k‖2∞〈ξ, ξ〉,∀ξ ∈ x .then ‖k‖−1∞ a〈k∗ξ,k∗ξ〉(‖k‖−1∞ a)∗ ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉b∗,∀ξ ∈ x . so {ti}i∈i is ∗-k-operator frame for x with bounds ‖k‖−1∞ a and b, respectively. in what follows, we introduce the analysis, the synthesis and the frame operator. we alsoestablish some properties.let {ti}i∈i be an ∗-k-operator frame for hom∗a(x ). define an operator r : x → l2(x ) by rξ = {tiξ}i∈i ,∀ξ ∈ x , then r is called the analysis operator. the adjoint of the analysis operator r, r∗ : l2(x )→ x is given by r∗({ξi}i) = ∑ i∈i t ∗ i ξi ,∀{ξi}i ∈ l2(x ). the operator r∗ is calledthe synthesis operator. by composing r and r∗, the frame operator s : x → x is given by sξ = r∗rξ = ∑ i∈i t ∗ i tiξ.note that s need not be invertible in general. but under some condition s will be invertible. theorem 3.6. let k be a surjective operators in hom∗a(x ). if {ti}i∈i is an ∗-k-operator frame for hom∗a(x ), then the frame operator s is positive, invertible and adjointable. in addition we have the reconstruction formula, ξ = ∑ i∈i t ∗ i tis −1ξ, ∀ξ ∈ x . proof. we start by showing that, s is a self-adjoint operator. by definition we have ∀ξ, η ∈ h 〈sξ, η〉 = 〈∑ i∈i t ∗i tiξ, η 〉 = ∑ i∈i 〈t ∗i tiξ, η〉 = ∑ i∈i 〈ξ, t ∗i tiη〉 https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 6 = 〈 ξ, ∑ i∈i t ∗i tiη 〉 = 〈ξ, sη〉. then s is a selfadjoint.the operator s is clearly positive.by (2) in remark 3.4 {ti}i∈i is an ∗-operator frame for hom∗a(x ).the definition of an ∗-operator gives a1〈ξ, ξ〉a∗1 ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉b∗. thus by the definition of norm in l2(x ) p̄x (rξ)2 = p̄x ( ∑ i∈i 〈tiξ, tiξ〉) ≤ p̄x (b)2p(〈ξ, ξ〉),∀ξ ∈ x . (3.3) therefore r is well defined and p̄x (r) ≤ p̄x (b). it’s clear that r is a linear a-module map. wewill then show that the range of r is closed. let {rξn}n∈n be a sequence in the range of r suchthat limn→∞rξn = η. for n,m ∈ n, we have p(a〈ξn − ξm, ξn − ξm〉a∗) ≤ p(〈r(ξn − ξm), r(ξn − ξm)〉) = p̄x (r(ξn − ξm))2. seeing that {rξn}n∈n is cauchy sequence in x , then p(a〈ξn − ξm, ξn − ξm〉a∗)→ 0, as n,m →∞.note that for n,m ∈ n, p(〈ξn − ξm, ξn − ξm〉) = p(a−1a〈ξn − ξm, ξn − ξm〉a∗(a∗)−1) ≤ p(a−1)2p(a〈ξn − ξm, ξn − ξm〉a∗). thus the sequence {ξn}n∈n is cauchy and hence there exists ξ ∈ x such that ξn → ξ as n →∞.again by (3.3), we have p̄x (r(ξn − ξm))2 ≤ p̄x (b)2p(〈ξn − ξ, ξn − ξ〉). thus p(rξn − rξ) → 0 as n → ∞ implies that rξ = η. it is therefore concluded that therange of r is closed. we now show that r is injective. let ξ ∈ x and rξ = 0. note that a〈ξ, ξ〉a∗ ≤ 〈rξ,rξ〉 then 〈ξ, ξ〉 = 0 so ξ = 0 i.e. r is injective.for ξ ∈ x and {ξi}i∈i ∈ l2(x ) we have 〈rξ, {ξi}i∈i〉 = 〈{tiξ}i∈i , {ξi}i∈i〉 = ∑ i∈i 〈tiξ, ξi〉 = ∑ i∈i 〈ξ, t ∗i ξi〉 = 〈ξ, ∑ i∈i t ∗i ξi〉. then r∗({ξi}i∈i) = ∑ i∈i t ∗ i ξi . since r is injective, then the operator r∗ has closed range and x = range(r∗), therefore s = r∗r is invertible � https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 7let k ∈ hom∗a(x ), in the following theorem we constructed an ∗-k-operator frame by using an ∗-operator frame. theorem 3.7. let {ti}i∈i be an ∗-k-operator frame in x with bounds a, b and k ∈ hom∗a(x ) be an invertible element such that both are uniformly bounded. then {tik}i∈i is an ∗-k∗-operator frame in x with bounds a, ‖k‖∞b. the frame operator of {tik}i∈i is s′ = k∗sk, where s is the frame operator of {ti}i∈i . proof. from a〈ξ, ξ〉a∗ ≤ ∑ i∈i 〈tiξ, tiξ〉 ≤ b〈ξ, ξ〉b∗,∀ξ ∈ x . we get for all ξ ∈ x , a〈kξ,kξ〉a∗ ≤ ∑ i∈i 〈tikξ, tikξ〉 ≤ b〈kξ,kξ〉b∗ ≤ ‖k‖∞b〈ξ, ξ〉(‖k‖∞b)∗. then {tik}i∈i is an ∗-k∗-operator frame in x with bounds a, ‖k‖∞b.by definition of s,we have skξ = ∑ i∈i t ∗ i tikξ. then k∗sk = k∗ ∑ i∈i t ∗i tikξ = ∑ i∈i k∗t ∗i tikξ. hence s′ = k∗sk. � corollary 3.8. let k ∈ hom∗a(x ) and {ti}i∈i be an ∗-operator frame. then {tis−1k}i∈i is an ∗-k∗-operator frame, where s is the frame operator of {ti}i∈i . proof. result of the theorem 3.7 for the ∗-operator frame {tis−1}i∈i . � 4. tensor product we denote by a⊗b, the minimal or injective tensor product of the pro-c∗-algebras a and b, itis the completion of the algebraic tensor product a⊗alg b with respect to the topology determinedby a family of c∗-seminorms. suppose that x is a hilbert module over a pro-c∗-algebra a and y is a hilbert module over a pro-c∗-algebra b. the algebraic tensor product x ⊗alg y of x and y is a pre-hilbert a⊗ b-module with the action of a⊗ b on x ⊗alg y defined by (ξ ⊗ η)(a ⊗ b) = ξa ⊗ ηb for all ξ ∈ x , η ∈ y, a ∈ a and b ∈ b and the inner product 〈·, ·〉 : ( x ⊗alg y)× (x ⊗alg y)→ a⊗alg b. defined by 〈ξ1 ⊗ η1, ξ2 ⊗ η2〉 = 〈ξ1, ξ2〉 ⊗ 〈η1, η2〉and we know that for z = ∑n i=1 ξi⊗ηi in x⊗algy we have 〈z, z〉a⊗b = ∑ i ,j〈ξi , ξj〉a⊗〈ηi , ηj〉b ≥ 0and 〈z, z〉a⊗b = 0 iff z = 0. https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 8the external tensor product of x and y is the hilbert module x ⊗y over a⊗b obtained by thecompletion of the pre-hilbert a⊗ b-module x ⊗alg y .if p ∈ m(x ) and q ∈ m(y) then there is a unique adjointable module morphism p ⊗ q : a⊗b → x ⊗y such that (p ⊗q)(a⊗ b) = p (a)⊗q(b) and (p ⊗q)∗(a⊗ b) = p ∗(a)⊗q∗(b)for all a ∈ a and for all b ∈ b (see, for example, cite the minimal or injective tensor product ofthe pro-c∗-algebras a and b, denoted by a⊗b, is the completion of the algebraic tensor product a⊗alg b with respect to the topology determined by a family of c∗-seminorms. suppose that xis a hilbert module over a pro-c∗-algebra a and y is a hilbert module over a pro-c∗-algebra b.the algebraic tensor product x ⊗alg y of x and y is a pre-hilbert a⊗b-module with the actionof a⊗ b on x ⊗alg y defined by (ξ ⊗ η)(a ⊗ b) = ξa ⊗ ηb for all ξ ∈ x , η ∈ y, a ∈ a and b ∈ b and the inner product 〈·, ·〉 : ( x ⊗alg y)× (x ⊗alg y)→ a⊗alg b. defined by 〈ξ1 ⊗ η1, ξ2 ⊗ η2〉 = 〈ξ1, ξ2〉 ⊗ 〈η1, η2〉 we also know that for z = ∑n i=1 ξi⊗ηi in x⊗algy we have 〈z, z〉a⊗b = ∑ i ,j〈ξi , ξj〉a⊗〈ηi , ηj〉b ≥ 0and 〈z, z〉a⊗b = 0 iff z = 0.the external tensor product of x and y is the hilbert module x ⊗y over a⊗b obtained by thecompletion of the pre-hilbert a⊗ b-module x ⊗alg y .if p ∈ m(x ) and q ∈ m(y) then there is a unique adjointable module morphism p ⊗ q : a⊗b → x ⊗y such that (p ⊗q)(a⊗ b) = p (a)⊗q(b) and (p ⊗q)∗(a⊗ b) = p ∗(a)⊗q∗(b)for all a ∈ a and for all b ∈ b (see, for example, [9])let i and j be countable index sets. theorem 4.1. let x and y be two hilbert pro-c∗-modules over unitary pro-c∗-algebras a and b, respectively. let {ti}i∈i ⊂ hom∗a(x ) be an ∗-k-operator frame for x with bounds a and b and frame operators st and {pj}j∈j ⊂ hom∗b(y) be an ∗-l-operator frame for k with bounds c and d and frame operators sl. then {ti ⊗ lj}i∈i,j∈j is an ∗-k⊗l-operator frame for hibert a⊗ b-module x ⊗ y with frame operator st ⊗ sp and bounds a⊗ c and b ⊗d. proof. the defintion of ∗-k-operator frame {ti}i∈i and ∗-l-operator frame {pj}j∈j gives a〈k∗ξ,k∗ξ〉aa∗ ≤ ∑ i∈i 〈tiξ, tiξ〉a ≤ b〈ξ, ξ〉ab∗,∀ξ ∈ x . c〈l∗η, l∗η〉bc∗ ≤ ∑ j∈j 〈pjη, pjη〉b ≤ d〈η, η〉bd∗,∀η ∈ y. https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 9therefore (a〈k∗ξ,k∗ξ〉aa∗)⊗ (c〈l∗η, l∗η〉bc∗) ≤ ∑ i∈i 〈tiξ, tiξ〉a ⊗ ∑ j∈j 〈pjη, pjη〉b ≤ (b〈ξ, ξ〉ab∗)⊗ (d〈η, η〉bd∗),∀ξ ∈ x ,∀η ∈ y.then (a⊗ c)(〈k∗ξ,k∗ξ〉a ⊗ 〈l∗η, l∗η〉b)(a∗ ⊗ c∗) ≤ ∑ i∈i,j∈j 〈tiξ, tiξ〉a ⊗ 〈pjη, pjη〉b ≤ (b ⊗d)(〈ξ, ξ〉a ⊗ 〈η, η〉b)(b∗ ⊗d∗),∀ξ ∈ x ,∀η ∈ y.consequently we have (a⊗ c)〈k∗ξ ⊗ l∗η,k∗ξ ⊗ l∗η〉a⊗b(a⊗ c)∗ ≤ ∑ i∈i,j∈j 〈tiξ ⊗ pjη, tiξ ⊗ pjη〉a⊗b ≤ (b ⊗d)〈ξ ⊗ η, ξ ⊗ η〉a⊗b(b ⊗d)∗,∀ξ ∈ x ,∀η ∈ y. then for all ξ ⊗ η in x ⊗ y we have (a⊗ c)〈(k ⊗ l)∗(ξ ⊗ η), (k ⊗ l)∗(ξ ⊗ η)〉a⊗b(a⊗ c)∗ ≤ ∑ i∈i,j∈j 〈(ti ⊗ pj)(ξ ⊗ η), (ti ⊗ pj)(ξ ⊗ η)〉a⊗b ≤ (b ⊗d)〈ξ ⊗ η, ξ ⊗ η〉a⊗b(b ⊗d)∗. the last inequality is true for every finite sum of elements in x ⊗alg y and then it’s true for all z ∈ x ⊗k. it shows that {ti ⊗ pj}i∈i,j∈j is an ∗-k ⊗ l-operator frame for hilbert a⊗b-module x ⊗ y with lower and upper bounds a⊗ c and b ⊗d, respectively.by the definition of frame operator st and sp we have st ξ = ∑ i∈i t ∗i tiξ, ∀ξ ∈ x . spη = ∑ j∈j p ∗j pjη, ∀η ∈ y. therefore (st ⊗ sp )(ξ ⊗ η) = st ξ ⊗ spη = ∑ i∈i t ∗i tiξ ⊗ ∑ j∈j p ∗j pjη = ∑ i∈i,j∈j t ∗i tiξ ⊗ p ∗j pjη = ∑ i∈i,j∈j (t ∗i ⊗ p ∗j )(tiξ ⊗ pjη) https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 10 = ∑ i∈i,j∈j (t ∗i ⊗ p ∗j )(ti ⊗ pj)(ξ ⊗ η) = ∑ i∈i,j∈j (ti ⊗ pj)∗(ti ⊗ pj)(ξ ⊗ η). then by the uniqueness of frame operator, the last expression is equal to st⊗p (ξ⊗η). consequentlywe have (st ⊗sp )(ξ⊗η) = st⊗p (ξ⊗η). the last equality is true for every finite sum of elementsin x ⊗alg y and then it’s true for all z ∈ x ⊗ y . it follows that (st ⊗ sp )(z) = st⊗p (z). thus st⊗p = st ⊗ sp . � references [1] n. haddadzadeh, g-frames in hilbert modules over pro-c*-algebras, int. j. ind. math. 9(4) (2017) 259-267.[2] m. azhini and n. haddadzadeh, fusion frames in hilbert modules over pro-c∗-algebras, int. j. ind. math. 5(2)(2013) article id ijim-00211.[3] r. j. duffin and a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952) 341-366.[4] m. fragoulopoulou, an introduction to the representation theory of topological ∗-algebras, schriftenreihe, univ.münster, 48 (1988) 1-81.[5] m. fragoulopoulou, tensor products of enveloping locally c∗-algebras, schriftenreihe, univ. münster (1997) 1-81.[6] m. fragoulopoulou, topological algebras with involution, north holland, amsterdam, 2005.[7] a. grossman, and y. meyer, painless nonorthogonal expansions, j. math. phys. 27 (1986) 1271-1283.[8] a. inoue, locally c∗-algebra, mem. fac. sci. kyushu univ. ser. a, math. 25(2) (1972) 197-235.[9] m. joita, on frames in hilbert modules over pro-c∗-algebras, topol. appl. 156 (2008) 83-92.[10] e. c. lance, hilbert c∗-modules, a toolkit for operator algebraists, london math. soc. lecture note series 210.cambridge univ. press, cambridge, 1995.[11] c. y. li and h. x. cao, operator frames for b(h), wavelet analysis and applications. birkhäuser basel, 2006.67-82.[12] a. mallios, topological algebras: selected topics, north holland, amsterdam, 1986.[13] m. naroei and a. nazari, some properties of –frames in hilbert modules over pro-c∗-algebras, sahand commun.math. anal. 16 (2019) 105–117.[14] n. c. phillips, inverse limits of c*-algebras, j. oper. theory. 19 (1988) 159-195.[15] n. c. phillips, representable k-theory for σ -c∗-algebras, k-theory, 3 (1989) 441-478.[16] m. rossafi and s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra. 8 (2019) 85-95. https://doi.org/10.28924/ada/ma.2.4 1. introduction 2. preliminaries 3. -k-operator frame for homa(x) 4. tensor product references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 15doi: 10.28924/ada/ma.2.15 quasi-likelihood estimation in fractional levy spdes from poisson sampling jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223-0001, usacorrespondence: j.bishwal@uncc.edu abstract. we study the quasi-likelihood estimator of the drift parameter in the stochastic partialdifferential equations driven by a cylindrical fractional levy process when the process is observed atthe arrival times of a poisson process. we use a two stage estimation procedure. we first estimatethe intensity of the poisson process. then we plug-in this estimate in the quasi-likelihood to estimatethe drift parameter. we obtain the strong consistency and the asymptotic normality of the estimators. 1. introduction parameter estimation in infinite dimensional stochastic differential equations was first studied byloges [20]. when the length of the observation time becomes large, he obtained consistency andasymptotic normality of the maximum likelihood estimator (mle) of a real valued drift parameterin a hilbert space valued sde. koski and loges [18] extended the work of loges [20] to minimumcontrast estimators. koski and loges [17] applied the work to a stochastic heat flow problem. seethe monograph bishwal [5] for asymptotic results on likelihood inference and bayesian inferencefor drift estimation of finite and infinite dimensional stochastic differential equations.huebner, khasminskii and rozovskii [12] started statistical investigation in spdes. they gavetwo contrast examples of parabolic spdes in one of which they obtained consistency, asymptoticnormality and asymptotic efficiency of the mle as noise intensity decreases to zero under thecondition of absolute continuity of measures generated by the process for different parameters (thesituation is similar to the classical finite dimensional case) and in the other they obtained theseproperties as the finite dimensional projection becomes large under the condition of singularity ofthe measures generated by the process for different parameters. the second example was extendedby huebner and rozovskii [13] and the first example was extended by huebner [11] to mle forgeneral parabolic spdes where the partial differential operators commute and satisfy differentorder conditions in the two cases. received: 19 feb 2022. key words and phrases. cylindrical fractional levy process, stochastic partial differential equations, space-time colornoise, convoluted levy field, infinite divisibility, poisson sampling, quasi maximum likelihood estimator, consistency,asymptotic normality. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 2huebner [10] extended the problem to the ml estimation of multidimensional parameter. lototskyand rozovskii [21] studied the same problem without the commutativity condition. small noiseasymptotics of the nonparmetric estimation of the drift coefficient was studies by ibragimov andkhasminskii [14].bishwal [3] proved the bernstein-von mises theorem (bvt) and obtained asymptotic properties ofregular bayes estimator of the drift parameter in a hilbert space valued sde when the correspond-ing ergodic diffusion process is observed continuously over a time interval [0, t ]. the asymptoticsare studied as t → ∞ under the condition of absolute continuity of measures generated by theprocess. results are illustrated for the example of an spde.bishwal [4] obtained bvt and spectral asymptotics of bayes estimators for parabolic spdeswhen the number of fourier coefficients becomes large. in that case, the measures generatedby the process for different parameters are singular. here we treat the case when the measuresgenerated by the process for different parameters are absolutely continuous under some conditionson the order of the partial differential operators. bishwal [9] studied the asymptotic properties ofthe posterior distributions and bayes estimators when one has either fully observed process orfinite-dimensional projections. the asymptotic parameter is only the intensity of noise. in thispaper we treat the more general model with non-gaussian noise with long memory.on the other hand, recently long memory processes, i.e. processes with slowly decaying auto-correlation and processes with jumps have received attention in finance, engineering and physics.the simplest continuous time long memory process is the fractional brownian motion discovered bykolmogorov [15] and later on studied by levy [19] and mandelbrot and van ness [27]. continuoustime long memory jump process is fractional levy process. hence fractional levy process can alsobe called the kolmogorov-levy process.we generalize fractional spde process to include non-normal innovations. we consider hurstparameter greater than half. this model is interesting as it preserves both jumps and long memory.a normalized fractional brownian motion {wh t , t ≥ 0} with hurst parameter h ∈ (0, 1) is acentered gaussian process with continuous sample paths whose covariance kernel is given by e(wh t w h s ) = 1 2 (s2h + t2h − |t − s|2h), s, t ≥ 0. the process is self similar (scale invariant) and it can be represented as a stochastic integralwith respect to standard brownian motion. for h = 1 2 , the process is a standard brownian motion.for h 6= 1 2 , the fbm is not a semimartingale and not a markov process, but a dirichlet process.the increments of the fbm are negatively correlated for h < 1 2 and positively correlated for for h < 1 2 and in this case they display long-range dependence. the parameter h which is alsocalled the self similarity parameter, measures the intensity of the long range dependence. thearima(p, d, q) with autoregressive part of order p, moving average part of order q and fractionaldifference parameter d ∈ (0, 0.5) process converge in donsker sense to fbm. see mishura [22].the fractional levy ornstein-uhlenbeck (fou) process, is an extension of fractional ornstein-uhlenbeck process with fractional levy motion (flm) driving term. in finance, it could be useful https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 3as a generalization of fractional vasicek model, as one-factor short-term interest rate model whichcould take into account the long memory effect and jump of the interest rate. the model parameteris usually unknown and must be estimated from data.fractional levy process (flp) is defined as mh,t = 1 γ(h + 1 2 ) ∫ r [(t − s) h−1/2 + − (−s) h−1/2 + ]dms , t ∈ r where {mt , t ∈ r} is a levy process on r with e(m1) = 0, e(m2 1 ) < ∞ and without browniancomponent.here are some properties of the fractional levy process:1) the covariance of the process is given by cov(mh,t ,mh,s) = e(m2 1 ) 2γ(2h + 1) sin(πh) [|t|2h + |s|2h − |t − s|2h]. 2) mh is not a martingale. for a large class of levy processes, mh is neither a semimartingale.3)mh is hölder continuous of any order β less than h − 1 2 .4) mh has stationary increments.5) mh is symmetric.6) m is self-similar, but mh is not self-similar.7) mh has infinite total variation on compacts.thus flp is a generalization and a natural counterpart of fbm. fractional stable motion is aspecial case of flp. first we discuss estimation in partially observed models and then we discussestimation in directly observed model in finite dimensional set up. in finance, the log-volatilityprocess can be modeled as a fractionally integrated moving average (fima) process which isdefined as yh(t) = ∫ t −∞ gh(t − u)dmu, t ∈ r where gh(t) = 1 γ(h − 1 2 ) ∫ t 0 g(t − s)sh− 3 2 ds, t ∈ r which is the riemann-liouville fractional integral of order h and the kernel g is the kernel of ashort memory moving average process. the log-volatility process will have slow (hyperbolic rate)decay of the auto-correlation function (acf).the process yh(t) can be written as yh(t) = ∫ t −∞ g(t − u)dmh,u, t ∈ r. we assume the following conditions on the kernel g : r → r, namely 1) g(t) = 0 for all t < 0(causality), 2) |g(t)| ≤ ce−ct for some constants c > 0 and c > 0 (short memory). https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 4the fima process is stationary and is infinite divisible. it has long memory and jumps whichagree empirically with stochastic volatility models. the asset return can be modeled as a coga-rch process dx(t) = √ eyh(t)dltwhere (lt , t ∈ r is another levy process and the initial value yh(0) is independent of l.consider the kernel g(t − s) = σe−θ(t−s)i(0,∞)(t − s), θ > 0 then gh(t) = σ γ(h − 1 2 ) ∫ ∞ 0 eθ(t−s)i(0,∞)(t − s)sh− 3 2 ds, t ∈ r. note that uh,θ,σt = ∫ r gh(t − u)dmu, t ∈ r is the fractional levy ornstein-uhlenbeck (flou) process satisfying the fractional langevin equa-tion dut = −θutdt + σdmh,t , t ∈ r.the process has long memory. levy driven processes of ornstein-uhlenbeck type have beenextensively studied over the last few years and widely used in finance, see barndorff-neilsenand shephard [1]. flou process generalizes fou process to include jumps. maximum quasi-likelihood estimation in fractional levy stochastic volatility model was studied in bishwal [6].berry-esseen inequalities for the discretely observed ornstein-uhlenbeck-gamma process wasstudied in bishwal [7]. minimum contrast estimation in fractional ornstein-uhlenbeck processbased on both continuous and discrete observations was studied in bishwal [8].consider the asset return driven by fractional levy process dsh,t = σt−dlh,t , t > 0, s0 = 0, with log-volatility logσ2 t = µ+xt , t ≥ 0 where the levy driven ou process x satisfies dxt = −θxtdt + dmt , t > 0 with θ ∈ r+ and the driving compound poisson process m is a levy process with levy symbol ψm(u) = − u2 2 + ∫ r (e iux − 1)φ0,1/λ(dx), where φ0,1/λ being a normal distribution with mean 0 and variance 1/λ. this means that m is thesum of a standard brownian motion w and a compound poisson process jt = ∑nt k=1 zk , j−t =∑−n−t k=1 z−k , t ≥ 0 where (nt , t ∈ r) is an independent poisson process with intensity λ > 0 and https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 5jump times (tk)k∈z, i.e., mt = wt + jt . the poisson process n is also independent from the i.i.d.sequence of jump sizes (zk)k∈z with z1 ∼ n(0, 1/λ). the levy process m in this case is given by mt = nt∑ k=1 (αzk + γ|zk |)− ct, t > 0 and c := γ ∫ r |x |λφ0,1/λ(dx) = √ 2λ π γ. {m−t , t ≥ 0} is defined analogously. the stationary log-volatility is given by logσ2 t = µ+ ∫ t −∞ e−θ(t−s)dms . we observe s at n consecutive jump times 0 = t0 < t1 < . . . < tn < t < tn+1, n ∈ z over the timeinterval [0, t ]. the state process x has then the following autoregressive representation xti = e−θ∆tixti−1 + nti∑ k=nti−1 +1 e−θ(ti−tk)[αzk + γ|zk |]− ∫ ti ti−1 e−θ(ti−s)cds = e−θ∆tixti−1 + αzi + ( |zi | − c θ (1− e−θ∆ti ) ) where ∆ti := ti − ti−1, i = 1, 2, . . . , n and nti−1 + 1 = nti = i .we do the parameter estimation in two steps. the rate λ of the poisson process n can beestimated given the jump times ti , therefore it is done at a first step. since we observe totalnumber of jumps n of the poisson process n over the t intervals of length one, the mle of λ isgiven by λ̂n := n t .to estimate the remaining parameters (α, θ, µ), we use the quasi maximum likelihood estimationprocedure in conditionally heteroscedastic time series models developed by straumann [25].assuming that s∆ti h,ti given s∆ti−1 h,ti−1 , . . . , s∆t1 h,t1 , x0 is conditionally normally distributed with meanzero and variance σ2 ti−/λ, the conditional log-likelihood given the initial value x0 has the repre-sentation l(ϑ|s∆ h, λ) := − n 2 log(2π)− 1 2 ( n∑ i=1 log(σ2 ti−/λ)− n∑ i=1 (s∆ti h,ti )2 σ2 ti−/λ ) . where s∆ti h,ti = sh,ti − sh,ti−1 is the return at time ti . since the volatility is unobservable, this log-likelihood can not be evaluated numerically. the quasi log-likelihood function for ϑ = (θ, α, γ, µ)given the data s∆ h := (s∆t1 h,t1 , s∆t2 h,t2 , . . . , s∆tn h,tn ) and the mle λ̂n is defined as l(ϑ|s∆ h, λ̂n) := − 1 2 n∑ i=1 log(σ̂2 h,ti (ϑ, λ̂n))− 1 2 n∑ i=1 (s∆ti h,ti )2 σ̂2 h,ti (ϑ, λ̂n)/λ̂nwhere the estimates of the volatility σ2 h,ti , i = 1, 2, . . . , n are given by σ̂2 h,ti (ϑ, λn) := exp(µ+ e−α∆tixh,ti−1 (ϑ, λ)− ĉ∆ti), ß = 1, 2, . . . , n and given the parameters ϑ and λ the estimates of the state process x are given by the recursion x̂h,ti = e−θ∆ti x̂h,ti−1 + α sh,ti σ̂ti (ϑ, λ) + ( sh,ti σ̂ti (ϑ, λ) − ĉ∆ti ) , i = 1, 2, . . . , n https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 6 note that e(|w |) = √ 2 πλ , w ∼ n(0, 1/λ).here the approximation (1−e−z) ≈ z for small z is used and sh,ti σ̂ti (ϑ,λ) approximates the innovation zi . the recursion needs a starting value x̂h,0 which will be set equal to the mean value of thestationary distribution of x which is zero. the mean value zero of the stationary distribution of x .qmle of ϑ is defined as ϑ̂n := arg max ϑ∈θ l(ϑ|s∆ h, λ̂n). let (ω,f , {ft}t≥0, p ) be the stochastic basis on which is defined the ornstein-uhlenbeck process xt satisfying the itô stochastic differential equation dxt = −θxtdt + dmh t , t ≥ 0, where {mh t } is a fractional levy motion with h > 1/2 with the filtration {ft}t≥0 and θ ∈ r+ isthe unknown parameter to be estimated on the basis of completely directly observed continuousobservation of the process {xt} on the time interval [0, t ]. observe that xt = ∫ t −∞ e−θ(t−s)dmh s . this process is stationary and is a process with long memory. it can be shown that xti is astationary discrete time ar(1) process with autoregression coefficient φ ∈ (0, 1) with the followingrepresentation xti = φxti−1 + εti−1where φ = e−θ∆ and εti−1 = ∫ ti ti−1 e−θ(ti−u)dmh u .then the problem is a ar(1) estimation with non-gaussian non-martingale error. for equidistantsampling, one can study the least squares estimator which boils down to the study of error distribu-tion for non-semimartingales. one can specialize to the case when m is a either a gamma processor an inverse gaussian process in order to have infinite number of jumps in a finite time inter-val unlike the compound poissoan case which have finite number of jumps in a finite time interval.these fractional gamma and fractional inverse gaussian ornstein-uhlenbeck (flou) processes arelou processes which include long memory. in the next section we deal with completely observedprocess.the rest of the paper is organized as follows : section 2 contains model, assumptions andpreliminaries. section 3 contains the asymptotic properties of quasi likelihood estimator. 2. flspde model and preliminaries in order to introduce fractional levy stochastic partial differential equation (flsode) we proceedas follows. let us fix θ0, the unknown true value of the parameter θ. let (ω,f , p ) be a complete https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 7probability space and w (t, x) be a process on this space with values in the schwarz space ofdistributions d′(g) such that for φ,ψ ∈ c∞0 (g), ‖φ‖−1 l2(g) 〈w (t, ·), φ(·)〉 is a one dimensionalwiener process and e(〈w (s, ·), φ(·)〉〈w (t, ·), ψ(·)〉) = (s ∧ t)(φ,ψ)l2(g).this process is usually referred to as the cylindrical brownian motion (c.b.m.).we assume that there exists a complete orthonormal system {hi}∞i=1 in l2(g)) such that forevery i = 1, 2, . . . , hi ∈ wm,2 0 (g) ∩ c∞(g) and λθhi = βi(θ)hi , and lθhi = µi(θ)hi for all θ ∈ θ where lθ is a closed self adjoint extension of aθ, λθ := (k(θ)i − lθ)1/2m, k(θ) is a constant andand the spectrum of the operator λθ consists of eigen values {βi(θ)}∞i=1 of finite multiplicities and µi = −β2m i + k(θ).cflp mh(t) can be expanded in the series mh(t, x) = ∞∑ i=1 mh,i(t)hi(x) where {mh,i(t)}∞i=1 are independent one dimensional flps, see peszat and zabczyk [24]. thelatter series converges p -a.s. in h−ν for ν > d/2. indeed ‖mh(t)‖2 −ν = ∞∑ i=1 m2 h,i(t)‖hi‖2 −ν = ∞∑ i=1 m2 h,i(t)β −2ν i and the later series converges p -a.s.consider the parabolic spde duθ(t, x) = θuθ(t, x) + ∂2 ∂x2 uθ(t, x)dt + dmh(t, x), t ≥ 0, x ∈ [0, 1] (2.1) u(0, x) = u0(x) ∈ l2([0, 1]) (2.2) uθ(t, 0) = uθ(t, 1), t ∈ [0, t ], (2.3)here θ ∈ θ ⊆ r is the unknown parameter to be estimated on the basis of the observationsof the field uθ(t, x), t ≥ 0, x ∈ [0, 1]. for x ∈ [0, 1], we observe the process {ut , t ≥ 0} attimes {t0, t1, t2, ....}. we assume that the sampling instants {ti , i = 0, 1, 2...} are generated bya poisson process on [0,∞), i.e., t0 = 0, ti = ti−1 + ξi , i = 1, 2, ... where ξi are i.i.d. positiverandom variables with a common exponential distribution f (x) = 1−exp(−λx). note that intensityparameter λ > 0 is the average sampling rate which is assumed to be known. it is also assumedthat the sampling process ti , i = 0, 1, 2, ... is independent of the observation process {xt , t ≥ 0}.we note that the probability density function of tk+i − tk is independent of k and is given by thegamma density fi(t) = λ(λt)i−1 exp(−λt)it/(i − 1)!, i = 0, 1, 2, .... (2.4)where it = 1 if t ≥ 0 and it = 0 if t < 0. https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 8consider the fourier expansion of the process u(t, x) = ∞∑ t=1 ui(t)φi(x) (2.5) corresponding to some orthogonal basis {φi(x)}∞i=1. note that the fourier coefficients {uθi (t), i ≥ 1} are independent one dimensional ornstein-uhlenbeck processes duθi (t) = µθi u θ i (t)dt + β−νi dmh,i(t) (2.6) uθi (0) = uθ0i ,recall that µi(θ) = k(θ)− β2m i . thus duθi (t) = (k(θ)− β2m i )uθi (t)dt + β−νi dmh,i(t) (2.7) the random field u(t, x) is observed at discrete times t and discrete positions x . equivalently, thefourier coefficients uθi (t) are observed at discrete time points.now we focus on the fundamental semimartingale behind the o-u model. define κh := 2hγ(3/2−h)γ(h + 1/2), kh(t, s) := κ−1 h (s(t − s)) 1 2 −h, ηh := 2hγ(3− 2h)γ(h + 1 2 ) γ(3/2−h) , vt ≡ vht := η−1 h t2−2h, mh t := ∫ t 0 kh(t, s)dmh s . for using girsanov theorem for brownian motion, since a radon-nikodym derivative process is al-ways a martingale, a central problem is how to construct an appropriate martingale which generatesthe same filtration, up to sets of measure zero, as the non-semimartingale called the fundamental martingale.extending norros et al. [23] it can be shown that mh t is a martingale, called the fundamen-tal martingale whose quadratic variation 〈mh〉t is vht . moreover, the natural filtration of themartingale mh coincides with the natural filtration of the flp mh since mh t := ∫ t 0 k(t, s)dmh s holds for h ∈ (1/2, 1) where kh(t, s) := h(2h − 1) ∫ t s rh− 1 2 (r − s)h− 3 2 dr, 0 ≤ s ≤ t and for h = 1/2, the convention k1/2 ≡ 1 is used.define qi(t) := d dvt ∫ t 0 kh(t, s)ui(s)ds, i ≥ 1. https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 9it is easy to see that qi(t) = ηh 2(2− 2h) { t2h−1zi(t) + ∫ t 0 r2h−1dzi(s) } . define the process zi = (zi(t), t ∈ [0, t ]) by zi(t) := ∫ t 0 kh(t, s)dui(s). extending kleptsyna and le breton [16], we have:(i) zi is the fundamental semimartingale associated with the process ui .(ii) zi is a (ft) -semimartingale with the decomposition zi(t) = µi(θ) ∫ t 0 qi(s)dvs + β−νi m h t . (iii) ui admits the representation ui(t) = ∫ t 0 kh(t, s)dzi(s). (iv) the natural filtration (zi(t)) of zi and (ui(t)) of ui coincide. we focus on our obserbations now. note that for equally spaced data (homoscedastic case) vtk − vtk−1 = η−1 h ( t n )2−2h [k2−2h − (k − 1)2−2h], k = 1, 2, · · · , n. (2.8) for h = 0.5, vtk − vtk−1 = η−1 h ( t n )2−2h [k2−2h − (k − 1)2−2h] = t n , k = 1, 2, . . . , n. we have qi(t) = d dvt ∫ t 0 kh(t, s)ui(s)ds = κ−1 h d dvt ∫ t 0 s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 d dt ∫ t 0 s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 ∫ t 0 d dt s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 ∫ t 0 s1/2−h(t − s)−1/2−hui(s)ds. (2.9) the process qi depends continuously on ui and therefore, the discrete observations of ui doesnot allow one to obtain the discrete observations of qi . the process qi can be approximated by q̃i(n) = κ−1 h ηhn 2h−1 n−1∑ j=0 j1/2−h(n − j)−1/2−hui(j). (2.10) https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 10 it is easy to show that q̃i(n)→ qi(t) almost surely as n →∞, see tudor and viens [26].define a new partition 0 ≤ r1 < r2 < r3 < · · · < rmk = tk , k = 1, 2, · · · , n. define q̃i(tk) = κ−1 h ηht 2h−1 k mk∑ j=1 r 1/2−h j (rmk − rj) −1/2−hui(rj)(rj − rj−1), (2.11) k = 1, 2, · · · , n.it is easy to show that q̃i(tk)→ qi(t) almost surely as mk →∞ for each k = 1, 2, · · · , n.we use this approximate observation in the calculation of our estimators. thus our observationsare ui(t) ≈ ∫ t 0 kh(t, s)dz̃i(s) where z̃i(t) = θ ∫ t 0 q̃i(s)dvs +mh t . (2.12) observed at poisson arrivals t1, t2, . . . , tn. we observe just one such approximate fourier coefficient ui(t) which we denote by u(t) and the corresponding observations are denoted by ut1 , ut2 , . . . , utnand let n →∞. ideally we are in a large time asymptotic framework.now we focus on the estimation methodology. define ρ := ρ(λ, θ) = λ λ− κ(θ) + β2m i . (2.13) the quasi likelihood estimator is the solution of the estimating equation: g∗n(θ) = 0 (2.14) where g∗n(θ) = β2ν i λ(ρ(λ, θ))2 ρ(λ, 2θ) n∑ i=1 uti−1 ( (uti−1 θρ(λ, θ))2 + λ )−1 (uti − ρ(λ, θ)uti−1 ) (2.15) we call the solution of the estimating equation the quasi likelihood estimator. there is no explicitsolution for this equation.the optimal estimating function for estimation of the unknown parameter θ is gn(θ) = β2ν i n∑ i=1 uti−1 [uti − ρ(λ, θ)uti−1 ]. (2.16) the martingale estimation function (mef) estimator of ρ is the solution of gn(θ) = 0 and isgiven by ρ̂n := ∑n i=1 uti−1 uti∑n i=1 u 2 ti−1 . (2.17) 3. main results we do the parameter estimation in two steps: the rate λ of the poisson process can be estimatedgiven the arrival times ti , therefore it is done at a first step. since we observe total number ofarrivals n of the poisson process over the t intervals of length one, the mle of λ is given by λ̂n := n t . (3.1) https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 11 theorem 3.1 we have λ̂n → λ a.s. as n →∞, √ n(λ̂n − λ)→d n (0, eλ(1− e−λ)) as n →∞. proof. let vi be the number of arrivals in the interval (i − 1, i ]. then vi , i = 1, 2, . . . , n are i.i.d.poisson distributed with parameter λ. since φ is continuous, we have i{0}(vi) = i{0}(u(ti)) a.s. i = 1, 2, . . . , n. note that 1 n n∑ i=1 i{0}(uti )→ a.s. e(i{0}v1) = p (v1 = 0) = e−λ as n →∞. lln and clt and delta method applied to the sequence i{0}(uti ), i = 1, 2, . . . , n give the results. the clt result above allows us to construct confidence interval for the jump rate λ. corollary 3.1 a 100(1− α)% confidence interval for λ is given by[ n t − z1−α 2 √ 1 n − 1 t , n t + z1−α 2 √ 1 n − 1 t ] where z1−α 2 is the (1− α 2 )-quantile of the standard normal distribution. we obtain the strong consistency and asymptotic normality of the mef estimator. theorem 3.2 we have ρ̂n → ρ a.s. as n →∞, √ n(ρ̂n − ρ)→d n (0, λ−i(1− e−ρ)) as n →∞. proof: by using the fact that every stationary mixing process is ergodic, it is easy to show thatif ut is a stationary ergodic o-u process and ti is a process with nonnegative i.i.d. incrementswhich is independent of ut , then {uti , i ≥ 1} is a stationary ergodic process. hence {uti , i ≥ 1} isa stationary ergodic process.observe that uθi (t) := vi is stationary ergodic and vi ∼ n (0, σ2) where σ2 is the variance of u0. thus by slln for zero mean square integrable martingales, we have as n →∞, 1 n n∑ i=1 uti−1 uti → a.s. e(ut0ut1 ) = ρe(u2 t0 ) 1 n n∑ i=1 u2 ti−1 →a.s. e(u2 t0 ) thus ∑n i=1 uti−1 uti∑n i=1 u 2 ti−1 →a.s. ρ. further, √ n(ρ̂n − ρ) = n−1/2 ∑n i=1 uti−1 (uti − θuti−1 ) n−1 ∑n i=1 u 2 ti−1 . https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 12since e(ut1ut2 |ut1 ) = θu2 t1it follows by lemma 3.1 in bibby and srensen [2] n−1/2 n∑ i=1 uti−1 (uti − θuti−1 ) converges in distribution to normal distribution with mean zero and variance equal to e[(ut1ut2 )− e(ut1ut2 |ut1 )]2 = 1− e2(θ−β1δ){2(β1 − θ)(βi + 1)}−1. applying delta method the result follows. in the next step, we use the estimator of λ to estimate θ.note that 1 ρ̂n = ∑n i=1 u 2 ti−1∑n i=1 uti−1 uti . hence 1 + β2m 1 − κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1 uti . thus β2m 1 − κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1 uti − 1 = − ∑n i=1 uti−1 [uti − uti−1 ]∑n i=1 uti−1 uti . now replace λ by its estimator mle λ̂n. β2m 1 − κ(θ) = − ∑n i=1 uti−1 [uti − uti−1 ] t n ∑n i=1 uti−1 uti . thus θ̂n = κ−1 ( β2m 1 + ∑n i=1 uti−1 [uti − uti−1 ] t n ∑n i=1 uti−1 uti ) . since the function κ−1(·) is a continuous function, by application of delta method, the followingresult is a consequence of theorem 3.2. theorem 3.3 θ̂n →a.s. θ as n →∞, √ n(θ̂n − θ)→d n (0, (κ′(θ))−2λ2(1− e−2λ−1(κ(θ)−β2m 1 ))) as n →∞.in the second stage, we plug-in λ by its estimator λ̂n. remark sub-fractional brownian motion, which has main properties of the fractional brownianmotion, excluding the stationarity of increments, has the covariance function ch(s, t) = s2h + t2h − 1 2 [ (s + t)2h + |s − t|2h ] , s, t > 0. https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 13one can gereneralize this to sub-fractional levy process by plug-in method which would havenonstationary increments and corresponding spde models could be used for modeling in financeand biology. references [1] o.e. barndorff-nielsen, n. shephard, non-gaussian ornstein-uhlenbeck-based models and some of their uses infinancial economics, j. r. stat. soc. b. 63 (2001) 167–241. https://doi.org/10.1111/1467-9868.00282.[2] b.m. bibby, m. sørensen, m. sorensen, martingale estimation functions for discretely observed diffusion processes,bernoulli. 1 (1995) 17-39. https://doi.org/10.2307/3318679.[3] j.p.n. bishwal, bayes and sequential estimation in hilbert space valued stochastic differential equations, j. koreanstat. soc. 28 (1999) 93-106.[4] j.p.n. bishwal, the bernstein-von mises theorem and 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https://doi.org/10.2478/s13540-011-0024-6 https://doi.org/10.1023/a:1009990504925 https://doi.org/10.1007/978-1-4615-7909-0_18 https://doi.org/10.1007/bf01204212 https://doi.org/10.1023/a:1021220818545 https://doi.org/10.1016/0167-7152(85)90015-x https://doi.org/10.1080/17442508608833374 eur. j. math. anal. 10.28924/ada/ma.2.15 14 [20] w. loges, girsanov’s theorem in hilbert space and an application to the statistics of hilbert spacevalued stochasticdifferential equations, stoch. processes appl. 17 (1984) 243–263. https://doi.org/10.1016/0304-4149(84) 90004-8.[21] s.v. lototsky, b.l. rosovskii, spectral asymptotics of some functionals arising in statistical inference for spdes,stoch. processes appl. 79 (1999) 69–94. https://doi.org/10.1016/s0304-4149(98)00079-9.[22] i.s. mishura, stochastic calculus for fractional brownian motion and related processes, springer-verlag, berlin,new york, 2008.[23] i. norros, e. valkeila, j. virtamo, an elementary approach to a girsanov formula and other 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https://doi.org/10.1214/009053606000001541 https://doi.org/10.1137/1010093 references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 10doi: 10.28924/ada/ma.2.10 weak and strong convergence theorems of modified projection-type ishikawa iteration scheme for lipschitz α-hemicontractive mappings imo kalu agwu∗, donatus ikechi igbokwe department of mathematics, micheal okpara university of agriculture, umudike, umuahia abia state, nigeria agwuimo@gmail.com, igbokwedi@yahoo.com ∗correspondence: agwuimo@gmail.com abstract. in this paper, we establish weak and strong convergence theorems of a two-step modifiedprojection-type ishikawa iterative scheme to the fixed point of α-hemicontractive mappings withoutany compactness assumption on the operator or the space. our results extend, improve and generalizeseveral previously known results of the existing literature. 1. introduction let h be a real hilbert space with inner product 〈, ., 〉 and induced norm ‖, ., ‖, k a nonemptyconvex and closed subset of h and t : k −→ k a selfmap on k. we use f (t ) to denote the setof fixed point of t , n to denote the set of natural numbers and xn → x (respectively xn ⇀ x ) todenote the strong (weak) convergence of the sequence {xn}∞n=0 to the point x . definition 1.1. let t : k −→ k be a maaping. then i. t is said to be l-lipschitizian if there exists l > 0 such that ‖ts − tz‖ ≤ ‖s − z‖,∀s, z ∈ k. (1.1) from the definition, it easy to observe that every nonexpansive mapping is lipschitizian with l = 1. ii. t is called k-strictly pseudocontraction (see, for example, [9]) if there exists k ∈ (0, 1] such that for all s, z ∈ k, the inequality ‖ts − tz‖2 ≤ ‖s − z‖2 + k‖(i − t )s − (i − t )z‖2 (1.2) hods. note that if k = 1 in (1.2), then t is a pseudocontraction. it well-known that in real hilbert spaces, the class of nonexpansive mapping is a proper subclass of the class of received: 1 nov 2021. key words and phrases. strong convergence; modified ishikawa iterative scheme; weak convergence; α-hemicontractive operator; fixed point; real hilbert space. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 2 k-strictly pseudocontive mapping. also, the class of k-strictly pseudocontive mapping is a proper subclass of the class of pseudocontive mapping. iii. t is called demicontractive mapping (see, for example, [?]) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s × q) ∈ (k × f (t )), there exists k ∈ [0, 1) such that the inequality ‖ts − tq‖2 ≤ ‖s − q‖2 + k‖s − ts‖2 (1.3) hods. iv. t is said to satisfy condition a (see, for example [?]) f (t ) = {x ∈ k : x = tx} 6= ∅ and there exists λ > 0 such that 〈s − ts, s − q〉 ≥ λ‖s − ts‖2,∀(s × q) ∈ (k × f (t )). (1.4) it is worthy to mention that the class of k-strictly pseudocontions with a nonempty fixed point set is a proper subclass of the class demicontractions. t is called hemicontraction (see, for example, [17]) if k = 1 in (1.3). the class of pseudocontractive maps is a proper subclass of the class of hemicontractive maps. again, the class of demicontractive maps is a proper subclass of the class of hemicontractive maps (see, for example, [?]). these two classes of mappings have been studied extensively by many researchers (see, for example, [?], [13], [17] and the references therein). v. t is called α-demicontraction (see, for examole, [13] ) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s × q) ∈ (k × f (t )), there exist λ > 0 and α ≥ 1 such that the inequality 〈s − ts, s − αq〉 ≥ λ‖s − ts‖2,∀(s × q) ∈ (k × f (t )). (1.5) holds. clearly, (1.5) is equivalent to ‖ts − αq‖2 ≤ ‖s − αq‖2 + k‖s − ts‖2, (1.6) where k = 1− 2λ ∈ [0, 1). v. t is called α-hemicontraction (see, for examole, [17] ) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s × q) ∈ (k × f (t )), there exists α ≥ 1 such that the inequality ‖ts − αq‖2 ≤ ‖s − αq‖2 + ‖s − ts‖2 (1.7) holds. observe that (1.7) is equivalent to 〈s − ts, s − αq〉 ≥ 0,∀(s × q) ∈ (k × f (t )). (1.8) in [ [17], example 2.2], osilike and onah gave an example of α-hemicontractive mapping with α > 1 which is not hemicontractive mapping, and also showed that there are hemicontractive (1-hemicontractive) mappings which are not α-hemicontraction for α > 1(see [ [17], example 2.1] for details). again, osilike and onah [17] presented an example of a mapping which is hemicontractive (1-hemicontractive) and alpha-hemicontractive mapping for α > 1 but https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 3 neither demicontractive (1-demicontractive) nor α-demicontractive mapping for α > 1(see [17], example 2.3 for details). for further cheracterisation of α-hemicontractive mapping, interested reader should consult [17]. a mapping t : h −→ h is called ν-strongly monotone if there exists ν > 0 such that 〈s − ts, s − z〉 ≥ ν‖s − z‖2,∀s, z ∈ h.. (1.9) iterative method for approximating fixed point of l-lipschitz pseudocontractive mapping has beenan active area of investigation in recent times (see, for example, [?], [?], [20], [14], [26], [27] and thereferences contained in them). in [24], voluhan introduced the modified projection-type ishikawaiterative method in the following way: let h be a hilbert space, k nonempty, closed and convexsubset of h and t : k −→ k be an l-lipshitz pseudocontractive mapping. for an arbitrary x0 ∈ k, define the sequence {xn}∞n=0 iteratively as follows. xn+1 = pk [(1− αn − γn)xn + γntyn] yn = (1− βn)xn + βntxn, n ≥ 1, (1.10) where {αn}∞n=0, {βn}∞n=0, {γn}∞n=0 ∈ (0, 1) and pk is a projection map from h onto k. using(1.10), she proved the following theorem. theorem 1.1. let h be a hilbert space, d a nonempty closed convex subset of h and t : d −→ d an l-lipschitz pseudocontractive mapping such that f (t ) 6= ∅. for any given x0 ∈ h, let {xn}∞n=0 be the sequence defined by (1.10). assume the sequences {αn}∞n=0, {βn}∞n=0, {γn}∞n=0 ∈ (0, 1) satisfy(1) βn(1− αn) > γn,∀n ≥ 1;(2) limn→∞ αn = 0 and ∑∞ n=0 αn =∞;(3) 0 < α ≤ γn ≤ βn ≤ β < 1√ 1 + l2 + 1 ,∀n ≥ 1. then, the sequence {xn}∞n=0 strongly converges to the fixed point of t . remark 1.1. if αn = 0,∀n ≥ 1, and pk is an identity, (1.10) reduces to the well-known ishikawa iteration method  xn+1 = (1− γn)xn + γntyn yn = (1− βn)xn + βntxn, n ≥ 1, (1.11) which has been used by several researchers to approximate the fixed points of different operators or operator equations in different spaces. motivated and inspired by the works in [17], [24] and some ongoing research in this direction, itis our purpose in this paper to extend the results in [24] and other related results from lipschitzpseudocontractive mapping to the more general α-hemicontractive mapping. our results is more https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 4general and also more applicable because fewer and simpler conditions are required to attainconvergence. 2. preliminary the following definitions and lemmas will be needed to prove our main results. definition 2.1. (see [27]) let h and k be as defined above. for each x ∈ h, there exists a unique nearest point of k, denoted by pkx , such that ‖x − pkx‖ ≤ ‖x − y‖,∀y ∈ k. such a pk is called metric projection from h onto k. it is well-known that pk is firmly nonexpansive mapping from h onto k; that is, ‖pkx − pky‖2 ≤ 〈pkx − pky , x − y〉,∀x, y ∈ h. also, for any x ∈ h and z ∈ k, z = pkx if and only if 〈x − z, z − y〉 ≥ 0,∀y ∈ k. definition 2.2. the banach space z is said to have opial property, if for each weakly convergent sequence {zn}∞n=0with weak limit z ∈ z, the following inequality holds: lim sup n→∞ ‖zn − z‖ < ‖zn − y‖,∀y ∈ zwithz 6= y . note that all finite dimensional banach spaces, all hilbert spaces and `p(0 ≤ p <∞) satisfy the opial property. but lp(1 < p <∞.p 6= 2) do not satisfies the opial property. definition 2.3. (see [27]) let e be a real banach space. a mapping t, with domain d(t ) ∈ e, is said to be demiclosed at 0 if for any sequence zn ⊂ e, zn � q ∈ d(t ) and ‖zn−tzn‖ → 0, then tq = q. lemma 2.1. (see [27]) let h be a real hilbert space. then, the following inequality holds: ‖λx + (1− λ)y‖2 ≤ λ‖x‖2 + (1− λ)‖y‖2 − λ(1− λ)‖x − y‖,∀λ ∈ [0, 1],∀x, y ∈ h. lemma 2.2. (see [27]) let {sn}n∈n be a sequence of nonnegative real numbers satisfying the inequality: sn+1 ≤ (1− γn)sn + δn,∀n ≥ 1, where {γn}n∈n and {δn}n∈n satisfy the following conditions:(i) {γn}n∈n ⊂ (0, 1);(ii) ∑∞ n=1 γn =∞. suppose ∑∞ n=1 δn <∞, then,limn→∞ sn = 0. https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 5 lemma 2.3. (see [4]) let e be a real hilbert space. then, for all x, y ∈ h, the following inequalities hold: i. ‖x − y‖2 ≤ ‖x‖2 − 2〈y , (x + y)〉+ ‖y‖2; ii. ‖x − y‖2 ≤ ‖x‖2 − 2〈y , (x + y)〉. lemma 2.4. (see [?]) let d be a sunset of a real hilbert space, t : d −→ h be a nonexpansive mapping and z a weak cluster point of the sequence {yn}∞n=0. if ‖tyn − yn‖ → 0, then z ∈ f (t ) proposition 2.5. (see [27]) let d be a nonempty subset of a real hilbert space amd γ : d −→ d an α-demicontractive mapping. assume that x ∈ d and α ≥ 1. then, γ is lipschitizian. theorem 2.6. (see [4]) a banach space e is reflexive if and only if every (normed) bounded sequence in e has a subsequence which converges weakly to an element of e. 3. convergence results now, we prove our main results. theorem 3.1. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k an l-lipschitz α-hemicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1− αn − γn)xn + γntyn] yn = (1− βn)xn + βntxn, n ≥ 1, (3.1) where the sequences {δn}∞n=0, {γn}∞n=0, {βn}∞n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1− δ 1 + l2 ; (i i) limn→∞ δn = 0 and ∑∞ n=0 δn =∞. then, the sequence {xn}∞n=0 generated by (3.1) weakly and strongly converges to the fixed point of t . proof. since f (t ) is nonempty, let αq ∈ f (t ) and x ∈ k. using (3.1), lemma 2.1 and the factthat t is l-lipschitizian, we estimate as follows: ‖xn+1 − αq‖2 = ‖pk[(1− δn − γn)xn + γntyn]− αq‖ ≤ ‖(1− δn − γn)xn + γntyn − αq‖ = ‖(1− δn − γn)(xn − αq) + γn(tyn − αq)− δnαq‖ ≤ ‖(1− δn − γn)(xn − αq) + γn(tyn − αq)‖+ δn‖αq‖. (3.2) set qn = ‖(1− δn − γn)(xn − αq) + γn(tyn − αq)‖2 and observe that qn = ‖(1− δn)(xn − αq)− (1− γn)(xn − αq) + γn(tyn − αq)‖2. (3.3) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 6since (1− δn)(xn − αq) = (1− δn)(1− γn)(xn − αq) + γn(1− δn))(xn − αq) (3.4) and γn(tyn − αq) = γn(1− δn)(tyn − αq) + γnδn(tyn − αq), (3.5) it follows from (3.3) that qn = ‖(1− δn)(1− γn)(xn − αq) + γn(1− δn))(xn − αq)− (1− γn)(xn − αq) +γn(1− δn)(tyn − αq) + γnδn(tyn − αq)‖2 = ‖(1− δn)[(1− γn)(xn − αq) + γn(tyn − αq)] + δnγn(tyn − xn)‖2. (3.6) (3.6) and lemma 2.1 imply that qn = (1− δn)‖(1− γn)(xn − αq) + γn(tyn − αq)‖2 + δn‖γn(tyn − xn)‖2 −δn(1− δn)‖xn − αq‖2. (3.7) if we denote vn = ‖(1− γn)(xn −αq) + γn(tyn −αq)‖2 and use similar technique as above, thenwe get vn = (1− γn)‖xn − αq‖2 + γn‖tyn − αq‖2 − γn(1− γn)‖xn − tyn‖2. (3.8) (3.7) and (3.8) imply qn = (1− δn)[(1− γn)‖xn − αq‖2 + γn‖tyn − αq‖2 − γn(1− γn)‖xn − tyn‖2] +δnγ 2 n‖tyn − xn‖2 − δn(1− δn)‖xn − αq‖2 = (1− δn)(1− γn)‖xn − αq‖2 + (1− δn)γn‖tyn − αq‖2 − γn(1− γn)(1− δn)‖xn − tyn‖2 +δnγ 2 n‖tyn − xn‖2 − δn(1− δn)‖xn − αq‖2 ≤ (1− δn)(1− γn)‖xn − αq‖2 + (1− δn)γnl 2‖yn − αq‖2 −(γn − δnγn − γ2n + γ2nδn)‖xn − tyn‖2 + δnγ 2 n‖tyn − xn‖2 − δn(1− δn)‖xn − αq‖2 = (1− δn)(1− γn)‖xn − αq‖2 + γnl 2‖yn − αq‖2 − δnγnl2‖yn − αq‖2 −(γn − δnγn − γ2n)‖xn − tyn‖2 − δn(1− δn)‖xn − αq‖2. (3.9) observr that |xn − tyn‖ ≤ (‖xn − αq‖+ l‖yn − αq‖)2 = ‖xn − αq‖2 + l(2‖xn − αq‖‖yn − αq‖) + l2‖yn − αq‖2 ≤ ‖xn − αq‖2 + l‖xn − αq‖2 + l‖yn − αq‖2 + l2‖yn − αq‖2 = (1 + l)‖xn − αq‖2 + l(1 + l)‖yn − αq‖2. (3.10) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 7(3.9) and (3.10) imply qn ≤ (1− δn)(1− γn)‖xn − αq‖2 + γnl 2‖yn − αq‖2 − δnγnl2‖yn − αq‖2 −(γn − δnγn − γ2n)[(1 + l)‖xn − αq‖2 + l(1 + l)‖yn − αq‖2]− δn(1− δn)‖xn − αq‖2 = (1− δn)(1− γn)‖xn − αq‖2 − (1 + l)(γn − δnγn − γ2n)‖xn − αq‖ −[(γn − δnγn − γ2n)l− l2γ2n ]‖yn − αq‖2 − δn(1− δn)‖xn − αq‖2 (3.11) again, from (3.1), we get ‖yn − αq‖2 = ‖(1− βn)(xn − αq) + βn(txn − αq)‖2 (3.12) since t is α-hemicontractive mapping, it follows from (3.12) and lemma 2.1 that ‖yn − αq‖2 ≤ (1− βn)‖xn − αq‖2 + βn[‖xn − αq‖2‖2 + ‖xn − txn‖2]− βn(1− βn)‖xn − txn‖2 = (1− βn)‖xn − αq‖2 + β2n‖xn − txn‖2. (3.13) putting (3.13) into (3.11), we have qn ≤ (1− δn)(1− γn)‖xn − αq‖2 − (1 + l)(γn − δnγn − γ2n)‖xn − αq‖ −[(γn − δnγn − γ2n)l− l2γ2n ]{(1− βn)‖xn − αq‖2 + β2n‖xn − txn‖2} −δn(1− δn)‖xn − αq‖2 ≤ (1− δn)(1− γn)‖xn − αq‖2 − [(γn − δnγn − γ2n)(1 + l) + δn(1− δn)− l2γ2n ]‖xn − αq‖2 −β2n [(γn − δnγn − γ2n)l− l2γ2n ]‖xn − txn‖2. (3.14) since from condition (i), (γn − δnγn − γ2n)− l2γ2n ≥ 0, it follows from (3.14) that qn ≤ (1− δn)2‖xn − αq‖2 (3.15) (3.2) and (3.15) imply |xn+1 − αq‖ ≤ (1− δn)‖xn − αq‖2 + δn‖αq‖ ≤ max{‖xn − αq‖2, ‖αq‖},∀n ∈ n. it is easy to see, using mathematical induction, that |xn+1 − αq‖ ≤ max{‖xn − αq‖2, ‖αq‖} = ‖x0 − αq‖2. (3.16) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 8hence, {xn}∞n=0 is bounded.furthermore, since from (3.1), ‖xn+1 − αq‖2 = ‖pk[(1− δn − γn)xn + γntyn]− αq‖2 ≤ ‖(1− δn − γn)xn + γntyn − αq‖2 = ‖xn − αq − γn(xn − tyn)− δnxn‖2, it follows from lemma 2.3(i) that ‖xn+1 − αq‖2 ≤ ‖xn − αq − γn(xn − tyn)‖2 − 2δn〈xn, xn+1 − αq〉. (3.17) since ‖xn − αq − γn(xn − tyn)‖2 = ‖(1− γn)(xn − αq) + γn(αq − tyn)‖2 = (1− γn)‖xn − αq‖2 + γn‖αq − tyn‖2 − γn(1− γn)‖tyn − xn‖2 ≤ (1− γn)‖xn − αq‖2 + γnl 2‖yn − αq‖2 −γn(1− γn)‖tyn − xn‖2, (3.18) it follows from (3.10) that ‖xn − αq − γn(xn − tyn)‖2 ≤ (1− γn)‖xn − αq‖2 + γnl 2‖yn − αq‖2 −γn(1− γn){(1 + l)‖xn − αq‖2 + l(1 + l)‖yn − αq‖2} = (1− γn)‖xn − αq‖2 + γnl 2‖yn − αq‖2 −γn(1− γn)(1 + l)‖xn − αq‖2 − γn(1− γn)l‖yn − αq‖2 −γnl2‖yn − αq‖2 + γ2nl 2‖yn − αq‖2 = (1− γn)‖xn − αq‖2 − γn(1− γn)(1 + l)‖xn − αq‖2 −[γn(1− γn)l− l2γ2n ]‖yn − αq‖2. (3.19) (3.13) and (3.19) imply ‖xn − αq − γn(xn − tyn)‖2 ≤ (1− γn)‖xn − αq‖2 − γn(1− γn)(1 + l)‖xn − αq‖2 −[γn(1− γn)l− l2γ2n ]{(1− βn)‖xn − αq‖2 + β2n‖xn − txn‖2} ≤ (1− γn)‖xn − αq‖2 − γnl[1− γn − γnl]{(1− βn)‖xn − αq‖2 +β2n‖xn − txn‖2}. (3.20) by condition (i), 1− γn − γnl > 0,∀n ≥ 0. consequently, ‖xn − αq − γn(xn − tyn)‖2 ≤ ‖xn − αq‖2 −(1− γn − γnl)β2nγnl‖xn − txn‖2. (3.21) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 9(3.17)and (3.21) imply ‖xn+1 − αq‖2 ≤ ‖xn − αq‖2 − (1− γn − γnl)β2nγnl‖xn − txn‖2 −2δn〈xn, xn+1 − αq〉. since {xn} is bounded, there exists a constant b > 0 such that −2〈xn, xn+1 − αq〉 ≤ b. thus, ‖xn+1 − αq‖2 ≤ ‖xn − αq‖2 − (1− γn − γnl)β2nγnl‖xn − txn‖2 δnb. the last inequality implies that ‖xn+1 − αq‖2 − ‖xn − αq‖2 + (1− γn − γnl)β2nγnl‖xn − txn‖2 ≤ δnb. (3.22) now, we consider the following two cases:case a: suppose there exists n0 ∈ n such that {‖xn −αq‖} is non-increasing. then, {‖xn −αq‖}is convergent. clearly, ‖xn+1−αq‖−‖xn−αq‖ → 0. in view, of condition (i i) and (3.22), we have ‖xn−txn‖ → 0. by lemma 2.4, it is obvious that ωω(xn) ⊂ f (t ), where ωω(xn){x : ∃xnk ⇀ αx?}is the weak limit set of {xn}. this implies that the sequence {xn} converges weakly to a fixed point αx? of t .suppose there exists some subsequences {xnk}∞k=0 ⊂ {xn}∞n=0 such that xnk ⇀ αy? weakly and αy? 6= αx?. since limn→∞ ‖xn − αv‖ exists for αv ∈ f (t ), by virtue of opial condition on h, wehave lim n→∞ ‖xn − αx?‖ = lim n→∞ ‖xnj − αx ?‖ < lim n→∞ ‖xnj − αy ?‖ = lim n→∞ ‖xnk − αy ?‖ < lim n→∞ ‖xnk − αx ?‖ = lim n→∞ ‖xnj − αy ?‖, which is a contradiction. consequently, αy? = αx?. this implies that {xnj}∞j=0 converges wealy toa common fixed point of t.next, we prove that {xn}∞n=0 converges strongly to x?/ let ξn = γntyn + (1− γnxn). then, from(3.1), we obtain xn+1 = pk[ξn − δnxn], n ≥ 0. this implies that xn+1 = pk[ξn + δnξn + δnξn − δnxn = pk[(1− δn)ξn + δn(ξn − xn)]. (3.23) observe that ‖ξn − αx?‖2 = ‖xn − αx? − γn(xn − tyn)‖2. (3.24)by using the same argument as in (3.20), with αx? = αq, we get, from (3.24), that ‖ξn − αx?‖ = ‖xn − αx?‖. (3.25) again, from (3.1), we obtain ‖yn − xn‖ = βn‖xn − txn‖ → 0 as n →∞, βn ∈ (0, 1). (3.26) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 10in addition, since t is lipschitz, it follows that ‖ξn − xn‖ = ‖γn[(tyn − txn)− (xn − txn)]‖ ≤ γn‖tyn − txn‖ − γn‖xn − txn‖ ≤ γnl‖yn − xn‖+ γn‖xn − txn‖ → 0 as n →∞. (3.27) now, using (3.23), we get ‖xn+1 − αx?‖2 ≤ ‖(1− δn)ξn + δn(ξn − xn)− αx?‖2 = ‖(1− δn)(ξn − αx?) + δn(ξn − xn)− δnαx?‖2, which by lemma 2.3 yields ‖xn+1 − αx?‖2 ≤ ‖(1− δn)(ξn − αx?) + δn(ξn − xn)‖2 − 2δn〈αx?, xn+1 − αx?〉 = (1− δn)‖ξn − αx?‖2 + δn‖ξn − xn‖2 − δn(1− δn)‖xn − αx?‖2 −2δn〈αx?, xn+1 − αx?〉 ≤ (1− δn)‖ξn − αx?‖2 + ‖ξn − xn‖2 − 2δn〈αx?, xn+1 − αx?〉 = (1− δn)‖ξn − αx?‖2 − 2δn〈αx?, xn+1 − αx?〉 ( by (3.27)) (3.28) ≤ (1− δn)‖ξn − αx?‖2 (3.29) (3.29) and lemma 2.2 imply that xn → αx? as n →∞.case b: assume that {‖xn − αq‖}∞n=0 is not a monotonically increasing sequence. set vn = ‖xn − αq‖2 and let τ : n −→ n be a mapping defined by τn = max{k ∈ n : k ≤ n, vn ≤ vn+1},∀n ≥ n0, for some n0 large enough. obviously, {τn}∞n=0 is a nondecreasing sequence given that τn → ∞ as n →∞ and vτn ≤ vτn+1 for all n ≥ n0. from (3.22), ‖xτ(n) − txτ(n)‖2 ≤ δτ(n)b (1− γτ(n) − γτ(n)l)β2 τ(n) γτ(n)l → 0 as n →∞. (3.30) therefore, limn→∞ ‖xτ(n) − txτ(n)‖ = 0. using similar argument as case a above, we concludethat {xτ(n)} → αx? →∞.from (3.28), we have 0 ≤ ‖xτ(n)+1 − αx?‖2 − ‖xτ(n) − αx?‖2 ≤ δτ(n)[2〈αx? − xτ(n)+1 − ‖xτ(n) − αx?‖2], (3.31) for δτ(n) ∈ (0, 1). hence, limn→∞ ‖xτ(n) − αx?‖2 = 0. this implies that limn→∞ vτ(n) = limn→∞ vτ(n)+1 = 0. in addition, for n ≥ n0, it is easy to see that vτ(n) = vτ(n)+1 if n 6= τ(n)(i.e., τ(n) < n) because vj > vj+1, f or τ(n) + 1 ≤ n. consequently. we obtain, for all n ≥ n0, 0 ≤ vτ(n)max{vτ(n), vτ(n)+1} = vτ(n)+1. hence, limn→∞ vn = 0. that is, {xn}∞n=0 converges https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 11strongly to αx?, and this completes the proof. � the following corollaries are immediate consequence of theorem 3.1. corollary 3.2. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k an l-lipschitz hemicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1− αn − γn)xn + γntyn] yn = (1− βn)xn + βntxn, n ≥ 1, (3.32) where the sequences {δn}∞n=0, {γn}∞n=0, {βn}∞n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1− δ 1 + l2 ; (i i) limn→∞ δn = 0 and ∑∞ n=0 δn =∞. then, the sequence {xn}∞n=0 generated by (3.32) weakly and strongly converges to the fixed point of t . corollary 3.3. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k is α-demicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1− αn − γn)xn + γntyn] yn = (1− βn)xn + βntxn, n ≥ 1, (3.33) where the sequences {δn}∞n=0, {γn}∞n=0, {βn}∞n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1− δ 1 + l2 ; (i i) limn→∞ δn = 0 and ∑∞ n=0 δn =∞. then, the sequence {xn}∞n=0 generated by (3.33) weakly and strongly converges to the fixed point of t . corollary 3.4. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k is demicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows:  xn+1 = pk [(1− αn − γn)xn + γntyn] yn = (1− βn)xn + βntxn, n ≥ 1, (3.34) where the sequences {δn}∞n=0, {γn}∞n=0, {βn}∞n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1− δ 1 + l2 ; (i i) limn→∞ δn = 0 and ∑∞ n=0 δn =∞. https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 12 then, the sequence {xn}∞n=0 generated by (3.34) weakly and strongly converges to the fixed point of t . competing interest. the authors declare that there is no conflict of interest. references [1] f.e. browder, nonlinear mappings of nonexpansive and accretive type in banach spaces, bull. amer. math. soc.73 (1967) 875-882.[2] f.e. browder, w.v. petryshyn, construction of fixed points of nonlinear mappings in hilbert space, j. math. anal.appl. 20 (1967) 197-228. https://doi.org/10.1016/0022-247x(67)90085-6.[3] c.e. 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https://doi.org/10.1090/s0002-9939-00-05573-8.[27] i. k. agwu, d. i. igbokwe, hybrid-type iteration scheme for approximating fixed point of lipschitz α-hemicontractivemappings, adv. fixed point theory, 10 (2020) 3. https://doi.org/10.28919/afpt/4442. https://doi.org/10.28924/ada/ma.2.10 https://doi.org/10.1006/jmaa.1993.1309 https://doi.org/10.1016/j.na.2009.03.075 https://doi.org/10.1016/j.jmaa.2008.01.045 https://doi.org/10.1016/j.na.2008.04.017 https://doi.org/10.1090/s0002-9939-00-05573-8 https://doi.org/10.28919/afpt/4442 1. introduction 2. preliminary 3. convergence results competing interest references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 3doi: 10.28924/ada/ma.2.3 on the ostrowski method for solving equations ioannis k. argyros1,∗, santhosh george2, christopher i. argyros3 1department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2 department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 3department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. in this paper, we revisited the ostrowski’s method for solving banach space valued equa-tions. we developed a technique to determine a subset of the original convergence domain and usingthis new lipschitz constants derived. these constants are at least as tight as the earlier ones leadingto a finer convergence analysis in both the semi-local and the local convergence case. these tech-niques are very general, so they can be used to extend the applicability of other methods withoutadditional hypotheses. numerical experiments complete this study. 1. introduction one of the most challenging tasks in computational mathematics is the problem of determininga solution x∗ of equation f (x) = 0, (1.1) where f : ω ⊂ b −→ b1 is an operator acting between banach spaces b and b1 with ω 6= ∅. theclosed form derivation of x∗ is possible only in rare cases. this leads practitioners and researchersin developing solution methods that are iterative.in this work, we consider ostrowski’s method defined for x0 ∈ ω and each n = 0, 1, 2, . . . by yn = xn − f ′(xn)−1f (xn) xk+1 = yn − a−1n f (yn), (1.2) received: 7 nov 2021. key words and phrases. ostrowski’s method; banach space; convergence criterion.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 2where an = 2[yn, xn;f ]− f ′(xn). the convergence order is four obtained under certain conditionson the initial data (ω, f, f ′, x0) and taylor expansion [17, 25]. so, the assumptions on the fourthderivative reduce the applicability of these schemes.for example: let b = b1 = r, ω = [−0.5, 1.5]. define λ on ω by λ(t) = { t3 log t2 + t5 − t4 i f t 6= 0 0 i f t = 0. then, we get t∗ = 1, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. obviously λ′′′(t) is not bounded on ω. so, the convergence of scheme (1.2) is not guaranteed bythe analyses in [17,24].we study two types of convergence called local and semi-local. in the first one based on thesolution x∗ we find the radii of the convergence balls. but in the second one based on the starter x0 we develop criteria that guarantee convergence of sequence {xn}. there is a plethora of thistypes of results [10,15,16,22,28,38]. but what all these results have in common is that the regionof accessibility (or convergence region) is limited in general reducing the applicability of newton’sand other methods [8,20,26,28,31]. moreover, the error bounds on distances ‖xk+1−xk‖ or ‖xk−x∗‖are pessimistic. the same is true for the uniqueness ball of these methods. these problems becomemore difficult when studying methods of convergence order three or higher [8, 17, 19, 31–33]. wehave developed different techniques to addres these problems.in technique 1, we determine a subset ω of ω also containing the iterates. but in this set ω thelipschitz-like parameters (or functions) are at least as tight as the original ones, so the resultingconvergence is finer. this technique does not depend on the convergence order of the method. butwe shall demonstrate it in case of fourth order methods. these methods require the evaluation ofthe second order fréchet derivative of operator f. notice that for a system (nonlinear) of i equationswith i unknowns, the first derivative is a matrix with i2 entries (values), whereas the second fréchetderivative has i3 entries. that is why there is a need for avoiding f ′′.the rest of the paper is organized as follows: in section 2 we develop the second technique basedon majorizing sequences. the local convergence analysis results appear in section 3. numericalexamples can be found in section 4. the paper ends with some concluding remarks. 2. semi-local convergence we base our semi-local convergence analysis on scalar parameters and functions. let η ≥ 0, k0 > 0, k > 0, k1 > 0, k2 > 0, k3 > 0, l0 > 0 with k0 ≤ k, l0 ≤ 2k1 and k4 = k2 + k3.define polynomials g1 and g2 on the interval [0, 1) by g1(t) = k1t 5 + (2k1 +k3)t 4 +k1t 3 + ( k 2 −k3)t2 − k 2 (2.1) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 3and g2(t) = l0t 4 + (l0 +k3)t 3 +k4t 2 −k3t −k4. (2.2) we have g1(0) = −k2 < 0, g1(1) = 4k1 > 0, g2(0) = −k4 < 0 and g2(1) = 2l0 > 0. it thenfollows from the intermediate value theorem that polynomials g1 and g2 have at least one root in (0, 1). denote by δ1 and δ2 the least such roots, respectively. moreover, it is convenient to definescalar sequences and parameters t0 = 0, s0 = η, t1 = s0 + k0 2 (s0 − t0)2 1− 2k1s0 , sn+1 = tn+1 + (k3(tn+1 − sn) +k4(sn − tn))(tn+1 − sn) 1− l0tn+1 tn+2 = sn+1 + k(sn+1 − tn+1)2 2(1− (k1(sn+1 + tn+1) +k3(sn+1 − tn+1)) , (2.3) αn = k(sn − tn) 2(1− (k1(sn+1 + tn+1) +k3(sn+1 − tn+1)) , γn = k3(tn − sn) +k4(sn − tn) 1− l0tn+1 , for all n = 0, 1, 2, . . . , δn = max{αn, γn}, λ = min{δ1, δ2} and µ = max{δ1, δ2}. next, we present a convergence result for sequences {tn} and {sn}. lemma 2.1. suppose: there exists δ satisfying 0 ≤ δ0 ≤ λ ≤ δ ≤ µ < 1−k1η. (2.4) then, sequences {tn}, {sn} are well defined nondecreasing, bounded from above bt s∗∗ = η 1−δ and as such they converge to their unique least upper bound s∗ ∈ [η, s∗∗]. moreover, the following error estimates hold for all n = 1, 2, . . . 0 ≤ tn+1 − sn ≤ δ(sn − tn) ≤ δ2n+1η, (2.5) 0 ≤ sn − tn ≤ δ(tn − sn−1) ≤ δ2nη (2.6) and tn ≤ sn ≤ tn+1. (2.7) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 4 proof. items (2.5)-(2.7) hold if 0 ≤ αm ≤ δ, (2.8) 0 ≤ γm ≤ δ (2.9) and tm ≤ sm ≤ tm+1 (2.10) are true for all m = 0, 1, 2, . . . . notice that by the definition of s0, t1 and (2.4), t1 ≥ 0. we alsohave (2.8) and (2.9) hold for m = 0. suppose (2.8)-(2.10) hold for m = 1, 2, . . . , n. then, we canobtain in turn that sm ≤ tm + δ2mη ≤ sm−1 + δ2m−1η + δ2mη ≤ η + . . .+ δη + . . .+ δ2mη = 1− δ2m+1 1− δ η ≤ η 1− δ = s∗∗, and tm+1 ≤ sm + δ2m+1η ≤ tm + δ2mη + δ2m+1η ≤ η + δη + . . .+ δ2m+1η = 1− δ2m+2 1− δ η ≤ η 1− δ . hence, by (2.7) and the induction hypotheses, we deduce that sequences {tm} and {sm} arenondecreasing. evidently, (2.8) holds if k 2 δ2nη + δk1 ( 1− δ2n+3 1− δ η ) +δk1 1− δ2n+2 1− δ η +k3δ 2(n+1)η − δ ≤ 0. (2.11) estimate (2.11) motivates us to introduce recurrent functions h(1)n (t) on the interval [0, 1) by h (1) n )t) = k 2 t2n−1η +k1(1 + t + . . .+ t2n+2)η +k1(1 + t + . . .+ t2n+1)η +k3t 2n+3η − 1. (2.12) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 5 we need a relationship between two consecutive functions f (1)n (t). by this definition, we have inturn that h (1) n+1(t) = k 2 t2n+1η +k1(1 + t + . . .+ t2n+4)η +k1(1 + t + . . .+ t2n+3)η +k3t 2n+3η − 1 − k 2 t2n−1η −k1(1 + t + . . .+ t2n+2)η −k1(1 + t + . . .+ t2n+1)η −k3t2n+1η + 1 + h (1) n (t) = h (1) n (t) + k 2 t2n+1η − k 2 t2n−1η +k1(t 2n+3 + t2n+4)η +k1(t 2n+2 + t2n+3)η +k3t 2n+3η −k3t2n+1η = h (1) n (t) + g1(t)t 2n−1η. (2.13) notice that by the definition of δ1 h (1) n+1(δ1) = h (1) n (δ1). by (2.11)-(2.13), estimate (2.11) shall be true if for 4k1 < k h (1) n (δ1) ≤ 0. (2.14) let h(1)∞ (t) = lim n−→∞ h (1) n (t). (2.15) but then h(1)∞ (δ) = 2k1η 1− δ − 1. (2.16) hence, instead of (2.13) we can show h(1)∞ (δ) ≤ 0, (2.17) which is true by (2.4). if 4k1 ≥ k then f∞(t) ≥ fn(t), so again f∞(δ) ≤ 0 holds. similarly, (2.9)holds if k3δ 2n+1η +k4δ 2nη + δl0 1− δ2n+2 1− δ η − δ ≤ 0 (2.18) or h (2) n (δ) ≤ 0, (2.19) where h (2) n (t) = k3t 2nη +k4t 2n−1η + l0(1 + t + . . .+ t2n+1)η − 1. (2.20) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 6this time we have h (2) n+1(t) = k3t 2n+2η +k4t 2n+1η + l0(1 + t + . . .+ t2n+3)η −1−k3t2nη −k4t2n−1η − l0(1 + t + . . .+ t2n+1)η + 1 + h (2) n (t) = h (2) n (t) +k3t 2n+2η −k3t2nη +k4t 2n+1 −k4t2n+1η +l0(t 2n+2 + t2n+3)η = h (2) n (t) + [k3t 3 −k3t +k4t 2 −k4 + l0t 2 + l− 0t4]t2n−1δ = h (2) n (t) + g2(t)t 2n−1η. (2.21) by the definition of δ2 h (2) n+1(δ2) = h (2) n (δ).let h(2)∞ (t) = limn−→∞ h (2) n (t). then, we get h(2)∞ (δ) = l0η 1− δ − 1. hence, instead of (2.19), we can show h(2)∞ (δ) ≤ 0,which is true by (2.4). the induction for (2.8)-(2.10) is completed. therefore, sequences {tn}, {sn}are nondecreasing, bounded from above by s∗∗ and as such they converge to s∗. �the semi-local convergence analysis shall be based on conditions (a).suppose:(a1) there exists x0 ∈ ω, η ≥ 0 such that f ′(x0)−1 ∈ l(e1, e) and ‖f ′(x0)−1f (x0)‖ ≤ η. (a2) for each x ∈ ω ‖f ′(x0)−1(f ′(x)− f ′(x0))‖ ≤ l0‖x − x0‖. set ω0 = u[x0, 1 l0 ] ∩ω.(a3) for each x, y ∈ ω0 ‖f ′(x0)−1(f ′(y)− f ′(x))‖ ≤ k‖y − x‖, ‖f ′(x0)−1([y , x ;f ]− f ′(x0))‖ ≤ k1(‖y − x0‖+ ‖x − x0‖), ‖f ′(x0)−1([z, y ;f ]− [y , x ;f ])‖ ≤ k2(‖z − y‖+ ‖y − x‖)and ‖f ′(x0)−1([z, y ;f ]− f ′(y))‖ ≤ k3‖z − y‖.(a4) u[x0, s∗] ⊂ ω and(a5) conditions of lemma 2.1 hold. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 7next, we present the semi-local convergence of method (1.2). theorem 2.2. suppose that conditions (a) hold. then, sequences {yn}, {xn} generated by method (1.2) are well defined in u[x0, s∗], remain in u[x0, s∗] for each n = 0, 1, 2, . . . and converge to a solution x∗ ∈ u[x0, s∗] of equation f (x) = 0. moreover, the following assertion holds ‖xn − x∗‖ ≤ s∗ − tn. (2.22) proof. mathematical induction on m shall be used to show(im) ‖ym − xm‖ ≤ sm − tmand(iim) ‖xm+1 − ym‖ ≤ tm+1 − sm.by the first substep of method (1.2) we have ‖y0 − x0‖ = ‖f ′(x0)−1f (x0)‖ ≤ η = s0 − t0 = s0 ≤ s∗. so (i0) holds and y0 ∈ u[x0, s∗]. by the first substep of method (1.2) we can write f (y0) = f (y0)− f (x0)− f ′(x0)(y0 − x0). (2.23) using (a2) and (2.23), we have ‖f ′(x0)−1f (y0)‖ ≤ k0 2 ‖y0 − x0‖2 ≤ k0 2 (s0 − t0)2. (2.24) we need to show the invertability of linear operator a. we have by (a3) ‖f ′(x0)−1(a0 − f ′(x0))‖ ≤ 2‖f ′(x0)−1([y0, x0;f ]− f ′(x0))‖ ≤ 2k1(‖y0 − x0‖+ ‖x0 − x0‖) ≤ 2k1(s0 + t0) < 1, so ‖a−10 f ′(x0)‖ ≤ 1 1− 2k1(s0 + t0) , (2.25) by the banach lemma on linear invertible operators [19, 25]. then, iterate x1 exists by the secondsubstep of method (1.2), and we can write x1 − y0 = (a−10 f ′(x0))(f ′(x0) −1f (y0)). (2.26) by (2.24)-(2.26), we get ‖x1 − y0‖ ≤ ‖a−10 f ′(x0)‖‖f ′(x0)−1f (y0)‖ ≤ k0(s0 − t0)2 2(1− 2k1(s0 + t0)) = t1 − s0, https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 8showing (ii0). moreover, we have ‖x1 − x0‖ ≤ ‖x1 − y0‖+ ‖y0 − x0‖ ≤ t1 − s0 + s0 − t0 = t1 ≤ s∗, so x1 ∈ u[x0, s∗]. suppose that (im) and (iim) hold, ym, xm+1 ∈ u[x0, s∗] and f ′(xm)−1, a−1m existfor each m = 1, 2, . . . , n. we shall prove they hold for m = n + 1. using the second substep ofmethod (1.2), we get ‖f (x0) −1f (xn+1)‖ = ‖f ′(x0)−1(f (xn+1)− f (yn))− an(xn+1 − yn))‖ = ‖f ′(x0)−1([xn+1, yn;f ]− an)(xn+1 − yn)‖ ≤ f ′(x0) −1([xn+1, yn;f ]− [yn, xn;f ])‖ +‖f ′(x0)−1([yn, xn;f ]− f ′(xn))‖ ≤ (k2(‖xn+1 − yn‖+ ‖yn − xn‖) +k3‖yn − xn‖)‖xn+1 − yn‖ (2.27) we need to show f ′(xn+1) is invertible. by (a2) and the induction hypotheses we obtain ‖f ′(x0)−1(f ′(xn+1)− f ′(x0))‖ ≤ l0‖xn+1 − x0‖ ≤ l0(tn+1 − t0) = l0tn=1 < 1, so ‖f ′(xn+1)−1f ′(x0)‖ ≤ 1 1− l0tn+1 . (2.28) hence, we get by (2.27), (2.28) and the first substep of method (1.20 that ‖yn+1 − xn+1‖ = ‖f ′(xn+1)−1f (x0)‖‖f ′(x0)−1f (xn+1)‖ ≤ (k2(tn+1 − sn) + (sn − tn)) +k3(sn − tn))(tn+1 − sn) 1− l0tn+1 = sn+1 − tn+1, (2.29) since k4 = k2 +k3, showing (im) for m = n + 1. then, we also have ‖yn+1 − x0‖ ≤ ‖yn+1 − xn+1‖+ ‖xn+1 − x0‖ ≤ sn+1 − tn+1 + tn+1 − s0 = sn+1 ≤ s∗, https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 9so yn+1 ∈ u[x0, s∗]. operator a−1n+1 shall be shown to exist ‖f ′(x0)−1(an+1 − f ′(x0))‖ ≤ ‖f ′(x0)−1([yn+1, xn+1;f ]− f ′(x0))‖ +‖f ′(x0)−1([yn+1; xn+1;f ]− f ′(xn+1))‖ ≤ k1(‖yn+1 − x0‖+ ‖xn+1 − x0‖) +k3‖yn+1 − xn+1‖ ≤ k1(sn+1 + tn+1) +k3(sn+1 − tn+1) < 1, so ‖a−1n+1f ′(x0)‖ ≤ 1 1− (k1(sn+1 + tn+1) +k3(sn+1 − tn+1)) . (2.30) by the first substep of method (1.2), we can write f (yn+1) = f (yn+1)− f (xn+1)− f ′(xn+1)(yn+1 − xn+1), so ‖f ′(x0)−1f (yn+1)‖ ≤ k 2 ‖yn+1 − xn+1‖2 ≤ k 2 (sn+1 − tn+1)2,so ‖xn+2 − yn+1‖ ≤ ‖a−1n+1f ′(x0)‖‖f ′(x0)−1f (yn+1)‖ ≤ k(sn+1 − tn+1)2 2(1− (k1(sn+1 + tn+1) +k3(sn+1 − tn+1))) = tn+2 − sn+1, showing (iim) for m = n + 1. we can get ‖xn+2 − x0‖ ≤ ‖xn+2 − yn+1‖+ ‖yn+1 − x0‖ ≤ tn+2 − sn+1 + sn+1 − t0 = tn+2 ≤ s∗, so xn+2 ∈ u[x0, s∗]. furthermore, we obtain ‖xn+1 − xn‖ ≤ ‖xn+1 − yn‖+ ‖yn − xn‖ = tn+1 − sn + sn − tn = tn+1 − tn, so sequence {xn} is fundamental in a banach space b, so it converges to some x∗ ∈ u[x0, s∗]. byletting n −→∞ in (2.27), we obtain ‖f ′(x0)−1f (xk+1)‖ ≤ (k2(tn+1 − sn) +k4(sn − tn))(sn+1 − sn) −→ 0, so f (x∗) = 0 by the continuity of f. �a uniqueness of the solution result is given next. proposition 2.3. suppose: https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 10(i) there exists a simple solution x∗ of equation f (x) = 0.(ii) there exists s̄ ≥ s∗ such that l0(s̄ + s∗) < 2. set ω1 = u[x0, s̄] ∩ω. then, the only solution of equation f (x) = 0 in the region ω1 is x∗. proof. let x̄ ∈ ω1 with f (x̄) = 0. let m = ∫ 1 0 f ′(x̄ + θ(x∗ − x̄))dθ. then, in view of (a2) and(ii), we obtain ‖f ′(x0)−1(m − f ′(x0))‖ ≤ l0 ∫ 1 0 [(1− θ)‖x̄ − x0‖+ θ‖x∗ − x0‖]dθ ≤ l0 2 (s̄ + s∗) < 1, so x̄ = x∗ since m−1 exists and m(x∗ − x̄) = f (x∗)− f (x̄) = 0− 0 = 0. � remark 2.4. notice that s∗∗ given in closed form can repalce s∗ in the conditions of theorem 2.2. 3. local convergence as in section 2 we develop some functions and parameters. let li , i = 0, 1, 2, 3, 4 be givenparameters. define function ϕ1 on the interval t = [0, 1l0 ) by ϕ1(t) = lt 2(1− l0t) . notice that parameter ra = 2 2l0 + l < 1 l0 (3.1)solves equation ϕ1(t) = 1.define functions on the interval t by q(t) = l0ϕ1(t)t − 1 and p(t) = (2l1(1 + ϕ1(t)) + l)t. suppose that these functions have smallest zeros rq and rp in (0, 1l0 ), respectively. let r1 = min{rq, rp} and t0 = [0, r1). define function ϕ2 on t0 by ϕ2(t) = [ lϕ1(t) 2(1− l0ϕ1(t)t) + l4(l2 + l3)(1 + ϕ1(t))ϕ1(t) (1− l0ϕ1(t)t)(1− p(t)) ] t. suppose that function ϕ2(t)− 1 https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 11has smallest zero r2 ∈ (0, r1). we shall show that parameter r = min{ra, r2} (3.2) is a convergence radius for method (1.2). let t1 = [0, r). then, it follows by these definitions thatfor each t ∈ t1 l0t < 1 (3.3) 0 ≤ ϕ1(t) < 1, (3.4) 0 ≤ ϕ1(t)t < 1 (3.5) 0 ≤ p(t) < 1 (3.6) and 0 ≤ ϕ2(t)t < 1 (3.7)hold.the conditions (h) to be used in the local convergence of method (1.2) are as follows.suppose:(h1) there exists a simple solution x∗ ∈ ω of equation f (x) = 0.(h2) for each x ∈ ω ‖f ′(x)−1(f ′(x)− f ′(x∗))‖ ≤ l0‖x − x∗‖.set ω0 = u[x∗, 1 l0 ] ∩ω.(h3) for each x, y ∈ ω0 ‖f ′(x∗)−1(f ′(y)− f ′(x))‖ ≤ l‖y − x‖, ‖f ′(x∗)−1(f ′(y)− f ′(x∗))‖ ≤ l1(‖y − x∗‖+ ‖x − x∗‖), ‖f ′(x∗)−1([y , x ;f ]− f ′(x))‖ ≤ l2‖y − x‖, ‖f ′(x∗)−1([y , x ;f ]− f ′(y))‖ ≤ l3‖y − x‖and ‖f ′(x∗)−1f ′(x)‖ ≤ l4‖x − x∗‖.and(h4) u[x∗, r ] ⊂ ω.in view of conditions (h) and the developed notation we can show the local convergence result formethod (1.2). theorem 3.1. under the conditions (h), further suppose that x0 ∈ u(x∗, r) − {x∗}. then, sequence {xk}, {yn} generated by method (1.2) is well defined in u(x∗, r), remains in u(x∗, r) for each k = 0, 1, 2, . . . and converges to x∗. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 12 proof. let u ∈ u(x∗, r)− {x∗}. by (h1) and (h2), we get in turn that ‖f ′(x∗)−1(f ′(u)− f ′(x∗))‖ ≤ l0‖u − x∗‖ ≤ l0r < 1, so f ′(u) is invertibale and ‖f ′(u)−1f ′(x∗)‖ ≤ 1 1− l0‖u − x∗‖ . (3.8) iterate y0 is well defined by the first substep of method (1.2) and (3.8) for u = x0. then, we canwrite y0 − x∗ = x0 − x∗ − f ′(x0)−1f (x0) = (f ′(x0) −1f ′(x∗)) ×( ∫ 1 0 f ′(x∗) −1(f ′(x∗ + θ(x0 − x∗))− f ′(x0))dθ(x0 − x∗). (3.9) by (3.2), (3.4), (h3), (3.8) and (3.9), we have in turn that ‖y0 − x∗‖ ≤ l0‖x0 − x∗‖2 2(1− l0‖x0 − x∗‖ ≤ l‖x0 − x∗‖2 2(1− l0‖x0 − x∗‖) ≤ ϕ1(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r (3.10) so y0 ∈ u(x∗, r). next, we show linear operator a) is invertible. indeed, using (3.2), (3.6), (h3) and(3.10), we get in turn that ‖f ′(x∗)−1(a0 − f ′(x∗))‖ ≤ ‖f ′(x∗)−1([y0, x0;f ]− f ′(x∗))‖ +‖f ′(x∗)−1([y0, x0;f ]− f ′(x∗))‖ +‖f ′(x∗)−1(f ′(x0)− f ′(x∗))‖ ≤ 2l1(‖y0 − x∗‖+ ‖x0 − x∗‖) + l0‖x0 − x∗‖ ≤ 2l1(1 + ϕ1(‖x0 − x∗‖))‖x0 − x∗‖+ l‖x0 − x∗‖ ≤ p(‖x0 − x∗‖) ≤ p(r) < 1, so ‖a−10 f ′(x∗)‖ ≤ 1 1− p(‖x0 − x∗‖) (3.11) and iterate x1 is well defined by the second substep of method (1.2) for n = 0. then, we can write x1 − x∗ = (y0 − x∗ − f ′(y0)−1f (y0)) + f ′(y0) −1(a) − f ′(y0))a−10 f (y0). (3.12) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 13using (3.2), (3.7), (h3), (3.8) (for u = x0, y0) and (3.10)–(3.13), we obtain in turn that ‖x1 − x∗‖ ≤ ‖y0 − x∗ − f ′(y0)−1f (y0)‖ +‖f ′(y0)−1f ′(x∗)‖‖f ′(x∗)−1(a0 − f ′(y0))‖‖a−10 f ′(x∗)‖‖f ′(x∗)−1f (y0)‖ ≤ l‖y0 − x∗‖2 2(1− l0‖x0 − x∗‖ + (l2 + l3)‖y0 − x0‖l4‖y0 − x∗‖ (1− l0‖y0 − x∗‖)(1− p(‖x0 − x∗‖)) ≤ ϕ2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r, so x1 ∈ u(x∗, r), where we also used ‖f ′(x∗)−1(a0 − f ′(y0))‖ ≤ ‖f ′(x∗)−1([y0, x0;f ]− f ′(x0))‖ +‖f ′(x∗)−1([y0, x0;f ]− f ′(y0))‖ ≤ (l2 + l3)‖y0 − x0‖ ≤ (l2 + l3)(‖y0 − x∗‖+ ‖x0 − x∗‖) ≤ (l2 + l3)(1 + ϕ1(‖x0 − x∗‖))‖x0 − x∗‖, and ‖f ′(x∗)−1f (y0)‖ = ‖ ∫ 1 0 f ′(x∗) −1f ′(x∗ + θ(y0 − x∗))dθ(y0 − x∗)‖ ≤ l4‖y0 − x∗‖2. so, far showed ‖y0 − x∗‖ ≤ ϕ1(‖x0 − x∗‖)‖x0 − x∗‖ < r and ‖x1 − x∗‖ ≤ ϕ2(‖x0 − x∗‖)‖x0 − x∗‖ < r. by simply replacing x0, y0, x1 by xm, ym, xm+1 in the preceding calculations, we get ‖ym − x∗‖ ≤ ϕ1(‖xm − x∗‖)‖xm − x∗‖ < r and ‖xm+1 − x∗‖ ≤ ϕ2(‖xm − x∗‖)‖xm − x∗‖ < r. then, from the estimation ‖xm+1 − x∗‖ ≤ α‖xm − x∗‖ < r, (3.13) where α = ϕ2(‖x0 − x∗‖) ∈ [0, 1), limm−→∞ xm = x∗ and ym, xm+1 ∈ u(x∗, r). � remark 3.2. by the definition of r, we see that r ≤ ra. (3.14) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 14 parameter ra was shown in [4] to be a convergence radius for newton’s method. notice the radius of convergence for newton’s method given independently by traub [35] and rheinbold [29] is rtr = 2 3m1 , where l1 is the lipschitz constant on ω. so, we have rtr ≤ ra, since l ≤ l1 and l0 ≤ m1. 4. numerical experiments we provide some examples, showing that the old convergence criteria are not verified but oursare. example 4.1. define function f (t) = θ0t + θ1 + θ2 sin θ3t, t0 = 0, where θj , j = 0, 1, 2, 3 are parameters. then, clearly for θ3 large and θ2 small, l0k can be small (arbitrarily). example 4.2. let b = b1 = u[0, 1] the domain of functions given on [0, 1] which are continuous. we consider the max-norm. choose ω = b(0, d), d > 1. define f on ω be f (x)(s) = x(s)− w(s)− ξ ∫ 1 0 k(s, t)x3(t)dt, (4.1) x ∈ b, s ∈ [0, 1], w ∈ b is given, ξ is a parameter and k is the green’s kernel given by k(s2, s1) = { (1− s2)s1, s1 ≤ s2 s2(1− s1), s2 ≤ s1. by (4.1), we have (f ′(x)(z))(s) = z(s)− 3ξ ∫ 1 0 k(s, t)x2(t)z(t)dt, t ∈ bs ∈ [0, 1]. consider x0(s) = w(s) = 1 and |ξ| < 8 3 . we get ‖i − f ′(x0)‖ < 3 8 |ξ|, f ′(x0)−1 ∈ l(b1b), ‖f ′(x0)−1‖ ≤ 8 8− 3|ξ| , η = |ξ| 8− 3|ξ| , l0 = 12|ξ| 8− 3|ξ| , k = 6d |ξ| 8−3|ξ| , k1 = l0 2 and k2 = k 2 = k3. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 15 example 4.3. let b = b1 = r3 and ω be as in the example 4.2. it is well known that the boundary value problem [16] ψ(0) = 0, ψ(1) = 1, ψ′′ = −ψ − τψ2 can be given as a hammerstein-like nonlinear integral equation ψ(s) = s + ∫ 1 0 k(s, t)(ψ3(t) + τψ2(t))dt where τ is a parameter. then, define f : ω −→ t2 by [f (x)](s) = x(s)− s − ∫ 1 0 k(s, t)(x3(t) + τx2(t))dt. choose x0(s) = s and ω = u(x0, r0). then, clearly u(x0, r0) ⊂ u(0, r0 + 1), since ‖x0‖ = 1. suppose 2τ < 5. then, by conditions (a) are satisfied for l0 = 2τ+3r0+6 8 , k = τ+6r0+3 4 , k1 = l0 2 and k2 = k 2 = k3. and η = 1+τ 5−2τ . notice that l0 < k. the rest of the examples are given for the local convergence study of newton’s method. example 4.4. let b = b1 = r3, ω = u[0, 1] and x∗ = (0, 0, 0)tr . define mapping e on ω for λ = (λ1, λ2, λ3) tr as e(λ) = (eλ1 − 1, e − 1 2 λ22 + λ1, λ3) tr . then, conditions (h) hold provided that l0 = e − 1, l = e 1 l0 and m1 = e, since f ′(x∗)−1 = f ′(x∗) = diag{1, 1, 1, }. notice that l0 < l < m1, l1 = l0 2 , l4 = l, l2 = l3 = l 2 . rtr = 0.2453 < ra = 0.3827, r = 0.2124. hence, our radius of convergence is larger. example 4.5. let b = b1 and ω be as in example 4.2. define f on ω as f (ϕ1)(x) = ϕ1(x)− ∫ 1 0 xϕ1(j) 3dj. then, we obtain f ′(ϕ1(ψ1))(x) = ψ1(x)− 3 ∫ 1 0 xjϕ1(j) 2ψ1(j)dj for all ψ1 ∈ ω. so, we can choose l0 = 1.5, l = m1 = 3. l1 = l0 2 , l4 = l, l2 = l3 = l 2 . but then, we get again rtr = 0.2222 < ra = 0.3333, r = 0.2663. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 165. conclusion ostrowski’s method was revisited and its applicability was extended in both the semi-localand local convergence case. in particular, the benefits in the semi-local convergence case include:weaker sufficient convergence criteria (i.e. more starters x0 become available); 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(2019).[38] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987) 671-684. https://doi.org/10.1080/01630568708816254. https://doi.org/10.28924/ada/ma.2.3 https://doi.org/10.1016/j.amc.2013.05.078 https://doi.org/10.1016/j.amc.2013.05.078 https://doi.org/10.1007/s10910-018-0856-y https://doi.org/10.1007/s10910-018-0856-y https://doi.org/10.1016/j.cam.2013.11.019 https://doi.org/10.1007/bf01385696 https://doi.org/10.1016/j.jco.2008.05.006 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/j.amc.2003.12.025 https://doi.org/10.1007/s11075-012-9585-7 https://doi.org/10.1007/s11590-013-0617-6 https://doi.org/10.1007/bf01400355 https://doi.org/10.1080/01630568708816254 1. introduction 2. semi-local convergence 3. local convergence 4. numerical experiments 5. conclusion references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 7doi: 10.28924/ada/ma.2.7 on the stratonovich estimator for the itô diffusion jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte, 376 fretwell bldg, 9201 university city blvd., charlotte, nc 28223-0001, usa correspondence: j.bishwal@uncc.edu abstract. for the parameter appearing non-linearly in the drift coefficient of homogeneous itô sto-chastic differential equation having a stationary ergodic solution, the paper obtains the strong con-sistency of an approximate maximum likelihood estimator based on stratonovich type approximationof the continuous girsanov likelihood, under some regularity conditions, when the corresponding dif-fusion is observed at equally spaced dense time points over a long time interval in the high frequencyregime. pathwise convergence of stochastic integral approximations and their connection to discretedrift estimators is studied. often it is shown that discrete drift estimators converge in probability. weobtain convergence of the estimator with probability one. ornstein-uhlenbeck process is consideredas an example. 1. introduction and preliminaries parameter estimation in diffusion processes based on discrete observations is being paid a lot ofattention now a days in view of its application in many fields such as biology, physics, oceanograpgyand especially in finance, see kutoyants (2004) and bishwal (2008, 2021).consider the itô stochastic differential equation dxt = f (θ,xt)dt + dwt , t ≥ 0 x0 = x0 (1.1) where {wt , t ≥ 0} is a one dimensional standard wiener process, θ ∈ θ, θ is a compact subsetof r, f is a known real valued function defined on θ × r, the unknown parameter θ is to beestimated on the basis of observation of the proces {xt , t ≥ 0}. let θ0 be the true value of theparameter which is in the interior of θ. we assume that the process {xt , t ≥ 0} is observed at 0 = t0 < t1 < . . . < tn = t with ∆ti := ti − ti−1 = t n = h, i = 1, 2, . . . , n and t = dn1/2for some fixed real number d > 0. we estimate θ from the observations {xt0 , xt1 , . . . , xtn}. this received: 5 dec 2021. key words and phrases. itô stochastic differential equation; stratonovich integral; diffusion process; discrete ob-servations; high frequency; approximate maximum likelihood estimators; conditional least squares estimator; strongconsistency; monte carlo methods. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 2model was first studied by dorogovcev (1976) who obtained weak consistency of the conditionalleast squares estimator (clse) under some regularity conditions as t →∞ and t n → 0. kasonga(1988) obtained the strong consistency of the clse under some regularity conditions as n → ∞assuming that t = dn1/2 for some fixed real number d > 0.note that the conditional least squares estimator (clse) of θ is defined as θn,t := arg min θ∈θ qn,t (θ) where qn,t (θ) = n∑ i=1 [ xti −xti−1 − f (θ,xti−1 )h ]2 ∆ti . note that the clse, the euler-maruyama estimator and the iamle are the same estimator(see shoji (1997)). for the ornstein-uhlenbeck process, bishwal and bose (2001) studied therates of weak convergence of approximate maximum likelihood estimators, which are of conditionalleast squares type. for the ornstein-uhlenbeck process bishwal (2010a) studied uniform rate ofweak convergence for the minimum contrast estimator, which has close connection to stratonovich-milstein scheme. bishwal (2009a) studied berry-esseen inequalities for conditional least squaresestimator discretely observed nonlinear diffusions. bishwal (2009b) studied stratonovich basedapproximate m-estimator of discretely sampled nonlinear diffusions. bishwal(2011a) studied mil-stein approximation of posterior density of diffusions. bishwal (2010b) studied conditional leastsquares estimation in nonlinear diffusion processes based on poisson sampling. bishwal (2011b)obtained some new estimators of integrated volatility using the stochastic taylor type schemeswhich could be useful for option pricing in stochastic volatility models. in mathematical finance,almost sure optimal hedging has received recent attention. gobet and landon (2014) studied theoptimal discretization error in the context of hedging error in a multidimensional itô model wherethe convergence is studied in an almost sure sense and the discrete trading dates are stoppingtimes which includes the sampling scheme of karandikar (1995) who studied pathwise convergenceof stochastic integrals. bishwal (2011c) studied higher order approximation of hedging error inthe mean square sense. almost sure hedging and optimality of discretization error motivates ouralmost sure consistency in estimation problem.florens-zmirou (1989) studied minimum contrast estimator, based on an euler-maruyama typefirst order approximate discrete time scheme of the sde (1.1) which is given by zti − zti−1 = f (θ, zti−1 )(ti − ti−1) +wti −wti−1 , i ≥ 1, z0 = x0. the log-likelihood function of {zti , 0 ≤ i ≤ n} is given by c n∑ i=1 [ zti − zti−1 − f (θ, zti−1 )h ]2 ∆ti . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 3where c is a constant independent of θ. a contrast for the estimation of θ is derived from the abovelog-likelihood by substituting {zti , 0 ≤ i ≤ n} with {xti , 0 ≤ i ≤ n}. the resulting contrast is hn,t = c n∑ i=1 [ xti −xti−1 − f (θ,xti−1 )h ]2 ∆ti . and the resulting minimum contrast estimator, called the euler estimator, is θ̌n,t := arg min θ∈θ hn,t (θ) florens-zmirou (1989) showed l2 consistency of the estimator as t →∞ and t n → 0.if continuous observation of {xt} on the interval [0, t ] were available, then the likelihood functionof θ would be lt (θ) = exp {∫ t 0 f (θ,xt)dxt − 1 2 ∫ t 0 f 2(θ,xt)dt } , (1.2) (see liptser and shiryayev (1977)). in our case we have discrete data and we have to approximatethe likelihood to get the mle. taking itô type approximation of the stochastic integral and rectanglerule approximation of the ordinary integral in (1.2) and obtain the approximate likelihood function ln,t (θ) = exp { n∑ i=1 f (θ,xti−1 )(xti −xti−1 )− h 2 n∑ i=1 f 2(θ,xti−1 ) } . (1.3) an approximate maximum likelihood estimate (amle) based on ln,t is defined as θ̂n,t := arg max θ∈θ ln,t (θ). weak consistency and other properties of this estimator were studied by yoshida (1992) as t →∞and t n → 0.note that the clse, the euler estimator and the amle1 are the same estimator (see shoji(1997)).in order to obtain a better estimator, which may have faster rate of convergence, we propose anew algorithm. note that the itô and the stratonovich integrals are connected by∫ t 0 f (θ,xt)dxt = ∫ t 0 f (θ,xt) o dxt − 1 2 ∫ t 0 ḟ (θ,xt)dt. (see ikeda and watanabe (1989)). we transform the itô integral in (1.2) to stratonovich integraland apply stratonovich type approximation of the stochastic integral and rectangular rule typeapproximation of the ordinary integrals and obtain the approximate likelihood l̃n,t (θ) = exp { 1 2 n∑ i=1 (f (θ,xti−1 ) + f (θ,xti ))(xti −xti−1 ) − h 2 n∑ i=1 (ḟ (θ,xti−1 ) + f 2(θ,xti−1 )) } . (1.4) https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 4 the stratonovich approximate maximum likelihood estimator (samle) based on ∼ln,t is defined as θ̃n,t := arg max θ∈θ ∼ ln,t (θ). this estimator is known to have faster rate of convergence (in the mean square sense) than theconditional least squares estimator, see bishwal (2009b).for monte carlo simulations in finance, one would be interested for pathwise convergence ofthe estimator. in this paper prove the strong consistency of the samle under some regularityconditions given below as n → ∞. we shall use the following notations : ∆xi = xti − xti−1 , ∆wi = wti − wti−1 , c is a generic constant independent of h, n and other variables (perhaps itmay depend on θ). prime denotes derivative w.r.t. θ and dot denotes derivative w.r.t. x . supposethat θ0 denote the true value of the parameter and θ0 ∈ θ. we assume the following conditions:(a1) the parameter space θ is compact.(a2) |f (θ, x)| ≤ k(θ)(1 + |x |), |f (θ, x)− f (θ, y)| ≤ k(θ)|x − y |. |f (θ, x)− f (φ, y)| ≤ c(x)|θ − φ| for all θ, φ ∈ θ, x, y ∈ r where sup θ∈θ |k(θ)| = k <∞, e|c(x0)|m = cm <∞ for some m > 16. (a3) the diffusion process x is stationary and ergodic with invariant measure ν, i.e., for any gwith e[g(·)] <∞ 1 n n∑ i=1 g(xti )→ eν [g(x0)] a.s. as t →∞ and h → 0. further e|x0|m <∞ for some m > 16.(a4) e|f (θ,x0)− f (θ,x0)|2 = 0 iff θ = θ0.(a5) f is twice continuously differentiable function in x with e sup t |ḟ (xt)|2 <∞, e sup t |f̈ (xt)|2 <∞. 2. main results we shall use the following theorem to prove the strong consistency of the samle. theorem 2.1 (frydman (1980). suppose the random function dn satisfy the following conditions: (c1) with probability one, dn(θ)→ d(θ) uniformly in θ ∈ θ as n →∞. (c2) the limiting nonrandom function d is such that d(θ0) ≥ d(θ) for all θ ∈ θ. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 5 (c3) d(θ) = d(θ0) iff θ = θ0. then θn → θ0 a.s. as n →∞, where θn = supθ∈θdn(θ). we need the following lemmas in order to prove our main result. lemma 2.1 under (a1)(a5), sup θ∈θ 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi − h 2 n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ]} → 0 a.s. as t →∞, tn → 0. proof. let v(θ, x) := f (θ, x)− f (θ0, x). the fourier expansion of v(θ, x) in l(θ) be given by v(θ, x) = ∞∑ m=1 am(x)eπjmθ, j = √ −1, x ∈ r where ak(x) are the fourier coefficients. thus 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi − h 2 n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ]} = 1 2t { ∞∑ m=1 n∑ i=1 [ am(xti−1 ) + am(xti ) ] eπjmθ∆wi − h 2 ∞∑ m=1 n∑ i=1 [ ȧm(xti−1 ) + ȧm(xti ) ] eπjmθ } where |am(x)| ≤ cm|x |, ∞∑ m=1 m1+γc4 m <∞. let am,n(s) := 1 2 n∑ i=1 [ am(xti−1 ) + am(xti ) ] i(ti−1−ti ](s) where i(ti−1−ti ], i = 1, 2, ..., n are indicator functions. then 1 2 n∑ i=1 [ am(xti−1 ) + am(xti ) ] ∆wi = ∫ t 0 am,n(s) o dws and h 2 n∑ i=1 [ ȧm(xti−1 ) + ȧm(xti ) ] = ∫ t o ȧm,nds. but ∫ t 0 am,n(s) o dws − 1 2 ∫ t o ȧm,nds = ∫ t 0 am,n(s)dws . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 6by exponential inequality for martingales, we have p {∫ t 0 am,n(s)dws − α 2 ∫ t o a2 m,nds > β } ≤ e−αβ for any α, β > 0. thus p { 1 t ∫ t 0 am,n(s)dws > β t + α 2t ∫ t o a2 m,nds } ≤ e−αβ and p {∣∣∣∣ 1 t ∫ t 0 am,n(s)dws ∣∣∣∣ > β t + αh 8t n∑ i=1 [ am(xti−1 ) + am(xti ) ]2} ≤ 2e−αβ. since h 2t n∑ i=1 [ am(xti−1 ) + am(xti ) ]2 ≤ c2 m h t n∑ i=1 [ (xti−1 )2 + (xti ) 2 ] and by (a3) h 2t n∑ i=1 [ (xti−1 )2 + (xti ) 2 ] → e(x2 0 ) > 0 a.s., there exists a random variable v such that h 2t n∑ i=1 [ (xti−1 )2 + (xti ) 2 ] < v a.s. for all t > 0, n = 1, 2, . . . . where p (v <∞) = 1.denote zm,n := 1 tn ∫ tn 0 am,n(s)dws . recall that t = tn. choose α := ma tδn , β := tγn mb , where δ < γ < 1 and 1 2 < b < 1+γ 2 .then p ( |zm,n| > 1 t1−γ n mb + mac2 mv 2tδn ) < 2e−m a−btγ−δn . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 7this p ( ∞∑ m=1 z2 m,n > ∞∑ m=1 ( 1 t1−γ n mb + mac2 mv 2tδn )2 ) ≤ ∞∑ m=1 p ( z2 m,n > ( 1 t1−γ n mb + mac2 mv 2tδn )2 ) = ∞∑ m=1 p ( |zm,n| > 1 t1−γ n mb + mac2 mv 2tδn ) ≤ 2 ∞∑ m=1 e−m a−btγ−δn ≤ 2e−t γ−δ n ∞∑ m=1 e−m a−b . hence ∞∑ n=1 p ( ∞∑ m=1 z2 m,n > ∞∑ m=1 ( 1 t1−γ n mb + mac2 mv 2tδn )2 ) ≤ 2 ∞∑ n=1 e−t 1−γ n ∞∑ m=1 e−m a−b <∞ since γ − δ > 0 and a − b > 0. the above implies ∞∑ n=1 p ( ∞∑ m=1 z2 m,n > 2 t 2(1−γ) n ∞∑ m=1 m−2b + v 2 t2δ n ∑ m m2ac4 m ) <∞. by borel-cantelli lemma, ∞∑ m=1 ( 1 2tn n∑ i=1 [ am(xti−1 ) + am(xti ) ] ∆wi − h 2tn n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ])2 −→ 0 a.s. as n →∞. this completes the proof of the lemma. lemma 2.2 under (a1)– (a5), with probability one, sup θ∈θ ∣∣∣∣∣ 1 t n∑ i=1 ∫ ti ti−1 [f (θ0, xs)− f (θ0, xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣→ 0. proof. for m > 0, we have e sup θ∈θ ∣∣∣∣∣ 1 t n∑ i=1 ∫ ti ti−1 [f (θ0, xs)− f (θ0, xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣ 2m  = e { sup θ∈θ ∣∣∣∣ 1 t ∫ t 0 gn(s)ds ∣∣∣∣2m } . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 8 where gn(s) = ∑n i=1 ∫ ti ti−1 [f (θ0, xs)− f (θ0, xti−1 )]v(θ,xti−1 ) if ti−1 ≤ s ≤ ti .hölder’s inequality implies that e { sup θ∈θ ∣∣∣∣ 1 t ∫ t 0 gn(s)ds ∣∣∣∣2m } ≤ t−2me { sup θ∈θ t 2m−1 ∫ t 0 |gn(s)|2mds } ≤ t−2me ( sup θ∈θ t 2m−1 n∑ i=1 ∫ ti ti−1 |f (θ0, xs)− f (θ0, xti−1 )|2m|v(θ,xti−1 )|2mds ) ≤ t−1um n∑ i=1 ∫ ti ti−1 e(|f (θ0, xs)− f (θ0, xti−1 )|2m|c(xti−1 )|2mds) by condition (a2) where um := supθ∈θ |θ − θ0|2m <∞.by cauchy-schwarz’s inequality the above term is ≤ t−1um n∑ i=1 ∫ ti ti−1 (e|f (θ0, xs)− f (θ0, xti−1 )|4m)1/2(e(c(xti−1 )|4m)1/2ds ≤ t−1umk 2m(θ0)(e|c(x0)|4m)1/2 n∑ i=1 ∫ ti ti−1 (e|xs −xti−1 )|4m)1/2ds by condition (a2). since e|xt − xs |2m ≤ m(t − s)m, from gikhman and skorohod (1975, p.48),the above term ≤ t−1umk 2m(θ0)(e|c(x0)|4m)1/2m1/2 n∑ i=1 ∫ ti ti−1 (s − ti−1)mds = umk 2m(θ0)(e|c(x0)|4mm)1/2t−1 n∑ i=1 (∆ti) m+1 m + 1 ≤ umk 2m(θ0) m + 1 (e|c(x0)|4mm)1/2hmn−m/2, m > 4. chebyshev’s inequality and the above implies that for any ε > 0, ∞∑ n=1 p { sup θ∈θ ∣∣∣∣∣ 1 t n∑ i=1 ∫ ti ti−1 [f (θ0, xs)− f (θ0, xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣ > ε } <∞. hence borel-cantelli lemma yields the result. lemma 2.3 under (a1)(a6), with probability one, 1 t n∑ i=1 [f (θ,xti−1 )− f (θ0, xti−1 )]2∆ti → e|v(θ,x0)|2 uniformly in θ as t →∞, tn → 0. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 9 proof. by the strong law of large numbers (ergodicity), 1 t ∫ t 0 |v(θ,xs) 2ds → e|v(θ,x0)|2. a.s. as t →∞ for each θ ∈ θ. the condition (a2) implies that 1 t ∫ t 0 |v(θ,xs) 2ds ≤ 1 t |θ − θ0|2 ∫ t 0 |c(xs)|2ds ≤ sup θ∈θ |θ − θ0|2 1 t ∫ t 0 |c(xs)|2ds ≤ b almost surely for some random variable b by (a1), (a2) and (a3). it also follows easily by (a1)-(a4)that ∣∣∣∣ 1 t ∫ t 0 |v(θ1, xs) 2ds − 1 t ∫ t 0 |v(θ2, xs) 2ds ∣∣∣∣ ≤ j|θ1 − θ2| almost surely for some random variable j and θ1, θ2 ∈ θ. thus the family of functions { 1 t ∫ t 0 |v(·, xs)|2ds, t ≥ 0 } is equicontinuous. hence by arzela-ascoli theorem, the convergence is uniform. denote g2 n(θ) := h 2 n∑ i=1 [ (xti−1 )2 + (xti ) 2 ] . now it is enough to show that 1 t ∫ t 0 |v(θ,xs)|2ds − 1 t g2 n(θ)→ 0 a.s. uniformly in θ. we have e { sup θ∈θ | ∫ t 0 |v(θ,xs) 2ds − g2 n(θ)|2m } e { sup θ∈θ | ∫ t 0 |v(θ,xs) 2ds − h n∑ i=1 |v(θ,xti−1 )|2|2m } = e { sup θ∈θ | n∑ i=1 ∫ ti ti−1 n∑ i=1 (v(θ,xs − v(θ,xti−1 ))(v(θ,xs + v(θ,xti−1 ))ds|2m } . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 10hölder inequality implies the above expectation ≤ t 2m−1e sup θ∈θ n∑ i=1 { ∫ ti ti−1 |v(θ,xs − v(θ,xti−1 )|2m|v(θ,xs + v(θ,xti−1 ))|2m} ≤ t 2m−1 n∑ i=1 ∫ ti ti−1 e[sup θ∈θ |v(θ,xs − v(θ,xti−1 )|2m sup θ∈θ |v(θ,xs + v(θ,xti−1 ))|2m]ds ≤ t 2m−1k2m22mum n∑ i=1 ∫ ti ti−1 e[|xs −xti−1 |2m(|c(xs)|2m + |c(xti−1 ))|2m]ds ≤ t 2m−1k2m22m+1um n∑ i=1 ∫ ti ti−1 (e|xs −xti−1 )|4m)1/2(e|c(xs)|4m + e|c(xti−1 ))|4m)1/2ds ≤ t 2m−1k2m22m+1umm 1/2(e|c(x0)|2m))1/2 n∑ i=1 ∫ ti ti−1 (s − ti−1)mds| (by stationarity) ≤ rmt 2m−1n(t/n)m+1 where um := supθ∈θ |θ − θ0|2m < ∞ and rm := k2m22m+2umm 1/2(e|c(x0)|4m)1/2. hence if m > 4, e { sup θ∈θ | 1 t ∫ t 0 |v(θ,xs)|2ds − 1 t g2 n(θ)|2m } ≤ rm(t/n)m ≤ rmhm/2n−m/2. borel-cantelli argument yields the result. now we are ready to present the main result of the paper: theorem 2.2 under the conditions (a1)-(a5), the samle is strongly consistent, i.e., θ̃n,t → θ0 a.s. as t →∞, t n → 0. proof. let ∼ l n,t (θ) := log ∼ ln,t (θ) and v(θ, x) := f (θ, x)− f (θ0, x). https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 11note that 1 t [∼ l n,t (θ)− ∼ l n,t (θ0) ] = 1 2t n∑ i=1 [f (θ,xti−1 ) + f (θ,xti )](xti −xti−1 ) − 1 2t n∑ i=1 [f (θ0, xti−1 ) + f (θ0, xti )](xti −xti−1 ) − 1 2n n∑ i=1 [ḟ (θ,xti−1 )− ḟ (θ0, xti−1 )] − 1 2n n∑ i=1 [f 2(θ,xti−1 )− f 2(θ0, xti−1 )] = 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi − h n∑ i=1 v̇(θ,xti−1 ) } − 1 2n n∑ i=1 v2(θ,xti−1 ) − 1 t n∑ i=1 ∫ ti ti−1 v(θ,xti−1 )[f (θ0, xt) + f (θ0, xti−1 )]dt − 1 t n∑ i=1 ∫ ti ti−1 [v(θ,xti )f (θ0, xt)− v(θ0, xti−1 )f (θ0, xti−1 )]dt =: i1 − i2 − i3 − i4. let dn,t (θ) := 1 t [∼ l n,t (θ)− ∼ l n,t (θ0) ] .below lemma 2.1-2.3 show that dn,t (θ)→ d(θ) a.s. as t →∞, t n → 0 where d(θ) := − 1 2 e|f (θ,x0)− f (θ0, x 0)|2.thus condition (c1) of theorem 2.1 is satisfied. the limiting function d(θ) satisfies the conditions(c2) and (c3) of theorem. hence as a consequence of theorem 2.1 we obtain the result. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 12 3. ornstein-uhlenbeck process consider the ornstein-uhlenbeck process satisfying dxt = θxtdt + dwt , t ≥ 0, x0 = 0, θ < 0. the euler estimator (conditional least squares estimator) is given by θ̌n,t = ∑n i=1xti−1 (xti −xti−1 ) h ∑n i=1x 2 ti−1 . strong consistency of this estimator is obtained in kasonga (1988). as a consequence of theorem2.2, we obtain the strong consistency of three estimators with θ̃n,t = (x2 t − t )/2 h ∑n i=1x 2 ti−1 , θ̄n,t,3 = x2 t /2 h ∑n i=1x 2 ti−1 , θ̂n,t,2 = −t/2 h ∑n i=1x 2 ti−1 . which are samle, yamle (young amle) and , amce respectively as t → ∞ and t/n → 0.samle is the linear combination of amce and yamle.define the continuous mle, ymle and mce respectively θt,1 = ∫ t 0 xtdxt∫ t 0 x2 t dt , θt,2 = x2 t /2∫ t 0 x2 t dt , θt,3 = −t/2∫ t 0 x2 t dt . interpreting ∫ t0 xtdxt to be the young (1936) integral, it equals x2 t /2. belfadli et al. (2011)(see also el machkouri et al. 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stochastic differential equations; stabilities of sdes; numerical schemes;vasicek and geometric brownian motion. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 2the choice of a suitable numerical scheme is based on the understanding and manipulation ofcertain qualitative properties as stability, consistency etc. the qualitative property like stability ofstochastic differential equations solutions, introduced by i.kats and n.krasovskii [2] and perfectedby i.i. gikhman, a.v. skorokhold [3] and a. friedman [4] plays a major role in the study of sdes andthe numerical schemes associated. thus, looking for numerical schemes that preserve qualitativeproperties as the stability of solutions constitutes and remains a very widespread problem innumerical analysis of sdes. in this article we establish and prove the conditions of numerical schemes stabilities in mean andmean-square. we apply the approach described by y.saito [5] to defined and demonstrate the sta-bilities of numericals sdes schemes as: euler-maruyama, milshtein and implicit euler-maruyamafor vasicek and geometric brownian motion models. to begin, let present some elementary notionsrelative to sdes and the numerical schemes adapted to the sdes. 2. preliminary notions 2.1. stochastic differential equation and stabilities. in this section,we present some definitions inconnection with stochastic differential equation and stabilities of solutions of sdes. definition 2.1. (stochastic differential equation (sde) [13]) let ( ω,f , (ft)t≥0 ,p ) be a filtered probability space, (bt)t≥0 a standard brownian motion on rd defines in a filtered probability space. a stochastic differential equation (sde) on rd with the drift coefficient: b (t, xt) ∈ [0, t ]× rn −→ rn and the diffusion: σ (t, xt) ∈ [0, t ]× rn −→ rn×d when xo is random variable independent of (bt)t≥0 is an equation of the form:{ dxt = b (t, xt) dt + σ (t, xt) dbt x (o) = xo (2.1) the white noise σ (t, xt) can be additive or multiplicative, depending on whether it does notinfluence or does influence the state of the system. theorem 2.1. (existence and uniqueness [14]) we assume that there is a positive constant k such that ∀ t ≥ 0, x, y ∈ rd(1) lipschitz condition: |b (t, x)− b (t, y ) |+ |σ (t, x)− σ (t, y ) | ≤ k|x − y | https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 3(2) linear growth condition: |b (t, x) | ≤ k (1 + |x|) , |σ (t, x) | ≤ k (1 + |x|) so the sde (2.1) admits, for any initial condition xo of square integrable ( e [ |xo |2 ] <∞ ) the strong solution (xt)t∈[0,t ],unique, almost surely continuous and satisfying the following condition: e ( sup 0≤t≤t |x2 t | ) <∞ definition 2.2. (asymptotic stability in probability in large sense [1], [24]) the solution is said to be asymptotically and stochastically stable in the large sense if ∀ xo ∈ l2 ft ([−t, 0] ,rn) , then p { lim t−→∞ x (t) = 0 } = 1. definition 2.3. (stability of pth moment [23], [25])(1) let p ≥ 2 we say that a solution of (2.1) is stable in pth moment if ∀ε > 0 it exists δ > 0 such as e [ sup t>0 |x (t) |p ] < ε avec |xo | < δ (2) let p ≥ 2, we say that a solution of (2.1) is stable asymptoticaly in pth moment if it is stable from peme moment ∀ xo ∈ l2 fto ([−t, 0] ,rn) then we have : lim t−→∞ e [ sup t>t |x (t) |p ] = 0 2.2. stochastic numerical schemes. in this section we present three numerical schemes as euler-maruyama, implicit euler-maruyama and milshtein schemes. definition 2.4. ( euler-maruyama scheme [10] , [11]) let {xt} the diffusion solution of the sde(2.1). let consider the interval [0, t ] and a regular subdivision t0 = 0 < t1 < t2 < t0 < · · · < tk = t with step ∆t = t n = t k , the euler-maruyama scheme of (2.1) is defined like:{ xemk+1 = xk + b(tk , xk)(tk+1 − tk) + σ(tk , xk)(bk+1 − bk) x(0) = x0 (2.2) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 4 definition 2.5. (implicit euler-maruyama scheme [10]) the implicit euler-maruyama scheme is a convergent scheme like the euler-maruyama scheme. to be reassured of the existence of the solutions of this scheme, only the term of the drift is implicit. for this fact: b(xk)∆tk which is in the euler-maruyama scheme is replaced by b(xk+1)∆tk and the diffusion term: σ(xk)∆bk remains unchanged. the implicit euler-maruyama scheme of the eds (2.1) has given by: xiemk+1 = xk + b(xk+1)∆tk + σ(xk)∆bk (2.3) definition 2.6. (milshtein scheme [7]) let consider the sde (2.1) and a regular subdivision of the intervalle et une subdivision of the interval [0, t ]: 0 = t0 < t1 < t2 < · · · < tn = t de [0, t ] the milshtein scheme is defined like: xmk+1 = xk + b(xk)∆tk + σ(xk)∆bk + 1 2 σ(xk)σ′(xk)(∆bk − ∆tk) x(0) = x0 (2.4) remark 2.1. it should be noted that the euler-maruyama scheme converges strongly up to the order 1 2 while that of milshtein converges up to the order 1. 3. numerical stabilities of vasicek model 3.1. explicit solution. the vasicek model (1977) is one of the first stochastic interest rate models.it is a gaussian process generalizing the ornstein-unlenbeck model and explains the observedempirical mean reversion effect on interest rate curves [15], this model looks like:dxt = (θ1 − θ2xt)dt + θ3dbt x(0) = x0 ∀θ1, θ2 et θ3 > 0 (3.1) with xt : the instant interest rate; θ2: mean reversion rate; θ1: the long-term average and θ3: thevolatility.the analytical solution of (3.1) model is: xt = θ1 θ2 + ( x0 − θ1 θ2 ) e−θ2t + θ3 ∫ +∞ 0 e−θ2(t−u)dbu (3.2) the model (3.1) is equivalent to the model:dxt = θ(µ−xt)dt + σdbt x(0) = x0 (3.3) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 5the solution of (3.3) has given by: xt = µ+ (x0 − µ) e−θt + θ ∫ t 0 e−θ2(t−u)dbu (3.4) considering the solution of (3.2), the mean and the mean-square give respectively: e[xt ] = θ1 θ2 ∀ θ2 > 0 and v (xt) = θ2 3 2θ2 ∀ θ2 > 0 which means that the stochastic process xt ' n ( θ1 θ2 , θ2 3 2θ2 ) by using some properties of brownian motion, the solution of the model (3.2) can be written asfollows: xt = θ1 θ2 + θ3e −2θ2t √ 2θ2 b(e2θ2t) (3.5) now, we present some numerical stabilities conditions for the system (3.1) of some numericalschemes (euler-maruyama, implicit euler-maruyama and milshtein) and the proofs of these basedon the approach described in [5]. 3.2. euler-maruyama scheme stabilities. the euler-maruyama scheme associated to the system(3.1) is : xemk+1 = xk + (θ1 − θ2xk) ∆t + θ3∆bk xemk+1 = θ1∆t + (1− θ2∆t)xk + θ3 √ ∆tzk (3.6) 3.2.1. mean stability of euler-maruyama scheme. theorem 3.1. (mean stability of euler-maruyama scheme) the euler-maruyama scheme (3.6) of the vasicek model (3.1) is mean asymptotically stable if: e [ xemk+1 ] = (1− θ2∆t)k+1e[x0] + θ1∆t [ k+1∑ i=0 (1− θ2∆t)i ] (3.7) with |1− θ2∆t| < 1 and lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ1 θ2 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 6 proof. to prove the theorem, we start by evaluating the mean of the (3.1) equation using theapproach defined in [5]. in effect, e [ xemk+1 ] = e [ θ1∆t +xk (1− θ2∆t) + θ3 √ ∆tzk ] = e [θ1∆t] + e [xk (1− θ2∆t)] + e [ θ3 √ ∆tzk ] = e [θ1∆t] + (1− θ2∆t)e [xk ] + 0] with zk ' n (0, 1) = θ1∆t + (1− θ2∆t)e[xk ] = θ1∆t + (1− θ2∆t) {(1− θ2∆t)e[xk−1] + θ1∆t} = θ1∆t + θ1∆t(1− θ2∆t) + (1− θ2∆t)2e[xk−1] = θ1∆t(1 + (1− θ2∆t)) + (1− θ2∆t)2e[xk−1] = θ1∆t(1 + (1− θ2∆t)) + (1− θ2∆t)2 {(1− θ2∆t)e[xk−2] + θ1∆t} = θ1∆t(1 + (1− θ2∆t)) + θ1∆(1− θ2∆t)2 + (1− θ2∆t)3e[xk−2] = θ1∆t(1 + (1− θ2∆t) + (1− θ2∆t)2) + (1− θ2∆t)3e[xk−2] = θ1∆t(1 + (1− θ2∆t) + (1− θ2∆t)2 + · · ·+ (1− θ2∆t))k+1 + (1− θ2∆t)k+1e[x0] = (1− θ2∆t)k+1e[x0] + θ1∆t [ k+1∑ i=0 (1− θ2∆t)i ] using the theory of geometric sequences and series, we get: e [ xemk+1 ] = (1− θ2∆t)k+1e[x0] + θ1∆t ( (1− (1− θ2∆t)k+1) 1− (1− θ2∆t) ) (3.8) as the identity (3.8) represents a geometric sequence, we have that it converges if |1− θ2∆t| < 1 by calculating the limit of the (3.8), for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ1 θ2 � 3.2.2. mean-square stability of euler-maruyama scheme. theorem 3.2. (mean-square stability of euler-maruyama scheme) the euler-maruyama scheme(3.6) of the vasicek model (3.1) is mean-square asymptotically stable if: e [∣∣xemk+1 ∣∣2] = (1− θ2∆t)2(k+1) e ( |x0|2 ) + ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=0 (1− θ2∆t)2i https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 7 and that the following two conditions are satisfied simultaneously: (1) |1− θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]2) = θ2 3 2θ2 proof. as in the previous theorem, we start by calculating the expression: e [∣∣xemk+1 ∣∣2] of vasicek model of the equation (3.1). in effect, e [∣∣xemk+1 ∣∣2] = |θ1∆t|2 + e (|xk(1− θ2∆t)|)2 + θ2 3∆t zk ' n(0, 1) = (1− θ2∆t)2e ( |xk |2 ) + (θ2 1∆t + θ2 3)∆t = (1− θ2∆t)2e ( |xk |2 ){ e ( |xk+1|2 ) (1− θ2∆t)2 + (θ2 3 + θ2 1∆t)∆t } + (θ2 3 + θ2 1∆t)∆t = (1− θ2∆t)4e ( |xk−1|2 ) + (θ2 3 + θ2 1∆t)∆t [ (1− θ2∆t)2 + 1 ] = (1− θ2∆t)6e ( |xk−2|2 ) + (θ2 3 + θ2 1∆t)∆t[(1− θ2∆t)4 + (1− θ2∆t)2 + 1] = (1− θ2∆t)8e ( |xk−3|2 ) + (θ2 3 + θ2 1∆t)∆t [ (1− θ2∆t)6 + (1− θ2∆t)4 + (1− θ2∆t)2 + 1 ] = (1− θ2∆t)2k+1e ( |x0|2 ) + ( θ2 3 + θ2 1∆t ) ∆t [ (1− θ2∆t)2k + ...+ (1− θ2∆t)4 +(1− θ2∆t)2 + (1− θ2∆t)0 ] = (1− θ2∆t)2(k+1) e ( |x0|2 ) + ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=0 (1− θ2∆t)2i = (1− θ2∆t)2k+2 e ( |x0|2 ) + ( θ2 3 + θ2 1∆t ) ∆t [ 1− |1− θ2∆t|2k+2 1− |1− θ2∆t|2 ] we get: e [∣∣xemk+1 ∣∣2] = (1− θ2∆t)2k+2 e ( |x0|2 ) + ( θ2 3 + θ2 1∆t ) ∆t [ 1 1− |1− θ2∆t|2 ] (3.9) the expression (3.9) as the geometric sequence, we have that it converges when |1− θ2∆t| < 1 passing to the limit of the equation (3.9), for ∆t → 0 and k → +∞, we find the desired result i.e: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ2 3 2θ2 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 83.3. milshtein’s scheme stabilities. the milshtein scheme associated to the system (3.1) is: xmk+1 = θ1∆t + (1− θ2∆t)xk + θ3 √ ∆tzk (3.10) with σ = θ3 σ′ = 0 then mean and mean-square stabilities gives the same results as in the euler-maruyama schemei.e theorem 3.3. (mean stability of milshtein scheme) the milshtein scheme (3.10) of vasicek model(3.1) is mean asymptotically stable if: e [ xmk+1 ] = θ1∆t [ k+1∑ i=0 (1− θ2∆t)i ] + (1− θ2∆t)k+1e[x0] and that the following two conditions are satisfied simultaneously: (1) |1− θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xmk+1 ]) = θ1 θ2 theorem 3.4. (mean-square stability of milshtein scheme) the milshtein scheme (3.10) of vasicek model (3.1) is mean-square asymptotically stable if: e [∣∣xmk+1 ∣∣2] = ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=0 (1− θ2∆t)2i + (1− θ2∆t)2(k+1) e ( |x0|2 ) and that the following two conditions are satisfied simultaneously: (1) |1− θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xmk+1 ]2) = θ2 3 2θ2 proof. the proofs of these theorems above is done in the same way as the result theorems of theeuler-maruyama scheme for vasicek model. � 3.4. implicit euler-maruyama scheme stabilities. the implicit euler-maruyama scheme associ-ated to the system (3.1) is : xiemk+1 = xk + (θ1 − θ2xk+1)∆t + θ3 √ ∆tzk xk+1 + θ2∆txk+1 = xk + θ1∆t + θ3 √ ∆tzk xk+1 (1 + θ2∆t) = θ1∆t +xk + θ3 √ ∆tzk xk+1 = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t xk + θ3 √ ∆t 1 + θ2∆t zk https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 9we get: xiemk+1 = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t xk + θ3 √ ∆t 1 + θ2∆t zk (3.11) 3.4.1. mean stability of implicit euler-maruyama scheme. theorem 3.5. (mean stability of implicit euler-maruyama scheme) the implicit euler-maruyama scheme (3.11) of vasicek model (3.1) is mean asymptotically stable if: e ( xiemk+1 ) = ( 1 1 + θ2∆t )k+1 e (x0) + θ1∆t k+1∑ i=0 ( 1 1 + θ2∆t )i and that the following two conditions are satisfied simultaneously: (1) |1 + θ2∆t| > 1 (2) lim ∆t→0 ( lim k→∞ e [ xiemk+1 ]) = θ1 θ2 proof. we start by evaluating the mean of the implicit euler-maruyama scheme of the expressiondefined in (3.11), in effect: e ( xiemk+1 ) = e ( θ1∆t 1 + θ2∆t ) + e ( 1 1 + θ2∆t xk ) + e ( θ3 √ ∆t 1 + θ2∆t zk ) = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t e (xk) = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t {( 1 1 + θ2∆t ) e (xk−1) + θ1∆t 1 + θ2∆t } = θ1∆t 1 + θ2∆t + θ1∆t (1 + θ2∆t)2 + 1 (1 + θ2∆t)2e (xk−1) = ( 1 1 + θ2∆t )2 e (xk−1) + θ1∆t ( 1 (1 + θ2∆t)2 + 1 (1 + θ2∆t) ) = ( 1 1 + θ2∆t )3 e (xk−2) + θ1∆t (( 1 1 + θ2∆t )3 + ( 1 1 + θ2∆t )2 + ( 1 1 + θ2∆t )) = ( 1 1 + θ2∆t )4 e (xk−3) + θ1∆t (( 1 1 + θ2∆t )4 + ( 1 1 + θ2∆t )3 + · · ·+ 1 ) by continuing the iterations until k + 1, we obtain: e ( xiemk+1 ) = ( 1 1 + θ2∆t )k+1 e (x0) + θ1∆t k+1∑ i=0 ( 1 1 + θ2∆t )i (3.12) the equation (3.12) is the geometric sum of geometric sequence and geometric series, the expres-sion: ( 1 1 + θ2∆t ) < 1 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 10or |1 + θ2∆t| > 1by using limit of (3.12), for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xiemk+1 ]) = θ1 θ2 � 3.4.2. mean-square stability of implicit euler-maruyama scheme. theorem 3.6. (mean-square stability of implicit euler-maruyama scheme) the implicit eulermaruyama of vasicek model (3.1) is mean-square asymptotically stable if: e (∣∣xiemk+1 ∣∣2) = ( 1 1 + θ2∆t )2(k+1) e (|x0|)2 + ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=0 ( 1 1 + θ2∆t )2i , with |1 + θ2∆t| > 1 and lim ∆t→0 ( lim k→∞ e ( |xk+1|2 )) = θ2 3 2θ2 proof. let us evaluate the mean-square of the implicit euler-maruyama scheme (3.11), in effect: e (∣∣xiemk+1 ∣∣2) = e (∣∣∣∣ θ1∆t 1 + θ2∆t ∣∣∣∣2 ) + e (∣∣∣∣ 1 1 + θ2∆t xk ∣∣∣∣)+ ∣∣∣∣ θ2 3∆t 1 + θ2∆t ∣∣∣∣2 = θ2 1(∆t)2 (1 + θ2∆t)2 + 1 (1 + θ2∆t)2e ( |xk |2 ) + θ2 3∆t (1 + θ2∆t)2 = θ2 1(∆t)2 + θ2 3∆t (1 + θ2∆t)2 + 1 (1 + θ2∆t)2e ( |xk |2 ) = ( θ2 3 + θ2 1∆t ) ∆t (1 + θ2∆t)2 + ( 1 1 + θ2∆t )2 {( 1 1 + θ2∆t )2 e (|xk−1|)2 + ( θ2 3 + θ2 1∆t ) ∆t (1 + θ2∆t)2 } = ( 1 1 + θ2∆t )4 e (|xk−1|)2 + ( θ2 3 + θ2 1∆t ) ∆t [( 1 1 + θ2∆t )4 + ( 1 1 + θ2∆t )2 ] ... = ( 1 1 + θ2∆t )2(k+1) e (|x0|)2 + ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=1 ( 1 1 + θ2∆t )2i by using the geometrical sequence and geometrical series, we get: e (∣∣xiemk+1 ∣∣2) = ( 1 1 + θ2∆t )2(k+1) e (|x0|)2 + ( θ2 3 + θ2 1∆t ) ∆t k+1∑ i=1 ( 1 1 + θ2∆t )2i we have the geometric sequence and series, converging when |1 + θ2∆t| > 1, by calculating limitof the equation below , for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xiemk+1 ]) = θ2 3 2θ2 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 11 � 4. numerical stabilities of geometric brownian motion 4.1. explicit solution of the model. geometric brownian motion known as exponential brownianmotion is a continuous stochastic process whose logarithm follows a brownian motion. it is appliedin the mathematical modeling of certain courses in the financial markets [26]. it represents areasonable approximation of the evolution of stock market prices, because a quantity which followsa geometric brownian motion takes all strictly positive values and only the elementary changesundergone by the random variable are significant. the geometric brownian motion xt is a process which is written in the form [13]:{ dxt = θ1xtdt + θ2xtdbt x(0) = x0 ∀θ1, θ2 ∈ r (4.1) this process admits as an explicit solution: xt = x0e {(θ1− 1 2 θ2 2)t+θ2bt} (4.2) the variable on the right hand follows a normal distribution, it can also be written in the form: xt = xse {(θ1− 1 2 θ2 2)t+θ2(bt−bs)} (4.3) the conditionnal mean is: e (xt |xs) = xse θ1(t−s) (4.4)the (4.3) process is often widely used to model the price of a financial asset the return on theasset between two dates is measured by the difference in the logarithms of the prices and is givenby the gaussian variable below:{ θ1 − 1 2 θ2 2 } (t − s) + θ2 (bt − bs) the mean and the mean-square give respectively: e(xt) = x0e θ1t e(x2 t ) = x2 0e (2θ1+θ2 2)t (4.5) remark 4.1. it should be noted that:(1) for mean if t →∞ and θ1 < 0 we have: lim t→∞ e(xt) = lim k→∞ x0e θ1t = 0 (2) for mean-square if ( 2θ1 + θ2 2 ) < 0 and t →∞ i.e lim t→∞ e(x2 t ) = 0 with ( 2θ1 + θ2 2 ) < 0. now, let’s analyze the stabilities of some numerical schemes (euler-maruyama, milshtein andimplicit euler-maruyama) in mean and mean-square. https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 124.2. euler-maruyama scheme stabilities. the euler-maruyama scheme associated to (4.1) is: xemk+1 = xk + θ1xk∆t + θ2xk∆bk xemk+1 = xk (1 + θ1∆t) + θ2xk √ ∆tzk (4.6) 4.2.1. mean stability of euler-maruyama scheme. theorem 4.1. (mean stability of euler-maruyama scheme) the euler-maruyama scheme (4.6) associated to (4.1) model is mean asymptotically stable if e [ xemk+1 ] = (1 + θ2∆t)k+1e (x0) with |1 + θ1∆t| < 1 and lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = 0 proof. by calculating the mean of the expression(4.6), we obtain: e [ xemk+1 ] = e [ xt(1 + θ1∆t) + θ2xt √ ∆tzt ] = e [xt(1 + θ1∆t)] + e [ θ2 √ ∆txtzt ] = e [(1 + θ1∆t)xt ] + e [ θ2 √ ∆t ] e [xt ]e [zt ]as zk ' n (0, 1) e(zk) = 0 e [xk+1] = (1 + θ1∆t)e(xt) = (1 + θ1∆t) ((1 + θ2∆t)e (xk−1)) = (1 + θ2∆t)2e (xk−1) = (1 + θ1∆t)2 ((1 + θ2∆t)e (xk−2))... = (1 + θ2∆t)k+1e (x0) we get the following geometric sequence: e [ xemk+1 ] = (1 + θ2∆t)k+1e (x0) which converges if |1 + θ2∆t| < 1 et nd passing to the limit for a ∆t → 0 and k → +∞, we obtain: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 134.2.2. mean-square stability of euler-maruyama scheme. theorem 4.2. (mean-square stability of euler-maruyama scheme) the euler-maruyama scheme(4.6) associated to (4.1) model is mean-square asymptotically stable if: e (∣∣xemk+1 ∣∣2) = ( |(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)2k+2 e ( |x0|2 ) , with ∣∣∣|1 + θ1∆t|2 + ∣∣θ2 √ ∆t ∣∣2∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e ( |xk+1|2 )) = 0 proof. the mean-square of the expression (4.6) gave: e [∣∣xemk+1 ∣∣2] = e [∣∣∣xk(1 + θ1∆t) + θ2xk √ ∆tzk ∣∣∣2] = e [ |xk(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆txkzk ∣∣∣2 + 2 ∣∣∣xk(1 + θ1∆t)θ2 √ ∆txkzk ∣∣∣] = e [ |xk(1 + θ1∆t)|2 ] + e [∣∣∣θ2 √ ∆txkzk ∣∣∣2]+ 2e [ |xk(1 + θ1∆t)| ∣∣∣θ2 √ ∆txkzk ∣∣∣] = |(1 + θ1∆t)|2 e [ |xk |2 ] + ∣∣∣θ2 √ ∆t ∣∣∣2 e [|xk |2] = ( |(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)e [|xt |2] = ( |(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)(|(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)e [|xk−1|2 ] ... = ( |(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)2k+2 e [ |x0|2 ] we get a geometric sequence: e (∣∣xemk+1 ∣∣2) = ( |(1 + θ1∆t)|2 + ∣∣∣θ2 √ ∆t ∣∣∣2)2(k+1) e ( |x0|2 ) for ∣∣∣|1 + θ1∆t|2 + ∣∣θ2 √ ∆t ∣∣2∣∣∣ < 1 the sequence converges, and passing to the limit, we obtain fora ∀∆t → 0 and k → +∞, lim ∆t→0 ( lim k→∞ e ( |xk+1|2 )) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 144.3. milshtein’s scheme stabilities. the milshtein schema associated to the expression(4.1) isgiven by : xmk+1 = xk + b(xk)∆t + σ(xk)∆bk + 1 2 σσ′(xk) { (∆bk)2 − ∆t } = xk + θ1xk∆t + θ2xk∆bk + 1 2 θ2xtθ2 { (∆bk)2 − ∆t } = xk + θ1xk∆t + θ2xk √ ∆tzk + 1 2 θ2 2xk ( ∆tz2 k − ∆t ) = ( 1 + θ1∆t − 1 2 θ2 2∆t ) xk + θ2xk √ ∆tzk + 1 2 θ2 2xk∆tz2 k = ( 1 + ( θ1 − 1 2 θ2 2 ) ∆t ) xk + θ2xk √ ∆tzk + 1 2 θ2 2xk∆tz2 kwe have after calculation: xmk+1 = xk ( 1 + ( θ1 − 1 2 θ2 2 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ2 2xk∆tz2 k (4.7) we now consider the same model of geometric brownian motion, we state some results on thestabilities following milshtein’s scheme and we prove these results. 4.3.1. mean stability of milshtein’s scheme. theorem 4.3. (mean stability of milshtein’s scheme) the milshtein’s scheme (4.7) assocated to(4.1) model is mean asymptotically stable if e ( xmk+1 ) = [1 + θ1∆t]k+1 e (x0) with |1 + θ1∆t| < 1 and lim ∆t→0 ( lim k→∞ e ( xmk+1 )) = 0 proof. applying the usual approach, let us evaluate the mean of gives: e ( xmk+1 ) = e ( xk ( 1 + ( θ1 − 1 2 θ2 2 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ2 2xk∆tz2 k ) = e ( xk ( 1 + ( θ1 − 1 2 θ2 2 ) ∆t )) + e ( θ2xk √ ∆tzk ) + e ( 1 2 θ2 2xk∆tz2 k ) = ( 1 + ( θ1 − 1 2 θ2 2 ) ∆t + 1 2 θ2 2∆t ) e (xk) = (1 + θ1∆t)e (xk)... = (1 + θ1∆t)k+1 e (x0) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 15ultimately we get that: e ( xmk+1 ) = [1 + θ1∆t]k+1 e (x0)as the previous expression has the form of a geometric sequence, we know that it converges if |1 + θ1∆t| < 1, passing to the limit, for all ∆t → 0 and k → +∞ we find the results searched i.e: lim ∆t→0 ( lim k→∞ e ( xmk+1 )) = 0 � 4.3.2. mean-square stability of milshtein’s scheme. theorem 4.4. (mean-square stability of milshtein’s scheme) the milshtein scheme (4.7) associated to (4.1) model is mean-square asymptotically stable if e (∣∣xmk+1 ∣∣) = [∣∣∣∣1 + ( θ1 − 1 2 θ2 2 ) ∆t ∣∣∣∣2 + ∣∣∣θ2 √ ∆t ∣∣∣2 + ∣∣∣∣12θ2 2∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) with ∣∣∣∣∣1 + ( θ1 − 1 2θ 2 2 ) ∆t ∣∣2 + ∣∣θ2 √ ∆t ∣∣2 + ∣∣1 2θ 2 2∆t ∣∣2∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e (∣∣xmk+1 ∣∣2)) = 0 proof. let’s start by calculating the mean-sqaure of the model expression, ie: e (∣∣xmk+1 ∣∣2) = e (∣∣∣∣xk (1 + ( θ1 − 1 2 θ2 2 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ2 2xk∆tz2 k ∣∣∣∣2 ) = e (∣∣∣∣xk (1 + ( θ1 − 1 2 θ2 2 ) ∆t )∣∣∣∣2 ) + e (∣∣∣θ2xk √ ∆tzk ∣∣∣2)+ e (∣∣∣∣12θ2 2xk∆tz2 k ∣∣∣∣2 ) = ∣∣∣∣1 + ( θ1 − 1 2 θ2 2 ) ∆t ∣∣∣∣2 e (|xk |2)+ ∣∣∣θ2 √ ∆t ∣∣∣2 e (|xk |2)+ ∣∣∣∣12θ2 2∆t ∣∣∣∣2 e (|xk |2) = [∣∣∣∣1 + ( θ1 − 1 2 θ2 2 ) ∆t ∣∣∣∣2 + ∣∣∣θ2 √ ∆t ∣∣∣2 + ∣∣∣∣12θ2 2∆t ∣∣∣∣2 ] e ( |xk |2 ) ... = [∣∣∣∣1 + ( θ1 − 1 2 θ2 2 ) ∆t ∣∣∣∣2 + ∣∣∣θ2 √ ∆t ∣∣∣2 + ∣∣∣∣12θ2 2∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) continuing with the iterations, we get: e (∣∣xmk+1 ∣∣) = [∣∣∣∣1 + ( θ1 − 1 2 θ2 2 ) ∆t ∣∣∣∣2 + ∣∣∣θ2 √ ∆t ∣∣∣2 + ∣∣∣∣12θ2 2∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie: lim ∆t→0 ( lim k→∞ e (∣∣xmk+1 ∣∣2)) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 164.4. implicit euler-maruyama scheme stabilities. the implicit euler-maruyama scheme (iem)gives: xiemk+1 = xk + b(xk+1)∆t + δ(xt)∆bk xk+1 = xk + θ1xk+1∆t + θ2xt∆bk xk+1 − θ1xk+1∆t = xk + θ2xk∆bk xk+1 (1− θ1∆t) = xk + θ2xk∆bkwe obtain: xiemk+1 = 1 1− θ1∆t xk + θ2 √ ∆t 1− θ1∆t xkzk zk ' n(0, 1) (4.8) 4.4.1. mean stability of implicit euler-maruyama scheme. theorem 4.5. (mean stability of implicit euler-maruyama scheme) the implicit euler-maruyama scheme (iem) (4.8) associated to (4.1) model is mean asymptotically stable if e ( xiemk+1 ) = ( 1 1− θ1∆t )k+1 e (x0) (4.9) with |1− θ1∆t| > 1 then, lim ∆t→0 ( lim k→∞ e (∣∣xiemk+1 ∣∣2)) = 0 proof. let’s evaluate the mean associated to the implicit euler-maruyama scheme e ( xiemk+1 ) = e ( 1 1− θ1∆t xk + θ2 1− θ1∆t xk √ ∆tzk ) = e ( 1 1− θ1∆t xk ) + e ( θ2 1− θ1∆t √ ∆t ) (xk) (zk) = e ( 1 1− θ1∆t xk ) = 1 1− θ1∆t e (xk) = ( 1 1− θ1∆t )2 e (xk−1) = ( 1 1− θ1∆t )3 e (xk−2) ... = ( 1 1− θ1∆t )k+1 e (x0) continuing with the iterations we get: e ( xiemk+1 ) = ( 1 1− θ1∆t )k+1 e (x0) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 17passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie lim ∆t→0 ( lim k→∞ e (∣∣xiemk+1 ∣∣2)) = 0 � 4.4.2. mean-square stability of implicit euler-maruyama scheme. theorem 4.6. (mean-square stability of implicit euler-maruyama scheme) the implicit eulermaruyama scheme associated to the model (4.1) is asymptotically mean-square stable if e (∣∣xiemk+1 ∣∣2) = [ 1 + ∣∣θ2 √ ∆t ∣∣ 1− θ1∆t ]2(k+1) e ( |x0|2 ) with ∣∣∣∣1+|θ2 √ ∆t| 1−θ1∆t ∣∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e (∣∣xemik+1 ∣∣2)) = 0 proof. : let us calculate the quadratic mean, in effect, e (∣∣xmk+1 ∣∣2) = e ∣∣∣∣∣ 1 1− θ1∆t xt + θ2 √ ∆t 1− θ1∆t xtzt ∣∣∣∣∣ 2  = ( 1 1− θ1∆t )2 e (∣∣∣xk + θ2 √ ∆txkzk ∣∣∣2) = ( 1 1− θ1∆t )2 [ e ( |xk |2 ) + e (∣∣∣+θ2 √ ∆txkzk ∣∣∣2)] = ∣∣∣∣ 1 1− θ1∆t ∣∣∣∣2 [e (|xk |2)+ ∣∣∣+θ2 √ ∆t ∣∣∣2 e (|xk |2)] = ( 1 + ∣∣θ2 √ ∆t ∣∣2) (1− θ1∆t)2 e ( |xk |2 ) = ( 1 + ∣∣θ2 √ ∆t ∣∣2) (1− θ1∆t)2 ( 1 + ∣∣θ2 √ ∆t ∣∣2) (1− θ1∆t)2 e ( |xk−1|2 ) continuing with the iterations, we get: e (∣∣xemik+1 ∣∣2) = [ 1 + ∣∣θ2 √ ∆t ∣∣ 1− θ1∆t ]2(k+1) e ( |x0|2 ) passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, i.e : lim ∆t→0 ( lim k→∞ e (∣∣xmk+1 ∣∣2)) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 185. numerical simulations and residual calculations in this section, we present some numerical simulations for vasicek and geometric brownianmotion models using matlab and we calculate the errors between the exact solution and thatobtained by applying the numerical schemes of euler-maruyama, milshtein and implicit euler-maruyama. 5.1. numerical simulation of vasicek and geometric brownian motion models. we present somesimulations of vasicek and brownian geometric motion models in the increasing and decreasingcases. figure 1. increasing vasicek model https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 19 figure 2. decreasing vasicek model figure 3. increasing geometric brownian motion model https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 20 figure 4. decreasing geometric brownian motion model 5.2. interpretation of results. 5.2.1. vasicek model. the figures 1 and 2 show the stability of the vasicek model in the increasingand decreasing cases, in these figures we see that the euler-maruyama scheme coincides with thatof milshtein. we have in the first two figures of figure 1 the following errors: emerr = 0.2280,milerr = 0.2280 and iemerr = 0.2258 in both figures of figure 1 emerr = 0.3007, milerr = 0.3007and iemerr = 0.2851in both figures of figure 2 emerr = 0.2268, milerr = 0.2268 and iemerr = 0.2237. 5.2.2. geometric brownian motion model. the figures figure 3 et figure 4 present the stability ofgeometric brownian motion in the increasing and decreasing cases. indeed, the first three figuresin figure 3 present the increasing stability of geometric motion and the last figure in figure 3 andthe two figures in figure 4 show the decreasing stability of the model. we have in the first twofigures and figure 3 the following errors: emerr = 0.0027, milerr = 0.0011 and iemerr = 0.0013and for the third figure in figure 3: emerr = 0.0177, milerr = 0.0111 and iemerr = 0.0128. in theboth figures of figure 4: emerr = 0.0054, milerr = 0.0022 and iemerr = 0.0026. remark 5.1. from the results bellow, in the cases of increasing and decreasing stabilities of vasicek et geometric brownian motion, we have that, the milshtein scheme is the best scheme because it’s the best approximates the exact solution. 6. conclusion we have presented in this article the analysis of the stability in mean and mean-square forvasicek and geometric brownian motion models. in these models, we established the conditionsof the numerical stabilities of euler-maruyama, implicit euler-maruyama and milshtein schemes.these conditions have been proved by using classical manner and y. saito’s approach. it should benoted that each case is different from the other depending on whether the models examined have https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 21additive (vasicek model) or multiplicative (geometric brownian motion) white noise type. finally, for these models, we found that the stability conditions of the vasicek model coincideswith the stability of the odes, on the other hand, for the stability conditions of the second modelto coincide with the stability of the odes, it is necessary that θ2 < 0. to support these results,numerical simulations were made and the calculations of the residuals (errors) comes in support ofthe results found. in the next work we will analyze the numerical stabilities of these two modelsby using non-standard euler-maruyama scheme. references [1] a.m. lyapunov, the general problem of the stability of motion, int. j. control. 55 (1992) 531?534. https://doi. org/10.1080/00207179208934253.[2] i. kats, on the stability in first approximation of systems with random lag, j. appl. math. mech. 31 (1967) 478?482. https://doi.org/10.1016/0021-8928(67)90030-5.[3] i.i. gihman, a.v. skorohod, stochastic differential equations, springer berlin heidelberg, 1972. https://doi.org/ 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e. platen, numerical solution of stochastic differential equations, springer berlin heidelberg, (2011).[12] x. mao, stochastic differential equations and application, horwood, chichester, (1997).[13] b. oksendal, stochastic differential equations, sixth edition, springer, berlin, heidelberg, (2003).[14] b. oksendal, s. agnes, applied stochastic control of jump diffusions, springer berlin heidelberg, (2007). https: //doi.org/10.1007/978-3-540-69826-5.[15] s.m. iacus, simulation and inference for stochastic differential equations, springer new york, (2008). https: //doi.org/10.1007/978-0-387-75839-8.[16] d. sondermann, introduction to stochastic calculus for finance, springer berlin heidelberg, (2006). https://doi. org/10.1007/3-540-34837-9.[17] y. komori, stahle row-type weak scheme for stochastic differential equations, monte carlo methods appl. 1(1995) 279-300. https://doi.org/10.1515/mcma.1995.1.4.279.[18] y. komori, y. saito, t. mitsui, some issues in discrete approximate solution 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numer.anal. 33 (1996) 2254-2267. https://doi.org/10.1137/s0036142992228409.[20] k. burrage, p. burrage, t. mitsui, numerical solutions of stochastic differential equations ? implementation and sta-bility issues, j. comput. appl. math. 125 (2000) 171?182. https://doi.org/10.1016/s0377-0427(00)00467-2.[21] y. saito, t. mitsui, t-stability of numerial scheme for stochastic differential equations, contribut. numer. math.(1993) 333?344. https://doi.org/10.1142/9789812798886_0026.[22] y. saito, t. mitsui, mean-square stability of numerical schemes for stochastic differential systems, vietnam j. math.30 (2002) 551-560.[23] r. sakthivel, p. revathi, n.i. mahmudov, asymptotic stability of fractional stochastic neutral differential equationswith infinite delays, abstr. appl. anal. 2013 (2013) 769257. https://doi.org/10.1155/2013/769257.[24] x. mao, exponential stability of large-scale stochastic differential equations, syst. control lett. 19 (1992) 71?81. https://doi.org/10.1016/0167-6911(92)90042-q.[25] r. khasminskii, stochastic stability of differential equations, springer berlin heidelberg, (2012). https://doi. org/10.1007/978-3-642-23280-0.[26] r.m. sheldon, variations sur le mouvement brownien:introduction aux modeles de probabilite (11e edition), elsevier,amsterdam, (2014). https://doi.org/10.28924/ada/ma.3.8 https://doi.org/10.1137/s0036142992228409 https://doi.org/10.1016/s0377-0427(00)00467-2 https://doi.org/10.1142/9789812798886_0026 https://doi.org/10.1155/2013/769257 https://doi.org/10.1016/0167-6911(92)90042-q https://doi.org/10.1007/978-3-642-23280-0 https://doi.org/10.1007/978-3-642-23280-0 1. introduction 2. preliminary notions 2.1. stochastic differential equation and stabilities 2.2. stochastic numerical schemes 3. numerical stabilities of vasicek model 3.1. explicit solution 3.2. euler-maruyama scheme stabilities 3.3. milshtein's scheme stabilities 3.4. implicit euler-maruyama scheme stabilities 4. numerical stabilities of geometric brownian motion 4.1. explicit solution of the model 4.2. euler-maruyama scheme stabilities 4.3. milshtein's scheme stabilities 4.4. implicit euler-maruyama scheme stabilities 5. numerical simulations and residual calculations 5.1. numerical simulation of vasicek and geometric brownian motion models 5.2. interpretation of results 6. conclusion references bibliographie ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 164-181doi: 10.28924/ada/ma.1.164 nonlinear differential problem with p-laplacian and via phi-hilfer approach: solvability and stability analysis hamid beddani1,∗, moustafa beddani2, zoubir dahmani3 1laboratory of complex systems of the higher school of electrical and energy engineering of oran, 31000, algeria beddanihamid@gmail.com 2department of mathematics, university of sidi bel-abbès 22000, algeria beddani2004@yahoo.fr 3laboratory of pure and applied mathematics, abdelhamid bni badis university, 27000, algeria zzdahmani@yahoo.fr ∗correspondence: beddanihamid@gmail.com abstract. this paper we consider a study of a general class of nonlinear singular fractional des withp-laplacian for the existence and uniqueness solution and the hyers-ulam (hu) stability. result via ϕ−hilfer derivative is studied. then, an existence of one solution is investigated. some illustrativeexamples are discussed at the end. 1. introduction recently, fractional differential equations with boundary conditions are being studied by manyinterested people. this is because fractional differential equations describe many more real op-erations than classical differential equations. therefore, partial differential equations appearin many engineering and technological disciplines that include several sciences; see for exam-ple [1, 3–6,8, 17,18,20,22,23,31]. currently there are several different definitions of fractional integrals and derivatives, from themost famous of which are the riemann-liouville and caputo fractional derivatives to other less wellknown definitions. a generalization of the derivatives of both riemann-liouville and caputo wasgiven by r. hilfer in [11], known as the fractional hilfer derivative of order α and type β ∈ [0, 1].some properties and applications of the helfer derivative are given in [12, 13] and the referencesmentioned therein. prime value problems involving fractional hilfer derivatives have been studiedby several authors, see [9,10,26]. however, in the literature there are few papers on the boundary received: 8 oct 2021. key words and phrases. ϕ−hilfer derivative; existence of solution; fixed point; hyers-ulam stability.164 https://adac.ee https://doi.org/10.28924/ada/ma.1.164 eur. j. math. anal. 1 (2021) 165value problems of the fractional hilfer derivatives. the authors set out in [2] non-local value prob-lems for derivatives of helfer’s fractions. for some recent work on boundary value problems withfractional hilfer derivatives, we refer to the papers in [28–30].some authors have worked on the eu of solutions for fractional des with p−laplacian operator.we cite, for example; li., wang., khan et al. [15, 19, 27] studieds a nonlinear fractional de withp-laplacian operator for the eu of solutions. h. khan, t. abdeljawad, m. aslam, r. a. khan and a. khan [16]. worked on the followingproposal for the existence of a positive solution (eps) and stability analysis:  dr1ψp [dr2 (u(t)− v1(t, u(t)))] = −a(t)v2(t, u(t − τ)), ψp [dr2 (u(t)− v1(t, u(t)))]|t=0 = ψp [ dr2 (u(t)− v1(t, u(t)))′ ]∣∣ t=0 = 0, u(0) = u(1) = 0,[ i2−r2 (u(t)− v1(t, u(t))) ]∣∣ t=0 = 0, where 0 < r1 < 1 < r2 < 2, and v1, v2 are continuous but singular at some points. thefractional derivatives dr1 and dr2 are taken in the caputo sense and in the riemann–liouvillesense, respectively, and ψp(z) = |z |p−2 z denotes the p−laplacian operator and satisfies 1 p + 1 q = 1, (ψp)−1 = ψq.a. devi, a. kumar, d. baleanu and a. khan [7]. worked on the eu and hu stability results, fornonliner fdes involving caputo fractional derivatives of distinct orders with ψp laplacian operator:  cdr1ψp [ cdr2 ( u(t)− ∑m i=1 vi(t) )] = −w(t, u(t)), t ∈ (0, 1] ψp [ cdr2 ( u(t)− ∑m i=1 vi(t) )]∣∣ t=0 = 0, u(0) = ∑m i=1 vi(0), u′(1) = ∑m i=1 v ′ i (1), uj(0) = ∑m i=1 v j i (0), for j = 2, 3, ..., n − 1, where 0 < r1 ≤ 1, n − 1 < r2 ≤ n, n ≥ 4, and vi , w are continuous functions. cdr1 and cdr2 denotes the derivative of fractional order r1 and r2 in caputo’s sense, respectively, and ψp(z) = |z |p−2 z denotes the p−laplacian operator and satisfies 1 p + 1 q = 1, (ψp)−1 = ψq. in the present research work, we study the existence and uniqueness of a solution (eps) andstability analysis which includes the ϕ−hilfer fractional-order of the form: eur. j. math. anal. 1 (2021) 166  hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) = h(t, u(t),rldµ;ϕ a+ u(t)), t ∈ j = (a, b] u(a) = 0, u(b) = n∑ i=1 λiu (ζi) , ψp ( hdα2,β2;ϕ a+ u ) (a) = 0,and ψp ( hdα2,β2;ϕ a+ u(b) ) = iρ;ϕ a+ u (ζ) , a < ζ, ζi < b, (1.1) here, we take hdα1,β;ϕ 0+ ,h dα2,β;ϕ 0+ , are the ϕ−hilfer fractional derivative of orders α1, α2, 1 < α1, α2 < 2 and β1, β2 two parameters 0 ≤ β1, β2 ≤ 1, rldκ;ϕ a+ the ϕ-riemann-liouville fractionalderivative of order µ where µ < α2, and iρ;ϕ 0+ the left-sided ϕ−riemann liouville fractional integralof order ρ, where ρ > 0, and ψp(z) = |z |p−2 z denotes the p−laplacian operator and satisfies 1 p + 1 q = 1, (ψp)−1 = ψq, and ϕ : j → r be an increasing function such that ϕ′(t) 6= 0, for all t ∈ j , and f : j × r× r→ r, is given function will be "well defined" later. 2. phi-hilfer derivatives calculus in this section, we introduce some notations and definitions of phi-hilfer derivatives calculusand present preliminary results needed in our proofs later, for details, see [17,24,25].let ϕ : [a, b] → r be an increasing function with ϕ′(t) 6= 0, for all t ∈ j , and let c([a, b] ,r) bethe banach space. for all υ > −1 and s, t ∈ [0,∞), (t ≥ s), we pose ϕυ(t, s) = (ϕ(t)− ϕ(s))υ. definition 1. let (a, b), (−∞ ≤ a < b ≤ ∞) be a finite or infinite interval of the half-axis (0,∞) and α > 0. in addition, let ϕ(t) be a positive increasing function on (a, b], which has a continuous derivative ϕ′(t) on (a, b). the ϕ−riemann–liouville fractional integral of a function u with respect to another function ϕ on [a, b] is defined by iα;ϕ a+ u(t) = 1 γ(α) t∫ a ϕ′(s)ϕα−1(t, s)u(s)ds, (2.1) where γ (.) is the gamma function. definition 2. let n ∈ n and let ϕ, u ∈ cn (j) be two functions such that ϕ is increasing and ϕ′(t) 6= 0, for all t ∈ (a, b]. the left-sided ϕ−riemann liouville fractional derivative of a function u of order α is defined by dα;ϕ a+ u(t) = ( 1 ϕ′(t) d dt )n in−α;ϕ a+ u(t) = 1 γ(n − α) ( 1 ϕ′(t) d dt )n t∫ a ϕ′(s)ϕn−α−1(t, s)u(s)ds, eur. j. math. anal. 1 (2021) 167 where n = [α] + 1, [α] represents the integer part of the real number α. definition 3. let n− 1 < α < n with n ∈ n, [a, b] is the interval such that −∞ ≤ a < b ≤ ∞ and ϕ, u ∈ cn ([a, b] ,r) two functions such that ϕ is increasing and ϕ′(t) 6= 0, for all t ∈ [a, b]. the ϕ-hilfer fractional derivative of a function u of order a and type 0 ≤ β ≤ 1 is defined by hdα,β;ϕ a+ u(t) = iβ(n−α);ϕ a+ ( 1 ϕ′(t) d dt )n i(1−β)(n−α);ϕ a+ u(t) = iγ−α;ϕ a+ dγ;ϕ a+ u(t), where n = [α] + 1, γ − α = β (n − α) . 2.1. auxiliary lemma. lemma 1. let α, ρ > 0. then, we have the following semigroup property given by iα;ϕ a+ iρ;ϕ a+ u(t) = iα+ρ;ϕ a+ u(t), t > a. next, we present the ϕ-fractional integral and derivatives of a power function. proposition 1. let α ≥ 0, σ > 0 and t > a. then, ϕ-fractional integral and derivative of a power function are given by(1) iα,ϕ a+ ϕσ−1(t, a)(t) = γ(σ) γ(α+σ)ϕσ+α−1(t, a).(2) hdα,β;ϕ a+ ϕσ−1(t, a)(t) = γ(σ) γ(σ−α)ϕσ−α−1(t, a), n − 1 < α < n, σ > n. lemma 2. if u ∈ cn([a, b],r), n − 1 < α < n, 0 ≤ β ≤ 1 and γ = α+ β(n − α). then iα,ϕ a+ (hdα,β;ϕ a+ u)(t) = u(t)− k=n∑ k=1 ϕγ−k(t, s) γ(γ − k + 1) ∇[n−k] ϕ i(1−β)(n−α);ϕ a+ u(a), t ∈ [a, b], where ∇[n] ϕ u(t) := ( 1 ψ′(t) d dt )n u(t). lemma 3. let u ∈ cn [a, b] and 0 < q < 1, we have∣∣iq;ϕ a+ u(t2)− iq;ϕ a+ u(t1) ∣∣ ≤ 2 ‖u‖ γ (q + 1) ϕq(t2, t1). lemma 4. ( [14]) for the p−laplacian operator ψp , the following conditions hold true: (1) if |δ1| , |δ2| ≥ ρ > 0, 1 < p ≤ 2, δ1δ2 > 0, then |ψp(δ1)− ψp(δ2)| ≤ (p − 1) ρp−2 |δ1 − δ2| . (2) if p > 2, |δ1| , |δ2| ≤ ρ∗ > 0, then |ψp(δ1)− ψp(δ2)| ≤ (p − 1) ρp−2 ∗ |δ1 − δ2| . lemma 5. [9] for nonnegative ai , i = 1, ..., k ,( k∑ i=1 ai )q ≤ kq−1 ( k∑ i=1 aqi ) , q ≥ 1. eur. j. math. anal. 1 (2021) 168 lemma 6. let a ≥ 0, 1 < α1, α2 < 2, 0 ≤ β1, β2 ≤ 1, and 2− γ1 = (1− β1) (2− α1) , 2− γ2 = (1− β2) (2− α2) . for f ∈ c(j, ,r,r), the unique solution of the sequential hilfer fractional boundary value problem hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) = f (t), t ∈ j = [a, b] , (2.2)  u(a) = 0, u(b) = n∑ i=1 λiu (ζi) , ψp ( hdα2,β2;ϕ a+ u ) (a) = 0, and ψp ( hdα2,β2;ϕ a+ u(b) ) = iρ;ϕ a+ u (ζ) , a < ζ, ζi < b, (2.3) is given by u(t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s)x(s, a)ds − ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)x(t, a)dt + ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) . where x(s, a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)f (z)dz + ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a)  iρ;ϕ 0+ u (ζ) = 1 γ(ρ) ζ∫ a ϕ′(s)ϕρ(ζ, s)u (s) ds, iα1;ϕ 0+ f (b) = 1 γ(α1) b∫ a ϕ′(s)ϕα1−1(b, s)f (s)ds. proof. assume that u is a solution of the sequential nonlocal boundary value problems (3.6) and(2.3). applying the two operators iα1;ϕ a+ , iα2;ϕ a+ to both sides of equation (3.6) and using lemma 2and proposition 1, we obtain ψp ( hdα2,β2;ϕ a+ u ) (t) = iα1;ϕ a+ f (t) + m0 γ (γ1 − 1) ϕγ1−2 (t, a) + m1 γ (γ1) ϕγ1−1 (t, a) , (2.4) where m0, m1 ∈ r, and 2− γ1 = (1− β1) (2− α1) . from the boundary condition ψp ( hdα2,β2;ϕ a+ u ) (a) = 0, and if t → a then ϕγ1−2 (t, a)→∞, we get m0 = 0. eur. j. math. anal. 1 (2021) 169 and by ψp (hdα2,β2;ϕ a+ u ) (b) = iρ;ϕ a+ u (ζ) , we obtain m1 = γ (γ1) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ) . so hdα2,β2;ϕ a+ u(t) = ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) )) , by (2.4)we have u(t) = iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ))] + m2 γ (γ2 − 1) ϕγ2−2 (t, a) + m3 γ (γ2) ϕγ2−1 (t, a) , where m2, m3 ∈ r, and 2− (1− β2) (2− α2) = γ2.and if t → a then ϕγ2−2 (t, a)→∞, we getby conditions u(a) = 0, and lim t→0 tγ2−2 =∞, we get m2 = 0. so u(t) = iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ))] + m3 γ (γ2) ϕγ2−1 (t, a) . by conditions u(b) = n∑ i=1 λiu (ζi) , we get m3 = γ (γ2) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) − γ (γ2) ϕγ2−1 (b, a) iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ))] t=b . then u(t) = iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ))] + ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) − ϕγ2−1 (t, a) ϕγ2−1 (b, a) iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ f (b) ))] t=b . this finishes the proof. � eur. j. math. anal. 1 (2021) 170 conjecture 1. rldµ;ϕ a+ u(t) = 1 γ(α2 − µ) t∫ a ϕ′(s)ϕα2−µ−1(t, s)x(s, a)ds + γ (γ2) γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) − γ (γ2) γ(α2)γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)x(t, a)dt. 3. main results in this section, we present to the reader our main results on the existence and stability for theabove problem. we begin by considering the space cµϕ = { u : u,rldµ;ϕ a+ u ∈ c ([a, b] ,r) } , with the norm ‖u‖cµϕ = ‖u‖c + ∥∥rldµ;ϕ a+ u ∥∥ c , such that ‖u‖c = sup t∈[a,b] |u(t)| , and ∥∥rldµ;ϕ a+ u ∥∥ c = sup t∈[a,b] ∣∣rldµ;ϕ a+ u(t) ∣∣ . 3.1. criteria for uniqueness solution. now, wee need to consider the following assumptions: h1) h is continuous function. h2) there exists a constant υ > 0, such that |h(t, u, v)− h(t, x, y)| ≤ υ (|u − x |+ |v − y |) , with t ∈ [a, b] , (u, v , x, y) ∈ r4. h3) there exists two continuous functions π1, π2 : [a, b]→ r+, such that |h(t, u, v)| ≤ π1(t) |u(t)|+ π2(t) |v(t)| , where π∗1 = sup t∈[a,b] |π1(t)| , and π∗2 = sup t∈[a,b] |π2(t)| . now, we define the following quantities: eur. j. math. anal. 1 (2021) 171 ϕq(b, a) = mq, ω = 3q−2 [( 2mα1 γ(α1 + 1) )q−1 ( (π∗1)q−1 + (π∗2)q−1 ) + ( mρ γ (ρ+ 1) )q−1 ] λ1 = 2.ω.mα2 γ(α2 + 1) , λ2 = ( n∑ i=1 |λi | ) , λ3 = ω.mα2−µ γ(α2 − µ+ 1) + ω.γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) , λ4 = γ (γ2)m−µ γ (γ2 − µ) λ2. based on the above hypotheses, we present to the reader the following result. theorem 1. under h2 and h3 the equation (1.1) has a solution. proof. firstly: we begin this proof by defining the operator g : cµϕ → cµϕ by: (gu) (t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s)xu(s, a)ds − ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt + ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) . where xu(s, a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz + ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ hu(b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a)  , where hu(t) = h(t, u(t),rldµ;ϕ a+ u(t)). we consider the set ur = { u ∈ cµϕ : ‖u‖cµϕ ≤ r } , so that max { (2 (λ1 + λ3)) 1 2−q , 2 (λ2 + λ4) } ≤ r. eur. j. math. anal. 1 (2021) 172we show that gur ⊂ ur . for any u ∈ ur , and by lemma 5 we have |xu(s, a)| = ∣∣∣∣∣∣ ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz + ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ hu(b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a) ∣∣∣∣∣∣ ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz + iρ;ϕ 0+ u (ζ) + iα1;ϕ 0+ hu(b) ∣∣∣∣∣∣ q−1 ≤ 3q−2 sup t∈[a,b]  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz q−1 + ( iρ;ϕ 0+ u (ζ) )q−1 + ( iα1;ϕ 0+ hu(b) )q−1  ≤ 3q−2 [ 2 ( π∗1m α1 γ(α1 + 1) ‖u‖c + π∗2m α1 γ(α1 + 1) ∥∥rldµ;ϕ a+ u ∥∥ c )q−1 + ( mρ γ (ρ+ 1) )q−1 (‖u‖c)q−1 ] ≤ 3q−2 [( 2.π∗1m α1 γ(α1 + 1) )q−1 (‖u‖c)q−1 + ( 2.π∗2m α1 γ(α1 + 1) )q−1 (∥∥rldµ;ϕ a+ u ∥∥ c )q−1 + ( mρ γ (ρ+ 1) )q−1 (‖u‖c)q−1 ] ≤ 3q−2 [( 2mα1 γ(α1 + 1) )q−1 ( (π∗1)q−1 + (π∗2)q−1 ) + ( mρ γ (ρ+ 1) )q−1 ] rq−1 ≤ ω.rq−1. then sup t∈[a,b] |(gu) (t)| (3.1) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s)xu(s, a)ds + ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt + ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) ∣∣∣∣∣ ≤ 2mα2 γ(α2 + 1) |xu|+ ( n∑ i=1 |λi | ) sup t∈[a,b] |u (t)| ≤ 2.ω.mα2 γ(α2 + 1) rq + ( n∑ i=1 |λi | ) r. ≤ λ1r q + λ2r. also, we have sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t) ∣∣ (3.2) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2 − µ) t∫ a ϕ′(s)ϕα2−µ−1(t, s)xu(s, a)ds + γ (γ2) γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) eur. j. math. anal. 1 (2021) 173 − γ (γ2) γ(α2)γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt ∣∣∣∣∣∣ ≤ [ mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ] |xu|+ γ (γ2)m−µ γ (γ2 − µ) ( n∑ i=1 |λi | ) sup t∈[a,b] |u (t)| ≤ [ ω.mα2−µ γ(α2 − µ+ 1) + ω.γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ] rq−1 + γ (γ2)m−µ γ (γ2 − µ) ( n∑ i=1 |λi | ) r ≤ λ3r q−1 + λ4r. by (3.1) and (3.2), we find ‖u‖cµϕ = sup t∈[a,b] |(gu) (t)|c + sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t) ∣∣ c (3.3) ≤ (λ1 + λ3) rq−1 + (λ2 + λ4) r ≤ r. that is gur belongs to ur on [a, b]. next, we prove that g is completely continuous. for any u ∈ ur and t1, t2 ∈ [a; b] such that t1 < t2, by lemma 3, we have sup t∈[a,b] |(gu) (t2)− (gu) (t1)| ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2) t2∫ a ϕ′(s)ϕα2−1(t2, s)xu(s, a)ds − 1 γ(α2) t1∫ a ϕ′(s)ϕα2−1(t1, s)xu(s, a)ds + ϕγ2−1 (t2, a)− ϕγ2−1 (t1, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt + ϕγ2−1 (t2, a)− ϕγ2−1 (t1, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) ∣∣∣∣∣ ≤ ω.rq−1 γ(α2 + 1) ϕα2 (t2, t1) + ω.mα2 .rq−1 + γ(α2 + 1)λ2r γ(α2 + 1)ϕγ2−1 (b, a) ϕγ2−1 (t2, t1) . hence, sup t∈[a,b] |(gu) (t2)− (gu) (t1)| → 0, as t2 → t1. eur. j. math. anal. 1 (2021) 174also, we can say that sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t2)− ( rldµ;ϕ a+ gu ) (t1) ∣∣ ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2 − µ) t2∫ a ϕ′(s)ϕα2−µ−1(t2, s)xu(s, a)ds − 1 γ(α2 − µ) t1∫ a ϕ′(s)ϕα2−µ−1(t1, s)xu(s, a)ds + γ (γ2) γ (γ2 − µ) ϕγ2−µ−1 (t2, a)− ϕγ2−µ−1 (t1, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) + γ (γ2) γ(α2)γ (γ2 − µ) ϕγ2−µ−1 (t2, a)− ϕγ2−µ−1 (t1, a) ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt ∣∣∣∣∣∣ ≤ ω.rq−1 γ(α2 − µ+ 1) ϕα2−µ(t2, t1) + ( γ (γ2) λ2r γ (γ2 − µ)ϕγ2−1 (b, a) + γ (γ2) .ω.rq−1.mα2 γ(α2 + 1)γ (γ2 − µ)ϕγ2−1 (b, a) ) ϕγ2−µ−1 (t2, t1) . hence, sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t2)− ( rldµ;ϕ a+ gu ) (t1) ∣∣→ 0, as t2 → t1. as a consequence of the above three steps and thanks to arzela–ascoli theorem, we conclude that g is completely continuous.the proof of theorem 1 is thus completely achieved. � 3.2. criteria for existence of a solution. theorem 2. assume that h2 and h3 are satisfied. suppose that υ1 + υ2 < 1, where υ1 = 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) + λ2, and υ2 = (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) )( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) + γ (γ2) λ2m −µ γ (γ2 − µ) . then, (1.1) has a uniqueness solution. eur. j. math. anal. 1 (2021) 175 proof. we pass to prove that g is a contraction. for any u, v ∈ ur , we have the following estimate |xu(s, a)−xv (s, a)| = ∣∣∣∣∣∣ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz + ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ hu(b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a)  − ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hv (z)dz + ( iρ;ϕ 0+ v (ζ)− iα1;ϕ 0+ hv (b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a) ∣∣∣∣∣∣ ≤ (q − 1) yq−2 ∣∣∣∣∣∣ 1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hu(z)dz − 1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hv (z)dz + ϕγ1−1 (s, a) ϕγ1−1 (b, a) ( iρ;ϕ 0+ u (ζ)− iρ;ϕ 0+ v (ζ) ) + ϕγ1−1 (s, a) ϕγ1−1 (b, a) ( iα1;ϕ 0+ hu(b)− iα1;ϕ 0+ hv (b) )∣∣∣∣ ≤ (q − 1) yq−2 ( 2mα1 γ(α1 + 1) sup t∈[a,b] |hu(t)− hv (t)|+ mρ γ (ρ+ 1) sup t∈[a,b] |u(t)− v(t)| ) ≤ (q − 1) yq−2 (( 2υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) sup t∈[a,b] |u(t)− v(t)| + 2υmα1 γ(α1 + 1) sup t∈[a,b] ∣∣rldµ;ϕ a+ u(t)−rl dµ;ϕ a+ v(t) ∣∣) ≤ (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) ‖u − v‖cµϕ . where  ∆ > 2υmα1 γ(α1+1) + mρ γ(ρ+1) , if q > 2,ou 0 < ∆ ≤ 2υmα1 γ(α1+1) + mρ γ(ρ+1) , if 1 < q ≤ 2. then sup t∈[a,b] |(gu) (t)− (gv) (t)| (3.4) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s) (xu −xv ) (s, a)ds ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣∣ ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t) (xu −xv ) (t, a)dt ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λi (u (ζi)− v(ζi)) ∣∣∣∣∣ eur. j. math. anal. 1 (2021) 176 ≤ 2mα2 γ(α2 + 1) sup t∈[a,b] |xu(t, a)−xv (t, a)|+ λ2 sup t∈[a,b] (|u (t)− v(t)|) ≤ ( 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) + λ2 ) ‖u − v‖cµϕ ≤ υ1 ‖u − v‖cµϕ .also sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t)− ( rldµ;ϕ a+ gv ) (t) ∣∣ (3.5) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2 − µ) t∫ a ϕ′(s)ϕα2−µ−1(t, s) (xu −xv ) (s, a)ds + γ (γ2) γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λi (u (ζi)− v(ζi)) + γ (γ2) γ(α2)γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t) (xu −xv ) (t, a)dt ∣∣∣∣∣∣ ≤ ( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) sup t∈[a,b] |xu −xv | + γ (γ2) λ2m −µ γ (γ2 − µ) sup t∈[a,b] (|u (t)− v(t)|) ≤ { (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) )( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) + γ (γ2) λ2m −µ γ (γ2 − µ) } ‖u − v‖cµϕ ≤ υ2 ‖u − v‖cµϕ .by (3.4) and (3.5), yields the following inequality ‖gu −gv‖cµϕ ≤ (υ1 + υ2) ‖u − v‖cµϕ .where υ1 + υ2 < 1. hence g is a contraction operator and the contraction mapping principleimplies that (1.1) has a unique solution. � 3.3. ulam type stability. we introduce the following two definitions definition 4. the problem (1.1) is ulam–hyers stable if ∃ λ ∈ r∗+, such that for each ε > 0, t ∈ j , and for each u ∈ cµϕ solution of the following inequality∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t)− h(t, u(t),rldµ;ϕ a+ u(t)) ∥∥∥ cµϕ < ε, (3.6) ∃v ∈ cµϕ solution of (1.1), i.e. hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ v ) (t) = h(t, v(t),rldµ;ϕ a+ v(t)), (3.7) eur. j. math. anal. 1 (2021) 177 such that, the inequality ‖u − v‖cµϕ ≤ λε, holds. definition 5. the equation (1.1) has the ulam–hyers stability in the generalized sense if ∃ ϕ ∈ c (j,r+), such that for each ε > 0, t ∈ j , and for each u ∈ cµϕ solution of:∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t)− h(t, u(t),rldµ;ϕ a+ u(t)) ∥∥∥ cµϕ < ε, (3.8) ∃v ∈ cµϕ solution of (1.1) that satisfies ‖u(t)− v(t)‖cµϕ ≤ εϕ(t). in the light of the first definition and using the above existence and uniqueness theorem, wepresent to the reader the following result. theorem 3. if the assumptions (h2) are satisfied, then eq (1.1) is ulam–hyers stable under the condition that n1 + n2 < 1, where n1 = 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) , and n2 = ( 4υ (q − 1) ∆q−2mα1 γ(α1 + 1) + (q − 1) ∆q−2mρ γ (ρ+ 1) )( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) . proof. let u ∈ cµϕ be a solution of the inequality (3.6), i.e.∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t)− h(t, u(t),rldµ;ϕ a+ u(t)) ∥∥∥ cµϕ < ε, ∀t ∈ j. (3.9) let v ∈ cµϕ be a unique solution of: hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ v ) (t) = h(t, v(t),rldµ;ϕ a+ v(t)), ∀t ∈ j, and  u(a) = v(a), u(b) = v(b)and ψp ( hdα2,β2;ϕ a+ u ) (a) = ψp ( hdα2,β2;ϕ a+ v ) (a), ψp ( hdα2,β2;ϕ a+ u ) (b) = ψp ( hdα2,β2;ϕ a+ v ) (b), eur. j. math. anal. 1 (2021) 178by using proof of lemma 6 v(t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s)xv (s, a)ds − ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xv (t, a)dt + ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) , where xv (s, a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s, z)hv (z)dz + ( iρ;ϕ 0+ u (ζ)− iα1;ϕ 0+ hv (b) ) ϕγ1−1 (b, a) ϕγ1−1 (s, a)  . by integration of inequality (3.9), for any t ∈ j, we have∥∥∥∥∥∥u(t)− 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s)xu(s, a)ds (3.10) + ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t)xu(t, a)dt − ϕγ2−1 (t, a) ϕγ2−1 (b, a) n∑ i=1 λiu (ζi) ∥∥∥∥∥ c ≤ iα2;ϕ a+ ψq ( iα1;ϕ a+ ε ) = mq−1ϕα1+α2 (t, a) γ (α1 + α2 + 1) ε. on the other hand, for any u, v ∈ cµϕ, we have the following estimate ‖u(t)− v(t)‖c (3.11) < mq−1ϕα1+α2 (t, a) γ (α1 + α2 + 1) ε + sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t, s) (xu −xv ) (s, a)ds ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣∣ ϕγ2−1 (t, a) γ(α2)ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t) (xu −xv ) (t, a)dt ∣∣∣∣∣∣ < mα1+α2+q1 γ (α1 + α2 + 1) ε+ 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ+ 1) ) ‖u − v‖cµϕ < mα1+α2+q−1 γ (α1 + α2 + 1) ε+ n1 ‖u − v‖cµϕ . eur. j. math. anal. 1 (2021) 179also, for any t ∈ j, we have∥∥rldµ;ϕ a+ (u(t)− v(t)) ∥∥ c (3.12) ≤ mq−1ϕα1+α2−µ (t, a) γ (α1 + α2 − µ+ 1) ε + sup t∈[a,b] ∣∣∣∣∣∣ 1 γ(α2 − µ) t∫ a ϕ′(s)ϕα2−µ−1(t, s) (xu −xv ) (s, a)ds + γ (γ2) γ(α2)γ (γ2 − µ) ϕγ2−µ−1 (t, a) ϕγ2−1 (b, a) b∫ a ϕ′(t)ϕα2−1(b, t) (xu −xv ) (t, a)dt ∣∣∣∣∣∣ ≤ mα1+α2+q−µ−1 γ (α1 + α2 − µ+ 1) ε+ ( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) sup t∈[a,b] |xu −xv | ≤ mα1+α2+q−µ−1 γ (α1 + α2 − µ+ 1) ε + ( 4υ (q − 1) ∆q−2mα1 γ(α1 + 1) + (q − 1) ∆q−2mρ γ (ρ+ 1) )( mα2−µ γ(α2 − µ+ 1) + γ (γ2)mα2−µ γ(α2 + 1)γ (γ2 − µ) ) ‖u − v‖cµϕ ≤ mα1+α2+q−µ−1 γ (α1 + α2 − µ+ 1) ε+ n2 ‖u − v‖cµϕ . so, by (3.11) and (3.12) we have ‖u − v‖cµϕ ≤ ε ( mα1+α2+q−1 γ (α1 + α2 + 1) + mα1+α2+q−µ−1 γ (α1 + α2 − µ+ 1) ) + (n1 + n2) ‖u − v‖cµϕ . therefore, we get ‖u − v‖cµϕ ≤ λε,such that λ = 1 1− (n1 + n2) ( mα1+α2+q−1 γ (α1 + α2 + 1) + mα1+α2+q−µ−1 γ (α1 + α2 − µ+ 1) ) , for any t ∈ j . this implies that the ulam-hyers stability condition is satisfied. � 3.4. illustrative exemple. consider the following problem hd 13 10 , 6 7 ;t2 0+ ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (t) = h(t, u(t),rld 1 2 ;t2 a+ u(t)), t ∈ j = [0, 2] , (3.13) u(a) = 0, u(2) = n∑ i=1 ( 3i 11 ) u ( i 2 + i ) , ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (0) = 0, ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (2) = i 3 2 ;t2 a+ u ( 4 3 ) f (t, u(t), v(t)) = exp ( 1 7 (1 + t2) ) u(t) + v(t) (1 + et) , eur. j. math. anal. 1 (2021) 180then assumptions (h1), (h2) and (h3) are satisfied with υ = π∗2 = 1 2 , π∗1 = e 1 7 , and m = 4. we conclude that (3.13) has an unique solution. references [1] b. ahmad, a. alsaedi, s. 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pantographfractional differential equations and inclusions, adv. differ. equ. 2020 (2020), 279. https://doi.org/10.1186/ s13662-020-02747-1.[30] a. wongcharoen, s. k. ntouyas, and j. tariboon, boundary value problems for hilfer fractional differential in-clusions with nonlocal integral boundary conditions, mathematics 8 (2020), 1905. https://doi.org/10.3390/ math8111905.[31] y. zhou, basic theory of fractional differential equations, world scientific, singapore, 2014. https://doi.org/10.1186/s13662-017-1172-8 https://doi.org/10.1515/math-2020-0122 https://doi.org/10.1016/j.amc.2015.05.144 https://doi.org/10.1155/2018/1462825 https://doi.org/10.1155/2018/1462825 https://doi.org/10.1155/2020/9606428 https://doi.org/10.1155/2020/9606428 https://doi.org/10.1186/s13662-020-02747-1 https://doi.org/10.1186/s13662-020-02747-1 https://doi.org/10.3390/math8111905 https://doi.org/10.3390/math8111905 1. introduction 2. phi-hilfer derivatives calculus 2.1. auxiliary lemma. 3. main results 3.1. criteria for uniqueness solution. 3.2. criteria for existence of a solution. 3.3. ulam type stability. 3.4. illustrative exemple. references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 11doi: 10.28924/ada/ma.2.11 on the α−ψ− contractive mappings in c∗-algebra valued b-rectangular metric spaces and fixed point theorems mohamed rossafi1,∗, abdelkarim kari2, hafida massit3 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco mohamed.rossafi@usmba.ac.ma 2ams laboratory faculty of sciences ben m’sik, hassan ii university, casablanca, morocco abdkrimkariprofes@gmail.com 3laboratory of partial differential equations, spectral algebra and geometry department of mathematics, faculty of sciences, university ibn tofail, kenitra, morocco massithafida@yahoo.fr ∗correspondence: mohamed.rossafi@usmba.ac.ma abstract. this present paper extends a version of α−ψ−contraction in c∗-algebra valued rectangu-lar b-metric spaces and establishing the existence and uniqueness of fixed point for them. non-trivialexamples are further provided to support the hypotheses of our results. 1. introduction a c∗-algebra valued metric spaces were introduced by ma et al. [6] as a generalization of metricspaces they proved certain fixed point theorems, by giving the definition of c∗-algebra valuedcontractive mapping analogous to banach contraction principle. many mathematicians worked onthis interesting space.various fixed point results were established on such spaces, see [1–3] and references therein.combining conditions used for definitions of c∗-algebra valued metric and generalized metricspaces, g kalapana and tasneem [4] announced the notions of c∗-algebra valued metric space andestablish nice results of fixed point on such space.in this paper, inspired by the work done in [9], we introduce the notion of α − ψ−contractionand establish some new fixed point theorems for mappings in the setting of complete c∗-algebravalued rectangular bmetric spaces.moreover, an illustrative examples is presented to support the obtained results. received: 18 dec 2021. key words and phrases. fixed point; c∗-algebra valued metric spaces; α − ψ− contraction; α − ψ − c∗ valuedcontraction. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 22. preliminaries throughout this paper, we denote a by an unital (i.e ,unity element i) c∗-algebra with linearinvolution ∗, such that for all x, y ∈ a, (xy)∗ = y∗x∗,and x∗∗ = x .we call an element x ∈ a a positive element, denote it by x � θif x ∈ ah = {x ∈ a : x = x∗} and σ(x) ⊂ r+,where σ(x) is the spectrum of x .using positiveelement ,we can define a partial ordering � on ah as follows : x � y if and only if y − x � θwhere θ means the zero element in a. we denote the set x ∈ a : x � θ by a+ and |x | = (x∗x) 1 2 .and a′ will denote the set {a ∈ a+; ab = ba, ∀b ∈ a} lemma 2.1. [8] suppose that a is a unital c∗-algebra with a unit i,(1) for any x ∈ a+ we have x � i ⇐⇒ ‖x‖ ≤ 1(2) if a ∈ a+ with ‖a‖ < 1 2 then i − a is unvertible and ‖a(1− a)−1‖ < 1(3) suppose that a, b ∈ a+ and ab = ba, then ab � θ(4) let a ∈ a′, if b, c ∈ a, with b � c � θ, and i − a ∈ a′+ is invertible operator, then (i − a)−1b � (i − a)−1c definition 2.2. [4] let x be a non-empty set and b ∈ a such that b � i . supposa the mapping d : x ×x → a+ satisfies:(i) d(x, y) = θ if and only if x = y ;(ii) d(x, y) = d(y , x) for all distinct points x, y ∈ x;(iii) d(x, y) � b[d(x, u) + d(u, v) + d(v , y)] for all x, y ∈ x and for all distinct points u, v ∈ x − {x, y}.then (x,a+, d) is called a c∗-algebra valued rectangular b−metric space. example 2.3. let x = r and a = m2(r). define d(x, y) = diag(|x − y |, 2|x − y |) where x, y ∈ r. it is easy to verify d is a c∗− algebra-valued rectangular b− metric and (x,m2(r), d)is a copmlete c∗-algebra valued rectangular b−metric space. definition 2.4. [9] if ψ : a → b is a linear mapping in c∗-algebra, it is said to be positive if ψ(a+) ⊆ b+. in this case ψ(ah) ⊆ bh, and the restriction map ψ : ah → bh is increasing. definition 2.5. [9] suppose that a and b are c∗-algebra .a mapping ψ : a→ b is said to be c∗homomorphism if :(i) ψ(ax + by) = aψ(x) + bψ(y) for all a, b ∈ c and x, y ∈ a(ii) ψ(xy) = ψ(x)ψ(y) for all x, y ∈ a https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 3(iii) ψ(x∗) = ψ(x)∗ for all x ∈ a(iv) ψ maps the unit in a to the unit in b. definition 2.6. [9] let a and b be c∗-algebra spaces and let ψ : a→ b be a homomorphismthen ψ is called an ∗− homomorphism if it is one to one ∗− homomorphism.a c∗-algebra a is ∗−isomorphic to a c∗-algebra b if there exists ∗− isomorphism of a onto b. definition 2.7. [9] let ψ be the set of positive functions ψ : a+ → a+ satisfying the followingconditions :(i) ψ is continous and nondecrasing(ii) ψ(a) = θ if and only if a = θ(iii) l imn−→∞ψn(a) = θ, (a � θ) ,∑∞n=1 ψn(a) <∞(iv) the series ∑∞k=1 bkψk(a) <∞ for a � θ is increasing and continuous at θ. corollary 2.8. [9] every c∗− homomorphism is contractive and hence bounded. lemma 2.9. every ∗− homomorphism is positive. definition 2.10. [9] let x be a nonempty set and α : x ×x → a′+ be a function, wesay that theself map t is α− admissible if (x, y) ∈ x ×x,α(x, y) � i ⇒ α(tx, t y) � i ,where i the unit of a. definition 2.11. [9] let (x,a, d) be a c∗-algebra valued b− metric space and t : x → x ismapping, we say that t is an α−ψ− contractive mapping if there exist two functions α : x×x → a+ and ψ ∈ ψ such that α(x, y)d(tx, t y) � ψ(d(x, y)), for all x, y ∈ x 3. main result in [9] introduced the concept of α−ψ− contractive mappings in a unital c∗-algebra valued b−metric space. in this paper we will develop the definitions in case of unital c∗-algebra valuedrectangular b− metric space and give some banach fixed point theorems. definition 3.1. let (x,a, d) be a c∗-algebra valued b− rectangular metric space and t : x → xis mapping, we say that t is an α − ψ− contractive mapping if there exist two functions α : x ×x → a+ and ψ ∈ ψ such that α(x, y)d(tx, t y) � ψ(d(x, y)), f oral lx, y ∈ x (3.1) theorem 3.2. let (x,a, d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping satisfying the following conditions:(i) t is α− admissible https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 4(ii) there exists x0 ∈ x such that α(x0, t x0) � i(iii) for all x, y ∈ x ,there exists z ∈ x such that α(x, z) � i and α(y , z) � i(iv) t is continuous then, t has a unique fixed point in x . proof. let x0 ∈ x such that α(x0, t x0) � i and define a sequence {xn} ∈ x such that xn+1 = txn, ∀n ∈ n. suppose that there exists n ∈ n such that xn = txn. then xn is a fixed point of t andthe proof is finished.hence, we assume that xn 6= txn+1, ∀n ∈ n, since t is α−admissible, we get α(x0, x1) = α(x0, t x0) � i ⇒ α(tx0, t 2x0) = α(x1, x2) � i . continuing this process, we have α(xn, xn+1) � i ∀n ∈ n. (3.2) by 3.1 and 3.2, we get d(xn, xn+1) = d(txn−1, t xn) � α(xn−1, xn)d(txn−1, t xn) � ψ(d(xn−1, xn)) � � ψn(d(x0, x1)). for m ≥ 1 and p ≥ 1, it follows that d(xm+p, xm) � b[d(xm+p, xm+p−1) + d(xm+p−1, xm+p−2) + d(xm+p−2, xm)] � bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b[b[d(xm+p−2, xm+p−3) + d(xm+p−3, xm+p−4) + d(xm+p−4, xm)]] = bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b2d(xm+p−2, xm+p−3) + b2d(xm+p−3, xm+p−4) + b2d(xm+p−4, xm) � bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b2d(xm+p−2, xm+p−3) + b2d(xm+p−3, xm+p−4) + ....+ b p−1 2 d(xm+3, xm+2) + b p−1 2 d(xm+2, xm+1) + b p−1 2 d(xm+1, xm) � bψm+p−1(d(x0, x1)) + bψm+p−2(d(x0, x1)) + ...+ b p−1 2 d(x0, x1)since b � i , using definition 2.6 we have d(xm, xm+p) � bψm+p−1(d(x0, x1))+bψm+p−2(d(x0, x1))+ ...+b p−1 2 d(x0, x1)→ θ as n → +∞therefore {xn} is a cauchy sequence in x . by the completeness of (x,a, d) there exists an x ∈ x such that l imn→∞xn = l imn→∞txn−1 = x . from continuity of t and by uniqueness of the limit, we get tx = x , ie. x is a fixed point of t .now suppose that y 6= x is another fixed point of t . https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 5from (i i i), there exists z ∈ x such that α(x, z) � i and α(y , z) � i .since t is α− admissible, we have α(x, t nz) � i and α(y , t nz) � i for all n ∈ n using (1), we obtain d(x, t nz) = d(tx, t (t n−1z)) � α(x, t n−1z)d(tx, t (t n−1z)) � ψn(d(x, z))→ θ as n →∞. thus, t nz = x . similary t nz = y as n →∞ so, the uniqueness of the limit we obtain x = y . � example 3.3. let x = r and a = m2(r) as given in example 2.3, define t : x → x , by tx = x 3and α : x ×x → m2(r) such that α(x, y) = ( |x − y | 0 0 0 ) thus, t is α− admissible, and ψ : m2(r)+ → m2(r)+ , ψ(a) = ( a2 0 0 a2 ) ∀a ∈ (r)+. this is clear that t is α− ψ− contractive mapping and satisfies α(x, y)d(tx, t y) � ψ(d(x, y)), for all x, y ∈ x theorem 3.4. let (x,a, d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping of kannan type ie, α(x, y)d(tx, t y) � ψ(d(tx, x) + d(ty, y)) (3.3) for all x, y ∈ x where ψ ∈ ψ and α : x ×x → a+ and the following conditions holds:(i) t is α− admissible(ii) there exists x0 ∈ x such that α(x0, t x0) � i(iii) t is continuous then, t has a fixed point in x . proof. by (3.3), we obtain d(xn, xn+1) = d(txn−1, t xn) � α(xn−1, xn)d(txn−1, t xn) � ψ(d(txn−1, xn−1) + d(txn, xn)) = ψ(d(xn, xn−1) + d(xn+1, xn)) = ψ(d(xn, xn−1)) + ψ(d(xn+1, xn)) (i − ψ)(d(xn, xn−1)) � ψ(d(xn, xn−1)) from lemma 2.1 and definition 2.6, we obtain https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 6 d(xn, xn+1) � (i − ψ)−1ψ(d(xn, xn−1)) = φ(d(xn, xn−1)) where φ = (i − ψ)−1ψ therefore d(xn, xn+1) � φn(d(x0, x1))∀n ∈ n for any m ≥ 1 and p ≥ 1 similary in theorem 3.1 we have d(xm, xm+p) � bψm+p−1(d(x0, x1))+bψm+p−2(d(x0, x1))+...+b p−1 2 d(x0, x1)→ θ as n → +∞.thus {xn} is a cauchy sequence in x . by the completeness of (x,a, d), there exists x ∈ xsuch that l imn→∞xn = l imn→∞txn−1 = x . the continuity of t gives that x is a fixed point of t .to prove that x is the unique fixed point, we suppose that y ∈ x is another fixed point of t .then θ � d(x, y) = d(tx, t y) � α(x, y)d(tx, t y) � ψ(d(tx, x) + d(ty, y)) = ψ(d(x, x) + d(y , y)) = θ hence x = y .therefore the fixed point is unique. � theorem 3.5. let (x,a, d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping of banach-kannan type ie, α(x, y)d(tx, t y) � ψ(d(x, y) + d(tx, x) + d(ty, y)) (3.4) for all x, y ∈ x where ψ ∈ ψ and α : x ×x → a+ such that ψ(1− ψ)−1 � 1 2i , and the following conditions holds:(i) t is α− admissible(ii) there exists x0 ∈ x such that α(x0, t x0) � i(iii) t is continuous then, t has a fixed point in x proof. using (3.4), we get d(xn, xn+1) = d(txn−1, t xn) � α(xn−1, xn)d(txn−1, t xn) � ψ(d(xn−1, xn) + d(txn−1, xn−1) + d(txn, xn)) = ψ(d(xn−1, xn)2i + d(xn, xn+1)) ⇒ (i − ψ)(d(xn, xn+1)) � 2iψ(d(xn, xn−1)) ⇒ d(xn, xn+1) � 2i(i − ψ)−1ψ(d(xn, xn−1)) � φ(d(xn, xn−1)).where https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 7 ϕ = 2i(i − ψ)−1ψ. then d(xn, xn+1) � φn(d(x0, x1). we refer to the proof of the theorem 3.1 we get that x is a fixed point of t . now, if y 6= x isanother fixed point of t , we have θ � d(x, y) = d(tx, t y) � α(x, y)d(tx, t y) � ψ(d(x, y) + d(tx, x) + d(ty, y)) = ψ(d(x, y) + d(x, x) + d(y , y)) = ψ(d(x, y). so d(x, y) = θ ; ie x = y . � 4. applications as application of α − ψ contractive in unital c∗-algebra valued rectangular b− metric spaces,existence and uniqueness results for a type of operator equation is given. example 4.1. suppose that h is a hilbert space, b(h) is the set of linear bounded operators on h. let a1, a2, ..., an, ... ∈ b(h)which satisfy ∑∞n=1 ‖an‖ < 1 and q ∈ b(h)+.then the operator equation x −∑∞n=1 a∗nxan = q has a unique solution in b(h). proof. set a = ( ∑∞ n=1 ‖an‖)p with p ≥ 1, then ‖a‖ < 1.without loss of generality, one can supposethat a > 0.choose a positive operator m ∈ b(h).for x, y ∈ b(h) and p ≥ 1, set d(x, y ) = ‖x − y ‖pm .then d(x, y ) is a c∗-algebra valued rectangular b− metric.suppose that x, y, z,w ∈ b(h) we have ‖x − y ‖p � 2p (‖x − z‖p + ‖z −w‖p + ‖w − y ‖p).which implies that d(x, y ) � a[d(x,z) + d(z,w ) + d(w, y )]where a = 2pi . consider the map t : b(h)→ b(h) such that t (x) = ∑∞ n=1 a ∗ nxan +q.then d(t (x), t (y )) = ‖t (x), t (y )‖pm = ‖ ∑∞ n=1 a ∗ n(x − y )an‖pm � ∑∞ n=1 ‖an‖2p‖x − y ‖pm � a2d(x, y ) https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 8let α : b(h)× b(h)→ b(h)+ defined by α(x, y ) = (x, y )iand ψ : b(h)+ → b(h)+ defined by ψ(x) = x .we get α(x, y )d(tx, ty ) � ψ(d(x, y )).using theorem 3.1, there exists a unique fixed point x in b(h). � 5. acknowledgments it is our great pleasure to thank the referee for his careful reading of the paper and for severalhelpful suggestions. references [1] h.h. alsulami, r.p. agarwal, e. karapınar, f. khojasteh, a short note on c∗-valued contraction mappings, j. inequal.appl. 2016 (2016) 50. https://doi.org/10.1186/s13660-016-0992-5.[2] s. chandok, d. kumar, c. park, c∗−algebra-valued partial metric space and fixed point theorems. proc. math. sci.129 (2019) 37. https://doi.org/10.1007/s12044-019-0481-0.[3] m. jleli, b. samet, a new generalization of the banach contraction principle. j. inequal. appl. 2014 (2014), 38. https://doi.org/10.1186/1029-242x-2014-38.[4] g. kalapana, z.s. tasneem c∗−algebra-valued rectangular b-metric spaces and some fixed point theorems, commun.fac. sci. univ. ank. ser. a1 math. stat. 68 (2019) 2198-2208. https://doi.org/10.31801/cfsuasmas.598146.[5] w. a. kirk, n. shahzad, generalized metrics and caristi’s theorem, fixed point theory appl. 2013 (2013) 129. https://doi.org/10.1186/1687-1812-2013-129.[6] z. ma, l. jiang, h. sun, c∗-algebra-valued metric spaces and related fixed point theorems, fixed point theoryappl. (2014) 2014, 206. https://doi.org/10.1186/1687-1812-2014-206.[7] h. massit, m. rossafi, fixed point for ψ− contractive mapping in c∗− algebra valued rectangular b-metric, j. math.comput. sci. 11(2021) 6507-6521. https://doi.org/10.28919/jmcs/6363.[8] g.j. murphy, c∗-algebras and operator theory, academic press, london, uk, 1990.[9] s. omran, i. masmali, on the (α− ψ)-contractive mappings in c∗-algebra valued b-metric spaces and fixed pointtheorems, j. math. 2021 (2021) 7865976. https://doi.org/10.1155/2021/7865976.[10] b. samet, c. vetro, p. vetro, fixed point theorems for α− ψ−contractive type mappings, nonlinear anal.: theorymethods appl. 75 (2012) 2154–2165. https://doi.org/10.1016/j.na.2011.10.014. https://doi.org/10.28924/ada/ma.2.11 https://doi.org/10.1186/s13660-016-0992-5 https://doi.org/10.1007/s12044-019-0481-0 https://doi.org/10.1186/1029-242x-2014-38 https://doi.org/10.31801/cfsuasmas.598146 https://doi.org/10.1186/1687-1812-2013-129 https://doi.org/10.1186/1687-1812-2014-206 https://doi.org/10.28919/jmcs/6363 https://doi.org/10.1155/2021/7865976 https://doi.org/10.1016/j.na.2011.10.014 1. introduction 2. preliminaries 3. main result 4. applications 5. acknowledgments references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 133-150doi: 10.28924/ada/ma.1.133 lie group analysis of a nonlinear coupled system of korteweg-de vries equations joseph owuor owino∗, benard okelo department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, box 210-40601, bondo, kenya bnyaare@yahoo.com, josephowuorowino@gmail.com ∗correspondence: josephowuorowino@gmail.com abstract. in this paper, we consider coupled korteweg-de vries equations that model the propagationof shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations alonginternal surfaces between layers of different densities in stratified fluids, for example propagation ofsolitons of long internal waves in oceans. the method of lie group analysis is used to on the systemto obtain symmetry reductions. soliton solutions are constructed by use of a linear combination oftime and space translation symmetries. furthermore, we compute conservation laws in two waysthat is by multiplier method and by an application of new conservation theorem developed by nailibragimov. 1. introduction the dynamics of shallow-water waves, ion-acoustic waves in plasmas, and long internal waves inoceans can be described by coupled kdv equations. the equations are derived from the classicalkdv equation. this section extends the previous study of kdv equations to that of a couplednonlinear system. from the kortweg-de vries equation qt + αqqx + βqxxx = 0, (1) for α and β as constants, we let q(t, x) = u(t, x) + iv(t, x), (2) where i2 = −1. then substituting (2) into (1) and separating the real and imaginary parts, weobtain ∆1 ≡ ut + αuux − αvvx + βuxxx = 0, ∆2 ≡ vt + αuvx + αvux + βvxxx = 0, (3) received: 3 sep 2021. key words and phrases. coupled kdv equations; lie group analysis; group-invariant solutions; stationary solutions;symmetry reductions; soliton; multipliers; conservation laws.133 https://adac.ee https://doi.org/10.28924/ada/ma.1.133 eur. j. math. anal. 1 (2021) 134which is a nonlinear system of coupled kdv equations. we perform lie symmetry analysis on (3),that is , we obtain lie point symmetries, invariant solutions and conservation laws of (3).this paperuses symmetry analysis method to construct exact solutions and conservation laws for a nonlinearcoupled kdv system (3). 2. preliminaries in this section, we outline preliminary concepts which are useful in the sequel. in euclideanspaces rn of x = x i independent variables and rm of u = uα dependent variables, we considerthe transformations tε : x̄ i = ϕi(x i , uα, ε), ūα = ψα(x i , uα, ε), (4) involving the continuous parameter ε which ranges from a neighbourhood n ′ ⊂ n ⊂ r of ε = 0where the functions ϕi and ψα differentiable and analytic in the parameter ε. definition 2.1. the set g of transformations given by (4) is a local lie group if it holds true that(1) (i). (closure) given tε1 , tε2 ∈ g, for ε1, ε2 ∈ n ′ ⊂ n , then tε1tε2 = tε3 ∈ g, ε3 = φ(ε1, ε2) ∈ n .(2) (ii). (identity) there exists a unique t0 ∈ g if and only if ε = 0 such that tεt0 = t0tε = tε.(3) (iii). (inverse) there exists a unique tε−1 ∈ g for every transformation tε ∈ g,where ε ∈ n ′ ⊂ n and ε−1 ∈ n such that tεtε−1 = tε−1tε = t0. remark 2.2. associativity of the group g in (4) follows from (1). in the system, ∆α ( x i , uα, u(1), . . . , u(π) ) = ∆α = 0, (5)the variables uα are dependent. the partial derivatives u(1) = {uαi }, u(2) = {uαij }, . . . , u(π) = {uαi1...iπ}, are of the first, second, . . . , up to the πth-orders.denoting di = ∂ ∂x i + uαi ∂ ∂uα + uαij ∂ ∂uαj + . . . , (6) the total differentiation operator with respect to the variables x i and δji , the kronecker delta, wehave di(x j) = δji , ′, uαi = di(u α), uαij = dj(di(u α)), . . . , (7) where uαi defined in (7) are differential variables [7].consider the local lie group g given by the transformations x̄ i = ϕi(x i , uα, ε), ϕi ∣∣∣ ε=0 = x i , ūα = ψα(x i , uα, ε), ψα ∣∣∣ ε=0 = uα, (8) where the symbol ∣∣∣ ε=0 means evaluated on ε = 0. eur. j. math. anal. 1 (2021) 135 definition 2.3. the construction of the group g given by (8) is an equivalence of the computationof infinitesimal transformations x̄ i ≈ x i + ξi(x i , uα)ε, ϕi ∣∣∣ ε=0 = x i , ūα ≈ uα + ηα(x i , uα)ε, ψα ∣∣∣ ε=0 = uα, (9) obtained from (4) by a taylor series expansion of ϕi(x i , uα, ε) and ψi(x i , uα, ε) in ε about ε = 0and keeping only the terms linear in ε, where ξi(x i , uα) = ∂ϕi(x i , uα, ε) ∂ε ∣∣∣ ε=0 , ηα(x i , uα) = ∂ψα(x i , uα, ε) ∂ε ∣∣∣ ε=0 . (10) remark 2.4. the symbol of infinitesimal transformations, x , is used to write (9) as x̄ i ≈ (1 +x)x i , ūα ≈ (1 +x)uα, (11) where x = ξi(x i , uα) ∂ ∂x i + ηα(x i , uα) ∂ ∂uα , (12) is the generator of the group g given by (8). remark 2.5. to obtain transformed derivatives from (4), we use a change of variable formulae di = di(ϕ j)d̄j , (13) where d̄j is the total differentiation in the variables x̄ i . this means that ūαi = d̄i(ū α), ūαij = d̄j(ū α i ) = d̄i(ū α j ). (14) if we apply the change of variable formula given in (13) on g given by (8), we get di(ψ α) = di(ϕ j), d̄j(ū α) = ūαj di(ϕ j). (15) expansion of (15) yields ( ∂ϕj ∂x i + uβi ∂ϕj ∂uβ ) ūβj = ∂ψα ∂x i + uβi ∂ψα ∂uβ . (16) the variables ūαi can be written as functions of x i , uα, u(1), that is ūαi = φα(x i , uα, u(1), ε), φα ∣∣∣ ε=0 = uαi . (17) definition 2.6. the transformations in the space of the variables x i , uα, u(1) given in (8) and (17)form the first prolongation group g[1]. definition 2.7. infinitesimal transformation of the first derivatives is ūαi ≈ uαi + ζαi ε, where ζαi = ζαi (x i , uα, u(1), ε). (18) remark 2.8. in terms of infinitesimal transformations, the first prolongation group g[1] is given by(9) and (18). eur. j. math. anal. 1 (2021) 136 definition 2.9. by using the relation given in (15) on the first prolongation group g[1] given bydefinition 2.6, we obtain [5] di(x j + ξjε)(uαj + ζαj ε) = di(u α + ηαε), which gives uαi + ζαj ε+ uαj εdiξ j = uαi +diη αε,(19) and thus ζαi =di(η α)− uαj di(ξj), (20) is the first prolongation formula. remark 2.10. similarly, we get higher order prolongations [8], ζαij = dj(ζ α i )− uαiκdj(ξκ), . . . , ζαi1,...,iκ = diκ(ζαi1,...,iκ−1 )− uαi1,i2,...,iκ−1j diκ(ξj). (21) remark 2.11. the prolonged generators of the prolongations g[1], . . . ,g[κ] of the group g are x[1] = x + ζαi ∂ ∂uαi , . . . , x[κ] = x[κ−1] + ζαi1,...,iκ ∂ ∂ζαi1,...,iκ , κ ≥ 1, (22) where x is the group generator given by (12). definition 2.12. a function γ(x i , uα) is called an invariant of the group g of transformations givenby (4) if γ(x̄ i , ūα) = γ(x i , uα). (23) theorem 2.13. a function γ(x i , uα) is an invariant of the group g given by (4) if and only if it solves the following first-order linear pde: [5] xγ = ξi(x i , uα) ∂γ ∂x i + ηα(x i , uα) ∂γ ∂uα = 0. (24) from theorem (2.13), we have the following result. theorem 2.14. the local lie group g of transformations in rn given by (4) [7] has precisely n− 1 functionally independent invariants. one can take, as the basic invariants, the left-hand sides of the first integrals ψ1(x i , uα) = c1, . . . , ψn−1(x i , uα) = cn−1, (25) of the characteristic equations for (24): dx i ξi(x i , uα) = duα ηα(x i , uα) . (26) eur. j. math. anal. 1 (2021) 137 definition 2.15. the vector field x (12) is a lie point symmetry of the pde system (5) if thedetermining equations x[π]∆α ∣∣∣ ∆α=0 = 0, α = 1, . . . , m, π ≥ 1, (27) are satisfied, where ∣∣∣ ∆α=0 means evaluated on ∆α = 0 and x[π] is the π-th prolongation of x . definition 2.16. the lie group g is a symmetry group of the pde system given in (5) if the pdesystem (5) is form-invariant, that is ∆α ( x̄ i , ūα, ū(1), . . . , ū(π) ) = 0. (28) theorem 2.17. given the infinitesimal transformations in (8), the lie group g in (4) is found by integrating the lie equations dx̄ i dε = ξi(x̄ i , ūα), x̄ i ∣∣∣ ε=0 = x i , dūα dε = ηα(x̄ i , ūα), ūα ∣∣∣ ε=0 = uα. (29) definition 2.18. a vector space vr of operators [5] x (12) is a lie algebra if for any two operators, xi , xj ∈ vr , their commutator [xi , xj ] = xixj −xjxi , (30) is in vr for all i , j = 1, . . . , r . remark 2.19. the commutator satisfies the properties of bilinearity, skew symmetry and the jacobiidentity [5]. theorem 2.20. the set of solutions of the determining equation given by (27) forms a lie algebra [5]. the methods of (g’/g)-expansion method [20], extended jacobi elliptic function expansion [21]and kudryashov [22] are usually applied after symmetry reductions. let a system of πth-orderpdes be given by (5). definition 2.21. the euler-lagrange operator δ/δuα is δ δuα = ∂ ∂uα + ∑ κ≥1 (−1)κdi1 , . . . , diκ ∂ ∂uαi1i2...iκ , (31) and the liebäcklund operator in abbreviated form [5] is x = ξi ∂ ∂x i + ηα ∂ ∂uα + . . . . (32) remark 2.22. the liebäcklund operator (32) in its prolonged form is x = ξi ∂ ∂x i + ηα ∂ ∂uα + ∑ κ≥1 ζi1...iκ ∂ ∂uαi1i2...iκ , (33) eur. j. math. anal. 1 (2021) 138where ζαi = di(w α) + ξjuαij , . . . , ζαi1...iκ = di1...iκ(wα) + ξjuαji1...iκ , j = 1, . . . , n. (34) and the lie characteristic function is wα = ηα − ξjuαj . (35) remark 2.23. the characteristic form of liebäcklund operator (33) is x = ξidi +wα ∂ ∂uα +di1...iκ(wα) ∂ ∂uαi1i2...iκ . (36) remark 2.24. noether’s theorem is applicable to systems from variational problems definition 2.25. a function λα ( x i , uα, u(1), . . . ) = λα, is a multiplier of the pde system given by(5) if it satisfies the condition that [16] λα∆α = dit i , (37) where dit i is a divergence expression. definition 2.26. to find the multipliers λα, one solves the determining equations (38) [3], δ δuα (λα∆α) = 0. (38) the technique [9] enables one to construct conserved vectors associated with each lie pointsymmetry of the pde system given by (5). definition 2.27. the adjoint equations of the system given by (5) are ∆∗α ( x i , uα, vα, . . . , u(π), v(π) ) ≡ δ δuα (vβ∆β) = 0, (39) where vα is the new dependent variable. definition 2.28. formal lagrangian l of the system (5) and its adjoint equations (39) is [9] l = vα∆α(x i , uα, u(1), . . . , u(π)). (40) theorem 2.29. every infinitesimal symmetry xof the system given by (5) leads to conservation laws [9] dit i ∣∣∣ ∆α=0 = 0, (41) where the conserved vector t i = ξil+wα [ ∂l ∂uαi −dj ( ∂l ∂uαij ) +djdk ( ∂l ∂uαijk ) − . . . ] + dj(w α) [ ∂l ∂uαij −dk ( ∂l ∂uαijk ) + . . . ] +djdk(wα) [ ∂l ∂uαijk − . . . ] . (42) eur. j. math. anal. 1 (2021) 1393. main results we now present our results in this section. an illustrative example with a simple kdv equationcan be found in [6]. the infinitesimal transformations of the lie group with parameter ε are t̄ = t + ξt(t, x, u, v)ε, x̄ = x + ξx(t, x, u, v)ε, ū = u + ηu(t, x, u, v)ε, v̄ = v + ηv (t, x, u, v)ε.(43) the vector field x = ξt(t, x, u, v) ∂ ∂t + ξx(t, x, u, v) ∂ ∂x + ηu(t, x, u, v) ∂ ∂u + ηv (t, x, u, v) ∂ ∂v , (44) is a lie point symmetry of (3) if x[3]∆1 ∣∣∣ ∆1=0, ∆2=0 = 0, x[3]∆2 ∣∣∣ ∆1=0, ∆2=0 = 0. (45) expanding (45) and and splitting on derivatives of v and u, we have an overdetermined system often pdes, namely, ξtu = 0, ξtv = 0, ξtx = 0, ξxu = 0, ξxv = 0, ξttt = 0, ξxtt = 0, 3ξxx − ξtt = 0, 3ηv + 2ξttv = 0, 3αηu + 2αξttu − 3ξxt = 0. (46) solving the system (46) yields ξt = a1 + 3a2t, ξx = a2x + αa3t + a4, η u = −2a2u + a3, η v = −2a2v , (47) for arbitrary constants a1, a2, a3, a4. hence from (47), the infinitesimal symmetries of the coupledkdv equations (3) is a lie algebra generated by the vector fields x1 = ∂ ∂t , x2 = ∂ ∂x , x3 = αt ∂ ∂x + ∂ ∂u , x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v . (48) the set of all infinitesimal symmetries of coupled kdv equations forms a lie algebra and yield thefollowing commutation relations in table 1. [xi , xj ] x1 x2 x3 x4 x1 0 0 αx2 3x1 x2 0 0 0 x2 x3 -αx2 0 0 -2x3 x4 -3x1 -x2 2x3 0table 1: a commutator table for the lie algebra generated by the symmetries of coupled kdvequation. eur. j. math. anal. 1 (2021) 140the following lie groups, for i = 1, 2, 3, 4, are obtained tε1 : t̄ = t + ε1, x̄ = x, ū = u, v̄ = v , (49) tε2 : t̄ = t, x̄ = x + ε2, ū = u, v̄ = v , (50) tε3 : t̄ = t, x̄ = x + αε3t, ū = u + ε3, v̄ = v , (51) tε4 : t̄ = te3ε4 , x̄ = xeε4 , ū = ue−2ε4 , v̄ = ve−2ε4 . (52) the symmetries obtained yield the following symmetry reductions. x1 = ∂ ∂t . (53) solving the characteristic equations dt 1 = dx c = du 0 = dv 0 , (54) associated to the operator x1 gives the invariants j1 = x, j2 = u, j3 = v . (55) hence, we have u = ϕ(x), v = ψ(x), (56) for arbitrary functions ϕ and ψ. substituting the expressions for u and v given by (56) into thesystem (3), we get a system of third order ordinary des namely, α [ ϕ(x)ϕ′(x)− ψ(x)ψ′(x) ] + βϕ′′′(x) = 0, α (ϕ(x)ψ(x))′ + βψ′′′(x) = 0. (57) integration of the system (57) yields; α 2 [ ϕ(x)2 − ψ(x)2 ] + βϕ′′(x) = c1, (58) α [ϕ(x)ψ(x)] + βψ′′(x) = c2, (59) for arbitrary constants c1 and c2. if we take c1 = c2 = 0, (60) the system (58)-(59) becomes α 2 [ ϕ(x)2 − ψ(x)2 ] + βϕ′′(x) = 0, (61) α [ϕ(x)ψ(x)] + βψ′′(x) = 0. (62) eur. j. math. anal. 1 (2021) 141to find more solutions of the system (61)-(62), we determine its lie point symmetries. using thelie’s algorithm for computing point symmetries, we see that the lie point symmetries of (61)-(62)are x∗1 = ∂ ∂x , x∗2 = x ∂ ∂x − 2ϕ ∂ ∂ϕ − 2ψ ∂ ∂ψ . (63) proceeding as above, we see that the symmetry x∗1 yields the trivial solution u = 0, v = 0. (64) the second symmetry x∗2 has the characteristic equations dx x = dϕ −2ϕ = dψ −2ψ , (65) which provides the invariants j1 = x2ϕ, j2 = x2ψ. (66) letting ϕ = λ x2 , ψ = µ x2 , (67) substituting the values of ϕ and ψ into (61)-(62) and solving the resulting equations yield: case one. taking µ = 0 (68) gives λ = 0 (69) or λ = − 12β α . (70) when λ = 0, and µ = 0, (71) we also get the trivial solution (64). one can easily see that if λ = − 12β α , and µ = 0, (72) then ϕ = − 12β αx2 , ψ = 0, (73) which is a solution of the system (61)-(62). hence u1(t, x) = − 12β αx2 , v1(t, x) = 0, (74) eur. j. math. anal. 1 (2021) 142is a solution of the coupled kdv system (3). case two. taking λ = − 6β α (75) gives µ = ± 6βi α , (76) with i2 = −1. consequently, u2(t, x) = − 6β αx2 , v2(t, x) = 6iβ αx2 , (77) and u3(t, x) = − 6β αx2 , v3(t, x) = − 6iβ αx2 , (78) are solutions of the coupled kdv system . hence lie group analysis has given us three steady-statesolutions for the coupled kdv system under the time translation symmetry x1 = ∂ ∂t . x2 = ∂ ∂x . (79) solving the characteristic equations dt 0 = dx 1 = du 0 = dv 0 , (80) associated to x2 gives the invariants j1 = t, , j2 = u j3 = v . (81) therefore, the group-invariant solution is u = φ(t), v = h(t), (82) for arbitrary functions h and φ. substitution of the solutions from (82) into (3), we get a system offirst order ordinary des, namely, φ′(t) = 0, h′(t) = 0, (83) which is integrated once with respect to t to yield, φ(t) = c1, h(t) = c2, (84) for arbitrary constants c1 and c2. consequently, the space translation group-invariant solution ofthe system (3) is u(t, x) = c1, v(t, x) = c2. (85) x3 = αt ∂ ∂x + ∂ ∂u . (86) eur. j. math. anal. 1 (2021) 143solving the characteristic equations dt 0 = dx αt = du 1 = dv 0 , (87) associated to galilean boost gives the invariants j1 = t, j2 = v , j3 = −u + x αt , t 6= 0. (88) thus the invariant solution of (3) is u = x αt − g(t), v = f (t), t 6= 0, (89) for arbitrary functions f and g. substitution of the values of u and v from (89) into the system (3),we get a nonlinear system of coupled first order ordinary des, namely, tg′(t) + g(t) = 0, tf ′(t) + f (t) = 0, (90) whose solutions are g(t) = c1 t f (t) = c2 t , (91) for arbitrary constants c1 and c2. hence the galilean boost group-invariant solution of the system(3) is u(t, x) = x + a αt , v(t, x) = c2 t (92) where a = −αc1 and t 6= 0. the scaling x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v (93) . by solving of the characteristic equations dt 3t = dx x = − du 2u = − dv 2v , (94) associated to this symmetry, we obtain the invariants j1 = x3 t , j2 = ux2, j3 = vx2. (95) generally, the group-invariant solution pair is u(t, x) = f (λ) x2 , v(t, x) = g(λ) x2 , where λ = x3 t , (96) and the functions f and g satisfy the system of third order nonlinear coupled ordinary des 2α(g2 − f 2)− λ2f ′ + 3αλ(f f ′ − gg′) + β(−24f + 24λf ′ + 27λ3f ′′′) =0, (97) −4αf g − λ2g′ + 3αλ(f g)′ + β(−24g + 24λg′ + 27λ3g′′′) =0. (98) x = x1 + cx2. (99) eur. j. math. anal. 1 (2021) 144we consider a symmetry x , which is a linear combination of the time and space translationssymmetries, that is, x = ∂ ∂t + c ∂ ∂x , (100) for a constant c . the invariants associated to this symmetry x are j1 = x − ct, j2 = u, j3 = v . (101) hence, the invariant solution for the symmetry x is u = f (x − ct), v = g(x − ct), (102) for arbitrary functions f and g. substitution of u and v from (102) into the system (3) yields asystem of nonlinear third order ordinary des, namely −cf ′(ξ) + α { f (ξ)f ′(ξ)− g(ξ)g′(ξ) } + βf ′′′(ξ) = 0, −cg′(ξ) + α(f (ξ)g(ξ))′ + βg′′′(ξ) = 0,(103)which on integrating once with respect to ξ yields −cf + 1 2 α(f 2 − g2) + βf ′′ + c1 = 0, −cg + αf g + βg′′ + c2 = 0, (104) for arbitrary constants c1 and c2. remark 3.1. if we take the constants c1 = c2 = 0, then when the wave velocity c = 0, we canrecover the stationary solutions given in (3). remark 3.2. traveling wave solutions of the system (3) must satisfy the system (104). computation of conservation laws for the coupled kdv equations (3) is done using two meth-ods; the method of multipliers and a theorem due to ibragimov. we seek local conservation lawmultipliers for the system (3), whose determining equations are δ δu [ λ1∆1 + λ2∆2 ] = 0, δ δv [ λ1∆1 + λ2∆2 ] = 0, (105) where δ δu = ∂ ∂u −dt ∂ ∂ut −dx ∂ ∂ux +d2 x ∂ ∂uxx −d3 x ∂ ∂uxxx + . . . , (106) δ δv = ∂ ∂v −dt ∂ ∂vt −dx ∂ ∂vx +d2 x ∂ ∂vxx −d3 x ∂ ∂vxxx + · · · , (107) are the euler-lagrange operators and dt = ∂ ∂t + ut ∂ ∂u + vt ∂ ∂v + utx ∂ ∂ux + vtx ∂ ∂vx + utt ∂ ∂ut + vtt ∂ ∂vt + · · · , (108) dx = ∂ ∂x + ux ∂ ∂u + vx ∂ ∂v + uxx ∂ ∂ux + vxx ∂ ∂vx + utx ∂ ∂ut + vtx ∂ ∂vt + · · · , (109) eur. j. math. anal. 1 (2021) 145are total derivatives operators. we look for second order multipliers, that is, λn = λn(t, x, u, ux , uxx , v , vx , vxx), n = 1, 2. (110) the determining equations (105) become δ δu [ λ1{ut + αuux − αvvx + βuxxx}+ λ2{vt + αuvx + αvux + βvxxx} ] = 0, (111) δ δv [ λ1{ut + αuux − αvvx + βuxxx}+ λ2{vt + αuvx + αvux + βvxxx} ] = 0. (112) expanding (111)-(112) and splitting on derivatives of u and v yields an overdetermined system of22 pdes, namely λ1 xx = 0, λ2 xx = 0 λ1 vx = 0, λ2 vx = 0, λ1 xvxx = 0, λ2 xvxx = 0, βλ1 vv − αλ2 vxx = 0, βλ2 vv + αλ1 vvxx = 0, λ1 vvxx = 0, λ2 vvxx = 0, λ1 vxxvxx = 0, λ2 vxxvxx = 0, λ1 u + λ2 v = 0, λ1 t + α ( λ2 xv + λ1 xu ) = 0, λ2 t + α ( λ2 xu − λ1 xv ) = 0, λ2 u − λ1 v = 0, λ1 ux = 0, λ2 ux = 0, λ1 uxx + λ2 vxx = 0, λ2 uxx − λ1 vxx = 0, λ2 vx = 0 λ1 vx = 0.(113)calculations reveal the solution of the system (113) as λ1 = α 2β ( c3{u2 − v2}+ 2c4uv ) + (c2t + c5)u + (c1t + c6)v + c3uxx + c4vxx + c7 − 1 α c2x, λ2 = α 2β ( c4{u2 − v2} − 2c3uv+ ) + (c1t + c6)u − (c2t + c5)v + c4uxx − c3vxx + c8 − 1 α c1x,(114)for arbitrary constants c1, . . . , c8. remark 3.3. essentially, the nonlinear coupled system of kdv equations (3) has eight sets of localconservation law multipliers. solving (105)„ we obtain conserved vectors corresponding to each set of multipliers as shownbelow.(i) the multiplier ( λ1 1,λ2 1 ) = ( tv , tu − x α ) , (115) has the conserved vectors t t1 = tuv − xv α , t x1 = β [ t{vuxx + uvxx − vxux}+ 1 α {vx − xvxx} ] + α [ t ( u2v − v3 3 )] (116) −xuv . (117) (ii) the multiplier ( λ1 2,λ2 2 ) = ( tu − x α ,−tv ) , (118) eur. j. math. anal. 1 (2021) 146has the conserved vectors t t2 = t 2 {u2 − v2} − xu α , t x2 = β [ t ( uuxx − vvxx + 1 2 {v2 x − u2 x} ) + 1 α {ux − xuxx} ] + αt [ u3 3 − uv2 ] + x 2 {v2 − u2}. (119) (iii) the multiplier ( λ1 3,λ2 3 ) = ( α 2β {u2 − v2}+ uxx ,−{ αuv β + vxx} ) , (120) has the conserved vectors t t3 = α 2β ( u3 3 − uv2 ) , t x3 = α 2 [ (u2 − v2)uxx − v2vxx ] − αuvvxx+ (121) β 2 [ u2 xx − v2 xx ] + utux − vtvx + α2 4β [ 1 2 {u4 + v4} − 3u2v2 ] . (122) (iv) the multiplier ( λ1 4,λ2 4 ) = ( { αuv β + vxx}, α[u2 − v2] 2β + uxx ) , (123) has the conserved vectors t t4 = α 2β ( u2v − v3 3 ) , (124) t x4 = α2 2β [ (u3v − uv3) ] + vtux + utvx + α 2 (u2 − v2)vxx + {αuv + βvxx}uxx . (125) (v) the multiplier ( λ1 5,λ2 5 ) = (u,−v) , (126) has the conserved vectors t t5 = 1 2 {u2 − v2}, t x5 = β ( uuxx − vvxx + v2 x − u2 x 2 ) + α ( u3 3 − uv2 ) . (127) (vi) the multiplier ( λ1 6,λ2 6 ) = (v , u) , (128) has the conserved vectors t t6 = uv, t x6 = β (vuxx + uvxx − uxvx) + α ( u2v − v3 3 ) . (129) (vii) the multiplier ( λ1 7,λ2 7 ) = (1, 0) , (130) has the conserved vectors t t7 = u, t x7 = α 2 {u2 − v2}+ βuxx . (131) eur. j. math. anal. 1 (2021) 147(viii) the multiplier has ( λ1 8,λ2 8 ) = (0, 1) , (132) the conserved vectors t t8 = v , t x8 = αuv + βvxx . (133) remark 3.4. it can be verified that dtt t i +dxt x i ∣∣∣ ∆1=0, ∆2=0 = 0, (134) for i = 1, . . . , 8. remark 3.5. the expressions in (134) are eight conservation laws for the coupled kdv system (3). remark 3.6. the presence of multipliers( λ1 7,λ2 7 ) = (1, 0) , ( λ1 8,λ2 8 ) = (0, 1) (135) manifest that the coupled kdv equations are themselves conservation laws. at this point, we derive conserved vectors for coupled kdv equations (3) by a new theorem due toibragimov. the adjoint equations for the nonlinear system coupled kdv equations (3) are ∆∗1 ≡ ft + α ufx + αvgx + βfxxx = 0, ∆∗2gt − αvfx + αugx + βgxxx = 0. (136) the formal lagrangian l for the nonlinear coupled system of the kdv equations (3) and its adjointequations (136) is given by l = f {ut + αuux − αvvx + βuxxx}+ g{vt + αuvx + αvux + βvxxx}, (137) where f and g are new variables. we shall use the lie point symmetries of the system (3) ,namely x1 = ∂t , x2 = ∂x , x3 = αt∂x + ∂u, x4 = 3t∂t + x∂x − 2u∂u − 2v∂v , (138) to derive conserved vectors corresponding to each symmetry below.case (i) the symmetry x1 = ∂ ∂t , yields lie characteristic functions given by w 1 1 = −ut , w 2 1 = −vt . (139) hence by ibragimov’s theorem [9], the associated conserved vector is given by t t1 =α [f {uux − vvx}+ g{vux + uvx}] + β{f uxxx + gvxxx}, t x1 =α [f {−uut + vvt} − g{vut + uvt}] + β{fxutx + gxvtx − ut fxx − vtgxx − f utxx − gvtxx}. (140) eur. j. math. anal. 1 (2021) 148case (ii) the symmetry x2 = ∂ ∂x , yields lie characteristic functions w 1 2 = −ux , w 2 2 = −vx . (141) therefore by ibragimov’s theorem [9], the associated conserved vector is t t2 = −ux f − vxg, t x2 = f ut + gvt + β{−ux fxx − vxgxx + fxuxx + gxvxx}. (142) case (iii) the symmetry x3 = αt ∂ ∂x + ∂ ∂u (143) yields lie characteristic functions given by w 1 3 = 1− αtux , w 2 3 = −αtvx . (144) hence by ibragimov’s theorem [9], the associated conserved vector is given by t t3 = f − αt{ux f + vxg} , t x3 = α [ f u + gv + t{ut f + vtg}+ βt{ fxx αt − ux fxx − vxgxx + fxuxx + gxvxx} ] . (145) case (iv) the symmetry x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v (146) yields the lie characteristic functions w 1 4 = −2u − 3tut − xux , w 2 4 = −2v − 3tvt − xvx . (147) consequently by ibragimov’s theorem [9], the corresponding conserved vector is given by t t4 = α [3t{f uux − f vvx + guvx + gvux}] + β [3t{f uxxx + gvxxx}] − 2{f u + gv} − x{f ux + gvx}, t x4 = x{f ut + gvt}+ β [ 3 ( fxux + gxvx + t{fxutx + gxvtx} )] − α [ 2 ( f {u2 − v2}+ 2guv ) + 3t ( f {uut − vvt}+ g{vut + uvt} )] − β [x{ux fxx + vxgxx − fxuxx − gxvxx}+ 2{ufxx + vgxx}] − β [3t{fxxut + gxxvt + f utxx + gvtxx}+ 4{f uxx + gvxx}] . (148) remark 3.7. the appearance of arbitrary functions f (t, x) and g(t, x) in the conserved vectorsproves the existence of infinite conservation laws for coupled kdv system obtained by ibagimov’smethod. eur. j. math. anal. 1 (2021) 1494. conclusion in this paper, lie group analysis was employed in studying a nonlinear coupled kdv system.a four-dimensional lie algebra of symmetries was found for the nonlinear coupled system kdvequations. this was spanned by space and time translations, galilean boost and scaling symmetrieswhere the scaling symmetry acts on four variables. associated to each symmetry, we obtainedsymmetry reductions that gave six nontrivial solutions for the coupled system. all the group-invariant solutions describe the various states of the system. the obtained solutions can be usedas a benchmark against numerical simulations. lastly, we constructed infinite conservation laws ofa nonlinear coupled kdv system by using multipliers and a theorem proposed by nail ibragimov. acknowledgement the first author acknowledges the financial support of aims-south africa and mastercard foun-dation. the authors are also grateful to the referees for their careful reading of the manuscript andvaluable comments. references [1] d. j. arigo, symmetry analysis of differential equations: an introduction, john wiley & sons, 2015.[2] g. bluman, s. anco, symmetry and integration methods for differential equations, springer science & businessmedia, 2008.[3] g. w. bluman, s. kumei, symmetries and differential equations, springer science & business media, 1989.[4] bluman, g. w., cheviakov, a. f., and anco, s. c, applications of symmetry methods to partial differential equations,springer, 2010.[5] n. h. ibragimov, elementary lie group analysis and ordinary differential equations, wiley, 1999.[6] j. owuor, m. khalique, lie group analysis of nonlinear partial differential equations, lambert academic publishers,2021[7] n. h. ibragimov, crc handbook of lie group analysis of differential equations, crc-press, 1994.[8] n. h. ibragimov, selected works, alga publications, blekinge institute of technology, selected works, 2009.[9] n. h. ibragimov, a new conservation theorem, j. math. anal. appl. 333(2007), 311-328. https://doi.org/10. 1016/j.jmaa.2006.10.078.[10] n. h. ibragimov, a practical course in differential equations and mathematical modelling: classical and newmethods. nonlinear mathematical models. symmetry and invariance principles, world scientific publishing com-pany, 2009.[11] c. m. khalique, s. a. abdallah, coupled burgers equations governing polydispersive sedimentation; a lie symmetryapproach. results phys. 16(2020), 76-90. https://doi.org/10.1016/j.rinp.2020.102967.[12] r. j. leveque, numerical methods for conservation laws, springer verlag, new york, 1992.[13] s lie, vorlesungen aber differentialgleichungen mit bekannten infinitesimalen transformationen. bg teubner, 1891.[14] i. mhlanga, c. khalique, travelling wave solutions and conservation laws of the korteweg-de vriesburgers equationwith power law nonlinearity. malays. j. math. sci. 11(2017), 1-8.[15] e. noether, invariant variations problem, nachr. konig. gissel. wissen, gottingen. math. phys. kl, 6(1918), 235-257.[16] p. j. olver, applications of lie groups to differential equations, springer science & business media, 1993. https://doi.org/10.1016/j.jmaa.2006.10.078 https://doi.org/10.1016/j.jmaa.2006.10.078 https://doi.org/10.1016/j.rinp.2020.102967 eur. j. math. anal. 1 (2021) 150 [17] l. ovsyannikov, lectures on the theory of group properties of differential equations, world scientific publishingcompany, 2013.[18] h. pie, symmetry methods for differential equations: a beginners guide, cambridge university pres, cambridge,2013.[19] a. m. wazwaz, partial differential equations and solitary waves theory, springer science & business media, 2010.[20] n. hasibun, l. abdullah, f. aini, the improved gg-expansion method to the (3 dimensional kadomstev-petviashviliequation, amer. j. appl. math. stat. 1(2013), 64-70.[21] b. hong, d. lu, f. sun, the extended jacobi elliptic functions expansion method and new exact solutions for thezakharov equations, world j. model. simul. 5(2009), 78-109.[22] s.m. ege, e. misirli, the modified kudryashov method for solving some fractional-order nonlinear equations, adv.difference equ. 2014 (2014), 135. https://doi.org/10.1186/1687-1847-2014-135. https://doi.org/10.1186/1687-1847-2014-135 1. introduction 2. preliminaries 3. main results 4. conclusion acknowledgement references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 3doi: 10.28924/ada/ma.3.3 different types of topological structures by graphs ali asghar, ather qayyum∗, noor muhammad institute of southern punjab, multan, pakistan ali.asghar190289283@gmail.com, atherqayyum@isp.edu.pk, noormustaffa681@gmail.com ∗correspondence: atherqayyum@isp.edu.pk abstract. in this paper, we will represent relation of graph which bring different type of topologicalstructure to the graph [2], then, consider certain properties of the graph. we will discuss mainlyblood circulation in lungs and some different diseases of it [4] and relate them with graph and maketopologies [8]. moreover, certain applications in medical field will be represent. we can also useresults in real life [11]. 1. introduction and preliminaries initially in eighteenth century swiss mathematician leonhard euler gave the basic idea aboutgraph [2]. he resolved famous problems. he drew any tenth spectral graph theory introducedin decade of 1950, while in 1980 introduced monograph spectra by cvetkovics, doob and sachs.recently graph theory has become very large field not only for mathematicians but also for otherfields of life [13]. in real life graph theory playing its vital role of life, very common example of it isall roads and motorways form a large network which is used by cruising services e.g. goggle mapswhen working on different routs between two points. graph theory is the study of graph, whichmathematically used to develop pairwise relationship between objects [13]. this is also a collectionof points and lines. points are known as vertices and lines are edges. the collection of verticesof any graph g is vertex set and collection of edges is known as edge set denominated as v(g)and e(g) respectively [4]. the number of vertices and edges in g is known as order and size of grespectively. if an edge has same end is loop. whenever more than one edges having same finalpoint than it will consider parallel edges [12]. mapping [14] play a specific role in graph theoryalso.notions on closure operations are helpful for algebra, topology, basic graph theory and alsofor many other fields [4]. topology is very advance field of mathematics. it deals with thinsindependently. it allow to increase or decrease things without cutting. consider [11] x might be received: 23 dec 2021. key words and phrases. topological space; graph; relation.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 2nonempty set furthermore φ is collection for x, than (1) φ,x belongs to τ (2) absolute unionfor number of τ belongs to τ (3) limited intersection for τ belongs to τ . than τ will be considertopology over x so, (x, τ) is called topological space. topology also helpful in different propertieslike convergence, existence, convexity and many other. all elements within topology known asopen set and complement might be close [13].consider g is any graph, than two adjacent vertices are called nbhd of each other n (v ) = {u ∈ v (g) | u be nbhd of v } is open nbhd for v and n[v ] = n (v ) ∪ {v } is closed nbhd for v. [10]loops and parallel edge free graph is simple graph. if any two distinct vertices joined by anedge is named as complete graph. [12]if vertices of two sets a and b joined by each edge between a and b is called bipartite graph. ifeach vertex from a connected with every vertices of b with only a single edge is complete bipartitegraph [8].if we delete any edge from a subgraph g is called spanning subgraph while deleting any vertexis induced subgraph [3].consider that if any subgraph do not contain their final point that channel p will be nominatedby topological open subgraph while having its initial and final point is topological closed graph [7].consider g = (v, e) be any connected graph. moreover, (v (g) , τ) be topology [7] generatewith βj = {v (g) , φ, {vj}, {n ( vj ) }} is basis moreover consider s1 and s2 be two open paths than (i) v (s1) ⊆ cl v (s1) (ii)s1 ⊆ s2and cl (v (s1)) ⊆ cl (v (s2)) 2. relation over graph suppose that u is vertex in any graph g having l∗ loop and m multiple edges than (degg (u))u = (2lu +mu)u while simple graph is (degg (u))u . [2] [4] here is relation r for any graph g is deformed by r = {((2lu +mu)u , (2lw +mw )w ), u, w ∈ v } https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 3while lu and lw are number for loops for vertices u ,w from each furthermore mu , mw are multipleedges for vertex u and w respectively. consider that g be simple graph, r = {((degg (u)u , degg (w)w )) ; u, w ∈ v }if l = 0 than r = {(lu)u , (mw )w , u, w ∈ w}if m = 1 and l = 0 than r = {(lu, lw ) u, w ∈ vconsider g is directed along with simple than r = {(lu, lw ) = (u,w ) u, w ∈ v } while if g is undirected than r = {(lu, lw ) = (u, w)or (w, u) u, w ∈ v } example 1. [4] suppose that g is undirected graph given above fig.1. r = {(11a, 8b) , (11a, 5c) , (11a, 8d) , (8b, 5c) , (8b, 8d) , (5c , 8d) , (11a, 11a) , (8b, 8b) , (8d , 8d) https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 4 example 2. [1] let g be a graph given in figure 02 r = {(3c , 3b) , (4a, 5e) , (3b, 3c) , (3c , 3d) , (3c , 5e) , (5e , 3d)} example 3. [2] let g be a graph in figure 3 r = {(3a, 2b) , (3a, 2d) , (3a, 3c) , (2b, 3c) , (3c , 2d) 3. topological structure on graph by previous illustration ( 1) created a topology. according to this example the vertices are givenas https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 5 (11a)r = {8b, 8d , 5c}, (8b)r = {11a, 5c , 8d}, (5c)r = {8b, 8d , 11a}, (8d)r = {11a, 8b, 5c}subbase sg = {{8b, 8d , 5c}, {11a, 5c , 8d}, {8b, 8d , 11a}, {11a, 8b, 5c}} topology τg = {x,φ, {8b, 8d , 5c}, {11a, 5c , 8d}, {8b, 8d , 11a}, {11a, 8b, 5c}, {8d , 5c}, {8b, 8d}, , {8b, 5c}, {11a, 8d}, {11a, 5c}, {11a, 8b}, {8b, 5c , 8d}, {11a, 5c , 8d}, {11a, 8b, 8d}, {11a, 8b, 5c} by previous illustration( 2) created a topology. according to this example the vertices are given as (4a)r = {3b, 5e}, (3b)r = {4a, 3c}, (5e)r = {4a, 3c , 3d}, (3c)r = {3b, 3d , 5e}, (3d)r = {3c , 5e} subbase sg = {{3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {4a, 3d , 5e}, {3c , 5e}} base βg = {x,φ, {3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {3b, 3d , 5e}, {3c , 5e}, {5e}, {3c}, {3d} topology τg = {x,φ, {3b, 5e}, {4a, 3c}, {4a, 3c , 3d}, {3b, 3d , 5e}, {3c , 5e}, {3c}, {3d}, {5e}, {4a, 3b, 3c , 5e}, {4a, 3c , 3d , 5e}, {3b, 3c , 5e}, {4a, 3c , 5e}, {4a, 3d}, {3b, 3c , 3d , 5e}, {3c , 3d , 5e}, {3c , 3d}, {3c , 5e}, {3d , 5e} https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 6by previous example (3) it is given as. (3a)r = {2b, 3c , 2d}, (2b)r = {3a, 3c}, (3c)r = {2b, 3a, 2d}, (2d)r = {3a, 3c} subbase sg = {{2b, 3c , 2d}, {3a, 3c}, {2b, 3a, 2d}, {3a, 3c} base βg = {x,φ, {2b, 3c , 2d}, {3a, 3c}, {3a, 2b, 2d}, {3a, 3c}, {3c}, {2b, 2d}, {3a} topology τg = {x,φ, {2b, 3c , 2d}, {3a, 3c}, {3a, 2b, 2d}, {3a, 3c}, {3c}, {2b, 2d}, {3a}, {2b, 3c , 2d} consider that g = (v ∗, e∗) is graph moreover h is induced subgraph for g. so, cl (v ∗ (h)) = v (h)u{x ∈ v∗ (g) ; xr ∩ v (h) 6= φ furthermore xr = { ( degg (ar )ar ) } ∀ r ∈ i and ar is set of every adjacent vertices vi .suppose that g = (v ∗, e∗) is graph. moreover h is induced subgraph for g and int (v ∗ (h)) = {x ∈ v ∗ (g) ; xr ⊆ v (h) , xr = { ( degg (ar )ar ) ∀ r ∈ i and ar adjacent for x. 4. some applications in this part we will give an example of blood circulation in lungs. we will also draw topologicalstructure of this circulation. we will relate mathematics with medical field. we will made graph ofit. moreover, we will discuss few reasons of disability in lungs and cause of dangerous diseases.we will explain these diseases mathematically. https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 7 here we will utilize our work discussed above in medical field. we will introduced the techniquein which connected graph is modifying condition in the medical field. diagram represent to graph.. https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 8we can notice the blood circulation in lungs is representation of set of vertices and edges. than, wecan define a topological structure τg on that. post classes for vertices in graph are the followinggiven below. (a1)r = {c3}, (b2)r = {c3}, (c3)r = {d4}, (d4)r = {f5}, (f5)r = {g6, h7}, (g6)r = {j9}, (h7)r = {i8}, (i8)r = {k10}, (j9)r = {k10}, (k10)r = {p11}, (p11)r = {q12}, (q12)r = {r13, s14}, (r13)r = {a1}, (s14)r = {b2}the subbase has a form sg = {{c3} , {d4} , {f5} , {g6, h7} , {j9} , {i8} , {k10} , {p11} , {q12} , {r13,s14} , {a1} , {b2}} base has a form βg = {x,φ, {c3}, {d4}, {f5}, {g6, h7}, {j9}, {i8}, {k10}, {p11}, {q12}, {r13, s14}, {a1}, {b2}} topology on a graph g have τg = {x,φ, {c3}, {d4}, {f5}, {g6, h7}, {j9}, {i8}, {k10}, {p11}, {q12}, {r13, s14}, {a1}, {b2}, {c3, d4}, {c3, f5}, {c3, g6, h7}, {c3, j9}, {c3, i8}, {c3, k10}, {c3, p11}, {c3, q12}, {c3, r13, s14}, {c3, a1}, {c3, b2}, {d4, f5}, {d4, g6, h7}, {d4, j9}, {d4, i8}, {d4, k10}, {d4, p11}, {d4, q12}, {d4, r13, s14}, {d4, a1}, {d4, b2}, {f5, g6, h7}, {f5, j9}, {i8}, {f5, k10}, {f5, p11}, {f5, q12}, {f5, r13, s14}, {f5, a1}, {f5, b2}, {g6, h7, j9}, {g6, h7, i8}, {g6, h7, k10}, {g6, h7, p11}, {g6, h7, q12}, {g6, h7, r13, s14}, {g6, h7, a1}, {g6, h7, b2}, {j9, i8}, {j9, k10}, {j9, p11}, {j9, q12}, {j9, r13, s14}, {j9, a1}, https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 9 {j9, b2}, {k10, p11}, {k10, q12}, {k10, r13, s14}, {k10, a1, b2}, {p11, q12}, {p11, r13, s14}, {p11, a1}, {p11, b2}, {q12, r13, s14}, {q12, a1, } {q12, b2}, {r13, s14, a1}, {r13, s14, b2}, {a1, b2} initially we get closure of graph. if h is any subgraph h = {b2, c3, e2, e3, e4} that is v (h) = {b2, c3}by definition of closure for subgraph h be cl (v (h)) = {b2, c3, d4} medically, here we will use that illustration for circulation of blood in lungs will be true. bloodflow in lungs by directed path to complete its cycle. but due to any fault flow of blood distributeand stop. it create serious diseases. moreover, we can find interior for graph over subgraph h = {f5, e5, g6, e7, h7} but from definition we can assume intv (h) = {f5, g6}in this example we note that end point does not include. this contradiction in heart but suitablefor lungs medically because due to some disorder people can also survive with only one lungs thisis gift of god. 5. some serious diseases in lungs there are some diseases in lungs due to some disorder. these diseases are divided in to somecategories. we will discuss reasons of these diseases and express them graphically. moreover, wewill also show topological structure. [3] https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 105.1. pulmonary arterial hypertension. heart problem autoimmune system can cause high bloodpressure in pulmonary arteries. (f5)r = {g6, h7}, (g6)r = {i8}, (h7)r = {j9} subbase sg = {{g6, h7}, {i8}, {j9}}base βg = {x,φ, {g6, h7}, {i8}, {j9}}topology τg = {x,φ, {g6, h7}, {i8}, {j9}, {g6, h7, i8}, {g6, h7, j9}, {h8, j9}} 5.2. pulmonary venous hypertension. any damage or self eating of mitral valve can cause higherblood pressure in pulmonary veins. (i8)r = {k10}, (j9)r = {k10}, (k10)r = {p11} subbase sg = {{k10}, {p11}}base βg = {x,φ, {k10}, {p11}} https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 11topology τg = {x,φ, {k10}, {p11}, {k10, p11}} 5.3. pulmonary embolism. any coagulation of blood or fat droplets can travel to lungs from heartand cause blockage of lungs blood vessels. (c3)r = {d4}, (d4)r = {f5}, (f5)r = {g6, h7}, (g6)r = {i8}, (h7)r = {j9}, (i8)r = {k10}, (j9)r = {k10} subbase sg = {{d4}, {f5}, {g6, h7}, {i8}, {j9}, {k10}} base βg = {x,φ, {d4}, {f5}, {g6, h7}, {i8}, {j9}, {k10}, } https://doi.org/10.28924/ada/ma.3.3 eur. j. math. anal. 10.28924/ada/ma.3.3 12topology τg = {x,φ, {d4}, {f5}, {g6, h7}, {i8}, {j9}, {k10}, {d4, f5}, {d4, g6, h7}, {d4, i8}, {d4, j9}, {d4, k10}, {f5, g6, h7}, {f5, i8}, {f5, j9}, {f5, k10}, {g6, h7, i8}, {g6, h7, j9}, {g6, h7, k10}, {i8, j9}, {i8, k10}, {j9, k10} 6. conclusion we derive topological structure by using different relations defined above. we also use differenttype of graphs. we also mentioned the method of general topology its graph and relationshipbetween both of them. we also represent medical field, blood circulation in lungs, its diseasesmade topologies by using graph. references [1] g. chartrand, l. lesniak, p. zhang, textbook in mathematics (graphs and diagraphs), sixth edition, taylor andfrancis, 2016.[2] m. shokry, r.e. aly, topological properties on graph vs medical application in human heart, int. j. appl. math. 15(2013) 1103-1109.[3] j. chen, j. li, an application of rough sets to graph theory, inform. sci. 201 (2012) 114–127. https://doi.org/ 10.1016/j.ins.2012.03.009.[4] m. shokray, y.y. yousif, closure operators on graph, aust. j. basic appl. sci. 5 (2011) 1856-1864.[5] g. birkhoff, lattic theory, amer math. soc. 1967.[6] z. bonikowski, a representation theorem for co-diagonalizable algebras, rep math logic, 38 (2004) 13-22.[7] d. dikranjan, w. tholen, catogrical structure of closure operator, mathematics and its application, kluwer academicpublisher, dordrecht, 1995.[8] c. kuratowski, topolgies, warsaw, 1952.[9] w. shi, k. liu, a fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitativelyin gis, computers geosci. 33 (2007) 898–915. https://doi.org/10.1016/j.cageo.2006.10.013.[10] c. largeron, s. bonnevay, a pretopological approach for structural analysis, inform. sci. 144 (2002) 169–185. https://doi.org/10.1016/s0020-0255(02)00189-5.[11] b.m.r. stadler, pf. stadler, generalized topological space in involuntarily and combinational chemistry, j. chem.inf. comput. sci. 42 (2002) 577-585.[12] a. galton, a generalized topological view of motion in discrete space, theor. computer sci. 305 (2003) 111-134.[13] s.a. morris, topology without tears, online e-book, 2017. https://www.topologywithouttears.net/topbook. pdf.[14] a. qayyum, m. shoaib, m.a. latif, a generalized inequality of ostrowski type for twice differentiable boundedmappings and applications, appl. math. sci. 8 (2014) 1889-1901. https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.1016/j.ins.2012.03.009 https://doi.org/10.1016/j.ins.2012.03.009 https://doi.org/10.1016/j.cageo.2006.10.013 https://doi.org/10.1016/s0020-0255(02)00189-5 https://www.topologywithouttears.net/topbook.pdf https://www.topologywithouttears.net/topbook.pdf 1. introduction and preliminaries 2. relation over graph 3. topological structure on graph 4. some applications 5. some serious diseases in lungs 5.1. pulmonary arterial hypertension 5.2. pulmonary venous hypertension 5.3. pulmonary embolism 6. conclusion references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 2doi: 10.28924/ada/ma.2.2 on geometric constants for discrete morrey spaces adam adam, hendra gunawan∗ analysis and geometry group, faculty of mathematics and natural sciences, bandung institute of technology, bandung 40132, indonesia adam_adam@students.itb.ac.id, hgunawan@math.itb.ac.id ∗correspondence: hgunawan@math.itb.ac.id abstract. in this paper we prove that the n-th von neumann-jordan constant and the n-th jamesconstant for discrete morrey spaces `pq where 1 ≤ p < q < ∞ are both equal to n. this resulttells us that the discrete morrey spaces are not uniformly non-`1, and hence they are not uniformly n-convex. 1. introduction let n ≥ 2 be a non-negative integer and (x, ‖ · ‖) be a banach space. the n-th von neumannjordan constant for x [6] is defined by c (n) nj (x) := sup {∑ ± ‖u1 ± u2 ± · · · ± un‖2x 2n−1 ∑n i=1 ‖ui‖x : ui 6= 0, i = 1, 2, . . . , n } and the n-th james constant for x [7] is defined by c (n) j (x) := sup{min ‖u1 ± u2 ± · · · ± un‖ : ui ∈ sx , i = 1, 2, . . . , n}.note that in the definition of c(n)nj (x), the sum ∑ ± is taken over all possible combinations of ±signs. similarly, in the definition of c(n)j (x), the minimum is taken over all possible combinationsof ± signs, while the supremum is taken over all ui ’s in the unit sphere sx := {u ∈ x : ‖u‖ = 1}.these constants measure some sort of convexity of a banach space.we say that x is uniformly n-convex [2] if for every ε ∈ (0, n] there exists a δ ∈ (0, 1) such thatfor every u1, u2, . . . , un ∈ sx with ‖u1 ± u2 ± · · · ± un‖ ≥ ε for all combinations of ± signs exceptfor ‖u1 + u2 + · · ·+ un‖, we have ‖u1 + u2 + · · ·+ un‖ ≤ n(1− δ). received: 31 aug 2021. key words and phrases. n-th von neumann-jordan constant; n-th james constant; discrete morrey spaces; uniformlynon-`1 spaces; uniformly n-convex spaces. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.2 https://orcid.org/0000-0001-7879-8321 eur. j. math. anal. 10.28924/ada/ma.2.2 2meanwhile, we say that x is uniformly non-`1n [1,5,8] if there exists a δ ∈ (0, 1) such that for every u1, u2, . . . , un ∈ sx we have min ‖u1 ± u2 ± · · · ± un‖ ≤ n(1− δ). note that for n = 2, uniformly non-`1n spaces are known as uniformly nonsquare spaces, while for n = 3 they are known as uniformly non-octahedral spaces. one may verify that if x is uniformly n-convex, then x is uniformly non-`1n [2].now a few remarks about the two constants, and their associations with the uniformly non-`1nand uniformly n-convex properties. • 1 ≤ c(n)nj (x) ≤ n and c(n)nj (x) = 1 if and only if x is a hilbert space [6]. • 1 ≤ c(n)j (x) ≤ n. if dim(x) = ∞, then √n ≤ c(n)j (x) ≤ n. moreover, if x is a hilbertspace, then c(n)j (x) = √n [7]. • x is uniformly non-`1n if and only if c(n)nj (x) < n [6]. • x is uniformly non-`1n if and only if c(n)j (x) < n [7]. the last two statements tell us that if c(n)nj (x) = n or c(n)j (x) = n, then x is not uniformly non-`1nand hence not uniformly n-convex.in this paper, we shall compute the value of the two constants for discrete morrey spaces. let ω := n ∪ {0} and m = (m1, m2, . . . , md) ∈ zd . define sm,n := {k ∈ zd : ‖k −m‖∞ ≤ n} where n ∈ ω and ‖m‖∞ = max{|mi | : 1 ≤ i ≤ d}. denote by |sm,n | the cardinality of sm,n for m ∈ zd and n ∈ ω. then we have |sm,n | = (2n + 1)d .now let 1 ≤ p ≤ q < ∞. define `pq = `pq(zd) to be the discrete morrey space as introducedin [3], which consists of all sequences x : zd → r with ‖x‖`pq := sup m∈zd ,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |xk |p ) 1 p <∞, where x := (xk) with k ∈ zd . one may observe that these discrete morrey spaces are banachspaces [3]. note, in particular, that for p = q, we have `pq = `q .from [4] we already know that cnj(`pq) = cj(` p q) = 2 for 1 ≤ p < q < ∞, which impliesthat `pq are not uniformly nonsquares for those p’s and q’s. in this paper, we shall show that c (n) nj (` p q) = c (n) j (` p q) = n for 1 ≤ p < q < ∞, which leads us to the conclusion that `pq arenot uniformly non-`1n for those p’s and q’s, which is sharper than the existing result. (if x is notuniformly non-`1n, then x is not uniformly non-`1n−1, provided that n ≥ 3.) https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 32. main results the value of the n-th von neumann-jordan constant and the n-th james constant for discretemorrey spaces are stated in the following theorems. to understand the idea of the proof, we firstpresent the result for n = 3. theorem 2.1. for 1 ≤ p < q <∞, we have c(3)nj (` p q(zd)) = c(3)j (` p q(zd)) = 3. proof. to prove the theorem, it suffices for us to find x (1), x (2), x (3) ∈ `pq such that∑ ± ‖x (1) ± x (2) ± x (3)‖2`pq 22 ∑3 i=1 ‖x (i)‖`pq = 3 for the von neumann-jordan constant, and min ‖x (1) ± x (2) ± x (3)‖`pq = 3 for the james constant. case 1: d = 1. let j ∈ z be a nonnegative, even integer such that j > 4 q q−p − 1, or equivalently (j + 1) 1 q − 1 p < 4− 1 p . construct x (1), x (2), x (3) ∈ `pq(z) as follows: • x (1) = (x (1)k )k∈z is defined by x (1) k = 1, k = 0, j, 2j, 3j, 0, otherwise; • x (2) = (x (2)k )k∈z is defined by x (2) k =  1, k = 0, j, −1, k = 2j, 3j, 0, otherwise; • x (3) = (x (3)k )k∈z is defined by x (3) k =  1, k = 0, 2j, −1, k = j, 3j, 0, otherwise. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 4the three sequences are in the unit sphere of `pq(z). indeed, for the first sequence, we have ‖x (1)‖`pq = sup m∈z,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k | p ) 1 p = sup m∈z∩[0,3j ],n∈z∩[0,3j/2] |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k | p ) 1 p = max{1, (j + 1) 1 q − 1 p 2 1 p , (2j + 1) 1 q − 1 p 3 1 p , (3j + 1) 1 q − 1 p 4 1 p }. since (3j + 1) 1q− 1p < (2j + 1) 1q− 1p < (j + 1) 1q− 1p < 4− 1p , we get ‖x (1)‖`pq = 1. similarly, one mayobserve that ‖x (2)‖`pq = ‖x (3)‖`pq = 1.next, we observe that x (1) k + x (2) k + x (3) k =  3, k = 0, 1, k = j, 2j, −1, k = 3j, 0, otherwise; x (1) k + x (2) k − x (3) k =  3, k = j, 1, k = 0, 3j, −1, k = 2j, 0, otherwise; x (1) k − x (2) k + x (3) k =  3, k = 2j, 1, k = 0, 3j, −1, k = j, 0, otherwise; x (1) k − x (2) k − x (3) k =  3, k = 3j, 1, k = j, 2j, −1, k = 0, 0, otherwise.we first compute that ‖x (1)+ x (2)+ x (3)‖`pq = max{3, (j +1) 1 q − 1 p (3p+1) 1 p , (2j +1) 1 q − 1 p (3p+2) 1 p , (3j +1) 1 q − 1 p (3p+3) 1 p }. notice that • (j + 1) 1 q − 1 p (3p + 1) 1 p < ( 3p+1p 4 ) 1 p < (3p) 1 p = 3. • (2j + 1) 1 q − 1 p (3p + 2) 1 p < (j + 1) 1 q − 1 p (3p + 2) 1 p < ( 3p+2 4 ) 1 p < 3. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 5 • (3j + 1) 1 q − 1 p (3p + 3) 1 p < (j + 1) 1 q − 1 p (3p + 3) 1 p < ( 3p+3 4 ) 1 p < 3.hence, we obtain ‖x (1) + x (2) + x (3)‖`pq = 3.similarly, we have ‖x (1) ± x (2) ± x (3)‖`pq = sup m∈z∩[0,3j ],n∈z∩[0,3j/2] |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k ± x (2) k ± x (3) k | p ) 1 p = 3 for every combination of ± signs.consequently, ∑± ‖x(1)±x(2)±x(3)‖2`pq 22 ∑3 i=1 ‖x(i)‖`pq = 3 and min ‖x (1) ± x (2) ± x (3)‖`pq = 3, so we come to theconclusion that c (3) nj (` p q(z)) = c (3) j (` p q(z)) = 3. case 2: d > 1. let j ∈ z be a nonnegative, even integer such that j > 4 q d(q−p) − 1, which isequivalent to (j + 1)d( 1 q − 1 p ) < 4− 1 p . we then construct x (1), x (2), x (3) ∈ `pq(zd) as follows: • x (1) = (x (1)k )k∈zd is defined by x (1) k = 1, k = (0, 0, . . . , 0), (j, 0, . . . , 0), (2j, 0, . . . , 0), (3j, 0, . . . , 0), 0, otherwise; • x (2) = (x (2)k )k∈zd is defined by x (2) k =  1, k = (0, 0, . . . , 0), (j, 0, . . . , 0), −1, k = (2j, 0, . . . , 0), (3j, 0, . . . , 0), 0, otherwise; • x (3) = (x (3)k )k∈zd is defined by x (3) k =  1, k = (0, 0, . . . , 0), (2j, 0, . . . , 0), −1, k = (j, 0, . . . , 0), (3j, 0, . . . , 0), 0, otherwise. as in the case where d = 1, one may observe that ‖x (1)‖`pq = sup m∈zd ,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k | p ) 1 p = max{1, (j + 1)d( 1 q − 1 p )2 1 p , (2j + 1)d( 1 q − 1 p )3 1 p , (3j + 1)d( 1 q − 1 p )4 1 p } = 1. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 6we also get ‖x (2)‖`pq = ‖x (3)‖`pq = 1. moreover, through similar observation as in the 1-dimensionalcase, we have ‖x (1) ± x (2) ± x (3)‖`pq = 3for every possible combinations of ± signs. it thus follows that c (3) j (` p q(zd)) = sup{min ‖x1 ± x2 ± x3‖`pq : x1, x2, x3 ∈ s`pq} = 3 and c (3) nj (` p q(zd)) = sup {∑ ± ‖x1 ± x2 ± x3‖2`pq 22 ∑3 i=1 ‖xi‖`pq : xi 6= 0, i = 1, 2, 3 } = 3. � we now state the general result for n ≥ 3. (the proof is also valid for n = 2, which amounts tothe work of [3].) theorem 2.2. for 1 ≤ p < q <∞, we have c(n)nj (` p q(zd)) = c(n)j (` p q(zd)) = n. proof. as for n = 3, we shall consider the case where d = 1 first, and then the case where d > 1later. case 1: d = 1. let j ∈ z be a nonnegative, even integer such that j > 2(n−1)( qq−p ) − 1, which isequivalent to (j + 1) 1 q − 1 p < 2− (n−1) p . we construct x (i) ∈ `pq ∈ z for i = 1, 2, . . . , n as follows: • x (1) = (x (1)k )k∈z is defined by x (1) k = 1, k ∈ s(1)1 , 0, otherwise, where s (1) 1 = {0, j, 2j, 3j, . . . , (2 n−1 − 1)j}; • x (i) = (x (i)k )k∈z for 2 ≤ i ≤ n is defined by x (i) k =  1, k ∈ s(i)1 , −1, k ∈ s(i)−1, 0, otherwise, with the following rules: write p = {0, j, 2j, . . . , (2n−1 − 1)j} as p = p (i) 1 ∪ p (i) 2 ∪ · · · ∪ p (i) 2i−1 https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 7 where p (i)1 consists of the first 2n−1 2i−1 terms of p , p (i)2 consists of the next 2n−1 2i−1 terms of p ,and so on. then s(i)1 and s(i)−1 are given by s (i) 1 = p (i) 1 ∪ p (i) 3 ∪ · · · ∪ p (i) 2i−1−1, s (i) −1 = p (i) 2 ∪ p (i) 4 ∪ · · · ∪ p (i) 2i−1 . for example, for i = 2, x (2) = (x (2)k )k∈z is defined by x (2) k =  1, k ∈ s(2)1 , −1, k ∈ s(2)−1 , 0, otherwise, where s (2) 1 = { 0, j, 2j, 3j, . . . , (2n−1 2 − 1 ) j } s (2) −1 = {(2n−1 2 ) j, (2n−1 2 + 1 ) j, . . . , (2n−1 − 1)j } ; note that the largest absolute value of the terms of x (i) in the above construction will beequal to 1 for each i = 1, . . . , n. next, since the number of possible combinations of ± signs in x (1) ± x (2) ± · · · ± x (n) is 2n−1, the above construction will give us 1 + 1 + · · · + 1 = n as thelargest absolute value of x (1)± x (2)±· · ·± x (n) for every combination of ± signs. this means that,if x (1) ± x (2) ± · · · ± x (n) = (xk)k∈z, then max k∈z |xk | = n.let us now compute the norms. for x (1), we have ‖x (1)‖`pq = sup m∈z,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k | p ) 1 p = sup m∈z∩[0,(2n−1−1)j ],n∈z∩[0,(2n−1−1)j/2] |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k | p ) 1 p = max{1, (j + 1) 1 q − 1 p 2 1 p , (2j + 1) 1 q − 1 p 3 1 p , . . . , ((2n−1 − 1)j + 1) 1 q − 1 p 2 n−1 p }. for each r = 1, 2, . . . , 2n−1 − 1, we have (r j + 1) 1q− 1p ≤ (j + 1) 1q− 1p and (r + 1) 1p ≤ 2 n−1p , so that (r j + 1) 1 q − 1 p (r + 1) 1 p ≤ (j + 1) 1 q − 1 p 2 n−1 p < 2− n−1 p 2 n−1 p = 1. hence we obtain ‖x (1)‖`pq = 1. similarly, one may verify that ‖x (2)‖`pq = ‖x (3)‖`pq = · · · = ‖x (n)‖`pq = 1. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 8next, we shall compute the norms of x (1)±x (2)±· · ·±x (n). write x (1)+x (2)+ · · ·+x (n) = (xk)k∈zwhere xk :=  a1, k = 0, a2, k = j, a3, k = 2j,... a2n−1 , k = (2n−1 − 1)j, 0, otherwise, with a1 = n and |ai | < n for i = 2, 3, . . . , (2n−1)j . accordingly, we have ‖x (1) + x (2) + · · ·+ x (n)‖`pq = sup m∈z,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |xk |p ) 1 p = sup m∈z∩[0,(2n−1−1)j ],n∈z∩[0,(2n−1−1)j/2] |sm,n | 1 q − 1 p ( ∑ k∈sm,n |xk |p ) 1 p =max { n, (j + 1) 1 q − 1 p (np + ap2) 1 p , (2j + 1) 1 q − 1 p (np + ap2 + a p 3) 1 p , . . . , ((2n−1 − 1)j + 1) 1 q − 1 p ( np + 2n−1∑ i=2 api ) 1 p } . since (r j + 1) 1q− 1p ≤ (j + 1) 1q− 1p for each r = 1, 2, . . . , 2n−1 − 1, we obtain (r j + 1) 1 q − 1 p ( np + r+1∑ i=2 api ) 1 p ≤ (j + 1) 1 q − 1 p ( np + r+1∑ i=2 api ) 1 p < 2− (n−1) p ( np + r+1∑ i=2 api ) 1 p < 2− (n−1) p (np + np + · · ·+ np︸ ︷︷ ︸ r + 1 times ) 1 p = 2− (n−1) p (r + 1) 1 p (np) 1 p ≤ 2− (n−1) p 2 (n−1) p n = n. it thus follows that ‖x (1) + x (2) + · · ·+ x (n)‖`pq = n.as we have remarked earlier, the largest absolute value of x (1) ± x (2) ± · · · ± x (n) is equal to n for every combination of ± signs. moreover, it is clear that for k /∈ {0, 2j, . . . , (2n−1 − 1)j}, the https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 9 k-th term of x (1) ± x (2) ± · · · ± x (n) is equal to 0. hence, we obtain ‖x (1) ± x (2) ± · · · ± x (n)‖`pq = sup m∈z,n∈ω |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k ± x (2) k ± · · · ± x (n) k | p ) 1 p = sup m∈z∩[0,(2n−1−1)j ],n∈z∩[0,(2n−1−1)j/2] |sm,n | 1 q − 1 p ( ∑ k∈sm,n |x (1)k ± x (2) k ± · · · ± x (n) k | p ) 1 p = n. consequently, we get ∑ ± ‖x (1) ± x (2) ± · · · ± x (n)‖2`pq 2n−1 ∑n i=1 ‖xi‖`pq = 2n−1n2 2n−1n = n and min ‖x (1) ± x (2) ± · · · ± x (n)‖`pq = n,whence c (n) nj (` p q(z)) = c (n) j (` p q(z)) = n. case 2: d > 1. here we choose j ∈ z to be a nonnegative, even integer such that j > 2( n−1 d )( q q−p ) − 1 or, equivalently, (j + 1)d( 1 q − 1 p ) < 2− (n−1) p . then, using the sequences x (i) = (x (i) k1 )k1∈z ∈ ` p q(z), i = 1, . . . , n, in the case where d = 1, we now define x (i) := (x (i)k )k∈zd ∈ `pq(zd) for i = 1, . . . , n, where x (i) k = x (i)k1 , k = (k1, 0, 0, . . . , 0), 0, otherwise. we shall then obtain c (n) nj (` p q(zd)) = c (n) j (` p q(zd)) = n,as desired. � corollary 2.2.1. for 1 ≤ p < q <∞, the space `pq is not uniformly non-`1n. corollary 2.2.2. for 1 ≤ p < q <∞, the space `pq is not uniformly n-convex. acknowledgement. the work is part of the first author’s thesis. both authors are supported byp2mi 2021 program of bandung institute of technology. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 10references [1] b. beauzamy, introduction to banach spaces and their geometry, 2nd ed., north holland, amsterdamnewyork-oxford, 1985. https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt= pascal82x0319279.[2] h. gunawan, d.i. hakim, a.s. putri, on geometric properties of morrey spaces, ufimsk. mat. zh. 13 (2021) 131–136. https://doi.org/10.13108/2021-13-1-131.[3] h. gunawan, e. kikianty, c. schwanke, discrete morrey spaces and their inclusion properties, math. nachr. 291(2018) 1283–1296. https://doi.org/10.1002/mana.201700054.[4] h. gunawan, e. kikianty, y. sawano, and c. schwanke, three geometric constants for morrey spaces, bull. korean.math. soc. 56 (2019) 1569-1575. https://doi.org/10.4134/bkms.b190010.[5] r.c. james, uniformly non-square banach spaces, ann. math. 80 (1964) 542-550. https://doi.org/10.2307/ 1970663.[6] m. kato, y. takahashi, and k. hashimoto, on n-th von neumann-jordan constants for banach spaces, bull. kyushuinst. tech. 45 (1998), 25-33. https://ci.nii.ac.jp/naid/110000079659.[7] l. maligranda, l. nikolova, l.-e. persson, t. zachariades, on n-th james and khintchine constants of banachspaces, math. inequal. appl. 1 (2007) 1–22. https://doi.org/10.7153/mia-11-01.[8] w.a. wojczynski, geometry and martingales in banach spaces, part ii, in: probability in banach spaces iv, j.kuelbs, ed., marcel-dekker, 1978, 267–517. https://doi.org/10.1201/9780429462153. https://doi.org/10.28924/ada/ma.2.2 https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt=pascal82x0319279 https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt=pascal82x0319279 https://doi.org/10.13108/2021-13-1-131 https://doi.org/10.1002/mana.201700054 https://doi.org/10.4134/bkms.b190010 https://doi.org/10.2307/1970663 https://doi.org/10.2307/1970663 https://ci.nii.ac.jp/naid/110000079659 https://doi.org/10.7153/mia-11-01 https://doi.org/10.1201/9780429462153 1. introduction 2. main results references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 106-132doi: 10.28924/ada/ma.1.106 new iterative algorithm for solving constrained convex minimization problem and split feasibility problem austine efut ofem1,∗ , unwana effiong udofia2, donatus ikechi igbokwe3 1department of mathematics, university of uyo, uyo, nigeria ofemaustine@gmail.com 2department of mathematics and statistics, akwa ibom state university, ikot akpaden, mkpatenin, nigeria unwanaudofia.aksu@yahoo.com 3department of mathematics, michael okpara university of agriculture, umudike, nigeria igbokwedi@yahoo.com ∗correspondence: ofemaustine@gmail.com abstract. the purpose of this paper is to introduce a new iterative algorithm to approximate the fixedpoints of almost contraction mappings and generalized α-nonexpansive mappings. also, we show thatour proposed iterative algorithm converges weakly and strongly to the fixed points of almost contrac-tion mappings and generalized α-nonexpansive mappings. furthermore, it is proved analytically thatour new iterative algorithm converges faster than one of the leading iterative algorithms in the liter-ature for almost contraction mappings. some numerical examples are also provided and used to showthat our new iterative algorithm has better rate of convergence than all of s, picard-s, thakur andm iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings.again, we show that the proposed iterative algorithm is stable with respect to t and data dependentfor almost contraction mappings. some applications of our main results and new iterative algorithmare considered. the results in this article are improvements, generalizations and extensions of severalrelevant results existing in the literature. 1. introduction fixed point theory is concerned with solution of the equation t` = `, (1.1) where t could be a nonlinear operator defined on a metric space. any ` that solves (1.1) is calledthe fixed point of t and the collection all such elements is denoted by f (t ). fixed point theory is received: 10 sep 2021. key words and phrases. stability; almost contraction map; generalized α-nonexpansive mapping; data dependence;iterative algorithm; constrained convex minimization problem; split feasibility problem.106 https://adac.ee https://doi.org/10.28924/ada/ma.1.106 https://orcid.org/0000-0001-8064-2326 eur. j. math. anal. 1 (2021) 107an area in nonlinear analysis that has become very attractive and interesting with a large numberof applications in various fields of mathematics and other branches of science. fixed point theoryhas remained not only a field with a huge development, but also a very helpful means for solvingvarious problems in different fields of mathematics. it is well known that fixed point theorems areused for proving the existence and uniqueness to various mathematical models like differential,integral and partial differential equations and variational inequalities problems etc., representingphenomena arising in different fields such as steady state temperature distribution, chemical equa-tions, neutron transport theory, economic theories, epidemics and flow of fluids. furthermore, itas also significant in the field of computer science, image processing, artificial intelligence, deci-sion making, population dynamics, computer science, operational research, industrial engineering,pattern recognition, medicine, group health underwriting, management and many others.existence theorem is concerned with establishing sufficient conditions in which the equation (1.1)will have solution, but does not necessarily show how to find such solution. on the other hand,iteration method of fixed points is concerned with approximation or computation of sequences whichconverge to the solution of (1.1). when existence of a fixed point of an operator is guaranteed,obtaining constructive technique for finding such a fixed point is also paramount.in 2003, berinde [6] introduced the concept of weak contraction mappings which is also knownas almost contraction mappings. he showed that the class of almost contraction mappings is moregeneral than the class of zamfirescu mappings [41] which includes contraction mappings, kannanmappings [22] and chatterjea mappings [10].throughout this paper, let ω denote a banach space and λ a nonempty closed convex subset of ω. let r stand for set of real numbers. definition 1.1. a mapping t : λ → λ is called almost contraction if there exists a constant γ ∈ (0, 1) and some constant l ≥ 0, such that ‖t`− tζ‖ ≤ γ‖`− ζ‖+ l‖`− t`‖, ∀ `, ζ ∈ λ. (1.2) definition 1.2. a mapping t : λ → λ is said to be suzuki generalized nonexpansive if for all `, ζ ∈ λ, we have 1 2 ‖`− t`‖ ≤ ‖`− ζ‖ =⇒ ‖t`− tζ‖ ≤ ‖`− ζ‖. suzuki generalized nonexpansive mappings is also known as mappings satisfying condition (c).in [33], suzuki showed that the class of suzuki generalized nonexpansive mappings is more generalthan the class of nonexpansive mappings and obtained some fixed points and convergence theorems. definition 1.3. a mapping t : λ→ λ is said to be α-nonexpansive if there exists α ∈ [0, 1) suchthat ‖t`− tζ‖2 ≤ α‖t`− ζ‖2 + α‖`− tζ‖2 + (1− 2α)‖`− ζ‖2, eur. j. math. anal. 1 (2021) 108for all `, ζ ∈ λ. the class of α-nonexpansive mappings was introduced in 2011 by aoyama and kohsaka [3]as generalization of nonexpansive mappings and further obtained some convergence results. itis worthy noting that nonexpansive mappings are continuous on their domains, but suzuki-typegeneralized nonexpansive mappings and α-nonexpansive mappings need not be continuous (see[33]). clearly, every nonexpansive mapping is an α-nonexpansive mapping with α = 0 (i.e., 0-nonexpansive) and every α-nonexpansive mapping with a nonempty fixed point set is quasinonex-pansive. definition 1.4. a mapping t : λ → λ is said to be generalized α-nonexpansive if there exists α ∈ [0, 1) such that 1 2 ‖`− t`‖ ≤ ‖`− ζ‖ implies ‖t`− tζ‖ ≤ α‖t`− ζ‖+ α‖tζ − `‖+ (1− 2α)‖`− ζ‖ for all `, ζ ∈ λ. in [26], pant and shukla introduced a wider class of nonexpansive mappings in banach spacesknown as generalized α-nonexpansive mappings which contains the class of suzuki generalizednonexpansive mappings.it is well known that the case of contraction mappings is simple and carries most of the goodbehavior using picard iterative algorithm. but when we move to the case of nonexpansive mappings,the picard iterative algorithm need not converge to a fixed point. apparently, the conclusion ofbanach contraction principle fails for nonexpansive mappings even if λ is compact. as an example,one may consider a geometric rotation on the unit circle in the plane r2.the limitation of picard iterative algorithm gave many researchers in nonlinear analysis the roomto construct more efficient iterative algorithms for approximating the fixed points of nonexpansivemappings and other classes of mappings which are more general than the class of nonexpansivemappings.some notable iterative algorithms in the existing literature are: mann [24], ishikawa [21], noor[25], argawal et al. [2], abbas and nazir [1], sp [27], s* [20], cr [12], normal-s [28], picard-s [17],thakur [36], thakur new [37], m [39], m* [38], garodia and uddin [16], two-step mann [35] iterativealgorithms and many others.in 2007, the s iterative algorithm was introduced by argawal et al. [2] as follows: ψ0 ∈ λ, µs = (1− βs)ψs + βstψs , ψs+1 = (1− δs)tψs + δstµs , ∀s ≥ 1, (1.3) where {δs} and {βs} are sequences in [0,1]. eur. j. math. anal. 1 (2021) 109in 2014, the picard-s iterative algorithm was introduced by gursoy and karakaya [17] as follows: u0 ∈ λ, ϕs = (1− βs)us + βstus , %s = (1− δs)tus + δstϕs , us+1 = t%s , ∀s ≥ 1, (1.4) where {δs} and {βs} are sequences in [0,1]. the authors showed with the aid of an example thatpicard-s iterative algorithm (1.4) converges at a rate faster than all of picard, mann, ishikawa,noor, sp, cr, s, s*, abbas and nazir, normal-s and two-step mann iterative algorithms forcontraction mappings.in 2016, thakur et al. [37] introduced the following three steps iterative algorithm: ω0 ∈ λ, ρs = (1− βs)ωs + βstωs , vs = t ((1− δs)ωs + δsρs), ωs+1 = tvs , ∀s ≥ 1, (1.5) where {δs} and {βs} are sequences in [0,1]. with the help of numerical example, they proved that(1.5) is faster than picard, mann, ishikawa, agarwal, noor and abbas iterative algorithm for suzukigeneralized nonexpansive mappings.in 2018, ullah and arshad [39] introduced m iterative algorithm as follows: m0 ∈ λ, cs = (1− δs)ms + δstms , ds = tcs , ms+1 = tds , ∀s ≥ 1, (1.6) where {δs} is a sequence in [0,1]. numerically they showed that m iterative algorithm (1.2)converges faster than s iterative algorithm (1.3) and picard-s iterative algorithm (1.4) for suzukigeneralized nonexpansive mappings. also, they noted that the speed of convergence of picard-siterative algorithm (1.4) and thakur iterative algorithm (1.5) are almost same.motivated by the above results, in this paper, we construct a new four step iterative algorithmwhich outperforms the iterative algorithm (1.6) in terms of convergence rate for almost contractionmappings as follows:  `0 ∈ λ, gs = (1− βs)`s + βst`s , ws = (1− δs)t`s + δstgs , ζs = tws , `s+1 = tζs , ∀s ≥ 1, (1.7) where {δs} and {βs} are sequences in [0,1]. eur. j. math. anal. 1 (2021) 110the purpose of this paper is to prove analytically that our new iterative algorithm convergesfaster than (1.6) for almost contraction mappings. in order to support our analytical proof, weuse some new examples to show that our iterative algorithm (1.7) converges faster than (1.6) anda number of other leading iterative algorithms in the literature. we also prove the weak andstrong convergence of new iterative algorithm (1.7) to the fixed points generalized α-nonexpansivemappings in a uniformly convex banach spaces. furthermore, we show that our new iterativealgorithm is t -stable and data dependent. finally, we use our new iterative algorithm (1.7) tosolve a constrained convex minimization problem and a split feasibility problem. 2. preliminaries the following definitions, propositions and lemmas will be useful in proving our main results. definition 2.1. a banach space ω is said to be uniformly convex if for each ε ∈ (0, 2], there exists δ > 0 such that for `, ζ ∈ ω satisfying ‖`‖ ≤ 1, ‖ζ‖ ≤ 1 and ‖`− ζ‖ > ε, we have ∥∥∥ `+ζ2 ∥∥∥ < 1− δ. definition 2.2. a banach space ω is said to satisfy opial’s condition if for any sequence {`s} in ω which converges weakly to ` ∈ ω implies lim sup s→∞ ‖`s − `‖ < lim sup s→∞ ‖`s − ζ‖, ∀ ζ ∈ ω with ζ 6= `. definition 2.3. let {`s} be a bounded sequence in ω. for ` ∈ λ ⊂ ω, we put r(`, {`s}) = lim sup s→∞ ‖`s − `‖. the asymptotic radius of {`s} relative to λ is defined by r(λ, {`s}) = inf{r(`, {`s}) : ` ∈ λ}. the asymptotic center of {`s} relative to λ is given as: a(λ, {`s}) = {` ∈ λ : r(`, {`s}) = r(λ, {`s})}. in a uniformly convex banach space, it is well known that a(λ, {`s}) consist of exactly one point. definition 2.4. [5] let {as} and {bs} be two sequences of real numbers that converge to a and brespectively, and assume that there exists k = lim s→∞ ‖as − a‖ ‖bs − b‖ . then,(r1) if k = 0, we say that {as} converges faster to a than {bs} does to b.(r2) if 0 < k <∞, we say that {as} and {bs} have the same rate of convergence. eur. j. math. anal. 1 (2021) 111 definition 2.5. [5] let {ηs} and {φs} be two fixed point iteration processes that converge to thesame point z , the error estimates ‖ηs − z‖ ≤ as , ∀ s ≥ 1, ‖φs − z‖ ≤ bs , ∀ s ≥ 1, are available where {as} and {bs} are two sequences of positive numbers converging to zero. thenwe say that {ηs} converges faster to z than {φs} does if {as} converges faster than {bs}. definition 2.6. [5] let t , t̃ : λ→ λ be two operators. we say that t̃ is an approximate operatorfor t if for some ε > 0, we have ‖t`− t̃ `‖ ≤ ε, ∀ ` ∈ λ. definition 2.7. [18] let {ys} be any sequence in λ. then, an iteration process `s+1 = f (t, ys),which converges to fixed point z , is said to be stable with respect to t , if for εs = ‖ys+1−f (t, ys)‖, ∀ s ∈ n, we have lim s→∞ εs = 0⇔ lim s→∞ ys = z. definition 2.8. [31] a mapping t : λ → λ is said to satisfy condition (i) if a nondecreasingfunction f : [0,∞) → [0,∞) exists with f (0) = 0 and for all r > 0 then f (r) > 0 such that ‖`− t`‖ ≥ f (d(`, f (t )))) for all ` ∈ λ, where d(`, f (t )) = infz∈f (t ) ‖`− z‖. proposition 2.9. [26] let λ be a nonempty subset of a banach space ω. suppose t : λ → λ is any mapping. then(i) if t is a suzuki generalized nonexpansive mapping, it follows that t is a generalized α-nonexpansive mapping.(ii) every generalized α-nonexpansive mapping with a nonempty fixed point set is quasinonexpansive mapping.(ii) if t is a generalized α-nonexpansive mapping, then f (t ) is closed. moreover, if ω is strictly convex and λ is convex, then f (t ) is also convex.(iv) if t is a generalized α-nonexpansive mapping, then the following inequality holds: ‖`− tζ‖ ≤ ( 3 + α 1− α ) ‖`− t`‖+ ‖`− ζ‖, ∀ `, ζ ∈ λ. lemma 2.10. [26] let t be a self mapping on a subset λ of a banach space ω which satisfies opial’s condition. suppose t is a generalized α-nonexpansive mapping. if {`s} converges weakly to z and lim s→∞ ‖t`s − `s‖ = 0, then tz = z . that is, i − t is demiclosed at zero. lemma 2.11. [33] let t be a self mapping on a weakly compact convex subset λ of a banach space ω with the opial’s property. if t is a suzuki generalized nonexpansive mapping, then t has a fixed point. eur. j. math. anal. 1 (2021) 112 lemma 2.12. [40] let {‘s} and {λs} be nonnegative real sequences satisfying the following inequalities: ‘s+1 ≤ (1− σs)‘s + λs , where σs ∈ (0, 1) for all s ∈ n, ∞∑ s=0 σs =∞ and lim s→∞ s σs = 0, then lim s→∞ ‘s = 0. lemma 2.13. [32] let {‘s} be a nonnegative real sequence and there exits an s0 ∈ n such that for all s ≥ s0 satisfying the following condition: ‘s+1 ≤ (1− σs)‘s + σsλs , where σs ∈ (0, 1) for all s ∈ n, ∞∑ s=0 σs =∞ and λs ≥ 0 for all s ∈ n, then 0 ≤ lim sup s→∞ ‘s ≤ lim sup s→∞ λs . lemma 2.14. [29] suppose ω is a uniformly convex banach space and {ιs} is any sequence satisfying 0 < p ≤ ιs ≤ q < 1 for all s ≥ 1. suppose {`s} and {ζs} are any sequences of ω such that lim sup s→∞ ‖`s‖ ≤ x , lim sup s→∞ ‖ζs‖ ≤ x and lim sup s→∞ ‖ιs`s + (1 − ιs)ζs‖ = x hold for some x ≥ 0. then lim s→∞ ‖`s − ζs‖ = 0. 3. rate of convergence in this section, we will prove that our new iterative algorithm (1.7) converges faster than theiterative algorithm (1.6) for almost contraction mappings. theorem 3.1. let ω be a banach space and let λ be a nonempty closed convex subset of ω. let t : λ→ λ be a mapping satisfying (1.2) with f (t ) 6= ∅. let {`s} be the iterative algorithm defined by (1.7) with sequences {δs}, {βs} ∈ [0, 1] such that ∞∑ s=0 δsβs = ∞, then {`s} converges strongly to a unique fixed point of t . proof. let z ∈ f (t ) and from (1.7), we have get ‖gs − z‖ = ‖(1− βs)`s + βst`s − z‖ ≤ (1− βs)‖`s − z‖+ βs‖t`s − z‖ ≤ (1− βs)‖`s − z‖+ βsγ‖`s − z‖ = (1− (1− γ)βs)‖`s − z‖. (3.1) eur. j. math. anal. 1 (2021) 113using (1.7) and (3.1), we have ‖ws − z‖ = ‖(1− δs)t`s + δstgs − z‖ ≤ (1− δs)‖t`s − z‖+ δs‖tgs − z‖ ≤ γ(1− δs)‖`s − z‖+ γδs‖gs − z‖ ≤ γ(1− δs)‖`s − z‖+ γδs(1− (1− γ)βs)‖`s − z‖ = γ(1− (1− γ)δsβs)‖`s − z‖. (3.2) from (1.7) and (3.2), we obtain ‖ζs − z‖ = ‖tws − z‖ ≤ γ‖ws − z‖ ≤ γ2(1− (1− γ)δsβs)‖`s − z‖. (3.3) using (1.7) and (3.3), we have ‖`s+1 − z‖ = ‖tζs − z‖ ≤ γ‖ζs − z‖ ≤ γ3(1− (1− γ)δsβs)‖`s − z‖. (3.4) from (3.4), we have the following inequalities: ‖`s+1 − z‖ ≤ γ3(1− (1− γ)δsβs)‖`s − z‖ ≤ γ3(1− (1− γ)δs−1βs−1)‖`s−1 − z‖... ‖`1 − z‖ ≤ γ3(1− (1− γ)δ0β0)‖`0 − z‖. (3.5) from (3.5), we get ‖`s+1 − z‖ ≤ ‖`0 − z‖γ3(s+1) s∏ t=0 (1− (1− γ)δtβt). (3.6) since γ ∈ (0, 1), δt , βt ∈ [0, 1] for all t ∈ n, it follows that (1− (1− γ)δtβt) ∈ (0, 1). since fromclassical analysis we know that 1− ` ≤ e−` for all ` ∈ [0, 1], thus from (3.6), we have ‖`s+1 − z‖ ≤ γ3(s+1)‖`0 − z‖ e (1−γ) s∑ t=0 δtβt . (3.7) if we take the limits of both sides of (3.7), we get lim s→∞ ‖`s − z‖ = 0. � eur. j. math. anal. 1 (2021) 114 theorem 3.2. let ω be a banach space and let λ be a nonempty closed convex subset of ω. let t : λ → λ be a mapping satisfying (1.2) with f (t ) 6= ∅. for given `0 = m0 ∈ λ, let {`s} and {ms} be the iterative algorithms defined by (1.7) and (1.6), respectively, with real sequences {δs} and {βs} in [0,1] such that δs ≤ δ < 1 and βs ≤ β < 1, for all s ∈ n and for some δ, β > 0. then {`s} converges to z faster than {ms} does. proof. from (3.6) in theorem 3.1 together with the assumptions αs ≤ α < 1 and βs ≤ β < 1, forall s ∈ n and for some α, β > 0, then we have ‖`s+1 − z‖ ≤ ‖`0 − z‖γ3(s+1) s∏ t=0 (1− (1− γ)αtβt) = ‖`0 − z‖γ3(s+1)(1− (1− γ)αβ)s+1. (3.8) similarly, from (1.6), we get ‖cs − z‖ = ‖(1− δs)ms + δstms − z‖ ≤ (1− δs)‖ms − z‖+ δs‖tms − z‖ ≤ (1− δs)‖ms − z‖+ δsγ‖mn − z‖ = (1− (1− γ)δs)‖ms − z‖. (3.9) using (1.6) and (3.9), we get ‖ds − z‖ = ‖tcs − z‖ ≤ γ‖cs − z‖ ≤ γ(1− (1− γ)δs)‖ms − z‖. (3.10) finally, from (1.6) and (3.10), we obtain ‖ms+1 − z‖ = ‖tds − z‖ ≤ γ‖ds − z‖ ≤ γ2(1− (1− γ)δs)‖ms − z‖. (3.11) from (3.11), we have the following inequalities: ‖ms+1 − z‖ ≤ γ2(1− (1− γ)δs)‖ms − z‖ ≤ γ2(1− (1− γ)δs−1)‖ms−1 − z‖... ‖m1 − z‖ ≤ γ2(1− (1− γ)δ0)‖m0 − z‖. (3.12) eur. j. math. anal. 1 (2021) 115from (3.12), we get ‖ms+1 − z‖ ≤ ‖m0 − z‖γ2(s+1) s∏ t=0 (1− (1− γ)δt). since δs ≤ δ < 1 and βs ≤ β < 1, for all s ∈ n and for some δ, β > 0, then we have ‖ms+1 − z‖ ≤ ‖m0 − z‖γ2(s+1) s∏ t=0 (1− (1− γ)δt) = ‖m0 − z‖γ2(s+1)(1− (1− γ)δ)s+1. set as = ‖`0 − z‖γ3(s+1)(1− (1− γ)δ)s+1, and bs = ‖`0 − z‖γ2(s+1)(1− (1− γ)δ)s+1. (3.13) hence, as bs = ‖`0 − z‖γ3(s+1)(1− (1− γ)δβ)s+1 ‖m0 − z‖γ2(s+1)(1− (1− γ)δ)s+1 → 0 as s →∞. this implies that our new iterative algorithm (1.7) converges faster to z than m iterative algorithm(1.6). � in order to support analytical prove in theorem 3.2 and demonstrate the advantage of our newiterative algorithm (1.7), we give the following example. example 3.3. let ω = < and λ = [1, 50]. let t : λ → λ be a mapping defined by t (`) = √ `2 − 8`+ 40. obviously, 5 is the fixed point of t . take δs = βs = 3 4 , with an initial value of `1 = 50. by writing all the codes in matlab (r2015a) for example 3.3, we obtain the following com-parison table 1 and figure 1. eur. j. math. anal. 1 (2021) 116 table 1. comparison of convergence behaviour of our new iterative algorithm withs, picard-s, thakur and m iterative algorithms.step s picard-s thakur m new1 50.00000000 50.00000000 50.00000000 50.00000000 50.000000002 44.16905011 40.46668490 40.46648707 39.77487312 36.794280913 38.40054569 31.13624438 31.13566491 29.79220887 24.079581494 32.71513008 22.15533283 22.15389446 20.25245189 12.593214715 27.14503094 13.88761070 13.88380778 11.71208997 5.609365616 21.74399379 7.46589475 7.45557218 6.06597569 5.003558697 16.60935306 5.14776230 5.14203305 5.02641919 5.000015698 11.93484164 5.00348330 5.00331403 5.00042732 5.000000009 8.12786414 5.00007676 5.00007301 5.00000684 5.0000000010 5.84725921 5.00000169 5.00000161 5.00000011 5.0000000011 5.12789697 5.00000004 5.00000004 5.00000000 5.0000000012 5.01483168 5.00000000 5.00000000 5.00000000 5.0000000013 5.00164168 5.00000000 5.00000000 5.00000000 5.00000000 iteration number s 2 4 6 8 10 12 14 s eq ue nc e va lu es 5 10 15 20 25 30 35 40 45 50 new iteration m iteration thakur iteration picard-s iteration s iteration figure 1. graph corresponding to table 1. eur. j. math. anal. 1 (2021) 1174. convergence results in this section, we will prove the weak and strong convergence of our new iterative algorithm (1.7)for generalized α–nonexpansive mappings in the framework of uniformly convex banach spaces.firstly, we will state and prove the following lemmas which will be useful in obtaining our mainresults. lemma 4.1. let ω be a banach space and λ be a nonempty closed convex subset of ω. let t : λ → λ be a generalized α–nonexpansive mapping with f (t ) 6= ∅. if {`s} is the iterative algorithm defined by (1.7), then lim s→∞ ‖`s − z‖ exists for all z ∈ f (t ). proof. let z ∈ f (t ). by proposition 2.9(ii), we know that every suzuki generalized nonexpansivemapping with f (t ) 6= ∅ is quasi-nonexpansive mapping. then, from (1.7), we have ‖gs − z‖ = ‖(1− βs)`s + βst`s − z‖ ≤ (1− βs)‖`s − z‖+ βs‖t`s − z‖ ≤ (1− βs)‖`s − z‖+ βs‖`s − z‖ = ‖`s − z‖. (4.1) using (1.7) and (4.1), we obtain ‖ws − z‖ = ‖(1− δs)t`s + δstgs − z‖ ≤ (1− δs)‖t`s − z‖+ δs‖tgs − z‖ ≤ (1− δs)‖`s − z‖+ δs‖gs − z‖ ≤ (1− δs)‖`s − z‖+ δs‖`s − z‖ = ‖`s − z‖. (4.2) again, using (1.7) and (4.2), we get ‖ζs − z‖ = ‖tws − z‖ ≤ ‖ws − z‖ ≤ ‖`s − z‖. (4.3) lastly, from (1.7) and (4.3), we have ‖`s − z‖ = ‖tζs − z‖ ≤ ‖ζs − z‖ ≤ ‖`s − z‖. (4.4) this implies that {‖`s − z‖} is bounded and nondecreasing for all z ∈ f (t ). hence, lim s→∞ ‖`s − z‖exists. � eur. j. math. anal. 1 (2021) 118 lemma 4.2. let ω be a uniformly convex banach space and λ be a nonempty closed convex subset of ω. let t : λ → λ be a generalized α–nonexpansive mapping. suppose {`s} is the iterative algorithm defined by (1.7). then, f (t ) 6= ∅ if and only if {`s} is bounded and lim s→∞ ‖t`s− `s‖ = 0. proof. suppose f (t ) 6= ∅ and let z ∈ f (t ). then, by lemma 4.1, lim s→∞ ‖`s − z‖ exists and {`s} isbounded. put lim s→∞ ‖`s − z‖ = x. (4.5) from (4.4) and (4.5), we obtain lim sup s→∞ ‖gs − z‖ ≤ lim sup s→∞ ‖`s − z‖ = x. (4.6) from proposition 2.9(ii), we know that every generalized α–nonexpansive mapping with f (t ) 6= ∅is quasi-nonexpansive mapping. so that we have lim sup s→∞ ‖t`s − z‖ ≤ lim sup s→∞ ‖`s − z‖ = x. (4.7) again, using (1.7), we get ‖`s+1 − z‖ = ‖tζs − z‖ ≤ ‖ζs − z‖ = ‖tws − z‖ ≤ ‖ws − z‖ = ‖(1− δs)t`s + δstgs − z‖ ≤ (1− δs)‖t`s − z‖+ δs‖tgs − z‖ ≤ (1− δs)‖`s − z‖+ δs‖gs − z‖ = ‖`s − z‖ − δs‖`s − z‖+ δs‖gs − z‖. (4.8) from (4.8), we have ‖`s+1 − z‖ − ‖`s − z‖ δs ≤ ‖gs − z‖ − ‖`s − z‖. (4.9) since δs ∈ [0, 1], then from (4.9), we have ‖`s+1 − z‖ − ‖`s − z‖ ≤ ‖`s+1 − z‖ − ‖`s − z‖ δs ≤ ‖gs − z‖ − ‖`s − z‖, which implies that ‖`s+1 − z‖ ≤ ‖gs − z‖. therefore, from (4.5), we obtain x ≤ lim inf s→∞ ‖gs − z‖. (4.10) eur. j. math. anal. 1 (2021) 119from (4.6) and (4.10) we obtain x = lim s→∞ ‖gn − z‖ = lim s→∞ ‖(1− βs)`s + βst`s − z‖ = lim s→∞ ‖(1− βs)(`s − z) + βs(t`s − z)‖ = lim s→∞ ‖βs(t`s − z) + (1− βs)(`s − z)‖. (4.11) from (4.5), (4.7), (4.11) and lemma 2.14, we obtain lim s→∞ ‖t`s − `s‖ = 0. (4.12) conversely, assume that {`s} is bounded and lim s→∞ ‖t`s−`s‖ = 0. let z ∈ a(λ, {`s}), by definition2.3 and proposition 2.9(iv), we have (tz, {`s}) = lim sup s→∞ ‖`s − tz‖ ≤ lim sup s→∞ ( (3 + α) (1− α) ‖t`s − `s‖+ ‖`s − z‖ ) = lim sup s→∞ ‖`s − z‖ = r(z, {`s}). (4.13) this implies that z ∈ a(λ, {`s}). since ω is uniformly convex, a(λ, {`s}) is singleton, thus wehave tz = z . � theorem 4.3. let ω, λ, t be same as in lemma 4.2. suppose tat ω satisfies opial’s condition and f (t ) 6= ∅. then, the sequence {`s} defined by (1.7) converges weakly to a fixed point of t . proof. let z ∈ f (t ), then by lemma 4.1, we have lim s→∞ ‖`s − z‖ exists. now we show that {`s}has weak sequential limit in f (t ). let ` and ζ be weak limits of the subsequences {`sj} and {`sk}of {`s}, respectively. by lemma 4.2, we have lim s→∞ ‖t`s − `s‖ = 0 and from lemma 2.10, i − t isdemiclosed at zero. it follows that (i − t )` = 0 implies ` = t`, similarly tζ = ζ.next we show uniqueness. suppose ` 6= ζ, then by opial’s property, we obtain lim s→∞ ‖`s − `‖ = lim sj→∞ ‖`sj − `‖ < lim sj→∞ ‖`sj − ζ‖ = lim s→∞ ‖`s − ζ‖ = lim sk→∞ ‖`sk − ζ‖ < lim sk→∞ ‖`sk − `‖ = lim s→∞ ‖`s − `‖, (4.14) which is a contradiction, so ` = ζ. hence, {`s} converges weakly to a fixed point of t . � eur. j. math. anal. 1 (2021) 120 theorem 4.4. let ω, λ, t be same as in lemma 4.2. then, the iterative algorithm {`s} defined by (1.7) converges strongly to a point of f (t ) if and only if lim inf s→∞ d(`s , f (t )) = 0, where d(`s , f (t )) = inf{‖`− z‖ : z ∈ f (t )}. proof. necessity is obvious. assume that lim inf s→∞ d(`s , f (t )) = 0. from lemma 4.1, we have lim s→∞ ‖`s − z‖ exists for all z ∈ f (t ), it follows that lim inf s→∞ d(`s , f (t )) exists. but by hypothesis, lim inf s→∞ d(`s , f (t )) = 0, thus lim s→∞ d(`s , f (t )) = 0. next we prove that {`s} is a cauchy sequencein λ. since lim inf s→∞ d(`s , f (t )) = 0, then given ε > 0, there exists s0 ∈ n such that, for all s, n ≥ s0,we have d(`s , f (t )) ≤ ε 2 , d(`n, f (t )) ≤ ε 2 . thus, we have ‖`s − `n‖ ≤ ‖`s − z‖+ ‖`n − z‖ ≤ d(`s , f (t )) + d(`n, f (t )) ≤ ε 2 + ε 2 = ε. hence {`s} is a cauchy sequence in λ. since λ is closed, therefore there exists a point `1 ∈ λsuch that lim s→∞ `s = `1. since lim s→∞ d(`s , f (t )) = 0, it implies that lim s→∞ d(`1, f (t )) = 0. hence, `1 ∈ f (t ) since f (t ) closed. � theorem 4.5. let ω, λ, t be same as in lemma 4.2. if t satisfies condition (i), then the iterative algorithm {`s} defined by (1.7) converges strongly to a fixed point of t . proof. we have shown in lemma 4.2 that lim s→∞ ‖t`s − `s‖ = 0. (4.15) using condition (i) in definition 2.8 and (4.15), we get lim s→∞ f (d(`s , f (t ))) ≤ lim s→∞ ‖t`s − `s‖ = 0, (4.16) i.e., lim s→∞ f (d(`s , f (t ))) = 0. since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f (0) = 0, f (r) > 0 for all r ∈ (0,∞), we have lim s→∞ d(`s , f (t )) = 0. (4.17) from theorem 4.4, then sequence {`s} converges strongly to a point of f (t ). � eur. j. math. anal. 1 (2021) 1215. numerical result in this section, we provide an example of generalized α-nonexpansive mapping which is notsuzuki generalized nonexpansive mapping. with the aid of the provided example, we will provethat our new iterative algorithm (1.7) outperforms a number of iterative algorithms in the existingliterature in terms of convergence. example 5.1. let λ = [0,∞) be endowed with the usual norm | · | and let t : λ → λ be definedas: t` = { 0, if ` ∈ [0, 15), 3` 4 , if ` ∈ [15 ,∞). (5.1) firstly, we show that t does not satisfy condition (c). to see this, let ` = 1 15 and ζ = 1 5 , then 1 2 |`− t`| = 1 30 < 2 15 = |`− ζ|. but |t`− tζ| = 3ζ 4 = 3 20 > 2 15 = |`− ζ|. hence, t does not satisfy condition (c), which implies that t is not a suzuki generalized nonex-pansive mapping.now we show that t is a generalized α-nonexpansive mapping with α = 1 3 (i.e., generalized 1 3-nonexpansive). we consider the following cases: case (a): when `, ζ ∈ [0, 15), we have 1 3 |t`− ζ|+ 1 3 |`− tζ|+ 1 3 |`− ζ| ≥ 0 = |t`− tζ|. case (b): when `, ζ ∈ [15 ,∞), we obtain 1 3 |t`− ζ|+ 1 3 |`− tζ|+ 1 3 |`− ζ| = 1 3 ∣∣∣∣3`4 − ζ ∣∣∣∣+ 1 3 ∣∣∣∣`− 3ζ 4 ∣∣∣∣+ 1 3 |`− ζ| ≥ 1 3 ∣∣∣∣(3` 4 − ζ ) + ( `− 3ζ 4 )∣∣∣∣+ 1 3 |`− ζ| = 7 12 |`− ζ|+ 1 3 |`− ζ| = 11 12 |`− ζ| ≥ 3 4 |`− ζ| = |t`− tζ|. eur. j. math. anal. 1 (2021) 122 case (c): when ` ∈ [15 ,∞) and ζ ∈ [0, 15), we get 1 3 |t`− ζ|+ 1 3 |`− tζ|+ 1 3 |`− ζ| = 1 3 ∣∣∣∣3`4 − ζ ∣∣∣∣+ 1 3 |`|+ 1 3 |`− ζ| ≥ 1 3 ∣∣∣∣3`4 − ζ ∣∣∣∣+ 1 3 |`− ζ| ≥ 7` 12 = |t`− tζ|. hence, t is generalized α-nonexpansive mapping with α = 1 3 (i.e., generalized 1 3-nonexpansive)with f (t ) = {0}.with the aid of matlab (r2015a), we obtain the following comparison table 2 and figure 2 forvarious iterative algorithms with control sequences δs = 0.65, βs = 0.8 and initial guess `1 = 50. table 2. comparison of convergence behaviour of our new iterative algorithm withs, picard-s, thakur and m iterative algorithms.step s picard-s thakur m new1 50.00000000 50.00000000 50.00000000 50.00000000 50.000000002 32.62500000 24.46875000 24.46875000 23.55468750 18.351562503 21.28781250 11.97439453 11.97439453 11.09646606 6.735596924 13.89029766 5.85996932 5.85996932 5.22747581 2.472174565 9.06341922 2.86772249 2.86772249 2.46263118 0.000000006 5.91388104 1.40339169 1.40339169 1.16013016 0.000000007 3.85880738 0.00000000 0.00000000 0.00000000 0.000000008 2.51787182 0.00000000 0.00000000 0.00000000 0.000000009 1.64291136 0.00000000 0.00000000 0.00000000 0.00000000 iteration number s 1 2 3 4 5 6 7 8 9 s eq ue nc e va lu es 0 5 10 15 20 25 30 35 40 45 50 new iteration m iteration thakur iteration picard-s iteration s iteration figure 2. graph corresponding to table 2. from the above table 2 and figure 2, it is clear that our new iterative algorithm (1.7) outperformsa number of existing iterative algorithms. eur. j. math. anal. 1 (2021) 1236. stability result our aim in this section is to show that our new iterative algorithm (1.7) is t–stable. theorem 6.1. let ω be a banach space and λ be a nonempty closed convex subset of ω. let t be a mapping satisfy (1.2). let {`s} be the iterative algorithm defined by (1.7) with sequences δs and βs ∈ [0, 1] such that ∑∞ s=0 δsβs =∞. then the iterative algorithm (1.7) is t–stable. proof. let {ys} ⊂ ω be an arbitrary sequence in λ and suppose that the sequence iterativelygenerated by (1.7) is `s+1 = f (g, ys) converging to a unique point z and that εs = ‖ys+1−f (t, ys)‖.to prove that (1.7) is t -stable, we have to show that lim s→∞ εs = 0⇔ lim s→∞ ys = z .let lim s→∞ εs = 0. then from (1.7) and (1.6), we obtain ‖ys+1 − z‖ = ‖ys+1 − f (t, ys) + f (t, ys)− z‖ ≤ ‖ys+1 − f (t, ys)‖+ ‖f (t, ys)− z‖ = εs + ‖f (t, ys)− z‖ = εs + ‖t (t ((1− δs)tys + δst ((1− βs)ys + βstys)))− z‖ = γ3(1− (1− γ)δsβs)‖ys − z‖+ εs . (6.1) for all s ≥ 1, put θs = ‖ys − z‖, σs = (1− γ)δsβs ∈ (0, 1), λs = εs . since lim s→∞ εs = 0, this implies that λs σs = εs (1−γ)δsβs → 0 as s →∞. apparently, all the conditionsof lemma 2.12 are fulfilled. hence, from lemma 2.12 we have lim s→∞ ys = z .conversely, let lim s→∞ ys = z . the we have εs = ‖ys+1 − f (t, ys)‖ = ‖ys+1 − z + z − f (t, ys)‖ ≤ ‖ys+1 − z‖+ ‖f (t, ys)− z‖ ≤ ‖ys+1 − z‖+ γ3(1− (1− γ)δsβs)‖ys − z‖. (6.2) from (6.2), it follows that lim s→∞ εs = 0. hence, our new iterative algorithm (1.7) is stable withrespect to t . � 7. data dependence result in this section, we obtain data dependence result for the mapping t satisfying (1.2) by utilizingour new iterative algorithm (1.7). eur. j. math. anal. 1 (2021) 124 theorem 7.1. let t̃ be an approximate operator of a mapping t satisfying (1.2). let {`s} be an iterative sequence generated by (1.7) for t and define an iterative algorithm as follows: ˜̀ 0 ∈ λ, g̃s = (1− βs)˜̀ s + βs t̃ ˜̀ s , w̃s = (1− δs)t̃ ˜̀ s + δs t̃ g̃s , ζ̃s = t̃ w̃s , ˜̀ s+1 = t̃ ζ̃s , ∀s ≥ 1, (7.1) where {δs} and {βs} are sequences in [0, 1] satisfying the following conditions:(i) 12 ≤ δsβs , ∀ s ∈ n,(ii) ∞∑ s=0 δsβs =∞. if tz = z and t̃ z̃ = z̃ such that lim s→∞ ˜̀ s = z̃ , we have ‖z − z̃‖ ≤ 7ε 1− γ , where ε > 0 is a fixed number. proof. using (1.7), (1.2) and (7.1), we have ‖`s+1 − ˜̀ s+1‖ = ‖tζs − t̃ ζ̃s‖ = ‖tζs − t ζ̃s + t ζ̃s − t̃ ζ̃s‖ ≤ ‖tζs − t ζ̃s‖+ ‖t ζ̃s − t̃ ζ̃s‖ ≤ γ‖ζs − ζ̃s‖+ l‖ζs − tζs‖+ ε. (7.2) from (1.7), (1.2) and (7.1), we have ‖ζs − ζ̃s‖ = ‖tws − t̃ w̃s‖ = ‖tws − t w̃s + t w̃s − t̃ w̃s‖ ≤ ‖tws − t w̃s‖+ ‖t w̃s − t̃ w̃s‖ ≤ γ‖ws − w̃s‖+ l‖ws − tws‖+ ε. (7.3) putting (7.3) into (7.2), we have ‖`s+1 − ˜̀ s+1‖ ≤ γ2‖ws − w̃s‖+ γl‖ws − tws‖ +γε+ l‖ζs − tζs‖+ ε. (7.4) eur. j. math. anal. 1 (2021) 125again, using (1.7), (1.2) and (7.1), we get ‖ws − w̃s‖ = (1− δs)‖t`s − t̃ ˜̀ s‖+ δs‖tgs − t̃ g̃s‖ ≤ (1− δs){‖t`s − t ˜̀ s‖+ ‖t ˜̀ s − t̃ ˜̀ s‖} +δs{‖tgs − t g̃s‖+ ‖t g̃s − t̃ g̃s‖} ≤ (1− δs){γ‖`s − ˜̀ s‖+ l‖`s − t`s‖+ ε} +δs{γ‖gs − g̃s‖+ l‖gs − tgs‖+ ε}. (7.5) using (1.7), (1.2) and (7.1), we get ‖gs − g̃s‖ ≤ (1− βs)‖`s − ˜̀ s‖+ βs‖t`s − t̃ ˜̀ s‖ ≤ (1− βs)‖`s − ˜̀ s‖+ βs{‖t`s − t ˜̀ s‖+ ‖t ˜̀ s − t̃ ˜̀ s‖} ≤ (1− βs)‖`s − ˜̀ s‖+ βs{γ‖`s − ˜̀ s‖+ l‖`s − t`s‖+ ε} = [1− (1− γ)βs ]‖`s − ˜̀ s‖+ βsl‖`s − t`s‖+ βsε (7.6) using (7.6) and (7.5), we have ‖ws − w̃s‖ ≤ (1− δs){γ‖`s − ˜̀ s‖+ l‖`s − t`s‖+ ε} +δs{γ[1− (1− γ)βs ]‖`s − ˜̀ s‖+ γβ‖`s − t`s‖+ γβsε} = γ[1− (1− γ)δsβs ]‖`s − ˜̀ s‖+ (1− δs)l‖`s − t`s‖ +(1− δs)ε+ γδsβsl‖`s − t`s‖+ γδsβsε. (7.7) substituting (7.7) into (7.4), we obtain ‖`s+1 − ˜̀ s+1‖ ≤ γ3[1− (1− γ)δsβs ]‖`s − ˜̀ s‖+ γ2(1− δs)l‖`s − t`s‖ +γ2(1− δs)ε+ γ3δsβsl‖`s − t`s‖+ γ3δsβsε +γl‖ws − tws‖+ γε+ l‖ζs − tζs‖+ ε. (7.8) since γ, γ2, γ3 ∈ (0, 1) and δs , βs ∈ [0, 1], then (7.8) becomes ‖`s+1 − ˜̀ s+1‖ ≤ [1− (1− γ)δsβs ]‖`s − ˜̀ s‖+ l‖`s − t`s‖ +δsβsl‖`s − t`s‖+ l‖ws − tws‖ +l‖ζs − tζs‖+ δsβsε+ 3ε. (7.9) by our assumption (i) that 12 ≤ δsβs , we have 1− δsβs ≤ δsβs ⇒ 1 = 1− δsβs + δsβs ≤ δsβs + δsβs = 2δsβs . eur. j. math. anal. 1 (2021) 126this yields ‖`s+1 − ˜̀ s+1‖ ≤ [1− (1− γ)δsβs ]‖`s − ˜̀ s‖+ 3δsβsl‖`s − t`s‖ +2δsβsl‖ws − tws‖+ 2δsβsl‖ζs − tζs‖+ 7δsβsε = (1− (1− γ)δsβs)‖`s − ˜̀ s‖ +δsβs(1− γ)× { 3l‖`s − t`s‖+ 2l‖ws − tws‖ (1− γ) + 2l‖ζs − tζs‖+ 7ε (1− γ) } . (7.10) set θs = ‖`s − ˜̀ s‖ σs = (1− γ)δsβs ∈ (0, 1) λs = { 3l‖`s − t`s‖+ 2l‖ws − tws‖+ 2l‖ζs − tζs‖+ 7ε (1− γ) } from theorem 3.1, we know that lim s→∞ `s = z and since tz = z , it follows that lim s→∞ ‖`s − t`s‖ = lim s→∞ ‖ws − tws‖ = lim s→∞ ‖ζs − gζs‖ = 0. using lemma 2.13, we get 0 ≤ lim sup s→∞ ‖`s − ˜̀ s‖ ≤ lim sup s→∞ 7ε (1− γ) . (7.11) since by theorem 3.1, we have that lim s→∞ `s = z and from our hypothesis lim s→∞ ˜̀ s = z̃ , it followsfrom (7.11) that ‖z − z̃‖ ≤ 7ε (1− γ) . this completes the proof. � 8. some applications in this section, we will prove that the sequence generated by our new iterative algorithm (1.7)converges strongly to solutions of the constrained convex minimization problem and split feasibilityproblem.now, we present the definitions of some operators that will we be important in proving our mainresults. let h be a hilbert space and let c be a nonempty closed and convex subset of h. definition 8.1. let t : c → c be a mapping. then t is said to be:(i) nonexpansive, if ‖t`− tζ‖ ≤ ‖`− ζ‖, for all `, ζ ∈ c; eur. j. math. anal. 1 (2021) 127(ii) lipschitz continuous, if there exists l > 0 such ‖t`− tζ‖ ≤ l‖`− ζ‖, for all `, ζ ∈ c; (iii) monotone if, 〈t`− tζ, `− ζ〉 ≥ 0, for all `, ζ ∈ c; (8.1) (iv) $-strongly monotone if there exists $ > 0, such that 〈`− ζ, t `− tζ〉 ≥ $‖`− ζ‖, for all `, ζ ∈ c. (8.2) for any ` ∈ h, we define the map pc : h → c satisfying ‖`− pc`‖ ≤ ‖`− ζ‖, for all ζ ∈ c. pc is called the metric projection of h onto c. it is well known that pc is nonexpansive. 8.1. application to constrained convex minimization problem.consider the following constrained convex minimization problem: minimize {f (`) : ` ∈ c}, (8.3) where f : c → r is a real-valued function. the minimization problem (8.3) is consistent if it hasa solution. throughout this paper, we shall use γ to stand for the solution set of the problem(8.3). it is worthy noting that f is (fréchect) differentiable, the gradient-projection method (gpm)generates a sequence {`s} by using the recursive formula:{ `0 ∈ c, `s+1 = pc(`s − λ∇f (`s)), for all s ≥ 1. (8.4) in more general form, (8.4) can be written as: { `0 ∈ c, `s+1 = pc(`s − λs∇f (`s)), for all s ≥ 1, (8.5) where λ and λs are positive real numbers.it is well known that if ∇f is $-strongly monotone and l-lipschitzian with $,l > 0, then theoperator t = pc(i − λ∇f ) (8.6) is a contraction; thus the sequence {`s} in (8.4) converges in norm to the unique minimizer of (8.3).from [14, 30], we know that z ∈ c solve the minimization problem (8.3) if and only if z solvesthe following fixed point equation: z = pc(i − λ∇f )z, (8.7) eur. j. math. anal. 1 (2021) 128where λ > 0 is any fixed positive number. the operator t = pc(i − λ∇f ) is well known tobe nonexpansive (see [14, 30] and the references therein). several authors have have considereddifferent iterative algorithm for constrained convex minimization problems (see [4, 9, 13, 19, 34] andthe references therein). we now give our main results theorem 8.2. let c be a nonempty closed convex subset of a real hilbert space h. supposed that the minimization problem (8.3) is consistent and let γ denote the solution set. supposed that the gradient ∇f is l-lipschitzian with constant l > 0. let {`s} be the sequence generated iteratively by  `0 ∈ c, gs = (1− βs)`s + βspc(i − λ∇f )`s ws = (1− δs)pc(i − λ∇f )`s + δspc(i − λ∇f )gs ζs = pc(i − λ∇f )ws `s+1 = pc(i − λ∇f )ζs , ∀s ≥ 1. (8.8) where {δs}, {βs} are sequences in [0,1] and λ ∈ ( 0, l2 ) . then the sequence {`s} converges strongly to a minimizer z of (8.3). 8.2. application to split feasibility problem.for modeling inverse problems which emanate from phase retrieval and medical image reconstruc-tion, in 1994, censor and elfving [11] firstly introduced the following split feasibility problem (sfp)in finite-dimensional hilbert spaces.let c and q be nonempty closed convex subsets of the hilbert spaces h1 and h2, respectivelyand a : h1 → h2 be a bounded linear operator. then the split feasibility problem (sfp) isformulated to find z ∈ c such that az ∈ q. (8.9)sfp has many applications, it has been found that sfp can been used in many areas such asimage restoration, computer tomograph, radiation therapy treatment planning. there exists someiterative several iterative methods for solving split feasibility problems, see, for instance [8, 15,30].in 2002, byrne [8] applied the forward-backward method, a type of projection gradient methodto approximate (8.9). the so called cq-iterative procedure is defined as follows: `s+1 = pc [i − γa∗(1− pq)a]`n, ∀ n ≥ 1, (8.10) where γ ∈ (0, 2 ‖a‖2 ) with λ being the spectral radius of the of operator a∗a, pc and pq denotethe projections onto sets c and q, respectively, and a∗ : h∗2 → h∗1 is the adjoint of a.we assume that the solution set γ of the sfp (8.10) is nonempty, let γ = {` ∈ c : a` ∈ q} = c ∩ a−1q, then γ is closed, convex and nonempty set. eur. j. math. anal. 1 (2021) 129 lemma 8.3. [15] let operator t = pc [i − γa∗(i − pq)a], where γ ∈ ( 0, 2 ‖a‖2 ) . then, t is said to be a nonexpansive map. since by our assumption γ 6= ∅, then it is clear that any z ∈ c solves (8.9) if and only if it solvesthe fixed point equation: t = pc[i − γa∗(i − pq)a]z = z, z ∈ c. thus, f (t ) = γ = c ∩ a−1q, i.e., the solution set γ is equal the set of fixed point of the map t .for more explicit explanation, the reader can see [42,43].now, to prove our main results in this part, we will consider the following scheme: `0 ∈ c, gs = (1− βs)`s + βspc [i − γa∗(i − pq)a]`s ws = (1− δs)pc [i − γa∗(i − pq)a]`s + δspc [i − γa∗(i − pq)a]gs ζs = pc [i − γa∗(i − pq)a]ws `s+1 = pc [i − γa∗(i − pq)a]ζs , (8.11) for all s ≥ 1, where {δs}, {βs} are sequences in [0,1] and γ ∈ (0, 2 ‖a‖2 ). theorem 8.4. let {`s} be the sequence iteratively generated by (8.11). then, {`s} converses weakly to an element in γ. proof. since t = pc [i − γa∗(i − pq)a] is a nonexpansive map and by proposition 2.9 we knowthat every generalized α-nonexpansive map is nonexpansive map with α = 0 (i.e., 0-nonexpansive),so the conclusion follows from theorem 4.3. � theorem 8.5. if {`s} is the sequence generated by the iterative scheme (8.11). then {`s} converges strongly the an element in γ if and only if lim inf s→∞ d(`s ,γ) = 0. proof. since t = pc [i − γa∗(i − pq)a] is nonexpansive map, then the conclusion of the prooffollows from theorem 4.4. � theorem 8.6. if t = pc [i − γa∗(i − pq)a] satisfies condition (i) and {`s} is the sequence iteratively defined by (8.11), then {`s} converges strongly to a point in γ. proof. the result follows from theorem 4.5. � 9. conclusion in this paper, we have shown numerically and analytically that our new iterative algorithm (1.7)has a better rate of convergence than m iterative algorithm and some other well known existingiterative algorithms in the literature for almost contraction mapping and generalized α-nonexpansivemappings. also, it is shown that our new iterative algorithm (1.7) is t–stable and data dependent eur. j. math. anal. 1 (2021) 130which make it reliable. as some applications of our new iterative algorithm (1.7), it is used tofind the solutions of constrained convex minimization problem and split feasibility problem. now,owing to the fact that the class of generalized α-nonexpansive mappings which is considered inour paper is more general than the class of suzuki generalized nonexpansive mappings which hasbeen considered by ullah and arshad [39] for m iteration, it implies that our results generalizeand improve the results in ullah and arshad [39] and several other related results existing in theliterature. references [1] m. abbas and t. nazir, a new faster iteration process applied to constrained minimization and feasibility problems,mat. vesn. 66(2014), 223–234.[2] r. p. agarwal, d. o. regan and d. r. sahu, iterative construction of fixed points of nearly asymptotically nonex-pansive mappings, j. nonlinear convex anal. 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(basel). 23 (1972), 292–298. https://doi.org/10.1186/1029-242x-2014-280 eur. j. math. anal. 1 (2021) 132 [42] h.k. xu, a variable krasnosel’skii-mann algorithm and the multiple-set split feasibility problem, inverse probl.22(6) (2006), 2021–2034.[43] h.k. xu, iterative methods for the split feasibility problem in infinite-dimensional hilbert spaces. inverse probl. 26(2010), 105018. 17 pp. 1. introduction 2. preliminaries 3. rate of convergence 4. convergence results 5. numerical result 6. stability result 7. data dependence result 8. some applications 8.1. application to constrained convex minimization problem 8.2. application to split feasibility problem 9. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 4doi: 10.28924/ada/ma.3.4 parameter estimation for spdes driven by cylindrical stable processes jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223-0001, usacorrespondence: j.bishwal@uncc.edu abstract. we consider infinite dimensional extension of affine models with heavy tails in finance. westudy several estimators of the drift parameter in the stochastic partial differential equation drivenby cylindrical stable processes. we consider several sampling schemes. we also consider randomsampling scheme, e.g, when the solution process is observed at the arrival times of a poisson process.we obtain the consistency and the asymptotic normality of the estimators. 1. introduction parameter estimation in stochastic partial differential equations is a very young area of researchin view of its applications in finance, physics, biology and oceanography. loges [32] initiated thestudy of parameter estimation in infinite dimensional stochastic differential equations. when thelength of the observation time becomes large, he obtained consistency and asymptotic normality ofthe maximum likelihood estimator (mle) of a real valued drift parameter in a hilbert space valuedsde. koski and loges [28] extended the work of loges [32] to minimum contrast estimators. koskiand loges [27] applied the work to a stochastic heat flow problem. martingale estimation functionfor discretely observed diffusions was studied in bibby and srensen [2]. bishwal [6] studied a newestimating function for discretely sampled diffusions by removing the stochastic integral in girsanovlikelihood. bishwal [7] contains asymptotic theory on likelihood method and bayesian method fordrift estimation of finite and infinite dimensional stochastic differential equations. bishwal [12]studied applications of levy processes in stochastic volatility models in finance.huebner, khasminskii and rozovskii [23] started statistical investigation in spdes. they gavetwo contrast examples of parabolic spdes in one of which they obtained consistency, asymptoticnormality and asymptotic efficiency of the mle as noise intensity decreases to zero under thecondition of absolute continuity of measures generated by the process for different parameters (the received: 12 apr 2022. key words and phrases. stochastic partial differential equations; space-time colored noise; cylindrical stable process;stable random field; super levy process; poisson sampling; martingale estimating function; quasi likelihood estimator;stable ornstein-uhlenbeck process; stable black-scholes model; stable cox-ingersoll-ross model; consistency; asymp-totic normality. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 2situation is similar to the classical finite dimensional case) and in the other they obtained theseproperties as the finite dimensional projection becomes large under the condition of singularity ofthe measures generated by the process for different parameters. the second example was extendedby huebner and rozovskii [24] and the first example was extended by huebner [22] to mle forgeneral parabolic spdes where the partial differential operators commute and satisfy differentorder conditions in the two cases.huebner [21] extended the problem to the ml estimation of multidimensional parameter. lototskyand rozovskii [33] studied the same problem without the commutativity condition. small noiseasymptotics of the nonparmetric estimation of the drift coefficient was studies by ibragimov andkhasminskii [29].based on continuous observations, usually there can be two asymptotic settings in spde: 1) t → ∞ 2) n → ∞ where t is the length of the observations and n is the number of fouriercoefficients of the spde solution.in a bayesian approach, using the first setting, bishwal [3] proved the bernstein-von misestheorem and asymptotic properties of regular bayes estimator of the drift parameter in a hilbertspace valued sde when the corresponding ergodic diffusion process is observed continuously overa time interval [0, t ]. the asymptotics are studied as t → ∞ under the condition of absolutecontinuity of measures generated by the process. results are illustrated for the example of anspde.using the second setting, bishwal [5] proved the bernstein-von mises theorem and spectralasymptotics of bayes estimators for parabolic spdes when the number of fourier coefficientsbecomes large. in this case, the measures generated by the process for different parameters aresingular.bishwal [10] studied bernstein-von mises theorem and small noise bayesian asymptotics for par-abolic stochastic partial differential equations. bishwal [9] studied hypothesis testing for fractionalstochastic partial differential equations with applications to neurophysiology and finance.in this paper we study the asymptotic properties of the quasi maximum likelihood estimatorwhen we have observations of finite-dimensional projections at poisson arrival time points. theasymptotic setting is only the large number of observations at random time points which are thearrivals of a poisson process.the rest of the paper is organized as follows: section 2 contains model, assumptions andpreliminaries. in section 3 we prove estimation results with additive noise. section 4 and 5, weprovide estimation results with multiplicative noise. in section 6, we give several examples. 2. model and preliminaries let h be a real separable hilbert space with inner product 〈·〉 and norm | · |. by l(h) we denotethe banach space of bounded linear operators from h into h endowded with the operator norm https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 3 ‖ · ‖l(h). we fix an orthonormal basis (en) in h. through the basis (en) we will often identify hin l2. more generally, for a given sequence ρ = (ρn) of real numbers we set l2ρ = {(xn) ∈ r∞ : ∑ n≥1 x2 nρ 2 n <∞}. where r∞ = rn. the space l2ρ becomes a separable hilbert space with the inner product: 〈x, y〉 =∑ n≥1 xnynρ 2 n for x = (xn), y = (yn) ∈ l2ρ . let us fix θ0, the unknown true value of the parameter θ.let (ω,f , p ) be a complete probability space and z(t, x) be a process on this space with valuesin the schwarz space of distributions d′(g) such that for φ,ψ ∈ c∞0 (g), ‖φ‖−1 l2(g) 〈w (t, ·), φ(·)〉is a one dimensional stable process.this process is usually referred to as the cylindrical α-stable process (c.s.p.), α ∈ (0, 2).we assume that there exists a complete orthonormal system {hi}∞i=1 in l2(g)) such that forevery i = 1, 2, . . . , hi ∈ zm,20 (g) ∩ c∞(g) and λθhi = βi(θ)hi , and lθhi = µi(θ)hi for all θ ∈ θ where lθ is a closed self adjoint extension of aθ, λθ := (k(θ)i − lθ)1/2m, k(θ) is a constant andand the spectrum of the operator λθ consists of eigenvalues {βi(θ)}∞i=1 of finite multiplicities and µi = −β2m i + k(θ).a levy process (zt) with values in h is an h-valued process defined on some stochastic basis (ω,f , (ft)t≥0, p ) having stationary independent increments, cadlag trajectories such that z0 = 0,p-a.s. one has that e[e i〈zt ,s〉] = exp(−tψ(s)), s ∈ hwhere ψ : h → c is sazonov continuous, negative definite function such that ψ(0) = 0. thefunction ψ is called the exponent of (zt).the exponent ψ can be expressed by the infinite dimensional levy-khintchine formula ψ(s) = 1 2 〈qs, s〉 − i〈a, s〉 − ∫ h ( e i〈s,y〉 − 1− i〈s, y〉 1 + |y |2 ) ν(dy), s ∈ h where q is the non-negative trace class operator on h, a ∈ h and ν is the levy measure or thejump intensity measure associated to (zt).cylindrical α-stable process (c.s.p.) is a levy process taking values in the hilbert space u = l2ρ ,with a properly chosen weight ρ.consider the linear spde dxt = θaxtdt + dzt , x ∈ hc.s.p. z(t) is a cylindrical α-stable process, α ∈ (0, 2) which can be expanded in the series z(t) = ∞∑ i=1 γizi(t)hi where {zi(t)}∞i=1 are independent, real valued, one dimensional, normalized, symmetric, α-stableprocesses and (γi) is a given sequence of, possibly unbounded, positive numbers, and hi is a fixed https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 4orthonormal basis in h. the latter series converges p -a.s. in h−α for α > d/2. indeed ‖z(t)‖2 −α = ∞∑ i=1 γ2 i z 2 i (t)‖hi‖2 −α = ∞∑ i=1 z2 i (t)β−2α i and the later series converges p -a.s.for any j ∈ n, t ≥ 0, e[e izj (t)h] = e−t|h| α . stable one-dimensional density :a one-dimensional, normalized, symmetric α-stable distribution µα, α ∈ (0, 2] has characteristicfunction µ̂α(s) = e−|s| α , s ∈ r.the density of µα with respect to lebesgue measure will be denoted by pα. this even functionis known in closed form only if α = 1 or 2. the precise asymptotic behavior of the density pα, α ∈ (0, 2) is as follows:for any α ∈ (0, 2), there exists cα such that pα(x) ∼ cα xα+1 as x →∞. stable measures on hilbert space :a random variable ξ on h is called α-stable (α ∈ (0, 2]) if for any n there exists avector an ∈ h such that for any independent copies ξ1, ξ2, . . . , ξn of ξ, the random variable n−1/α(ξ1 + ξ2, . . .+ ξn)− an has the same distribution as ξ. a borel probability measure µ on his said to be α-stable if it is the distribution of a stable random variable with vales in h. stable ou process: dxt = −θxtdt + σdzt , x0 = x0the solution is xt = e−θtx0 + ∫ t 0 e−θ(t−s)σdzs .the stochastic integral can be defined as the limit in probability of riemann sums.let yt = ∫ t 0 e−θ(t−s)σdzs .then e[e ihyt ] = exp [ −σα|h|α ∫ t 0 e−αθsds ] = e−|h| αcα(t) where cα(t) = σ ( 1− e−αθt αθ ) .we show that the process x is stochastically continuous.first we show that y is stochastically continuous, i.e., lim h→0+ sup t≥0 p (|yt+h − yt | > ε) = 0 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 5note that for any t ≥ 0, h ≥ 0, yt+h − yt = ∫ t+h t e(t+h−s)adzs + eha ∫ t 0 e(t−s)adzs − ∫ t 0 e(t−s)adzs = ehayt − yt + ∫ t+h t e(t+h−s)adzslet us choose p ∈ (0, α). we have p (|yt+h − yt | > ε) ≤ p ( |ehayt − yt | > ε 2 ) + p (∣∣∣∣∫ t+h t e(t+h−s)adzs ∣∣∣∣ > ε 2 ) ≤ 2p e|ehayt − yt |p εp + 2p e| ∫ h 0 e sadzs |p εp = i1(t, h) + i2(h).but e|yt |p ≤ cp ( ∞∑ n=1 1− e−αθt αθ )p/α and so [i2(h)]α/p → 0 as h → 0. concerning i1, by khintchine inequality |ehayt − yt | = ∑ n≥1 ∣∣(e−θh − 1)y nt ∣∣21/2 ≤ cp ẽ ∣∣∣∣∣∣∑n≥1 rn(e−θh − 1)y nt ∣∣∣∣∣∣ p1/p . where ẽ denotes expectation w.r.t. to the measure p̃ p̃ (rn = 1) = p̃ (rn = 1) = 1/2 where arademacher sequence (rn) with rn : ω̃ → {−1, 1} is defined on the probability space (ω̃, f̃ , p̃ ).hence e|ehayt − yt |p ≤ cpp ẽe ∑ n≥1 ∣∣(e−θh − 1)y nt ∣∣21/2 ≤ cp ∑ n≥1 ∣∣(1− e−θht)βn ∣∣α (1− e−θht) αθ p/α ≤ cp αp/α ∑ n≥1 ∣∣(1− e−θht)βn ∣∣α θ p/α since lim h→0+ ∑ n≥1 ∣∣(1− e−θht)βn ∣∣α θ p/α = 0, we get lim h→0+ sup t≥0 2p e|ehayt − yt |p εp = 0. since e|yt |p ≤ cp ∑ n≥1 |βn|α (1− e−θht) αθ p/α , hence lim h→0 ∑ n≥1 |βn|α (1− e−θht) αθ p/α = 0 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 6hence lim h→0+ 2p e| ∫ h 0 e sadzs |p εp → 0.thus lim h→0+ sup t≥0 i1(t, h) = 0 this proves stochastic continuity of yt . using the stochastic continuity and ft-adaptedness of x , we conclude that the process x hasa predictable version. time change: let l be a one dimensional α-stable process, α ∈ (0, 2). then there exists an α-stable process, α ∈ (0, 2) z = (zt) such that∫ t 0 e−θsdls = z(u(t)) where u(t) = 1− e−αθt αθ . recall that u ∈ c∞([0,∞]) with u′(t) 6= 0, t ≥ 0.in the limiting gaussian case of α = 2, it becomes time change for brownian motion. infinite dimensional stable ou process dxnt = −θxnt dt + σdznt , x n 0 = xn, n ∈ nwith x = (xn) ∈ l2 = h. the solution is a stochastic process x = xxt with values in r∞ withcomponents xxt = e−θtxn + ∫ t 0 e−θ(t−s)σdzns .(the stochastic integral can be defined as the limit in probability of riemann sums.) xxt = ∞∑ n=1 xnt en = etax + za(t) where za(t) = ∫ t 0 e(t−s)adzs = ∞∑ n=1 (∫ t 0 e−θ(t−s)σdzns ) en. the process xxt is an ft-adapted irreducible markov process and its transition semigroup isstrong feller.let y nt = zna(t) = ∫ t 0 e−θ(t−s)σdzns , n ∈ n, t ≥ 0.then e[e ihy n t ] = exp [ −σα|h|α ∫ t 0 e−αθsds ] = e−|h| αcαn (t) https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 7where cn(t) = σ ( 1− e−αθt αθ )1/α . it follows that e[e ihy n t ] = e[e ihcn(t)ln ], h ∈ r where (ln) are independent α-stable random variables having the same law µα. thus xxt is ft-adapted.the markov property easily follows from the identity za(t + h)− ehaza(t) = ∫ t+h t e(t+h−s)adzs , t, h ≥ 0. if the cylindrical levy process z takes values in hilbert space h, the by the kotelenez regularityresults trajectories of the process x are cadlag with values in h. moments of the process the ou process is stochastically continuous and trajectories in lp([0, t ];h) for any 0 < p < αa.s. set yt := za(t). then we have e|yt |p ≤ c̃pσp ( ∞∑ n=1 1− e−αθt αθ )p/α where c̃p depends on p. moments of the stochastic integral suppose (zt) is an α-stable levy process with 0 ≤ α ≤ 2 and y(t) is a predictable processsatisfying ∫ t0 |y(t)|αdt <∞. then for any 0 < r < α, there exists a constant c such that e [ sup t≤t ∣∣∣∣∫ t 0 y(s)dzs ∣∣∣∣r] ≤ e [(∫ t 0 |y(t)|αdt )r/α] . equivalence of transition probabilities assume sup n≥1 e−γntγ 1/α n βn = ct <∞, e ∫ t 0 ∑ n≥1 |y nt |2 p/2 dt <∞. let pα be the density of the one dimensional stable measure. then the laws µxt and µyt of xxt and xyt respectively are equivalent for any t > 0, x, y ∈ h,α ∈ (0, 2). moreover, the density dµxt dµxt of https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 8 µxt with respect to µyt is given by dµxt dµyt = lim n→∞ n∏ k=1 pα ( zk−e−θtxk c(t) ) pα ( zk−e−θtyk c(t) ) . the corresponding mle is denoted as θ̂n. priola et al. [38] obtained exponential convergence to the invariant measure, in the total variationnorm, for solutions to sdes driven by α-stable noises in finite and infinite dimensions using twoapproaches: lyapounov’s function approach by harris and doeblin’s coupling argument. in bothapproaches irreducibility and uniform strong feller property play crucial role.first we consider the method of moments estimation in modified tempered stable-ornstein-uhlenbeck model. masuda and uehara [36] studied two-step estimation in ergodic levy drivensde dxt = a(θ,xt)dt + b(β,xt−)dzt , x0 = x0.masuda [35] studied multi-step estimation in stable ou model: dxt = −θxtdt + σdzt , x0 = x0.for the least squares estimator (lse) θ̃n of θ, hu and long [20] obtained( t log n )1/α (θ̃n − θ0)→d s′α s′′+ α/2where sα is stable distribution of order β.while in gaussian ou case, for different parts θ > 0, θ < 0 and θ = 0, lan, lamn and labfhold respectively (see bishwal [11]), in stable case entirely different phenomena occur.the solution of the sde is given by xt = e−θ(t−s)xs + σ ∫ t s e−θ(t−s)dzu, t ≥ 0. due to the stable integral property, l (∫ t s e−θ(t−s)dzu ) = sα(κ∆(θ)) where κ∆(θ) = { 1− e−θ∆ θα }1/α ∼ ∆1/α. for each j ≤ n, the transition probability is given by l(xtj |xtj−1 = x) = δx exp(−θ∆) ? sα(κ∆(θ)). lamn holds for θ ∈ r when t is fixed. n1/α−1/2(θ̂n − θ)→d mn(0, iθ(t )−1). where iθ(t ) is the fisher information of the process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 9we study estimation in mts-ou sv model. the inverse gaussian-ou and gamma-ou modelsare special cases.an infinitely divisible distribution is said to be α-modified tampered stable distribution (α-mts)distribution if its levy triplet is given by σ2 = 0, ν(dx) = c λα+ 1 2 + kα+ 1 2 λ+x xα+ 1 2 ix>0 + λ α+ 1 2 + kα+ 1 2 λ−x xα+ 1 2 ix<0  dx, γ = µ+ c ( γ( 1 2 − α) 2α+ 1 2 (λ2α−1 + − λ2α−1 − )− λα− 1 2 + kα− 1 2 (λ+) + λ α− 1 2 − kα− 1 2 (λ−) ) where c > 0, λ+, λ− > 0, µ ∈ r, α ∈ (−∞, 1)\{1 2} and kp(x) is the modified bessel functionof second kind. we denote the mts random variable by x ∼ mts(α,c, λ+, λ−, µ). the levymeasure ν(dx) is called the mts levy measure with parameter (α,c, λ+, λ−).the mts distribution is obtained by taking a symmetric α-stable distribution with α ∈ (0, 1)and multiplying by a levy measure with √|x |λα+ 1 2kα+ 1 2 (λ|x |) on each half of the real axis. themeasure can be extended to the case α ≤ 0. if α = 1 2 , then γ may not be defined, so it is removed.the mts distribution was introduced by kim, rachev and chung [25].the tails of the α-mts distribution are thinner than those of the 2α-stable and fatter (heavier)than those of the 2α-ts distribution. at the zero neighborhood, all three have the same asymptoticbehavior.if λ+ > λ−, then the distribution is skewed to the left. if λ+ < λ−, then the distribution is skewed to the right. if λ+ = λ−, then the distribution is symmetric. c controls the kurtosis of the distribution. if c increases, the peakedness of the distributionincreases.as α decreases, the distribution has fatter tails and increased peakedness. the levy processcorresponding to the mts distribution has finite activity if α < 0 and infinite activity if α > 0. ithas finite variation if α < 1 2 and infinite variation if α > 1 2 .with proper choice of c and µ, mts distribution has zero mean and unit variance, and thedistribution is called standard mts distribution and denoted x ∼ stdmts(α, λ+, λ−).cgmy process proposed in carr et al. [14] is a tempered stable process. in order to obtain aclosed form solution of the european option price, cgmy used the generalised fourier transformof the distribution of the stock price under the assumption of markov property.the stochastic volatility model is given by d yt = (µ+ βxt) dt + √ xt dwt + ρ dzt dxt = −θ xt dt + dztwhere µ is the drift parameter, β is the risk premium, θ > 0 is the drift of the volatility and zt isa mts process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 10we estimate θ from the observations of {yt} at the time points tk = k∆, k = 0, 1, 2, . . . , n, ∆ > 0. cm(z) := dm dum logφts(u)|u=0for the tempered stable distribution ts(b, δ, γ) where 0 < b < 1, δ > 0, γ ≥ 0, the m-th cumulantis given by cm(z) = −δ(−2)mγ(b−m)/bb(b − 1) . . . (b − (m − 1))for γ > 0. when γ = 0, it is positive b-stable distribution for which the moments of only order k < b exist. for b = 1/2, ts distribution reduces to inverse gaussian (ig) distribution.the infinite divisibility of this distribution allows one to construct the corresponding levy process.a levy process z = (zt)t≥0 is said to be a tempered stable process if z1 follows a temperedstable distribution. the tempered stable process is of finite activity if α < 0 and infinite activity if 0 < α < 2. the tempered stable process is of finite variation if 0 < α < 1 and infinite variation if 1 < α < 2.the mts-garch model is given by log st st−1 = rt − dt + λtσt − g(σt ;α, λ+, λ−) + σtεt σ2 t = (α0 + α1σ 2 t−1ε 2 t−1 + β1σ 2 t−1) ∧ ρ, ε0 = 0 α0, α1, β1 ≥ 0, α1 + β1 < 1, 0 < ρ < λ2 +, εt ∼ stdmts(α, λ+, λ−), rt is the risk-free rate, dtis the dividend rate, λt is the market price of risk, g is the characteristic exponent of the laplacetransform for the distribution stdmts(α, λ+, λ−), i.e., g(x ;α, λ+, λ−) = log(e(exp(xεt)).the characteristic function of z is given by φz(u) = exp(iuµ+ gr(u;α,c, λ+, λ−) + gi(u;α,c, λ+, λ−)) where for u ∈ r, gr(u;α,c, λ+, λ−) = 2− α+3 2 √ πcγ ( 1− α 2 ) [ (λ2 + + u2) α 2 − λα+ + (λ2 − + u2) α 2 − λα− ] , gi(u;α,c, λ+, λ−) = iuc2− α+1 2 γ ( 1− α 2 )[ λα−1 + f ( 1, 1− α 2 ; 3 2 , ;− u2 λ2 + ) − λα−1 − f ( 1, 1− α 2 ; 3 2 , ;− u2 λ2 − )] where f is the hyper-geometric function. the value of gi for symmetric mts distribution is alwayszero.the m-th cumulant is given by cm(z) = µ if m = 1, cm(z) = 2m− α+3 2 ( m − 1 2 ) !cγ ( m − α 2 ) (λα−m+ − λα−m− ) i f m = 3, 5, 7, . . . cm(z) = 2− α+3 2 √ π ( m! m 2 ! ) cγ ( m − α 2 ) (λα−m+ + λα−m− ) i f m = 2, 4, 6, . . . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 11the mean, variance, skewness and excess kurtosis are given by e(z) = c1(z) = µ+ 2− α+1 2 cγ ( 1− α 2 ) (λα−1 + − λα−1 − ), v (z) = c2(z) = 2− α+1 2 √ πcγ ( 1− α 2 ) (λα−2 + + λα−2 − ), s(z) = c3(z) c2(z)3/2 = 2 α+9 4 γ ( 3−α 2 ) (λα−3 + − λα−3 − ) π3/4c1/2(γ( 1−α 2 )(λα−2 + + λα−2 − ))3/2 , κ(z) = c4(z) c2(z)2 = 3 · 2 α+3 2 cγ ( 2− α 2 ) (λα−4 + + λα−4 − ) √ πc(γ( 1−α 2 )(λα−2 + + λα−2 − ))2 . if α ∈ (0, 2)\{1}, the levy measure of α-stable, α-ts and α-mts have the same asymptotic behav-ior at the zero neighborhood. however, the tails of the levy measures for the α-mts distributionare thinner than those of α-stable and heavier than those of α-ts distribution.when z is a ig process, the moment estimators of ρ and θ are given by θ̂n := γȳ ∆δρ̂n , ρ̂n := γ(γs2 y − ∆δ) 2ȳwhere ȳ := 1 n n∑ j=1 yj , yj := yj∆ − y(j−1)∆, s2 y := 1 n n∑ j=1 (yj − ȳ)2 = 1 n n∑ j=1 y2 j − (ȳ)2. when z is a gamma process, the moment estimators are given by θ̂n := 1 n2 [∑n i=1(yi∆ − y(i−1)∆) ]2 1 n2 ∑n i=1(yi∆ − y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ − y(i−1)∆) ] 2a3(a + 1) b4∆ , ρ̂n := 1 n2 ∑n i=1(yi∆ − y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ − y(i−1)∆) ] 1 n2 [∑n i=1(yi∆ − y(i−1)∆) ] b3∆ 2a2(a + 1) . for the mts-ou model, the estimating functions are given by c1(y1) = λρ∆c1(z), c2(y1) = ∆c1(z) + 2λρ2∆c1(z), c3(y1) = ∆c1(z) + 2λρ2∆c2(z), c4(y1) = ∆c1(z) + 2λρ2∆c3(z)which give e(y1) = c1(y1) = µ+ 2− α+1 2 cγ ( 1− α 2 ) (λα−1 + − λα−1 − ), v (y1) = c2(y1) = 2− α+1 2 √ πcγ ( 1− α 2 ) (λα−2 + + λα−2 − ).this gives the moment estimators for the sou model θ̂n := 1 n2 [∑n i=1(yi∆ − y(i−1)∆) ]2 1 n2 ∑n i=1(yi∆ − y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ − y(i−1)∆) ] https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 12 × [2− α+1 2 cγ ( 1− α 2 ) (λα−1 + − λα−1 − )]2[2− α+1 2 √ πcγ ( 1− α 2 ) (λα−2 + + λα−2 − )]2∆−1. ρ̂n := 1 n2 ∑n i=1(yi∆ − y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ − y(i−1)∆) ] 1 n2 [∑n i=1(yi∆ − y(i−1)∆) ] × [2− α+1 2 cγ ( 1− α 2 ) (λα−1 + − λα−1 − )2− α+1 2 √ πcγ ( 1− α 2 ) (λα−2 + + λα−2 − )]−12−1∆. let ϑ = (ρ, θ) and ϑ̂n = (ρ̂n, θ̂n). by using theorem 2.2 in masuda [34] (see also theorem 4.1 vander vaart [41]), we obtain the strong consistency and asymptotic normality of the mm estimators: proposition 2.1 for fixed ∆ > 0 as n →∞, (a) ϑ̂n → ϑ0 a.s. as n →∞. (b) √ n(ϑ̂n − ϑ0)→d n2(0, (j−1(ϑ0)) as n →∞. where j(ϑ0) is the fisher information. 3. spdes with additive noise consider the parabolic spde duθ(t, x) = θuθ(t, x) + ∂2 ∂x2 uθ(t, x)dt + dz(t, x), t ≥ 0, x ∈ [0, 1] (3.1) u(0, x) = u0(x) ∈ l2([0, 1]), (3.2) uθ(t, 0) = uθ(t, 1), t ∈ [0, t ]. (3.3) here θ ∈ θ ⊆ r is the unknown parameter to be estimated on the basis of the observations ofthe field uθ(t, x), t ≥ 0, x ∈ [0, 1].let s3 and s4 be independent stable random variables, s3 is positive α/2-stable with distri-bution sα/2(σ1, 1, 0) and s4 is symmetric α-stable random variable with distribution sα(σ2, 0, 0), σ1 = c −2/α α/2 , σ2 = c −1/α α , cα = ( ∫∞ 0 x−α sin xdx)−1 = [γ(1− α) cos(πα/2)]−1. in this case, in the limiting distribution, s3 and s4 are independent stable random variables witha rate faster than the cylindrical brownian motion case.for x ∈ [0, 1], we observe the process {ut , t ≥ 0} at times {t0, t1, t2, . . .}. we assume thatthe sampling instants {ti , i = 0, 1, 2, . . .} are generated by a poisson process on [0,∞), i.e., t0 = 0, ti = ti−1 + αi , i = 1, 2, ... where αi are i.i.d. positive random variables with a commonexponential distribution f (x) = 1 − exp(−λx). note that intensity parameter λ > 0 is theaverage sampling rate which is assumed to be known. it is also assumed that the sampling process https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 13 ti , i = 0, 1, 2, ... is independent of the observation process {xt , t ≥ 0}. we note that the probabilitydensity function of tk+i − tk is independent of k and is given by the gamma density fi(t) = λ(λt)i−1 exp(−λt)it/(i − 1)!, i = 0, 1, 2, .... (3.4) where it = 1 if t ≥ 0 and it = 0 if t < 0.consider the fourier expansion of the process uθ(t, x) = ∞∑ t=1 uθi (t)φi(x) (3.5) corresponding to some orthogonal basis {φi(x)}∞i=1. note that uθi (t), i ≥ 1 are independent onedimensional stable ornstein-uhlenbeck processes duθi (t) = µθi u θ i (t)dt + β−αi dzi(t) (3.6) uθi (0) = uθ0i ,recall that µi(θ) = k(θ)− β2m i . thus duθi (t) = (k(θ)− β2m i )uθi (t)dt + β−αi dzi(t) (3.7) the random field u(t, x) is observed at discrete times t and discrete positions x . equivalently, thefourier coefficients uθi (t) are observed at discrete time points.define ρ := ρ(λ, θ) = λ λ− κ(θ) + β2m i . the quasi-likelihood estimator is the solution of the estimating equation: g∗n(θ) = 0 (3.8) where g∗n(θ) = β2α i λ(ρ(λ, θ))2 ρ(λ, 2θ) n∑ i=1 uti−1 ( (uti−1 θρ(λ, θ))2 + λ )−1 (uti − ρ(λ, θ)uti−1 ). (3.9) we call the solution of the estimating equation the quasi-likelihood estimator. there is no explicitsolution for this equation.the optimal estimating function for estimation of the unknown parameter θ is gn(θ) = β2α i n∑ i=1 uti−1 [uti − ρ(λ, θ)uti−1 ]. (3.10) the martingale estimation function (mef) estimator of ρ is the solution of gn(θ) = 0 (3.11)and is given by ρ̂n := ∑n i=1 uti−1 uti∑n i=1 u 2 ti−1 . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 14 we do the parameter estimation in two steps: the rate λ of the poisson process can be estimatedgiven the arrival times ti , therefore it is done at a first step. since we observe total number ofarrivals n of the poisson process over the t intervals of length one, the mle of λ is given by λ̂n := n t . theorem 3.1 we have λ̂n → λ a.s. as n →∞. √ n(λ̂n − λ)→d n (0, eλ(1− e−λ)) as n →∞. proof. let vi be the number of arrivals in the interval (i − 1, i ]. then vi , i = 1, 2, . . . , n are i.i.d.poisson distributed with parameter λ. since φ is continuous, we have i{0}(vi) = i{0}(u(ti)) a.s. i = 1, 2, . . . , n. note that 1 n n∑ i=1 i{0}(uti )→ a.s. e(i{0}v1) = p (v1 = 0) = e−λ as n →∞. lln and clt and delta method applied to the sequence i{0}(uti ), i = 1, 2, . . . , n give the results. the clt result above allows us to construct confidence interval for the jump rate λ. corollary 3.1 a 100(1− α)% confidence interval for λ is given by[ n t − z1−α 2 √ 1 n − 1 t , n t + z1−α 2 √ 1 n − 1 t ] where z1−α 2 is the (1− α 2 )-quantile of the standard normal distribution. we obtain the strong consistency and asymptotic normality of the mef estimator. theorem 3.2 when α = 2, we have ρ̂n →a.s. ρ as n →∞ √ n(ρ̂n − ρ)→d n (0, λ−i(1− e−ρ)) as n →∞. proof: by using the fact that every stationary mixing process is ergodic, it is easy to show that if utis a stationary ergodic o-u markov process and ti is a process with nonnegative i.i.d. incrementswhich is independent of ut , then {uti , i ≥ 1} is a stationary ergodic markov process. hence {uti , i ≥ 1} is a stationary ergodic markov process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 15observe that uθi (t) := vi is a stationary ergodic markov chain and vi ∼ n (0, σ2) where σ2 isthe variance of u0. thus by slln for zero mean square integrable martingales, we have 1 n n∑ i=1 uti−1 uti → a.s. e(ut0ut1 ) = ρe(u2 t0 ) 1 n n∑ i=1 u2 ti−1 →a.s. e(u2 t0 ) thus ∑n i=1 uti−1 uti∑n i=1 u 2 ti−1 →a.s. ρ. further, √ n(ρ̂n − ρ) = n−1/2 ∑n i=1 uti−1 (uti − θuti−1 ) n−1 ∑n i=1 u 2 ti−1 . since e(ut1ut2 |ut1 ) = θu2 t1it follows by lemma 3.1 in bibby and srensen [2] n−1/2 n∑ i=1 uti−1 (uti − θuti−1 ) converges in distribution to normal distribution with mean zero and variance equal to e[(ut1ut2 )− e(ut1ut2 |ut1 )]2 = 1− e2(θ−βiδ){2(βi − θ)(βi + 1)}−1. applying delta method the result follows. in the next step, we use the estimator of λ to estimate θ.note that 1 ρ̂n = ∑n i=1 u 2 ti−1∑n i=1 uti−1 uti . thus 1 + β2m 1 − κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1 utiwhich gives β2m 1 − κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1 uti − 1 = − ∑n i=1 uti−1 [uti − uti−1 ]∑n i=1 uti−1 utinow replace λ by its estimator mle λ̂n. β2m 1 − κ(θ) = − ∑n i=1 uti−1 [uti − uti−1 ] t n ∑n i=1 uti−1 utithus θ̂n = κ−1 ( β2m i + ∑n i=1 uti−1 [uti − uti−1 ] t n ∑n i=1 uti−1 uti ) . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 16since the function κ−1(·) is a continuous function, by application of delta method, the followingresult is a consequence of theorem 3.2. theorem 3.3 when α = 2, a) θ̂n → θ a.s. as n →∞ b) √ n(θ̂n − θ)→d n (0, (κ′(θ))−2λ2(1− e−2λ−1(κ(θ)−β2m 1 ))) as n →∞.in the second stage, we substitute λ by its estimator λ̂n. theorem 3.4 when 0 < α < 2, a) θ̂n →a.s. θ as n →∞ b) n(α−1)/α2 (θ̂n − θ)→d ( κ′(θ))−2λ2(1− e−2λ−1(κ(θ)−β2m 1 ) )1/α s4 s3 as n →∞.where s4 and s3 are independent stable random variables. in the second stage, we substitute λ by its estimator λ̂n. the limit distribution is normal onlyin the gaussian case α = 2. 4. spdes with linear multiplicative noise consider the spde with multiplicative noise: duθ(t, x) = (a0 + θa1)uθ(t, x)dt +muθ(t, x)dz(t, x), t ≥ 0, x ∈ [0, 1] (4.1) where m is a known linear operator.equation (4.1) is called diagonalizable if a0, a1 and m have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. denote by ρk , νk and µk , the eigenvalues of the operators a0, a1 and m respectively.then uθ(t, x) = ∑ j≥1 uj,thj . the fourier coefficients have the dynamics duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dzk(t), k ≥ 1 which is the stable black-scholes model whose solution is geometric stable process.let θνk + ρk =: µk(θ), ṽk,t := ln(uk,t /uk,0).conditional characteristic function (ccf) estimator is given by µ̂k(θ) = ṽk,t t 2(α−1)/α2 + σ2 k b1t−((α−1)2+1)/α2 . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 17since µk(θ) is strictly monotone function of θ, by invariance principle of ccfe, under invertibletransformations, we can find the ccfe of the parameter θ θ̂k,t = ṽk,t νkt 2(α−1)/α2 + σ2 k b1νkt−((α−1)2+1)/α2 − ρk νkwhich can be represented as θ̂k,t = θ0 + σkmt νkt 2(α−1)/α2where mt is a square-integrable martingale. due to the lln for martingales, we have strongconsistency.note that in the standard black-scholes case where α = 2, σk = σ, the mle of the driftcoefficient of the geometric brownian motion is given by θ̂t = ln(ut /u0) t + σ2 2 = θ0 + σ wt t . due to the law of iterated logarithm for brownian motion, the mle is strongly consistent as t →∞. theorem 4.1 when 0 < α < 2, a) θ̂k,t is an unbiased estimator of θ.b) θ̂k,t → θ a.s. as t →∞.c) t (α−1)/α2 (θ̂k,t − θ)→d ( σ2 k ν2 k )1/α s4 s3 as t →∞ where s4 and s3 are independent stable random variables.d) if in addition, lim k→∞ ∣∣∣∣σkνk ∣∣∣∣ = 0, then for every fixed t > 0, θ̂k,t → θ a.s. as k →∞and ∣∣∣∣νkσk ∣∣∣∣ (θ̂k,t − θ)→d ( t (α−1)/α2 )1/α s4 s3 as k →∞. remark: the parabolicity condition and the mle consistency condition in general are notconnected. in terms of operator’s order, parabolicity states that the order of operator m from thediffusion term is smaller than half of the order of operators a0 and a1 from deterministic part.the consistency condition assumes that the order of the operator m from the diffusion part doesnot exceed the order of the operator a1 from the deterministic part that contains the parameter ofinterest θ. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 18 5. spdes with nonlinear multiplicative noise consider the spde with multiplicative noise: duθ(t, x) = (a0 + θa1)uθ(t, x)dt +muθ(t, x)dz(t, x), t ≥ 0, x ∈ [0, 1] (5.1) where m is a known nonlinear operator.equation (5.1) is called diagonalizable if a0, a1 and m have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. denote by ρk , νk and µk , the eigenvalues of the operators a0, a1 and m respectively.then uθ(t, x) = ∞∑ j=1 uj,thj . we consider stable cir model as example. here s1 and s2 are dependent stable randomvariables unlike the linear case where s3 and s4 are independent stable random variables. the existence and pathwise uniqueness of solutions to the sdes with non-lipschitz coefficientdriven by spectrally positive levy processes were studied in fu and li [17].consider the nonlinear spde dx(t, x) = θ 2 xxx(t, x)dt + √ x(t, x)dw (t, x) where w (t, x) is a space-time white noise. konno and shiga [26] studied the existence and weakuniqueness of the above equation as a martingale problem for the associated super-brownian mo-tion. the pathwise uniqueness of nonnegative solution still remains open. the main difficultycomes from the unbounded drift coefficient and non-lipschitz diffusion coefficient. wang et al. [42]studied proved a comparison theorem and showed that the solution of the nonlinear spde is distri-bution function valued. they also established pathwise uniqueness. as application they obtainedwell-posedness of martingale problems for two classes of measure-valued diffusions: interactingsuper-brownian motions and interacting fleming-viot processes. he et al. [18] obtained pathwiseunique solution to nonlinear spde with super levy process, which is a combination of space-time gaussian white noises and poisson random measures which is a generalization of work ofxiong [43] where the result for a super-brownian motion with binary branching mechanism wasobtained. using an extended yamada-watanabe argument, xiong [43] established strong exis-tence and uniqueness of the solution to the spde. super-brownian motion (smb), also calledthe dawson-watanabe process introduced by sawson and watanabe is a measure valued processarising as the limit of empirical measure process of a branching particle system. sbm satisfiesa martingale problem. when the state space is r, sbm has a density w.r.t. lebesgue measureand this density valued process x(t, x) satisfies the above spde. when the space r is s single https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 19point, the spde becomes an sde which is cir diffusion dxt = √ xtdwt whose uniqueness isestablished using the yamada-watanabe argument. xiong and yang (2019) studied existence andpathwise uniqueness to an spde with hölder continuous coefficient driven by α-stable colorednoise. the existence of the solution is shown by considering the weak limit of a sequence of sdesystem which is obtained by replacing the laplacian operator in the spde by its discrete version.the pathwise uniqueness is shown by using a backward doubly stochastic differential equation totake care of the laplacian. in the case of d = 1, the pathwise uniqueness of a nonnegative solutionto the corresponding equation was established by yang and zhou [45] for 1 < α < √ 5 − 1 andpathwise uniqueness for √5− 1 < α < 2 is still open.consider spde model with multiplicative noise and mean reversion, where the j-th fouriercoefficient is the stable cox-ingersoll-ross (scir) model: duj,t = (a − θuj,t)dt + σu 1/α j,t−dzj,t , j ≥ 1 (5.2) where a is the mean reverting level and θ is mean reverting speed. recall that for α = 2, for every j ≥ 1, the process zj,t is a standard brownian motion, this is the famous cox-ingersoll-ross (cir)model used for modeling interest rate, which is also used a stochastic volatility process in hestonmodel. note that there are brownian cir models with additive compound poisson type jumps.when 1 < α < 2, zj,t is stable process with levy measure να(dz) = 1{z>0}dz αγ(−α)zα+1 . (5.3) the discontinuous scir model captures the heavy tailed property in the sense of infinite variance.there is empirical evidence from high frequency data available in support of application of purejump models in financial modeling.the scir model has the unique stationary distribution µ with laplace transform given by lµ(λ) = ∫ ∞ 0 e−λxµ(dx) = exp { − ∫ λ 0 αa αθ + σαzα−1 dz } , λ ≥ 0. (5.4) applying itô’s formula, for t ≥ r ≥ 0, we obtain uj,t = e−θ(t−r)uj,r + a ∫ t r e−θ(t−s)ds + σ ∫ t r e−θ(t−s)u 1/α j,s−dzj,s , j ≥ 1. (5.5) let the process be observed at {kh, k = 0, 1, . . . , n} from a single realization {uj,t , t ≥ 0} for fixed h. for simplicity, we take h = 1. this equation can be considered as a first order autoregressive(ar(1)) equation uj,k = ρ+ γuj,k−1 + εj,k , j ≥ 1 (5.6) where γ = e−θ, ρ = aθ−1(1− γ) and εj,k = σ ∫ k k−1 e−θ(k−s)u 1/α j,s−dzj,s , k ≥ 1, j ≥ 1. (5.7) https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 20for b ∈ b(r+), let s2,j,n(b) = n∑ k=1 uj,k−1εk ib(|uj,k−1εj,k |), s1,j,n(b) = n∑ k=1 u2 j,k−1ib(uj,k−1), j ≥ 1. (5.8) it is easy to see that εj,k = uj,k − e(uj,k |fk−1), k ≥ 1, j ≥ 1. (5.9)is a sequence of martingale differences for every fixed j .let s1,j,n := s1,j,n(0,∞), s2,j,n := s2,j,n(0,∞) and recall that γ = e−θ .then θ̂j,n − θ = s2,j,n s1,j,n (5.10) where θ̂n is the conditional least squares estimator (clse) which minimizes n∑ k=1 ε2 j,k = n∑ k=1 [uj,k − e(uj,k |fk−1)]2 = n∑ k=1 [uj,k − ρ− γuj,k−1]2 (5.11) and are given by γ̂j,n = ∑n k=1 uj,k−1 ∑n k=1 uj,k − n ∑n k=1 uj,k−1uj,k ( ∑n k=1 uj,k−1)2 − n ∑n k=1 u 2 j,k−1 , ρ̂j,n = 1 n n∑ k=1 uj,k − γ̂n 1 n n∑ k=1 uj,k−1, θ̂j,n = − log γ̂j,n, âj,n = ρ̂nθ̂n 1− γ̂n .let (s1, s2) have the characteristic function given by e[exp{iλ1s1 + iλ2s2}] := exp { − σα θ2γ(−α) ∫ ∞ 0 e ( 1− exp{iλ1y 2 + iλ2y (α+1)/αvj,1} ) × e ( exp { ie−2θλ1y 2 1− e−2θ + ie−θ(α+1)/αλ2y (α+1)/αvj,2 (1− eθ(α+1))1/α }) dy yα+1 } (5.12) and vj,k := σ ∫ k k−1 e−θ(k−s)e−θ(s−k+1)/αdzj,s , k = 1, 2, j ≥ 1 (5.13) which are i.i.d. with the same distribution as σ ( e−θ − 1 (α− 1)θ )1/α zj,1 which is regularly varying with index α. the limit distribution is normal only in the gaussian case α = 2. following li and ma [31] it can be shown that for every fixed j , if we have 1 < α < (1 + √ 5)/2,then we have as n →∞ (d−2 n s1,j,n, c −1 n s2,j,n) d→(s1, s2) on r2 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 21 where dn = n1/α and cn = n(α+1)/α2 = d (α+1)/α n .for the stable spde model, we have the following result on the consistency and the limitdistribution of the clse: theorem 5.1 if we have 1 < α < (1 + √ 5)/2, then for every fixed j ≥ 1a) θ̂j,n →p θ as n →∞.b) n(α−1)/α2 (θ̂j,n − θ)→d ( σ2 ν2 j )1/α s2 s1 as n →∞. c) if in addition, limj→∞ ∣∣νj ∣∣ =∞, then for every fixed n ≥ 1, θ̂j,t →p θ as j →∞ and ∣∣νj ∣∣ (θ̂j,n − θ)→d σ ( n−(α−1)/α2 )1/α s2 s1 as j →∞.where s2 and s1 are defined in (5.12). remarks1) the limit distribution in the case (1 + √ 5)/2 < α < 2 is still open.2) the process (xj) is exponentially ergodic and hence strongly mixing.3) for the gaussian case (α = 2), the limit results are based on ergodic theory and martingaleconvergence theorem. for the non-gaussian case (1 < α < 2), limit results are obtained by thetheory of regular variation and convergence of point processes.4) let 0 < α < 2 and let zt be a one dimensional α-stable process with levy measure ν(dz).then as n →∞, np (n−1/αzt ∈ ·)→v tν(·). 6. examples (a) consider the linear stochastic heat equation with additive noise du(t, x) = θuxx(t, x)dt + dz(t, x) for 0 ≤ t ≤ t and x ∈ (0, 1) and θ > 0 with periodic boundary conditions.here 2m = m1 = 2 and µj = −θπ2j2, γ > 1/2. the eigenfunctions are hj(x1, . . . , xn) = ( √ 2/π)d(sin(n1x1), . . . , sin(ndxd)), x = (x1, . . . , xn) ∈ rd , j = (n1, . . . , nd) ∈ nd . the corre-sponding eigenvalues are −νj where νj = (n2 1 + . . .+ n2 d).as n →∞, h → 0, nh1+α/ log n → 0, nh2α−1 log n →∞, nh2−α/2+ρ →∞ for some ρ > 0,( n log n )1/α h1/α(θ̂n − θ0)→d 2θ0(αθ0)−1/αs4 s3 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 22where s3 and s4 are independent stable random variables, s3 is positive α/2-stable with distri-bution sα/2(σ1, 1, 0) and s4 is symmetric α-stable random variable with distribution sα(σ2, 0, 0), σ1 = c −2/α α/2 , σ2 = c −1/α α , cα = ( ∫∞ 0 x−α sin xdx)−1 = [γ(1− α) cos(πα/2)]−1. observe the rate of convergence (nh)1/α(log n)−1/α = (t )1/α(log n)−1/α. for α = 2, this rate is t 1/2(log n)−1/2. (b)consider the linear stochastic heat equation with multiplicative noise du(t, x) = θuxx(t, x)dt + u(t, x)dz(t, x) for 0 ≤ t ≤ t and x ∈ (0, 1) and θ > 0 with zero boundary conditions and nonzero initial value u(0) ∈ l2(0, 1). here a1 is the laplace operator on (0, 1) with zero boundary conditions that hasthe eigenfunctions hk(x) = √ 2/π sin(kx), k > 0 and the eigenvalues νk = −k2, ρk = 0, σk = 1, k > 0. uk(t) = ∫ 1 0 hk(x)u(t, x)dx, duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dzk(t).recall that ṽk,t := ln(uk,t /uk,0).the ccfe has the form θ̂k,t = ṽk,t − 1 k2 .(c) consider the following spde du(t, x) = [∆u(t, x) + θu(t, x)]dt + (1− ∆)ru(t, x)dz(t, x). in this case a0 = ∆, a1 = i,m = (1− ∆)r with the eigenvalues νk = 1, ρk = σk , µk = (1 + σk)r .it has a unique solution for any r ≤ 1/2.the ccfe has the form̂ θk,t = ṽk,t k2t 2(α−1)/α2 − (1− σk)2r k2t−((α−1)2+1)/α2 − 1 σk(d) stable cox-ingersoll-ross model xiong and yang [44] studied existence and strong uniqueness of the following spde: duk(t) = (θνk + ρk)uk(t)dt + σk(uk(t))1/αdzk(t), k ≥ 1.the existence of the solution in the case of space-time white noise is shown by consideringthe weak limit of a sequence of sde systems which is obtained by replacing the laplacianoperator in the spde by its discrete version. the weak uniqueness follows from the uniqueness https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 23of solution to the martingale problem for the associated super-brownian motion. in the case of α-stable noise the existence and pathwise uniqueness of the solution is studied in xiong and yang [44]. concluding remark we considered levy process driving term in this paper. using fractionallevy process as the driving term, maximum quasi-likelihood estimation in fractional levy stochasticvolatility model was studied in bishwal [8]. recently, sub-fractional brownian (sub-fbm) motionwhich is a centered gaussian process with covariance function ch(s, t) = s2h + t2h − 1 2 [ (s + t)2h + |s − t|2h ] , s, t > 0 for 0 < h < 1 introduced by bojdecki, gorostiza and talarczyk [13] has received some attentionrecently in finite dimensional models. the interesting feature of this process is that this processhas some of the main properties of fbm, but the increments of the process are nonstationary,more weakly correlated on non-overlapping time intervals than that of fbm, and its covariancedecays polynomially at a higher rate as the distance between the intervals tends to infinity. itwould be interesting to see extension of this paper to sub-fbm case. we generalize sub-fbm tosub-fractional levy process (sub-flp).sub-fractional levy process (sflp) is defined as sh,t = 1 γ(h + 1 2 ) ∫ r [(t − s) h−1/2 + − (−s) h−1/2 + ]dms , t ∈ r where mt , t ∈ r is a levy process on r with e(m1) = 0, e(m2 1 ) < ∞ and without browniancomponent. sflp has the following properties:1) the covariance of the process is given by cov(sh,t , sh,s) = s2h + t2h + e[l(1)2] 2γ(2h + 1) sin(πh) [|t|2h + |s|2h − |t − 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see [?][12], john t [9], [7]and the references therein. recently, xu et al [3] proposed the following viscosity implicit midpointrule (vimr) for nonexpansive mappings: un+1 = αnψ(un) + (1− αn)t (un + un+1 2 ) , n ≥ 0, (1.3) received: 23 aug 2021. key words and phrases. viscosity; hilbert space; convex minimization; asymptotically nonexpansive mapping; varia-tional inequality; fixed point. 19 https://adac.ee https://doi.org/10.28924/ada/ma.1.19 https://orcid.org/0000-0002-3774-0761 eur. j. math. anal. 1 (2021) 20in 2015, ke and ma [4] proposed the generalized viscosity implicit rules of nonexpansive mappingsin hilbert spaces as follows: un+1 = αnψ(un) + (1− αn)t (snun + (1− sn)un+1), n ≥ 0, (1.4) and un+1 = αnun + βnψ(un) + γnt (snun + (1− sn)un+1), n ≥ 0, (1.5) they proved that the generalized viscosity implicit rules 1.4 and 1.5 converge strongly to a fixedpoint of t under certain assumptions, which also solved the v i(1.1). in 2016, motivated by the work of xu [3], zhao et al [5] proposed the following implicit midpointrule for asymptotically nonexpansive mappings: un+1 = αnψ(un) + (1− αn)t n (un + un+1 2 ) , n ≥ 0, (1.6) where t is an asymptotically nonexpansive mapping. they proved that the sequence {un} con-verges strongly to a fixed point of t , which, in addition, also solves the v i(1.1). in 2017, he et la [14] studied the following iterative un+1 = αnψ(un) + (1− αn)t n(βnun + (1− βn)un+1), n ≥ 0 (1.7) in the setting of a hilbert space and proved that the sequence {un} converges strongly to u∗ = pf (t )ψ(u∗) which is also the unique solution of the following v i 〈(i − ψ)u, v − u〉 ≥ 0,∀v ∈ f (t ) (1.8) in this paper, we introduce and study the generalized viscosity implicit rules of asymptoticallynonexpansive mappings in hilbert spaces. more precisely, we consider the following implicititerative algorithm: u1 ∈m un+1 = αnun + βnψ(un) + γnt n ( snun + (1− sn)un+1 ) ∀n ∈ n (1.9) under suitable conditions, we proved that the sequence {un} converge strongly to a fixed point ofthe asymptotically nonexpansive mapping t , which also solves the variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). as applications, we apply our results to solve convexly constrained minimization problem. this wayresults in 1.5 are complemented, extended and generalized. eur. j. math. anal. 1 (2021) 212. preliminaries in the sequel, we always assume that h is a real hilbert space and m is a nonempty, closed,and convex subset of h. the nearest point projection from h onto m, pm, is defined by pm(u) := arg min z∈m ∥∥∥u − z∥∥∥2, u ∈ h. (2.1) namely, pm(u) is the only point in m that minimizes the objective ∥∥∥u − z∥∥∥ over z ∈ m. and pm(u) is characterized as follows: pm(u) ∈m and 〈 u − pm(u), z − pm(u) 〉 ≤ 0 f or al l z ∈m. (2.2) definition 2.1. . a mapping t :m→m is said to be: a): α-inverse strongly monotone if there exists α > 0 satisfying 〈u − v , t u − t v〉 ≥ α‖au − av‖2 ∀u, v ∈m; (2.3) b): l-lipschitz continuous if there exists l ≥ 0 satisfying ‖t u − t v‖ ≤ l‖u − v‖ ∀u, v ∈m; (2.4) c): nonexpansive if ‖t u − t v‖ ≤ ‖u − v‖ ∀u, v ∈m; (2.5) d): asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1,∞) with lim n→∞ kn = 1 suchthat ‖t nu − t nv‖ ≤ kn‖u − v‖ ∀u, v ∈m and ∀n ∈ n; (2.6) e): contraction if there exists the contractive constant α ∈ [0, 1) such that ‖t u − t v‖ ≤ α‖u − v‖ ∀u, v ∈m; (2.7) lemma 2.2. (the demiclosedness principle [10]) . let h be a hilbert space, m be a nonempty closed convex subset of h, and t : m → m be a asymptotically nonexpansive mapping with f ix(t ) 6= ∅. if {un} is a sequence in m such that {un} weakly converges to u and {(i − t )un} converges strongly to 0, then u = t (u) lemma 2.3. let h be a hilbert space. then for all θ, u, v ∈ h, the following inequality holds ‖u − θ‖2 ≤ ‖v − θ‖2 + 2〈u − v , u − θ〉 lemma 2.4. [11]). assume that {αn} is a sequence of nonnegative real numbers such that αn+1 ≤ (1− λn)αn + δn for all n ∈ n, where {λn} ⊆ (0, 1) and {δn} ⊆ r are two sequences satisfying the following conditions: eur. j. math. anal. 1 (2021) 22 (i): ∞∑ n=1 λn =∞ (ii): lim sup n→∞ δn λn ≤ 0 or ∞∑ n=1 |δn| <∞ then lim n→∞ αn = 0 then the sequence {αn} converges to 0. 3. main result we now prove the following new result. theorem 3.1. let m be a nonempty closed convex subset a real hilbert space h, t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞ kn = 1, f ix(t ) 6= ∅ and ψ : m → m be a contraction mapping with the contractive constant α ∈ [0, 1). define a sequence {un} in m as follows: u1 ∈m un+1 = αnun + βnψ(un) + γnt n ( snun + (1− sn)un+1 ) ∀n ∈ n (3.1) where αn, βn, γn, sn ∈ (0, 1) satisfying the following conditions, a1: αn + βn + γn = 1 a2: ∞∑ n=0 αn =∞ a3: 0 < ε ≤ sn ≤ sn+1 < 1 for all n ≥ 0 a4: lim n→∞ γn = 1 and lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 lim n→∞ ‖un − t nun‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). we now show that algorithm 3.1 is well posed. letting bn(u) = αnun + βnψ(un) + γnt n ( snun + (1− sn)un ) ‖bn(u)− bn(v)‖ = ‖γnt n ( snun + (1− sn)u ) − γnt n ( snun + (1− sn)v ) ‖ = ‖γnt n(1− sn)u − γnt n(1− sn)v‖ ≤ γnkn(1− sn)‖u − v‖ since lim n→∞ sn = 0, lim n→∞ kn = 1, lim n→∞ γn = 1 and 0 < ε ≤ sn ≤ sn+1 < 1 for all n > 0, we mayassume that γnkn(1 − sn) ≤ 1 − ε for all n > 0. this implies that bn is a contraction for each eur. j. math. anal. 1 (2021) 23 n. therefore there exists a unique fixed point for bn by banach contraction principle, which alsoimplies that (3.1) is well-defined. we now show that the sequence {un} is bounded. rewriting 3.1, we have un+1 = βnψ(un) + αnun + (1− βn)vn (3.2) where vn = γnt n(snun + (1− sn)un+1) 1− βn remark 3.2. the real sequences that satisfies the above conditions are αn = 1 n , βn = 1 n and γn = 1− 2 n proof. our prove are in six steps. first we prove that the sequence {un} defined by 3.1 is bounded. step 1: letting p ∈ f ix(t ), we have the following estimates ‖un+1 − p‖ = ‖βnψ(un) + αnun + (1− βn)vn − p‖ ≤ βn‖ψ(un)− ψ(p)‖+ βn‖ψ(p)− p‖+ αn‖un − p‖+ (1− βn)‖vn − p‖ ≤ (αβn + αn)‖un − p‖+ βn‖ψ(p)− p‖+ (1− βn)‖vn − p‖ (3.3) ‖vn − p‖ = ‖ γnt n(snun + (1− sn)un+1) 1− βn − p‖ = γnt nsn(un − p) 1− βn + γnt n(1− sn)(un+1 − p) 1− βn ‖ ≤ γnknsn 1− βn ‖un − p‖+ γnkn(1− sn) 1− βn ‖un+1 − p‖ (3.4) putting 3.4 in 3.3, gives the following ‖un+1 − p‖ ≤ (αβn + αn)‖un − p‖+ βn‖ψ(p)− p‖ + γnknsn‖un − p‖+ γnkn(1− sn)‖un+1 − p‖ (1− γnkn(1− sn))‖un+1 − p‖ ≤ (αβn + αn + γnknsn)‖un − p‖+ βn‖ψ(p)− p‖ ‖un+1 − p‖ ≤ (αβn + αn + γnknsn) 1− γnkn(1− sn) ‖un − p‖ + βn 1− γnkn(1− sn) ‖ψ(p)− p‖ eur. j. math. anal. 1 (2021) 24since γn, sn ∈ (0, 1), 1 − γnkn(1 − sn) > 0 and lim n→∞ kn = 1. from the condition (a1), wehave ‖un+1 − p‖ ≤ 1− 1− αβn − αn − γnkn 1− γnkn(1− sn) ‖un − p‖ + βn 1− γnkn(1− sn) ‖ψ(p)− p‖ ] ‖un+1 − p‖ ≤ 1− βn(1− α) 1− γnkn(1− sn) ‖un − p‖ + βn(1− α) 1− γnkn(1− sn) 1 (1− α) ‖ψ(p)− p‖ ] ‖un+1 − p‖ ≤ max { ‖un − p‖, 1 (1− α) ‖ψ(p)− p‖ } therefore by mathematical induction, we have ‖un+1 − p‖ ≤ max { ‖u0 − p‖, 1 (1− α) ‖ψ(p)− p‖ } for all n ≥ n . therefore {un} is bounded. consequently, {ψ(un)} and {vn} are also bounded. step 2: we now prove that the sequence {un+1} converges to {un} as n →∞. that is lim n→∞ ‖un+1− un‖ = 0 ‖un+1 − un‖ = ‖un+1 − t nun + t nun − un‖ = ‖βnψ(un) + αnun + (1− βn)vn − (βn + αn + γn)t n + t nun − un‖ ≤ ‖βnψ(un)− βnt nun‖+ ‖αnun − αnt nun‖ +‖(1− βn)vn − γnt n + t nun − un‖ ≤ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖ +(1− βn)‖vn − γnt n‖+ ‖t nun − un‖ (3.5) ‖vn − γnt nun‖ = ‖ γnsn 1− βn t nun + γn(1− sn) 1− βn t nun+1 − γnt nun‖ ≤ ‖ γnsn 1− βn ‖t nun − t nun‖+ γn(1− sn) 1− βn ‖t nun+1 − t nun‖ ≤ γn(1− sn)kn 1− βn ‖un+1 − un‖ (3.6) eur. j. math. anal. 1 (2021) 25now putting 3.6 in 3.5, we have the following ‖un+1 − un‖ ≤ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖ +(1− βn) [γn(1− sn)kn 1− βn ‖un+1 − un‖ ] + ‖t nun − un‖ ≤ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖ +γn(1− sn)kn‖un+1 − un‖ ] + ‖t nun − un‖ ≤ βn‖ψ(un)− t nun‖+ (αn + 1)‖un − t nun‖ + γn(1− sn)kn‖un+1 − un‖[ 1− γn(1− sn)kn ] ‖un+1 − un‖ ≤ βn‖ψ(un)− t nun‖+ (αn + 1)‖un − t nun‖ ‖un+1 − un‖ ≤ βn 1− γn(1− sn)kn ‖ψ(un)− t nun‖ + (αn + 1) 1− γn(1− sn)kn ‖un − t nun‖ let m :> max { ‖ψ(un)− t nun‖ }, then we have ‖un+1 − un‖ ≤ βnm 1− γn(1− sn)kn + (αn + 1) 1− γn(1− sn)kn ‖un − t nxn‖ ‖un+1 − un‖ ≤ βnm 1− γn(1− sn)(1 + εαn) + (αn + 1) 1− γn(1− sn)(1 + εαn) ‖un − t nun‖ since lim n→∞ αn = lim n→∞ βn = lim n→∞ ‖un − t nun‖ = 0, we then conclude that lim n→∞ ‖un+1 − un‖ = 0 step 3: again we then show that lim n→∞ ∥∥∥un − t (un) ∥∥∥ = 0. estimating as follows we have ‖un − t nun‖ = ‖un − un+1 + un+1 − t nun‖ ≤ ‖un − un+1‖+ ‖un+1 − t nun‖ ≤ ‖un − un+1‖+ ‖βnψ(un) + αnun + (1− βn)vn − t nun ∥∥∥ ≤ ‖un − un+1‖+ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖+ (1− βn)‖vn − γnt nun‖(3.7) ‖vn − γnt nun‖ = ‖ γnt n(snun + (1− sn)un+1) 1− βn − γnt nun‖ ≤ ‖ γnsn 1− βn ‖t nun − t nun‖+ (1− sn)γn 1− βn ‖t nun+1 − t nun‖ ≤ (1− sn)γnkn 1− βn ‖un+1 − un‖ (3.8) eur. j. math. anal. 1 (2021) 26now substituting 3.8 into 3.7, gives the following estimation ‖un − t nun‖ ≤ ‖un − un+1‖+ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖ + (1− βn) ((1− sn)γnkn 1− βn ‖un+1 − un‖ ) ≤ ( 1 + (1− sn)γnkn ) ‖un − un+1‖+ βn‖ψ(un)− t nun‖+ αn‖un − t nun‖ ≤ ( 1 + (1− sn)γnkn ) 1− αn ‖un − un+1‖+ βn 1− αn ‖ψ(un)− t nun‖ ‖un − t nun‖ ≤ ( 1 + (1− sn)γnkn ) 1− αn ‖un+1 − xn‖+ βnm 1− αn therefore from 3.1 condition a4, with lim n→∞ ‖un+1 − un‖ = 0, we can conclude that lim n→∞ ‖un − t nun‖ = 0 (3.9) but we know that from the following fact lim n→∞ ‖un − t (un)‖ ≤ lim n→∞ ‖un − t nun‖+ lim n→∞ ‖t nun − t xn‖ ≤ lim n→∞ ‖un − t nun‖+ lim n→∞ k1‖t n−1un − un‖ (3.10) proving that lim n→∞ ‖t n−1un − un‖ = 0, we have the following estimation ‖t n−1(un)− un‖ = ‖un − t n−1(un)‖ = ‖βn−1ψ(un−1) + αn−1un−1 + (1− βn−1)vn−1 −(βn−1 + αn−1 + γn−1)t n−1un‖ = ‖βn−1ψ(un−1)− βn−1t n−1un + αn−1un−1 − αn−1t n−1un +(1− βn−1)vn−1 − γn−1t n−1un‖ ≤ βn−1‖ψ(un−1)− t n−1un‖+ αn−1‖un−1 − t n−1un‖ +(1− βn−1)‖vn−1 − γn−1t n−1xn‖ (3.11) ‖vn−1 − γn−1t n−1un‖ = ‖ γn−1t n−1(sn−1un−1 + (1− sn−1)un) 1− βn−1 − γn−1t n−1un‖ ≤ γn−1kn−1sn−1 1− βn−1 ‖|un − un−1‖ (3.12) eur. j. math. anal. 1 (2021) 27combining 3.12 and 3.11 we have the following ‖t n−1(un)− un‖ ≤ βn−1‖ψ(un−1)− t n−1un‖+ αn−1‖un−1 − t n−1un‖ +(1− βn−1) [γn−1kn−1sn−1 1− βn−1 ‖|un − un−1‖ ] ≤ βn−1‖ψ(un−1)− t n−1un‖+ αn−1‖un−1 − t n−1un‖ +γn−1kn−1sn−1‖un − un−1‖ ≤ βn−1‖ψ(un−1)− t n−1un‖+ αn−1‖un−1 − t n−1un‖ +γn−1kn−1sn−1‖|un − un−1‖ with the assumption of {αn}, {βn} and lim n→∞ ‖un+1 − un‖ = 0, we can conclude that lim n→∞ ‖t n−1un − un‖ = 0 (3.13) therefore from 3.9 and 3.13, we can see from inequality 3.10, that lim n→∞ ‖un − t (un)‖ = 0 (3.14) step 4: in this step, we will show that wω(xn) ⊆ f ix(t ), where wω(un) := {u ∈ h : there exist a subsequence of {un} converges weakly to u}.suppose that u ∈ wω(un). then there exists a subsequence {uni} of {un} such that uni ⇀ xas i →∞ . from 3.14, we have lim i→∞ ∥∥∥(i − t )xni ∥∥∥ = lim n→∞ ∥∥∥uni − t uni∥∥∥ = 0 . this implies that {(i − t )uni} converges strongly to 0. by using lemma 2.2, we have t u = u, and so u ∈ f ix(t ). step 5: in this step, we will show that lim sup n→∞ 〈q − ψ(q), q − un〉 ≤ 0, (3.15) where q ∈ f (t ) is the unique fixed point of pf (t ) ◦ ψ, that is, q = pf (t )(ψ(z)). since {un} is bounded, there exists a subsequence {uni} of {un} such that uni ⇀ u as i →∞ forsome u ∈ h and lim sup n→∞ 〈q − ψ(q), q − un〉 = lim i→∞ 〈q − ψ(q), q − uni 〉 (3.16) from step 4, we get x ∈ f (t ). by using inequality 2.2, we obtain lim sup n→∞ 〈q − ψ(q), q − un〉 = lim i→∞ 〈q − ψ(q), q − uni 〉 = 〈q − ψ(q), q − u〉 ≤ 0 eur. j. math. anal. 1 (2021) 28 step 6: finally, setting ϕn = βnq + αnq + (1 − βn)vn we show that un → q as n → ∞. again,take q ∈ f (t ) to be the unique fixed point of the contraction pf (t ) ◦ ψ. for each n ∈ n,consider ‖un+1 − q‖2 ≤ ‖ϕn − q‖2 + 2〈un+1 − ϕn, un+1 − q〉 = (1− βn)2‖vn − q‖2 + 2〈βn(ψ(un)− q) + αn(un − q), un+1 − q〉 ≤ (1− βn)‖vn − q‖2 + 2〈βn(ψ(un)− ψ(q)) + βn(ψ(q)− q) + αn(un − q), un+1 − q〉 ≤ (1− βn)2‖vn − q‖2 + 2βn‖ψ(xn)− ψ(q)‖‖un+1 − q‖+ 2αn‖un − q‖‖un+1 − q‖ +2βn〈ψ(q)− q, un+1 − q〉 ≤ (1− βn)2‖vn − q‖2 + 2βnα‖un − q‖‖un+1 − q‖+ 2αn‖un − q‖‖un+1 − q‖ +2βn〈ψ(q)− q, un+1 − q〉 ≤ (1− βn)2‖vn − q‖2 + (2βnα+ 2αn)‖un − q‖‖un+1 − q‖ +2βn〈ψ(q)− q, un+1 − q〉 (3.17) for the fact that ‖vn − q‖2 = ∥∥∥γnt n(snun + (1− sn)un+1) (1− βn) − q ∥∥∥2 ≤ γ2ns 2 nk 2 n (1− βn)2 ‖un − q‖2 + γ2n(1− sn)2k2n (1− βn)2 ‖un+1 − q‖2 + γ2nsn(1− sn)k2n (1− βn)2 〈un − q, un+1 − q〉 ≤ γ2ns 2 nk 2 n (1− βn)2 ‖un − q‖2 + γ2n(1− sn)2k2n (1− βn)2 ‖un+1 − q‖2 + γ2nsn(1− sn)k2n (1− βn)2 ‖un − q‖‖un+1 − q‖ +2αnβn 〈 ψ(un)− ψ(q), t n (un + un+1 2 ) − q 〉 (3.18) now substituting 3.18 into 3.17, we have the following estimation ‖un+1 − q‖2 ≤ γ2ns 2 nk 2 n‖un − q‖2 + γ2n(1− sn)2k2n‖un+1 − q‖2 +γ2nsn(1− sn)k2n‖un − q‖‖un+1 − q‖ +2(βnα+ αn)‖un − q‖‖un+1 − q‖+ 2βn〈ψ(q)− q, un+1 − q〉 ≤ γ2ns 2 nk 2 n‖un − q‖2 + γ2n(1− sn)2k2n‖un+1 − q‖2 + [ γ2nsn(1− sn)k2n + 2(βnα+ αn) ] ‖un − q‖‖un+1 − q‖ +2βn〈ψ(q)− q, un+1 − q〉 (3.19) eur. j. math. anal. 1 (2021) 29again using the fact that( ‖un − q‖ − ‖un+1 − q‖ )2 ≤ ‖un − q‖2 − 2‖un − q‖‖un+1 − q‖ +‖un+1 − q‖2 setting the left hand to zero, we have the following estimate 2‖un − q‖‖un+1 − q‖ ≤ ‖un − q‖2 + ‖un+1 − q‖2 ‖un − q‖‖un+1 − q‖ ≤ 1 2 ‖un − q‖2 + 1 2 ‖un+1 − q‖2 (3.20) putting inequality 3.20 in inequality 3.19, gives the following ‖un+1 − q‖2 ≤ γ2ns 2 nk 2 n‖un − q‖2 + γ2n(1− sn)2k2n‖un+1 − q‖2 + γ2nsn(1− sn)k2n 2 ‖un − q‖2 + (βnα+ αn)‖un − q‖2 + γ2nsn(1− sn)k2n 2 ‖un+1 − q‖2 + (βnα+ αn)‖un+1 − q‖2 +2βn〈ψ(q)− q, un+1 − q〉 ‖un+1 − q‖2 ≤ [γ2nsnk2n (sn + 1) + 2(βnα+ αn) 2 ] ‖un − q‖2 + [γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) 2 ] ‖un+1 − q‖2 +2βn〈ψ(q)− q, un+1 − q〉 thus we have( 1− [γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) 2 ]) ‖un+1 − q‖2 ≤ [γ2nsnk2n (sn + 1) + 2(βnα+ αn) 2 ] ‖un − q‖2 + 2βn〈ψ(q)− q, un+1 − q〉 ‖un+1 − q‖2 ≤ γ2nsnk 2 n (sn + 1) + 2(βnα+ αn) 2− [ γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) ]‖un − q‖2 + 4βn 2− [ γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) ]〈ψ(q)− q, un+1 − q〉 ‖un+1 − q‖2 ≤ ( 1− 2− γ2n(1− sn)2k2n (2− sn)− γ2nsnk2n (sn + 1) 2− [ γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) ])‖un − q‖2 + 4βn 2− [ γ2n(1− sn)2k2n (2− sn) + 2(βnα+ αn) ]〈ψ(q)− q, un+1 − q〉 eur. j. math. anal. 1 (2021) 30therefore from condition lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 in 3.1, we concludes that ‖un+1 − q‖2 ≤ ( 1− 2− 2γ2nk 2 n 2− 2γ2nk 2 n ) ‖un − q‖2 lim n→∞ ‖un+1 − q‖2 = 0 this complete the proof. � theorem 3.3. let m be a nonempty closed convex subset a real hilbert space h, t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞ kn = 1, f ix(t ) 6= ∅ and ω be a constant. define a sequence {un} in m as follows: u1 ∈m un+1 = αnun + βnω + γnt n ( snun + (1− sn)un+1 ) ∀n ∈ n (3.21) where αn, βn, γn, sn ∈ (0, 1) satisfying conditions a1− a4 and ψ(un) = ω lim n→∞ ‖t nun − un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). taking sn = 0the following corollaries holds: corollary 3.4. let m be a nonempty closed convex subset a real hilbert space h, t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞ kn = 1, f ix(t ) 6= ∅ and ψ : m → m be a contraction mapping with the contractive constant α ∈ [0, 1). define a sequence {un} in m as follows:{ u1 ∈m un+1 = αnun + βnψ(un) + γnt n(un+1) ∀n ∈ n (3.22) where αn, βn, γn ∈ (0, 1) satisfying conditions a1− a4 without lim n→∞ sn = 0 lim n→∞ ‖t nun − un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). eur. j. math. anal. 1 (2021) 31 corollary 3.5. let m be a nonempty closed convex subset a real hilbert space h, t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞ kn = 1, f ix(t ) 6= ∅ and u ∈m be a constant. define a sequence {un} in m as follows:{ u1 ∈m un+1 = αnun + βnω + γnt n(un+1) ∀n ∈ n (3.23) where αn, βn, γn ∈ (0, 1) satisfying conditions a1− a4 without lim n→∞ sn = 0 lim n→∞ ‖t nun − un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). 4. application to convex minimization problems in this section, we study the problem of finding a minimizer of a convex function φ defined from areal hilbert space m to r.consider the optimization problem min x∈c φ(x) (4.1)where φ : m → r is a convex and differentiable function. assume 4.1 is consistent, and let ω 6= ∅ be its set of solutions. the gradient projection algorithm generates a sequence {un} via theiterative procedure: un+1 = pm(un − δ∇φ(u)) (4.2)if ∇φ is θ−inverse strongly monotone mapping and δ(0, 2θ). the following basic results are wellknown. remark 4.1. it is well known that if φ : m → r be a real-valued differentiable convex functionand u∗ ∈m, then the point u∗ is a minimizer of φ on m if and only if dφ(u∗) = 0. definition 4.2. a function φ :m→ r is said to be strongly convex if there exists α > 0 such thatfor every u, v ∈m and λ ∈ (0, 1), the following inequality holds: φ(λu + (1− λ)v) ≤ λφ(u) + (1− λ)φ(v)− α‖u − v‖2. (4.3) lemma 4.3. let e be normed linear space and φ : m → r a real-valued differentiable convex function. assume that φ is strongly convex. then the differential map dψ : m→m is strongly monotone, i.e., there exists a positive constant k such that 〈dφ(u)− dφ(v), u − v〉 ≥ k‖u − v‖2 ∀ u, v ∈m. (4.4) the prove of the following theorem follows from 3.1 eur. j. math. anal. 1 (2021) 32 theorem 4.4. let m be a nonempty closed convex subset a real hilbert space h. for the minimization problem 4.1, assume that φ is (gateaux) differentiable and the gradient ∇φ is a θ−inverse-strongly monotone mapping for some positive real number θ. let ψ : m → m be a contraction with coefficient α ∈ [0, 1). for a given u1 ∈m, let {un} be a sequence generated by:{ u1 ∈m un+1 = αnun + βnψ(un) + γnpm(1− δ∇φ)(snun + (1− sn)(un+1)) ∀n ∈ n (4.5) where αn, βn, γn, sn ∈ (0, 1) satisfying the following conditions a1: αn + βn + γn = 1 a2: lim n→∞ k2n − 1 αn = 0 a3: ∞∑ n=0 αn =∞ a4: lim n→∞ γn = 1 and lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 then {un} converges strongly to a solution (u∗) of the minimization problem 4.1, which is also the unique solution of the variational inequality 〈(i − ψ)u, p − u〉 ≥ 0 p ∈ f (t ). conflict of interest:the authors declare that they have no competing interests. availability of data and materials:no data were used to support this study. funding:no funding was given towards this manuscript. authors contributions:all authors have contributed equally and significantly in writing this paper and also readand approved the final manuscript. acknowledgement:the authors are very grateful to the editor and anonymous referees for their helpfulcomments. references [1] h. attouch, viscosity approximation methods for minimization problems, siam j. optim. 6 (3) (1996) 769-806. https://doi.org/10.1137/s1052623493259616.[2] a. moudafi, viscosity approximation methods for fixed-points problems, j. math. anal. appl. 241 (1) (2000) 46-55. https://doi.org/10.1006/jmaa.1999.6615.[3] h.k. xu, m.a. alghamdi, n. shahzad, the viscosity technique for the implicit midpoint rule of nonexpansive mappingsin hilbert spaces, fixed point theory appl. 2015 (2015) 41. https://doi.org/10.1186/s13663-015-0282-9. https://doi.org/10.1137/s1052623493259616 https://doi.org/10.1006/jmaa.1999.6615 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asymptotically nonexpansivemappings in hilbert spaces, appl. math. sci. 11 (2017), 549-560. https://doi.org/10.12988/ams.2017.718. https://doi.org/10.1186/s13663-015-0439-6 http://doi.org/10.22436/jnsa.009.06.86 http://doi.org/10.22436/jnsa.009.06.86 https://doi.org/10.12988/ams.2017.718 http://doi.org/10.30538/oms2017.0011 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1112/s0024610702003332 https://dx.doi.org/10.1073/pnas.53.5.1100 https://doi.org/10.12988/ams.2017.718 1. background 2. preliminaries 3. main result 4. application to convex minimization problems references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 2doi: 10.28924/ada/ma.3.2 analysis of neuronal oscillations of fractional-order morris-lecar model tahmineh azizi department of mechanical engineering, florida state university, usa correspondence: tazizi@fsu.edu abstract. fractional calculus is a new approach for modeling biological and physical phenomena withmemory effects. fractional calculus uses differential and integral operators including non-integerorders to study the non-linear behavior of physical and biological systems with some degrees offractionality or fractality. since the long memory properties of neuronal responses can be betterexplained using fractional derivative, in this study we generalize the integer-order morris-lecarmodel in the fractional-order domain to better modeling of neuron dynamics. to investigate thecomplex spiking patterns of fractional-order morris-lecar neural system the fractional calculus hasbeen applied to build this new mathematical model. we compare the results with integer-order morris-lecar model. the analytical solutions of these equations cannot explicitly be obtained. therefore, tofind the dynamical behaviors of solutions, we used approximation and numerical schemes. dependingon the different parameters values for 0 < η ≤ 1, the fractional-order morris-lecar reproducesquiescent, spiking and bursting activities the same as its original model but for higher input current.we numerically discover the hopf bifurcation, saddle node bifurcation of limit cycle and homoclinicbifurcation for this model for different input current and derivative orders. taking the advantages ofthe fractional order derivative, for a variety of orders, we define different classes of this model whichhelps to better extract all the complicated dynamics of this single neuron model. 1. introduction recently, fractional calculus has been frequently used by many researchers in biology, physics,chemistry and biochemistry, hydrology, medicine, and finance and its application in modelingcomplex phenomena has increased its popularity and the number of publications in above area [1–6].the main characteristic of fractional order derivative is called the "memory effect" and it has beenexperimentally proved that the fractional order differential or integral equations models are morerealistic to demonstrate the complex behavior of some biological or physical systems includingfractality and memory compared to their odes of integer-order systems [2, 6]. complexity in thiscontext combined the recent advances in neuroscience with the concepts from fractal geometry andnonlinear dynamics to form a new approach within the life sciences which is useful in controlling received: 17 apr 2022. key words and phrases. fractional calculus; nsfd method; hopf bifurcation; homoclinic bifurcation; grunwald-letinkov method. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 2the dynamics of fractal processes in this area [7, 8].understanding the complicated functioning of the neuronal cells and exploring the molecular andcellular mechanisms of their network have been one of the greatest challenges in different fieldsof science . the progress and advances in computational neuroscience could help scientists tobetter understanding of the performance of brain and neuron cells and better fighting with diseasesrelated to neuron cells such as parkinson’s and depression.non-linear dynamical system theory has a very important role in the computational neuroscienceresearch [9–13]. in 1948 hodgkin by injecting a dc-current of varying amplitude discovered thatsome preparations could show repetitive spiking activities with arbitrarily low frequencies, whilethe others discharged in a narrow frequency band [9, 13–15]. his finding motivated rinzel andermentrout to discover that different bifurcation mechanisms of excitability may cause the differencein neuronal behavior [9,16,17]. basically, if assume the applied current iapp as a control parameter,we can easily see the transition in behavior of a neuron which corresponds to a bifurcation fromequilibrium to a limit cycle attractor. that is when iapp is small, the cell remains quiescent andwith increasing the injected current, the cell starts to fire repetitive spikes [9–13,18,19].according to moaddy. k, et. al [20], the fractional-order models can better explain the long memorydependence of the neuron response. one of the most interesting properties of neural system isadaptation to changes in stimulus. it has been shown that a single neuron has a single time scaleadaptation, however, there are some neurons with multiple time scale adaptation to responsesconsistent with fractional-order derivatives, means that the firing rate for these neurons acts asfractional derivative of slowly varying stimulus parameters [21,22]. therefore, the other advantagesof neuronal fractional derivatives is their ability to adapt to changes in stimulus in different timescales. moreover, shi. m, et. al [23] proved that the fractional-order derivative demonstrates thereal dielectric behaviors and the history memory property of membranes, cells and so on. accordingto their finding, non integer derivative activates the slow ion channel with higher speed, and helpsto activate fast spiking modulation which forms different kinds of bursting behaviors. one importantfact about using fractional order model their ability to display different dynamical behaviors such aschaotic and periodic firing for the same parameter values as the order of derivative is varying [24].on the other hand, using these new parameters as the order of fractional derivative operatorsenhances the controllability of behavior of the neuron cells [25]. the fractional order models playan important role in determining the firing properties of neuronal models and these importantproperties of models with fractional order derivatives including depiction of long term memory andthe multiple time scale adaptation have motivated many efforts to use them as a perfect frameworkto cover the complicated dynamics of many different neuronal cells such as fractional cable model,izhikevich neuron model, and fitzhugh-rinzel bursting neuron model [26–28].due to the complexity of nerve systems, it is impossible to fully understand the various phenomena https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 3in neuroscience only using the integer order models, since it does not meet all neuronal propertiesand complicated behaviors of neurons. thus, in this study we use the fractional calculus to explorethe dynamics of fractal processes using the fractional calculus and apply this dynamical approach onmorris-lecar model to catch all the spiking properties of this neuron and to simulate the fluctuationsof this single neuron cell and obtain biological physiological characteristics of it. we have selectedmorris-lecar model because it is a reduced and simpler version of the hodgkin-huxley equationsand preserves many important characteristics of neuronal dynamics such as generation of actionpotentials, threshold for firing spike, and sustained oscillations with increasing the applied current.because the solutions of fractional morris lecar model (fml) may not be explicitly obtained, weuse numerical methods to approximate the solutions of this model. we compare these results withits original integer order model using phase portrait analysis. by considering this fractional ordermodel, we can explain all the possible geometric mechanisms underlying each of neuronal activitiesof morris-lecar model. 2. grünwald-letinkov approximation we define the fractional differential as the following form [29,30] dγy (t) = f (t, y (t)), y (t0) = y0 where γ > 0 represents the order of derivative and dγ denotes the fractional derivative which isgiven by: dγy (t) = jk−γdky (t) where γ ∈ (k − 1, k ], for k = 1, 2, . . . and integral operator jk called the riemann-liouville of kth-order which is obtained by the following formula jky (t) = 1 γ(k) ∫ t 0 (t − τ)(k−1)y (τ) dτ, t > 0 where γ(.) denotes the gamma function.to apply the micken’s (nsfd) [31–33], we need to find the fractional order derivative using thegrünwaldletinkov (g-l) approximation for model equations as the form dγy (t) = lim s→0 s−γ t∑ i=0 (−1)i ( γ i ) y (t − i s) (1) where t = [t]/s and [.] used to show the integer value and s represents the step size. thus,equation (1) would be discretized as t∑ i=0 cγi y (tk−i) = f (tk , y (tk)) (2) https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 4where tk = k s and cγi are the coefficients for (g-l) approximation written as cγi = [ i − 1− γ i ] cγi−1, cγ0 = s−γ i = 1, 2, . . . next we introduce the non-standard finite difference schemes.to discretize a systems of differential equations, both ordinary differential equations (odes) andpartial differential equations (pdes), one may apply the mickens nsfd discretization methodwhich is more flexible in construction rather than standard finite difference method and thereforehas better performance. this method checks the positivity of solutions and is concerned aboutboundedness and monotonicity of them. another advantage of using nsfd schemes is their abilityto preserve the structure and properties of the systems of differential equations and therefore, weapply nsfd schemes on the general compartmental model in the form: d y dt = f (y ) (3) however, to use the non-standard scheme we need to check that if non-local approximation is usedand or we need to have a non traditional discretization of derivatives and also we may need to use anon-negative function φ(h) = s+o(s2). to apply nsfd scheme, we consider a grid tk = t0+k s ,such that s > 0, and we approximately write the discretized function y as yk ≈ y (tk). next, wediscretize (3): d y dt = yk+1 − yk φ(s) +o(φ(s)) (4) when s → 0 we have d y dt ≈ yk+1 − yk φ(s) (5) where real valued φ(s) as a function of the step size s need to satisfy the following properties [34]:(i) φ(s) = s +o(s2),(ii) φ(s) ∈ (0, 1), ∀ s ∈ (0,∞)here, the equality (4) is equivalent with the integer order derivative as follow: d y dt = lim s→0 [ y (t + s)− y (t) φ(s) +o(φ(s)) ] = lim s→0 [ y (t + s)− y (t) s ] lim s→0 [ s φ(s) ] + lim s→0 o(φ(s)) = ẏ (t) as s → 0 the discrete form in (4) converges to its associated continuous derivative. nsfd methodsare convergent without any restriction related to step size s but this is not always true for sfdmethods which depend on the step size s . moreover, when we discretize a system using nsfd https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 5method, if the original system is persistent, and solutions are stable and convergent, these proper-ties remain the same after discretization, but not for the case we use sfd to discretize the systemof differential equations. 3. description of model equations to demonstrate the generation of action potential, kathleen morris and harold lecar proposeda simple model, morris-lecar model, in 1981 [35] that is a reduction version of the four dimensionalhodgkin-huxley model preserving the main properties of spike generations with much simplermathematical and computational analysis [35, 36]. this model describes the electrical activities ofneurons using a system of non-linear ordinary differential equations and includes three channelsa potassium channel, a leak and a calcium channel and has the following form cm dv dt = iapp − gl(v − el)− gkw(v − ek)− gcam∞(v )(v − eca) = iapp − iion(v, w), dw dt = φ(n∞(v )− w)/τw (v ), (6) where m∞(v ) = 1 2 [1 + tanh((v − v1)/v2)], (7) τn(v ) = 1/cosh((v − v3)/(2v4)), (8) n∞(v ) = 1 2 [1 + tanh((v − v3)/v4)]. (9) and iion(v, w) = gl(v − el) + gkw(v − ek) + gcam∞(v )(v − eca) (10) where v demonstrates membrane potential, and w the activation variable of the persistent k+current, so it is a two-dimensional vector (v, w). ek , eca, and el denote the nernst equilibriumpotentials. iapp demonstrates the injected current and iion the ionic current. parameter φ is atemperature factor. gl is leak membrane conductance, gk is potassium membrane conductance and gca is calcium membrane conductance. moreover, cm is the total membrane capacitance. also, thevoltage-sensitive steady-state activation function m∞(v ) and n∞(v ), and the time constant τw (v )can be measured experimentally. the non-linear dynamics of the original morris-lecar model havebeen studied by different researchers during recent decades [18,37–43]. in the next section we willlook at the fractional-order morris-lecar model and its spiking patterns. 3.1. fractional morris-lecar model. now, we apply the basic theorems of the fractional calculuson model (6). in morris-lecar model, we write the total membrane current to being the sum of ioniccurrents and the capacitive current: iapp = iion + icm https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 6for the fractional-order model, we define: icm = cm dη v where, 0 < η ≤ 1 and dη is defined in the following form [44]: dη v (t) = lim h→0 h−η [t] h∑ i=0 (−1)i ( η i ) v (t − i h) (11) we do the same for the second equation: dη w(t) = lim h→0 h−η [t] h∑ i=0 (−1)i ( η i ) w(t − i h) (12) where [t] denotes the integer part of t and h is the step size.after discretization, (11) and (12) become: ∑ [t] h i=0 c η i v (tk−i) = f (tk , v (tk)), ∑ [t] h i=0 c η i w(tk−i) = g(tk , w(tk)), where tk = kh for k = 1, 2, 3, . . . and cηi are the grunwald-letinkov coefficients as: cηi = ( 1− 1 + η i ) cηi−1, cη0 = h−η i = 1, 2, . . . then, we apply the non-standard finite difference (nsfd) schemes proposed by mickens [31–33]and replace the step size h by a function ψ(h). next, we discretize the equations (11) and (12)following the grunwald-letinkov discretization, using v (tk) = vk , w(tk) = wk , we have: cm ∑k+1 i=0 c η i vk+1−i = iapp − gl(vk − el)− gkwk(vk − ek)− gcam∞(vk)(vk − eca), ∑k+1 i=0 c η i wk+1−i = φ(n∞(vk)− wk)/τwk (vk), (13) where m∞(vk) = 1 2 [1 + tanh((vk − v1)/v2)], (14) τn(vk) = 1/cosh((vk − v3)/(2v4)), (15) n∞(vk) = 1 2 [1 + tanh((vk − v3)/v4)]. (16) and iion(vk , wk) = gl(vk − el) + gkwk(vk − ek) + gcam∞(vk)(vk − eca) (17) https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 7 table 1. parameter values for the fractional-order morris-lecar model. after some algebra, vk+1 = iapp − ∑k+1 i=1 c η i vk+1−i − gl(vk − el)− gkwk(vk − ek)− gcam∞(vk)(vk − eca) cm c η 0 wk+1 = φ(n∞(vk)− wk)− τwk (vk) ∑k+1 i=1 c η i wk+1−i cη0 τwk (vk) (18) where cη0 = ψ(h)−η, ψ(h) = sin(h) the fractional-order morris-lecar model displays different ranges of dynamics such as hopfbifurcation, saddle node on invariant limit cycles (snlc) and homoclinic bifurcation. we keepthe same biological parameters as the original morris-lecar model. we have represented theseparameters for these three different dynamics in table (1) [18]. we assume iapp as a controlparameter for numerical simulations. 3.2. local stability analysis of fractional order morris-lecar model. in neuroscience, it’s usuallyhard to extract analytically the dynamics of the neuronal systems and we may need to use somegeometrical and qualitative techniques such as phase portrait analysis. phase portraits demonstratethe evolution of state variables in time with different initial states. by looking at the phase portrait,we can observe the qualitative behavior of the system without knowing the model equations. toanalyze the local dynamics of fractional order morris-lecar model, we apply a useful theoremin dynamical systems theory, called the hartman-grobman theorem [45–47]. according to this https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 8theorem non-linear fractional order morris-lecar system{ vk+1 = f (vk , wk) wk+1 = g(vk , wk) (19) sufficiently near equilibrium (v, w) = (v ∗, w∗) is locally topologically equivalent to the linear partof the system. first we transform the fixed point (v ∗, w∗) of the system (19) to the origin by thetranslation v = v ∗ + v̄ and w = w∗ + w̄ . if we split off the linear part of the system from itsnon-linear part, we have [ v̄ w̄ ] =  ∂f (v̄ , w̄) ∂v̄ ∂f (v̄ , w̄) ∂w̄ ∂g(v̄ , w̄) ∂v̄ ∂g(v̄ , w̄) ∂w̄  [ v̄ w̄ ] + f̃ (v̄ , w̄) g̃(v̄ , w̄)  (20) where f̃ and g̃ represent the non-linear part of the system (19) and ∂f (v̄ , w̄) ∂v̄ = − ∑k+1 i=1 c η i − gl − gkw̄ − gcam∞(v̄ ) cm c η 0 ≡ a ∂f (v̄ , w̄) ∂w̄ = −gk v̄ cm c η 0 ≡ b ∂g(v̄ , w̄) ∂v̄ = φn′∞(v̄ ) cη0 τwk (v̄ ) ≡ c ∂g(v̄ , w̄) ∂w̄ = −φ− τwk (v̄ ) ∑k+1 i=1 c η i cη0 τwk (v̄ ) ≡ d (21) to find the stability of the interior equilibrium point of the system, we need to look at the linearpart of (20) which is given by (21). at first, we assume that for the voltage ek < v̄ < eca. thenwe have b < 0, c > 0, and d < 0 and a can be either positive or negative. m∞(v̄ ) which definesthe slope of the calcium activation function can make a > 0. on the other hand, for a < 0, theequilibrium point is asymptotically stable because a + d < 0 and ad − bc > 0. moreover, for b < 0 the equilibrium point is stable because the negativity of slope of v̄ -nullcline, −ab < 0.for the case that a > 0 the equilibrium point is a saddle point and unstable because of thepositivity of slope of the w̄-nullcline −cd > 0.for −ab > −c d , the equilibrium point is a saddle point and unstable because ad − bc < 0.however, for −ab < −c d and a + d < 0, the equilibrium point is stable and for −ab < −c d and a+d > 0, the equilibrium point is unstable.finally, for the case that a > 0 and the speed of potassium dynamics φ is small, then the equilibriumpoint is unstable. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 94. numerical results in this section, we use some numerical simulations to study the qualitative behavior such aslocal bifurcations of the fractional order morris-lecar model for different fractional order η andapplied current iapp . one of the most common types of bifurcation in neuroscience is saddle nodebifurcation of limit cycle or snlc, and this bifurcation occurs when with increasing the controlparameter, here, applied current iapp , two stable and unstable limit cycles which are associated tothe stable node and saddle point respectively, close to each other, collide and at the bifurcationtime, a limit cycle appears. with increasing iapp further, this limit cycle disappears. figures(1)-(4) exhibit different spiking behaviors for fractional morris-lecar model (13), when we increase iapp = 5, 30, 45, 100 using snlc parameters value in table (1). the solution of the integer ordermodel (6) has been demonstrated in the third row to compare with fractional-order morris-lecarmodel of different order. for the case of hopf bifurcation in figures (5)-(9), with increasing the applied current iapp , themodel (13) displays the occurrence of limit cycle corresponding to hopf bifurcation like the originalmodel (6) but for orders η = 0.3, .0.5, 0.7, 0.9 the fractional order model needs greater value forinput current to start the bifurcation.the topological normal form of the model (13) in polar coordinate for the case of hopf bifurcationhas the form:  ∑k+1 i=0 c η i rk+1−i = αr + a r3 ∑k+1 i=0 c η i θk+1−i = ω0 + βr2 (22) after simplification, the fractional-order system which is linear and time-invariant has the followingform:  rk+1 = αr + a r3 − ∑k+1 i=1 c η i rk+1−i cη0 θk+1 = ω0 + βr2 − ∑k+1 i=1 c η i θk+1−i cη0 (23) where, α and ω0 represent the real part and imaginary part of the eigenvalues of the jacobian matrixfor the model (13) around its equilibrium point respectively, a is called first lyapunov coefficient.for a > 0 there should exist an unstable limit cycle, bifurcating from the equilibrium and it indicatesthe appearance of subcritical hopf bifurcation and for a < 0 we have stable limit cycle solutionand supercritical hopf bifurcates from the equilibrium. here, β does not have any dynamical effect. θ represents the angle of oscillations. if θ̇ > 0, it means the frequency of damped or sustained https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 10 figure 1. occurrence of saddle node bifurcation of limit cycle or snlc in fractionalmorris-lecar model (13) for iapp = 5, third row displays the trajectory of the originalmodel (6) with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 11 figure 2. occurrence of saddle node bifurcation of limit cycle or snlc in fractionalmorris-lecar model (13) for iapp = 30, third row displays the trajectory of theoriginal model (6) with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 12 figure 3. occurrence of saddle node bifurcation of limit cycle or snlc in fractionalmorris-lecar model (13) for iapp = 45, third row displays the trajectory of theoriginal model (6) with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 13 figure 4. occurrence of saddle node bifurcation of limit cycle or snlc in fractionalmorris-lecar model (13) for iapp = 100, third row displays the trajectory of theoriginal model (6) with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 14 figure 5. occurrence of hopf bifurcation in fractional morris-lecar model (13) for iapp = 20, third row displays the trajectory of the original model (6) with the sameapplied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 15 figure 6. occurrence of hopf bifurcation in fractional morris-lecar model (13) for iapp = 88, third row displays the trajectory of the original model (6) with the sameapplied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 16 figure 7. occurrence of hopf bifurcation in fractional morris-lecar model (13) for iapp = 90, third row displays the trajectory of the original model (6) with the sameapplied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 17 figure 8. occurrence of hopf bifurcation in fractional morris-lecar model (13) for iapp = 95, third row displays the trajectory of the original model (6) with the sameapplied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 18 figure 9. occurrence of hopf bifurcation in fractional morris-lecar model (13) for iapp = 220, third row displays the trajectory of the original model (6) with the sameapplied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 19oscillations around ω0 is increasing. on the other hand, for θ̇ < 0 the frequency of damped orsustained oscillations around ω0 is decreasing.in neuroscience point of view, the hopf bifurcation happens when the behaviors of neuron changefrom resting to spiking (the stable constant solutions are corresponding to the resting state andspiking state shows the existence of periodic solutions).the other common type of dynamical behavior for a neuron cell occurs when with increasing thecontrol parameter, a saddle point and a limit cycle collide, this bifurcation called saddle-homoclinicbifurcation. the period of the periodic orbit that appears at the moment of bifurcation goes to infinityand with further increasing of control parameter this periodic orbit disappears. figures (10)-(14),demonstrate the appearance and disappearance of saddle-homoclinic bifurcation in the model (13)with increasing the applied current iapp = 23, 40, 50, 60, 70 like the original model (6) but liketwo previous bifurcations, for fractional order model of orders η = 0.3, .0.5, 0.7, 0.9 the neuronneeds higher input current iapp to bifurcate.in neuroscience point of view, when saddle homoclinic bifurcation happens, we expect theappearance or disappearance of spiking behavior. 5. discussion fractional-order excitable systems can be physically considered as a memory dependent phe-nomenon which display oscillatory behaviors for certain types of neuron models. in this research,we have studied the neuronal spiking patterns of fractional morris-lecal neuron model where thefractional-orders could change the responses of the model from periodic to non-periodic, and wehave compared its dynamics to the original morris-lecar model. the original morris-lecal neuronmodel which is a reduction version of hodgkin-huxley model includes two equations with integerorder derivatives and three ionic channels, a potassium channel, a leak and a calcium channel.we have preserved the same ionic channels and to find the fractional order morris-lecar model,we have applied the non-standard finite difference (nsfd) schemes on this system of equationssince they have a better performance than standard finite difference methods. then we have dis-cretized the model using the grunwald-letinkov discretization. we use effective numerical methodsto display the solution of fractional order morris-lecar model. to explore the exciting behaviorsof fractional morris-lecar model, we have conducted different numerical simulations with changinginput currents. it was obvious that the solutions depends on the fractional-order parameters. wehave shown that the fractional morris-lecar model with the same biological parameters values asthe original model, displays the same firing patterns such as quiescent and spiking behaviors butfor different values of input currents. we have noticed that in this case, the saddle node bifurcationof limit cycle (snlc), hopf bifurcation and saddle-homoclinic bifurcation happen at larger values forinjected current compare to the original model and we have derived these bifurcations analyticallyusing rigorous normal form theory. similar to the original morris-lecar model, its fractional-order https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 20 figure 10. occurrence of saddle-homoclinic bifurcation in fractional morris-lecarmodel (13) for iapp = 23, third row displays the trajectory of the original model (6)with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 21 figure 11. occurrence of saddle-homoclinic bifurcation in fractional morris-lecarmodel (13) for iapp = 40, third row displays the trajectory of the original model (6)with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 22 figure 12. occurrence of saddle-homoclinic bifurcation in fractional morris-lecarmodel (13) for iapp = 50, third row displays the trajectory of the original model (6)with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 23 figure 13. occurrence of saddle-homoclinic bifurcation in fractional morris-lecarmodel (13) for iapp = 60, third row displays the trajectory of the original model (6)with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 24 figure 14. occurrence of saddle-homoclinic bifurcation in fractional morris-lecarmodel (13) for iapp = 70, third row displays the trajectory of the original model (6)with the same applied current. https://doi.org/10.28924/ada/ma.3.2 eur. j. math. anal. 10.28924/ada/ma.3.2 25model undergoes a transition between integrator and resonator. when saddle-node bifurcationhappens, the neuron is called an integrator means that there is no damped subthreshold oscilla-tions. on the other hand, when hopf bifurcation happens the neuron is called a resonator withdamped subthreshold oscillations. using the fractional order derivative, we have added a newparameter as the order of derivatives that helped us to control the spiking patterns of the neuroncell. taking the advantages of this type modeling, we investigated how the classical order systemschanges its complex dynamics such as firing patterns and also firing frequency, when they turn tobe fractional order systems. this work improved the preceding 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https://doi.org/10.4236/am.2020.113017 https://doi.org/10.4236/am.2020.117044 https://doi.org/10.9734/bpi/mono/978-93-91312-16-9 https://doi.org/10.1142/s0218127421501704 https://doi.org/10.1016/j.chaos.2022.111959 https://doi.org/10.1016/j.chaos.2022.111959 https://doi.org/10.1016/j.cam.2004.01.033 https://doi.org/10.1090/s0002-9939-1960-0121542-7 1. introduction 2. grünwald-letinkov approximation 3. description of model equations 3.1. fractional morris-lecar model 3.2. local stability analysis of fractional order morris-lecar model 4. numerical results 5. discussion references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 9doi: 10.28924/ada/ma.2.9 efficient numerical schemes for computations of european options with transaction costs md. shorif hossan1 , md. shafiqul islam1 , md. kamrujjaman2,∗ 1department of applied mathematics, university of dhaka, dhaka, bangladesh shorif@du.ac.bd, mdshafiqul@du.ac.bd 2department of mathematics, university of dhaka, dhaka, bangladesh kamrujjaman@du.ac.bd ∗correspondence: kamrujjaman@du.ac.bd abstract. this paper aims to find numerical solutions of the non-linear black-scholes partial dif-ferential equation (pde), which often appears in financial markets, for european option pricing inthe appearance of the transaction costs. here we exploit the transformations for the computationalpurpose of a non-linear black-scholes pde to modify as a non-linear parabolic type pde with reli-able initial and boundary conditions for call and put options. several schemes are derived rigorouslyusing the finite volume method (fvm) and finite difference method (fdm), which is the novelty of thispaper. stability and consistency analysis assure the convergence of these schemes. we apply theseschemes to various volatility models, such as the leland, boyle and vorst, barles and soner, andrisk-adjusted pricing methodology (rapm). all the schemes are tested numerically. the convergenceof the obtained results is observed, and we find that they are also reliable. finally, we display allthe approximate results together with the exact values through graphical and tabular representations. 1. introduction understanding and accurately evaluating transaction costs in a financial market is vital forsecurity trading, asset pricing, stock market regulation, and many other issues. during the last fewdecades, pricing options more accurately after including realistic assumptions-such as transactioncost, getting more importance from both the traders and the investors.the literature’s [1–6], contains descriptive discussions of options. fischer black and myronscholes [7] worked jointly, and first disclosed the concept of the black-scholes model for options pricing and corporate liabilities, and was published in 1973, while robert merton [8] advanced thismodel in the article "theory of rational option pricing" in the same year. their derived equation isbased on the assumption that there are no fees for buying and selling options and stocks, as wellas no trade barriers (i.e., no commissions and transaction costs). in other words, this model makes received: 20 dec 2021. key words and phrases. nonlinear black-scholes pde; option pricing; volatility model; finite volume method; finitedifference method. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.9 https://orcid.org/0000-0003-4115-9002 https://orcid.org/0000-0001-7121-3386 https://orcid.org/0000-0002-4892-745x eur. j. math. anal. 10.28924/ada/ma.2.9 2a friction-less assumption (which is indispensable, as actual costs correlated with practical marketapplications) to implement a hedging plan for any contingent claim of the european type.various studies have been conducted about the linear black-scholes model [9–15] though itadopts the unrealistic assumption of no transaction costs. several studies have been attempted toevaluate the price of european options [16–23], american options [24–28], asian options [29, 30],and barrier options [31] in a completely friction-less market. recently, the fractional black-scholesmodel [32–34] received some attention.contrastingly, the non-linear black-scholes pde, where the non-linear term denotes the pres-ence of transaction costs, is of great importance to our contemporary world over some time both interms of approach and applicability. several models [35] consider transaction costs: leland model,paras, and avellaneda model, boyle and vorst model, hodges and neuberger model, barles andsoner model, and rapm (risk-adjusted pricing methodology) model. if the transaction cost pa-rameters are equal to zero, all of these non-linear transaction cost models are unvarying with thelinear model.soner et al. [36] showed that there is no nontrivial hedging portfolio for option pricing withtransaction costs. they also suggested that the best hedging strategy is buying an asset andtaking on it for a certain period as a call or put option. leland [37] inaugurates the idea of usingtransaction costs at discrete times. he also indicated that the hedging error could be minimized ifthe length of re-balancing frequency approaches zero. later, boyle and vorst [38] demonstrated fur-ther in a discrete-time framework with a binomial tree model for the option prices with proportionaltransaction costs, and it is pretty accurate for possible parameter values. besides, dewynne etal. [39] considered path-dependent and exotic options with transaction costs. recently, asymptoticanalysis [40] and markov chain approximation [41] were also studied for pricing european optionswith transaction costs in some previous literature.on the other hand, few researchers [42–46, 49–51] paid their attention to solve the non-linearblack-scholes equation numerically. for example, the exponential time differencing (etd) method[44] was applied to solve the non-linear black–scholes model for pricing american options with ahighly stable and efficient transaction cost. lesmana and wang [45] developed the numerical methodbased on an upwind finite difference scheme for a non-linear parabolic pde, and they attemptedto pricing european options under transaction costs. ankudinova and ehrhardt [46] focused on thenon-linear black-scholes equation for european call options using several transaction cost modelsas well as crank–nicolson and rigal compact schemes. r. l. valkov [47] has solved the non-linearblack–scholes-bellman model numerically as well as discuss the monotonicity and consistency ofhis suggested scheme in considerable detail. a monotone finite volume spatial discretization and asecond-order predictor-corrector scheme in time are considered by radoslas valkov [48] to handlethe black–scholes equation with uncertain volatility and dividend. the applicability of implicit https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 3numerical schemes for the valuation of contingent claims in non-linear black–scholes models hasbeen discussed by pascal heider [49]. he also studied the practical implications of the derivedstability criteria on relevant numerical examples. he claimed that if certain stability requirementsare satisfied, it is possible to construct convergent implicit algorithms for non-linear black–scholesequations. ekaterina dremkova and matthias ehrhardt [50] have solved non-linear black–scholesequations for american options with a non-linear volatility function using various compact finitedifference techniques to improve the order of the accuracy. the existence and uniqueness ofsolutions to the well-known non-linear black-scholes equation have been demonstrated by naoyukiishimura [51] for both in the classical and weak senses.however, in this paper, we work on approximating non-linear black-scholes pde for valuingeuropean options when there are transaction costs. for this, we organize the present research workas follows: we modify the original model into parabolic type pde exploiting the transformations [46]which are written in section 2. a brief description of different volatility models is given in section3 subsequently. section 4 is devoted to discretize the transformed parabolic type equation byusing some numerical schemes. stability and consistency analysis are included in sections 5 and6, respectively. in section 7, numerical examples are given to show the efficacy of the proposedschemes. subsequently, a general conclusion is drawn in section 8. finally, all relevant referencesare included. 2. the model equation this section considers a non-linear black-scholes pde and modifies it to a non-linear parabolictype equation with appropriate and available transformations, which would be easy to computenumerically. let us consider the non-linear black-scholes pde [46], ∂f ∂t + rs ∂f ∂s + 1 2 σ̃2s2 ∂ 2f ∂s2 − rf = 0, 0 < s <∞, t ∈ (0, t ) (1) subject to the terminal and boundary conditions for european call and put options: f (s, t ) = max(s − k, 0), f (s, t) = 0 when s = 0, f (s, t) = s − ke−r(t−t), when s → ∞and f (s, t ) = max(k − s, 0), f (s, t) = ke−r(t−t), when s = 0, f (s, t) = 0, when s → ∞respectively. throughout this paper, we use the notations: f = f (s, t) = the option price, s =stock price, k = strike price, t = maturity time, r = interest rate, t = time in years, and σ̃ = σ̃ ( t, s, ∂f∂s , ∂2f ∂s2 ) depends on the volatility model. now consider the transformations [46] as given below, y = ln ( k−1s ) , τ = 1 2 σ2(t − t) and u(y , t) = k−1e−yf (s, t) and substituting these into equation (1) to obtain the following non-linear parabolic pde https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 4 ∂u ∂τ = 2r σ2 ∂u ∂y + ( σ̃ σ )2(∂2u ∂y2 + ∂u ∂y ) , ymin < y < ymax, τ ∈ ( 0, σ2 2 t ) (2) with the modified initial and boundary conditions for european call and put options: u(y , 0) = max ( 1− e−y , 0 ) as y ∈ (−∞,∞), u(y , τ) = 0 as y → −∞, u(y , τ) = 1− e−(y+2rτ/σ2)as y →∞, and u(y , 0) = max ( e−y − 1, 0 ) as y ∈ (−∞,∞), u(y , τ) = e−(y+2rτ/σ2) as y → −∞, u(y , τ) = 0 as y →∞, respectively. 3. volatility models this section concerns four stochastic volatility models to discretize the non-linear black-scholespde, whose solution provides the option price for transaction fees. we give a short description,but details are available in some previous literature [46]. leland volatility model (lvm). leland [37] developed a technique for replicating options in thepresence of transactions costs for a small time interval. he proposed that the option price isthe solution of the non-linear black-scholes equation (1) but with the adjusted volatility [46] asfollows: σ̃ = σ √ 1 + √ 2 π µ σ √ ∆t sign (fss) (3) where, σ is the original volatility, µ is the round-trip transaction cost per unit dollar of the trans-action, and ∆t is the transaction frequency. in this formula, both µ and ∆t are assumed to be smallwhile keeping the ratio µ√ ∆t of order one. boyle and vorst volatility model (bvvm). boyle and vorst [38] derived a method for calculatingoption prices in a discrete-time where option price meets to black-scholes price with the modifiedvolatility [46] given by σ̃ = σ √ 1 + µ σ √ ∆t sign (fss) (4) where, σ, µ, and ∆t represents the same meaning as leland. barles and soner volatility model (bsvm). barles and soner [43] evolved a model using theutility function approach of hodges and neuberger [52] along with asymptotic analysis of partialdifferential equations. for this case, the formula for the modified volatility [46] is given by σ̃ = σ √ 1 + er(t−t)a2s2fss (5) where, µ = a √ ∈ is the round-trip transaction cost per unit dollar of the transaction for someconstant a > 0 and ∈→ 0. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 5 rapm volatility model (rapmvm). kratka [53] took the first step for this model and laterimproved by jandačka and ševčovič [54]. here the modified volatility is of the form [46] σ̃ = σ √ 1 + 3× 3 √ c2m 2π sfss (6) where, m ≥ 0 is the transaction cost measure and c ≥ 0 is the risk premium measure. 4. derivations of computational schemes in this section, we derive five computational schemes, in detail, for equation (2) using twowell-known numerical methods. 4.1. dufort-frankel finite difference scheme. the dufort-frankel fd scheme [55] can be appliedto solve various kinds of problems which occur in finance. this scheme is a multi-step method, andrequires another scheme for simulating the first temporal vector. in this formulation ∂u ∂τ , ∂u ∂y , and ∂2u ∂y2are discretized by central difference and uji is replaced by (uj+1 i + uj−1 i ) /2. thus, discretizingequation (2) by dufort-frankel fdm, we obtain 1 2∆τ ( uj+1 i − uj−1 i ) = ( σ̃ σ )2 [ 1 (∆y)2 ( uji−1 − ( uj+1 i + uj−1 i ) + uji+1 )] + ( σ̃ σ )2 [ 1 2∆y ( uji+1 − u j i−1 )] + r σ2∆y ( uji+1 − u j i−1 ) or, equivalently uj+1 i =uj−1 i + 2r(∆τ)(∆y) σ2 ( uji+1 − u j i−1 ) + ∆τ ∆y ( σ̃ σ )2 ( uji+1 − u j i−1 ) + 2(∆τ) (∆y)2 ( σ̃ σ )2 ( uji−1 − u j+1 i − uj−1 i + uji+1 ) which can be written as uj+1 i = aiu j i−1 + biu j i+1 + ciu j−1 i ; i = 0, 1, 2, . . . , n − 1; j = 0, 1, 2, . . . , m − 1 (7) where ai = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2 ]−1 × [ (∆τ) ( σ̃ σ )2 (2− ∆y)− (∆y)(∆τ) 2r σ2 ] , bi = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2 ]−1 × [ (∆τ) ( σ̃ σ )2 (2 + ∆y) + (∆y)(∆τ) 2r σ2 ] , and ci = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2 ]−1 × [ (∆y)2 − 2(∆τ) ( σ̃ σ )2 ] which is our proposed dufort-frankel finite difference scheme (dffds). https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 64.2. laasonen finite difference scheme. the laasonen finite difference scheme [55] can be ap-plied to solve linear and non-linear partial differential equations. this method metamorphosedpartial differential equations into a system of linear algebraic equations. in this formulation ∂u ∂τis approximated by a central differencing at a step ∆τ 2 , and ∂u ∂y , ∂2u ∂y2 are approximated by centraldifferences at time levels j + 1. now the discretized form of equation (2) is as follows 1 ∆τ ( uj+1 i − uji ) = 1 2(∆y)2 ( σ̃ σ )2 [ 2 ( uj+1 i−1 − 2uj+1 i + uj+1 i+1 ) + ∆y ( uj+1 i+1 − u j+1 i−1 )] + r σ2∆y ( uj+1 i+1 − u j+1 i−1 ) after simplification, we get[ r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y − 2) ] uj+1 i−1 + [ 1 + 2∆τ (∆y)2 ( σ̃ σ )2 ] uj+1 i − [ r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y + 2) ] uj+1 i+1 = uji the above equation reduces to diu j+1 i−1 + (1 + ei) u j+1 i + fiu j+1 i+1 = uji ; i = 0, 1, 2, . . . , n − 1; j = 0, 1, 2, . . . , m − 1 (8) where di = r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y − 2), ei = 2∆τ (∆y)2 ( σ̃ σ )2 and fi = − r∆τ ∆yσ2 − ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y + 2) 4.3. finite volume schemes. the finite volume scheme is a scheme of solving different kinds oftime-dependent or independent partial differential equations in algebraic equations. in this scheme,we divide the physical space into a finite number of control volumes. in this section, we describeit in a few lines, but details are available in the previous study [56] conducted by malalasekera etal. applying the finite volume integration in equation (2) over a control volume (cv) with a finite timestep ∆τ , we obtain ∫ τ+∆τ τ ∫ cv ∂u ∂τ dv dτ = ( 2r σ2 + ( σ̃ σ )2 )∫ τ+∆τ τ ∫ cv ∂u ∂y dv dτ + ( σ̃ σ )2 ∫ τ+∆τ τ ∫ cv ∂2u ∂y2 dv dτ https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 7after rearranging, we get∫ cv [∫ τ+∆τ τ ∂u ∂τ dτ ] dv = ( 2r σ2 + ( σ̃ σ )2 )∫ τ+∆τ τ [∫ cv ∂u ∂y dv ] dτ + ( σ̃ σ )2 ∫ τ+∆ τ [∫ cv ∂2u ∂y2 dv ] dτ applying gauss’s divergence theorem, the above equation leads ( up − u0 p ) ∆v = 1 2 ( 2r σ2 + ( σ̃ σ )2 ) a ∫ τ+∆τ τ (ue − uw ) dτ + ( σ̃ σ )2 ∫ τ+∆τ τ [( a ue − up δype ) − [( a up − uw δywp )]] dτ (9) for 0 ≤ θ ≤ 1, we assume ∫ τ+∆τ τ up dτ = [ θup + (1− θ)u0 p ] ∆τ (10) applying equation (10) into equation (9) and dividing by we get ( up − u0 p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2 )[ θ (ue − uw ) + (1− θ) ( u0 e − u0 w )] + ( σ̃ σ )2 θ ( ue − up δype − up − uw δywp ) + ( σ̃ σ )2 (1− θ) ( u0 e − u0 p δype − u0 p − u0 w δywp ) (11) for convenience, we put δywp = δype = ∆y on the following three schemes. explicit scheme. substitution of θ = 0 into equation (11) gives the following explicit discretizedequation, ( up − u0 p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2 )( u0 e − u0 w ) + 1 ∆y ( σ̃ σ )2 ( u0 e − 2u0 p + u0 w ) this equation may be re-writtens as up = αiu 0 w + (1 + βi) u 0 p + γiu 0 e (12) where αi = ∆τ (∆y)2 ( σ̃ σ )2 − ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2 ) , βi = − 2∆τ (∆y)2 ( σ̃ σ )2 and γi = ∆τ (∆y)2 ( σ̃ σ )2 + ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2 ) which is the desired finite volume explicit scheme (fves). https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 8 crank-nicolson scheme. putting θ = 1 2 into equation (11), we get the following crank-nicolsondiscretized equation,( up − u0 p ) ∆y ∆τ = 1 4 ( 2r σ2 + ( σ̃ σ )2 )( ue − uw + u0 e − u0 w ) + 1 2∆y ( σ̃ σ )2 ( ue − 2up + uw + u0 e − 2u0 p + u0 w ) after simplification, we get the following equation λiuw + (1 + ξi) up + ηiue = −λiu0 π + (1− ξi) u0 p − ηiu0 e (13) where λi = ∆τ 4∆y ( 2r σ2 + ( σ̃ σ )2 ) − ∆τ 2(∆y)2 ( σ̃ σ )2 , ξi = ∆τ (∆y)2 ( σ̃ σ )2 and ηi = − [ ∆τ 4∆y ( 2r σ2 + ( σ̃ σ )2 ) + ∆τ 2(∆y)2 ( σ̃ σ )2 ] which is the proposed finite volume crank-nicolson scheme (fvcns). fully implicit scheme. substitution of θ = 1 into equation (11) leads to the following form:( up − u0 p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2 ) (ue − uw ) + 1 ∆y ( σ̃ σ )2 (ue − 2up + uw ) and the reduced formula is then qiuw + (1 + ri) up + siue = u0 p (14) where qi = ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2 ) − ∆τ (∆y)2 ( σ̃ σ )2 , ri = 2∆τ (∆y)2 ( σ̃ σ )2 and si = − ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2 ) − ∆τ (∆y)2 ( σ̃ σ )2 which is our proposed finite volume fully implicit scheme (fvfis). 5. stability of the numerical schemes to test the stability of the derived schemes in section 4, with the help of the von-neumannstability method [55], let us consider a fourier component for uji and u0 p as uji = u jeiθi and u0 p = u jeiθi (15) where i = √ −1, i.e., imaginary unit, u j is the amplitude at a time level j, θ(= r∆y) is the phaseangle, r is the wave number in the x-direction, and i represents the index of the node. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 9similarly, uj∓1 i±1 = u j∓1eiθ(i±1) u0 w = u jeiθ(i−1) u0 e = u jeiθ(i+1) up = u j+1eiθi uw = u j+1eiθ(i−1) ue = u j+1eiθ(i+1) (16) for convenience, let us suppose that g = u j+1 u j . thus, the stability requirement is |g|2 ≤ 1.applying equation (15) and equation (16) into equation (7), and dividing by eiθi , we get |g|2 = 1 4 {4∆τ ( σ̃ σ )2 cos θ ± √ a }2 + 4(∆τ)2(∆y)2 ( 2r σ2 + ( σ̃ σ )2 )2 ( 1− cos2 θ ) × ( (∆y)2 + 2∆τ ( σ̃ σ )2 )−2 (17) where a =16(∆τ)2 ( σ̃ σ )4 cos2 θ − 4(∆τ)2(∆y)2 ( 2r σ2 + ( σ̃ σ )2 )2 ( 1− cos2 θ ) + 4(∆y)4 − 16(∆τ)2 ( σ̃ σ )4 + i16(∆τ)2(∆y) ( σ̃ σ )2 ( 2r σ2 + ( σ̃ σ )2 ) cos θ √ 1− cos2 θ for extremum value of |g|2, solving d |g|2 d(cos θ) = 0 for cos θ, and substituting it into d2|g|2 d(cos θ)2 < 0. thenfrom equation (17), we cannot confirm that the maximum value of |g|2 would occur. however, theextreme values of cos θ must yet be investigated. for cos θ = 1, equation (17) gives |g|2 = 1, andthe stability requirement is satisfied. for cos θ = −1, equation (17) also yields |g|2 = 1 and, andthe stability requirement is satisfied. thus, the dffds proposed in equation (7) is unconditionallystable. similarly, we can show that lfds and fvfis wrote in equation (8) and equation (14), respectively,both are unconditionally stable. again, applying equation (15) and equation (16) into equation (12) and dividing by eiθi , we get g = 1 + 2∆τ (∆y)2 ( σ̃ σ )2 (cos θ − 1) + i ∆τ ∆y ( 2r σ2 + ( σ̃ σ )2 ) sin θ then we may obtain easily, |g|2 = { 1 + 2∆τ (∆y)2 ( σ̃ σ )2 (cos θ − 1) }2 + ( ∆τ ∆y )2 ( 2r σ2 + ( σ̃ σ )2 )2 ( 1− cos2 θ ) (18) for extremum value of |g|2 such that d |g|2 d(cos θ) = 0, we can find cos θ = 1 ∆τ × [ 2 ( σ̃ σ )2 − 4 ∆τ (∆y)2 ( σ̃ σ )4 ] × (2r σ2 + ( σ̃ σ )2 )2 − 4 ∆τ (∆y)2 ( σ̃ σ )4 −1 (19) https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 10 considering d2|g|2 d(cos θ)2 < 0, and substituting the value of cos θ from equation (19) into equation (18),which does not provide us the maximum value of |g|2. but, the extreme values of cos θ must beinvestigated. for cos θ = 1, equation (18) gives |g|2 = 1, and the stability requirement is satisfied.for cos θ = −1, equation (18) yields |g|2 = { 1− 4∆τ (∆y)2 ( σ̃ σ )2 }2 and, imposing the requirement of |g|2 ≤ 1, yields, fves in equation (12) is conditionally stable and the condition is( σ̃ σ )2 ≤ (∆y)2 2∆τ (20) similarly, we can state that fvcns, equation (13) is also conditionally stable and the conditionis ( σ̃ σ )2 ≤ (∆y)2 ∆τ (21) 6. consistency of the numerical schemes for consistency, the finite difference equation (fde) approximation of a pde must reduce tothe original pde as the step sizes approach zero [55]. now expanding each u(y , τ) in a taylor series expansion about uji , we get uj+1 i = uji + ∆τ ∂u ∂τ + (∆τ)2 2! ∂2u ∂τ2 + (∆τ)3 3! ∂3u ∂τ3 +o(∆τ)4 (22) uj+1 i+1 =uji + ∆τ ∂u ∂τ + ∆y ∂u ∂y + 1 2! ( ∆τ ∂ ∂τ + ∆y ∂ ∂y )2 u + 1 3! ( ∆τ ∂ ∂τ + ∆y ∂ ∂y )3 u +o [ (∆τ)4, (∆y)4 ] (23) uj+1 i−1 =uji + ∆τ ∂u ∂τ − ∆y ∂u ∂y + 1 2! ( ∆τ ∂ ∂τ −∆y ∂ ∂y )2 u + 1 3! ( ∆τ ∂ ∂τ − ∆y ∂ ∂y )3 u +o [ (∆τ)4, (∆y)4 ] (24) applying equations (22), (23), and (24) into equation (8) yields (di + ei + fi) u j i + (1 + di + ei + fi) ∆τ ∂u ∂τ + (1 + di + ei + fi) (∆τ)2 2 ∂2u ∂τ2 + (−di + fi) ∆y ∂u ∂y + (−di + fi) ∆τ∆y ∂2u ∂τ∂y + (di + fi) (∆y)2 2 ∂2u ∂y2 +o [ (∆τ)3, (∆y)3 ] = 0from which we get ∂u ∂τ + ∆τ 2 ∂2u ∂τ2 − ( 2r σ2 + ( σ̃ σ )2 ) ∂u ∂y − ∆τ ( 2r σ2 + ( σ̃ σ )2 ) ∂2u ∂y∂τ − ( σ̃ σ )2 ∂2u ∂y2 +o [ (∆τ)2, (∆y)2 ] = 0 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 11it is obvious that if ∆τ,∆y → 0, then the original pde (2) is recovered. therefore, the laasonenfinite difference scheme, equation (8), is consistent. now according to lax’s equivalence theorem,[55], lfds is convergent for all values of the parameters. similar arguments hold for dffds andfvfis. on the other hand, fves and fvcns are also convergent if the conditions (20) and (21)respectively, are satisfied. table 1. call option prices using the leland volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.04734 0.04893 0.00000 0.00054 0.0005047.00 0.00182 0.30914 0.31368 0.00006 0.01340 0.0129757.00 0.05078 1.09349 1.09949 0.00036 0.10771 0.1058867.00 0.45226 2.93576 2.94472 0.00335 0.65191 0.6479577.00 1.97686 6.06630 6.07104 0.01709 2.26632 2.2623087.00 5.46222 10.56460 10.56947 1.25649 5.63768 5.6368897.00 11.17037 16.34912 16.34757 6.49278 11.07370 11.07714107.00 18.71972 23.33442 23.33541 16.15278 18.66210 18.66616117.00 27.48006 31.19401 31.19406 26.33380 27.42028 27.42359127.00 36.91158 39.65579 39.65486 36.72393 36.83405 36.83640137.00 46.67034 48.63231 48.63285 46.68664 46.59745 46.59938147.00 56.57397 57.89847 57.89955 56.61703 56.45705 56.45879157.00 66.53723 67.45340 67.45432 66.59903 66.46595 66.46767167.00 76.52370 77.08904 77.08984 76.50332 76.39766 76.39940177.00 86.51886 86.88858 86.88931 86.48557 86.40846 86.41023187.00 96.51716 96.75424 96.75478 96.48843 96.43330 96.43508197.00 106.51657 106.59706 106.59728 106.40288 106.36023 106.36202207.00 116.51636 116.67091 116.67078 116.55261 116.52319 116.52499217.00 126.51629 126.48888 126.48906 126.39973 126.37558 126.37738227.00 136.51627 136.46008 136.46065 136.39791 136.37870 136.38049237.00 146.51626 146.57401 146.57489 146.53847 146.52419 146.52597247.00 156.51626 156.41475 156.41484 156.38607 156.37369 156.37545257.00 166.51626 166.40053 166.39979 166.37904 166.36868 166.37039267.00 176.51626 176.52571 176.52411 176.51173 176.50347 176.50515 7. results and discussions in this section, we choose the same parameters: r = 0.1, σ = 0.2, k = 100, t = 1, µ = 0.05, ∆t = 0.01, a = 0.02, m = 0.01, and c = 30, as illustrated in the literature [46]. then wecalculate the call option values using the proposed schemes, described in previous section 4, fordifferent volatility models. we compare the approximate results with the exact value of the linearblack-scholes model and among themselves also. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 12 figure 1. approximate results of equation (1) by using (a) boyle and vorst volatilitymodel, and (b) barles and soner volatility model. from table 1 and figure 8.7 (see appendix), we observe that fully implicit fvs and crank-nicolson fvs provide comparatively better results than the other methods. note that all of themethods provide poor results when the initial stock price is less than the strike price (here strikeprice, in comparison with the exact value of the linear black-scholes model.from table 8.3 in appendix 8 and figure 1 (a), we can make similar comments, but here thefves gives a very poor approximation than the other methods when the initial stock price is lessthan the strike price (k = 100). table 8.4 in appendix 8 and figure 1(b) show that all of themethods provide a closer approximation to the exact value of the linear black-scholes model forall of the initial stock price, whether it is greater than the strike price, k = 100. table 2. call option prices using rapm volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.09975 0.10198 0.00075 0.00054 0.0005047.00 0.00182 0.64320 0.64859 0.02029 0.01340 0.0129757.00 0.05078 2.01164 2.01581 0.16518 0.10771 0.1058867.00 0.45226 4.58788 4.59820 0.97606 0.65191 0.6479577.00 1.97686 8.35995 8.36458 3.28335 2.26632 2.2623087.00 5.46222 13.24558 13.25523 7.65193 5.63768 5.6368897.00 11.17037 19.14277 19.14296 13.90023 11.07370 11.07714107.00 18.71972 25.95302 25.96004 21.59995 18.66210 18.66616117.00 27.48006 33.49710 33.50138 30.05441 27.42028 27.42359127.00 36.91158 41.58132 41.58229 39.02457 36.83405 36.83640137.00 46.67034 50.16133 50.16454 48.32523 46.59745 46.59938147.00 56.57397 59.07941 59.08280 57.80665 56.45705 56.45879157.00 66.53723 68.31796 68.31997 67.49623 66.46595 66.46767167.00 76.52370 77.71942 77.72089 77.20105 76.39767 76.39941177.00 86.51886 87.32671 87.32812 87.03280 86.40846 86.41023187.00 96.51716 97.04281 97.04399 96.91429 96.43330 96.43509197.00 106.51657 106.79810 106.79881 106.75046 106.36023 106.36202207.00 116.51636 116.77932 116.77954 116.81690 116.52319 116.52498217.00 126.51629 126.56491 126.56478 126.62193 126.37555 126.37734227.00 136.51627 136.50685 136.50637 136.57895 136.37864 136.38041237.00 146.51626 146.59198 146.59115 146.67807 146.52410 146.52585247.00 156.51626 156.42589 156.42489 156.50633 156.37350 156.37521257.00 166.51626 166.40454 166.40336 166.47896 166.36838 166.37004267.00 176.51626 176.52231 176.52094 176.59035 176.50307 176.50468 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 13finally, from table 2 and the corresponding figure 8.8 in appendix 8, we may observe thatfor the rapm volatility model, the fvcns and fvfis give better approximation than the othernumerical schemes when the initial stock price is closer to and/or greater than the strike price.on the other hand, from figures 2, 3, 4, it is clear that fvfis and fvcns produce comparativelybetter results than the other schemes for all of the volatility models. figures 5, 6 depict the optionprices at various time periods (from initial time t = 0 to maturity time, t = t ) with different initialstock values. the similar results of solution surface for option price by using barles and sonervolatility model and rapm volatility model are presented in appendix 8, see figures 8.9,8.10. figure 2. approximate results of equation (1) using (a) dufort-frankel finite dif-ference scheme, and (b) laasonen finite difference scheme. figure 3. approximate results of equation (1) using (a) finite volume explicitscheme, and (b) finite volume fully implicit scheme. figure 4. approximate results of equation (1) using finite volume crank-nicolsonscheme. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 14 (a) lvm (dffds) (b) lvm (lds) (c) lvm (fves) (d) lvm (fvcns) (e) lvm (fvfis) figure 5. solution surface for option price by using leland volatility model. (a) bvvm (dffds) (b) bvvm (lds) (c) bvvm (fves) (d) bvvm (fvcns) (e) bvvm (fvfis) figure 6. solution surface for option price by using boyle and vorst volatilitymodel. 8. conclusion in this research work, we have derived some numerical schemes using the fvm and fdm tosolve the non-linear black-scholes pde for european option pricing with the transaction costs byexploiting the transformations available in the existing literature [46]. thus we have modified themodel equation accordingly to a non-linear parabolic pde. for the convergence of these schemes,stability and consistency have been shown rigorously. then these schemes have been applied tovarious volatility models. according to the visible results, as presented in the earlier sections, it https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 15is noted that all of the proposed schemes provide the best approximation to the exact value of thelinear black-scholes model for all initial stock prices, regardless of 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https://doi.org/10.1080/00207160.2012.688115 https://doi.org/10.1080/00207160.2012.688115 https://doi.org/10.1016/j.amc.2012.12.077 https://doi.org/10.1016/j.amc.2012.12.077 https://doi.org/10.1016/j.camwa.2008.02.005 https://doi.org/10.1080/00207160.2015.1069818 https://doi.org/10.1515/cmam-2015-0029 https://doi.org/10.1080/13504860903075670 https://doi.org/10.1080/00207160.2011.558574 https://doi.org/10.1080/00207160.2011.558574 https://doi.org/10.1007/s10690-010-9115-3 https://doi.org/10.1155/jam.2005.235 eur. j. math. anal. 10.28924/ada/ma.2.9 18appendix this section contains the supporting figures and tables to observe the accuracy of the solutionmethodologies. figure 8.7. approximate results of equation (1) by using leland volatility model. figure 8.8. approximate results of equation (1) by using rapm volatility model. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 19 table 8.3. call option prices using boyle and vorst volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.08130 0.08362 0.00000 0.00054 0.0005047.00 0.00182 0.44919 0.45475 0.00016 0.01340 0.0129757.00 0.05078 1.43187 1.43827 0.00137 0.10771 0.1058867.00 0.45226 3.52436 3.53389 0.00181 0.65191 0.6479577.00 1.97686 6.88032 6.88512 -0.01095 2.26632 2.2623087.00 5.46222 11.52515 11.53065 -0.03314 5.63768 5.6368897.00 11.17037 17.36380 17.36252 1.18226 11.07370 11.07714107.00 18.71972 24.30605 24.30805 14.49730 18.66210 18.66616117.00 27.48006 32.07120 32.07189 26.66264 27.42028 27.42359127.00 36.91158 40.41514 40.41449 37.98090 36.83405 36.83640137.00 46.67034 49.26375 49.26507 47.25031 46.59745 46.59938147.00 56.57397 58.41457 58.41649 56.80881 56.45705 56.45879157.00 66.53723 67.86188 67.86337 66.66025 66.46595 66.46767167.00 76.52370 77.41353 77.41510 76.50821 76.39767 76.39940177.00 86.51886 87.14056 87.14251 86.47640 86.40846 86.41023187.00 96.51716 96.94695 96.94884 96.47214 96.43330 96.43509197.00 106.51657 106.75019 106.75144 106.38908 106.36023 106.36202207.00 116.51636 116.78198 116.78255 116.54146 116.52319 116.52500217.00 126.51629 126.57938 126.58055 126.39087 126.37558 126.37738227.00 136.51627 136.53059 136.53250 136.39143 136.37869 136.38048237.00 146.51626 146.62463 146.62715 146.53436 146.52418 146.52596247.00 156.51626 156.45839 156.45983 156.38285 156.37367 156.37541257.00 166.51626 166.43684 166.43715 166.37675 166.36863 166.37034267.00 176.51626 176.55434 176.55347 176.51042 176.50341 176.50508 table 8.4. call option prices using barles and soner volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.00056 0.00060 0.00047 0.00054 0.0005047.00 0.00182 0.01448 0.01496 0.01255 0.01340 0.0129757.00 0.05078 0.11766 0.11964 0.10402 0.10771 0.1058867.00 0.45226 0.70853 0.71289 0.64387 0.65191 0.6479577.00 1.97686 2.42414 2.42846 2.25831 2.26632 2.2623087.00 5.46222 5.90671 5.90840 5.63756 5.63768 5.6368897.00 11.17037 11.38951 11.38681 11.08596 11.07370 11.07714107.00 18.71972 18.90018 18.89694 18.68219 18.66210 18.66616117.00 27.48006 27.57177 27.56903 27.44285 27.42028 27.42359127.00 36.91158 36.90926 36.90718 36.85630 36.83405 36.83640137.00 46.67034 46.63600 46.63420 46.61611 46.59745 46.59938147.00 56.57397 56.47483 56.47316 56.47209 56.45705 56.45879157.00 66.53723 66.47486 66.47318 66.47729 66.46595 66.46767167.00 76.52370 76.40243 76.40071 76.40662 76.39766 76.39940177.00 86.51886 86.41173 86.40997 86.41573 86.40846 86.41022187.00 96.51716 96.43561 96.43381 96.43938 96.43329 96.43508197.00 106.51657 106.36234 106.36053 106.36582 106.36022 106.36202207.00 116.51636 116.52509 116.52327 116.52827 116.52319 116.52499217.00 126.51629 126.37745 126.37563 126.38047 126.37557 126.37737227.00 136.51627 136.38055 136.37873 136.38340 136.37869 136.38048237.00 146.51626 146.52603 146.52421 146.52870 146.52418 146.52596247.00 156.51626 156.37561 156.37380 156.37798 156.37368 156.37543257.00 166.51626 166.37066 166.36886 166.37271 166.36866 166.37038267.00 176.51626 176.50552 176.50374 176.50725 176.50346 176.50514 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 20 (a) bsvm (dffds) (b) bsvm (lds) (c) bsvm (fves) (d) bsvm (fvcns) (e) bsvm (fvfis) figure 8.9. solution surface for option price by using barles and soner volatilitymodel. (a) rapm (dffds) (b) rapm (lds) (c) rapm (fves) (d) rapm (fvcns) (e) rapm (fvfis) figure 8.10. solution surface for option price by using rapm volatility model. https://doi.org/10.28924/ada/ma.2.9 1. introduction 2. the model equation 3. volatility models leland volatility model (lvm) boyle and vorst volatility model (bvvm) barles and soner volatility model (bsvm) rapm volatility model (rapmvm) 4. derivations of computational schemes 4.1. dufort-frankel finite difference scheme 4.2. laasonen finite difference scheme 4.3. finite volume schemes 5. stability of the numerical schemes 6. consistency of the numerical schemes 7. results and discussions 8. conclusion conflicts of interest funding statement references appendix ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 6doi: 10.28924/ada/ma.3.6 fractionalization of hankel type integral transforms and their relevance b. b. waphare∗, r. z. shaikh mit arts, commerce and science college, alandi(d), pune, maharashtra, india balasahebwaphare@gmail.com, shaikhrahilanaz@gmail.com ∗correspondence: balasahebwaphare@gmail.com abstract. in this paper, the fractionalization of certain types of hankel transforms is suggested.barut-girardello type transforms are then introduced along with the relevant fractional order forms.finally some further generalizations are suggested. 1. introduction the theory of hankel transforms is very vast and it is studied by many researchers in recent aswell as in past. the forward and inverse transforms are completely symmetric and resemble thefourier transform, with the complex exponent as kernal being replaced by the bessel function offirst kind jα−β of order α− β ≥ −12 .the formal equivalence between the zeroth-order hankel transform and the abel transform fol-lowed by the fourier transform is used as basis for developing fast algorithms for the computationof the zeroth-order hankel transform [5]. algorithms for the computation of the hankel transformof integer order n > 0 have been proposed. on the basis of the general relation involving thehankel transform of integer order n > 0 and the abel transform, whose kernel is modulated by thechebyshev polynomial of the first kind of order n, followed by the fourier sine or cosine transformaccording to whether n is odd or even [5, 13].it has been evidenced in [18] the formal equivalence between the hankel transform of order α− βand the erdelyi-kober fractional integral of order (α − β + 1 2) followed by the fourier cosinetransform, with both acted on function and the resulting transform being modulated by properly α− β dependent power functions of the inherent variables.notably such as equivalence suggests a tool for the optical computation of erdelyi-kober typefractional integrals of order (n + 1 2), n integer, through the optical implementation of the hankeland id fourier transforms. in a sense, the erdelyi-kober type fractional integrals of order n + 1 2 received: 4 jun 2022. key words and phrases. hankel type transform; barut-girardello transform; fractional transform; erdelyi-kobertransform. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 2can be regarded as the (2n + 1)-plane abe1 transform on rm, with 0 < 2n + 1 < m.various forms of hankel like integral transforms have been considered in detail in a series of pa-pers [6,7,10–12,22]. we will be concerned here with certain hankel type transform having relevanceamong others, in connection with the solution to evolution problems involving the bessel type differ-ential operators xα+3β−1 ( ∂∂x ) x2(α−β)+1 ( ∂∂x ) x−3α−β and x−3α−β ( ∂∂x ) x2(α−β)+1 ( ∂∂x ) xα+3β−1with α− β ≥ −12 and −2(α+ β) real parameter [6, 11]. we introduced the fractional order formsof such hankel-type transforms by following the lines of the fractionalization of the conventionalhankel transform [9, 16].work of torre [17] motivated us to prepare this paper. 2. hankel type transforms: the first and second hankel-type transforms of bessel order α− β, depending on an arbitraryreal parameter −2(α+ β), respectively defined by the operational relations [6, 11]. f̃1,α,β(y) = [ h1,α−β,−2(α+β)f ] (y) = y1−4(α+β) ∫ ∞ 0 (xy)2(α+β)jα−β(xy)f (x)dx (1) f̃2,α,β(y) = [h2,α−β,−2(α+β)f ](y) = ∫ ∞ 0 x1−4(α+β)(xy)2(α+β)jα−β(xy)f (x)dx (2) where jα−β is the bessel type function of the first kind and order (α−β) ≥ −12 . here f ∈ l2(r+)the space of the complex-valued functions which are lebesgue integrable on r+ = (0,+∞).thetransform [h2,α−β,−2(α+β)f ](y) for α = −1 3 β was originally considered in [15], where the relevantcondition for its inversion were established. also (1) and (2) relate to the hankel type cliffordtransforms [10].for suitable values of α, β the able transforms can be framed within the formalism, developedin [20, 21] concerning the integral transforms associated with complex linear transformations inquantum mechanics, which maps the position and momentum operators to canonically conjugate,but not necessarily hermitian operator. thus according to that formalism, the able transformscan be seen as the radial parts of n-dimentional linear cannonical transformations, specificallyrepresenting a π/2-rotation for each pair of the cannonically conjugate operators in the respec-tive n-component position and momentum operator vectors. precisely, n = 4(α + β) for (1) and n = 2[1− 2(α+ β)] = 2− 4(α+ β) for (2).the order α−β of the bessel type function relates to the eigen value λ = −l(l+n−2), l = 0, 1, 2, ...of the angular momentum; specifically, it turns out that l = α−β− n 2+ 1, and so l = −(α+ 3β−1)for (1) and l = 3α+ β for (2), thus respectively yeilding λ = 3(α2 + β2) + 10αβ − 4(α+ β) + 1and λ = 3(α2 + β2) + 10αβ. when α+ β = 1 4 , n = 1 in both cases, and accordingly both trans-forms yield the conventional hankel type transform. the symmetry of (1) and (2) reflects into therelation between the respective integral kernels k1,α−β,−2(α+β)(x, y) and k2,α−β,−2(α+β)(y , x);i.e. k1,α−β,−2(α+β)(x, y) = y1−2(α+β) x2(α+β) jα−β(xy) = k2,α−β,−2(α+β)(y , x). https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 3as a consequence of well known orthogonality relation of the bessel functions, both transforms (1)and (2) are self reciprocal. h−1 1,α−β,−2(α+β) = h1,α−β,−2(α+β), h−1 2,α−β,−2(α+β) = h2,α−β,−2(α+β). (3) interestingly, the adjoint operator of h1,α−β,−2(α+β) is h2,α−β,−2(α+β) and so h∗1,α−β,−2(α+β) = h2,α−β,−2(α+β), h∗2,α−β,−2(α+β) = h1,α−β,−2(α+β). (4) also one can prove for the operator h1,α−β,−2(α+β) and h2,α−β,−2(α+β) the parsevel equalities [12]∫ ∞ 0 x−1+4(α+β) f ∗(x) g(x)dx = ∫ ∞ 0 x−1+4(α+β) f̃ ∗1,α−β,−2(α+β) g̃1,α−β,−2(α+β)(x)dx,∫ ∞ 0 x1−4(α+β) f ∗(x) g(x)dx = ∫ ∞ 0 x1−4(α+β) f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx (5) both containing a weight function i.e. x−1+4(α+β) and x1−4(α+β) respectively. a mixed parsevelrelation holds as well, which writes as∫ ∞ 0 f ∗(x) g(x)dx = ∫ ∞ 0 f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx. (6) note that it does not contain any weight function and involves both transforms [12].relations (4) and (6) express the complementary of (1) and (2).for α+ β = 1 4 , we recover the conventional hankel type transform of bessel order α− β; ĥ1,α−β,−1 2 = ĥ2,α−β,−1 2 ≡ ĥα−β with [ ĥα−β,−2(α+β)f ] (y) ≡ f̃α−β(y) = ∫ ∞ 0 (xy)2(α+β)jα−β(xy)f (x)dx = ∫ ∞ 0 kα,β(x, y)f (x)dx (7) the latter expressing the transform in terms of the kernel kα,β(x, y) = (xy)2(α+β) jα−β(xy) = kα,β(y , x).now we evidence the similarity transformations like structure of the transforms (1) and (2), beingindeed [ ĥ1,α−β,−2(α+β)f ] (y) = (y)−2(α+β)+1/2 ∫ ∞ 0 (x)2(α+β)−1/2 kα,β(x, y) f (x)dx = (y)−2(α+β)+1/2 [ ĥα,β(x)2(α+β)−1/2f ] (y)[ ĥ2,α−β,−2(α+β)f ] (y) = (y)2(α+β)−1/2 ∫ ∞ 0 (x)−2(α+β)+1/2 kα,β(x, y) f (x)dx = (y)2(α+β)−1/2 [ ĥα,β(x)−2(α+β)+1/2f ] (y) (8) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 4and so for the kernels k1,α−β,−2(α+β)(x, y) = (y)−2(α+β)+1/2 kα,β(x, y)x2(α+β)−1/2 = k2,α−β,−2(α+β)(y , x). it can be easily verified that [6, 11][ ĥ1,α−β,−2(α+β) b̂∗α−β,−2(α+β)f ] (y) = −y2 [ ĥ1,α−β,−2(α+β)f ] (y)[ ĥ2,α−β,−2(α+β) b̂α−β,−2(α+β)f ] (y) = −y2 [ ĥ2,α−β,−2(α+β)f ] (y) (9) where b̂α−β,−2(α+β) is the bessel type differential operator b̂α−β,−2(α+β) = xα+3β−1 dxx 2(α−β)+1 dxx −3α−β = d2x + [1− 4(α+ β)] 1 x dx + (3α+ β)(α+ 3β) x2 (10) whose adjoint is then b̂∗α−β,−2(α+β) = x−3α−β dxx 2(α−β)+1 dxx α+3β−1 = d2x + [4(α+ β)− 1] 1 x dx + (α+ 3β − 1)(3α+ β − 1) x2 (11) notice that for α+ β = 1 4 both operators turn into the self-adjoint operator b̂α,β , being b̂α−β,− 1 2 = b̂∗ α−β,− 1 2 = d2x + 64αβ + 3 16 x2 ≡ b̂α−β. (12) from (9) it follows, for instance, that the solution of the differential equation [6, 11]. k ∂ ∂τ h(x, τ) = b̂∗α−β,−2(α+β) (13) satisfying the initial condition h(x, 0) = f (x) can be written into the transform conjugate y-space as ĥ1,α−β,−2(α+β)(y , τ) = e− y2 kτ f̂1,α,β(y) for any value of the arbitrary constant k . then transformingback to the x-space, one obtains h(x, τ) = k 2τ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β)−( k 4τ )(x2+y2) iα−β ( k 2τ xy ) f (y)dy (14) under the condition that |arg(τk )| ≤ π 4 , which for both τ and k real turns into τ k > 0. 3. hankel-type transforms of fractional order: it is well known that equation (13) has the formal solution h(x, τ) = e τ k b̂∗α,βf (x). equation(14) yields an explicit functional representation of the exponential operator eb b̂∗α−β,−2(α+β) :[ e b b̂∗ α−β,−2(α+β) ] f (y) = 1 2b y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−( 14b )(x2+y2) iα−β (xy 2b ) f (x)dx (15) where iα−β denotes the modified bessel function of the first kind of order α − β : jα−β(ix) = iα−β iα−β(x).in particular, setting b = i 2 we obtain a representation of ĥ1,α−β,−2(α+β) in the form of a symmetricfractional product of the exponential of the generators of the su(1, 1) algebra: ĥ1,α−β,−2(α+β) = iα−β+1e−( i 2 )x2e ( i 2 ) b̂∗ α−β,−2(α+β)e−( i 2 )x2 (16) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 5in this connection, we may note that the operators k̂ (1) + = 1 2 x2, k̂ (1) − = − 1 2 b̂∗α−β,−2(α+β), k̂ (1) 3 = − i 2 ( x d dx + 2(α+ β) ) (17) conjugate a non self-adjoint one variable realization of the su(1, 1) algebra generators accordingto the inherent commutation relations[ k̂ (1) + , k̂ (1) − ] = 2i k̂ (1) 3 , [ k̂ (1) ± , k̂ (1) 3 ] = ±i k̂(1)± . (18) thus (16) can be formally be rewritten in terms of the operators k̂(1)+ and k̂(1)− , and further recastin the single exponential form ĥα−β,−2(α+β) = iα−β+1 e−i π 2 [ k̂ (1) + + k̂ (1) − ] . (19) this is on account of disentanglement relation for the su(1, 1) algebra generators e −iφ [ k̂ (1) + +k̂ (1) − ] = e−i tan(φ/2)k̂ (1) + e−i sinφk̂ (1) − e−i tan(φ/2)k̂ (1) + (20) holding for −π < φ < π. expressing (19) corresponds to the value φ = (π/2).exploiting the integral transform representation (15) of the centred operator in equation (20), weobtain an expression for the operator e−iφ[k̂(1)+ +k̂(1)− ] in the form of a hankel -type integral transform.then writing φ = a(π/2) and multiplying both sides by e i(aπ/2)(α−β+1), one ends up on the l.h.s.with the ath power of the operator iα−β+1e−i π2 [k̂(1)+ +k̂(1)− ] and corresponding on the r.h.s. with ath power of the first hankel-type transform, ĥa1,α−β,−2(α+β), or the first hankel-type transform offractional order a. accordingly, we can write[ ĥa1,α−β,−2(α+β)f ] (y) = e i(α−β+1)(φ−π/2) sinφ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e i 2 cotφ[x2+y2] jα−β ( xy sinφ ) f (x)dx (21) = [e iφ(α−β+1) e −iφ [ k̂ (1) + +k̂ (1) − ] f ](y) = f̃ (a) 1,α,β(y) where φ = a(π/2).we can interpret it as the functional representation of the operator associated with the equation i ∂ ∂τ h(x, τ) = − 1 2 { ∂2 ∂x2 − (1− 4(α+ β)) 1 x ∂ ∂x + [1− 2(α+ β)2 − (α− β)2] 1 x2 − x2 + 2(α− β + 1) } h(x, τ) (22) with the relevant initial condition h(x, 0) = f (x).the evolution variable is here measured in unitsof (π/2) : τ = a(π/2), and conventionally denoted by φ.we can also develop similar results in relation with the second hankel-type transform ĥ2,α−β,−2(α+β),the inherent su(1, 1) algebra generators being the adjoint of equation (17), namely, k̂ (2) + = 1 2 x2, k̂ (2) − = − 1 2 b̂α−β,−2(α+β), k̂ (2) 3 = − i 2 ( x d dx − 2(α+ β) + 1 ) . (23) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 6 now we introduce the second hankel-type transform of fractional order a, ĥa2,α−β,−2(α+β) as[ ĥa2,α−β,−2(α+β)f ] (y) = e i(α−β+1)(φ−π/2) sinφ ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) e i 2 cotφ[x2+y2] jα−β ( xy sinφ ) f (x)dx (24) = [ e iφ(α−β+1) e −iφ [ k̂ (2) + +k̂ (2) − ] f ] (y) = f̃ (a) 2,α−β,−2(α+β)(y), φ = a(π/2). this yields the functional representation of the evolution operator for the equation i ∂ ∂τ h(x, τ) = − 1 2 { ∂2 ∂x2 + (1− 2(α+ β)) 1 x ∂ ∂x + [−2(α+ β)2 − (α− β)2] 1 x2 − x2 + 2(α− β + 1) } h(x, τ) (25) with the initial condition h(x, 0) = f (x).the evolution variable parameterized as τ = a(π/2). for α+β = 1 4 , both equations (21) and (24) yields the expression of the conventional hankel transform offractional order, originally introduced by namia [9] and further investigated in [16,21].in particular,for α+ β = 1 4 , equations (17) and (23) turn into the same set of self-adjoint operators k̂+ = 1 2 x2, k̂− = − 1 2 b̂α−β, k̂3 = − i 2 ( x d dx + 1 2 ) (26) which pertain to the conventional hankel transform (7).thus we have ĥα−β = iα−β+1e i( π2 ) [ d2 dx2 −(α−β)2− (1/4) x2 − x2 2 ] (27) and corresponding for the transform of fractional order a ĥaα−β = e i( aπ 2 )(α−β+1)e i( aπ2 ) [ d2 dx2 −(α−β)2− (1/4) x2 − x2 2 ] (28) whose fractional equation is [9, 16,21][ ĥaα−βf ] (y) ≡ f̃ (a)α−β(y) = e i(α−β+1)(φ−π/2) sinφ ∫ ∞ 0 (xy)( 1 2 )e i 2 cotφ[x2+y2] jα−β ( xy sinφ ) f (x)dx (29) since (1) and (2), also ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) can be framed for suitable values of α, β,within the formalism of [20, 21], the relevant canonical transformation being now the rotation bythe angle φ for each pair of corresponding canonically conjugate position and momentum operatorsin the relevant n-component operator vectors. 4. properties of ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β):the fractionalization of order a of an integral transform t̂ is intended to produce an integraltransform t̂ a which satisfy specific properties; more precisely we need that(1) t̂ a is continuous with respect to the order,i.e. t̂ b −→ t̂ a as b −→ a https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 7 (2) t̂ a obeys the semigroup property, so that composing two fractional transform of order a1and a2 yeilds the fractional transform of order a1 + a2 t̂ a1 t̂ a2 = t̂ a2 t̂ a1 = t̂ a1+a2 , (3) t̂ a reduces to the identity operator with a = 0 and to the ordinary transform for a = 1; insymbols : t̂ 0 = 1̂ and t̂ 1 = t̂ .from the procedure we followed to introduced the fractional order transforms ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β), it is clear that both transforms satisfy the above properties.thus the additivityproperties follows from ĥa1,α−β,−2(α+β) = [ ĥ1,α−β,−2(α+β) ]a , ĥa2,α−β,−2(α+β) = [ ĥ2,α−β,−2(α+β) ]a (30) which in turn implies that ĥ01,α−β,−2(α+β) = 1̂, ĥ02,α−β,−2(α+β) = 1̂. (31) in our case, the ordinary transform ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are recovered for a = ±1: ĥ11,α−β,−2(α+β) = ĥ1,α−β,−2(α+β), ĥ−1 1,α−β,−2(α+β) = ĥ1,α−β,−2(α+β) ĥ12,α−β,−2(α+β) = ĥ2,α−β,−2(α+β), ĥ−1 2,α−β,−2(α+β) = ĥ2,α−β,−2(α+β). (32) this confirms to the self-reciprocal property (3) of the ordinary transform. we have ĥ−a 1,α−β,−2(α+β) = [ ĥa1,α−β,−2(α+β) ]−1 , ĥ−a 2,α−β,−2(α+β) = [ ĥa2,α−β,−2(α+β) ]−1 . (33) in fact, both transforms are periodic with respect to order parameter a i.e. ĥa+2j 1,α−β,−2(α+β) = ĥa1,α−β,−2(α+β), ĥa+2j 2,α−β,−2(α+β) = ĥa2,α−β,−2(α+β) (34) so that a can be taken in [-1,1].intrestingly, for the adjoint operator, we find the cross-relations:[ ĥa1,α−β,−2(α+β) ]∗ = ĥ−a 2,α−β,−2(α+β), [ ĥa2,α−β,−2(α+β) ]∗ = ĥ−a 1,α−β,−2(α+β) (35) which reproduce (4) for a = ±1.parsevel’s equalities separately pertaining to ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are easily proceedi.e.∫ ∞ 0 x−1+4(α+β)f (x)∗g(x)dx = ∫ ∞ 0 x1+2(α+β) [ f̃ (a) 1,α−β,−2(α+β)(x) ]∗ g̃ (a) 1,α−β,−2(α+β)(x)dx∫ ∞ 0 x1−2(α+β)f (x)∗g(x)dx = ∫ ∞ 0 x1−4(α+β) [ f̃ (a) 2,α−β,−2(α+β)(x) ]∗ g̃ (a) 2,α−β,−2(α+β)(x)dx (36) as well as the mixed parsevel’s relation:∫ ∞ 0 f (x)∗g(x)dx = ∫ ∞ 0 [ f̃ (a) 1,α−β,−2(α+β)(x) ]∗ g̃ (a) 2,α−β,−2(α+β)(x)dx. (37) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 8for α + β = 1 4 , the above equalities turn into the energy preserving relation of the conventionaltransform: ∫ ∞ 0 f (x)∗g(x)dx = ∫ ∞ 0 [ f̃ (a) α−β(x) ]∗ g̃ (a) α−β(x)dx. operational rules similar to (9) can be stated for the fractional transforms, involving of courseappropriate bessel-type differential operators. indeed we see that[ ĥa1,α−β,−2(α+β) b̂ ∗ α−β,−2(α+β),af ] (y) = − y2 sin2 φ [ ĥa1,α−β,−2(α+β)f ] (y),[ ĥa2,α−β,−2(α+β) b̂ ∗ α−β,−2(α+β),af ] (y) = − y2 sin2 φ [ ĥa2,α−β,−2(α+β)f ] (y), (38) with b̂α−β,−2(α+β),a = −2k̂ (2) − − 2 cot2 φk̂ (2) + − 4 cotφk̂ (2) 3 , (39) and in particular b̂α−β,a ≡ b̂α−β,−( 1 2 ),a = b̂∗ α−β,−( 1 2 ),a = −2k̂− − 2 cot2 φk̂+ − 4 cotφk̂3. (40) relation (38) gives the relevance of the fractional transforms for the solution of differential equationsinvolving the operators b̂∗α−β,−2(α+β),a and b̂α−β,−2(α+β),a like, for instance, the evolution equation k ∂ ∂τ h(x, τ) = b̂∗α−β,−2(α+β),a h(x, τ), (41) or, more in general, the following k ∂ ∂τ h(x, τ) = p (b̂∗α−β,−2(α+β),a) h(x, τ) (42) involving polynomial function of b̂∗α−β,−2(α+β),a (or the adjoint involving a polynomial of b̂α−β,−2(α+β),a).then considering equation (41), we note that the solution turns out to be h(x, τ) = [ ĥ−a 1,α−β,−2(α+β) ĥ a α−β,−2(α+β),a(y , τ) ] (x, τ) (43) before giving details of the expression of h(x, τ) from the above scheme, let us note that theoperators b̂α−β,−2(α+β),a and b̂∗α−β,−2(α+β),a arise from the adjoint transformation respectively of b̂α−β,−2(α+β),a and b̂∗α−β,−2(α+β),a through the operator k̂(2)+ = k̂ (1) + = ( x 2 2 ). in other words: b̂α−β,−2(α+β),a = e −i cot (φ) ( x2 2 ) b̂α−β,−2(α+β)e i cot (φ) ( x2 2 ) b̂∗α−β,−2(α+β),a = e −i cot (φ) ( x2 2 ) b̂∗α−β,−2(α+β)e i cot (φ) ( x2 2 ) (44) which can be recast as b̂α−β,−2(α+β),a = xα+3β−1d̂ax 2(α−β)+1d̂ax −3α−β b̂∗α−β,−2(α+β),a = x−3α−βd̂ax 2(α+β)+1d̂ax α+3β−1 (45) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 9 where d̂a being a linear combination of the heisenberg operators x and −i ∂∂x as d̂a = e −i cot (φ) ( x2 2 ) ∂ ∂x e i cot (φ) ( x2 2 ) = i sinφ [ cos(φ)x − i sin(φ) ∂ ∂x ] (46) the exponential operators ebb̂α−β,−2(α+β),a and ebb̂∗α−β,−2(α+β),a arise from the same adjoint transfor-mations of ebb̂α−β,−2(α+β) and ebb̂∗α−β,−2(α+β) respectively; viz ebb̂α−β,−2(α+β),a = e −i cot (φ) ( x2 2 ) ebb̂α−β,−2(α+β)e i cot (φ) ( x2 2 ) e bb̂∗ α−β,−2(α+β),a = e −i cot (φ) ( x2 2 ) e bb̂∗ α−β,−2(α+β)e i cot (φ) ( x2 2 ) (47) which on account of (15) yields an explicit functional expression for both operators.accordingly, thesolution of equation (41) can be written as h(x, τ) = k 2τ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β)e−( k 4τ )(x2+y2)e−( i 2 ) cot(φ)(x2−y2)iα−β ( k 2τ xy ) f (y)dy (48) under the same condition specified in connection with equation (14). one can also note thatthe similarity transformation like link between the operators b̂α−β,−2(α+β),a and b̂α−β,−2(α+β)suggests to recover equation(48) from(41) transforming h(x, τ) to h(x, τ) = h(x, τ)e i cot (φ) ( x2 2 ).in fact the fractional transforms ĥa1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are linked to ĥaα−β through thesame similarity transformation holding between the ordinary transforms; viz. ĥa1,α−β,−2(α+β) = x−2(α+β)+ 1 2 ĥaα−β x2(α+β)− 1 2 ĥa2,α−β,−2(α+β) = x2(α+β)− 1 2 ĥaα−β x−2(α+β)+ 1 2 as a straightforward consequence of the relations b̂α−β,−2(α+β),a = x2(α+β)− 1 2 b̂α−β,a x −2(α+β)+ 1 2 b̂∗α−β,−2(α+β),a = x−2(α+β)+ 1 2 b̂α−β,a x 2(α+β)− 1 2 . 5. barut-girardello-type transformations: the functional expression (15) of ebb̂∗α−β,−2(α+β) resembles the barut-girardello-type transform.the barut-girardello-type transform of bessel order α− β is defined by [3].[ ĝα−βf ] (y) = √ 2 ∫ ∞ 0 (xy)1/2e−(1/2)(x 2+y2)iα−β( √ 2xy)f (x)dx. (49) as a straightforward generalization of the notion of coherent stakes associated with the heisenbergalgebra, such generalized coherent stakes were introduced as eigenstates of the lowering operator ofthe aforementioned algebra in the relative discrete representations d±(k), k = −1/2,−1,−3/2, ... https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 10then taking 2b = 1√ 2 in equation (15) and multiplying the integrand function by e−(1/2)(x 2+y2)e(1/2)(x 2+y2), we end up expression[ e−(1/ √ 2)k̂ (1) − f ] (y) = e−(y 2/2)( √ 2−1)√2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2)(x 2+y2) iα−β( √ 2xy) e−(x 2/2)( √ 2−1)f (x)dx. this allows us to define the first barut-girardello-type transform of bessel order α−β, dependingon a real parameter −2(α+ β), through the expression[ ĝ1,α−β,−2(α+β)f ] (y) = √ 2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2)(x 2+y2) iα−β( √ 2xy)f (x)dx. (50) thus we may write ĝ1,α−β,−2(α+β) = e( √ 2−1)k̂(1)+ ek̂ (1) − / √ 2 e( √ 2−1)k̂(1)+ which can eventually be imposed into the single exponential form: ĝ1,α−β,−2(α+β) = e (π/4) [ k̂ (1) + −k̂ (1) − ] . (51) it clearly states that ĝ1,α−β,−2(α+β) can be regarded as the evolution operator e−iτĥ , associatedwith the dynamical problem ruled by the hamiltonian operator ĥ = k̂ (1) + − k̂ (1) − = 1 2 [ x2 + b̂∗α−β,−2(α+β) ] and evaluated at the purely imaginary value τ = i(π/4) of the evolution variable.now we can define the second barut-girardello-type transform of bessel order α− β as[ ĝ2,α−β,−2(α+β)f ] (y) = √ 2 ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) e−(1/2)(x 2+y2)iα−β( √ 2xy)f (x)dx (52) for which the following operational relatoins can be stated as: ĝ2,α−β,−2(α+β) = e( √ 2−1)k̂(2)+ ek̂ (2) − / √ 2 e( √ 2−1)k̂(2)+ = e (π/4) [ k̂ (2) + −k̂ (2) − ] , (53) involving of course the operators (23). accordingly, ĝ2,α−β,−2(α+β) can be interpreted as theevolution operator operator e−iτĥ , associated with the dynamical problem ruled by the hamiltonianoperator ĥ = k̂ (2) + − k̂ (2) − = 1 2 [ x2 + b̂α−β,−2(α+β) ] and evaluated at the same complex value τ = i(π/4) of the evolution variable as ĝ1,α−β,−2(α+β) .both definitions (50) and(52) can be recast into the comprehensive expression[ ĝj,α−β,−2(α+β)f ] (y) = ∫ ∞ 0 k (bg) j,α−β,−2(α+β)(x, y)f (x)dx, j = 1, 2. (54) in terms of the kernels k (bg) 1,α−β,−2(α+β)(x, y) = √ 2y1−2(α+β)(x)2(α+β)iα−β( √ 2xy)e−(1/2)(x 2+y2) = k (bg) 2,α−β,−2(α+β)(y , x) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 11 which relate to the kernel k (bg) α−β (x, y) = √ 2iα−β( √ 2xy)e−(1/2)(x 2+y2) of the conven-tional transform (49) through the same similarity transformation like relation holding between k1,α−β,−2(α+β)(x, y), k2,α−β,−2(α+β)(y , x) and kα−β(x, y) .as the hankel-type transforms, the transforms ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β) are adjoint toeach other: ĝ∗1,α−β,−2(α+β) = ĝ2,α−β,−2(α+β), ĝ∗2,α−β,−2(α+β) = ĝ1,α−β,−2(α+β). (55) however, they are not self reciprocal; the respective inverse transforms can be easily obtained fromthe corresponding factored representation in equations (51) and (53) which yield[ ĝ−1 1,α−β,−2(α+β)f ] (y) = (−1)α−β+1 √ 2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e(1/2)(x 2+y2) iα−β( √ 2xy)f (x)dx = {[ ĝ−1 2,α−β,−2(α+β) ]∗ f } (y). operational relations similar to (9) can be deduced for ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β). in fact:[ ĝ1,α−β,−2(α+β) î∗α−β,−2(α+β)f ] (y) = 2y2 [ ĝ1,α−β,−2(α+β)f ] (y),[ ĝ2,α−β,−2(α+β) îα−β,−2(α+β)f ] (y) = 2y2 [ ĝ2,α−β,−2(α+β)f ] (y), (56) with the differential operator îα−β,−2(α+β) being îα−β,−2(α+β) = 2 [ k̂ (2) + − k̂ (2) − + 2i k̂ (2) 3 ] . (57) it can be easily seen that îα−β,−2(α+β) = e−(x 2/2) b̂α−β,−2(α+β) e −(x2/2) = x2(α+β)−(α−β)−1 × ( x + ∂ ∂x ) x2(α−β)+1 ( x + ∂ ∂x ) x−(3α+β) (58) even though equation (56) correspond to equation (9) , pertaining to the hankel transform, in-volve operators îα−β,−2(α+β) and î∗α−β,−2(α+β) comprise also the operators k̂+ and k̂3 of thecorresponding algebras. 6. barut-girardello-type transforms of fractional order: we may introduce fractional order versions of the transforms ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β). let us consider, the disentanglement relation for the su(1, 1) algebra generators eζ [k̂+−k̂−] = etan(ζ/2) k̂+ e− sin(ζ) k̂− etan(ζ/2) k̂+ , (59) holding for −π < ζ < π. the expressions above obtained for ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β)correspond to value ζ = (π/4) with appropriate set of operations (21) and (23) being respectively https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 12involved.let us refer in particular to the operators (17) so that on account of (15) one ends up with[ eζ [k̂ (1) + −k̂ (1) − ]f ] = 1 sin ζ y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2) cot(ζ)(x 2+y2) iα−β ( xy sin ζ ) f (x)dx then,writing ζ = (aπ/4), one can obtains the ath power of (51), with the first barut-girardello-typetransforms of fractional order a being accordingly defined by the functional expression:[ ĝa1,α−β,−2(α+β)f ] (y) = 1 sin (φ/2) y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e− cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx(60)with φ = (aπ/2), as beforethe second barut-girardello-type transforms of fractional order a is similarly introduced through e (aπ/4) [ k̂ (2) + −k̂ (2) − ] = ĝa2,α−β,−2(α+β), (61) the relevant functional expression being then:[ ĝa2,α−β,−2(α+β)f ] (y) = 1 sin (φ/2) ∫ ∞ 0 x1−2(α+β)(xy)2(α+β) e−(1/2) cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx.(62)the ordinary transforms are recovered, of course with a = 1, while for α + β = 1 4 , we obtainthe conventional barut-girardello-type transforms of fractional order a, ĝaα−β , introduced in [16]. ĝa1,α−β,1/4 = ĝa2,α−β,1/4 ≡ ĝ a α−β , with [ ĝaα−βf ] (y) = 1 sin (φ/2) ∫ ∞ 0 √ xy e−(1/2) cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx. (63) the fractional transforms ĝa1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) are cyclic with respect to order a,being ĝa+8j 1,α−β,−2(α+β) = ĝa1,α−β,−2(α+β), ĝa+8j 2,α−β,−2(α+β) = ĝa2,α−β,−2(α+β), (64) which allows us to limit the values of a to the interval a ∈ [−4, 4].the operational relations (55)can be generalized to ĝa1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) for which we obtain[ ĝa1,α−β,−2(α+β) î ∗ α−β,−2(α+β),af ] (y) = y2 sin2 (φ/2) [ ĝ1,α−β,−2(α+β)f ] (y),[ ĝa2,α−β,−2(α+β) îα−β,−2(α+β),af ] (y) = y2 sin2 (φ/2) [ ĝ2,α−β,−2(α+β)f ] (y). (65) the differential operator îα−β,−2(α+β),a is given by îα−β,−2(α+β),a = 2 cot2(φ/2) k̂ (2) + − k̂ (2) − + 4i cot(φ/2) k̂ (2) 3 = e[1−cot(φ/2)](x 2/2) îα−β,−2(α+β) e −[1−cot(φ/2)](x2/2). (66) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 13by using equation (57) we have îα−β,−2(α+β),a = e− cot(φ/2)(x 2/2) b̂α−β,−2(α+β) e cot(φ/2)(x2/2) = e2(α+β)−(α−β)−1îax 2(α−β)+1îax −2(α+β)−(α−β) (67) with îa = e− cot(φ/2)(x 2/2) ∂ ∂x ecot(φ/2)(x 2/2) = 1 sin2 (φ/2) [ cos(φ/2)x + sin(φ/2) ∂ ∂x ] . (68) therefore, ĝa1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) are of relevance in connection with evolution equa-tions like k ∂ ∂τ h(x, τ) = p (î∗α−β,−2(α+β),a) h(x, τ), or k ∂ ∂τ h(x, τ) = p (îα−β,−2(α+β),a) h(x, τ) involving polynomial function of îα−β,−2(α+β),a and î∗α−β,−2(α+β),a respectively. 7. generalized hankel transforms: the h and g transform discussed above are associated with hamiltonian operators involvinga linear combination of the generators k̂+ and k̂− of the relevant su(1, 1) algebra realizationsin a form that naturally suggests an arbitrary respectively with the attractive and repulsive radialquantum mechanics oscillator.the dynamical symmetry of the linear quantum mechanical oscillator is that of the su(1, 1) algebra,whose generator are defined in terms of the position and momentum operators are defined in terms ofthe position and momentum operators x̂ and p̂ = i ( d dx ) (h = 1) through the self-adjoint quadranticexpressions k̂+ = 1 2 x̂2 = 1 2 x2, k̂− = − i 2 p̂2 = − 1 2 d2 dx2 , k̂3 = 1 4 (x̂ p̂ + p̂x̂) = − i 2 ( x d dx + 1 2 ) . thus the conventional hankel transform of any order a, being associated with the sum operator k̂1 = k̂+ + k̂−, turns out to be linked to the dynamics of the alternative radial oscillator, therelevant k̂− generator (12) being the radical part of the 2d laplacian operator. in fact, as notedearlier, the hankel transform of integer bessel order can be regarded as the radial part of the 2dfourier transform of rotationally symmetric function, when polar co-ordinates are adopted.in otherwords we have[ f̂a f (ζ, η) ] (x, y) = e−imφ e−imθ [ ĥam ρ1/2 g(ρ) ] (r), m = 0, 1, 2, ... (69) where (ρ, φ) and (r, θ) are polar co-ordinates respectively in the function and transform domainand f is a rotationally symmetric function: f (ζ, η) = g(ρ) e imφ. https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 14evidently (69) can be generalised to the transform (21) and (24) as[ f̂a f (ζ, η) ] (x, y) = e−imφ e−imθ r−1+2(α+β) [ ĥa1,m ρ1−2(α+β) g(ρ) ] (r),[ f̂a f (ζ, η) ] (x, y) = e−imφ e−imθ r−2(α+β) [ ĥa2,m,−2(α+β) ρ 2(α+β) g(ρ) ] (r) for non-negative integers m and rotationally symmetric functions.likewise, the barut-girardello transform resorting to the difference operator k̂2 = k̂+ + k̂−,can be associated with dynamics of the repulsive radical oscillator.therefore barut-girardellotransform of integer bessel order can be regarded as the radial part of the 2d-bargman transform b̂a [2, 16, 20], the inherent relation being similar to (69) i.e. [ b̂a f (ζ, η) ] (x, y) = e−imθ r−1/2 [ ĝam ρ1/2 g(ρ) ] (r), m = 0, 1, 2, ... (70)with the same meaning of the symbols as in equation (69).the generalization of (70) to the transforms of the first and second type is obvious.following the correspondence of the hankel to the fourier transform, we may introduce a gener-alized fractional hankel transforms as the operator associated with evolution equation driven by ageneric operator belonging to the su(1, 1) algebra namely ĥ(1,2) = ak̂ (1,2) + + bk̂ (1,2) − + ck̂ (1,2) 3 + d(v + 1)1̂ being its pertinent to the algebra realization (17) or (23).we exploit the disentanglement scheme e−iτ [ak̂ (1,2) + +bk̂ (1,2) − +ck̂ (1,2) 3 +d(v+1)1̂] = e−idτ(α−β+1) eak̂ (1,2) + eck̂ (1,2) 3 e−iφk̂ (1,2) 1 , giving the operator e−iτĥ(1,2) in three-term factored form, apart from the phase factor e−idτ(v+1).we may introduce the generalized hankel-type transforms of first and second type ,depending onthe parameter p, m and γ, [ ĥa,p,m,γ 1,α−β,−2(α+β)f ] (y) = e i(α−β+1)(γ−π/2) m sin(φ) e−ip(y 2/2) y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) × e(i/2) cot(φ)(x2+(y2/m2)) jα−β ( xy m sin (φ/2) ) f (x)dx (71) [ ĥa,p,m,γ 2,α−β,−2(α+β)f ] (y) = e i(α−β+1)(γ−π/2) m sin(φ) e−ip(y 2/2) ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) × e(i/2) cot(φ)(x2+(y2/m2)) jα−β ( xy m sin (φ/2) ) f (x)dx. the above relations reproduce (21) and (24) respectively, for b = a, d = −a, c = 0 and aτ = φ;also relations like (9) can be deduced for the generalized transform.in addition , generalized borut-girardello type transforms can be introduced on the basis of a https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 15 disentanglement scheme involving the operators k̂(1,2)2 instead of k̂(1,2)1 .as a conclusion, we note that from equation (71) we may recover with p = ib/m2, m2 = 1 − b2, tan(φ) = −ib and γ = 0, the integral transformations corresponding to the exponential forms e bb̂∗ α−β,−2(α+β) and ebb̂α−β,−2(α+β) (see (15) and the relevant adjoint expression, easily deducible)which can respectively be regarded as the first and second weiestrass-gauss integral transformsof bessel order α−β depending on the real parameter −2(α+β). they generalize to α+β = 1/4the expression of the radical weiestrass-gauss integral transform, for which we have [16][ ŵα−β,bf ] (y) = [ e−(b/2)b̂α−β f ] (y) = 1 b ∫ ∞ 0 (xy)(1/2) e−(1/2b)(x 2+y2) iα−β (xy b ) f (x)dx for any real parameter β > 0. ŵα−β,b arises as the transfer operator associated with the radial part of heat conduction likeequations.it can be in fact considered as the radial part of 2d fresnel transform for real parametersis the optical operator for free propagation. a linear weiestrass-gauss integral transform has alsobeen introduced [19] the relation of ŵm,b, m = 0, 1, 2, ... to it is evidently similar to (69), whenrationally symmetric function are involved. remark. [(i)](1) if we take α = ν 2 − µ 4 , β = −µ4 − ν 2 throught this paper then all the results studied in this paper reduce to the results studied in torre [17].(2) authors claim that results of this paper are stronger than that of torre [17]. 8. conclusions: following the scheme already applied to other type of transforms, like for instance, the fouriertransform , we have introduced the fractional forms of two adjoint self-reciprocal variants of thehankel type transform, which, as noted are of interest in connection with evolution problems ruledby the bessel-type differential operators, b̂α−β,−2(α+β) = xα+3β−1dxx 2(α−β)+1 dxx −3α−β and b̂α−β,−2(α+β) = x−3α−βdxx 2(α−β)+1dxx α+3β−1. the fractional order transform relate to evolution problems ruled the operators b̂α−β,−2(α+β),a = xα+3β−1 la ( x, ( ∂ ∂x )) x2(α−β)+1 la ( x, ( ∂ ∂x )) x−3α−β and b̂∗α−β,−2(α+β),a = x−3α−β la ( x, ( ∂ ∂x )) x2(α−β)+1 la ( x, ( ∂ ∂x )) xα+3β−1 https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 16where la (x, ( ∂∂x )) is linear a-depending combination of x and ( ∂∂x ). since a ranges from -1 to 1,it is evident that the set of evolution problems inherent in the transforms has been greatly enlargeby the fractionalization.in general we have shown that introduced transforms can be regarded as the evolution operatorsassociated with evolution problems having an underlying su(1, 1) symmetry, the specific realizationof the algebra resorting to b̂α−β,−2(α+β) and b̂∗α−β,−2(α+β) as the relative ladder operators.evidently, transforms of complex fractional order can be considered although of course the space offunctions on which they can meaningfully be applied must be carefully investigated. disregardinghere this aspect of the question, we simply note that due to the additivity with respect to the order,we may write ĥar+ial j,α−β,−2(α+β) = ĥar j,α−β,−2(α+β) + ĥial j,α−β,−2(α+β), j = 1, 2. where ar and al respectively denote the real and imaginary part of the order a = ar + ial .thus ĥar j,α−β,−2(α+β), j = 1, 2 have just the expression considered in this paper, while ĥial j,α−β,−2(α+β), j = 1, 2 are from then easily deducible replacing φ with iφ.as earlier mentioned, the hankel transform is of interest within the context of the fractional calculus[17]. following the arguments in [18], for the transforms of our concern we find that[ ĥ1,α−β,−2(α+β)f ] (y) = 21−(α−β) x−α−3β+1√ π ∫ ∞ 0 cos(yτ) [ k̂α−β+1/2 g1 ] (τ)dτ, [ ĥ2,α−β,−2(α+β)f ] (y) = 21−(α−β) x−α−3β+1√ π ∫ ∞ 0 cos(yτ) [ k̂α−β+1/2 g2 ] (τ)dτ (72) where k̂α−β+1/2 is the left hand sided erdelyi-kober fractional integral operator, represented by[ k̂bg ] (y) = 1 γ(b) ∫ ∞ y (y2 − x2)b−1 x g(x) dx, r(b) > 0, y ∈ r, and the functions g1(x) and g2(x) on which it acts in (72) involve f as g1(x) = xα+3β−1f (x), g2(x) = x−3α−βf (x).relations (72) holds under the assumption that both xg1(x) and xg2(x) are inegrable. note thataccording to sonine’s first integral for bessel functions, we may say that the right hand sidederdelyi-kober operator of order b acts on the function xα−β jα−β(x) as a rising operator turningit into xα−β+b jα−β+b(x).expressions similar to (72) can be deduced for the fractional order transforms, of course.in addition (72) as paralleled by similar expressions involving the barut-girardello transform ofthe first and second type. as an example, we deduce here the relation involving the conventionaltransform (49). on account of the integral representation of the modified bessel function of thefirst kind, iα−β , i.e. iα−β(xy) = 21−(α−β) yα−β x−(α−β)√ π γ(α− β + 1/2) ∫ x 0 (x2 − τ2)α−β−1/2 cosh(yτ)dτ, r(α− β + 1/2) > 0, https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 17it is easy to rewrite (49) in the form[ ĝα−βf ] (y) = √ 2 π 21+(α−β2 ) e−(1/2)y 2 ∫ ∞ 0 cosh( √ 2yτ) [k̂α−β+1/2 h](τ)dτ, with the function h(x) being h(x) = x−1−(α−β2 ) e−(1/2)y 2 f (x). the above relation holds under the assumption that xh(x) is integrable.we finally note 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hankel transform and its application for studying the propagation ofcylindrical electromagnetic fields, opt. express. 10 (2002), 521-525. https://doi.org/10.1364/oe.10.000521. https://doi.org/10.28924/ada/ma.3.6 https://doi.org/10.1016/j.sigpro.2005.10.001 https://doi.org/10.1090/s0002-9939-1969-0243294-0 https://doi.org/10.1090/s0002-9939-1969-0243294-0 https://doi.org/10.1016/s0377-0427(02)00637-4 https://doi.org/10.1080/10652460701827848 https://doi.org/10.1006/jmaa.1997.5351 https://doi.org/10.1111/j.1365-2478.1986.tb00481.x https://doi.org/10.1063/1.1666811 https://doi.org/10.1063/1.1666811 https://doi.org/10.1063/1.1666590 https://doi.org/10.1063/1.1666590 https://doi.org/10.1364/oe.10.000521 1. introduction 2. hankel type transforms: 3. hankel-type transforms of fractional order: 4. properties of a1,-,-2(+) and a2,-,-2(+): 5. barut-girardello-type transformations: 6. barut-girardello-type transforms of fractional order: 7. generalized hankel transforms: 8. conclusions: references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 1-18doi: 10.28924/ada/ma.1.1 existence and stability results for second-order neutral stochastic differential equations with random impulses and poisson jumps k. ravikumar1, k. ramkumar1, dimplekumar chalishajar2,∗ 1department of mathematics, psg college of arts and science, coimbatore, 641 046, india; ravikumarkpsg@gmail.com, ramkumarkpsg@gmail.com2department of applied mathematics, mallory hall, virginia military institute, lexington, va 24450, usa ∗correspondence: chalishajardn@vmi.edu abstract. the objective of this paper is to investigate the existence and stability results of second-order neutral stochastic functional differential equations (nsfdes) in hilbert space. initially, weestablish the existence results of mild solutions of the aforementioned system using the banachcontraction principle. the results are formulated using stochastic analysis techniques. in the laterpart, we investigate the stability results through the continuous dependence of solutions on initialconditions. 1. introduction stochastic differential equations (sdes) captures disturbances from random factors. mathe-matical models obtained by integrating stochastic process provide a better understanding of thereal-world system [12]. for elementary study of stochastic differential equations, the reader mayrefer to [7, 12,14,24].impulsive differential equations also attracted the attention of researchers (see [4, 11, 13, 21, 22]etc.). impulse in general occurs as deterministic or random models. nevertheless by naturalphenomena, the impulses often occur at random time points. many researches have been undergonesolving various differential equations with fixed time impulses [1, 9, 16, 23]. random impulsivedifferential equations involving fractional derivative are also studied see [20,25].it is known that impulsive stochastic differential equations play a vital role in modelling practicalprocesses. not only from guassian white noise there are certain other factors that results in therise of random effects. random impulsive stochastic differential equations (isdes) are widely used received: 20 aug 2021. key words and phrases. existence; stability; banach contraction principle; second-order neutral stochastic functionaldifferential equations; random impulse; stochastic differential system.1 https://adac.ee https://doi.org/10.28924/ada/ma.1.1 https://orcid.org/0000-0002-6146-5544 eur. j. math. anal. 1 (2021) 2in the fields of medicine, biology, economy, finance and so on. for example, the classical stockprice model [28]. d[s(t)] = fs(t)dt + σs(t)dw(t), t ≥ 0, t 6= τk , s(τk ) = aks(τ−k ), k = 1, 2, ..., s(0) = s0, is described using an isdes. here wt is a brownian motion or wiener process, s(t) representsthe price of the stock at time t, and {τk} represents the release time of the important informationrelating to the stock. s(τ−k ) = limt→τk−0 s(t) and s0 ∈ r. in reality, {τk} is a sequence ofrandom variables, which satisfies 0 < τ2 < τ3 < · · · . recently, in [10] the authors have contributedthe existence and hyers-ulam stability of mild solutions for random impulsive stochastic functionalordinary differential equations which are studied using krasnoselskii’s fixed point theorem.solving second-order differential equations has been observed by many scholars. many authorssolved second-order stochastic differential equations see [5, 6, 8, 19]. however, there are not manypapers considering the existence and stability results on stochastic differential equations withrandom impulse. anguraj et.al [3], considered the sdes with random impulses and poisson jumpsof the form d[x(t)] = f(t, xt) + g(t, xt)dw(t) + ∫ u h(t, xt , u)ñ(dt, du), t ≥ t0, t 6= τk , x(ζk ) = bk (τk )x(ζ−k ), k = 1, 2, ..., xt0 = ζ = {ζ(θ) : −τ ≤ θ ≤ 0} . the authors studied the existence, uniqueness, and stability through continuous dependence oninitial conditions for sdes with random impulses and poisson jumps by using banach fixed pointtheorem. very recently, anguraj et.al [2] investigated the existence and hyers ulam stability ofrandom impulsive stochastic functional integrodifferential equations with finite delays.motivated by the above discussion, here we consider the following second-order nsfdes withrandom impulses and poisson jumps. d [ x ′(t)− h(t, xt)] = [ax(t) + f(t, xt)]dt + g(t, xt)dω(t) + ∫ u σ (t, xt , u)ñ(dt, du), , t ≥ t0, t 6= ξk, x(ξk) = bk(δk)x(ξ−k ), x ′(ξk) = bk(δk)x ′(ξ−k ), k = 1, 2, ..., (1.1) xt0 = φ, x ′(t0) = φ, where a : d(a) ⊂ h → h is the infinitesimal generator of a strongly continuous cosine family {c(t), t ≥ 0}. w(t) is a given q-wiener process with a finite trace nuclear covariance operator q > 0. δk is a random variable defined from ω to d ≡ (0, dk) for k = 1, 2 · · · . suppose that δi and δj are independent of each other as i 6= j, (i, j = 1, 2, · · · ). the impulsive moments ξk are randomvariables and satisfy ξk = ξk−1 + δk, k = 1, 2, · · · . obviously, {ξk} is a process with independentincrements. 0 < t0 = ξ0 < ξ2 < ξ3 < · · · < limk→∞ ξk = ∞, and x(ξ−k ) = lim t→ξk−0 x(t). bk : dk → h, eur. j. math. anal. 1 (2021) 3for each k = 1, 2, · · · . the time history xt(θ) = {x(t + θ) : −δ ≤ θ ≤ 0} with some given δ > 0.moreover, h, f, g, σ , and φ, φ will be specified later.to the best of authors knowledge, up to now, no work has been reported to derive the second-order nsfdes with random impulses and poisson jumps. the main contributions are summarizedas follows:(1) second-order nsfdes with random impulses and poisson jumps is formulated.(2) initially, we establish the existence results of mild solutions of the aforementioned system usingbanach contraction principle.(3) next, we investigate the stability results through continuous dependence of solutions on initialconditions.(4) an example is provided to illustrate the obtained theoretical results.the rest of the paper is organised as follows. section 2 is devoted to basic definitions, notions andlemma. in section 3, existence of mild solutions of the aforementioned system (1.1) is investigatedusing banach contraction principle. eventually in section 4, the stability of mild solution is obtainedthrough continuous dependence of solutions on initial conditions. 2. preliminaries let (ω,=,p) be a complete probability space equipped with the normal filtration {=}t≥t0 . =t0containing all p-null sets. h and k be two real hilbert spaces. l(h,k) denotes the space of allbounded linear operators from k to h.we may assume that, {n (t), t ≥ t0} be a counting process generated by {ξk, k ≥ 0}. =(1) t denotethe minimal σ algebra denoted by {n (r), r ≤ t} and denote =(2) t the σ-algebra generated by {ω(s), s ≤ t}. we assume that =(1) ∞ ,=(2) ∞ and ξ are mutually independent and =t = =(1) t ∨ = (2) t .we assume that there exist a complete orthonormal system {en}∞n=1 in k, a bounded sequenceof non-negative real numbers λn such that, qen = λnen, n = 1, 2, · · · . let {βn(t)}(n = 1, 2, 3...) bea sequence of real valued one dimensional standard brownian motion mutually independent over(ω,=,p). a q-wiener process can be defined by ω(t) = ∞∑ n=1 √ λnβn(t)en, (t ≥ 0). set φ ∈ l(k,h) we define, ∥∥φ∥∥2 q = t r(φqφ∗) = ∞∑ n=1 ∥∥∥√λnφen∥∥∥2 if ∥∥φ∥∥2 q <∞, then φ is called a q-hilbert-schmidt operator. let lq(k,h) denote the space ofall q-hilbert-schmidt operator φ : k → h. the completion lq(k,h) of l(k,h) with respect tothe topology induced by the norm ∥.∥q , where ∥∥φ∥∥2 q = 〈φ,φ〉 is a hilbert space.let t ∈ (t0,+∞), j := [t0, t ], jk = [ξk, ξk+1) , k = 0, 1, · · · , j̃ = {t : t ∈ j, t 6= ξk, k = 1, 2, · · · }. l2(ω,h) be the collection of square integrable =t-measurable, h-valued random variables definedby the norm ∥x∥l2 = (e∥x∥2) 12 , the expectation being expressed by the form e∥x∥2 = ∫ω ∥x∥2 dp.let pc (j,l2(ω,h)) = {x : j → l2(ω,h)}, x is continuous on every jk, and the left limits x(ξ−k ), x ′(ξ−k ) exist k = 1, 2, · · · be a piecewise continuous space. eur. j. math. anal. 1 (2021) 4we may define the space c = c ([−δ, 0],h) which contains all piecewise continuous functionsmapping from [−δ, 0] to h with the norm ∥x∥t = sup t−δ≤s≤t ∥∥x(s)∥∥ for each t ≥ t0. b be the banachspace, b ([t0−δ, t ],l2(ω,h)) consists of continuous, =t-measurable, c-valued processes. the normis defined by ∥x∥b = (sup t∈j e∥x∥2 t ) 12 . in (1.1), ñ(dt, du) = n(dt, du)− dtv (du) denotes the compensated poisson measure independentof ω(t) and n(dt, du) represents the poisson counting measure associated with a characteristicmeasure v . for a basic study on the poisson jumps we refer to the book by [27].subsequently, we introduce certain definitions of sine and cosine operators.a bounded linear operators family {c(t), t ∈ r} is called a strongly continuous cosine family ifand only if(i) c(0) = i (i is the identity operator in h);(ii) c(t)x is continuous in t, for all x ∈ h;(iii) c(t + s) + c(t − s) = 2c(t)c(s) for all t, s ∈ r.the corresponding strongly continuous sine family {s(t), t ∈ r} is defined by s(t)x = ∫ t 0 c(s)xds, x ∈ h, t ∈ r then the following property holds: a ∫ t t0 s(s)xds = [c(t)− c(t0)] x lemma 2.1. [18] let {c(t), t ∈ r} be a strongly continuous cosine family in h, then for all s, t ∈ r, the following results are true: (i) c(t) = c(−t); (ii) s(s+ t) + s(s− t) = 2s(s)c(t); (iii) s(s+ t) = s(s)c(t) + s(t)c(s); (iv) s(t) = −s(−t); (v) c(t + s) + c(s− t) = 2c(s)c(t); (vi) c(t + s)− c(t − s) = 2as(t)s(s). before investigating mild solution (1.1), we consider the second-order neutral functional differ-ential equation, which is given byd[u′(t)− g(t, u(t))] = autdt + f(t, ut)dt, t ≥ 0, u0 = φ ∈ c, u′(0) = φ ∈ h, t ∈ (−r, 0], (2.1) where a is the infinitesimal generator of a strongly continuous cosine family {c(t), t ∈ r+} andthe functions g, f ∈ l1(0, t ; h). eur. j. math. anal. 1 (2021) 5 lemma 2.2. [15] a continuously differentiable function u(t) : [0, t ]→ h is called the mild solution for the cauchy problem (2.1), if it satisfies, u(t) = c(t)φ(0) + s(t)[φ − g(0, φ)] + ∫ t 0 c(t − s)g(s, us)ds+ ∫ t 0 s(t − s)f(s, xs)ds, t ≥ 0, where s(t) = 12πi ∫ γ eλtr(λ2;a)dλ; c(t) = 12πi ∫ γ eλtλr(λ2;a)dλ, and γ is a suitable path. consider the linear second-order linear differential equation with impulse conditions, u′′(t) = au(t) + f(t), t ≥ 0, t 6= tk , u(0) = u0, u′(0) = v0, u(tk) = bku(t−k ), u′(tk) = bku′(t−k ), k = 1, 2, · · · , (2.2) where 0 = t0 < t1 < t2 < · · · < tk < · · · , {tk, k ≥ 1} is a sequence of fixed impulsive points, f(t) : [0, t )→ h is an integrable function. lemma 2.3. the piecewise continuous differentiable function u(t) : [0, t ]→ h is a mild solution of (2.2), if and only if x(t) satisfies the integral equation u(t) = k∏ i=1 bic(t)u0 + k∏ i=1 bis(t)v0 + k∑ i=1 k∏ j=i bj ∫ ti ti−1 s(t − s)f(s)ds × ∫ t tk s(t − s)f(s)ds, t ∈ [tk, tk+1), k = 0, 1, · · · . (2.3) proof . (i)for t ∈ [0, t1), the mild solution is studied in [17], u(t) = c(t)u0 + s(t)v0 + ∫ t 0 s(t − s)f(s)ds, t ∈ [0, t1). (ii) for t ∈ [t1, t2), we set u(t) = c(t − t1)u(t1) + s(t − t1)u′(t1) + ∫ t t1 s(t − s)f(s)ds, t ∈ [t1, t2). (2.4) since, u(t1) = b1u(t−1 ), u′(t1) = b1u′(t−1 ),and from (i) we know u(t−1 ) = c(t1)u0 + s(t1)v0 + ∫ t1 0 s(t1 − s)f(s)ds; (2.5) u′(t−1 ) = as(t1)u0 + c(t1)v0 + ∫ t1 0 c(t1 − s)f(s)ds. (2.6) eur. j. math. anal. 1 (2021) 6thus, u(t) = b1c(t − t1)c(t1)u0 + b1s(t − t1)as(t1)u0 + b1c(t − t1)s(t1)v0 + b1s(t − t1)c(t1)v0 + b1c(t − t1)∫ t1 0 s(t1 − s)f(s)ds+ b1s(t − t1)∫ t1 0 s(t1 − s)f(s)ds+ ∫ t t1 s(t − s)f(s)ds, t ∈ [t1, t2). applying lemma 2.1, we get u(t) = b1c(t)u0 + b1s(t)v0 + b1 ∫ t1 0 s(t1 − s)f(s)ds+ ∫ t t1 s(t − s)f(s)ds, t ∈ [t1, t2). (iii) for t ∈ [t2, t3), u(t) = c(t − t2)u(t2) + s(t − t2)u′(t2) + ∫ t t2 s(t − s)f(s)ds= c(t − t2)b2u(t−2 ) + s(t − t2)b2u′(t−2 ) + ∫ t t2 s(t − s)f(s)ds. (2.7) from the conclusions of (ii), it is known that, u(t−2 ) = b1c(t2)u0 + b1s(t2)v0 + b1 ∫ t2 0 s(t2 − s)f(s)ds+ ∫ t2 t1 s(t2 − s)f(s)ds; (2.8) u′(t−2 ) = b1as(t2)u0 + b1c(t2)v0 + b1 ∫ t2 0 c(t2 − s)f(s)ds+ ∫ t2 t1 c(t2 − s)f(s)ds (2.9) along with (2.7) and using lemma 2.1, we have u(t) = b2b1c(t)u0 + b2b1s(t)v0 + b2b1 ∫ t1 0 s(t − s)f(s)ds+ b2 ∫ t2 t1 s(t − s)f(s)ds+ ∫ t2 t1 s(t − s)f(s)ds, t ∈ [t2, t3) similarly, for all t ∈ [tk, tk−1). x(t) = k∏ i=1 bic(t)u0 + k∏ i=1 bis(t)v0 + k∑ i=1 k∏ j=i bj ∫ ti ti−1 s(t − s)f(s)ds+ ∫ t ξk s(t − s)f(s)ds. �by lemma 2.2, lemma 2.3 the mild solution of the system (1.1) applying index function for t ∈ j. definition 2.1. for a given t ∈ (t0,+∞), a =-adapted process function {x ∈ b , t0 − δ ≤ t ≤ t} is called a mild solution of system (1.1), if (i) xt0(s) = φ(s) ∈ l02(ω, b ) for δ ≤ s ≤ 0; (ii) x ′(t0) = φ(t) ∈ l02(ω,h) for t ∈ j; (iii) the functions f(s, xt), g(s, xt), h(s, xt) and σ (s, xs, u) are integrable, and for a.e. t ∈ j, the eur. j. math. anal. 1 (2021) 7 following integral equation is satisfied. x(t) = +∞∑ k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ − h(0, φ)] + k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 c(t − s)h(s, xs)ds+ ∫ t ξk c(t − s)h(s, xs)ds+ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s) × f(s, xs)ds+ ∫ t ξk s(t − s)f(s, xs)ds+ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)g(s, xs)dω(s) + ∫ t ξk s(t − s)g(s, xs)dω(s) + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 ∫ u s(t − s)σ (s, xs, u)n(ds, du) + ∫ t ξk ∫ u s(t − s)σ (s, xs, u)n(ds, du)]i[ξk ,ξk+1)(t), t ∈ [t0, t ]. (2.10) where, k∏ j=i bj (δj ) = bk (δk )bk−1(δk−1) · · ·bi(δi), and i(a)(.) is the index function, i.e., ia(t) = 1, if t ∈ a,0, if t /∈ a. lemma 2.4. for any p ≥ 1, and for lq(k,h)-valued predictable process u(.) such that, sup s∈[0,t ]e ∥∥∥∥∫ s 0 u(η)dω(η)∥∥∥∥2p ≤ (p(2p− 1))p(∫ t 0 ( e ∥∥u(s)∥∥2p q )1/p ds )p , t ∈ j. 3. existence results of mild solution to prove the existence of mild solutions of random impulsive stochastic differential equations,the following assumptions are to be made.(h1) c(t), s(t)(t ∈ j) are equicontinuous and there exist positive constants m, m̃ such that sup t∈j ∥∥c(t)∥∥ ≤m, sup t∈j ∥∥s(t)∥∥ ≤ m̃. (3.1) (h2) the functions f : j× c→ h; h : j× c→ h; g : j× c→ lq(k,h) and σ : j× c× u→ h e ∥∥f(t, xt)− f(t, yt)∥∥2 ≤ lf ∥∥x − y∥∥2 t , e ∥∥g(t, xt)− g(t, yt)∥∥2 ≤ lg ∥∥x − y∥∥2 t , e ∥∥h(t, xt)− h(t, yt)∥∥2 ≤ lh ∥∥x − y∥∥2 t ,∫ u e ∥∥σ (t, xt , u)− σ (t, yt , u)∥∥2 v (du)ds ∨ ∫ u ( e ∥∥σ (t, xt , u)− σ (t, yt , u)∥∥4 v (du)ds) 12 ≤ lσ ∥∥x − y∥∥2 t ,∫ u ( e ∥∥σ (t, xt , u)− σ (t, yt , u)∥∥4 v (du)ds) 12 ≤ lσ ∥x∥2 t . eur. j. math. anal. 1 (2021) 8(h3) for all t ∈ j, there exist constants κf, κg, κh, κσ ∈ l′(j,r+) such that, e ∥∥f(t, 0)∥∥2 ≤ κf, e∥∥g(t, 0)∥∥2 ≤ κg, e ∥∥h(t, 0)∥∥2 ≤ κh, e∥∥σ (t, 0, u)∥∥2 ≤ κσ . (h4) e max i,k { k∏ j=i ∥∥bj (δj )∥∥}  is uniformly bounded then there exist constant n for all δj ∈ dj such that e max i,k { k∏ j=i ∥∥bj (δj )∥∥}  ≤ n . theorem 3.1. if assumptions (h1)-(h4) gets satisfied then there exist a unique continuous mild solution of the system (1.1). proof . we define an operator φ : b → b by φx such that, (φx)(t) =  φ(t), t ∈ [t0 − δ, t0],+∞∑ k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ − h(0, φ)] + k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 c(t − s)h(s, xs)ds+ ∫ t ξk c(t − s)h(s, xs)ds+ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s) × f(s, xs)ds+ ∫ tξk s(t − s)f(s, xs)ds+ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)g(s, xs)dω(s) +∫ t ξk s(t − s)g(s, xs)dω(s) + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 ∫ u s(t − s)σ (s, xs, u)n(ds, du) +∫ t ξk ∫ u s(t − s)σ (s, xs, u)n(ds, du)]i[ξk ,ξk+1)(t), t ∈ [t0, t ]. we need to prove that φ maps b into itself. e ∥∥(φx)(t)∥∥2 ≤ e ∥∥∥∥ +∞∑ k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ − h(0, φ)] + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 c(t − s)h(s, xs)ds+ ∫ t ξk c(t − s)h(s, xs)ds eur. j. math. anal. 1 (2021) 9 + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)f(s, xs)ds+ ∫ t ξk s(t − s)f(s, xs)ds + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)g(s, xs)dω(s) + ∫ t ξk s(t − s)g(s, xs)dω(s) + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 ∫ u s(t − s)σ (s, xs, u)n(ds, du) + ∫ t ξk ∫ u s(t − s)σ (s, xs, u)n(ds, du)]i[ξk ,ξk+1)(t)∥∥∥∥2 ≤ 6e[+∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥c(t − t0)∥∥∥∥φ(0)∥∥] i[ξk ,ξk+1)(t)]2 + 6e[ +∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥s(t − t0)∥∥ × ∥∥φ − h(0, φ)∥∥ ]i[ξk ,ξk+1)(t)]2 + 6e[ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥∫ ξi ξi−1 ∥∥c(t − s)∥∥∥∥h(s, xs)∥∥ds + ∫ t ξk ∥∥c(t − s)∥∥∥∥h(s, xs)∥∥ds]i[ξk ,ξk+1)(t)]2 + 6e[ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥∫ ξi ξi−1 ∥∥s(t − s)∥∥ × ∥∥f(s, xs)∥∥ds+ ∫ t ξk ∥∥s(t − s)∥∥∥∥f(s, xs)∥∥ds]i[ξk ,ξk+1)(t)]2 + 6e[ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥ × ∫ ξi ξi−1 ∥∥s(t − s)∥∥∥∥g(s, xs)∥∥dω(s) + ∫ t ξk ∥∥s(t − s)∥∥∥∥g(s, xs)∥∥dω(s)]i[ξk ,ξk+1)(t) + 6e[ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥× ∫ ξi ξi−1 ∫ u ∥∥s(t − s)∥∥∥∥σ (s, xs, u)∥∥ ñ(ds, du) + ∫ t ξk ∫ u ∥∥s(t − s)∥∥∥∥σ (s, xs, u)∥∥ ñ(ds, du)]i[ξk ,ξk+1)(t)]2 , = 6 6∑ i=1 gi. where, g1 ≤ e [+∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥c(t − t0)∥∥∥∥φ(0)∥∥] i[ξk ,ξk+1)(t)]2 ≤ m2e max i,k { k∏ j=i ∥∥bj (δj )∥∥}  2 e ∥∥φ(0)∥∥2 ≤ m2n 2e∥∥φ(0)∥∥2 , g2 ≤ e [ +∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥s(t − t0)∥∥∥∥φ − h(0, φ)∥∥ ]i[ξk ,ξk+1)(t)]2 eur. j. math. anal. 1 (2021) 10 ≤ m̃2e max i,k { k∏ j=i ∥∥bj (δj )∥∥}  2 e ∥∥φ − h(0, φ)∥∥2 ≤ m̃2n 2e∥∥φ − h(0, φ)∥∥2 , g3 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥∫ ξi ξi−1 ∥∥c(t − s)∥∥∥∥h(s, xs)∥∥ds+ ∫ t ξk ∥∥c(t − s)∥∥∥∥h(s, xs)∥∥ds]i[ξk ,ξk+1)(t)]2 ≤ m2e max i,k {1, k∏ j=i ∥∥bj (δj )∥∥}  2 (t − t0)∫ t t0 e ∥∥h(s, xs)∥∥2 ds ≤ 2m2 max{1,n 2}(t − t0) [∫ t t0 e ∥∥h(s, xs)− h(s, 0)∥∥2 ds+ ∫ t t0 ∥∥h(s, 0)∥∥2 ds] ≤ 2m2 max{1,n 2}(t − t0)∫ t t0 [ lhe ∥x∥2 s + κh ] ds ≤ 2m2 max{1,n 2}(t − t0)∫ t t0 lhe ∥x∥2 s ds+ 2m2 max{1,n 2}(t − t0)2κh, g4 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥∫ ξi ξi−1 ∥∥s(t − s)∥∥∥∥f(s, xs)∥∥ds+ ∫ t ξk ∥∥s(t − s)∥∥ ∥∥f(s, xs)∥∥ds]i[ξk ,ξk+1)(t)]2 ≤ m̃2e max i,k {1, k∏ j=i ∥∥bj (δj )∥∥}  2 (t − t0)∫ t t0 e ∥∥f(s, xs)∥∥2 ds ≤ 2m̃2 max{1,n 2}(t − t0) [∫ t t0 e ∥∥f(s, xs)− f(s, 0)∥∥2 ds+ ∫ t t0 ∥∥f(s, 0)∥∥2 ds] ≤ 2m̃2 max{1,n 2}(t − t0)∫ t t0 [ lfe ∥x∥2 s + κf ] ds ≤ 2m̃2 max{1,n 2}(t − t0)∫ t t0 lfe ∥x∥2 s ds+ 2m̃2 max{1,n 2}(t − t0)2κf, g5 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥∫ ξi ξi−1 ∥∥s(t − s)∥∥∥∥g(s, xs)∥∥dω(s) + ∫ t ξk ∥∥s(t − s)∥∥∥∥g(s, xs)∥∥dω(s)]i[ξk ,ξk+1)(t)]2 ≤ m̃2e max i,k {1, k∏ j=i ∥∥bj (δj )∥∥}  2 ∫ t t0 e ∥∥g(s, xs)∥∥2 ds ≤ 2m̃2 max{1,n 2}t r(q) [∫ t t0 e ∥∥g(s, xs)− g(s, 0)∥∥2 ds+ ∫ t t0 ∥∥g(s, 0)∥∥2 ds] ≤ 2m̃2 max{1,n 2}t r(q)∫ t t0 [ lge ∥x∥2 s + κg ] ds ≤ 2m̃2 max{1,n 2}t r(q)∫ t t0 lge ∥x∥2 s ds+ 2m̃2 max{1,n 2}(t − t0)t r(q)κg. eur. j. math. anal. 1 (2021) 11 g6 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj (δj )∥∥× ∫ ξi ξi−1 ∫ u ∥∥s(t − s)∥∥∥∥σ (s, xs, u)∥∥ ñ(ds, du) + ∫ t ξk ∫ u ∥∥s(t − s)∥∥∥∥σ (s, xs, u)∥∥ ñ(ds, du)]i[ξk ,ξk+1)(t)]2 ≤ 2m̃2 max{1,n 2} ∫ t t0 ∫ u [ e ∥∥σ (s, xs, u)− σ (s, 0, u)∥∥2 + ∥∥σ (s, 0, u)∥∥2]ds + 2m̃2 max{1,n 2}(∫ t t0 ∫ u e ∥∥σ (s, xs, u)∥∥4 v (du)ds) 12 ≤ 4m̃2 max{1,n 2} ∫ t t0 lσe ∥x∥2 s ds+ 2m̃2 max{1,n 2}(t − t0)κσ , thus we would obtain, e ∥∥(φx)(t)∥∥2 t ≤ 6m2n 2e∥∥φ(0)∥∥2 + 6m̃2n 2e∥∥φ − h(0, φ)∥∥2 + 12m2 max{1,n 2}(t − t0) × ∫ t t0 lhe ∥x∥2 s ds+ 12m2 max{1,n 2}(t − t0)2κh + 12m̃2 max{1,n 2}(t − t0) × ∫ t t0 lfe ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)2κf + 12m̃2 max{1,n 2}t r(q) × ∫ t t0 lge ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)t r(q)κg + 24m̃2 max{1,n 2}∫ t t0 lσe ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)κσ . taking supremum over t, sup t0≤t≤t e ∥∥(φx)(t)∥∥2 t ≤ 6m2n 2e∥∥φ(0)∥∥2 + 6m̃2n 2e∥∥φ − h(0, φ)∥∥2 + 12m2 max{1,n 2}(t − t0) × ∫ t t0 lh sup t0≤t≤t e ∥x∥2 s ds+ 12m2 max{1,n 2}(t − t0)2κh + 12m̃2max{1,n 2} × (t − t0)∫ t t0 lf sup t0≤t≤t e ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)2κf + 12m̃2 × max{1,n 2}t r(q)∫ t t0 lg sup t0≤t≤t e ∥x∥2 s ds+ 12m̃2max{1,n 2}(t − t0)t r(q)κg + 24m̃2 max{1,n 2} ∫ t t0 lσ sup t0≤t≤t e ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)κσ eur. j. math. anal. 1 (2021) 12 ≤ 6m2n 2e∥∥φ(0)∥∥2 + 6m̃2n 2e∥∥φ − h(0, φ)∥∥2 + 12m2 max{1,n 2}(t − t0)2 × lh sup t0≤t≤t e ∥x∥2 t + 12m2 max{1,n 2}(t − t0)2κh + 12m̃2 max{1,n 2} × (t − t0)2lf sup t0≤t≤t e ∥x∥2 t + 12m̃2 max{1,n 2}(t − t0)2κf + 12m̃2 max{1,n 2}t r(q)(t − t0)lg sup t0≤t≤t e ∥x∥2 t + 12m̃2 max{1,n 2}(t − t0)t r(q)κg+ 24m̃2 max{1,n 2}(t − t0)lσ sup t0≤t≤t e ∥x∥2 s ds+ 12m̃2 max{1,n 2}(t − t0)κσ ≤ 6[n 2 [m2e∥∥φ(0)∥∥2 + m̃2e∥∥φ − h(0, φ)∥∥2] ]+ 12 max{1,n 2}(t − t0) × [ m2(t − t0)κh + m̃2(t − t0)κf + m̃2t r(q)κg + m̃2κσ ]+ 12 max{1,n 2} × (t − t0) [m2(t − t0)lh + m̃2(t − t0)lf + m̃2t r(q)lg + 2m̃2lσ] ∥x∥2 t∥∥φx∥∥2 b ≤ c1 + c2 ∥x∥2 b .where, c1 = 6[n 2 [m2e∥∥φ(0)∥∥2 + m̃2e∥∥φ − h(0, φ)∥∥2] ]+ 12 max{1,n 2}(t − t0) × [ m2(t − t0)κh + m̃2(t − t0)κf + m̃2t r(q)κg + m̃2κσ ] , c2 = 12 max{1,n 2}(t − t0) [m2(t − t0)lh + m̃2(t − t0)lf + m̃2t r(q)lg + 2m̃2lσ] . where c1 and c2 are constants.hence φ is bounded.now we need to prove that φ is a contraction mapping. for any x, y ∈ b we have,∥∥(φx)(t)− (φy)(t)∥∥2 ≤ ∥∥∥∥ +∞∑ k=0 [ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 c(t − s)h(s, xs)ds+ ∫ t ξk c(t − s)h(s, xs)ds + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)f(s, xs)ds+ ∫ t ξk s(t − s)f(s, xs)ds+ k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 s(t − s)g(s, xs)dω(s) + ∫ t ξk s(t − s)g(s, xs)dω(s) + k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 ∫ u s(t − s)σ (s, xs, u)ñ(ds, dt) + ∫ t ξk ∫ u s(t − s)σ (s, xs, u)ñ(ds, dt)]i[ξk ,ξk+1)(t)∥∥∥∥2 − ∥∥∥∥ +∞∑ k=0 [ k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 c(t − s)h(s, ys)ds+ ∫ t ξk c(t − s)h(s, ys)ds eur. j. math. anal. 1 (2021) 13 + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s)f(s, ys)ds+ ∫ t ξk s(t − s)f(s, ys)ds+ k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 s(t − s)g(s, ys)dω(s) + ∫ t ξk s(t − s)g(s, ys)dω(s) + k∑ i=1 k∏ j=i bj (δj ) × ∫ ξi ξi−1 ∫ u s(t − s)σ (s, ys, u)ñ(ds, dt) + ∫ t ξk ∫ u s(t − s)σ (s, ys, u)ñ(ds, dt)]i[ξk ,ξk+1)(t)∥∥∥∥2 ≤ 4 max{1,n 2}m2(t − t0)∫ t t0 ∥∥h(t, xs)− h(t, ys)∥∥2 ds+ 4 max{1,n 2} × m̃2(t − t0)∫ t t0 ∥∥f(t, xs)− f(t, ys)∥∥2 ds+ 4 max{1,n 2}m̃2 × ∫ t t0 ∥∥g(t, xs)− g(t, ys)∥∥2 ds+ 4 max{1,n 2}m̃2 × [ ∫ t t0 ∫ u ∥∥σ (t, xs, u)− σ (t, ys, u)∥∥2 v (du)ds + (∫ t t0 ∫ u ∥∥σ (t, xs, u)− σ (t, ys, u)∥∥4 v (du)ds) 12 ] moreover, sup t0≤t≤t e ∥∥(φx)(t)− (φy)(t)∥∥2 ≤ 4 max{1,n 2}m2(t − t0)2lh sup t0≤t≤t e ∥∥x − y∥∥2 s ds+ 4 max{1,n 2} × m̃2(t − t0)2lf sup t0≤t≤t e ∥∥x − y∥∥2 s ds+ 4 max{1,n 2}m̃2t r(q) × (t − t0)lg sup t0≤t≤t e ∥∥x − y∥∥2 s ds+ 4 max{1,n 2}m̃2 × (t − t0)lσ sup t0≤t≤t e ∥∥x − y∥∥2 s ds ≤ [4 max{1,n 2}m2(t − t0)2lh + 4 max{1,n 2}m̃2(t − t0) × [(t − t0)lf + t r(q)lg + lσ ] ] sup t0≤t≤t e ∥∥x − y∥∥2 t hence, ∥∥(φx)− (φy)∥∥2 b ≤ γ(t )∥∥x − y∥∥2 b .where, γ(t ) = 3max{1,n 2}m2(t − t0)2lh + 3 max{1,n 2}m̃2(t − t0) [(t − t0)lf + t r(q)lg + lσ ] . eur. j. math. anal. 1 (2021) 14by taking suitable 0 < t1 < t sufficiently small such that, γ(t1) < 1.hence φ is a contraction on b . by banach contraction principle, a unique fixed point x ∈ b isobtained for the operator φ and therefore φx = x is a mild solution of the system.the solution can be extended to the entire interval (−δ, t ] in finitely many steps. thus the existenceand uniqueness of the mild solution on (−δ, t ] is proved. � 4. stability the stability through continuous dependence of solutions on initial conditions are established. definition 4.1. a mild solution xξ,x (t) of the system (1.1) with the initial value (ξ, x) is said to be stable in mean square if for all ε > 0 such that e ( sup0≤s≤t ∥∥∥xξ,x (s)− yξ,x (t)∥∥∥2) ≤ ε, when e ∥∥ξ − ζ∥∥2 + e ∥∥x − y∥∥2 < δ, where xζ,y(t) is another solution of the system (1.1) with initial value (ζ, y). theorem 4.1. let x(t) and x(t) be mild solution of the system (1.1) with the initial condition φ1 and φ2 respectively. if the assumptions of theorem 3.1 gets satisfied, the mean solution of the system (1.1) is stable in the mean square. proof . we may assume that x(t) and x(t) be the mild solutions of the system (1.1) with initialvalues φ1 and φ2 respectively. x(t)− x(t) = +∞∑ k=0 [ k∏ i=1 bi(δi)c(t − t0)[φ1 − φ2] + k∏ i=1 bi(δi)s(t − t0)[(φ1 − φ2)− [(h(0, φ1)− (h(0, φ2))]] + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 c(t − s) [h(s, xs)− h(s, x(s))]ds+ ∫ t ξk c(t − s) [h(s, xs)− h(s, xs)]ds + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s) [f(s, xs)− f(s, xs)]ds+ ∫ t ξk s(t − s) [f(s, xs)− f(s, xs)]ds + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 s(t − s) [g(s, xs)− g(s, xs)]dω(s) + ∫ t ξk s(t − s) [g(s, xs)− g(s, xs)]dω(s) + k∑ i=1 k∏ j=i bj (δj ) ∫ ξi ξi−1 ∫ u s(t − s) [σ (s, xs, u)− σ (s, xs, u)] ñ(ds, du) + ∫ t ξk ∫ u s(t − s) [σ (s, xs, u)− σ (s, xs, u)] ñ(ds, du)]i[ξk ,ξk+1)(t) eur. j. math. anal. 1 (2021) 15 e ∥∥x(t)− x(t)∥∥2 ≤ 6n 2m2e∥∥φ1 − φ2∥∥2 + 12n 2m̃2e∥∥φ1 − φ2∥∥2 + 12n 2m̃2e∥∥h(0, φ1)− h(0, φ2)∥∥2 + 6m2 max{1,n 2} ∫ t t0 e ∥∥h(s, xs)− h(s, xs)∥∥2 ds+ 6m̃2 max{1,n 2} × ∫ t t0 e ∥∥f(s, xs)− f(s, xs)∥∥2 ds+ 6m̃2 max{1,n 2} ∫ t t0 e ∥∥g(s, xs)− g(s, xs)∥∥2 ds + 6 max{1,n 2}m̃2 × [ ∫ t t0 ∫ u ∥∥σ (t, xs, u)− σ (t, xs, u)∥∥2 v (du)ds + (∫ t t0 ∫ u ∥∥σ (t, xs, u)− σ (t, xs, u)∥∥4 v (du)ds) 12 ] ≤ 6n 2m2e∥∥φ1 − φ2∥∥2 + 10n 2m̃2e∥∥φ1 − φ2∥∥2 + 12n 2m̃2lhe∥∥φ1 − φ2∥∥2 + 6m2 max{1,n 2} ∫ t t0 lhe ∥∥x − x∥∥2 s ds+ 6m̃2 max{1,n 2} ∫ t t0 lfe ∥∥x − x∥∥2 s ds + 6m̃2 max{1,n 2}t r(q) ∫ t t0 lge ∥∥x − x∥∥2 s ds + 6m̃2 max{1,n 2} ∫ t t0 lσe ∥∥x − x∥∥2 s ds furthermore, sup t0≤t≤t e ∥∥x − x∥∥2 t ≤ 6n 2m2e∥∥φ1 − φ2∥∥2 + 12n 2m̃2e∥∥φ1 − φ2∥∥2 + 12n 2m̃2lhe∥∥φ1 − φ2∥∥2 + 6m2 max{1,n 2}(t − t0)lh sup t0≤t≤t e ∥∥x − x∥∥2 t + 6m̃2 max{1,n 2}(t − t0) × lf sup t0≤t≤t e ∥∥x − x∥∥2 t + 6m̃2 max{1,n 2}(t − t0)t r(q)lg sup t0≤t≤t e ∥∥x − x∥∥2 t + 6m̃2 max{1,n 2}(t − t0)lσ sup t0≤t≤t e ∥∥x − x∥∥2 t sup t0≤t≤t e ∥∥x − x∥∥2 t ≤ 6n 2 [m2 + m̃2lh]1− 6 max{1,n 2}(t − t0) [m2lh + m̃2 [lf + t r(q)lg + lσ ]]e ∥∥φ1 − φ2∥∥2 + 12n 2m̃2 1− 6 max{1,n 2}(t − t0) [m2lh + m̃2 [lf + t r(q)lg + lσ ]]e ∥∥φ1 − φ2∥∥2 ≤ ρe ∥∥φ1 − φ2∥∥2 + υe∥∥φ1 − φ2∥∥2 where, ρ = 5n 2 [m2 + m̃2lh]1− 5 max{1,n 2}(t − t0) [m2lh + m̃2 [lf + t r(q)lg + lσ ]] υ = 10n 2m̃2 1− 5 max{1,n 2}(t − t0) [m2lh + m̃2 [lf + t r(q)lg + lσ ]] eur. j. math. anal. 1 (2021) 16given ε > 0, µ > 0 choose, λ = ε ρ , µ = συ such that, e ∥∥φ1 − φ2∥∥2 ≤ λ and e ∥∥φ1 − φ2∥∥2 ≤ µ therefore, ∥∥x − y∥∥2 b ≤ ε.thus the proof is complete. 5. illustration in this section, the results obtained are applied to a stochastic partial differential equations withrandom impulses. let us consider a space h = l2([0, π]). the infinitesimal generator a is definedto be a : d(a) ⊂ h→ h by a = ∂2 ∂x2 , with the domain, d(a) = {z ∈ h | z and ∂z ∂x are absolutely continuous, ∂2z ∂x2 ∈ h, z(0) = z(π) = 0} . for z ∈ d(a),az = − ∞∑ n=1 n 2 < z, zn > zn, where {zn : n ∈ z} is an orthonormal basis of h, zn(x) := 1√2πeinx , n ∈ z+, x ∈ [0, π]. it is known that a generates strongly continuous operators c(t) and s(t) in a hilbert space h, such that c(t)z = ∞∑ n=1 cos(nt) < z, zn > zn, and s(t)z = ∞∑ n=1 sin(nt)/n < z, zn > zn, for t ∈ r. and we assume that s(t) is not a compact semigroup and θ (s(t)d) ≤ θ (d), where d ∈ h denotes a bounded set, θ is the hausdroff measure of non-compactness.in the sequel, we may consider second-order neutral stochastic functional differential equation ofthe form, ∂ ∂t [ ∂ ∂t z(t, x)− m15 ∫ 0 −r ε1(s)z(t + s, x)ds] (5.1) = [ ∂2 ∂x2 z(t, x) + m25 ∫ 0 −r ε1(s)z(t + s)ds]dt + m35 ∫ 0 −r ε3(s)z(t + s)dω(t) + m45 ∫ u ∫ 0 −r ε4(s)z(t + s)ñ(dt, du), t ≥ t0, t 6= ξk, x ∈ [0, π], z(ξk, x) = ρ(k)δkz(ξ−k , x), k = 1, 2, 3..., (5.2) ∂ ∂t z(ξk, x) = ρ(k)δk ∂∂t z(ξ−k , x), z(t0, x) = φ(θ, x), θ ∈ [−r, 0], x ∈ [0, π], r > 0, ∂ ∂t z(t0, x) = φ(x), x ∈ [0, π], z(t, 0) = z(t, π) = 0. eur. j. math. anal. 1 (2021) 17let δk be a random variable defined on dk ≡ (0, dk) where, 0 < dk < +∞, for k = 1, 2, · · · . ξ0 = t0 > 0 and ξk = ξk−1 + δk for k = 1, 2, · · · . ω(t) denotes a standard cylindrical weiner process inh. furthermore, let ρ be a function of k. εi : [−r, 0] → r are positive functions and mi > 0 for i = 1, 2, 3, 4. ∥∥c(t)∥∥ ,∥∥s(t)∥∥ are bounded on r. ∥∥c(t)∥∥ ≤ e−π2t and ∥∥s(t)∥∥ ≤ e−π2t(t ≥ 0).we may assume that,(i)the function ε(θ) ≥ 0 is continuous on [−r, 0],∫ 0 −r ε2 i (θ)dθ <∞(i = 1, 2, 3, 4.) (ii)max i,k = { k∏ j=i e[∥∥ρ(j)δj∥∥2]} < n . using above assumptions and functions ε1, ε2, ε3, ρ we can show that lg = rm125 ∫ 0 −r ε 21(θ)dθ,lf = rm225 ∫ 0 −r ε 21(θ)dθ, lh = rm325 ∫ 0 −r ε 21(θ)dθ and lσ = rm425 ∫ 0 −r ε 21(θ)dθ. hence stability in mean squareof mild solution (5.1) is obtained. � 6. conclusion in this paper, the existence and stability results of second-order neutral stochastic functionalsystems with random impulse is presented. the existence results of aforementioned system is estab-lished using banach contraction principle. then the stability of mild solutions through continuousdependence of solutions on initial conditions are calculated. references [1] a. anguraj, m. mallika arjunan, e. hernández m, existence results for an impulsive neutral functional dif-ferential equation with state-dependent delay, appl. anal. 86 (2007) 861–872. https://doi.org/10.1080/ 00036810701354995.[2] a. anguraj, k. ramkumar, k. ravikumar, existence and hyers-ulam stability of random impulsive stochastic func-tional integrodifferential equations with finite delays, comput. methods differ. equ. 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lipschitz conditions, chinese. j. appl. probab. statist. 26 (2010), 347-356.[27] d. applebaum, levy process and stochastic calculus, cambridge university press, cambridge, 2009.[28] t. wang, s wu, random impulsive model for stock prices and its application for insurers, master thesis (in chinese),shanghai, east china normal university, 2008. https://doi.org/10.1080/17442508.2018.1551400 https://doi.org/10.1080/17442508.2018.1551400 https://doi.org/10.11948/20190089 https://doi.org/10.11948/20190089 https://doi.org/10.1016/j.na.2010.11.007 https://doi.org/10.1016/j.camwa.2006.04.026 https://doi.org/10.1007/s10255-004-0157-z https://doi.org/10.3934/mbe.2018069 https://doi.org/10.1002/asjc.918 https://doi.org/10.1186/s13662-018-1779-4 1. introduction 2. preliminaries 3. existence results of mild solution 4. stability 5. illustration 6. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 34-44doi: 10.28924/ada/ma.1.34 katugampola fractional calculus with generalized k−wright function ahmad y. a. salamooni1,∗ , d. d. pawar2 1department of mathematics, faculty of education zabid, hodeidah university, al-hodeidah, yemen ayousss83@gmail.com 2school of mathematical sciences, swami ramanand teerth marathwada university, nanded-431606, india dypawar@yahoo.com ∗correspondence: ayousss83@gmail.com abstract. in this article, we present some properties of the katugampola fractional integrals andderivatives. also, we study the fractional calculus properties involving katugampola fractional inte-grals and derivatives of generalized k−wright function nφk m(z). 1. introduction and preliminaries in recent years, researchers have introduced new fractional integral and differential operatorswhich are generalizations of the famous definitions of riemann-liouville, caputo, hadamard, hilfer,etc. they have made a qualitative contribution to fractional differential equations. for more details,see [1, 5-7,9-14] and references therein. definition 1.1. [9] let ω = [a, b], the katugampola fractional integrals ρi γ 0+ϕ and ρi γ −ϕ of order γ ∈ c(r(γ) > 0) are defined for ρ > 0, a = 0 and b =∞ as (ρi γ 0+ϕ)(s) = ρ1−γ γ(γ) ∫ s 0 τρ−1ϕ(τ) (sρ − τρ)1−γ dτ (s > 0), (1.1) and (ρi γ −ϕ)(s) = ρ1−γ γ(γ) ∫ ∞ s τρ−1ϕ(τ) (τρ − sρ)1−γ dτ (s > 0), (1.2) the corresponding katugampola fractional derivatives ρd γ 0+ϕ and ρd γ −ϕ are defined with (n = 1 + [r(γ)] ) as (ρd γ 0+ϕ)(s) := ( s1−ρ d ds )1+[r(γ)]( ρ i 1−γ+[r(γ)] 0+ ϕ ) (s) received: 30 aug 2021. key words and phrases. katugampola fractional integral and derivative; k−gamma function; k−wright function.34 https://adac.ee https://doi.org/10.28924/ada/ma.1.34 https://orcid.org/0000-0001-8227-3093 https://orcid.org/0000-0001-8986-5243 eur. j. math. anal. 1 (2021) 35 = ργ−[r(γ)] γ(1− γ + [r(γ)]) ( s1−ρ d ds )1+[r(γ)] ∫ s 0 τρ−1ϕ(τ) (sρ − τρ)γ−[r(γ)] dτ (s > 0), (1.3) and (ρd γ −ϕ)(s) := ( − s1−ρ d ds )1+[r(γ)]( ρ i 1−γ+[r(γ)] − ϕ ) (s) = ργ−[r(γ)] γ(1− γ + [r(γ)]) ( − s1−ρ d ds )1+[r(γ)] ∫ ∞ s τρ−1ϕ(τ) (τρ − sρ)γ−[r(γ)] dτ (s > 0). (1.4) definition 1.2. [2] the generalized k−gamma function γk(y) is defined by γk(y) = lim n→∞ n!kn(nk) y k −1 (y)n,k (k > 0; y ∈ c \ kz−), (1.5) where (y)n,k is the k−pochhammer symbol given as (y)n,k :=  γk(y+nk) γk(y) (k ∈ r; y ∈ c \ {0}) y(y + k)(y + 2k)...(y + (n − 1)k) (n ∈ n+; y ∈ c) (1.6) and for r(y) > 0, the k−gamma function γk(y) is defined by the integral γk(y) = ∫ ∞ 0 xy−1e− xk k dx. (1.7) this gives a relation with euler’s gamma function as γk(y) = k y k −1γ( y k ). (1.8) also, in [8], we have γ(1− y)γ(y) = π sin(yπ) . (1.9) definition 1.3. [14] the beta function b(υ, ω) is defined as b(υ, ω) = ∫ 1 0 zυ−1(1− z)ω−1dz, r(υ) > 0, r(ω) > 0, = γ(υ)γ(ω) γ(υ + ω) (1.10) furthermore, we have∫ ∞ x̂ (z − x̂)υ−1(z − ŷ)ω−1dz = (x̂ − ŷ)υ+ω−1b(υ, 1− υ − ω), x̂ > ŷ , 0 < r(υ) < 1−r(ω). (1.11) recently, the generalized k−wright function introduced by (gehlot and prajapati [3]) is definedas follows: eur. j. math. anal. 1 (2021) 36 definition 1.4. for k ∈ r+; z ∈ c; pi , qj ∈ c, αi , βj ∈ r (αi , βj 6= 0; i = 1, 2, ..., n; j = 1, 2, ..., m) and (pi + αi r), (qj + βj r) ∈ c \ kz−, the generalized k−wright function nφk m isdefined by nφk m(z) = nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣z] = ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) z r r ! , (1.12) with the convergence conditions described as ∆ = m∑ j=1 (βj k ) − n∑ i=1 (αi k ) ;µ = n∏ i=1 ∣∣αi k ∣∣−αi k m∏ j=1 ∣∣βj k ∣∣ βjk ; ν = m∑ j=1 (qj k ) − n∑ i=1 (pi k ) + n −m 2 . lemma 1.1. [3] for k ∈ r+; z ∈ c; pi , qj ∈ c, αi , βj ∈ r (αi , βj 6= 0; i = 1, 2, ..., n; j = 1, 2, ..., m) and (pi + αi r), (qj + βj r) ∈ c \ kz− (1) if ∆ > −1, then series (1.12) is absolutely convergent for all z ∈ c and generalized k−wrightfunction nφk m(z) is an entire function of z. (2) if ∆ = −1, then series (1.12) is absolutely convergent for all |z | < µ and of |z | = µ,r(µ) > 1 2 . 2. properties of katugampola fractional integral and derivative in this section, we investigate some properties of the katugampola fractional integrals andderivatives (1.1), (1.2) and (1.3), (1.4) for the power function ϕ(s) = sα−1 and the exponentialfunction e−λ sρ . lemma 2.1. let ρ > 0,r(γ) = 0 and n = 1 + [r(γ)] (1) if r(α) > 0, then (ρi γ 0+τ α−1)(s) = ρ−γγ(1 + α−1 ρ ) γ(1 + α−1 ρ + γ) sργ+(α−1) (r(γ) ≥ 0; r(α) > 0) (2.1) (ρd γ 0+τ α−1)(s) = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ − γ) s(α−1)−ργ (r(γ) = 0; r(α) > 0). (2.2) (2) if α ∈ c, then (ρi γ −τ α−1)(s) = ρ−γγ( 1−α ρ − γ) γ( 1−α ρ ) sργ+(α−1) (r(γ) ≥ 0; r(γ + α) < 1) (2.3) (ρd γ −τ α−1)(s) = ργ−nγ( 1−α ρ + γ) γ( 1−α ρ ) s(α−1)−ργ (r(γ) = 0; r(γ + α− [r(γ)]) < 1). (2.4) (3) if r(λ) > 0, then (ρi γ −e −λτρ)(s) = (λρ)−γe−λ s ρ (r(γ) ≥ 0) (2.5) eur. j. math. anal. 1 (2021) 37 (ρd γ −e −λτρ)(s) = (λρ)γe−λ s ρ (r(γ) = 0). (2.6) proof. to prove this lemma, let the substitution x = τρ sρ in parts (1) and (2). (1) firstly, by the equation (1.1) and the given substitution, we have (ρi γ 0+τ α−1)(s) = ρ−γsργ+α−1 γ(γ) ∫ 1 0 x α−1 ρ (1− x)1−γ dx = ρ−γsργ+α−1 γ(γ) b ( γ, 1 + α− 1 ρ ) . now, using equation (1.10), we obtain the result (2.1).secondly, by the equation (1.3), the given substitution and by using the result (2.1), we have (ρd γ 0+τ α−1)(s) = ( s1−ρ d ds )n( ρ in−γ0+ τα−1 ) (s) = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ + n − γ) ( s1−ρ d ds )n sρ(n−γ)+α−1 = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ − γ) s(α−1)−ργ . (2) firstly, by the equation (1.2) and the given substitution, we have (ρi γ −τ α−1)(s) = ρ−γsργ+α−1 γ(γ) ∫ ∞ 1 x α−1 ρ (x − 1)γ−1dx. now, using the equation (1.11) with x̂ = 1 and ŷ = 0, we obtain (ρi γ −τ α−1)(s) = ρ−γsργ+α−1 γ(γ) b ( γ, 1− γ − (1 + α− 1 ρ ) ) . by using equation (1.10), we obtain the result (2.3).secondly, by the equation (1.4), the given substitution and by using the result (2.3), we have (ρd γ −τ α−1)(s) = ( − s1−ρ d ds )n( ρ in−γ− τα−1 ) (s) = (−1)nργ−nγ( 1−α ρ + γ − n) γ( 1−α ρ ) ( s1−ρ d ds )n sρ(n−γ)+α−1 = (−1)nργ−n γ( 1−α ρ ) γ( 1−α ρ + γ − n)γ(1− [ 1−α ρ + γ − n]) γ(1− [γ − α−1 ρ ]) . (2.7) also, by using (1.9), we have γ( 1− α ρ + γ − n)γ(1− [ 1− α ρ + γ − n]) = π sin([ 1−α ρ + γ − n]π) = (−1)nπ sin([γ − α−1 ρ ]π) (2.8) and 1 γ(1− [γ − α−1 ρ ]) = γ(γ − α−1 ρ ) γ(γ − α−1 ρ )γ(1− [γ − α−1 ρ ]) = γ(γ − α−1 ρ ) π sin([γ − α− 1 ρ ]π) (2.9) substituting relations (2.8) and (2.9) in (2.7), we obtain (2.4). eur. j. math. anal. 1 (2021) 38 (3) for this part, let the substitution x = τρ − sρ.firstly, by the equation (1.2) and the given substitution in this part, we have (ρi γ −e −λτρ)(s) = ρ−γ γ(γ) e−λ s ρ ∫ ∞ 0 e−λ xxγ−1dx, then by use the substitution ϑ = λ x, we obtain (ρi γ −e −λτρ)(s) = ρ−γ γ(γ) e−λ s ρ λ−γ ∫ ∞ 0 e−ϑϑγ−1dϑ, since ∫∞ 0 e−ϑϑγ−1dϑ = γ(γ) [8], then the result is satisfied.secondly, by the equation (1.4) and by using the result (2.5), we have (ρd γ −e −λτρ)(s) = ( − s1−ρ d ds )n( ρ in−γ− e−λτ ρ) (s) = (−1)n ( s1−ρ d ds )n( (λρ)γ−ne−λ s ρ) = (−1)n s(1−ρ)n (λρ)γ−n ( dn dsn e−λ s ρ) = (λρ)γe−λ s ρ . � remark 2.1. (a) in lemma 2.1, if the power function is ϕ(s) = ( sρ ρ )α−1 , then (1) if r(α) > 0, then( ρi γ 0+ (τρ ρ )α−1 ) (s) = γ(α) γ(α+ γ) (sρ ρ )α+γ−1 (r(γ) ≥ 0; r(α) > 0) ( ρd γ 0+ (τρ ρ )α−1 ) (s) = γ(α) γ(α− γ) (sρ ρ )α−γ−1 (r(γ) = 0; r(α) > 0). (2) if α ∈ c, then( ρi γ − (τρ ρ )α−1 ) (s) = γ(1− γ − α) γ(1− α) (sρ ρ )α+γ−1 (r(γ) ≥ 0; r(γ + α) < 1) ( ρd γ − (τρ ρ )α−1 ) (s) = γ(1 + γ − α) γ(1− α) (sρ ρ )α−γ−1 (r(γ) = 0; r(γ + α− [r(γ)]) < 1). (b) if r(α) > r(γ) > 0, then (ρi γ −τ −α)(s) = ρ−γγ(αρ − γ) γ(αρ ) sργ−α. (2.10) eur. j. math. anal. 1 (2021) 393. katugampola fractional integration for generalized k−wright function in this section, we establish the katugampola fractional integration for generalized k−wrightfunction (1.12). theorem 3.1. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional integration ρi γ 0+ for generalized k−wright function nφk m(z)is given as( ρi γ 0+ ( τ α k −1 nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ ν k ])) (s) = ( k ρ )γ s α k +ργ−1 n+1φk m+1 [ ( pi , αi ) 1,n , ( 1 ρ(α+ (ρ− 1)k), νρ )( qj , βj ) 1,m , ( 1 ρ(α+ (ρ(γ + 1)− 1)k), νρ )∣∣∣∣∣ λ s νk ] . (3.1) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (3.1)exists for s > 0. we consider that m ≡ ( ρi γ 0+ ( τ α k −1 nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ ν k ])) (s). using (1.12), we can write the above equation as m ≡ ( ρi γ 0+ ( τ α k −1 ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) (λ τ ν k )r r ! )) (s). now, using the integration of the series term by term, we obtain m ≡ ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) (λ)r r ! ( ρi γ 0+ ( τ α k + νr k −1 )) (s). applying (2.1), the above equation is reduced to m ≡ ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) (λ)r r ! ρ−γγ(1 + α k + νr k −1 ρ ) γ(1 + α k + νr k −1 ρ + γ) s α+νr k +ργ−1. using (1.8), we obtain m ≡ ( k ρ )γ s α k +ργ−1 n+1φk m+1 [ ( pi , αi ) 1,n , ( 1 ρ(α+ (ρ− 1)k), νρ )( qj , βj ) 1,m , ( 1 ρ(α+ (ρ(γ + 1)− 1)k), νρ )∣∣∣∣∣ λ s νk ] . � theorem 3.2. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional integration ρi γ − for generalized k−wright function nφk m(z) isgiven as ( ρi γ − ( τ− α k nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ− ν k ])) (s) eur. j. math. anal. 1 (2021) 40 = ( k ρ )γ sργ− α k n+1φk m+1 [( pi , αi ) 1,n , ( α ρ − kγ, ν ρ )( qj , βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s− ν k ] . (3.2) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (3.2)exists for s > 0. we consider that n ≡ ( ρi γ − ( τ− α k nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ− ν k ])) (s). using (1.12), we can write the above equation as n ≡ ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) (λ)r r ! ( ρi γ − ( τ− α+νr k )) (s). applying (2.10), the above equation is reduced to n ≡ ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r) (λ)r r ! ρ−γγ( α+νr k ρ − γ) γ( α+νr k ρ ) sργ− α+νr k . using (1.8), we obtain n ≡ ( k ρ )γ sργ− α k n+1φk m+1 [( pi , αi ) 1,n , ( α ρ − kγ, ν ρ )( qj , βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s− ν k ] . � 4. katugampola fractional differentiation for generalized k−wright function this section deals with the katugampola fractional differentiation for generalized k−wrightfunction (1.12). theorem 4.1. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, thenfor ∆ > −1, the katugampola fractional differentiation ρd γ 0+ for generalized k−wright function nφk m(z) is given as( ρd γ 0+ ( τ α k −1 nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ ν k ])) (s) = ( k ρ )−γ s α k −ργ−1 n+1φk m+1 [ ( pi , αi ) 1,n , ( 1 ρ(α+ (ρ− 1)k), νρ )( qj , βj ) 1,m , ( 1 ρ(α+ (ρ(1− γ)− 1)k), νρ )∣∣∣∣∣ λ s νk ] . (4.1) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (4.1)exists for s > 0. let n = 1 + [r(γ)]. then, we consider that p ≡ ( ρd γ 0+ ( τ α k −1 nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ ν k ])) (s). eur. j. math. anal. 1 (2021) 41using (1.3), we have p ≡ ( s1−ρ d ds )n( ρi n−γ 0+ ( τ α k −1 nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ ν k ])) (s). using theorem 3.1, we obtain p ≡ ( s1−ρ d ds )n( ( k ρ )n−γ s α k +ρ(n−γ)−1 n+1φk m+1 [ ( pi , αi ) 1,n , ( 1 ρ (α+ (ρ− 1)k), νρ )( qj , βj ) 1,m , ( 1 ρ (α+ (ρ(n − γ + 1)− 1)k), νρ )∣∣∣∣∣ λ s νk ]) . using (1.12), we can write the above equation as p ≡ ( k ρ )n−γ ∞∑ r=0 ∏n i=1 γk(pi + αi r)γk( 1 ρ (α+ (ρ− 1)k) + ν ρ r)∏m j=1 γk(qj + βj r)γk( 1 ρ (α+ (ρ(n − γ + 1)− 1)k) + ν ρ r) (λ)r r ! ( s1−ρ d ds )n( s α k + ν k +ρ(n−γ)−1 ) . also, the above equation can be written as p ≡ kn−γ ργ ∞∑ r=0 ∏n i=1 γk(pi + αi r)γk( 1 ρ(α+ (ρ− 1)k) + ν ρ r)∏m j=1 γk(qj + βj r)γk( 1 ρ(α+ (ρ(n − γ + 1)− 1)k) + ν ρ r) (λ)r r ! × γ( 1 ρ(αk + νr k + (n − γ)ρ+ ρ− 1) γ( 1 ρ(αk + νr k − γρ+ ρ− 1) s α k + ν k −ργ−1. using (1.8), we obtain p ≡ ( k ρ )−γ s α k −ργ−1 n+1φk m+1 [ ( pi , αi ) 1,n , ( 1 ρ(α+ (ρ− 1)k), νρ )( qj , βj ) 1,m , ( 1 ρ(α+ (ρ(1− γ)− 1)k), νρ )∣∣∣∣∣ λ s νk ] . � theorem 4.2. let γ, α ∈ c such that r(γ) > 0, r(α) > 1+[r(γ)]−r(γ); λ ∈ c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional differentiation ρd γ − for generalized k−wrightfunction nφk m(z) is given as( ρd γ − ( τ− α k nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ− ν k ])) (s) = ( k ρ )−γ s−ργ− α k n+1φk m+1 [( pi , αi ) 1,n , ( α ρ + kγ, νρ )( qj , βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s− ν k ] (4.2) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (4.2)exists for s > 0. let n = 1 + [r(γ)]. then, we consider that q ≡ ( ρd γ − ( τ− α k nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ− ν k ])) (s). using (1.4), we have q ≡ ( − s1−ρ d ds )n( ρi n−γ − ( τ− α k nφk m [ (pi , αi)1,n (qj , βj)1,m ∣∣∣ λ τ− ν k ])) (s). eur. j. math. anal. 1 (2021) 42using theorem 3.2, we obtain q ≡ ( − s1−ρ d ds )n ( k ρ )n−γ sρ(n−γ)−α k n+1φk m+1 [( pi , αi ) 1,n , ( α ρ − k(n − γ), νρ )( qj , βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s− ν k ] . using (1.12), we can write the above equation as q ≡ (−1)n( k ρ )n−γ ∞∑ r=0 ∏n i=1 γk(pi + αi r)γk(αρ − (n − γ)k + ν ρ r)∏m j=1 γk(qj + βj r)γk(αρ + ν ρ r) (λ)r r ! ( s1−ρ d ds )n( sρ(n−γ)−α k − ν k ) . on simplifying the above equation, we obtain q ≡ (−1)nkn−γργ ∞∑ r=0 ∏n i=1 γk(pi + αi r)γk(αρ − (n − γ)k + ν ρ r)∏m j=1 γk(qj + βj r)γk(αρ + ν ρ r) (λ)r r ! × γ(1 + (n − γ)− α ρk − ν ρk r) γ(1− γ − α ρk − ν ρk r) ( s−ργ− α k − ν k ) . using (1.8), we obtain q ≡ (−1)nργ ∞∑ r=0 ∏n i=1 γk(pi + αi r)∏m j=1 γk(qj + βj r)γ( αρk + ν ρk r) (λ)r r ! × γ(γ − n + α ρk + ν ρk r)γ(1− (γ − n + α ρk + ν ρk r)) γ(1− (γ + α ρk + ν ρk r)) ( s−ργ− α k − ν k ) . (4.3) using (1.9), we have γ(γ − n + α ρk + ν ρk r)γ(1− (γ − n + α ρk + ν ρk r)) = π sin[(γ + α ρk + ν ρk r)π − nπ] = π sin[(γ + α ρk + ν ρk r)π] cos(nπ) = (−1)nπ sin[(γ + α ρk + ν ρk r)π] (4.4) and 1 γ(1− (γ + α ρk + ν ρk r)) = γ(γ + α ρk + ν ρk r) sin[(γ + α ρk + ν ρk r)π] π . (4.5) substituting (4.4) and (4.5) in (4.3) and finally by using (1.8), we obtain q ≡ ( k ρ )−γ s−ργ− α k n+1φk m+1 [( pi , αi ) 1,n , ( α ρ + kγ, νρ )( qj , βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s− ν k ] . � eur. j. math. anal. 1 (2021) 435. concluding remarks • if ρ = 1, thentheorems 3.1, 3.2, 4.1 and 4.2, are reduced to theorems 2, 3, 4 and 5 respectively(see [4]). • some general properties of the katugampola fractional integrals and derivatives for thepower function ϕ(s) = sα−1 and the exponential function e−λ sρ are investigated. • the katugampola fractional integration ρi γ 0+ and ρi γ − for generalized k−wright function nφk m(z) are established. • the katugampola fractional differentiation ρd γ 0+ and ρd γ − for generalized k−wright func-tion nφk m(z) are established. acknowledgment the authors are would like to thank the reviewers for their important remarks and suggestions. references [1] r. almeida, a.b. malinowska, t. odzijewicz, fractional differential equations with dependence 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pawar, existence and stability results for hilfer-katugampola-type fractional implicit dif-ferential equations with nonlocal conditions, j. nonlinear sci. appl. 14 (3) (2021), 124-138. http://dx.doi.org/ 10.22436/jnsa.014.03.02.[14] a.y.a. salamooni, d.d. pawar, existence and uniqueness of nonlocal boundary conditions for hilfer-hadamard-type fractional differential equations, adv. differ. equations, 2021 (2021), 198. https://doi.org/10.1186/ s13662-021-03358-0.[15] s.g. samko, a.a. kilbas, o.i. marichev, fractional integrals and derivatives: theory and applications, gordon andbreach, new york (1993). http://dx.doi.org/10.22436/jnsa.014.03.02 http://dx.doi.org/10.22436/jnsa.014.03.02 https://doi.org/10.1186/s13662-021-03358-0 https://doi.org/10.1186/s13662-021-03358-0 1. introduction and preliminaries 2. properties of katugampola fractional integral and derivative 3. katugampola fractional integration for generalized k-wright function 4. katugampola fractional differentiation for generalized k-wright function 5. concluding remarks acknowledgment references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 1doi: 10.28924/ada/ma.2.1 convergence and stability of new approximation algorithms for certain contractive-type mappings imo kalu agwu∗, donatus ikechi igbokwe department of mathematics, micheal okpara university of agriculture, umudike, umuahia abia state, nigeria agwu.imoh@mouau.edu.ng, igbokwedi@yahoo.com ∗correspondence: agwu.imoh@mouau.edu.ng abstract. we present new fixed points algorithms called multistep h-iterative scheme and multistepsh-iterative scheme. under certain contractive-type condition, convergence and stability results wereestablished without any imposition of the ’sum conditions’, which to a large extent make some existingiterative schemes so far studied by other authors in this direction practically inefficient. our resultscomplement and improve some recent results in literature. 1. introduction there is an intimate connection existing between nonlinear problems and fixed point problemsof related contractive-type operators. as a result, researchers have focused more attention onfinding approximate fixed points of different contractive-type mappings in recent times; see, forexample, [5], [6], [7], [8], [9], [10], [11], [18], [25], [28], etc. and the reference contained in them. let xbe a normed linear space and γ : x −→ x a given of x . we represent the set of fixed points of γby f (γ) = {q ∈ x : q = γ(q)}.for the past forty years or so, some investigation of fixed points via iterative schemes have beena flourishing area of research for many mathematicians. mann [21], ishikawa [22] and noor [19]iterative schemes, with their modifications, have been studied by different authors and differentinteresting results were obtained. however, to meet up with the demand of the modern fixedpoint theory, researchers have continually renewed their efforts toward constructing more efficientiterative schemes. in this direction, following kirk’s introduction of his remarkable iterative schemein 1971, the results below have found thier place in the current literature.let x and γ be as earlier stated. received: 1 nov 2021. key words and phrases. strong convergence; multistep h-iterative scheme; multistep sh-iterative scheme; stability;contractive operator; fixed point; normed linear space. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 2 (a) for arbitrarily y0 ∈ x , let the sequence {yn}∞n=0 be defined iteratively as follows: yn+1 = ∑̀ j=0 αjγ jyn, ∑̀ j=0 αj = 1, n ≥ 0. (1.1) the iteration method defined by (1.1) is due to kirk [20]. (b) in [17], olatinwo presented the algorithms below:(i) for an arbitrary point y0 ∈ x and for αn,t ≥ 0, αn,0 6= 0, αn,t ∈ [0, 1] and ` as a fixedinteger, define the sequence {yn}∞n=0 by yn+1 = ∑̀ t=0 αn,tγ tyn, ∑̀ t=0 αn,t = 1, n ≥ 0 (1.2) (i i) for an arbitrary point y0 ∈ x and for ` ≥ m,αn,t βn,t ≥ 0, αn,0, βn,0 6= 0, αn,t , βn,t ∈ [0, 1] and `,m as fixed integers, define the sequence {yn}∞n=0 by yn+1 = αn,0yn + ∑̀ t=0 αn,tγ jzn, ∑̀ t=0 αn,t = 1; zn = m∑ t=0 βn,tγ tyn, ∑̀ t=0 βn,t = 1, n ≥ 0, (1.3) and called them kirk-mann and kirk-ishikawa algorithms, respectively. (c) chugh and kumar [25] presented the following iterative scheme: for an arbitrary point y0 ∈ x and for ` ≥ m ≥ p, αn,s , γn,r , βn,t ≥ 0, γn,0, αn,0, βn,0 6= 0, αn,s , γn,r , βn,t ∈ [0, 1] and `,m, p as fixed integers, define the sequence {yn}∞n=0 by yn+1 = γn,0yn + ∑̀ r=1 γn,rγ rzn, ∑̀ r=0 γn,r = 1; zn = αn,0yn + m∑ s=1 αn,sγ szn, m∑ s=0 αn,s = 1; zn = p∑ t=0 βn,tγ tyn, p∑ t=0 βn,t = 1, n ≥ 0, (1.4) (d) very recently, akewe, okeke and olayiwola [26] presented the following general iterativescheme in the sense of kirk [20]: (i) for an arbitrary point y0 ∈ x , for `1 ≥ `2 ≥ `3 ≥ · · · ≥ `u , for each i , αtn,s , γn,t ≥ 0, γn,0, αn,0, 6= 0, for each i , αin,s , γn,t ∈ [0, 1] and `1, `u as fixed integers for each u, https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 3define the sequence {yn}∞n=0 by yn+1 = γn,0yn + `1∑ r=1 γn,rγ rz1n , `1∑ k=0 αn,r = 1; z tn = αtn,0yn + `t+1∑ s=1 αtn,sγ jz t+1n , `t+1∑ s=0 αtn,s = 1, t = 1, 2, · · · , u − 2; zu−1n = `u∑ s=0 αu−1n,t γsyn, `u∑ s=0 αu−1n,t = 1, u ≥ 2, n ≥ 0, (1.5) (i i) for an arbitrary point y0 ∈ x , retaining the conditions in (i), define the sequence {yn}∞n=0 by yn+1 = γn,0z 1 n + `1∑ r=1 γn,kγrz1n , `1∑ r=0 αn,r = 1; z tn = αtn,0z t+1 n + `t+1∑ s=1 αtn,sγ sz t+1n , `t+1∑ s=0 αtn,s = 1, t = 1, 2, · · · , u − 2; zu−1n = `u∑ s=0 αu−1n,t γsyn, `u∑ s=0 αu−1n,t = 1, u ≥ 2, n ≥ 0, (1.6) it is worthy to mention that in application, the stability of the iterative schemes studied aboveis quite invaluable. the first researcher to demonstrate this respecting the banach contractionconditions is ostrowski [13]. afterwards, several authors have developed this subject basicallybecause of its indispensable position in the current trend of computer programing. some recentworks in this direction could be seen in [1], [2], [3], [4], [12], [13], [14],[23], [24], [26] and the references therein. remark 1.1. the stability and the convergence results in the papers studied were made possible due to the sum conditions imposed on the control parameters; see, for example, [20], [17], [25], [26], etc and the references therein. but in application, especially for n large enough, the iterative schemes defined by (1.1), (1.2), (1.3), (1.4) (1.5) and (1.6) become practically inefficient due to the difficulties involved in generating a family of such control parameters, the windy process involved for each sum and the computational cost. base on the problems mentioned in remark 1.1, it becomes necessary to ask the followingquestions: question 1.1. is it possible to construct an alternative iterative scheme that would address the problems generated by the sum conditions imposed on the control parameters while maintaining, in particular, the results in [26], which in a larger sense contains the results of the other papers studied? https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 4following the same argument as in [27] regarding the linear combination of the products ofcountably finite family of control parameters and the problems mentioned in remark 1.1, in thispaper, we provide an affirmative answer to question 1.1. 2. preliminary throughout the remaining sections, φ : r+ −→ r+,r+,n and h will denote monotone in-creasing subadditive function, the set of positive integers, the set of natural numbers and a realhilbert space, respectively. also, the following definition, lemmas and propositions will be neededestablish our results. definition 2.1. ( [13]) suppose y is a metric space and let γ : y −→ y be a self-map of y . let {xn}∞n=0 ⊆ y be a sequence generated by an iteration scheme xn+1 = g(γ, xn), (2.1) where x0 ∈ y is the initial approximation and g is some function. suppeose {xn}∞n=0 converges to a fixed point q of γ. let {tn}∞n=0 ⊆ y be an arbitrary sequence and set εn = d(tn, g(γ, tn)), n = 1, 2, · · · then, (2.1) is said to be γ-stable if and only if limn→∞ εn = 0 implies limn→∞ yn = q. note that in practice, the sequence {tn}∞n=0 could be obtained using the following approach: let x0 ∈ y . set xn+1 = g(γ, xn) and let t0 = x0. since, x1 = g(γ, x0) following the rounding in thefunction γ, the value t1 (which is estimated to be equal to x1) could be calculated to give t2, anapproximate value of g(γ, t1). the procedure is continued to yield the sequence {tn}∞n=0, which isapproximately tha same as the sequence {xn}∞n=0. lemma 2.1. (see, e.g., [26]) let {τn}∞n=0 ∈ r+ : τn → 0 as n →∞. for 0 ≤ δ < 1, let {wn}∞n=0 be a sequence of positive numbers satisfying wn+1 ≤ δwn+τn, n = 0, 1, 2, · · · then, wn → 0 as n →∞. lemma 2.2. (see, e.g., [17]) let (y, ‖ .‖) be a normed space, the self-map γ : y −→ y satisfies (1.13) and φ : r+ −→ r+ (retaining its usual meaning) be such that ψ(0) = 0, φ(mt) = mφ(t),m ≥ 0, t ∈ r+. then, ∀i ∈ n and ∀s, t ∈ y, we have ‖γjs − γj t‖ ≤ ρj‖s − t‖+ j∑ i=0 ( j i ) ρj−1φ(‖s − γs‖). (2.2) proposition 2.3. (see,e.g., [27]) let {αi}∞i=1 ⊆ n, where k ∈ [0,r+] is fixed and n ∈ n is any integer with k + 1 ≤ n. then, the following holds: αk + n∑ i=k+1 αi i−1∏ j=k (1− αj) + n∏ j=k (1− αj) = 1. (2.3) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 5 proposition 2.4. (see,e.g., [27]) let t, u, v ∈ h. let k ∈ [0,r+] be fixed and n ∈ n be such that k + 1 ≤ n. let {vi}n−1i=1 ⊆ h and {αi}ni=1 ⊆ [0, 1]. define y = αkt + n∑ i=k+1 αi i−1∏ j=k (1− αj)vi−1 + n∏ j=k (1− αj)v . then, ‖y − u‖2 = αk‖t − u‖2 + n∑ i=k+1 αi i−1∏ j=k (1− αj)‖vi−1 − u‖2 + n∏ j=k (1− αj)‖v − u‖2 −αk [ n∑ i=k+1 αi i−1∏ j=k (1− αj)‖t − vi−1‖2 + i−1∏ j=k (1− αj)‖t − v‖2 ] −(1− αk) [ n∑ i=k+1 αi i−1∏ j=k (1− αj)‖vi−1 − (αi+1 + wi+1)‖2 +αn i−1∏ j=k (1− αj)‖v − vn−1‖2 ] , where wk = ∑n i=k+1 αi ∏i−1 j=k(1− αj)vi−1 + ∏i−1 j=k(1− αj)v , k = 1, 2, · · · , n and wn = (1− cn)v . 3. main results i let h be a hilbert space and let γ : h −→ h be a self-map of x. for arbitrary x0 ∈ h definethe sequence {xn+1}∞n=0 iteratively, for s = 1, 2, · · · , k − 2, as follows: xn+1 = δn,1xn + ∑`1 j=2 δn,j ∏j−1 i=1(1− δn,i)γj−1y1n + ∏`1 i=1(1− δn,i)γ`1y1n ; y sn = αsn,1xn + ∑`s+1 j=2 α s n,j ∏j−1 i=1(1− αsn,i)γj−1y s+1n + ∏`s+1 i=1 (1− αsn,i)γ`1y s+1n ; y k−1n = ∑`k j=1 α k−1 n,j ∏j−1 i=1(1− αk−1n,i )γj−1xn + ∏`k i=1(1− αk−1n,i )γ`k xn, k ≥ 2, n ≥ 1, (3.1) where `1 ≥ `2 ≥ `3 ≥ · · · ≥ `k , for each s , {{δn,i}∞n=0}`kj=1, {{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1, `2, · · · , `k are fixed integers (for each k). we shall call the iteration scheme defined by (3.1)the multistep ih-iteration scheme.again, for any x0 ∈ x , we shall call the sequence {xn}∞n=0 defined recursively, for s = 1, 2, · · · , k − 2, by xn+1 = δn,1y 1 n + ∑`1 j=2 δn,j ∏j−1 i=1(1− δn,i)γj−1y1n + ∏`1 i=1(1− δn,i)γ`1y1n ; y sn = αsn,1y s+1 n + ∑`s+1 j=2 α s n,j ∏j−1 i=1(1− αsn,i)γj−1y s+1n + ∏`s+1 i=1 (1− αsn,i)γ`1y s+1n ; y k−1n = ∑`k j=1 α k−1 n,j ∏j−1 i=1(1− αk−1n,i )γj−1xn + ∏`k i=1(1− αk−1n,i )γ`k xn, k ≥ 2, n ≥ 1, (3.2) where `1 ≥ `2 ≥ `3 ≥ · · · ≥ `k , for each s , {{δn,i}∞n=0}`kj=1, {{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1, `2, · · · , `k are fixed integers (for each k), the multistep di-iteration scheme. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 6 theorem 3.1. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖ ≤ ρj‖x − y‖+ j∑ i=0 ( j i ) ρj−iφ(‖x − γx‖), (3.3) where x, y ∈ h, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0, t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep h-iteration scheme defined by (3.1). then, (i) γ defined by (3.3) has a fixed point q; (i i) the multistep ih-iteration scheme converges strongly to q ∈ γ. proof. firstly, we show that γ satisfying condition of (3.3) has a fixed point. assume there existstwo points q1, q2 ∈ f (γ) with 0 < ‖q1 − q2‖. then, we have 0 < ‖q1 − q2‖ = ‖γjq1 − γjq2‖ ≤ ρj‖q1 − q2‖+ j∑ i=0 ( j i ) ρj−iφ(‖q √ 1− γq1‖) = ρj‖q1 − q2‖+ j∑ i=0 ( j i ) ρj−iφ(0) ⇒ (1− ρj)ρj‖q1 − q2‖ ≤ 0. using the fact that ρj ∈ [[0, 1), we get 0 < 1− ρj and ‖q1 − q2‖ ≤ 0.since the norm is a nonnegative function, we get ‖q1 − q2‖ = 0; q1 = q2 = q(say). therefore, γconverges uniquely to a point of f (γ).now, we show that the sequence defined by (3.1) converges strongly to q ∈ f (γ). using (3.3)and proposition 2.4 with xn+1 = y , u = q, xn = t, j = i , k = 1,γj−1y1n = vj−1 and γ`1y1n = v , wehave ‖xn+1 − q‖2 ≤ δn,1‖xn − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1y1n − γj−1q‖2 + `1∏ i=1 (1− δn,i)‖γ`1y1n − γ`1q‖2 (3.4) but from (3.3), with y = y1n , we have ‖γj−1y1n − γj−1q‖ ≤ ρj‖y1n − q‖+ j∑ i=0 ( j i ) ρj−1φ(‖q − γq‖) = ρj‖y1n − q‖ (3.5) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 7proposition 2.3, (3.4) and (3.5) imply ‖xn+1 − q‖2 ≤ δn,1‖xn − q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i)‖y1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖y1n − q‖2 = δn,1‖xn − q‖2 + ( 1− δ1n,1 − `1∏ i=1 (1− δn,i)(ρj)2 ) ‖y1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖y1n − q‖2 = δn,1‖xn − q‖2 + ( 1− δ1n,1 ) ‖y1n − q‖2 (3.6) since `1, `k are fixed integers and αsn,i ∈ [0, 1] for each s , we have, using proposition 2.3, thefollowing estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n − q‖2 ≤ αn,1‖xn − q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1− αn,i)‖γj−1y2n − γj−1q‖2 + `2∏ i=1 (1− αn,i)‖γ`2y2n − γ`2q‖2 ≤ α1n,1‖xn − q‖2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1− αn,i)‖y2n − q‖2 + `2∏ i=1 (1− αn,i)(ρj)2‖y2n − q‖2 ≤ α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) [ α2n,1‖xn − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)‖y3n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖y3n − q‖2 ] + `2∏ i=1 (1− α1n,i)(ρj)2 [ α2n,1‖xn − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)‖y3n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖y3n − q‖2 ] https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 8 = α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ‖y3n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖y3n − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) ‖y3n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) ‖y3n − q‖2 ≤ α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) [α3n,1‖xn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖y4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖y4n − q‖2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖xn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖y4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖y4n − q‖2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )[ α3n,1‖xn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖y4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖y4n − q‖2 ] + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )[ α3n,1‖xn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖y4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖y4n − q‖2 ] https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 9 = α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖y4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖y4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 10 = α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2 ) ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2) ) ‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2) ) ‖y4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2 ) ‖y4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖y4n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 11 = α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  (1− α3n,1)(ρj)2‖y4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) (1− α3n,1)(ρj)2)‖y4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) (1− α3n,1)(ρj)2)‖y4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) (1− α3n,1)(ρj)2‖y4n − q‖2 ≤ α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 12 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1(1− α3n,1)(ρj)2‖xn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2)‖xn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2)‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2‖xn − q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1− α2n,i) × · · · ×`s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) αsn,1‖xn − q‖2 + (ρj)2 ( `2∏ i=1 (1− α1n,i)(ρj)2 ) ×(ρj)2 ( `3∏ i=1 (1− α2n,i)(ρj)2 ) (ρj)2 ( `4∏ i=1 (1− α3n,i)(ρj)2 ) × · · · × (ρj)2 `s−1∏ i=1 (1− α`s−2n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ‖xn − q‖2 < α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) α3n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i) ) ‖xn − q‖2 + ( 1− α2n,1 − `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1‖xn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 13 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + ( 1− α2n,1 − `3∏ i=1 (1− α2n,i)) )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + ( `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + · · · + ( 1− α1n,1 − `2∏ i=1 (1− α2n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) × ( 1− α3n,1 − `4∏ i=1 (1− α3n,i) ) × · · · × 1− α`s−2n,1 − `s−1∏ i=1 (1− α`s−2n,i )  × ( 1− α`s−1n,1 − `s∏ i=1 (1− α`s−1n,i ) ) αsn,1‖xn − q‖2 + ( `2∏ i=1 (1− α1n,i) ) × ( `3∏ i=1 (1− α2n,i) )( `4∏ i=1 (1− α3n,i) ) × · · · × `s−1∏ i=1 (1− α`s−2n,i )  × ( `s∏ i=1 (1− α`s−1n,i ) ) ‖xn − q‖2 (3.7) < α1n,1‖xn − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) ) α3n,1 ( 1− α2n,1 ) ‖xn − q‖2 +α3n,1 ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) ‖xn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 ) α3n,1(1− α3n,1)‖xn − q‖2 + ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + · · ·+ + ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) αsn,1‖xn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 14 = α1n,1‖xn − q‖2 + α2n,1 ( 1− α1n,1 − `2∏ i=1 (1− α1n,i) ) ‖xn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖xn − q‖2 + ( (1− α1n,1 − `2∏ i=1 (1− α1n,i) ) α3n,1 ( 1− α2n,1 ) ‖xn − q‖2 +α3n,1 ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) ‖xn − q‖2 + ( (1− α1n,1 − `2∏ i=1 (1− α1n,i) )( 1− α2n,1 ) α3n,1(1− α3n,1)‖xn − q‖2 + ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖xn − q‖2 + · · ·+ + ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) αsn,1‖xn − q‖2 < [α1n,1 + α2n,1 ( 1− α1n,1 ) + (1− α1n,1)α3n,1 ( 1− α2n,1 ) + (1− α1n,1) ( 1− α2n,1 ) α3n,1(1− α3n,1) + · · ·+ ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) ]‖xn − q‖2 (3.8) (3.6) and (3.8) imply that ‖xn+1 − q‖2 ≤ {δn,1 + (1− δn,1) [α1n,1 + α2n,1 ( 1− α1n,1 ) + (1− α1n,1)α3n,1 ( 1− α2n,1 ) +(1− α1n,1) ( 1− α2n,1 ) (1− α3n,1) + · · ·+ ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) ]}‖xn − q‖2 (3.9) using lemma 2.3, we obtain (from (3.9)) that the sequence {xn}∞n=0 converges strongly to q ∈ f (γ);and this completes the proof. � theorem 3.2. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖ ≤ ρj‖x − y‖+ j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (3.10) where x, y ∈ h, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0, t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep di-iteration scheme defined by (3.2). then, (i) γ defined by (3.10) has a fixed point q; https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 15 (i i) the multistep sh-iteration scheme converges strongly tp q ∈ γ. proof. we first show that γ satisfying condition of (3.10) has a fixed point. assume there existstwo points q1, q2 ∈ f (γ) with 0 < ‖q1 − q2‖. then, we have 0 < ‖q1 − q2‖ = ‖γjq1 − γjq2‖ ≤ ρj‖q1 − q2‖+ j∑ i=0 ( j i ) ρj−1φ(‖q √ 1− γq1‖) = ρj‖q1 − q2‖+ j∑ i=0 ( j i ) ρj−1φ(0) ⇒ (1− ρj)ρj‖q1 − q2‖ ≤ 0. using the fact that ρj ∈ [[0, 1), we get 0 < 1− ρj and ‖q1 − q2‖ ≤ 0.since the norm is a nonnegative function, we get ‖q1 − q2‖ = 0; q1 = q2 = q(say). therefore, γconverges uniquely to a point of f (γ).now, we show that the sequence defined by (3.1) converges strongly to q ∈ f (γ). using (3.2)and proposition 2.4 with xn+1 = y , u = q, y1n = t, j = i , k = 1,γj−1y1n = vj−1 and γ`1y1n = v , weget ‖xn+1 − q‖2 = δn,1‖y1n − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1y1n − γj−1q‖2 + `1∏ i=1 (1− δn,i)‖γ`1y1n − γ`1q‖2 (3.11) but from (3.10), with y = y1n , we have ‖γj−1y1n − γj−1q‖ ≤ ρj‖y1n − q‖+ j∑ i=0 ( j i ) ρj−1φ(‖q − γq‖) = ρj‖y1n − q‖ (3.12) proposition 2.3, (3.11) and (3.12) imply ‖xn+1 − q‖2 ≤ δ1n,1‖y1n − q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i)‖y1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖y1n − q‖2 = δ1n,1‖y1n − q‖2 + ( 1− δ1n,1 − `1∏ i=1 (1− δn,i)(ρj)2 ) ‖y1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖y1n − q‖2 = ‖y1n − q‖2 (3.13) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 16since `1, `k are fixed integers and αsn,i ∈ [0, 1] for each s , we have (using proposition 2.3, (3.2)and (3.12)) the following estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n − q‖2 ≤ α1n,1‖y2n − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)‖γj−1y2n − γj−1q‖2 + `2∏ i=1 (1− α1n,i)‖γ`2y2n − γ`2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  ‖y2n − q‖2 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2 [α2n,1‖y3n − q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1− α2n,i)‖γj−1y3n − γj−1q‖2 + `3∏ i=1 (1− α2n,i)‖γ`3y3n − γ`3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2 [α2n,1‖y3n − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i)‖y2n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖y3n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖y3n − q‖2 (3.14) ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖y4n − q‖2 + `4∑ j=2 α3n,j j−1∏ i=1 (1− α3n,i)‖γj−1y4n − γj−1q‖2 + `4∏ i=1 (1− α3n,i)‖γ`4y4n − γ`4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 17 × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖y4n − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖y4n − q‖2 + `4∏ i=1 (1− α3n,i)(ρj)2‖y4n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) ‖y4n − q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 )[ α4n,1‖y5n − q‖2 + `5∑ j=2 α4n,j j−1∏ i=1 (1− α4n,i)‖γj−1y5n − γj−1q‖2 + `5∏ i=1 (1− α4n,i)‖γ`5y5n − γ`5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 )[ α4n,1‖y5n − q‖2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i)‖y5n − q‖2 + `5∏ i=1 (1− α4n,i)(ρj)2‖y5n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 18 × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) ‖y5n − q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ×‖xn − q‖2 (3.15) (3.13) and (3.15) imply that ‖xn+1 − q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ×‖xn − q‖2 (3.16) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 19since ρj ∈ [0, 1], we obtain using proposition 2.3, for j = 1, 2, 3, · · · , s − 1, that q ≤ p = 1, (3.17) where q = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) and p = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i) ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i ) ) applying (3.17) in (3.16), we obtain, using lemma 2.3 that the sequence {xn}∞n=0 defined by (3.2)converges strongly to the fixed point q in f (γ). thus, the proof is completed. � example 3.1. let the operator γ : [0, 1] −→ [0, 1] be defined as γz = z 3 ,∀z ∈ [0, 1]. clearly, γ is quasi-contractive satisfying (2.2) with a unique fixed point 0; see, for example, [26] for details. set α1n,1 = δ1n,1 = 1√ n + 1 , n = 1, 2, · · · , n0, f or n0 ∈ n; δn,i = 1− δ1n,1, f or i = 1, 2, · · · , `1 and αsn,i = 1− 2α1n,1, f or i = 1, 2, · · · , `s+1, s = 1, 2, · · · , n0. it is not hard to see that all the conditions of theorem 3.1 and theorem 3.2 has been satisfied by example 3.1. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 204. main results ii here, we consider stablity results for the multistep ih-iteration scheme and the multistep di-iteration scheme defined by (3.1) and (3.2) for operators satisfying (2.2), respectively. theorem 4.1. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖ ≤ ρj‖x − y‖+ j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (4.1) where x, y ∈ h, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0, t ∈ r+. for arbitrary x0 ∈ h, let {xn}∞n=0 be the multistep di-iteration scheme defined by (3.2). assume f (γ) 6= ∅, q ∈ f (γ). then, the multisetp di-iterative scheme is γ-stable. proof. let {vn}∞n=0, be a real sequences in h. suppose {tn}∞n=0 ⊂ x is an arbitrary sequence, set εn = ‖tn+1 − δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 (4.2) where, for s = 1, 2, · · · , k − 2, v sn = αsn,1v s+1 n + `s+1∑ j=2 αsn,j j−1∏ i=1 (1− αsn,i)γj−1v s+1n + `s+1∏ i=1 (1− αsn,i)γ`1v s+1n (4.3) and, for k ≥ 2, v k−1n = `k∑ j=1 αk−1n,j j−1∏ i=1 (1− αk−1n,i )γj−1tn + `k∏ i=1 (1− αk−1n,i )γ`k tn, n ≥ 1, (4.4) now, suppose εn → 0 as n →∞. then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).indeed, using proposition 2.4 with u = q, v1n = t, j = i , k = 1,γj−1v1n = vj−1 and γ`1v1n = v „we obtain ‖tn+1 − q‖2 = ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q −[δn,1v 1 n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − tn+1]‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 21 ≤ ‖ − [tn+1 − δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ]‖2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 = ‖tn+1 − δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 = εn + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 ≤ εn + δn,1‖v1n,1 − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1v1n − q‖2 + `1∏ i=1 (1− δn,i)‖γ`1v1n − q‖2 ≤ εn + δn,1‖v1n,1 − q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i)‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 ≤ εn + ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i) + `1∏ i=1 (1− δn,i)(ρj)2 ) ×‖v1n − q‖2 (4.5) since `1, `k are fixed integers and αsn,i ∈ [0, 1] for each s , using (3.2) and (3.12), the estimationsbelow are obtained, for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 :, ‖v1n − q‖2 ≤ α1n,1‖v2n − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)‖γj−1v2n − γj−1q‖2 + `2∏ i=1 (1− α1n,i)‖γ`2v2n − γ`2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  ‖v2n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 22 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2 [α2n,1‖v3n − q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1− α2n,i)‖γj−1v3n − γj−1q‖2 + `3∏ i=1 (1− α2n,i)‖γ`3v3n − γ`3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2 [α2n,1‖v3n − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i)‖v3n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖v3n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖v3n − q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖v4n − q‖2 + `4∑ j=2 α3n,j j−1∏ i=1 (1− α3n,i)‖γj−1v4n − γj−1q‖2 + `4∏ i=1 (1− α3n,i)‖γ`4v4n − γ`4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖v4n − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖v4n − q‖2 + `4∏ i=1 (1− α3n,i)(ρj)2‖v4n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) ‖v4n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 23 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 )[ α4n,1‖v5n − q‖2 + `5∑ j=2 α4n,j j−1∏ i=1 (1− α4n,i)‖γj−1v5n − γj−1q‖2 + `5∏ i=1 (1− α4n,i)‖γ`5v5n − γ`5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 )[ α4n,1‖v5n − q‖2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i)‖v5n − q‖2 + `5∏ i=1 (1− α4n,i)(ρj)2‖v5n − q‖2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) ‖v5n − q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 24 × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ×‖tn − q‖2 (4.6) (4.5) and (4.6)imply that ‖tn+1 − q‖2 ≤ ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i) + `1∏ i=1 (1− δn,i)(ρj)2 ) × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ×‖tn − q‖2 + εn (4.7) note that (4.7) is valid since γq = q and φ(0) = 0.now, since ρj ∈ [0, 1], we obtain using proposition 2.3, for j = 1, 2, 3, · · · , s − 1, that τn < ηn = 1, (4.8) where τn = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 25 × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) and ηn = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i) ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i ) ) putting (4.8) in (4.7), we obtain, using lemma 2.3 that the sequence {tn}∞n=0 converges strongly tothe point q in f (γ).on the other hand, suppose tn → q as n → ∞. then, we show that ε → 0 as n → ∞. indeed,from (3.5) with v1n = y1n , (4.2) and proposition 2.4 with u = q, v1n = t, j = i , k = 1,γj−1v1n = vj−1and γ`1v1n = v „ we have εn = ‖tn+1 − δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 = ‖tn+1 − q − δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q  ‖2 ≤ ‖tn+1 − q‖2 + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 ≤ ‖tn+1 − q‖2 + δn,1‖v1n,1 − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1v1n − γj−1q‖2 + `1∏ i=1 (1− δn,i)‖γ`1v1n − γ`1q‖2 ≤ ‖tn+1 − q‖2 + δn,1‖v1n,1 − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 26 = ‖tn+1 − q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)(ρj)2 + `1∏ i=1 (1− δn,i)(ρj)2  ×‖v1n − q‖2 (4.9) putting (4.6) into (4.9), and using (4.8), we get εn ≤ ‖tn+1 − q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)(ρj)2 + `1∏ i=1 (1− δn,i)(ρj)2  × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) + `2∏ i=1 (1− α1n,i)(ρj)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− αn,i) + `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i) + `4∏ i=1 (1− α3n,i)(ρj)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1− α4n,i) + `5∏ i=1 (1− α4n,i)(ρj)2 ) × · · · × ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i ) + `s−1∏ i=1 (1− α`s−2n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) + `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ×‖tn − q‖2 ≤ ‖tn+1 − q‖2 + τn‖tn − q‖2 (4.10) thus, from our assumption, we obtain from (4.10) that εn → 0 as n → ∞. hence, the multistep di-iteration scheme (3.2) is γ-stable. thus, tje proof is completed. � theorem 4.2. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖ ≤ ρj‖x − y‖+ j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (4.11) where x, y ∈ h, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0, t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep ih-iteration scheme defined by (3.1). assume f (γ) 6= ∅, q ∈ f (γ). then, the multisetp ih-iteration scheme is γ-stable. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 27 proof. let {tn}∞n=0 and {vn}∞n=0, for i = 1, 2, · · · , s − 1, be two real sequences in h. set εn = ‖tn+1 − δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 (4.12) where, for s = 1, 2, · · · , k − 2, v sn = αsn,1tn + `s+1∑ j=2 αsn,j j−1∏ i=1 (1− αsn,i)γj−1v s+1n + `s+1∏ i=1 (1− αsn,i)γ`1v s+1n (4.13) and, for k ≥ 2, v k−1n = `k∑ j=1 αk−1n,j j−1∏ i=1 (1− αk−1n,i )γj−1tn + `k∏ i=1 (1− αk−1n,i )γ`k tn, n ≥ 1, (4.14) now, suppose εn → 0 as n →∞. then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).indeed, using proposition 2.4 with u = q, tn = t, j = i , k = 1,γj−1v1n = vj−1 and γ`1v1n = v „we obtain ‖tn+1 − q‖2 = ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q −[δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − tn+1]‖2 ≤ ‖ − [tn+1 − δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ]‖2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 = ‖tn+1 − δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 = εn + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 ≤ εn + δn,1‖tn − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1v1n − q‖2 + `1∏ i=1 (1− δn,i)‖γ`1v1n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 28 ≤ εn + δn,1‖tn − q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1− δn,i)‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 = εn + δn,1‖tn − q‖2 + ( 1− δn,1 − `1∏ i=1 (1− δn,i) ) (ρj)2‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 < εn + δn,1‖tn − q‖2 + (1− δn,1) ‖v1n − q‖2 (4.15) since `1, `k are fixed integers and αsn,i ∈ [0, 1] for each s , the estimations below are obtained for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖v1n − q‖2 ≤ αn,1‖tn − q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1− αn,i)‖γj−1v2n − γj−1q‖2 + `2∏ i=1 (1− αn,i)‖γ`2v2n − γ`2q‖2 ≤ α1n,1‖tn − q‖2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1− αn,i)‖v2n − q‖2 + `2∏ i=1 (1− αn,i)(ρj)2‖v2n − q‖2 ≤ α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) [ α2n,1‖tn − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)‖v3n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖v3n − q‖2 ] + `2∏ i=1 (1− α1n,i)(ρj)2 [ α2n,1‖tn − q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)‖v3n − q‖2 + `3∏ i=1 (1− α2n,i)(ρj)2‖v3n − q‖2 ] = α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ‖v3n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 29 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖v3n − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) ‖v3n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) ‖v3n − q‖2 ≤ α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) [α3n,1‖tn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖v4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖v4n − q‖2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )[ α3n,1‖tn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖v4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖v4n − q‖2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )[ α3n,1‖tn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖v4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖v4n − q‖2 ] + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )[ α3n,1‖tn − q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)‖v4n − q‖2 + `4∏ i=1 (1− α4n,i)(ρj)2‖v4n − q‖2 ] = α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 30 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖v4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1− α3n,i)  ‖v4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 = α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 31 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2 ) ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2) ) ‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) × ( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2) ) ‖v4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( (1− α3n,1 − `4∏ i=1 (1− α3n,i))(ρj)2 ) ‖v4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 )( `4∏ i=1 (1− α4n,i)(ρj)2 ) ‖v4n − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 32 = α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  (1− α3n,1)(ρj)2‖v4n − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) (1− α3n,1)(ρj)2)‖v4n − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) (1− α3n,1)(ρj)2)‖v4n − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) (1− α3n,1)(ρj)2‖v4n − q‖2 ≤ α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)(ρj)2α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) ‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 33 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) α3n,1(1− α3n,1)(ρj)2‖tn − q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2)‖tn − q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i) ( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2)‖tn − q‖2 + ( `3∏ i=1 (1− α2n,i)(ρj)2 )( `2∏ i=1 (1− α1n,i)(ρj)2 ) α3n,1(1− α3n,1)(ρj)2‖tn − q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1− α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1− α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1− α2n,i) × · · · ×`s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1− α`s−2n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1− α`s−1n,i ) αsn,1‖tn − q‖2 + (ρj)2 ( `2∏ i=1 (1− α1n,i)(ρj)2 ) ×(ρj)2 ( `3∏ i=1 (1− α2n,i)(ρj)2 ) (ρj)2 ( `4∏ i=1 (1− α3n,i)(ρj)2 ) × · · · × (ρj)2 `s−1∏ i=1 (1− α`s−2n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1− α`s−1n,i )(ρj)2 ) ‖tn − q‖2 < α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i) ) ‖tn − q‖2 + ( 1− α2n,1 − `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1‖tn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 34 + ( `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( `3∏ i=1 (1− α2n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + ( 1− α2n,1 − `3∏ i=1 (1− α2n,i)) )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + ( `3∏ i=1 (1− α2n,i) )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + · · · + ( 1− α1n,1 − `2∏ i=1 (1− α2n,i) )( 1− α2n,1 − `3∏ i=1 (1− α2n,i) ) × ( 1− α3n,1 − `4∏ i=1 (1− α3n,i) ) × · · · × 1− α`s−2n,1 − `s−1∏ i=1 (1− α`s−2n,i )  × ( 1− α`s−1n,1 − `s∏ i=1 (1− α`s−1n,i ) ) αsn,1‖tn − q‖2 + ( `2∏ i=1 (1− α1n,i) ) × ( `3∏ i=1 (1− α2n,i) )( `4∏ i=1 (1− α3n,i) ) × · · · × `s−1∏ i=1 (1− α`s−2n,i )  × ( `s∏ i=1 (1− α`s−1n,i ) ) ‖tn − q‖2 (4.16) < α1n,1‖tn − q‖2 + `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) ) α3n,1 ( 1− α2n,1 ) ‖tn − q‖2 +α3n,1 ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) ‖tn − q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1− α1n,i) )( 1− α2n,1 ) α3n,1(1− α3n,1)‖tn − q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 35 + ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + · · ·+ + ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) αsn,1‖tn − q‖2 = α1n,1‖tn − q‖2 + α2n,1 ( 1− α1n,1 − `2∏ i=1 (1− α1n,i) ) ‖tn − q‖2 + `2∏ i=1 (1− α1n,i)α2n,1‖tn − q‖2 + ( (1− α1n,1 − `2∏ i=1 (1− α1n,i) ) α3n,1 ( 1− α2n,1 ) ‖tn − q‖2 +α3n,1 ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) ‖tn − q‖2 + ( (1− α1n,1 − `2∏ i=1 (1− α1n,i) )( 1− α2n,1 ) α3n,1(1− α3n,1)‖tn − q‖2 + ( 1− α2n,1 )( `2∏ i=1 (1− α1n,i) ) α3n,1(1− α3n,1)‖tn − q‖2 + · · ·+ + ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) αsn,1‖tn − q‖2 < [ α1n,1 + α2n,1 ( 1− α1n,1 ) + (1− α1n,1)α3n,1 ( 1− α2n,1 ) + (1− α1n,1) ( 1− α2n,1 ) α3n,1(1− α3n,1) + · · ·+ ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) ] ‖tn − q‖2 (4.17) (4.15) and (4.17) imply that ‖tn+1 − q‖2 ≤ {δn,1 + (1− δn,1) [α1n,1 + α2n,1 ( 1− α1n,1 ) + (1− α1n,1)α3n,1 ( 1− α2n,1 ) +(1− α1n,1) ( 1− α2n,1 ) (1− α3n,1) + · · ·+ ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) ]}‖tn − q‖2 (4.18) using lemma 2.3, we obtain (from (4.18)) that the sequence {xn}∞n=0 converges strongly to q ∈ f (γ).conversely, suppose tn → q as n → ∞. then, we show that ε → 0 as n → ∞. indeed, from(3.5) with v1n = y1n , (4.12) and proposition 2.4 with u = q, tn = t, j = i , k = 1,γj−1v1n = vj−1 and https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 36 γ`1v1n = v „ we have εn = ‖tn+1 − δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n − `1∏ i=1 (1− δn,i)γ`1v1n ‖2 = ‖tn+1 − q − δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q  ‖2 ≤ ‖tn+1 − q‖2 + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)γj−1v1n + `1∏ i=1 (1− δn,i)γ`1v1n − q‖2 ≤ ‖tn+1 − q‖2 + δn,1‖tn − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)‖γj−1v1n − γj−1q‖2 + `1∏ i=1 (1− δn,i)‖γ`1v1n − γ`1q‖2 ≤ ‖tn+1 − q‖2 + δn,1‖tn − q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 = ‖tn+1 − q‖2 + δn,1‖tn − q‖2 + ( 1− δn,1 − `1∏ i=1 (1− δn,i) ) (ρj)2‖v1n − q‖2 + `1∏ i=1 (1− δn,i)(ρj)2‖v1n − q‖2 = ‖tn+1 − q‖2 + δn,1‖tn − q‖2 + (1− δn,1) ‖v1n − q‖2 (4.19) (4.17) and (4.19) imply εn ≤ ‖tn+1 − q‖2 + { δn,1 + (1− δn,1) [ α1n,1 + α2n,1 ( 1− α1n,1 ) +(1− α1n,1)α3n,1 ( 1− α2n,1 ) + (1− α1n,1) ( 1− α2n,1 ) α3n,1(1− α3n,1) + · · ·+ ( 1− α1n,1 ) ( 1− α2n,1 ) ( 1− α3n,1 ) × · · · × ( 1− α`s−2n,1 ) × ( 1− α`s−1n,1 ) ]} ‖tn − q‖2 (4.20) again, from our assumption, we obtain from (4.20) that εn → 0 as n → ∞. hence, the multistep ih-iteration scheme (3.1) is γ-stable, and this completes the proof. � remark 4.1. the following areas are still open: https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 37(i) to reconstruct, approximate the fixed points and the stability results of some existing iterative schemes in the current literature, other than the ones under study, for finite family of certain class of contractive-type map;(ii) to compare convergent rates of the iterative schemes defined by (3.1) and (3.2) with those of (1.5) and (1.6). competing interestthe authors declare that there is no conflict of interest. references [1] b. e. rhoade, fixed point theorems and stability results for fixed point iteration procedures, indian j. pure appl.math. 24(11) (1993) 691-03.[2] b. e. rhoade, fixed point theorems and stability results for fixed point iteration procedures, indian j. pure appl.math. 21 (1990) 1-9.[3] m. o. osilike, a. udoemene, a short proof of stability resultsfor fixed point iteration 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demicontractive multivalued mappings, (in press). https://doi.org/10.28924/ada/ma.2.1 https://doi.org/10.2307/2032162 https://doi.org/10.2307/2032162 https://doi.org/10.1090/s0002-9939-1974-0336469-5 https://doi.org/10.1090/s0002-9939-1974-0336469-5 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1155/2020/3287968 1. introduction 2. preliminary 3. main results i 4. main results ii references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 16doi: 10.28924/ada/ma.2.16 a new approximate birkhoff orthogonality type chuanjiang zhou, qi liu, yongjin li∗ department of mathematics, sun yat-sen university, guangzhou, 510275, p. r. china 1090871744@qq.com, liuq325@mail2.sysu.edu.cn, stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. in this note, we introduce a new approximate birkhoff orthogonality type and give a char-acterization for inner product spaces using the approximate orthogonality. we show some generalproperties of the approximate birkhoff orthogonality type as well as applications. in particular, westudy the relationship between the new approximate birkhoff orthogonality type and other approx-imate orthogonality types that have been defined before. furthermore we study the approximatepreserving mapping and give some properties. 1. introduction one of the important ideas playing a fundamental role in geometry of normed spaces is the con-cept of orthogonality. many mathematicians have introduced different types of orthogonality for thenormed linear spaces, cf. [2, 20,24]. in 1934 [23], the first orthogonality type:roberts orthogonalitywas introduced by roberts. after that in 1935 [5], birkhoff introduced one of the most importantorthogonality types: x is said to be birhoff orthogonal to y (x ⊥b y ) if ‖x+ty‖ ≥ ‖x‖ for all t ∈ r.then james in 1945 [15] introduced the pythagorean orthogonality and isosceles orthogonality: x is said to be isosceles orthogonal to y (x ⊥i y ) if ‖x + y‖ = ‖x − y‖. there are also otherorthognality types related to norm limit such as ρ-orthogonality and g-orthogonality [10,18].let x be inner product spaces (x, 〈·|·〉), all the orthogonality types are equivalent to x ⊥ y orequivalently, 〈x |y〉 = 0. in inner product spaces a natural way to generalize orthogonality is todefine the approximate orthogonality by: x ⊥ε y if and only if |〈x |y〉| ≤ ε‖x‖‖y‖, x, y ∈ x [9, 26].inspired by the approximate orthogonality, dragomir [13] gave the definition of the approximatebirkhoff orthogonality xε ⊥b y : ‖x + ty‖ ≥ (1 − ε)‖x‖ for all t ∈ r. it is easy to see thatthis type of approximate orthogonality is equivalent to ⊥ε in inner product spaces [13]. after thatjacek chmieliński [21] introduced the approximate birkhoff orthogonality x ⊥εb y : ‖x + ty‖2 ≥ ‖x‖2 − 2ε‖x‖‖ty‖ for all t ∈ r, the approximate isosceles orthogonality [11] x ⊥εi y : |‖x + y‖2 − received: 20 feb 2022. key words and phrases. birkhoff orthogonality; isosceles-orthogonality; approximate orthogonality; orthogonalitypreserving mappings. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 2 ‖x − y‖2| ≤ 4ε‖x‖‖y‖ for all t ∈ r, and xε ⊥i y : |‖x + y‖ − ‖x − y‖| ≤ ε‖x + y‖‖x − y‖ forall t ∈ r. many meaningful results have been found about approximate orthogonality through thetireless efforts of mathematicians, see [12,14].in this paper we will introduce a new approximate birkhoff othogonality type and investigateits properties and its relationship with other approximate orthogonality types. moreover we give acharacterization of inner product spaces by approximate orthogonality types and some propertiesabout approximately orthogonality preserving mapping.throughout the paper we will only consider normed spaces with dimx ≥ 2, we use 〈·|·〉 denotingthe inner product and (·|·) denoting the angle between x and y , i,e, in inner product spaces (x, y) = ‖x+y‖2−‖x‖2−‖y‖2 2‖x‖‖y‖ . 2. approximate birkhoff orthogonality ⊥bε let ε ∈ [0, 1) and x, y be elements of inner product spaces x , we have the vertical relationship: x ⊥ y ⇐⇒ |〈x |y〉| = 0. to generalize the orthogonality, it is natural to consider the approximateorthogonality (ε-orthogonality: x ⊥ε y ) defined by: x ⊥ε y ⇐⇒ |〈x |y〉| ≤ ε‖x‖‖y‖ ⇐⇒ |cos(x, y)| ≤ ε. now we consider normed spaces, many mathematicians have introduced different types of orthogo-nality to represent orthogonality such as birkhoff orthogonality [5] and isosceles orthogonality [15].as an extension for the orthogonality, approximately orthogonality such as approximate birkhofforthogonalty [13,21]: xε ⊥b y ⇐⇒ ‖x + ty‖ ≥ (1− ε)‖x‖ t ∈ r. x ⊥εb y ⇐⇒ ‖x + ty‖2 ≥ ‖x‖2 − 2ε‖x‖‖ty‖ t ∈ r,and approximate isosceles orthogonality [11]: x ⊥εi y ⇐⇒ |‖x + y‖2 − ‖x − y‖2| ≤ 4ε‖x‖‖y‖ t ∈ r. xε ⊥i y ⇐⇒ |‖x + y‖ − ‖x − y‖| ≤ ε‖x + y‖‖x − y‖ t ∈ r,have been defined and studied. notice that the definition of ε ⊥b is quadratic while the definitionof ⊥εb is of first order, we give a new approximate birkhoff orthogonality type: x ⊥bε y ⇐⇒ ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖, which is also of first order but different from ε ⊥b . it is easy to see that the inequality is alwayscorrect if t ≥ ‖x‖ ε‖y‖ . example 2.1. let x = (r2, ‖ · ‖1), assume that x = (1, 0), y = (z, 1 − z), z ∈ [0, 1). if we want xε ⊥b y , then the inequality ‖x + ty‖ ≥ (1− ε)‖x‖ should hold for all t ∈ r, thus we have: ‖(1 + tz, t(1− z))‖ ≥ 1− ε t ∈ r. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 3 if t ≥ 0, the inequality is always correct. for t < 0, if 1 + tz ≥ 0, we have: 1 + tz − t + tz ≥ 1− ε. thus z ≤ 1 2 − ε 2t . by 1 + tz ≥ 0, we get z ≤ 1 2−ε . if 1 + tz < 0, similarly we need t ≤ ε− 2. since z ≤ 1 2−ε , from 1 + tz < 0, we get t ≤ ε− 2. thus xε ⊥b y iff z ≤ 1 2−ε . on the other hand, if we want x ⊥bε y , the inequality ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖ should hold for all t ∈ r, thus we have: ‖(1 + tz, t(1− z))‖ ≥ 1− ε|t| t ∈ r. if t ≥ 0, the inequality is also always correct. for t < 0, if 1 + tz ≥ 0, we have: 1 + tz − t + tz ≥ 1 + εt. thus z ≤ 1+ε 2 . similarly we can get ‖(1 + tz, t(1− z))‖ ≥ 1− ε|t| for 1 + tz < 0 if z ≤ 1+ε 2 . thus x ⊥bε y iff z ≤ 1+ε 2 . we have the result that ⊥bε is not always equivalent to ε ⊥b in x . since the definition of approximate birkhoff orthogonality comes from the notion of approximateorthogonality ⊥ε in inner product spaces, it is natural to require the equivalence: x ⊥bε y ⇐⇒ x ⊥ε y in inner product spaces. now we give some basic properties about ⊥bε before prove theequivalence. proposition 2.2. let x be normed spaces, then ⊥bε is homogeneous., this is x ⊥bε y implies αx ⊥bε βy (x, y ∈ x,α, β ∈ r). proof. since x ⊥bε y , we have ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖ for any t ∈ r. if α = 0 , αx ⊥bε βy is always correct; if α 6= 0, we have ‖αx + tβy‖ = |α|‖x + β α ty‖ ≥ |α|{‖x‖ − ε‖ β α ty‖} = ‖ax‖ − ε‖tβy‖. thus αx ⊥bε βy . � recall that the limits [16] : n±(x ; y) = lim n→±∞ ‖nx + y‖ − |nx‖ = lim h→0± ‖x + hy‖ − ‖x‖ h , exist and satisfy the weakened linearity condition [4]. x, y are said to be gateaux differentiable [1]at 0 i f n−(x, y) = n+(x, y). moreover we have [16]: n±(x ; rx + sy) = r‖x‖+ s · n±(x ; y), f or s ≥ 0 and al l r. we then give a characterization of x ⊥bε y using the definition of n±(x, y). https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 4 proposition 2.3. let x be normed spaces, then x ⊥bε y if and only if n+(x, y) + ε‖y‖ ≥ 0 ≥ n−(x, y)− ε‖y‖. proof. let x ⊥bε y and t ∈ r\{0} then ‖x + ty‖ − ‖x‖ |t| ≥ ε‖y‖. let t → 0+, we have n+(x, y) ≥ −ε‖y‖. similarly, let t → 0−, we have n−(x, y) ≤ ε‖y‖. to sum up, n+(x, y) + ε‖y‖ ≥ 0 ≥ n−(x, y)− ε‖y‖. conversely, if n+(x, y) ≥ −ε‖y‖, for ∀η > 0, there ∃δ such that if 0 < t ≤ δ, we have: ‖x + ty‖ − ‖x‖ t ≥ −(ε+ η)‖y‖, or equivalently ‖x + ty‖ − ‖x‖ ≥ −t(ε+ η)‖y‖ f or t ∈ (0, δ]. because of the convexity of ‖x + ty‖, we have ‖x + ty‖ − ‖x‖ ≥ −t(ε+ η)‖y‖ f or t > 0. let δ → 0, we have ‖x + ty‖ − ‖x‖ ≥ −ε‖ty‖ f or t > 0. similarly, using n−(x, y) ≤ ε‖y‖, we have:‖x + ty‖ − ‖x‖ ≥ t(ε)‖y‖ f or t < 0. if t = 0, ‖x + ty‖ − ‖x‖ ≥ ε‖ty‖ is obvious. to conclude, we have: ‖x + ty‖ − ‖x‖ ≥ ε‖ty‖ f or t ∈ r. thus x ⊥bε y . � to verify the validity of the new approximate birkhoff orthogonality, we have the followingproposition: proposition 2.4. let x be normed spaces, we have: x ⊥bε y if and only if x ⊥ε y . proof. since x ⊥bε y , for 0 < t ≤ ‖x‖ ε‖y‖ we have ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖. square both sides we get ‖x‖2 + t2‖y‖2 + 2t(x, y) ≥ ‖x‖2 + ε2t2‖y‖2 − 2ε‖x‖‖ty‖. thus (1− ε2)t‖y‖ ≥ −2‖x‖(ε+ cos(x, y)) when t tends to 0, (1− ε2)t‖y‖ tends to 0, so we have ε+ cos(x, y) ≥ 0 ⇐⇒ cos(x, y) ≥ −ε. similarly for 0 > t ≥ − ‖x‖ε‖y‖ , we have cos(x, y) ≤ ε, thus x ⊥ε y . https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 5 conversely, if |cos(x, y)| ≤ ε, we have ‖x + ty‖ − ‖x‖ ≥ ε‖ty‖ f or |t| ∈ [0, ‖x‖ ε‖y‖ ]. on the other hand, ‖x + ty‖ − ‖x‖ ≥ ε‖ty‖ is always correct for |t| ≥ ‖x‖ ε‖y‖ . to conclude, ‖x + ty‖ − ‖x‖ ≥ ε‖ty‖ f or t ∈ r. thus x ⊥bε y . � from n±(x ; rx + sy) = r‖x‖+ s · n±(x ; y), f or s ≥ 0 and al l r, we have the following: proposition 2.5. in the normed space x, if x ⊥bε y , then we have x ⊥bε rx + sy for s ≥ 0, r satisfying ε(‖rx + sy‖ − s‖y‖) ≥ r‖x‖ ≥ ε(s‖y‖ − ‖rx + sy‖). proof. since x ⊥bε y , we have n+(x, y) ≥ −ε‖y‖ and n−(x, y) ≤ ε‖y‖, so n+(x, rx + sy) ≥ r‖x‖+ (−sε‖y‖) ≥ −ε‖rx + sy‖. similarly we have n−(x, rx + sy) ≤ ε‖rx + sy‖, thus x ⊥bε rx + sy . � let x be normed spaces, it is known that [16] for any x, y ∈ x there exists a real number asuch that x ⊥b ax + y , moreover, such a number satisfies |a| ≤ ‖y‖‖x‖ . on this basis, chmieliński [9]discovered that x ⊥εb y if and only if there exists a real number |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax + y .in fact, in inner product spaces, it is easy to see that x ⊥ε y if and only if there exists |a| ≤ ‖y‖‖x‖εsuch that x ⊥b ax + y by taking a = − 〈x |y〉‖x‖2 x + y for x 6= 0.in the following we will prove that it is also true for ⊥bε, that is, in normed spaces, x ⊥bε y if and only if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax + y . before the proof, we need some lemma. lemma 2.6. [16] let x be normed spaces, n−(x, y) ≤ n+(x, y). lemma 2.7. [16] let x be normed spaces, a ≤ b, a, b ∈ x , if x ⊥b ax + y , x ⊥b bx + y , then x ⊥b cx + y f or c ∈ [a, b]. lemma 2.8. [16] let x be normed spaces, x ⊥b ax + y ⇐⇒ n−(x, y) ≤ −a‖x‖ ≤ n+(x, y). https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 6 theorem 2.9. let x be normed spaces, x ⊥bε y if and only if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax + y . proof. if x ⊥bε y , first we have n−(x, y) ≤ ε‖y‖ , n+(x, y) ≥ −ε‖y‖. on the other hand, from lemma 2.8, we have if − n+(x, y) ‖x‖ ≤ a ≤ − n−(x, y) ‖x‖ , then x ⊥b ax + y . if there exists no |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax + y , then − n+(x, y) ‖x‖ > ‖y‖ ‖x‖ε or − n−(x, y) ‖x‖ < − ‖y‖ ‖x‖ε. thus n+(x, y) < ‖y‖ or n(x, y) > ‖y‖ε. contradict to x ⊥bε y , so there must exists |a| ≤ ‖y‖‖x‖ε, such that x ⊥b ax + y . conversely, if there exists |a| ≤ ε‖y‖ ‖x‖ such that x ⊥b ax + y , we have: x ⊥b ax + y =⇒ −n+(x, y) ≤ a‖x‖ ≤ −n−(x, y) =⇒ n+(x, y) + ε‖y‖ ≥ 0 ≥ n−(x, y)− ε‖y‖ =⇒ x ⊥bε y . to conclude, in normed spaces, x ⊥bε y if and only if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax+y . � since both x ⊥bε y and x ⊥bε y are equivalent to there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥b ax+y .we have x ⊥bε y if and only if x ⊥εb y in normed spaces. now we give a direct proof for this. theorem 2.10. let x be normed spaces, x ⊥bε y if and only if x ⊥εb y . proof.we can assume that |t| ≤ ‖x‖ ε‖y‖ and x 6= 0. if x ⊥bε y , we have: ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖ ≥ 0. take square on both sides, we get ‖x + ty‖2 ≥ ‖x‖2 − ε2‖ty‖2 − 2ε‖x‖‖ty‖. thus ‖x + ty‖2 ≥ ‖x‖2 − 2ε‖x‖‖ty‖, which means that x ⊥εb y . conversely, if x ⊥εb y , we have ‖x + ty‖2 ≥ ‖x‖2 − 2ε‖x‖‖ty‖ ≥ 0. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 7 let both sides be divided by ‖x + ty‖+ ‖x‖, we get: ‖x + ty‖ − ‖x‖ ≥ −2ε‖x‖‖ty‖ ‖x + ty‖+ ‖x‖ . since ‖x + ty‖ tends to ‖x‖ when t → 0, for every 2‖x‖ > δ > 0, we can find η > 0, such that ‖x + ty‖+ ‖x‖ ≥ 2‖x‖ − δ i f |t| ≤ η. we then have ‖x + ty‖ − ‖x‖ ≥ −2‖x‖‖ty‖ 2‖x‖ − δ . let δ → 0 we have ‖x + ty‖ − ‖x‖ ≥ −2‖x‖‖ty‖ 2‖x‖ when t → 0. thus n+(x, y) + ε‖y‖ ≥ 0 ≥ n−(x, y)− ε‖y‖ =⇒ x ⊥bε y . � recall that dragomir gave the following definition about approximate birkhoff orthogonality: xε ⊥b y ⇐⇒ ‖x + ty‖ ≥ (1− ε)‖x‖. it is known that [19] in normed spaces, x ⊥εb y implies xδ ⊥b, where δ = 1− √ 1− 4ε. now wegive a more accurate estimate of δ as an application of the above proposition. proposition 2.11. let x be normed spaces, let x, y ∈ x, then: x ⊥εb y =⇒ xδ ⊥b y where δ = 2ε. proof. let f (t) = ‖x + ty‖ and assume that f (t) attains its minimum at t0, hence ‖x + t0y + ty‖ ≥ ‖x + t0y‖ f or al l t ∈ r. choose t = −t0 we have ‖x‖ ≥ ‖x + t0y‖ ≥ |‖x‖ − |t0|‖y‖|, thus we get |t0| ≤ 2‖x‖ ‖y‖ , then ‖x + ty‖ ≥ ‖x + t0y‖ ≥ ‖x‖ − ε|t0|‖y‖ ≥ (1− 2ε)‖x‖ f or al l t ∈ r. thus xδ ⊥b y , where δ = 2ε. by the equivalence between ⊥bε and ⊥εb, we have the result that x ⊥εb y implies xδ ⊥b y . since 2ε ≤ 1− √ 1− 4ε, 2ε can be seen as a more accurate estimate. � https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 83. approximate isosceles orthogonality and approximate birkhoff orthogonality in the following we will use the notion of approximate isosceles orthogonality [11], recall thatthe approximate isosceles orthogonality is defined by: x ⊥εi y : |‖x + y‖2 − ‖x − y‖2| ≤ 4ε‖x‖‖y‖. xε ⊥i y : |‖x + y‖ − ‖x − y‖| ≤ ε(‖x + y‖+ ‖x − y‖). it is easy to see that in inner product spaces we have: x ⊥εi y ⇐⇒ |cos(x, y)| ≤ ε ⇐⇒ x ⊥bε y , and [11] xε ⊥i y ⇐⇒ |cos(x, y)| ≤ ε 1 + ε2 (‖x‖2 + ‖y‖2). in the following we give some simple properties about approxiamte isosceles orthogonality. proposition 3.1. let x be normed spaces, if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥i ax + y , then xε ⊥i y . proof. since x ⊥i ax + y we have ‖x + ax + y‖ = ‖x − ax − y‖, then |‖x + y‖ − ‖x − y‖| = |‖x + ax + y − ax‖ − ‖x − ax − y + ax‖|. on the other hand, by trigonometric inequality we have: ‖x + ax + y‖ − ‖ax‖ − (‖x − ax − y‖+ ‖ax‖) ≤ ‖x + ax + y − ax‖ − ‖x − ax − y + ax‖, and ‖x + ax + y − ax‖ − ‖x − ax − y + ax‖ ≤ ‖x + ax + y‖+ ‖ax‖ − (‖x − ax − y‖ − ‖ax‖). thus |‖x + ax + y − ax‖ − ‖x − ax − y + ax‖| ≤ 2‖ax‖, then |‖x + y‖ − ‖x − y‖| ≤ 2‖ax‖ ≤ ε(‖x + y‖+ ‖x − y‖), thus xε ⊥i y . proposition 3.2. let x be normed spaces, if for every ‖x‖ = ‖y‖ = 1, there is no 0 ≤ ε < 1 such that x ⊥εi y , then x is a strictly convex space. proof. for any ‖x‖ = ‖y‖ = ‖x+y‖ 2 = 1, if x 6= y ,we have |‖x + y‖2 − ‖x − y‖2| = |4− ‖x − y‖| < 4, https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 9 thus there must exist a 0 ≤ ε < 1 such that |4 − ‖x − y‖2| ≤ 4ε which means that x ⊥εi y , contradict to the condition. so there must be x = y . from the equivalent characterization of strictly convex space [23]. we get the result that x must be a strictly convex space. it is known that in inner product spaces, different orthogonality types such as isosceles,pythagorean, and birkhoff orthogonality is equivalent [3]. using the notions of orthogonality innormed linear spaces it is possible to give different characterizations for inner product spaces. forinstance [17], if x ⊥i y =⇒ x ⊥b y in a normed space x , then x must be inner product spaces.inspired by this, now we give a characterization for inner product spaces using approximate or-thogonality. theorem 3.3. let x be normed spaces, then x is inner product spaces iff the following two conditions are satisfied. (1) if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥i ax + y , then x ⊥εi y . (2) x ⊥εi y impliesx ⊥bε y . proof. if x is an inner product space, we have x ⊥i y ⇐⇒ x ⊥b y and x ⊥εi y ⇐⇒ x ⊥bε y . thus (2) is satisfied. if there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥i ax + y , we have: x ⊥b ax + y , |a| ≤ ‖y‖ ‖x‖ε. thus x ⊥bε y which implies x ⊥εi y . thus both (1) and (2) are satisfied. conversely, assume that both (1) and (2) are satisfied, let x ⊥i y , x 6= 0. if |a| ≤ ε ‖ax+y‖ ‖x‖ , let b = −a, then |b| ≤ ε‖ax+y‖‖x‖ and x ⊥i bx + ax + y , thus form (1) we have x ⊥εi ax + y . to conclude we have: x ⊥εi ax + y i f |a| ≤ ε ‖ax + y‖ ‖x‖ . now define: f (t) = ‖tx+y‖ ‖x‖ ε, we have f (t) = ‖tx + y‖ ‖x‖ ε ≤ ‖tx‖+ ‖y‖ ‖x‖ ε = ε|t|+ ‖y‖ ‖x‖ε. since 0 ≤ ε < 1, when |t| tends to infinite, f (t) < |t|. when t = 0, f (0) = ‖y‖ ‖x‖ε > 0. by the convexity of f (t) we have x ⊥εi t1x + y , t1 < 0, ‖t1x + y‖ ‖x‖ ε = −t1. x ⊥εi t2x + y , t2 > 0, ‖t2x + y‖ ‖x‖ ε = t2. from (2) we have x ⊥bε t1x + y and x ⊥bε t2x + y . from proposition 2.9,there must exist |a1| ≤ ‖t1x + y‖ ‖x‖ ε = −t1 and |a2| ≤ ‖t2x + y‖ ‖x‖ ε = t2 https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 10 such that x ⊥b a1 + t1 + y , x ⊥b a2 + t2 + y . by a1 + t1 ≤ 0 , a2 + t2 ≥ 0 and lemma 2.7, we have x ⊥b y . thus we have x ⊥i y =⇒ x ⊥b y , which means that x is inner product spaces. example 3.4. let x = (r2, ‖ · ‖∞), that is, ‖(x1, x2)‖ = max(|x1|, |x2|), assume that x = (1, 0), y = (z, 1), |z | < 1. in order to satisfy x ⊥bε y or equivalently ‖x + ty‖ ≥ ‖x‖ − ε‖ty‖, the following inequality shuold hold for al l t ∈ r: ‖(1 + tc, t)‖ ≥ 1− ε|t|. since ‖(1 + tc, t)‖ ≥ ‖t‖, we know the above inequality is always correct if |t| ≥ 1 1+ε , then we may assume that |t| < 1 1+ε ≤ 1. when 1 > t ≥ 0, if 1 + tc ≥ t which means that t ≤ 1 1−c , we have 1 + tc ≥ 1− εt which implies that c ≥ −ε. if 1 + tc < t or equivalently t > 1 1−c , we have t ≥ 1 − εt that is t ≥ 1 1+ε . so there must be 1 1− c ≥ 1 1 + ε which implies that c ≥ −ε. similarly when −1 < t ≤ 0, we can get c ≤ ε. to conclude we have: (1, 0) ⊥bε (c, 1) if |c | ≤ ε. in order to satisfy x ⊥iε y or equivalently |‖x+y‖2−‖x−y‖2| ≤ 4ε‖x‖‖y‖, the following inequality shuold hold: |‖(1 + c, 1)‖2 − ‖(1− c,−1)‖2| ≤ 4ε. if c ≥ 0, we have (1 + c)2 − 1 ≤ 4ε =⇒ 0 ≤ c < −1 + √ 1 + 4ε. if c < 0, we have (1− c)2 − 1 ≤ 4ε =⇒ 1− √ 1 + 4ε ≤ c < 0. to conclude we have (1, 0) ⊥εi (c, 1) if |c | ≤ −1 + √ 1 + 4ε. since ε ≤ −1 + √ 1 + 4ε, it can be seen as an example that x ⊥εi y does not imply x ⊥bε y . example 3.5. let x = (r2, ‖ · ‖∞), we assume that x = (1, 1), y = (−1− √ 2 2 ε, 1− √ 2 2 ε), z = (−1, 1). we have x ⊥i z , and z = − ε√ 2 x + y , ε ≤ ‖y‖ ‖x‖ε. thus | − ε√ 2 | ≤ ‖y‖‖x‖ε, which implies that there exists |a| ≤ ‖y‖‖x‖ε such that x ⊥i ax + y . on the other hand, since ‖x + y‖ = ‖(− √ 2 2 ε, 2− √ 2 2 ε)‖ = 2− √ 2 2 ε. ‖x − y‖ = ‖(2 + √ 2 2 ε, √ 2 2 ε)‖ = 2 + √ 2 2 ε. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 11 we have |‖x + y‖2 − ‖x − y‖2| = 4 √ 2ε ≥ 4 √ 2. so x 6⊥εi y , thus it can be seen as an example that condition (1) is not satisfied. a mapping t : h→ k which satisfies the condtion x ⊥ y =⇒ t (x) ⊥ t (y). is called orthogonality preserving(o.p.) [7,8], and t is said to be an isometry maping [22] if ‖tx‖ = ‖x‖. to promote the concept, jacek chmielinski [6] introduced the notion of approximately orthogo-nality preserving (a.o.p.) mapping and have studied the properties of mapping that is approximarelyisosceles orthogonality preserving(t: x ⊥i y =⇒ t (x) ⊥εi t (y)). after that many mathematicianshave show great interest in the a.o.p mapping [25], and aleksej turnšek [19] studied the mappingthat is approximately birkhoff orthogonality preserving(t : x ⊥b y =⇒ t (x) ⊥εb t (x))innormed spaces. now we try to study the approximarely orthogonality preserving mapping types: t : x ⊥i y =⇒ tx ⊥εb ty, and t : x ⊥i y =⇒ tx εx ⊥b ty. proposition 3.6. let t : x → y be a nontrivial linear mapping satisfying x ⊥i y =⇒ tx ⊥εb ty, x, y ∈ x. then t is a bounded and bounded from below, ‖tx‖ ≥ (1−ε)2 3−ε2+2 √ 2−ε2 . proof. take two arbitrary unit vectors x and y and note that x+y 2 ⊥i x−y 2 , it follows that t (x + y) ⊥εb t (x − y), hence for all λ ∈ r we have ‖t (x + y) + λt (x − y)‖2 ≥ ‖t (x + y)‖2 − 2ε‖t (x + y)‖‖λt (x − y)‖, by the triangle inequality and the linearity of t it follows that ‖t (x + y)‖2 ≤ ‖(1 + λ)tx + (1− λ)ty‖2 + 2ε|λ|‖tx + ty‖2. on the other hand we have ‖t (x + y)‖2 ≥ (‖tx‖ − ‖ty‖)2, thus we get: ‖tx‖2+‖ty‖2−2‖tx‖‖ty‖ ≤ (1+λ)2‖tx‖2+(1−λ)2‖ty‖2+2(1−λ2)‖tx‖‖ty‖+2ε|λ|‖tx+ty‖2. if tx = 0, then let y ∈ x such that ty 6= 0, substitute x, y into the above formula,we have 0 ≤ λ2 − 2λ+ 2ε|λ| f or al l λ ∈ r. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 12 it is impossible cause 0 ≤ ε < 0, so we can divide both sides of the inequality by ‖tx‖2 and denote z = ‖ty‖ ‖tx‖ . we get: (z − 1)2λ2 + ( 2− 2z2 + 2ε(1 + z)2 ) λ+ 4z ≥ 0 f or al l λ ≥ 0. the inequality is satisfied when − b 2a ≤ 0 or ∆ = b2 − 4ac ≤ 0. ∆ = 4 ( (1− z2)2 + ε2(1 + z)4 + 2ε(1 + z)2(1− z2)− (4z)(z − 1)2 ) . (1− z2)2 + ε2(1 + z)4 + 2ε(1 + z)2(1− z2)− (4z)(z + 1)2 ≤ 0 =⇒ ∆ ≤ 0. thus we get z ≤ 3−ε2+2 √ 2−ε2 (1−ε)2 which implies ‖ty‖ ≤ 3−ε2+2 √ 2−ε2 (1−ε)2 ‖tx‖. since x, y are arbitrary, t is bounded and ‖tx‖ ≥ (1− ε)2 3− ε2 + 2 √ 2− ε2 ‖t‖‖x‖. theorem 3.7. let t : x → y be a nontrivial linear mapping satisfying x ⊥i y =⇒ tx ε ⊥b ty, x, y ∈ x. then t is a scalar mutiple of a isometric mapping, i.e., for some γ > 0, ‖tx‖ = γ‖x‖. proof. take two arbitrary unit vectors x and y and note that x+y 2 ⊥i x−y 2 , it follows that t (x + y)ε ⊥b t (x − y). hence for all λ ∈ r we have ‖t (x + y) + λt (x − y)‖ ≥ (1− ε)‖t (x + y)‖, thus ( ‖(1 + λ)tx‖+ ‖(1− λ)ty‖ )2 ≥ (1− ε)2‖tx + ty‖2 ≥ (1− ε)2(‖tx‖ − ‖ty‖)2. if tx = 0, then let y ∈ x such that ty 6= 0, substitute x, y into the above formula, we have ‖(1− λ)ty‖2 ≥ (1− ε)2‖ty‖2 =⇒ (1− λ)2 ≥ (1− ε)2 f or al l λ ∈ r, it is impossible, then we can divide both sides by ‖tx‖2 like before and denote z = ‖ty‖ ‖tx‖ , we get (1− z)2λ2 + 2(1− z2)λ+ (z + 1)2 − (1− ε)2(1− z)2 ≥ 0 f or al l λ ∈ r. if z 6= 1, then ∆ = 4 · (1 − z)4(1 − ε)2 > 0. thus the inequality is satisfied only when z = 1, so z = 1 which means that ‖tx‖ = ‖ty‖, thus t must be a scalar mutiple of a isometric mapping. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 13references [1] m. abbasi, a.y. kruger, m. théra, gateaux differentiability revisited, appl. math. optim. 84 (2021) 3499–3516. https://doi.org/10.1007/s00245-021-09754-y.[2] j. alonso, h. martini, s. wu, on birkhoff orthogonality and isosceles orthogonality in normed linear spaces, aequat.math. 83 (2011) 153–189. https://doi.org/10.1007/s00010-011-0092-z.[3] d. amir, characterizations of inner product spaces, birkhäuser basel, 1986. https://doi.org/10.1007/ 978-3-0348-5487-0.[4] g. ascoli, sugli spazi lineari metrici e le loro varietà lineari, ann. mat. 10 (1932) 203–232. https://doi.org/10. 1007/bf02417142.[5] g. birkhoff, orthogonality in linear metric spaces, duke math. j. 1 (1935) 169-172. https://doi.org/10.1215/ s0012-7094-35-00115-6.[6] j. chmieliński, linear mappings approximately preserving orthogonality, j. math. anal. appl. 304 (2005) 158–169. https://doi.org/10.1016/j.jmaa.2004.09.011.[7] j. chmieliński, stability of the orthogonality preserving property in finite-dimensional inner product spaces, j. math.anal. appl. 318 (2006) 433–443. https://doi.org/10.1016/j.jmaa.2005.06.016.[8] j. chmieliński, j. chmieliński, orthogonality preserving property and its ulam stability. in: rassias, t., brzdek, j.(eds) functional equations in mathematical analysis. springer optimization and its applications, vol 52. springer,new york, ny. 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https://doi.org/10.1016/j.jmaa.2007.03.016 https://doi.org/10.1007/s00010-013-0233-7 https://doi.org/10.1007/s00010-013-0233-7 1. introduction 2. approximate birkhoff orthogonality b 3. approximate isosceles orthogonality and approximate birkhoff orthogonality references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 1doi: 10.28924/ada/ma.4.1 frame operators for frames in krein spaces shah jahan1 , p. sam johnson2,∗ 1department of mathematics, central university of haryana, haryana 123029, india shahjahan@cuh.ac.in 2department of mathematical and computational sciences, national institute of technology karnataka, surathkal 575025, india sam@nitk.edu.in ∗correspondence: sam@nitk.edu.in abstract. in recent years, frames in krein spaces and several generalizations have been extensivelystudied. in this paper, we propose an alternative way of looking at the notion of frames in kreinspaces and give a necessary and sufficient condition for a sequence in a krein space to be a besselsequence. we observe that a subsequence of a frame in a krein space need not be a frame. also,two complementary subsequences are considered in which one of them is a frame for a krein space.we obtain necessary and sufficient conditions under which the other one is also a frame for the kreinspace. 1. introduction hilbert space frames were originally introduced by duffin and schaeffer [7] to deal with someproblems in non-harmonic fourier analysis. the linear independence property for a (hamel) basis,which allows every vector to be uniquely represented as a linear combination is very restrictive forpractical problems. frames allow each element in the space to be written as a linear combinationof the elements in the frame, but linear independence is not required. frames can be viewed asredundant bases which are generalization of riesz bases. this redundancy property sometimes isextremely important in applications such as sampling theory [9], filter banks [3], signal and imageprocessing [6] and so on. definition 1.1. [4] let h be a hilbert space and i be a countable index set. a collection {fn}n∈i in a hilbert space h is said to be a frame for h if there exist a, b > 0 such that a‖f ‖2 ≤ ∑ n∈i |〈f , fn〉|2 ≤ b‖f ‖2, ∀f ∈ h. received: 19 nov 2023.2020 mathematics subject classification. 42c15, 46c05, 46c20. key words and phrases. krein space; bessel sequence; frame sequence; frame operator.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.1 https://orcid.org/0000-0002-5966-9185 https://orcid.org/0000-0003-3461-5380 eur. j. math. anal. 10.28924/ada/ma.4.1 2we now look at the definition of frame which is equivalent to perceive as the map h 3 f 7→ ∑ n∈i 〈f , fn〉fn ∈ h (1) which is a well-defined bounded positive invertible operator.the bounded linear operator s : h −→ h defined by sf = ∑ n∈i 〈f , fn〉fn, f ∈ h, is known as the frame operator associated to the frame {fn}n∈i . this operator s is bounded invert-ible, positive and self adjoint. it allows to reconstruct each vector in terms of the sequence {fn}n∈ias follows: f = ∑ n∈i 〈f , s−1fn〉fn = ∑ n∈i 〈f , fn〉s−1fn. (2) the formula (2) is known as reconstruction formula associated to {fn}n∈i and if s = i, then thereconstruction formula resembles the fourier series of f associated with the orthonormal sequence {fn}n∈i .the concept of indefinite inner product was first found in a paper on quantum field theory bydirac in 1942 [5]. pontrjagin gave the mathematical interpretation of indefinite inner product.giribet et al. have introduced and studied frames for krein spaces [8]. motivated by the equivalentdefinition of frame as given in (1), in this paper, we propose an alternative way of looking at thenotion of frames in krein spaces by decomposing the index set i in a natural way and obtain somenew results on frames sequences.the paper is organized as follows. standard definition of krein space is given in section 2along with some notations and examples which will be used in the sequel. in section 3, we definethe concept of bessel sequence in krein spaces and give a necessary and sufficient condition fora sequence to be a bessel sequence in krein spaces. in section 4, we give the definition of framefor krein space and study operators associated to the frame. in the last section, we study framesequences in krein spaces. in general, if {fn}n∈i is a frame in a krein space and {nk} is any infiniteincreasing sequence in i , then {fnk} need not be a frame sequence. we provide some sufficientconditions under which subsequences become frame sequence for the krein space. 2. preliminaries let k be a complex vector space with a hermitian sesquilinear form defined on it. then wecall (k, [., .]) an inner product space. an element x ∈ k is called neutral, positive, or negativeif [x, x ] = 0, [x, x ] > 0, or [x, x ] < 0 respectively. if k contains positive as well as negativeelements, then it is called an indefinite inner product space, otherwise it is called a semi-definiteinner product space. we refer [1, 2] for basics on indefinite inner product spaces. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 3an indefinite inner product space (k, [., .]) is decomposable if it can be written as an orthogonaldirect sum of a neutral subspace k0, a positive definite subspace k+ and a negative definitesubspace k−: k = k0[+̇]k+[+̇]k−. (3) then (3) is known as a fundamental decomposition of k.an indefinite inner product space (k, [., .]) is a krein space if it can be written as an orthogonaldirect sum of a positive definite subspace k+ and a negative definite subspace k− such that (k+, [., .]) and (k−,−[., .]) are hilbert spaces. let a fundamental decomposition of a krein space k be given by k = k+[+̇]k− (4) and p± be the orthogonal projections onto k±. the linear map j = p+ − p− is called the fundamental symmetry corresponding to (4). then (f , g)j = [jf , g] is a positive definite inner product on k, called j-inner product corresponding to the fundamentaldecomposition (4). we can write (f , f )j = [jf , f ] = [(2p + − i)f , f ] = 2[p+f , p+f ]− [f , f ]. (5) the corresponding norm (called j-norm) is denoted by ‖f ‖j = (f , f ) 1 2 j = [jf , f ] 1 2 . example 2.1. consider k = `2(n), the linear space of square-summable sequences, with [f , g] = ∞∑ n=1 (−1)nfngn for f = (fn) ∞ n=1, g = (gn) ∞ n=1 ∈ k. let k+ = { (fn) ∞ n=1 : fn = 0 if n is odd } and k− = { (fn) ∞ n=1 : fn = 0 if n is even } . then k = k+[+̇]k−, where k+ and k− are complete with respect to the induced norm and hence k is a krein space. theorem 2.1. [2] let k be a krein space. then the following are equivalent:(1) there exists a fundamental decomposition of k.(2) there exists a maximal uniformly positive ortho-complemented subspace.(3) there exists a maximal uniformly negative ortho-complemented subspace.(4) there exists a mapping j in k such that j = j∗ = j−1. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 4different fundamental decompositions induce different j-norms. hence various norms can bedefined on a krein space by choosing different underlying fundamental decompositions. example 2.2. let k be a two-dimensional vector space with basis {e1, e2} and an indefinite inner product defined by [e1, e1] = 1, [e2, e2] = −1 and [e1, e2] = 0. if we take y = span{e1}, then it is a maximal uniformly positive definite subspace and hence there exists a fundamental decomposition of k with k+ = y and k− = span{e2}. choosing k+n = span{(n, 1)} and k−n = span{(1, n)} where n > 1, we get several fundamental decompositions. the corresponding fundamental symmetries jn are given by jn = ( n2+1 n2−1 −2n n2−1 2n n2−1 −(n2+1) n2−1 ) . here we can see that the fundamental symmetries jn satisfy j2n = in, [jnf , g] = [f , jng] and [jnf , jng] = [f , g] for all f , g ∈ k. 3. frame operator for frames in krein spaces let k be a krein space and let {fn}n∈i be a sequence in k. in relation to the sequence {fn}n∈i ,the index set i is decomposed as i+ = {n ∈ i : [fn, fn] ≥ 0} and i− = {n ∈ i : [fn, fn] < 0}. it iseasy to observe that `2(i) is the orthogonal direct sum of `2(i+) and `2(i−). definition 3.1. a sequence {fn}n∈i in a krein space k is called a bessel sequence if there exists a constant b > 0 such that ∑ n∈i |[fn, f ]|2 ≤ b‖f ‖2, for all f ∈ k. (6) the constant b in the inequality (6) is called a bessel bound for {fn}n∈i . theorem 3.1. let {fn}n∈i be a sequence in a krein space k. then {fn}n∈i is a bessel sequence with a bessel bound b if and only if the operators t+ : `2(i+) −→ k+ defined by t+{cn}n∈i+ =∑ n∈i+ cnfn and t− : `2(i−) −→ k− defined by t−{cn}n∈i− = ∑ n∈i− cnfn are well defined bounded operators and ‖t‖ ≤ √ b, where t = t+ + t−. proof. suppose first that {fn}n∈i is a bessel sequence with a bessel bound b. let {cn}n∈i+ ∈ `2(i+)and {cn}n∈i− ∈ `2(i−). let `,m ∈ i− such that ` > m. then∥∥∥∥∥∑̀ n=1 cnfn − m∑ n=1 cnfn ∥∥∥∥∥ = ∥∥∥∥∥ ∑̀ n=m+1 cnfn ∥∥∥∥∥ = sup ‖g‖=1 |[ ∑̀ n=m+1 cnfn, g]| ≤ sup ‖g‖=1 ∑̀ n=m+1 |[cnfn, g]| https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 5 ≤ ( ∑̀ n=m+1 |cn|2 ) 1 2 sup ‖g‖=1 ( ∑̀ n=m+1 |[fn, g]|2 ) 1 2 ≤ √ b ( ∑̀ n=m+1 |cn|2 ) 1 2 . since {cn}n∈i− ∈ `2(i−), {∑`n=1 |cn|2} is a cauchy sequence in c. the above calculation showsthat {∑`n=1 cnfn}`∈i− is a cauchy sequence in k−, and so it is convergent. hence t− is welldefined. with similar arguments and by considering `,m ∈ i+ with ` > m one may prove that { ∑` n=1 cnfn}`∈i+ is a cauchy sequence in k+ and so it is convergent. thus t− and t+ arewell defined and boundedness follows from the above calculation. clearly t− and t+ are linear.conversely, suppose that t− and t+ are well defined bounded linear operators with their adjoints t−∗ : k− −→ `2(i−) and t+∗ : k+ −→ `2(i+) defined by t−∗f = {−[f , fn]}n∈i− and t+∗f = {[f , fn]}n∈i+ respectively. since the adjoint of a bounded operator is bounded, ‖t−∗‖ = ‖t−‖ and ‖t+∗‖ = ‖t+‖. also we have ‖t+∗f ‖2 ≤ ‖t+‖2‖f ‖2, for all f ∈ k+ and ‖t−∗f ‖2 ≤ ‖t−‖2‖f ‖2, for all f ∈ k−.so ‖t ∗f ‖2 ≤ ‖t‖2‖f ‖2, for all f ∈ k. hence {fn}n∈i is a bessel sequence. � corollary 3.2. let {fn}n∈i be a sequence in a krein space k such that both ∑ n∈i+ cnfn and∑ n∈i− cnfn are convergent for all {cn}n∈i− ∈ `2(i−) and {cn}n∈i+ ∈ `2(i+). then {fn}n∈i is a bessel sequence. the condition (6) remains unchanged regardless how the elements of {fn}n∈i are numbered. corollary 3.3. let {fn}n∈i be a bessel sequence in a krein space k. then ∑ n∈i+ cnfn and∑ n∈i− cnfn converge unconditionally for all {cn}n∈i+ ∈ `2(i+) and {cn}n∈i− ∈ `2(i−) respectively. the following example illustrates that how the norm of a single element actually depends uponthe choice of fundamental decomposition. if a frame is defined relative to fundamental decom-position, then frame bounds will vary arbitrarily when difference fundamental decompositions areconsidered. example 3.2. consider the two dimensional minkowski space k = r2 with the inner product [f , g] = f1g1−f2g2 where f = (f1, f2), g = (g1, g2) ∈ r2. consider the fundamental decompositions with k+n = span{(n+1n , n−1 n )} and k−n = span{(n−1n , n+1 n )} where n > 1. then we get ‖f ‖2jn = 1 4 [(2n + 2/n)(f 21 + f 2 2 ) + 4f1f2(1/n − n)]. let f = (1, 1) and g = (1, 0). then ‖f ‖2jn = 2 n and ‖g‖2jn = 1 2(n + 1 n ). https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 6let {fn}n∈i be a bessel sequence in k. then both {fn}n∈i+ ⊂ k+ and {fn}n∈i− ⊂ k− arebessel sequences. define t+ : `2(i+) −→ k+ and t− : `2(i−) −→ k− by t+{cn} = ∑ n∈i+ cnfn and t−{cn} = ∑ n∈i− cnfn respectively. then t+ and t− are both bounded linear operators and are called synthesis opera-tors. define t+∗ : k+ −→ `2(i+) and t−∗ : k− −→ `2(i−) by t+∗f = {[f , fn]}n∈i+ and t−∗f = {[f , fn]}n∈i−are called analysis operators. thus s+ = t+t+∗ : k+ −→ k+ given by s+f = t+t+∗f =∑ n∈i+ [f , fn]fn and s− = t−t−∗ : k− −→ k− given by s−f = t−t−∗f = ∑ n∈i− [f , fn]fn.therefore the frame operator s = s+ + s− : k −→ k is defined by sf = (s+ + s−)f = ∑ n∈i+ [f , fn]fn + ∑ n∈i− [f , fn]fn. esmeral et al. [5] have given a defintion of frame which involves fundamental symmetry of thekrein space k. as shown in the example 3.2, for sufficiently large values on n, j-norms of elementsof k can be too small or too large. thus we propose the following definition for frame in kreinspaces. definition 3.3. let {fn}n∈i be a bessel sequence in k. the sequence {fn}n∈i} is said to be a frame if the frame operator s = s+ + s− : k −→ k defined by sf = (s+ + s−)f = ∑ n∈i+ [f , fn]fn + ∑ n∈i− [f , fn]fn is a bounded positive invertible operator. the following is the frame decomposition theorem for the krein space k, which states that if {fn}n∈i is a frame for a krein space k, then every element can be written as a linear combinationof frame elements. theorem 3.4. let {fn}n∈i be a frame for a krein space k with the frame operator s. then f = ∑ n∈i+ [f , s+ −1 fn]fn + ∑ n∈i− [f , s− −1 fn]fn, for all f ∈ k, (7) and both the series converges unconditionally for all f ∈ k. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 7 proof. since the operator s is self adjoint and invertible, we have f = ss−1f = ∑ n∈i [s−1f , fn]fn = ∑ n∈i+ [s+ −1 f , fn]fn + ∑ n∈i− [s− −1 f , fn]fn = ∑ n∈i+ [f , s+ −1 fn]fn + ∑ n∈i− [f , s− −1 fn]fn since {fn}n∈i is a frame with [f , s+−1fn] ∈ `2(i+) and [f , s−−1fn]fn ∈ `2(i−), the unconditionalconvergence follows from corollary 3.3. � 4. frame sequences in krein spaces we begin this section with the following definitions for frame sequence definition 4.1. let k be a krein space. a sequence {fn}n∈i ∈ k is called a(a) frame sequence if it is a frame for [fn] = span{fn : n ∈ i}.(b) exact if removal of an arbitrary fn render the collection {fn} no longer a frame for the krein space k.(c) near exact if it can be made exact by removing finitely many elements from it. example 4.2. let {fn}n∈i be a sequence of unit orthonormal vectors in krein space k and {nk} be any infinite increasing subset of i. then {fnk} is a frame sequence. let {fn}n∈i be a frame for a krein space k and {nk} be any infinite increasing sequence in i .then {fnk} need not be a frame sequence. example 4.3. let {fn}n∈i be a sequence of unit orthonormal vectors in a krein space k. define a sequence {hn}n∈i ∈ k by hn = 1√ n fn, n ∈ i. let nk = nk−1 + (k − 1), k ∈ i and n0 = ±1. then {nk} is an infinite increasing sequence in i . define another sequence {gn}n∈i ∈ k by g1 = h1, gnk = gnk+1 = gnk+2 = · · · = gnk+1 − 1 = hk , k ≥ 2. then {gn}n∈i is a tight frame for k. but note that {gnk} = {hk} is not a frame sequence. the following theorem gives a necessary and sufficient condition for a existence of a subsequenceto be a frame for a krein space k. theorem 4.1. let {fn}n∈i be a frame for a krein space k and let {mk} and {nk} be two infinite increasing sequences in i with {m+k }∪{n + k } = i+ and {m−k }∪{n − k } = i−. if {fmk}mk∈i is a frame, then {fnk}nk∈i is a frame if and only if there exists a bounded linear operator t : `2(i) −→ `2(i) such that t = t++t−, where t+ : `2(i+) −→ `2(i+) defined by t+{[f +nk , f +]} = {[f +mk , f +]}, f + ∈ k+ and t− : `2(i−) −→ `2(i−) is defined by t−{[f −nk , f −]} = {[f −mk , f −]}, f − ∈ k−. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 8 proof. suppose that {fmk}mk∈i is a frame with lower frame bounds a and a′ . then∑ mk∈i+ |[fmk , f +]|2 = ∑ nk∈i+ ‖[t+{[f +nk , f +]}‖ ≤ ‖t+‖ ∑ nk∈i+ |[fnk , f +]|2. so, we have ∑ nk∈i+ |[fnk , f +]|2 ≥ ∑ mk∈i+ |[fmk , f +]|2 ‖t+‖ ≥ a ‖t‖‖f +‖2. similarly, we have ∑ nk∈i− |[fnk , f −]|2 ≥ a ′ ‖t‖‖f −‖2. hence {fnk}nk∈i is a frame for the krein space k.conversely, suppose that {fnk}nk∈i is a frame for the krein space k. then there exist operators t1 + : `2(i+) −→ k1 given by t1+{[f +nk , f +]} −→ f + and t ∗1 + : k1 −→ `2(i+) given by t ∗1 +f + = {[f +nk , f +]} and similarly there exist operators t1− : `2(i−) −→ k2 given by t1−{[f −nk , f −]} = f −and t−1 ∗ : k2 −→ `2(i−) given by t−1 ∗ f − = {[f −nk , f −]}. also, since {fmk}mk∈i is a frame forthe krein space k, there exist operators t+2 : `2(i+) −→ k1 given by t2+{[f +mk , f +]} = f + and t+2 ∗ : k1 −→ `2(i+) given by t+2 ∗f + = {[f +mk , f +]}.similarly, there exist operators t2− : `2(i−) −→ k2 given by t2−{[f −mk , f −]} = f − and t−2 ∗ : k2 −→ `2(i−) given by t−2 ∗f − = {[f −mk , f −]}. then t+ = t+2 ∗ t1 + : `2(i+) −→ `2(i+) is abounded linear operator such that t+{[f +nk , f +]} = {[f +mk , f +]}, f + ∈ k+ and t− = t−2 ∗ t1 − : `2(i−) −→ `2(i−) is a bounded linear operator such that t−{[f −nk , f −]} = {[f −mk , f −]}, f − ∈ k−. � next, we give a sufficient condition for two subsequences of a frame for k to be a frame sequence. theorem 4.2. let {fn}n∈i be a frame for a krein space k. let {mk} and {nk} be two infinite increasing sequences in i with {m+k } ∪ {n + k } = i+ and {m−k } ∪ {n − k } = i−. let k1 = [f +mk ] ∩ [f + nk ]. if k1 is a finite dimensional space, then {f +mk} and {f +nk } are frame sequences for k+. further, if k2 = [f −mk ] ∩ [f − nk ] is finite dimensional, then {f −mk} and {f −nk } are frame sequences for k−. proof. let {`+k } be a finite subsequence of {n+k } such that k1 = [f`k ]`k∈i+ . since k1 is finitedimensional, {f +`k } is a frame for k1. let a′ and b ′ be the frame bounds for {f +`k }. consider {fnk}nk∈i+ , let f + ∈ {fnk}nk∈i+ be any element. now, if f +[⊥]k1, then∑ n∈i+ [f +, fn] 2 = ∑ nk∈i+ [f +, fnk ] 2 ≥ a‖f +‖2. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 9also, if f + ∈ k1, then ∑ nk∈i+ [f +, fnk ] 2 ≥ ∑ `k∈i+ [f +, flk ] 2 ≥ a ′‖f +‖2. otherwise, we have f + = ∑ αk fnk = ∑ αi fn + ∑ αj fj , i ∈ {nk}\{`k}, j ∈ {`k} = (f +) ′ + (f +) ′′ ,where (f +)′ [⊥]k1 and (f +)′′ ∈ k1. thus ∑ nk∈i+ [f +, fnk ] 2 = ∑ [f +, fn] 2 + ∑ [f +, fj ] 2, i ∈ {nk}\{`k}, j ∈ {`k} = ∑ [(f +) ′ + (f +) ′′ , fn] 2 + ∑ [(f +) ′ + (f +) ′′ , fj ] 2 = ∑ [(f +) ′ , fn] 2 + ∑ [(f +) ′′ , fj ] 2 ≥ a‖(f +)′‖2 + a′‖(f +)′′‖ ≥ min {a 2 , a ′ 2 } ‖f +‖2. hence {fnk}nk∈i+ is a frame sequence for k+. similarly we can show that {fmk}mk∈i+ is a framesequence for k+. also, in a similar way, one can prove that {fnk}nk∈i− and {fmk}mk∈i− are framesequences for k−. � corollary 4.3. let {fn}n∈i be a frame for a krein space k. let {mk} and {nk} be two infinite increasing sequences in i with {m+k } ∪ {n + k } = i+ and {m−k } ∪ {n − k } = i−. let {fmk}mk∈i+ and {fnk}nk∈i+ be frames for [fmk ]mk∈i+ and [fnk ]nk∈i+ respectively and let {fmk}mk∈i− and {fnk}nk∈i− be frames for [fmk ]mk∈i− and [fnk ]nk∈i− respectively. if {g+i } = {fmk}mk∈i+∪{fnk}nk∈i+ and {g−i } = {fmk}mk∈i− ∪ {fnk}nk∈i− , then {g+i } and {g−i } are frame sequences. proof. the proof of the corollary follows from the theorem 4.2 and the fact that {fmk}mk∈i+ and {fnk}nk∈i+ are frames for the [fmk ]mk∈i+ and [fnk ]nk∈i+ respectively. � finally, we give a sufficient condition for the exactness of frames in a krein space k. theorem 4.4. let {fn}n∈i be a frame for a krein space (k, [., .]) with bounds a , a′ and b, b′ such that fn 6= 0, for all n ∈ i. if for every infinite increasing sequence {nk} ∈ i+ and {mk} ∈ i−, {fnk}nk∈i+ and {fmk}mk∈i− are frame sequences with bounds a, b and a′ , b′ respectively, then {fn}n∈i is an exact frame. https://doi.org/10.28924/ada/ma.4.1 eur. j. math. anal. 10.28924/ada/ma.4.1 10 proof. suppose on the contrary that {fn}n∈i is not an exact frame. then, there existsm ∈ i such that fm ∈ [fn], i 6= m. let {nk} be an increasing sequence in i , given by nk = k, k = 1, 2, 3, . . . , m−1and nk = k + 1, k = m,m + 1, . . . . since {fnk}nk∈i+ is a frame for k+ and {fnk}nk∈i− is a framefor k− with bounds a, b and a′ , b′ respectively, we have a‖f ‖2 ≤ ∑ n 6=m n∈i+ [f , fn] 2 ≤ b‖f ‖2, for all f ∈ k+ (8) and a ′‖f ‖2 ≤ ∑ n 6=m n∈i− |[f , fn]|2 ≤ b ′‖f ‖2, for all f ∈ k−. (9) since {fn}n∈i is a frame for the krein space (k, [., .]), by (8), we have [f , fm] = 0 for all f ∈ k+. inparticular, [fm, fm] = 0. this gives fm = 0. also, by (8), |[f −, fm]| = 0, for all f ∈ k−. in particular |[fm, fm]| = 0. this implies that fm = 0 which is a contradiction. hence {fnk}nk∈i is an exactframe. � acknowledgementsthe present work of the first author is partially supported by university grants commission(ugc), government of india. the present work of the second author is partially supported byscience and engineering research board (serb), government of india (reference number :tar/2022/000219). references [1] t. y. azizov and i. s. iokhvidov, linear operators in spaces with an indefinite metric, john wiley & sons, incorporated,1989.[2] j. bognár, indefinite inner product spaces, vol. 78, springer science & business media, 2012. https://doi.org/ 10.1007/978-3-642-65567-8.[3] p. g. casazza and g. kutyniok, finite frames: theory and applications, springer science & business media, 2012. https://doi.org/10.1007/978-0-8176-8373-3.[4] i. daubechies, a. grossmann and y. meyer, painless nonorthogonal expansions, j. math. phys. 27 (1986) 1271-1283. https://doi.org/10.1063/1.527388.[5] p. a. m. dirac, the physical interpretation of the quantum dynamics, proc. r. soc. lond. ser. a, 113 (1927) 621-641. https://doi.org/10.1098/rspa.1927.0012.[6] d. l. donoho and m. elad, optimally sparse representation in general (nonorthogonal) dictionaries via `1 mini-mization, proc. nat. acad. sci. 100 (2003) 2197-2202. https://doi.org/10.1073/pnas.0437847100.[7] r. j. duffin and a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952) 341-366. https://doi.org/10.1090/s0002-9947-1952-0047179-6.[8] j. giribet, a. maestripieri, f. m. pería and p. g. massey, on frames for krein spaces, j. math. anal. appl. 393 (2012)122-137. https://doi.org/10.1016/j.jmaa.2012.03.040.[9] d. han, k. kornelson, d. larson and e. weber, frames for undergraduates, vol. 40. american mathematical soc.,2007. https://doi.org/10.28924/ada/ma.4.1 https://doi.org/10.1007/978-3-642-65567-8 https://doi.org/10.1007/978-3-642-65567-8 https://doi.org/10.1007/978-0-8176-8373-3 https://doi.org/10.1063/1.527388 https://doi.org/10.1098/rspa.1927.0012 https://doi.org/10.1073/pnas.0437847100 https://doi.org/10.1090/s0002-9947-1952-0047179-6 https://doi.org/10.1016/j.jmaa.2012.03.040 1. introduction 2. preliminaries 3. frame operator for frames in krein spaces 4. frame sequences in krein spaces references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 12doi: 10.28924/ada/ma.3.12 coordination of classical and dynamic inequalities complying on time scales muhammad jibril shahab sahir department of mathematics and statistics, the university of lahore, lahore, pakistan correspondence: jibrielshahab@gmail.com abstract. in this research article, we present extensions of some classical inequalities such asschweitzer, pólya–szegö, kantorovich and greub–rheinboldt inequalities of fractional calculus ontime scales. to investigate generalizations of such types of classical inequalities, we use the timescales riemann–liouville type fractional integrals. we explore dynamic inequalities on delta calcu-lus and their symmetric nabla versions. a time scale is an arbitrary nonempty closed subset of thereal numbers. the theory of time scales is applied to combine results in one comprehensive form.the calculus of time scales unifies and extends continuous versions and their discrete and quantumanalogues. by using the calculus of time scales, results are presented in more general form. thishybrid theory is also widely applied on dynamic inequalities. 1. introduction the calculus of time scales was initiated by stefan hilger [11]. the three most popular examplesof calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e.,when t = r, t = n and t = qn0 = {qt : t ∈ n0} where q > 1. the time scales calculus is studiedas delta calculus, nabla calculus and diamond-α calculus. during the last two decades, manyresearchers investigated several dynamic inequalities [1–4, 7, 16–18]. the basic work on dynamicinequalities is done by ravi agarwal, george anastassiou, martin bohner, allan peterson, donalo’regan, samir saker and many other authors.there have been recent achievements of the theory and applications of dynamic inequalities ontime scales. from the theoretical point of view, the study provides a harmonious reconciliation andextension of commonly known differential, difference and quantum equations. moreover, it is animportant tool in many computational, biological, economical and numerical applications.in this paper, it is assumed that all considerable integrals exist and are finite and t is a timescale, a, b ∈ t with a < b and an interval [a, b]t means the intersection of a real interval with thegiven time scale. received: 22 aug 2021. key words and phrases. time scales; fractional riemann–liouville integral; schweitzer, pólya–szegö, kantorovichand greub–rheinboldt inequalities. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 22. preliminaries we need here basic concepts of delta calculus. the results of delta calculus are adopted frommonographs [7, 8].for t ∈ t, the forward jump operator σ : t→ t is defined by σ(t) := inf{s ∈ t : s > t}. the mapping µ : t → r+ 0 = [0,+∞) such that µ(t) := σ(t) − t is called the forward graininessfunction. the backward jump operator ρ : t→ t is defined by ρ(t) := sup{s ∈ t : s < t}. the mapping ν : t→ r+ 0 = [0,+∞) such that ν(t) := t − ρ(t) is called the backward graininessfunction. if σ(t) > t , we say that t is right-scattered, while if ρ(t) < t , we say that t is left-scattered. also, if t < supt and σ(t) = t , then t is called right-dense, and if t > inf t and ρ(t) = t , then t is called left-dense. if t has a left-scattered maximum m , then tk = t − {m},otherwise tk = t.for a function f : t→ r, the delta derivative f ∆ is defined as follows:let t ∈ tk . if there exists f ∆(t) ∈ r such that for all ε > 0, there is a neighborhood u of t ,such that |f (σ(t))− f (s)− f ∆(t)(σ(t)− s)| ≤ ε|σ(t)− s|,for all s ∈ u , then f is said to be delta differentiable at t , and f ∆(t) is called the delta derivativeof f at t .a function f : t→ r is said to be right-dense continuous (rd-continuous), if it is continuous ateach right-dense point and there exists a finite left-sided limit at every left-dense point. the setof all rd-continuous functions is denoted by crd(t,r).the next definition is given in [7, 8]. definition 2.1. a function f : t → r is called a delta antiderivative of f : t → r, provided that f∆(t) = f (t) holds for all t ∈ tk . then the delta integral of f is defined by∫ b a f (t)∆t = f (b)− f (a). the following results of nabla calculus are taken from [6–8].if t has a right-scattered minimum m, then tk = t − {m}, otherwise tk = t. a function f : tk → r is called nabla differentiable at t ∈ tk , with nabla derivative f ∇(t), if there exists f ∇(t) ∈ r such that given any ε > 0, there is a neighborhood v of t , such that |f (ρ(t))− f (s)− f ∇(t)(ρ(t)− s)| ≤ ε|ρ(t)− s|, for all s ∈ v . https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 3a function f : t→ r is said to be left-dense continuous (ld-continuous), provided it is continuousat all left-dense points in t and its right-sided limits exist (finite) at all right-dense points in t.the set of all ld-continuous functions is denoted by cld(t,r).the next definition is given in [6–8]. definition 2.2. a function g : t→ r is called a nabla antiderivative of g : t→ r, provided that g∇(t) = g(t) holds for all t ∈ tk . then the nabla integral of g is defined by∫ b a g(t)∇t = g(b)− g(a). the following definition is taken from [2, 4]. definition 2.3. for α ≥ 1, the time scale ∆-riemann–liouville type fractional integral for a function f ∈ crd is defined by iαa f (t) = ∫ t a hα−1(t, σ(τ))f (τ)∆τ, (1) which is an integral on [a, t)t, see [9] and hα : t × t → r, α ≥ 0 are the coordinate wiserd-continuous functions, such that h0(t, s) = 1, hα+1(t, s) = ∫ t s hα(τ, s)∆τ, ∀s, t ∈ t. (2) notice that i1 a f (t) = ∫ t a f (τ)∆τ, which is absolutely continuous in t ∈ [a, b]t, see [9]. the following definition is taken from [3, 4]. definition 2.4. for α ≥ 1, the time scale∇-riemann–liouville type fractional integral for a function f ∈ cld is defined by j αa f (t) = ∫ t a ĥα−1(t, ρ(τ))f (τ)∇τ, (3) which is an integral on (a, t]t, see [9] and ĥα : t × t → r, α ≥ 0 are the coordinate wiseld-continuous functions, such that ĥ0(t, s) = 1, ĥα+1(t, s) = ∫ t s ĥα(τ, s)∇τ, ∀s, t ∈ t. (4) notice that j 1 a f (t) = ∫ t a f (τ)∇τ, which is absolutely continuous in t ∈ [a, b]t, see [9]. https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 4we will generalize the following classical inequalities [13] by using the calculus of time scales.first we consider the inequality given by schweitzer [19] such that( 1 p p∑ k=1 xk )( 1 p p∑ k=1 1 xk ) ≤ (m +m)2 4mm , (5) where 0 < m ≤ xk ≤ m for k = 1, . . . , p.in the same paper, schweitzer has also shown that if functions y 7→ f (y) and y 7→ 1 f (y) areintegrable on [a, b] and 0 < m ≤ f (y) ≤ m on [a, b], then∫ b a f (y)dy ∫ b a 1 f (y) dy ≤ (m +m)2 4mm (b − a)2. (6) pólya and szegö [15] proved that( p∑ k=1 x2 k )( p∑ k=1 y2 k ) ( p∑ k=1 xkyk )2 ≤ √mn mn + √ mn mn 2 2 , (7) where 0 < m ≤ xk ≤ m and 0 < n ≤ yk ≤ n for k = 1, . . . , p.kantorovich [12] proved that( p∑ k=1 xky 2 k )( p∑ k=1 1 xk y2 k ) ≤ 1 4 (√ m m + √ m m )2( p∑ k=1 y2 k )2 , (8) where 0 < m ≤ xk ≤ m and yk ∈ r for k = 1, . . . , p, and he pointed out that inequality (8) is aparticular case of (7).greub and rheinboldt [10] proved that( p∑ k=1 x2 k z 2 k )( p∑ k=1 y2 k z 2 k ) ≤ (mn +mn)2 4mnmn ( p∑ k=1 xkykz 2 k )2 , (9) where 0 < m ≤ xk ≤ m , 0 < n ≤ yk ≤ n and zk ∈ r for k = 1, . . . , p with p∑ k=1 z2 k <∞. 3. main results in order to present our main results, first we give a simple proof for an extension of pólya–szegö’sinequality by using the time scale ∆-riemann–liouville type fractional integral. theorem 3.1. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 5 let α, β ≥ 1 and hα−1(., .), hβ−1(., .) > 0. then we have the following inequality iαa ((f1f2)(x)|w(x)|) iβa ((g1g2)(x)|w(x)|) iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ){ iαa (f1(x)|(wf )(x)|) iβa (g1(x)|(wg)(x)|) + iαa (f2(x)|(wf )(x)|) iβa (g2(x)|(wg)(x)|) }2 ≤ 1 4 . (10) proof. using the given conditions, for y , z ∈ [a, x ]t, ∀x ∈ [a, b]t, we have( f2(y) g1(z) − |f (y)| |g(z)| ) ≥ 0, and ( |f (y)| |g(z)| − f1(y) g2(z) ) ≥ 0, which imply that ( f1(y) g2(z) + f2(y) g1(z) ) |f (y)| |g(z)| ≥ |f (y)|2 |g(z)|2 + f1(y)f2(y) g1(z)g2(z) . multiplying both sides by g1(z)g2(z)|g(z)|2, we have f1(y)g1(z)|f (y)g(z)|+ f2(y)g2(z)|f (y)g(z)| ≥ g1(z)g2(z)|f (y)|2 + f1(y)f2(y)|g(z)|2. (11) multiplying both sides of (11) by hα−1(x, σ(y))|w(y)|hβ−1(x, σ(z))|w(z)| and double integratingover y and z from a to x , respectively, we have iαa (f1(x)|w(x)f (x)|) iβa (g1(x)|w(x)g(x)|) + iαa (f2(x)|w(x)f (x)|) iβa (g2(x)|w(x)g(x)|) ≥ iαa ( |w(x)||f (x)|2 ) iβa (g1(x)g2(x)|w(x)|) + iαa (f1(x)f2(x)|w(x)|) iβa ( |w(x)||g(x)|2 ) . (12) applying the am-gm inequality √ζη ≤ ζ+η 2 , ζ ≥ 0, η ≥ 0, the inequality (12) takes the form iαa (f1(x)|w(x)f (x)|) iβa (g1(x)|w(x)g(x)|) + iαa (f2(x)|w(x)f (x)|) iβa (g2(x)|w(x)g(x)|) ≥ 2 √ iαa (|w(x)||f (x)|2) iβa (g1(x)g2(x)|w(x)|) iαa (f1(x)f2(x)|w(x)|) iβa (|w(x)||g(x)|2). (13) inequality (13) directly yields inequality (10). the proof of theorem 3.1 is completed. � now, we give an extension of pólya–szegö’s inequality by using the time scale ∇-riemann–liouville type fractional integral. theorem 3.2. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). let α, β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality j αa ((f1f2)(x)|w(x)|)j βa ((g1g2)(x)|w(x)|)j αa ( |w(x)||f (x)|2 ) j βa ( |w(x)||g(x)|2 ){ j αa (f1(x)|(wf )(x)|)j βa (g1(x)|(wg)(x)|) + j αa (f2(x)|(wf )(x)|)j βa (g2(x)|(wg)(x)|) }2 ≤ 1 4 . (14) https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 6 proof. similar to the proof of theorem 3.1. � corollary 3.3. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α, β ≥ 1 and hα−1(., .), hβ−1(., .) > 0. then we have the following inequality iαa (|w(x)|) iβa (|w(x)|) iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ){ iαa (|(wf )(x)|) iβa (|(wg)(x)|) }2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (15) proof. putting f1 = m, f2 = m , g1 = n and g2 = n in theorem 3.1, we get the inequality (15). � corollary 3.4. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α, β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality j αa (|w(x)|)j βa (|w(x)|)j αa ( |w(x)||f (x)|2 ) j βa ( |w(x)||g(x)|2 ){ j αa (|(wf )(x)|)j βa (|(wg)(x)|) }2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (16) proof. similar to the proof of corollary 3.3. � remark 3.1. let t = r, α, β > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (10) reduces to iα0 ((f1f2)(x)) iβ0 ((g1g2)(x)) iα0 ( f 2(x) ) iβ0 ( g2(x) ){ iα0 ((f1f )(x)) iβ0 ((g1g)(x)) + iα0 ((f2f )(x)) iβ0 ((g2g)(x)) }2 ≤ 1 4 , (17) as given in [14]. theorem 3.5. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). let α, β ≥ 1 and hα−1(., .), hβ−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ) ≤ iαa ( f2(x) g1(x) |(wf g)(x)| ) iβa ( g2(x) f1(x) |(wf g)(x)| ) . (18) proof. using the given condition, for y ∈ [a, x ]t, ∀x ∈ [a, b]t, we have |f (y)|2 ≤ f2(y) g1(y) |f (y)g(y)|. multiplying both sides of the last inequality by hα−1(x, σ(y))|w(y)| and integrating over y from a to x , we have∫ x a hα−1(x, σ(y))|w(y)||f (y)|2∆y ≤ ∫ x a hα−1(x, σ(y)) f2(y) g1(y) |w(y)||f (y)g(y)|∆y . (19) https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 7the inequality (19) takes the form iαa ( |w(x)||f (x)|2 ) ≤ iαa ( f2(x) g1(x) |w(x)||f (x)g(x)| ) . (20) similarly, we have that iβa ( |w(x)||g(x)|2 ) ≤ iβa ( g2(x) f1(x) |w(x)||f (x)g(x)| ) . (21) multiplying (20) and (21), we get the desired inequality (18). � theorem 3.6. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). let α, β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality j αa ( |w(x)||f (x)|2 ) j βa ( |w(x)||g(x)|2 ) ≤ j αa ( f2(x) g1(x) |(wf g)(x)| ) j βa ( g2(x) f1(x) |(wf g)(x)| ) .(22) proof. similar to the proof of theorem 3.5. � corollary 3.7. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions. assume that there exist four positive constants m, m , n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α, β ≥ 1 and hα−1(., .), hβ−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ) iαa (|(wf g)(x)|) iβa (|(wf g)(x)|) ≤ mn mn . (23) proof. putting f1 = m, f2 = m , g1 = n and g2 = n in theorem 3.5, we get the desired inequality. � corollary 3.8. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions. assume that there exist four positive constants m, m , n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α, β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality j αa ( |w(x)||f (x)|2 ) j βa ( |w(x)||g(x)|2 ) j αa (|(wf g)(x)|)j βa (|(wf g)(x)|) ≤ mn mn . (24) proof. similar to the proof of corollary 3.7. � remark 3.2. let t = r, α > 0, α = β, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (23) reducesto iα0 ( f 2(x) ) iα0 ( g2(x) ){ iα0 (f (x)g(x)) }2 ≤ mn mn . (25) inequality (25) may be found in [5]. https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 8 theorem 3.9. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). let α ≥ 1 and hα−1(., .) > 0. then we have the following inequality iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) {iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}2 ≤ 1 4 . (26) proof. using the given conditions, for y ∈ [a, x ]t, ∀x ∈ [a, b]t, we have( f2(y) g1(y) − |f (y)| |g(y)| ) ≥ 0, and ( |f (y)| |g(y)| − f1(y) g2(y) ) ≥ 0. multiplying the last two inequalities, we have( f2(y) g1(y) − |f (y)| |g(y)| )( |f (y)| |g(y)| − f1(y) g2(y) ) ≥ 0, which implies ( f1(y) g2(y) + f2(y) g1(y) ) |f (y)| |g(y)| ≥ |f (y)|2 |g(y)|2 + f1(y)f2(y) g1(y)g2(y) . multiplying both sides by g1(y)g2(y)|g(y)|2, we have f1(y)g1(y)|f (y)g(y)|+ f2(y)g2(y)|f (y)g(y)| ≥ g1(y)g2(y)|f (y)|2 + f1(y)f2(y)|g(y)|2. (27) multiplying both sides of (27) by hα−1(x, σ(y))|w(y)| and integrating over y from a to x , we have iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|) ≥ iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) + iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) . (28) applying the am-gm inequality, we get iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|) ≥ 2 √ iαa (g1(x)g2(x)|w(x)||f (x)|2) iαa (f1(x)f2(x)|w(x)||g(x)|2). (29) analogously, we have that iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) ≤ 1 4 {iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}2 . (30) this directly yields the desired inequality (26). � https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 9 theorem 3.10. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) <∞ and 0 < g1(y) ≤ |g(y)| ≤ g2(y) <∞, (y ∈ [a, x ]t,∀x ∈ [a, b]t). let α ≥ 1 and ĥα−1(., .) > 0. then we have the following inequality j αa ( g1(x)g2(x)|w(x)||f (x)|2 ) j αa ( f1(x)f2(x)|w(x)||g(x)|2 ) {j αa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}2 ≤ 1 4 . (31) proof. similar to the proof of theorem 3.9. � remark 3.3. let t = r, α > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (26) reduces to iα0 ( g1(x)g2(x)f 2(x) ) iα0 ( f1(x)f2(x)g2(x) ){ iα0 ((f1(x)g1(x) + f2(x)g2(x)) f (x)g(x)) }2 ≤ 1 4 . (32) inequality (32) may be found in [14]. corollary 3.11. let w, f , g ∈ crd ([a, b]t,r− {0}) be ∆-integrable functions. assume that there exist four positive constants m, m , n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α ≥ 1 and hα−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iαa ( |w(x)||g(x)|2 ) {iαa (|w(x)||f (x)g(x)|)}2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (33) proof. putting f1 = m, f2 = m , g1 = n and g2 = n in theorem 3.9, we get the desired inequality(33). � corollary 3.12. let w, f , g ∈ cld ([a, b]t,r− {0}) be ∇-integrable functions. assume that there exist four positive constants m, m , n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a, x ]t, ∀x ∈ [a, b]t. let α ≥ 1 and ĥα−1(., .) > 0. then we have the following inequality j αa ( |w(x)||f (x)|2 ) j αa ( |w(x)||g(x)|2 ) {j αa (|w(x)||f (x)g(x)|)}2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (34) proof. similar to the proof of corollary 3.11. � remark 3.4. we have the following:(i) let α = 1, t = z, a = 1, x = b = p + 1, xk > 0, w(k) = wk = 1 xk , f (k) = xk for k = 1, . . . , p and n = g = n = 1. then inequality (33) reduces to inequality (5).(ii) let α = 1, t = r, x = b, 0 < m ≤ f (y) ≤ m < ∞, w(y) = 1 f (y) on [a, b] and n = g = n = 1. then inequality (33) reduces to inequality (6).(iii) let α = 1, t = z, a = 1, x = b = p + 1, w ≡ 1, f (k) = xk > 0 and g(k) = yk > 0 for k = 1, . . . , p. then inequality (33) reduces to inequality (7). https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 10(iv) let α = 1, t = z, a = 1, x = b = p + 1, xk > 0, yk ∈ r, w(k) = wk = 1 xk y2 k , f (k) = xk for k = 1, . . . , p and n = g = n = 1. then inequality (33) reduces to inequality (8).(v) let α = 1, t = z, a = 1, x = b = p + 1, zk ∈ r, w(k) = wk = z2 k , f (k) = xk > 0 and g(k) = yk > 0 for k = 1, . . . , p. then inequality (33) reduces to inequality (9). 4. conclusion the subject of dynamic inequalities on time scales has become a crucial field of pure and appliedmathematics. many researchers developed interesting results concerning fractional calculus on timescales. due to utility of dynamic inequalities in many branches of mathematics, this field is given aprominent importance. this field has a wide scope. recently, interesting results have obtained byusing specht’s ratio and kantorovich’s ratio on time scales as given in [18]. by using these ratios,we can explore further results.dynamic inequalities may be extended by applying other techniques such as diamond-α inte-gral, which is defined as a linear operator of delta and nabla integrals on time scales. quantumcalculus, α, β-symmetric quantum calculus, functional generalization, fractional derivatives and n-tuple diamond-alpha integral are some other developed techniques and we will continue them toinvestigate other dynamic inequalities in future research. references [1] r.p. agarwal, d. o’regan, s.h. saker, dynamic inequalities on time scales, springer, cham, switzerland, 2014. https://doi.org/10.1007/978-3-319-11002-8.[2] g.a. anastassiou, principles of delta fractional calculus on time scales and inequalities, math. comp. model. 52(2010) 556–566. https://doi.org/10.1016/j.mcm.2010.03.055.[3] g.a. anastassiou, foundations of nabla fractional calculus on time scales and inequalities, comp. math. appl. 59(2010) 3750–3762. https://doi.org/10.1016/j.camwa.2010.03.072.[4] g.a. anastassiou, integral operator inequalities on time scales, int. j. diff. equ. 7 (2012) 111–137.[5] a. anber, z. dahmani, new integral results using pólya–szegö inequality, acta comment. univ. tart. math. 17(2013) 171–178. https://doi.org/10.12697/acutm.2013.17.15.[6] d. anderson, j. bullock, l. erbe, a. peterson, h. tran, nabla dynamic equations on time scales, pan-amer. math.j. 13 (2003) 1–47.[7] m. bohner, a. peterson, dynamic equations on time scales, birkhäuser boston, inc., boston, ma, 2001.[8] m. bohner, a. peterson, advances in dynamic equations on time scales, birkhäuser boston, boston, ma, 2003. https://doi.org/10.1007/978-0-8176-8230-9.[9] m. bohner, h. luo, singular second-order multipoint dynamic boundary value problems with mixed derivatives, adv.diff. equ. (2006) 1–15. https://doi.org/10.1155/ade/2006/54989.[10] w. greub, w. rheinboldt, on a generalization of an inequality of l.v. kantorovich, proc. amer. math. soc. 10 (1959)407–415. https://doi.org/10.1090/s0002-9939-1959-0105028-3.[11] s. hilger, ein maβkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, ph.d. thesis, universität würzburg,1988. https://doi.org/10.28924/ada/ma.3.12 https://doi.org/10.1007/978-3-319-11002-8 https://doi.org/10.1016/j.mcm.2010.03.055 https://doi.org/10.1016/j.camwa.2010.03.072 https://doi.org/10.12697/acutm.2013.17.15 https://doi.org/10.1007/978-0-8176-8230-9 https://doi.org/10.1155/ade/2006/54989 https://doi.org/10.1090/s0002-9939-1959-0105028-3 eur. j. math. anal. 10.28924/ada/ma.3.12 11 [12] l.v. kantorovich, functional analysis and applied mathematics (russian), uspehi mat. nauk (n.s.). 3 (1948) 89–185(in particular, pp. 142–144) [also translated from russian into english by c.d. benster, nat. bur. standards rep.no. 1509. 1952, 202 pp. (in particular, pp. 106–109)].[13] d.s. mitrinović, analytic inequalities, springer-verlag, berlin, 1970. https://doi.org/10.1007/ 978-3-642-99970-3.[14] s.k. ntouyas, p. agarwal, j. tariboon, on pólya–szegö and chebyshev types inequalities involving the riemann–liouville fractional integral operators, j. math. ineq. 10 (2016) 491–504. https://doi.org/10.7153/jmi-10-38.[15] g. pólya, g. szegö, aufgaben und lehrsätze aus der analysis, berlin. 1 (1925) 213–214. https://doi.org/10. 1007/978-3-662-38381-0.[16] m.j.s. sahir, formation of versions of some dynamic inequalities unified on time scale calculus, ural math. j. 4(2018) 88–98. https://doi.org/10.15826/umj.2018.2.010.[17] m.j.s. sahir, symmetry of classical and extended dynamic inequalities unified on time scale calculus, turk. j. ineq.2 (2018) 11–22.[18] m.j.s. sahir, parity of classical and dynamic inequalities magnified on time scales, bull. int. math. virtual inst. 10(2020) 369–380.[19] p. schweitzer, an inequality concerning the arithmetic mean (hungarian), math. phys. lapok. 23 (1914) 257–261. https://doi.org/10.28924/ada/ma.3.12 https://doi.org/10.1007/978-3-642-99970-3 https://doi.org/10.1007/978-3-642-99970-3 https://doi.org/10.7153/jmi-10-38 https://doi.org/10.1007/978-3-662-38381-0 https://doi.org/10.1007/978-3-662-38381-0 https://doi.org/10.15826/umj.2018.2.010 1. introduction 2. preliminaries 3. main results 4. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 86-105doi: 10.28924/ada/ma.1.86 some properties on the [p,q]-order of meromorphic solutions of homogeneous and non-homogeneous linear differential equations with meromorphic coefficients mansouria saidani, benharrat belaïdi∗ department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria saidaniman@yahoo.fr, benharrat.belaidi@univ-mosta.dz ∗correspondence: benharrat.belaidi@univ-mosta.dz abstract. in the present paper, we investigate the [p, q]-order of solutions of higher order lineardifferential equations ak (z) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = 0 and ak (z) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = f (z) ,where a0 (z) , a1 (z) , ..., ak (z) 6≡ 0 and f (z) 6≡ 0 are meromorphic functions of finite [p, q]-order.we improve and extend some results of the authors by using the concept [p, q]-order. 1. introduction and main results in this paper, we assume that the reader is familiar with the fundamental results and the standardnotations of the nevanlinna’s value distribution theory of meromorphic functions (see [7] , [9] , [14] , [24]) . in addition, for any integers p ≥ q ≥ 1 and a meromorphic function f in the whole complexplane, we will use ρ[p,q] (f ) , µ[p,q] (f ) to denote respectively the [p, q]-order and the lower [p, q]-order, λ[p,q] (f − a) (or λ[p,q] (f − a)) to denote the [p, q]-convergence exponent of the sequence ofdistinct a-points (or of a-points) and λ[p,q] (1f ) to denote the [p, q]-exponent of convergence of thepoles, we refer the reader to see [12] , [15] , [16] and [25] . in particular for q = 1, ρ[p,1] (f ) = ρp (f )is the iterated p-order, µ[p,1] (f ) = µp (f ) is the iterated lower p-order, λ[p,1] (f − a) = λp (f , a)(or λ[p,1] (f − a) = λp (f , a)) is the iterated convergence exponent of the sequence of distinct a-points (or of a-points), λ[p,1] (1f ) = λp ( 1 f ) is the iterated exponent of convergence of the poles, see [7] , [11] , [13] , [14] and [24] for notations and definitions. received: 6 sep 2021. key words and phrases. linear differential equations; meromorphic functions; [p, q]-order; [p, q]-exponent of conver-gence of zeros. 86 https://adac.ee https://doi.org/10.28924/ada/ma.1.86 https://orcid.org/0000-0002-6635-2514 eur. j. math. anal. 1 (2021) 87 several authors have investigated the growth of solutions of second order and higher orderhomogeneous and non-homogeneous linear differential equations with analytic, entire or meromor-phic coefficients, see ([1−3], [6], [8], [11], [13−16], [18] , [20−21], [23], [25]). in the recent years,many authors have studied the complex linear differential equations f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = 0, (1.1) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = f (z) , (1.2)where a0 (z) 6≡ 0, a1 (z) , ..., ak−1 (z) and f (z) 6≡ 0 are meromorphic functions of finite iterated p-order. in [2] , belaïdi considered the growth of meromorphic solutions of equations (1.1) and (1.2) with meromorphic coefficients of finite iterated p−order and obtained some results whichimprove and generalize some previous results. theorem a ([2]) let h ⊂ [0,+∞) be a set with a positive upper density, and let aj (z) (j = 0, 1, ..., k − 1) be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0, α > 0 such that ρ = max { ρp ( aj ) : j = 1, ..., k − 1 } < σ and |a0 (z) | ≥ expp (αrσ) as |z | = r ∈ h, r → +∞, then every meromorphic solution f 6≡ 0 of equation (1.1) satisfies µp (f ) = ρp(f ) = +∞, ρp+1(f ) ≥ σ. furthermore, if λp ( 1 f ) <∞, then i (f ) = p + 1 and σ ≤ ρp+1 (f ) ≤ ρp (a0) . theorem b ([2]) let h ⊂ [0,+∞) be a set with a positive upper density, and let aj (z) (j = 0, 1, ..., k − 1) and f (z) 6≡ 0 be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0, α > 0 such that |a0 (z) | ≥ expp (αrσ) as |z | = r ∈ h, r → +∞, and ρ = max { ρp ( aj ) (j = 1, ..., k − 1), ρp (f ) } < σ, then every meromorphic solution of equation (1.2) with λp ( 1 f ) < σ satisfies λp (f ) = λp(f ) = ρp(f ) =∞, λp+1 (f ) = λp+1(f ) = ρp+1(f ). furthermore, if λp ( 1 f ) < min {µp (f ) , σ} , then i (f ) = p + 1 and λp+1 (f ) = λp+1(f ) = ρp+1 (f ) ≤ ρp (a0) . recently, in [18] the authors have studied the growth of solutions of the equations (1.1) and (1.2) when as(z) to dominate all other coefficients and they got some results about ρp+1 (f ) asfollows. theorem c ([18]) let h ⊂ (1,+∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ..., k − 1) be meromorphic functions with finite iterated p-order. eur. j. math. anal. 1 (2021) 88 if there exist positive constants σ > 0, α > 0 and an integer s, 0 ≤ s ≤ k − 1, such that |as (z) | ≥ expp (αrσ) as |z | = r ∈ h, r → +∞, and ρ = max { ρp ( aj ) (j 6= s) } < σ, then every non-transcendental meromorphic solution f 6≡ 0 of (1.1) is a polynomial with deg f ≤ s − 1 and every transcendental meromorphic solution f of (1.1) with λp ( 1 f ) < µp (f ) satisfies i (f ) = p+ 1 µp (f ) = ρp(f ) = +∞ and σ ≤ ρp+1 (f ) ≤ ρp (as) . theorem d ([18]) let h ⊂ (1,+∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ..., k−1) and f (z) 6≡ 0 be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0, α > 0 and an integer s, 0 ≤ s ≤ k − 1, such that |as (z) | ≥ expp (αrσ) as |z | = r ∈ h, r → +∞, and max { ρp ( aj ) (j 6= s), ρp (f ) } < σ, then every non-transcendental meromorphic solution f of (1.2) is a polynomial with deg f ≤ s − 1 and every transcendental meromorphic solution f of (1.2) with λp ( 1 f ) < min {σ, µp(f )} satisfies i (f ) = p + 1 λp (f ) = λp(f ) = ρp(f ) = µp (f ) = +∞ and σ ≤ λp+1 (f ) = λp+1(f ) = ρp+1 (f ) ≤ ρp (as) .thus, the following question arises: can we have the same properties as in theorems c andd for the solutions of equations ak (z) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = 0 (1.3) and ak (z) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = f (z) , (1.4)when the coefficients aj (j = 0, 1, ..., k) are of [p, q]−order? in this paper, we proceed this wayand we obtain the following results. theorem 1.1 let h ⊂ (1,+∞) be a set with a positive upper logarithmic density (or ml (h) = +∞) and let aj (z) (j = 0, 1, ..., k) with ak (z) 6≡ 0 be meromorphic functions with finite [p, q]order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ − ε) logq r } as |z | = r ∈ h, r → +∞ and ρ = max { ρ[p,q] ( aj ) (j 6= s) } < σ, then every non-transcendental meromorphic solution f 6≡ 0 of (1.3) is a polynomial with deg f ≤ s−1 and every transcendental meromorphic solution f of (1.3) with λ[p,q] ( 1 f ) < µ[p,q] (f ) satisfies ρ[p,q](f ) = µ[p,q] (f ) = +∞, σ ≤ ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . remark 1.1 putting ak (z) ≡ 1 and q = 1 in theorem 1.1, we obtain theorem c. eur. j. math. anal. 1 (2021) 89 corollary 1.1 under the hypotheses of theorem 1.1, suppose further that ϕ is a transcendental meromorphic function satisfying ρ[p+1,q] (ϕ) < σ. then, every transcendental meromorphic solution f of equation (1.3) with λ[p,q] ( 1 f ) < µ[p,q] (f ) satisfies σ ≤ λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = ρ[p+1,q] (f − ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . considering the non-homogeneous linear differential equation (1.4), we obtain the followingresults. theorem 1.2 let h ⊂ (1,+∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ..., k) with ak (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions with finite [p, q]-order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ − ε) logq r } as |z | = r ∈ h, r → +∞ and max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) } < σ, then every non-transcendental meromorphic solution f of (1.4) is a polynomial with deg f ≤ s − 1 and every transcendental meromorphic solution f of (1.4) with λ[p,q] ( 1 f ) < min { σ, µ[p,q](f ) } satisfies λ[p,q] (f ) = λ[p,q](f ) = ρ[p,q](f ) = µ[p,q] (f ) = +∞ and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . remark 1.2 putting ak (z) ≡ 1 and q = 1 in theorem 1.2, we obtain theorem d. corollary 1.2 let aj (z) (j = 0, 1, ..., k) , f (z) , h satisfy all the hypotheses of theorem 1.2, and let ϕ be a transcendental meromorphic function satisfying ρ[p+1,q] (ϕ) < σ. then, every transcendental meromorphic solution f with λ[p,q] ( 1 f ) < min{σ, µ[p,q] (f )} of equation (1.4) satisfies σ ≤ λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = ρ[p+1,q] (f − ϕ) ≤ ρ[p,q] (as) . remark 1.3 in [17, 19] , the authors have studied the growth and the oscillation of solutionsof equations (1.3) and (1.4) when the coefficients aj (z) (j = 0, 1, ..., k) and f (z) are entirefunctions of iterated p-order or of [p, q]-order. however, in the present paper the coefficients aj (z) (j = 0, 1, ..., k) and f (z) are meromorphic functions with reduction of the hypotheses in theorems1.1 and 1.2. so, this article may be understood as an extension and an improvement of [17, 19] . eur. j. math. anal. 1 (2021) 902. some auxiliary lemmas in order to prove our theorems, we need the following definition, proposition and lemmas. thelebesgue linear measure of a set e ⊂ [0,+∞) is m (e) = ∫ e dt, and the logarithmic measure of a set f ⊂ [1,+∞) is ml (f ) = ∫ f dt t . the upper density of e ⊂ [0,+∞) is given by dens (e) = lim sup r→∞ m (e ∩ [0, r ]) rand the upper logarithmic density of the set f ⊂ [1,+∞) is defined by log dens (f ) = lim sup r−→+∞ ml (f ∩ [1, r ]) log r . proposition 2.1 ([2]) for all h ⊂ (1,+∞) the following statements hold: (i) if ml (h) = +∞, then m (h) = +∞; (ii) if dens (h) > 0, then m (h) = +∞; (iii) if log dens (h) > 0, then ml (h) = +∞. lemma 2.1 ([5]) let f be a transcendental meromorphic function in the plane, and let α > 1 be a given constant. then, there exist a set e1 ⊂ (1,+∞) that has a finite logarithmic measure, and a constant b > 0 depending only on α and (i , j) ((i , j) positive integers with i > j) such that for all z with |z | = r 6∈ [0, 1] ∪ e1, we have∣∣∣∣∣ f (i)(z) f (j)(z) ∣∣∣∣∣ ≤ b ( t (αr, f ) r (logα r) logt (αr, f ) )i−j . lemma 2.2 ([4]) let p ≥ q ≥ 1 be integers and g be an entire function such that ρ[p,q] (g) < +∞. then, there exist entire functions u(z) and v(z) such that g (z) = u(z)ev(z), ρ[p,q] (g) = max { ρ[p,q] (u) , ρ[p,q] ( ev(z) )} and ρ[p,q] (u) = lim sup r→+∞ logp n ( r, 1g ) logq r . moreover, for any given ε > 0, we have |u(z)| ≥ exp { − expp {( ρ[p,q] (u) + ε ) logq r }} (r /∈ e2) , where e2 ⊂ (1,+∞) is a set of r of finite linear measure. eur. j. math. anal. 1 (2021) 91 lemma 2.3 let p ≥ q ≥ 1 be integers. suppose that f is a meromorphic function such that ρ[p,q] (f ) < +∞. then, there exist entire functions u1 (z) , u2 (z) and v (z) such that f (z) = u1 (z) ev(z) u2 (z) (2.1) and ρ[p,q](f ) = max { ρ[p,q](u1), ρ[p,q](u2), ρ[p,q](e v(z)) } . (2.2) moreover, for any given ε > 0, we have exp { − expp { (ρ(p,q) (f ) + ε) logq r }} ≤ |f (z)| ≤ expp+1 { (ρ(p,q) (f ) + ε) logq r } (r /∈ e3) , (2.3) where e3 ⊂ (1,+∞) is a set of r of finite linear measure. proof. when p ≥ q = 1, the lemma is due to tu and long [21]. thus, we assume that p > q > 1 or p = q > 1. by hadamard factorization theorem, we can write f as f (z) = g(z) d(z) , where g (z) and d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) < +∞ and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q] ( 1 f ) < µ. by lemma 2.2, there exist entire functions u(z) and v(z) such that g (z) = u(z)ev(z), ρ[p,q] (g) = max { ρ[p,q] (u) , ρ[p,q] ( ev(z) )} . so, there exist entire functions u(z), v(z) and d (z) such that f (z) = u(z)ev(z) d (z)and ρ[p,q](f ) = max { ρ[p,q] (u) , ρ[p,q](d), ρ[p,q] ( ev(z) )} . thus (2.1) and (2.2) hold. set f (z) = u1(z)e v(z) u2(z) , where u1 (z) , u2 (z) are the canonical productsformed with the zeros and poles of f respectively. by the definition of [p, q]-order, for sufficientlylarge r and any given ε > 0, we have |u1 (z)| ≤ expp+1 {( ρ[p,q] (u1) + ε 3 ) logq r } , |u2 (z)| ≤ expp+1 {( ρ[p,q] (u2) + ε 3 ) logq r } . (2.4) since max { ρ[p,q](u1), ρ[p,q](u2), ρ[p,q](e v(z)) } = ρ[p,q](f ), then we obtain |u1 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } , (2.5) |u2 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } , (2.6) eur. j. math. anal. 1 (2021) 92∣∣∣ev(z)∣∣∣ ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } . (2.7) by lemma 2.2, there exists a set e3 ⊂ (1,+∞) of r with a finite linear measure such that for anygiven ε > 0, we have |u1 (z)| ≥ exp { − expp {( ρ[p,q] (u1) + ε 3 ) logq r }} ≥ exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} , (r /∈ e3) , (2.8) |u2 (z)| ≥ exp { − expp {( ρ[p,q] (u2) + ε 3 ) logq r }} ≥ exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} , (r /∈ e3) . (2.9) then, by using (2.5) , (2.7) and (2.9), we obtain for sufficiently large r /∈ e3 and any given ε > 0 |f (z)| = |u1 (z)| ∣∣ev(z)∣∣ |u2 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} ≤ expp+1 {( ρ[p,q] (f ) + ε ) logq r } . (2.10) on the other hand, we have ρ[p−1,q] (v) = ρ[p,q] ( ev(z) ) ≤ ρ[p,q] (f ) and ∣∣ev(z)∣∣ ≥ e−|v(z)|. makinguse of the definition of [p, q]-order, we obtain |v (z)| ≤ m(r, v) ≤ expp {( ρ(p−1,q) (v) + ε 3 ) logq r } ≤ expp {( ρ[p,q] (f ) + ε 3 ) logq r } . then, for sufficiently large r and any given ε > 0, we have∣∣∣ev(z)∣∣∣ ≥ e−|v(z)| ≥ exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} . (2.11) by (2.6) , (2.8) and (2.11), we can easily obtain |f (z)| = |u1 (z)| ∣∣ev(z)∣∣ |u2 (z)| ≥ exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} exp { − expp {( ρ[p,q] (f ) + ε 3 ) logq r }} expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } . = exp { −3 expp {( ρ[p,q] (f ) + ε 3 ) logq r }} ≥ exp { − expp {( ρ[p,q] (f ) + ε ) logq r }} . thus, we complete the proof of lemma 2.3. lemma 2.4 under the assumptions of theorem 1.1 or theorem 1.2, we have ρ[p,q] (as) = β ≥ σ. eur. j. math. anal. 1 (2021) 93 proof. assume that ρ[p,q] (as) = β < σ. according to the hypotheses of theorems 1.1 or 1.2, thereexists a positive constant σ > 0 such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ − ε) logq r } (2.12) as |z | = r ∈ h, r → +∞, where h ⊂ (1,+∞) is a set with a positive upper logarithmic density (by proposition 2.1, we have ml (h) = +∞). by lemma 2.3, we can find a set e3 ⊂ (1,+∞) thathas finite linear measure (and so of finite logarithmic measure) such that when |z | = r /∈ e3, wehave for any given ε (0 < 2ε < σ − β) |as (z) | ≤ expp+1 { (β + ε) logq r } . (2.13) by (2.12) and (2.13) , we obtain for |z | = r ∈ h r e3, r → +∞ expp+1 { (σ − ε) logq r } ≤ |as (z) | ≤ expp+1 { (β + ε) logq r } and by ε (0 < 2ε < σ − β) this is a contradiction. hence ρ[p,q] (as) = β ≥ σ. lemma 2.5 (wiman-valiron, [10] , [22]) let f be a transcendental entire function, and let z be a point with |z | = r at which |f (z)| = m (r, f ). then the estimation f (j) (z) f (z) = ( νf (r) z )j (1 + o (1)) (j ≥ 1 is an integer) holds for all |z | outside a set e4 of r of finite logarithmic measure, where νf (r) is the central index of f . lemma 2.6 ([12]) let f be an entire function of [p, q]-order and let νf (r) be the central index of f . then ρ[p,q] (f ) = lim sup r→+∞ logp νf (r) logq r , µ[p,q] (f ) = lim inf r→+∞ logp νf (r) logq r . the following two lemmas were given in [4] without proof, so for the convenience of the reader,we prove them. lemma 2.7 let f (z) = g(z) d(z) be a meromorphic function, where g (z) , d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ and λ[p,q] (d) = ρ[p,q] (d) = β = λ[p,q] ( 1 f ) < µ. then, there exists a set e5 ⊂ (1,+∞) of finite logarithmic measure such that for all |z | = r /∈ [0, 1] ∪ e5 and |g (z) | = m (r, g) , we have f (n) (z) f (z) = ( νg (r) z )n (1 + o (1)) , n ∈ n, where νg (r) denote the central index of g. eur. j. math. anal. 1 (2021) 94 proof. by mathematical induction, we obtain f (n) = g(n) d + n−1∑ j=0 g(j) d ∑ (j1...jn) cj j1...jn ( d ′ d )j1 × · · · × ( d (n) d )jn , (2.14) where cj j1...jn are constants and j + j1 + 2j2 + · · ·+ njn = n. hence f (n) f = g(n) g + n−1∑ j=0 g(j) g ∑ (j1...jn) cj j1...jn ( d ′ d )j1 × · · · × ( d (n) d )jn . (2.15) from lemma 2.5, there exists a set e4 ⊂ (1,+∞) with finite logarithmic measure such that for apoint z satisfying |z | = r /∈ e4 and |g (z)| = m (r, g), we have g(j)(z) g(z) = ( νg (r) z )j (1 + o (1)) (j = 1, 2, ..., n) , (2.16) where νg (r) is the central index of g. substituting (2.16) into (2.15) yields f (n) (z) f (z) = ( νg (r) z )n [(1 + o (1)) + n−1∑ j=0 ( νg (r) z )j−n (1 + o (1)) ∑ (j1...jn) cj j1...jn ( d ′ d )j1 × · · · × ( d (n) d )jn . (2.17) since ρ[p,q] (d) = β < µ, then for any given ε (0 < 2ε < µ− β) and sufficiently large r , we have t (r, d) ≤ expp {( β + ε 2 ) logq r } by using lemma 2.1, for α = 2, there exist a set e1 ⊂ (1,+∞) with ml(e1) <∞ and a constant b > 0, such that for all z satisfying |z | = r /∈ [0, 1] ∪ e1, we have∣∣∣∣∣d (m) (z) d (z) ∣∣∣∣∣ ≤ b [t (2r, d)]m+1 ≤ b [ expp {( β + ε 2 ) logq (2r) }]m+1 ≤ expp { (β + ε) logq r }m , m = 1, 2, ..., n. (2.18)by lemma 2.6 and µ[p,q] (g) = µ[p,q] (f ) = µ, it follows that νg (r) > expp { (µ− ε) logq r } for sufficiently large r . thus, by using j1 + 2j2 + · · ·+ njn = n − j, we obtain∣∣∣∣∣∣ ( νg (r) z )j−n (d ′ d )j1 × · · · × ( d (n) d )jn ∣∣∣∣∣∣ ≤ [ expp { (µ− ε) logq r } r ]j−n × [ expp { (β + ε) logq r }]n−j = [ r expp { (β + ε) logq r } expp { (µ− ε) logq r } ]n−j → 0 (2.19) as r → +∞, where |z | = r /∈ [0, 1] ∪ e5, e5 = e1 ∪ e4 and |g (z)| = m (r, g) . from (2.17) and (2.19), we obtain our assertion. eur. j. math. anal. 1 (2021) 95 lemma 2.8 let f (z) = g(z) d(z) be a meromorphic function, where g (z), d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q] ( 1 f ) < µ. then, there exists a set e6 ⊂ (1,+∞) of finite logarithmic measure such that for all |z | = r /∈ [0, 1] ∪ e6 and |g (z) | = m (r, g), we have∣∣∣∣ f (z) f (s) (z) ∣∣∣∣ ≤ r2s , (s ∈ n) . proof. by lemma 2.7, there exists a set e5 of finite logarithmic measure such that the estimation f (s)(z) f (z) = ( νg (r) z )s (1 + o (1)) (s ≥ 1 is an integer) (2.20) holds for all |z | = r /∈ [0, 1]∪e5 and |g (z)| = m (r, g), where νg (r) is the central index of g. onthe other hand, by lemma 2.6, for any given ε (0 < ε < 1), there exists r > 1 such that for all r > r, we have νg (r) > expp { (µ− ε) logq (r) } . (2.21) if µ = +∞, then µ− ε can be replaced by a large enough real number m . set e6 = [1, r] ∪ e5, lm (e6) < +∞. hence from (2.20) and (2.21), we obtain∣∣∣∣ f (z) f (s) (z) ∣∣∣∣ = ∣∣∣∣ z νg (r) ∣∣∣∣s 1 |1 + o (1)| ≤ r s( expp { (µ− ε) logq (r) })s ≤ r2s , where |z | = r /∈ [0, 1] ∪ e6, r → +∞ and |g (z)| = m (r, g) . lemma 2.9 ([6]) let ϕ : [0,+∞)→ r and ψ : [0,+∞)→ r be monotone nondecreasing functions such that ϕ(r) ≤ ψ(r) for all r /∈ (e7 ∪ [0, 1]) , where e7 is a set of finite logarithmic measure. let α > 1 be a given constant. then, there exists an r1 = r1(α) > 0 such that ϕ(r) ≤ ψ(αr) for all r > r1. lemma 2.10 ([19]) let f (z) = g(z) d(z) be a meromorphic function, where g (z), d (z) are entire functions. if 0 ≤ ρ[p,q] (d) < µ[p,q] (f ) , then µ[p,q] (g) = µ[p,q] (f ) and ρ[p,q] (g) = ρ[p,q] (f ) . moreover, if ρ[p,q] (f ) = +∞, then ρ[p+1,q] (g) = ρ[p+1,q] (f ) . lemma 2.11 assume that k ≥ 2 and a0, a1, ..., ak 6≡ 0, f are meromorphic functions. let ρ = max { ρ[p,q] ( aj ) (j = 0, 1, ..., k), ρ[p,q] (f ) } < ∞ and let f be a meromorphic solution of infinite [p, q]-order of equation (1.4) with λ[p,q] ( 1 f ) < µ[p,q] (f ) . then, ρ[p+1,q](f ) ≤ ρ. proof. let f be a meromorphic solution of infinite [p, q]-order of equation (1.4) with λ[p,q] (1f ) < µ[p,q] (f ) . so, we can use hadamard factorization theorem and write f as f (z) = g(z) d(z) , where g(z)and d(z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ eur. j. math. anal. 1 (2021) 96and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q] ( 1 f ) < µ. by lemma 2.3, there exists a set e3 ⊂ (1,+∞)of r with a finite linear measure such that for all |z | = r /∈ e3 and any given ε (0 < 2ε < µ[p,q] (f )− ρ[p,q] (d)), we have |aj (z) | ≤ expp+1 { (ρ(p,q) ( aj ) + ε) logq r } ≤ expp+1 { (ρ+ ε) logq r } , j = 0, 1, ..., k − 1, (2.22) |ak (z) | ≥ exp { − expp { (ρ(p,q) (ak) + ε) logq r }} ≥ exp { − expp { (ρ+ ε) logq r }} (2.23)and |f (z)| ≤ expp+1 { (ρ(p,q) (f ) + ε) logq r } ≤ expp+1 { (ρ+ ε) logq r } . (2.24)by (2.24), for all z satisfying |z | = r /∈ e3 at which |g(z)| = m(r, g) and any given ε ( 0 < 2ε < µ[p,q] (f )− ρ[p,q] (d) ) , we obtain∣∣∣∣f (z) f (z) ∣∣∣∣ = |f (z)| |g(z)| |d (z)| ≤ expp+1 { (ρ[p,q] (d) + ε) logq r } expp+1 { (ρ+ ε) logq r } expp+1 { (µ[p,q] (f )− ε) logq r } ≤ expp+1 { (ρ+ ε) logq r } . (2.25)by lemma 2.7, there exists a set e5 ⊂ (1,+∞) of finite logarithmic measure such that for all |z | = r /∈ [0, 1] ∪ e5 and |g (z) | = m (r, g) , we have f (j) (z) f (z) = ( νg (r) z )j (1 + o (1)) , j = 1, ..., k. (2.26) we can rewrite (1.4) as∣∣∣∣∣ f (k) (z) f (z) ∣∣∣∣∣ ≤ 1 |ak (z) | |a0 (z) |+ ∣∣∣∣f (z) f (z) ∣∣∣∣+ k−1∑ j=1 |aj (z) | ∣∣∣∣∣ f (j) (z) f (z) ∣∣∣∣∣  . (2.27) by substituting (2.22) , (2.23) , (2.25) and (2.26) into (2.27), we obtain∣∣∣∣νg (r) z ∣∣∣∣k |1 + o (1)| ≤ 1 exp { − expp { (ρ+ ε) logq r }}×1 + k−1∑ j=1 ∣∣∣∣νg (r) z ∣∣∣∣j |1 + o (1)|  expp+1 { (ρ+ ε) logq r } + expp+1 { (ρ+ ε) logq r }) = 2 + k−1∑ j=1 ∣∣∣∣νg (r) z ∣∣∣∣j |1 + o (1)|  exp { 2 expp { (ρ+ ε) logq r }} . hence |νg (r)| |1 + o (1)| ≤ (k + 1) r |1 + o (1)| exp { 2 expp { (ρ+ ε) logq r }} (2.28) eur. j. math. anal. 1 (2021) 97holds for all z satisfying |z | = r /∈ [0, 1] ∪ e3 ∪ e5 and |g (z) | = m (r, g) , r → +∞. by (2.28),we get lim sup r→+∞ logp+1 νg (r) logq r ≤ ρ+ ε. (2.29) since ε > 0 is arbitrary, by (2.29) and lemma 2.6, we obtain ρ[p+1,q] (g) ≤ ρ. since ρ[p,q] (d) < µ[p,q] (f ) , so by lemma 2.10, we have ρ[p+1,q] (g) = ρ[p+1,q] (f ) . thus, ρ[p+1,q] (f ) ≤ ρ. therefore,lemma 2.11 is proved. lemma 2.12 ([19]) let aj (z) (j = 0, 1, ..., k) , ak (z) (6≡ 0) , f (z) (6≡ 0) be meromorphic functions and let f be a meromorphic solution of (1.4) of infinite [p, q]-order satisfying the following condition b = max { ρ[p+1,q] (f ) , ρ[p+1,q] ( aj ) (j = 0, 1, ..., k) } < ρ[p+1,q] (f ) . then λ[p+1,q](f ) = λ[p+1,q](f ) = ρ[p+1,q] (f ) . lemma 2.13 let h ⊂ (1,+∞) be a set with a positive upper logarithmic density (or infinite logarithmic measure), and let aj (z) (j = 0, 1, ..., k) with ak (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions with finite [p, q]-order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ − ε) logq r } as |z | = r ∈ h, r → +∞ and max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) } < σ, then every transcendental meromorphic solution f of equation (1.4) satisfies ρ[p,q](f ) ≥ σ. proof. assume that f is a transcendental meromorphic solution of equation (1.4) with ρ[p,q](f ) < σ.from (1.4) , we have as = f f (s) − k∑ j=0 j 6=s aj f (j) f (s) . (2.30) since max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) } < σ and ρ[p,q] (f ) < σ, then from (2.30) we obtainthat ρ1 = ρ[p,q] (as) ≤ max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) , ρ[p,q] (f ) } < σ.by lemma 2.3, for any ε (0 < 2ε < σ − ρ1) , there exists a set e3 ⊂ (1,+∞) with a finite linearmeasure such that |as (z)| ≤ expp+1 { (ρ(p,q) (as) + ε) logq r } = expp+1 { (ρ1 + ε) logq r } (2.31) holds for all z satisfying |z | = r /∈ e3. from the hypotheses of lemma 2.13, there exists a set hwith log densh > 0 (or ml (h) = +∞) such that |as (z)| ≥ expp+1 { (σ − ε) logq r } (2.32) eur. j. math. anal. 1 (2021) 98holds for all z satisfying |z | = r ∈ h, r → +∞. by (2.31) and (2.32), we conclude that for all zsatisfying |z | = r ∈ h r e3, r → +∞, we have expp+1 { (σ − ε) logq r } ≤ expp+1 { (ρ1 + ε) logq r } and by ε (0 < 2ε < σ − ρ1) this is a contradiction as r → +∞. consequently, any transcendentalmeromorphic solution f of equation (1.4) satisfies ρ[p,q] (f ) ≥ σ. lemma 2.14 ([23]) let p ≥ q ≥ 1 be integers. let f be a meromorphic function for which ρ[p,q] (f ) = β < +∞, and let k ≥ 1 be an integer. then for any ε > 0, m ( r, f (k) f ) = o ( expp−1 { (β + ε) logq r }) , holds outside of a possible exceptional set e8 of finite linear measure. lemma 2.15 let a0, a1, ..., ak 6≡ 0, f 6≡ 0 be finite [p, q]-order meromorphic functions. if f is a meromorphic solution with ρ[p,q] (f ) = +∞ and ρ[p+1,q] (f ) = ρ < +∞ of equation (1.4) , then λ[p,q] (f ) = λ[p,q](f ) = ρ[p,q](f ) = +∞ and λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ) = ρ. proof let f be a meromorphic solution of (1.4) with infinite [p, q]-order and ρ[p+1,q] (f ) = ρ < +∞. note first that by definition, we have λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ ρ[p+1,q] (f ) . then, itremains to show that ρ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) . we rewrite (1.4) as 1 f = 1 f ( ak (z) f (k) f + ak−1 (z) f (k−1) f + · · ·+ a1 (z) f ′ f + a0 (z) ) . (2.33) by using lemma 2.14 and (2.33), for |z | = r outside a set e8 of a finite linear measure and anygiven ε > 0, we get m ( r, 1 f ) ≤ m ( r, 1 f ) + k∑ j=1 m ( r, f (j) f ) + k∑ j=0 m ( r, aj ) +o (1) ≤ m ( r, 1 f ) + k∑ j=0 m ( r, aj ) +o ( expp { (ρ+ ε) logq r }) . (2.34) on the other hand, by (1.4), if f has a zero at z0 of order α (α > k), and a0, a1, ..., ak are allanalytic at z0, then f must have a zero at z0 of order at least α− k . hence, n ( r, 1 f ) ≤ kn ( r, 1 f ) + n ( r, 1 f ) + k∑ j=0 n ( r, aj ) eur. j. math. anal. 1 (2021) 99and n ( r, 1 f ) ≤ kn ( r, 1 f ) + n ( r, 1 f ) + k∑ j=0 n ( r, aj ) . (2.35) therefore, by (2.34) and (2.35), for all sufficiently large r /∈ e8 and any given ε > 0, we have t (r, f ) = t (r, 1 f ) +o (1) ≤ t (r, f ) + k∑ j=0 t ( r, aj ) + kn ( r, 1 f ) +o ( expp { (ρ+ ε) logq r }) . (2.36) noting c = max { ρ[p,q] ( aj ) (j = 0, 1, ..., k), ρ[p,q] (f ) } . then, by using the definition of the [p, q]-order, for the above ε and sufficiently large r , we have t (r, f ) ≤ expp { (c + ε) logq r } , (2.37) t ( r, aj ) ≤ expp { (c + ε) logq r } , j = 0, 1, ..., k. (2.38) replacing (2.37) and (2.38) into (2.36) , for r /∈ e8 sufficiently large and any given ε > 0, weobtain t (r, f ) ≤ kn ( r, 1 f ) + (k + 2) expp { (c + ε) logq r } +o ( expp { (ρ+ ε) logq r }) . (2.39) hence, for any f with ρ[p,q] (f ) = +∞ and ρ[p+1,q](f ) = ρ, by (2.39) , we have λ[p,q] (f ) ≥ ρ[p,q] (f ) = +∞, λ[p+1,q] (f ) ≥ ρ[p+1,q] (f ) , so ρ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) . and the fact that λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ ρ[p+1,q] (f ) , we obtain λ[p+1,q] (f ) = λ[p+1,q] (f ) = ρ[p+1,q] (f ) = ρ. 3. proof of theorem 1.1 assume that f 6≡ 0 is a rational solution of (1.3). first, we will prove that f must be a polynomialwith deg f ≤ s − 1. for this, if f is a rational function, which has a pole at z0 of degree m ≥ 1,or f is a polynomial with deg f ≥ s, then f (s)(z) 6≡ 0. by (1.3) and lemma 2.4, we obtain σ ≤ ρ[p,q](as) = ρ[p,q](as f (s)) = ρ[p,q] − k∑ j=0, j 6=s aj f (j)  ≤ max j=0,1,...,k, j 6=s { ρ[p,q] ( aj )} which is a contradiction. therefore, f must be a polynomial with deg f ≤ s − 1. eur. j. math. anal. 1 (2021) 100now, we assume that f is a transcendental meromorphic solution of (1.3) such that λ[p,q] (1f ) < µ[p,q](f ). by lemma 2.3, for any given ε (0 < 2ε < σ − ρ) , there exists a set e3 ⊂ (1,+∞) witha finite linear measure (and so of finite logarithmic measure) such that |aj (z) | ≤ expp+1 { (ρ+ ε) logq r } , j = 0, 1, ..., k, j 6= s (3.1) holds for all z satisfying |z | = r /∈ e3. in view of lemma 2.8, there exists a set e6 ⊂ (1,+∞) offinite logarithmic measure such that |z | = r /∈ [0, 1] ∪ e6, |g (z) | = m (r, g) and for r sufficientlylarge, we have ∣∣∣∣ f (z) f (s) (z) ∣∣∣∣ ≤ r2s (s ≥ 1 is an integer) . (3.2) according to lemma 2.1, there exist a set e1 ⊂ (1,+∞) with ml(e1) <∞ and a constant b > 0,such that for all z satisfying |z | = r /∈ [0, 1] ∪ e1, we have∣∣∣∣∣ f (j) (z) f (z) ∣∣∣∣∣ ≤ b [t (2r, f )]k+1 , j = 1, 2, ..., k, j 6= s. (3.3) from the hypotheses of theorem 1.1, there exists a set h ⊂ (1,+∞) with ml (h) = +∞, suchthat for all z satisfying |z | = r ∈ h, r → +∞ and sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ − ε) logq r } . (3.4) now, by rewriting equation (1.3) in the form |as | ≤ ∣∣∣∣ ff (s) ∣∣∣∣ |a0|+ k∑ j=1 j 6=s ∣∣aj ∣∣ ∣∣∣∣∣ f (j)f ∣∣∣∣∣  (3.5) and substituting (3.1) , (3.2) , (3.3) and (3.4) into (3.5), for all z satisfying |z | = r ∈ hr ([0, 1]∪ e1 ∪ e3 ∪ e6), r → +∞, we obtain expp+1 { (σ − ε) logq r } ≤ bkr2s expp+1 { (ρ+ ε) logq r } [t (2r, f )]k+1 . since 0 < 2ε < σ − ρ, then we have exp { (1− o (1)) expp { (σ − ε) logq r }} ≤ bkr2s [t (2r, f )]k+1 . (3.6) from (3.6) and lemma 2.9, for any given γ > 1 and sufficiently large r > r, we get exp { (1− o (1)) expp { (σ − ε) logq r }} ≤ bk (γr)2s [t (2γr, f )]k+1 which gives ρ[p,q](f ) = µ[p,q] (f ) = +∞, σ ≤ ρ[p+1,q] (f ) . (3.7) by using lemma 2.4, we have max { ρ[p,q] ( aj ) : j = 0, 1, ..., k } = ρ[p,q] (as) = β < +∞. eur. j. math. anal. 1 (2021) 101since f is of infinite [p, q]-order meromorphic solution of equation (1.3) satisfying λ[p,q] (1f ) < µ[p,q] (f ), then by lemma 2.11, we obtain ρ[p+1,q] (f ) ≤ max { ρ[p,q] ( aj ) : j = 0, 1, ..., k } = ρ[p,q] (as) . (3.8) by (3.7) and (3.8) , we conclude that µ[p,q] (f ) = ρ[p,q] (f ) = +∞ and σ ≤ ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . 4. proof of corollary 1.1 assume that ϕ is a transcendental meromorphic function such that ρ[p+1,q] (ϕ) < σ. noting g = f − ϕ, then ρ[p+1,q] (g) = ρ[p+1,q] (f ), so by theorem 1.1, σ ≤ ρ[p+1,q] (g) ≤ ρ[p,q] (as) . bysubstituting f = g + ϕ into (1.3), we obtain ak (z) g(k) + ak−1 (z) g(k−1) + · · ·+ a1 (z) g′ + a0 (z) g = − ( ak (z)ϕ(k) + ak−1 (z)ϕ(k−1) + · · ·+ a1 (z)ϕ′ + a0 (z)ϕ ) = g (z) . (4.1)it is clear that the right side g of equation (4.1) is non-zero, because by theorem 1.1, ϕ is not asolution of equation (1.3). moreover, the [p + 1, q]-order of g satisfies ρ[p+1,q] (g) ≤ max { ρ[p+1,q] (ϕ) , ρ[p+1,q] ( aj ) (j = 0, 1, ..., k) } < σ, which implies max { ρ[p+1,q] (g) , ρ[p+1,q] ( aj ) (j = 0, 1, ..., k) } < σ ≤ ρ[p+1,q] (g) . then by lemma 2.12, we obtain σ ≤ λ[p+1,q] (g) = λ[p+1,q] (g) = ρ[p+1,q] (g) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) ,that is σ ≤ λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = ρ[p+1,q] (f − ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . 5. proof of theorem 1.2 assume that f is a rational solution of (1.4). first, we will prove that f must be a polynomial with deg f ≤ s − 1. for this, if f is a rational function, which has a pole at z0 of degree m ≥ 1, or f isa polynomial with deg f ≥ s, then f (s)(z) 6≡ 0. by (1.4) and lemma 2.4, we obtain σ ≤ ρ[p,q](as) = ρ[p,q](as f (s)) = ρ[p,q] f − k∑ j=0 j 6=s aj (z) f (j)  ≤ max j=0,1,...,k, j 6=s { ρ[p,q] ( aj ) , ρ[p,q] (f ) } , eur. j. math. anal. 1 (2021) 102which is a contradiction. therefore, f must be a polynomial with deg f ≤ s − 1. now, we assume that f is a transcendental meromorphic solution of (1.4) such that λ[p,q] (1f ) < µ[p,q](f ). from lemma 2.13, we know that f satisfies ρ[p,q] (f ) ≥ σ. by the hypothesis λ[p,q] (1f ) < min{µ[p,q](f ), σ} and hadamard factorization theorem, we can write f as f (z) = g(z) d(z) , where g (z)and d (z) are entire functions satisfying µ[p,q](g) = µ[p,q](f ) = µ ≤ ρ[p,q](g) = ρ[p,q](f ), ρ[p,q](d) = λ[p,q] ( 1 f ) = β < min{µ[p,q](f ), σ}. the definition of the lower [p, q]-order assures us that |g (z)| = m(r, g) ≥ expp+1 { (µ[p,q] (g)− ε) logq r } . (5.1) putting ρ1 = max { ρ[p,q] ( aj ) (j 6= s) , ρ[p,q] (f ) } < σ.then, by lemma 2.3 and (5.1), for any given ε satisfying 0 < 2ε < min{σ − ρ1, µ[p,q] (g)− ρ[p,q] (d)}, there exists a set e3 ⊂ (1,+∞) with a finite logarithmic measure such that for all z satisfying |z | = r /∈ e3 at which |g (z) | = m(r, g), we obtain∣∣∣∣f (z) f (z) ∣∣∣∣ = |f (z)| |g(z)| |d (z)| ≤ expp+1 { (ρ[p,q] (d) + ε) logq r } expp+1 { (ρ1 + ε) logq r } expp+1 { (µ[p,q] (g)− ε) logq r } ≤ expp+1 { (ρ1 + ε) logq r } . (5.2)by using the same arguments as in the proof of theorem 1.1, for any given ε ( 0 < 2ε < min{σ − ρ1, µ[p,q] (g)− ρ[p,q] (d)} ) and all z satisfying |z | = r ∈ hr(e1 ∪ e3 ∪ e6) , r → +∞ at which |g (z) | = m(r, g), we have (3.2) , (3.3) , (3.4) hold and |aj (z) | ≤ expp+1 { (ρ1 + ε) logq r } , j = 0, 1, ..., k, j 6= s. (5.3) by (1.4) , we have |as | ≤ ∣∣∣∣ ff (s) ∣∣∣∣ |a0|+ k∑ j=1 j 6=s ∣∣aj ∣∣ ∣∣∣∣∣ f (j)f ∣∣∣∣∣+ ∣∣∣∣ff ∣∣∣∣  . (5.4) hence, by substituting (3.2) , (3.3) , (3.4) , (5.2) and (5.3) into (5.4) , for all z satisfying |z | = r ∈ h r (e1 ∪ e3 ∪ e6), r → +∞, at which |g (z) | = m (r, g) and any given ε ( 0 < 2ε < min{σ − ρ1, µ[p,q] (g)− ρ[p,q] (d)} ) , we obtain expp+1 { (σ − ε) logq r } ≤ r2s ( expp+1 { (ρ1 + ε) logq r } eur. j. math. anal. 1 (2021) 103 + k∑ j=1,j 6=s expp+1 { (ρ1 + ε) logq r } b [t (2r, f )]k+1 + expp+1 { (ρ1 + ε) logq r }) ≤ b (k + 1) r2s [t (2r, f )]k+1 expp+1 { (ρ1 + ε) logq r } . (5.5) since 0 < 2ε < σ − ρ1, then we can use lemma 2.9 with (5.5) such that for any given γ > 1 andsufficiently large r > r, we obtain exp { (1− o (1)) expp { (σ − ε) logq r }} ≤ b (k + 1) (γr)2s [t (2γr, f )]k+1 which gives ρ[p,q](f ) = µ[p,q] (f ) = +∞, ρ[p+1,q](f ) ≥ σ. (5.6) making use of lemma 2.4, we have max { ρ[p,q] ( aj ) (j = 0, 1, ..., k) , ρ[p,q] (f ) } = ρ[p,q] (as) = β < +∞. by lemma 2.11 and since f is of infinite [p, q]-order meromorphic solution of equation (1.4)satisfying λ[p,q] (1f ) < µ[p,q] (f ) , we get ρ[p+1,q] (f ) ≤ max { ρ[p,q] ( aj ) (j = 0, 1, ..., k) , ρ[p,q] (f ) } = ρ[p,q] (as) . (5.7) since f 6≡ 0, then by lemma 2.15, we have λ[p,q] (f ) = λ[p,q](f ) = µ[p,q] (f ) = ρ[p,q](f ) = +∞ (5.8) and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ). (5.9) by (5.7) , (5.8) and (5.9) , we conclude that λ[p,q] (f ) = λ[p,q](f ) = µ[p,q] (f ) = ρ[p,q](f ) = +∞ and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ) ≤ ρ[p,q] (as) . 6. proof of corollary 1.2 assume that ϕ is a transcendental meromorphic function such that ρ[p+1,q] (ϕ) < σ. noting h = f − ϕ, then ρ[p+1,q] (h) = ρ[p+1,q] (f ), so by theorem 1.2, σ ≤ ρ[p+1,q] (h) ≤ ρ[p,q] (as) . bysubstituting f = h + ϕ into (1.4), we obtain ak (z) h(k) + ak−1 (z) h(k−1) + · · ·+ a1 (z) h′ + a0 (z) h = f (z)− ( ak (z)ϕ(k) + ak−1 (z)ϕ(k−1) + · · ·+ a1 (z)ϕ′ + a0 (z)ϕ ) = ψ (z) . (6.1) eur. j. math. anal. 1 (2021) 104it is clear that the right side ψ of the equation (6.1) is non-zero, because by theorem 1.2, ϕ isnot a solution of equation (1.4). moreover, the [p + 1, q]-order of ψ verifies ρ[p+1,q] (ψ) ≤ max { ρ[p+1,q] (ϕ) , ρ[p+1,q] ( aj ) (j = 0, 1, ..., k) } < σ, which leads to max { ρ[p+1,q] (ψ) , ρ[p+1,q] ( aj ) (j = 0, 1, ..., k) } < σ ≤ ρ[p+1,q] (h) . therefore, by lemma 2.12, we obtain σ ≤ λ[p+1,q] (h) = λ[p+1,q] (h) = ρ[p+1,q] (h) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) ,that is σ ≤ λ[p+1,q] (f − ϕ) = λ[p+1,q] (f − ϕ) = ρ[p+1,q] (f − ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . acknowledgements. this paper was supported by the directorate-general for scientific researchand technological development (dgrsdt). references [1] b. belaïdi, growth and oscillation theory of [p, q]-order analytic solutions of linear differential equations in theunit disc. j. math. anal. 3 (2012), 1–11. http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf.[2] b. belaïdi, iterated order of meromorphic solutions of homogeneous and non-homogeneous linear differentialequations. romai j. 11 (2015), 33–46. http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf.[3] b. belaïdi, differential polynomials generated by meromorphic solutions of [p, q]-order to complex linear differentialequations. rom. j. math. comput. sci. 5 (2015), 46–62. http://rjm-cs.ro/belaidi-2015.pdf.[4] a. ferraoun and b. belaïdi, on the (p, q)−order of solutions of some complex linear differential equations. commun.optim. theory, 2017 (2017), article id 17, 1–23. https://doi.org/10.23952/cot.2017.17.[5] g. g. gundersen, estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. j.london math. soc. 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https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817 https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817 https://doi.org/10.1080/1726037x.2017.1413065 https://www.math.u-szeged.hu/ejqtde/p453.pdf https://www.math.u-szeged.hu/ejqtde/p453.pdf https://doi.org/10.1155/2013/243873 https://doi.org/10.1155/2013/243873 1. introduction and main results 2. some auxiliary lemmas 3. proof of theorem 1.1 4. proof of corollary 1.1 5. proof of theorem 1.2 6. proof of corollary 1.2 references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 14doi: 10.28924/ada/ma.3.14 on the kolmogorov distance for the least squares estimator in the fractional ornstein-uhlenbeck process jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223, usacorrespondence: j.bishwal@uncc.edu abstract. the paper shows that the distribution of the normalized least squares estimator of the driftparameter in the fractional ornstein-uhlenbeck process observed over [0, t ] converges to the standardnormal distribution with an uniform optimal error bound of the order o(t−1/2) for 0.5 ≤ h ≤ 0.63and of the order o(t 4h−3) for 0.63 < h < 0.75 where h is the hurst exponent of the fractionalbrownian motion driving the ornstein-uhlenbeck process. for the normalized quasi-least squaresestimator, the error bound is of the order o(t−1/4) for 0.5 ≤ h ≤ 0.69 and of the order o(t 4h−3)for 0.69 < h < 0.75. 1. introductionthe fractional ornstein-uhlenbeck process, is an extension of ornstein-uhlenbeck process withfractional brownian motion (fbm) driving term. in finance it is known as fractional vasicek model,and is being extensively used these days as one-factor short-term interest rate model which takesinto account the long memory effect of the interest rate. the model parameter is usually unknownand must be estimated from data.parameter estimation in stochastic differential equations is studied in bishwal [1]. for thestandard ornstein-uhlenbeck process, sufficiency and rao-blackwellization was studied in bish-wal [4] where also a time transformation to reduce the general problem to a fixed time case andthe asymptotics were studied in large parameter case. for the fractional ornstein-uhlenbeck pro-cess, berry-esseen inequalities of minimum contrast estimators based on continuous and discreteobservations was studied in bishwal [2]. hu et al. [11] studied parameter estimation for the frac-tional ornstein-uhlenbeck process of general hurst parameter. bishwal [5] studied berry-esseeninequalities for the fractional black-karasinski model of term structure of interest rates. usingfractional levy process as the driving term which include jumps, maximum quasi-likelihood estima-tion in fractional levy stochastic volatility model was studied in bishwal [3]. parameter estimationin partially observed stochastic differential system was studied in bishwal [6]. received: 12 nov 2022. key words and phrases. itô stochastic differential equation; fractional brownian motion; fractional ornstein-uhlenbeck process; long-memory; least squares estimator; quasi-least squares estimator; rate of weak convergence;kolmogorov distance; wiener chaos; fourier method; analytic continuation.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 2let (ω,f , {ft}t≥0, p ) be a stochastic basis on which is defined the ornstein-uhlenbeck process xt satisfying the itô stochastic differential equation dxt = θxtdt + dwh t , t ≥ 0, x0 = 0 (1.1) where {wh t } is a fractional brownian motion with h > 1/2 with the filtration {ft}t≥0 and θ < 0is the unknown parameter to be estimated on the basis of continuous observation of the process {xt} on the time interval [0, t ].recall that a fractional brownian motion (fbm) has the covariance c̃h(s, t) = 1 2 [ s2h + t2h − |s − t|2h ] , s, t > 0. (1.2) for h > 0.5 the process has long range dependence or long memory and the process is self-similar.for h 6= 0.5, the process is neither a markov process nor a semimartingale. for h = 0.5, theprocess reduces to standard brownian motion.note that the solution of the equation (1.1) is given by xt = ∫ t 0 eθ(t−s)dwh s . (1.3) let the realization {xt , 0 ≤ t ≤ t} be denoted by xt0 . let p tθ be the measure generatedon the space (ct , bt ) of continuous functions on [0, t ] with the associated borel σ-algebra btgenerated under the supremum norm by the process xt0 and p t0 be the standard wiener measure.applying girsanov type formula for fbm, when θ is the true value of the parameter, p tθ is absolutelycontinuous with respect to p t0 and the radon-nikodym derivative (likelihood) of p tθ with respectto p t0 based on xt0 is given by lt (θ) := dp tθ dp t0 (xt0 ) = exp { θ ∫ t 0 qtdzt − θ2 2 ∫ t 0 q2t dvt } . (1.4) consider the score function, the derivative of the log-likelihood function, which is given by yt (θ) := ∫ t 0 qtdzt − θ ∫ t 0 q2t dvt . (1.5) a solution of yt (θ) = 0 provides the maximum likelihood estimate (mle) θt := ∫ t 0 qtdzt∫ t 0 q 2 t dvt . (1.6) kleptsyna and le breton [13] showed that θt is strongly consistent. using the fourier method,bishwal [2] proved a berry-esseen type theorem for the estimator θt which gives the rate of weakconvergence in asymptotic normality.using the fractional itô formula, the score function yt (θ) can be written as yt (θ) = 1 2 [ λh (2− 2h) zt ∫ t 0 t2h−1dzt − t ] − θ ∫ t 0 q2t dvt . (1.7) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 3consider the contrast function kt (θ) := − thγ(h) 2 − θ ∫ t 0 q2t dvt (1.8) and the minimum contrast estimate (mce) θ̄t := −thγ(h) 2 ∫ t 0 q 2 t dvt . (1.9) the least squares estimator (lse) of θ minimizes∫ t 0 |ẋt − θxt |2dt (1.10) and is given by θ̂t := ∫ t 0 xtdxt∫ t 0 x 2 t dt. = θ − ∫ t 0 xtdw h t∫ t 0 x 2 t dt. (1.11) based on ergodicity, quasi least squares estimate (qlse) θ̃t := ( −thγ(2h)∫ t 0 x 2 t dt ) 1 2h (1.12) the lse and the qlse are strongly consistent and asymptotically norma as t →∞ √ t (θ̂t − θ)→d n (0, θσ2h), √ t (θ̃t − θ)→d n (0, θσ2h 4h2 ) (1.13) where σ2h := (4h − 1) ( 1 + γ(3− 4h)γ(4h − 1) γ(2− 2h)γ(2h) ) . (1.14) observe that h = 1/2, σ2h = 2. in this case the lse and the mle are identical. since θ̃t is aconsistent estimator of θ, we can derive the self normalized limit distributions immediately: ( t σ2h θ̃t )1/2(θ̂t − θ)→d n (0, 1), 2h( t σ2h θ̃t )1/2(θ̃t − θ)→d n (0, 1). (1.15) define mt := ∫ t 0 xt dw h t and it := ∫ t 0 x2t dt, nt := θ2hit − thγ(2h). (1.16) vh,θ := θ−2hhγ(2h). (1.17)observe that ( t −σ2hθ )1/2 (θ̂t − θ) = ( −σ2hθ t )1/2 mt( σ2hθ t ) it (1.18) applying taylor’s formula to the function x− 1 2h at the point vh,θ , we have( it t )− 1 2h = v − 1 2h h,θ − 1 2h v − 1+2h 2h h,θ ( it t − vh,θ ) + 1 + 2h 8h2 $ − 1+4h 2h t ( it t − vh,θ )2 (1.19) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 4 where $t is a random point between vh,θ and it t . further θ̃t − θ = − θ1+2h 2h2γ(2h) ( it t − vh,θ ) + (1 + 2h)(hγ(2h)) 1 2h 8h2 $ − 1+4h 2h t ( it t − vh,θ )2 . (1.20) thus 2h ( t −σ2hθ )1/2 (θ̃t − θ) = ( −σ2hθ 4th2 )1/2 nt( σ2hθ 4th2 ) it . (1.21) we study the large deviations, moderate deviations and berry-esseen bounds of the lse andthe qlse in this paper. we will use the following optimal fourth moment theorem from nourdinand peccati [14] in the sequel. see also douissi et al. [15]. theorem 1.1 (skewness kurtosis inequality) let (xn)n≥1 be a sequence of random variables in fixed wiener chaos of order q ≥ 2 such that v ar(xn) = 1. assume xn converges to normal distribution which is equivalent to limn e(xn)4 = 3, which is also known as the fourth moment theorem. then we have the following optimal rate for dtv (xn,n ) known as the optimal fourth moment theorem: there exist two constants c, c > 0 depending only on the sequence (xn)n≥1 but not on n, such that c max{e(x4n)− 3, |e(x3n)|} ≤ dtv (xn,n ) ≤ cmax{e(x4n)− 3, |e(x3n)|}. (1.22) let φ(·) denote the standard normal distribution function. throughout the paper, c denotes ageneric constant (which does not depend on t and x ). we have not tried to estimate the constantin the bound on normal approximation.hu et al. [11] obtained limiting normal distribution of the lse and the qlse for the memoryrange 12 < h ≤ 3 4 with the rate √t for 12 < h < 3 4 and √t (logt )−1/2 for the case h = 3 4 , andlimiting rosenblatt distribution for the memory range 34 < h < 1.we only consider the memory range 12 < h < 3 4 . jiang et al. [12] used self-normalization alongwith the splitting method for the lse and the qlse in fractional ornstein-uhlenbeck process andobtained the rate t−1/2 logt for the range 12 ≤ h ≤ 5 8 for the lse and t−1/4 logt for the range 1 2 ≤ h ≤ 11 16 for the qlse. they obtained the rate t 4h−3 for the range 58 < h < 3 4 for the lseand the same rate t 4h−3 for the range 1116 < h < 3 4 for the qlse.in this paper we improve the first rate to t−1/2 for the mle for the range 1 2 ≤ h ≤ 5 8 and t−1/4 for the range 1 2 ≤ h ≤ 11 16 for the qlse using the squeezing method as in chapter 1 inbishwal [1]. the main contribution of the paper is thus improvement in the rate my removing the logt term. note the critical points: 1 2 = 0.50, 58 = 0.63, 23 = 0.67, 1116 = 0.69, 34 = 0.75. also 0.63 + 0.06 = 0.69, 0.69 + 0.06 = 0.75. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 5 remark on the critical point 58 : for the discrete observations case, es-sebaiy and viens [7]pointed out that if 0 < h < 5 8 , then the fourth moment is of the order n−1 and if 58 < h < 3 4 ,then the fourth moment is of the order n2(4h−3) where n is the number of observations. theberry-esseen rate for θ̂ is shown to be of the order n−1/4 for 0 < h < 5 8 and of the order n−(4h−3)/2 if 58 < h < 3 4 . for h = 3 4 , the rate is (log n)−1/4. the proofs also need large deviation results for the stochastic integral and the energy integral.these integrals can be represented by multiple wiener integrals. then their expectations andvariances as well as the fourth moment of their malliavin derivatives can be estimated.first we calculate bounds on the moments. let ϕt (s, t) := e−θ|t−s|, ψt (s, t) := e−2θt+θ(s+t), gt (s, t) := e−θ(t−s)i[0,t](s), (1.23) vh,θ := θ−2hhγ(2h), ch,θ := θ1−4h(4h − 1)h2 ( γ2(2h) + γ(2h)γ(3− 4h)γ(4h − 1) γ(2− 2h) ) . (1.24)observe that xt = i1(gt (·, t)), (1.25) mt = ∫ t 0 xtdw h t = ∫ t 0 ∫ t 0 eθ(t−s)dwh s dw h t = 1 2 eθ|t−s|dwh s dw h t = 1 2 i2(ϕt ), (1.26) it = ∫ t 0 x2t dt = 1 2θ i2(ϕt ) + 1 2θ i2(ψt ) + ∫ t 0 ‖gt (·, t)‖2hdt (1.27)where i1 and i2 are first and second wiener chaos respectively. furthermore,∫ t 0 ‖gt (·, t)‖2hdt = vh,θt + o(t ). (1.28) for 12 < h < 3 4 , e(xtxs) ≤ c|t − s|2h−2, (1.29) ‖ϕt ‖2h = 2t (ch,t + (o(1)), ‖ψt ‖2h = o(1). (1.30)for h = 1 2 , by the isometry of the itô integral, we obtain ‖ϕt ‖2h = 2 ∫ t 0 ∫ t 0 e2θ(t−s)dtds = t θ + e2θt − 1 2θ2 = 2t (ch,t + (o(1)). (1.31) ‖ψt ‖2h = e−4θt ∫ t 0 ∫ t 0 e−2θ(t+s)dtds = (e−2θt − 1)2 4θ2 = o(1). (1.32) for 12 < h < 3 4 ,, using lemma 5.3 in hu and nualart [10], we have ‖ψt ‖2h ≤ γ2(2h) (2h − 1)2 θ−4h. (1.33) let υt = t for h = 1 2 and υt = t 8h−4 for 12 < h < 3 4 . we obtain the variances bounds on the malliavin derivative of mt and it . e(‖dmt ‖2h − e‖dmt ‖2h) ≤ cυt , (1.34) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 6 e(‖dit ‖2h − e‖dit ‖2h) ≤ cυt (1.35)where d is the malliavin derivative operator.we have the bound on the fourth moment e(‖di2(ϕt )‖2h − e‖di2(ϕt )‖2h)2 ≤ cυt . (1.36) for 12 < h < 3 4 ,, we have the bound on the fourth moment e(‖di2(ϕt )‖2h − e‖di2(ϕt )‖2h)2 ≤ ct 8h−4. (1.38) we have the bound on the fourth moment e(‖di2(ψt )‖2h − e‖di2(ψt )‖2h)2 ≤ c. (1.39) ds i2(ψt ) = −2e−2θt+θs ∫ t 0 eθtdwh t . (1.40) e‖ds i2(ψt )‖4h = 16e−8θt (∫ t 0 eθtdwh t )4(∫ t 0 e2θtdt )2 = 48e−8θt (∫ t 0 e2θtdt )4 . (1.41) e‖ds i2(ψt )‖2h = 4e−4θt (∫ t 0 e2θtdt )2 . (1.42) therefore e(‖di2(ψt )‖2h − e‖di2(ψt )‖2h)2 = e‖di2(ψt )|4h − (e‖di2(ψt )‖2h)2 = 2(1− e−2θt )4 θ4 . (1.43)similarly for the case 12 < h < 3 4 , it can be shown that e(‖di2(ψt )‖2h − e‖di2(ψt )‖2h)2 ≤ 32γ4(2h) (2h − 1)4 θ−8h. (1.44)first we have the berry-esseen bounds for the stochastic integral and adjusted energy integral. byusing the optimal fourth moment theorem (skewness-kurtosis inequality) from stein-malliavintheory, we have:for 12 ≤ h ≤ 5/8, we have sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2 mt ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 mt ‖2h − e‖d ( c−1h,θ t )1/2 mt ‖2h )2 1/2 ≤ ct−1/2. (1.45) for 58 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2 mt ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 mt ‖2h − e‖d ( c−1h,θ t )1/2 mt ‖2h )2 1/2 ≤ c t 4h−3. (1.46) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 7for 12 ≤ h ≤ 5/8, we have sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2( θ̃t it − t −σ2h ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 ( θ̃t it − t −σ2h ) ‖2h − e‖d ( c−1h,θ t )1/2 ( θ̃t it − t −σ2h ) ‖2h )2 1/2 ≤ ct−1/2. (1.47)for 58 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2( θ̃t it − t −σ2h ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 ( θ̃t it − t −σ2h ) ‖2h − e‖d ( c−1h,θ t )1/2 ( θ̃t it − t −σ2h ) ‖2h )2 1/2 ≤ c t 4h−3. (1.48)for 12 ≤ h ≤ 5 8 , we have for |x | ≤ 2(logt )1/2, sup y∈r ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 mt − (( −σ2h θ̃t t ) it − 1 ) x ≤ y −φ(y) ∣∣∣∣∣∣ ≤ ct−1/2. (1.49) for 58 < h < 3 4 , we have for |x | ≤ 2(logt )1/2, sup y∈r ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 mt − (( −σ2h θ̃t t ) it − 1 ) x ≤ y −φ(y) ∣∣∣∣∣∣ ≤ ct 4h−3. (1.50) for 12 ≤ h ≤ 11 16 , we have sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2 nt ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 nt ‖2h − e‖d ( c−1h,θ t )1/2 nt ‖2h )2 1/2 ≤ ct−1/2. (1.51) for 1116 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1h,θ t )1/2 nt ≤ x −φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1h,θ t )1/2 nt ‖2h − e‖d ( c−1h,θ t )1/2 nt ‖2h )2 1/2 ≤ c t 4h−3. (1.52) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 8for 12 ≤ h ≤ 11 16 , we have for |x | ≤ 2(logt )1/2, sup y∈r ∣∣∣∣∣∣p 2h ( −σ2h θ̃t t )1/2 nt − (( −σ2h θ̃t t ) it − 1 ) x ≤ y −φ(y) ∣∣∣∣∣∣ ≤ ct−1/4. (1.53) for 1116 < h < 3 4 , we have for |x | ≤ 2(logt )1/2, sup y∈r ∣∣∣∣∣∣p 2h ( −σ2h θ̃t t )1/2 nt − (( −σ2h θ̃t t ) it − 1 ) x ≤ y −φ(y) ∣∣∣∣∣∣ ≤ ct 4h−3. (1.54) 2. main results we need the next two lemmas from jiang et al. [12] on large deviations to obtain bounds onthe tail probabilities of the estimators. the first lemma is on large deviations for stochastic integral. lemma 2.1 for every δ > 0, p {∣∣∣∣mt t ∣∣∣∣ ≥ δ} ≤ c exp ( − t 1/2δ 4c 1/2 h,θ ) . remark for the case h = 0.5, there is a long history of work: for every δ > 0, p {∣∣∣∣mt t ∣∣∣∣ ≥ δ} ≤ c0 exp ( −c1tδ2 ) . see gao and jiang [9].for any 0 ≤ α ≤ θ2/4, there exist constants c3 and c4 such that e(eαit ) ≤ c3ec4αt . (2.1)see gao and jiang [9]. by chebyshev inequality, we have p (|xt − e(xt )| ≥ δ) ≤ 2 exp(−θδ2). (2.2) the second lemma is on large deviations in the ergodic theorem. lemma 2.2 for every δ > 0, p {∣∣∣∣ itt − vh,θ ∣∣∣∣ ≥ δ} ≤ c exp ( − t 1/2δ 4c 1/2 h,θ ) . observe that by (1.11) θ̂t = θ − mt it . https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 9using the elementary inequality p (| ξ η | ≥ u) ≤ p (|ξ| ≥ uv) + p (η − 2v | ≥ v), (2.3) we have p (|θ̂t − θ| ≥ δ) ≤ p (|it − vh,θt | ≥ 1 2vh,θt ) + p (|θ̂t − θ| ≥ δ, |it − vh,θt | < 1 2vh,θt ) ≤ p (|it − vh,θt | ≥ 1 2vh,θt ) + p (|mt | ≥ 1 2vh,θtδ). (2.4) combining lemma 2.1 and lemma 2.2, we obtain lemma 2.3 for every δ > 0 and large t > 0, we have a) p (|θ̂t − θ| ≥ δ) ≤ c0 exp(−c1t 1/2δ) b) p (|θ̃t − θ| ≥ δ) ≤ c0 exp(−c1t 1/2δ1/2). to obtain the rate of normal approximation for the lse and the qlse, we need the followingtail probability estimate of the estimators. lemma 2.4 (a) p  ( t −σ2h θ̃t )1/2 |θ̂t − θ| ≥ 2(logt )1/2  ≤ ct−1/2. (b) p 2h ( t −σ2h θ̃t )1/2 |θ̃t − θ| ≥ 2(logt )1/2  ≤ ct−1/4. proof : observe that p  ( t −σ2h θ̃t )1/2 |θ̂t − θ| ≥ 2(logt )1/2  = p  ∣∣∣∣∣∣∣∣∣ ( −σ2h θ̃t t )1/2 mt ( −σ2h θ̃t t )it ∣∣∣∣∣∣∣∣∣ ≥ 2(logt )1/2  ≤ p  ∣∣∣∣∣∣ ( −σ2h θ̃t t )1/2 mt ∣∣∣∣∣∣ ≥ (logt )1/2 + p {∣∣∣∣∣−σ2h θ̃tt it ∣∣∣∣∣ ≤ 1 2 } ≤ ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 |mt | ≥ (logt )1/2 − 2φ(−(logt )1/2) ∣∣∣∣∣∣ +2φ(−(logt )1/2) + p {∣∣∣∣∣σ2h θ̃tt it − 1 ∣∣∣∣∣ ≥ 1 2 } https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 10 ≤ sup x∈r ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 |mt | ≥ x − 2φ(−x) ∣∣∣∣∣∣ +2φ(−(logt )1/2) + p {∣∣∣∣∣ ( −σ2h θ̃t t ) it − 1 ∣∣∣∣∣ ≥ 1 2 } ≤ ct−1/2 + c(t logt )−1/2 + c exp ( − t 1/2 8c 1/2 h,θ ) ≤ ct−1/2.the bounds for the first and the third terms come from lemma 2.2 and lemma 2.1 respectively andthat for the middle term comes from feller [8] (p. 166). proof of (b) is similar. now we are ready to obtain the uniform rate of normal approximation of the distribution of thelse and the qlse.recall that σ2h := (4h − 1) ( 1 + γ(3− 4h)γ(4h − 1) γ(2− 2h)γ(2h) ) . (2.5) theorem 2.5a) if 12 ≤ h ≤ 5 8 sup x∈r ∣∣∣∣∣∣p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct−1/2. b) if 58 < h < 3 4 sup x∈r ∣∣∣∣∣∣p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct 4h−3. c) if 12 ≤ h ≤ 11 16 sup x∈r ∣∣∣∣∣∣p 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct−1/4. d) if 1116 < h < 3 4 sup x∈r ∣∣∣∣∣∣p 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct 4h−3. proof : first we prove (a). we shall consider two possibilities (i) and (ii). (i) |x | > 2(logt )1/2. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 11we shall give a proof for the case x > 2(logt )1/2. the proof for the case x < −2(logt )1/2 runssimilarly. note that∣∣∣∣∣∣p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≥ x +φ(−x). (2.6) but from feller [8] (p. 166) we have φ(−x) ≤ φ(−2(logt )1/2) ≤ ct−1. (2.7) moreover, by lemma 2.4 (a), we have p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≥ 2(logt )1/2  ≤ ct−1/2. (2.8) hence ∣∣∣∣∣∣p  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct−1/2. (2.9) (ii) |x | ≤ 2(logt )1/2. let at :=  ( t −σ2h θ̃t )1/2 |θ̂t − θ| ≤ 2(logt )1/2  and bt := { it t > c0 } (2.10) where 0 < c0 < 1 −σ2hθ . by lemma 2.4, we have p (act ) ≤ ct−1/2. (2.11) by lemma 2.1, we have p (bct ) = p {( −σ2h θ̃t t ) it − 1 < σ2hθc0 − 1 } < p {∣∣∣∣∣ ( −σ2h θ̃t t ) it − 1 ∣∣∣∣∣ > 1− σ2hθc0 } ≤ c exp ( − t 1/2(1− σ2hθc0) 4c 1/2 h,θ ) . (2.12) let b0 be some positive number. on the set at∩bt for all t > t0 with 4b0(logt0) 1/2 ( σ2hθ t )1/2 ≤ c0, we have ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x ⇒ it + b0t (θ̂t − θ) < it + ( t −σ2h θ̃t )1/2 σ2hb0θx ⇒ ( t −σ2h θ̃t )1/2 (θ̂t − θ)[it + b0t (θt − θ)] < x [it + ( t −σ2h θ̃t )1/2 σ2hb0θx ] ⇒ (θ̂t − θ)it + b0t (θt − θ)2 < ( −σ2h θ̃t t )1/2 it x + σ2hb0θx 2 https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 12 ⇒ −mt + (θ̂t − θ)it + b0t (θ̂t − θ)2 < −mt + ( σ2h θ̃t t )1/2 it x + σ2hb0θx 2 ⇒ 0 < −mt + ( −σ2h θ̃t t )1/2 it x + σ2hb0θx 2 since it + b0t (θ̂t − θ) > tc0 + b0t (θ̂t − θ) > 2σ2hb0(logt )1/2 ( −σ2h θ̃t t )1/2 − σ2hb0(logt )1−h ( −σ2h θ̃t t )1/2 = σ2hb0(logt )1/2 ( −σ2h θ̃t t )1/2 > 0. on the other hand, on the set at ∩bt for all t > t0 with 4b0(logt0) 1/2 ( −σ2h θ̃t t0 )1/2 ≤ c0, wehave ( t −σ2h θ̃t )1/2 (θ̂t − θ) > x ⇒ it − b0t (θ̂t − θ) < it − ( t σ2h θ̃t )1/2 2b0θx ⇒ ( t −σ2h θ̃t )1/2 (θ̂t − θ)[it − b0t (θ̂t − θ)] > x [it − ( t −σ2h θ̃t )1/2 σ2hb0θx ] ⇒ (θ̂t − θ)it − b0t (θ̂t − θ)2 > ( t −σ2h θ̃t )−1/2 it x − σ2hb0θx2 ⇒ −mt + (θ̂t − θ)it − b0t (θ̂t − θ)2 > −mt + ( t −σ2h θ̃t )−1/2 it x − σ2hb0θx2 ⇒ 0 > −mt + ( −σ2h θ̃t t )1/2 it x − σ2hb0θx2 since it − b0t (θ̂t − θ) > tc0 − b0t (θ̂t − θ) > 2σ2hb0(logt )1/2 ( −σ2h θ̃t t )1/2 − σ2hb0(logt )1/2 ( −σ2h θ̃t t )1/2 = σ2hb0(logt )1/2 ( −σ2h θ̃t t )1/2 > 0. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 13hence 0 < −mt + ( t −σ2h θ̃t )1/2 it x − σ2hb0θx2 ⇒ ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x. letting d±t,x := −mt + ( σ2h θ̃t t )1/2 it x ± σ2hb0θx2 > 0  , we obtain d−t,x ∩ at ∩ bt ⊆ at ∩ bt ∩  ( t −σ2h θ̃t )1/2 (θ̂t − θ) ≤ x  ⊆ d+t,x ∩ at ∩ bt . (2.13) if it is shown that ∣∣p {d±t,x}−φ(x) ∣∣ ≤ ct−1/2 (2.14)for all t > t0 and |x | ≤ 2(logt )1/2, then the theorem would follow from (2.11) (2.14).we shall prove (2.4) for d+t,x . the proof for d−t,x is analogous. observe that∣∣p {d+t,x}−φ(x) ∣∣ = ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 mt − (( −σ2h θ̃t t ) it − 1 ) x < x + σ2h ( −σ2h θ̃t t )1/2 b0θx 2 −φ(x) ∣∣∣∣∣∣ ≤ sup y∈r ∣∣∣∣∣∣p  ( −σ2h θ̃t t )1/2 mt − (( −σ2h θ̃t t ) it − 1 ) x ≤ y −φ(y) ∣∣∣∣∣∣ + ∣∣∣∣∣∣φ x + ( −σ2h θ̃t t )1/2 b0θx 2 −φ(x) ∣∣∣∣∣∣ =: ∆1 + ∆2. (2.15)(1.50) immediately yields ∆1 ≤ ct−1/2. (2.16)on the other hand, for all t > t0, ∆2 ≤ 2 ( −σ2h θ̃t t )1/2 b0θx 2(2π)−1/2 exp(−x2/2) where |x − x | ≤ 2 ( −σ2h θ̃t t )1/2 b0θx 2. since |x | ≤ 2(logt )1/2, it follows that |x̄ | > |x |/2 for all t > t0 and consequently ∆2 ≤ 2 ( −σ2h θ̃t t )1/2 b0θx 2(2π)−1/2x2 exp(−x2/8) ≤ ct−1/2. (2.17) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 14from (2.15) (2.17), we obtain ∣∣p {d+t,x}−φ(x) ∣∣ ≤ ct−1/2. this completes the proof of part (a) of the theorem. next we prove (c). again we shall consider two possibilities (i) and (ii). (i) |x | > 2(logt )1−/2. we shall give a proof for the case x > 2(logt )1/2. the proof for the case x < −2(logt )1/2runs similarly. note that∣∣∣∣∣∣p 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ p 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≥ x + φ(−x). by (2.7) and lemma 2.4 (b), we have p 2h ( t σ2h θ̃t )1/2 (θ̃t − θ) ≥ 2(logt )1/2  ≤ ct−1/4. hence ∣∣∣∣∣∣p 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ ct−1/4.(ii) |x | ≤ 2(logt )1/2. let a1,t := 2h ( t −σ2h θ̃t )1/2 |θ̃t − θ| ≤ 2(logt )1/2  and b1,t := { it t > c0 } where 0 < c0 < 1 −σ2hθ . by lemma 2.4, we have p (ac1,t ) ≤ ct−1/4. (2.18) by lemma 2.1, we have p (bc1,t ) = p {( −σ2hθ 4th2 ) it − 1 < σ2hθc0 − 1 } < p {∣∣∣∣(−σ2hθ4th2 ) it − 1 ∣∣∣∣ > 1− σ2hθc0 } ≤ ct−1. (2.19)let b0 be some positive number. on the set a1,t ∩ b1,t for all t > t0 with 4b0(logt0) 1/2 ( −σ2hθ 4t0h2 )1/2 ≤ c0, we have 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x ⇒ it + b0t (θ̃t − θ) < it + 2h ( t −σ2h θ̃t )1/2 σ2hb0θx https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 15 ⇒ 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ)[it + b0t (θt − θ)] < x it + 2h ( t −σ2h θ̃t )1/2 σ2hb0θx  ⇒ (θ̃t − θ)it + b0t (θt − θ)2 < ( −σ2hθ 4th2 )1/2 it x + σ2hb0θx 2 ⇒ −nt + (θ̃t − θ)it + b0t (θ̃t − θ)2 < −nt + ( −σ2hθ 4th2 )1/2 it x + σ2hb0θx 2 ⇒ 0 < −nt + ( −σ2hθ 4th2 )1/2 it x + σ2hb0θx 2 since it + b0t (θt − θ) > tc0 + b0t (θt − θ) > 4b0(logt )1/2 ( −σ2hθ 4th2 )1/2 − σ2hb0(logt )1−h ( −σ2hθ 4th2 )1/2 = σ2hb0(logt )1/2 ( −σ2hθ 4th2 )1/2 > 0. on the other hand, on the set a1,t ∩ b1,t for all t > t0 with 4b0(logt0) 1/2 ( −σ2hθ 4t0h2 )1/2 ≤ c0,we have 2h ( t σ2h θ̃t )1/2 (θt − θ) > x ⇒ it − b0t (θ̃t − θ) < it − 2h ( t −σ2h θ̃t )1/2 σ2hb0θx ⇒ 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ)[it − b0t (θt − θ)] > x it − 2h ( t −σ2h θ̃t )1/2 2b0θx  ⇒ (θ̃t − θ)it − b0t (θ̃t − θ)2 > ( −σ2hθ 4th2 )1/2 it x − σ2hb0θx2 ⇒ −nt + (θ̃t − θ)it − b0t (θt − θ)2 > −nt + ( −σ2hθ 4th2 )1/2 it x − σ2hb0θx2 ⇒ 0 > −nt + ( −σ2hθ 4th2 )1/2 it x − σ2hb0θx2 since it − b0t (θ̃t − θ) > tc0 − b0t (θ̃t − θ) > 2σ2hb0(logt )1/2 ( −σ2hθ 4th2 )1/2 − σ2hb0(logt )1/2 ( −σ2hθ 4th2 )1/2 = σ2hb0(logt )1/2 ( −σ2hθ 4th2 )1/2 > 0. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 16hence 0 < −nt + ( −σ2hθ 4th2 )1/2 it x − σ2hb0θx2 ⇒ 2h( t −σ2h θ̃t )1/2(θt − θ) ≤ x. letting d±1,t,x := { −nt + ( −σ2hθ 4th2 )1/2 it x ± σ2hb0θx2 > 0 } , we obtain d−1,t,x ∩ a1,t ∩b1,t ⊆ a1,t ∩b1,t ∩ 2h ( t −σ2h θ̃t )1/2 (θ̃t − θ) ≤ x  ⊆ d+1,t,x ∩ a1,t ∩b1,t . (2.20)if it is shown that ∣∣p {d±1,t,x}−φ(x) ∣∣ ≤ ct−1/4 (2.21)for all t > t0 and |x | ≤ 2(logt )1/2, then the theorem would follow from (2.18) (2.21).we shall prove (2.21) for d+1,t,x . the proof for d−1,t,x is analogous.observe that∣∣∣p {d+1,t,x}−φ(x) ∣∣∣ = ∣∣∣∣∣p {( −σ2hθ 4th2 )1/2 nt − (( −σ2hθ 4th2 ) it − 1 ) x < x + 2 ( −σ2hθ 4th2 )1/2 b0θx 2 } −φ(x) ∣∣∣∣∣ ≤ sup y∈r ∣∣∣∣∣p {( −σ2hθ 4th2 )1/2 nt − (( −σ2hθ 4th2 ) it − 1 ) x ≤ y } −φ(y) ∣∣∣∣∣ + ∣∣∣∣∣φ ( x + ( −σ2hθ 4th2 )1/2 b0θx 2 ) −φ(x) ∣∣∣∣∣ =: ∆11 + ∆12. (2.22)(1.53) immediately yields ∆11 ≤ ct−1/4. (2.23)on the other hand, for all t > t0, ∆12 ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2(2π)−1/2 exp(−x2/2) where |x − x | ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2. since |x | ≤ 2(logt )1/2, it follows that |x̄ | > |x |/2 for all t > t0 and consequently ∆12 ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2(2π)−1/2x2 exp(−x2/8) ≤ ct−1/4. (2.24) from (2.12) (2.14), we obtain ∣∣p {d+1,t,x}−φ(x) ∣∣ ≤ ct−1/4. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 17this completes the proof of part (c) of the theorem. next we demonstrate the proof of (b) and (d).if 58 < h < 3 4 by following similar steps, one can show that sup x∈r ∣∣∣∣∣∣p  ( t σ2h θ̃t )1/2 (θ̂t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ cθt 4h−3. if 1116 < h < 3 4 by following similar steps, one can show that sup x∈r ∣∣∣∣∣∣p 2h ( t σ2h θ̃t )1/2 (θ̃t − θ) ≤ x −φ(x) ∣∣∣∣∣∣ ≤ cθt 4h−3.this completes the proof of the theorem. concluding remark for the case 12 ≤ h ≤ 5 8 , our rate is o(t−1/2) is optimal. references [1] j.p.n. bishwal, parameter estimation in stochastic differential equations, springer-verlag, berlin, (2008).[2] j.p.n. bishwal, minimum contrast estimation in fractional ornstein-uhlenbeck process: continuous and discretesampling, fract. calc. appl. anal. 14 (2011) 375–410. https://doi.org/10.2478/s13540-011-0024-6.[3] j.p.n. bishwal, maximum quasi-likelihood estimation in fractional levy stochastic volatility model, j. math. finance.1 (2011) 58–62. https://doi.org/10.4236/jmf.2011.13008.[4] j.p.n. bishwal, sufficiency and rao-blackwellization of vasicek model, theory stoch. processes. 17 (2011) 12-15.[5] j.p.n. bishwal, berry–esseen inequalities for the fractional black–karasinski model of term structure of interestrates, monte carlo methods appl. 28 (2022) 111–124. https://doi.org/10.1515/mcma-2022-2111.[6] j.p.n. bishwal, parameter estimation in stochastic volatility models, springer nature, cham. 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total asymptotically nonexpansive nonself mapping;hybrid mixed type iteration scheme; common fixed point; uniformly convex banach space; weak convergence.45 https://adac.ee https://doi.org/10.28924/ada/ma.1.45 eur. j. math. anal. 1 (2021) 46 definition 1.2. a mapping t is said to be total asymptotically nonexpansive [1] if ‖t n(x)− t n(y)‖ ≤ ‖x − y‖+ µnφ(‖x − y‖) + νn,∀x, y ∈ k,∀n ∈ n, (1.2) where {µn}and {νn} are nonnegative real sequences such that µn → 0 and νn → 0 as n →∞ and φ is a strictly increasing continuous function φ : [0,∞)→ [0,∞) with φ(0) = 0. from the above definitions, we see that the class of total asymptotically nonexpansive mappingsincludes the class of asymptotically nonexpansive mapping as a special case; see [4] for moredetails. each asymptotically nonexpansive mapping is total asymptotically nonexpansive mappingwith νn = 0, µn = kn − 1 f or al l n ≥ 1, φ(t) = t, t ≥ 0. definition 1.3. a subset k of a banach space e is said to be a retract of e if there exists a continuous mapping p : e −→ k (cal led retraction) such that p (x) = x for all x ∈ k. if, in addition p is nonexpansive, then p is said to be nonexpansive retraction of e. if p : e −→ k is a retraction, then p 2 = p. a retract of a hausdorff space must be a closed subset. every closed convex subset of a uniformly convex banach space is a retract. in 2012, yolacan and kiziltune [18] defined the following: definition 1.4. let k be a nonempty and closed convex subset of a banach space e. a nonself mapping t : k → e is said to be total asymptotically nonexpansive mapping if there exist sequences k(1)n and k(2)n in [0,∞) with k(1)n → 0 and k(2)n → 0 as n →∞ and a strictly increasing function φ : [0,∞)→ [0,∞) with φ(0) = 0 such that ‖t (pt )n−1(x)− t (pt )n−1(y)‖ ≤ ‖x − y‖+ k1(n)φ(‖x − y‖) + k (2) n ,∀x, y ∈ k, n ∈ n. (1.3) chidume et al. [3] studied the following iterative scheme in 2004: x1 = x ∈ k xn+1 = p (αnt (pt )n−1xn + (1− αn)xn), n ≥ 1, (1.4) where {αn} is a sequence in (0, 1), k is a nonempty closed convex subset of of a real uniformlyconvex banach space e, p is a nonexpansive retraction of e onto k, and proved some strong andweak convergence theorems for asymptotically nonexpansive nonself mappings in the intermediatesense in the framework of uniformly convex banach spaces.ln 2006, wang [17] generalised the iteration process (1.4) as follows: x1 = x ∈ k, xn+1 = p ((1− αn)xn + αnt1(pt1) n−1yn), yn = p ((1− βn)xn + βnt2(pt2) n−1xn), n ≥ 1, (1.5) eur. j. math. anal. 1 (2021) 47where t1, t2 : k −→ e are two asymptotically nonexpansive nonself mappings, {αn} and {βn} arereal sequences in [0, 1), and proved some weak and strong convergence theorems for asymptoticallynonexpansive nonself mappings.ln 2012, guo et al [8] generalised the iteration process (1.5) as follows: x1 = x ∈ k, xn+1 = p ((1− αn)sn1xn + αnt1(pt1) n−1yn), yn = p ((1− βn)sn2xn + βnt2(pt2) n−1xn), n ≥ 1, (1.6) where t1, t2 : k −→ e are two asymptotically nonexpansive nonself mappings, s1, s2 : k −→ eare two asymptotically nonexpansive self mappings and {αn}, {βn} are real sequences in [0, 1),and proved some strong and weak convergence theorems for mixed-type asymptotically nonexpan-sive mappings. hybrid mixed-type iteration schemelet e be a real uniformly convex banach space, k a nonempty closed convex subset of e and p : e −→ k a nonexpansive retraction of e onto k. let s1, s2, s3 : k −→ k be three totalasymptotically nonexpansive self mappings and t1, t2, t3 : k −→ e be three total asymptoti-cally nonexpansive nonself mappings. then, the hybrid iteration scheme for the above mentionedmappings is as follows: x1 = x ∈ k; xn+1 = p ((1− αn)sn1xn + αnt1(pt1) n−1yn); yn = p ((1− βn)sn2xn + βnt2(pt2) n−1zn); zn = p ((1− γn)sn3xn + γnt3(pt3) n−1xn), (1.7) where {αn}, {βn}, and {γn} are real sequences in [0, 1).the aim of this paper is to study this new hybrid mixed-type iteration scheme (1.7), prove demi-closedness principle for total asymptotically nonexpansive nonself map and establish some conver-gence theorems for mixed-type mappings in the setting of uniformly convex banach spaces. 2. preliminary for the sake of convenience, we restate the following concepts and results:let e be a banach space with its dimension greater than or equal to 2. the modulus of convexityof e is a function δe(ε) : (0, 2] −→ (0, 2] defined by δe(ε) = inf{1− ‖ 1 2 (x + y)‖ : ‖x‖ = 1, ‖y‖ = 1, ε = ‖x − y‖}. eur. j. math. anal. 1 (2021) 48a banach space e is uniformly convex if and if δe(ε) > 0, for all ε ∈ (0, 2].we recall the following: definition 2.1. (see [19]: let % = {x ∈ e : ‖x‖ = 1} and let e? be the dual of e. the space e has gateaux differentiable norm if limn→∞ ‖x+ty‖−‖x‖ t exists ∀x, y ∈ %. definition 2.2. (see [19]: the space e has frechet differentiable norm [15] if for each x ∈ %, the limit of the norm above exists and is attained uniformly for all y ∈ %, and in this case, it is also well known that 〈h, j(x)〉+ 1 2 ‖x‖2 ≤ 1 2 ‖x + h‖2 ≤ 〈h, j(x)〉+ 1 2 ‖x‖2 + b(‖x‖), (2.1) ∀x, y ∈ e, where j is the frechet derivative of the functional 12‖ · |2 at x ∈ e, 〈·〉 is the pairing between e and e? and b is an increasing function defined on [0,∞) such that limt→∞ b(t) t = 0. definition 2.3. : the space e has opial condition [10] if for any sequence {xn} in e, xn converges to x weakly, then it follows that lim supn→∞ ‖xn − x‖ < lim supn→∞ ‖xn − y‖ for all y ∈ e with x 6= y . examples of banach spaces satisfying opial conditions are hilbert spaces and all spaces lp(1 < p <∞). on the other hand, lp[0, π] with 1 < p 6= 2 fails to satify opial condition. definition 2.4. : a mapping t : k −→ k is said to be demiclosed at 0, if for any sequence {xn} in k, the condition that xn converges weakly to x ∈ k and txn converges strongly to 0 implies tx = 0. definition 2.5. : a banach space has the kadec-klec property [14] if for every sequence xn in e, xn → x weakly and ‖xn‖ → ‖x‖, then it follows that ‖xn − x‖ → 0. next, we state the following useful lemmas which will be needed in order to prove our mainresults. lemma 2.1. (see [16]): let {αn}∞n=1, {βn}∞n=1 and {γn}∞n=1 be sequences of nonnegative numbers satisfying the inequality: αn+1 ≤ (1 + βn)αn + γn,∀n ≥ 1. (2.2) if ∑∞ n=1 βn <∞ and ∑∞ n=1 γn <∞, then(1) limn→∞ αn exists(2) ln particular, if {αn}∞n=1 has a subsequence which converges strongly to 0, then limn→∞ αn = 0. eur. j. math. anal. 1 (2021) 49 lemma 2.2. (see [14]): let e be a uniformly convex banach space and 0 < p ≤ tn ≤ q < 1 for each n ≥ 1. suppose that {xn} and {yn} are sequences in e such that lim sup n→∞ ‖xn‖ ≤ r, lim sup n→∞ ‖yn‖ ≤ r and lim n→∞ ‖tnxn + (1− tn)yn‖ = r, (2.3) hold for some r ≥ 0. then limn→∞ ‖xn − yn‖ = 0. lemma 2.3. (see [14]): let e be a real reflexive banach space such that its dual e? has the kadec-klec property. let {xn} be a bounded sequence in e and p, q ∈ ωω(xn) ( where ωω(xn) denotes the set of all weak subsequential limits of {xn}). suppose limn→∞ ‖txn + (1 − t)p − q‖ exists for all t ∈ [0, 1]. then, p = q. lemma 2.4. (see [14]): let k be a nonempty convex subset of a uniformly convex banach space e. then, there exists a strictly incraesing continous convex function φ : [0,∞)→ [0,∞) with φ(0) = 0 such that for each lipshitizian mapping t : c −→ c with the lipschiz constant l, ‖tt x − (1− t)ty − t (tx − (1− t)y)‖ ≤ lφ−1(‖x − y‖ − 1 l ‖tx − ty‖) (2.4) for all x, y ∈ k and for all t ∈ [0.1]. lemma 2.5. (see [2]) let e be a uniformly convex banach space, k a nonempty bounded close convex subset of e. then, there exists a strictly increasing continous convex function φ : [0,∞) −→ [0,∞) with φ(0) = 0 such that for any lipschitizian mapping t : k −→ e with lipschitz constant l ≥ 1 and elements {xn}nj=i in k and any nonnegative numbers {tj}nj=1 with ∑n j=1 tj = 1, the following inequality holds: ‖t ( n∑ j=1 tjxj)− n∑ j=1 tjtxj‖ ≤ lφ−1{max1≤j,k≤n(‖xj − xk‖ − l−1‖txj − txk‖)} lemma 2.6. (see [21]) if the sequence {xn}∞n=1 converges weakly to x , then there exists a sequence of convex combination yj = ∑n(j) k=1 λ (j) k xk+j , λ (j) k ≥ 0 and ∑n(j) k=1 λ (j) = 1, such that ‖yj − x‖ → 0. as n →∞. 3. main results lemma 3.1. ( demiclosedness p r inciple f or nonself total asymptotical ly nonexpansive maps ) let k be a nonempty closed convex and bounded subset of a uniformly convex banach space e and t : k −→ e be l-lipschitz continuous and total asymptotically nonexpansive mapping with the function φ : [0,∞) −→ [0,∞) (such that φ(0) = 0) and nonnegative sequences {k(1)n }, {k(2)n } such that k(1)n , k (2) n → 0 as n →∞. then, i − t is demiclosed at 0. proof. let {xn} converge weakly to ω ∈ k and {xn − txn} converge strongly to 0. we prove that (i − t )ω = 0. clearly, {xn} is bounded. so, there exists ρ > 0 such that {xn} ⊂ c = k ∩ bρ(0),where bρ(0) is a closed ball in e with centre 0 and radius ρ. thus, c is nonempty, closed , eur. j. math. anal. 1 (2021) 50bounded and convex subset in k.claim: t (pt )n−1ω → ω as n → ∞. in fact, since {xn} converges weakly to ω, by lemma6(see [21]), we have for all n > 1, there exists a convex combination yn = m(n)∑ i=1 t (n) i xi+n, t (n) i ≥ 0 and m(n)∑ i=1 t (n) i = 1 such that ‖yn − ω‖ → 0 as n →∞. (3.1) also, since {xn−txn} converges to 0, then for any ε > 0 and a positive integer m ≥ 1, there exists n1 = n(ε) > 0 such that ‖(i − t )xn‖ < ε 1 +m ,∀n ≥ n1. (3.2) hence, ∀n ≥ n1, using definition 1.4 and the fact that p is nonexpansive , we have the followingestimates:for arbitrary but fixed j ≥ 1, we have ‖xn − t (pt )(j−1)xn‖ ≤ ‖(i − t )xn‖+ ‖(t − t (pt ))xn‖ +‖(t (pt )− t (pt )2)xn‖ +‖(t (pt )2 − t (pt )3)xn‖ + · · ·+ ‖(t (pt )j−2 − t (pt )j−1))xn‖ ≤ ‖(i − t )xn‖+ (‖(i − t )xn‖+ µ (1) n φ(‖(i − t )xn‖) +ξ (1) n ) + (‖(i − t )xn‖+ µ (2) n φ(‖(i − t )xn‖) + ξ (2) n ) +(‖(i − t )xn‖+ µ (3) n φ(‖(i − t )xn‖) + ξ (3) n ) + · · ·+ (‖(i − t )xn‖+ µ (j−1) n φ(‖(i − t )xn‖) + ξ (j−1) n ) = ‖(i − t )xn‖+ m−1∑ j=1 ‖(i − t )xn‖+ m−1∑ j=1 µ (j) n φ(‖(i − t )xn‖) + m−1∑ j=1 ξ (j) n ≤ m‖xn − txn‖+mµnφ(‖(i − t )xn‖) +mξn, (3.3) where µn = max1≤j≤m−1{µ(j)n } and ξn = max1≤j≤m−1{ξ(j)n }.from (3.2) and (3.3), we get ‖xn − t (pt )j−1xn‖ < ε. (3.4) now, since t : k −→ e is l-lipschitizian and total asymptotically nonexpansive , so is t : c −→ e. therefore, ∀j ≥ 1, t (pt )j−1 : c −→ e is lipschitizian mapping with the lipschitz constant µj ≥ 1. eur. j. math. anal. 1 (2021) 51in addition, ‖t (pt )j−1yn − yn‖ = ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n + m(n)∑ i=1 t (n) i t (pt )j−1xi+n − m(n)∑ i=1 t (n) i xi+n‖ ≤ ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖ + m(n)∑ i=1 t (n) i ‖t (pt )j−1xi+n − xi+n‖. (3.5) using (3.4), we get m(n)∑ i=1 t (n) i ‖t (pt )j−1xi+n − xi+n‖ < ε,∀n ≥ n. (3.6) furthermore, by lemma 2.5, there exists a strictly increasing continous function φ : [0,∞) −→ [0,∞) with φ(0) = 0 such that for all n ≥ n , we have ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖ = ‖t (pt )j−1( m(n)∑ i=1 t (n) i xi+n)− m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖ ≤ µjφ −1{max1≤j,k≤n(‖xi+n − xi+k‖ −µ−1j ‖t (pt )j−1xi+n − t (pt )j−1xk+n‖)} = µjφ −1{max1≤j,k≤n(‖xi+n − t (pt )j−1xi+n +t (pt )j−1xi+n − t (pt )j−1xk+n +t (pt )j−1xk+n − xi+k‖ −µ−1j ‖t (pt )j−1xi+n − t (pt )j−1xk+n‖)} ≤ µjφ −1{max1≤j,k≤n(‖xi+n − t (pt )j−1xi+n‖ +‖t (pt )j−1xi+n − t (pt )j−1xk+n‖ +‖t (pt )j−1xk+n − xi+k‖ −µ−1j ‖t (pt )j−1xi+n − t (pt )j−1xk+n‖)} ≤ µjφ −1{max1≤j,k≤n(ε+ ε+ (1− µ−1j ) ×‖t (pt )j−1xi+n − t (pt )j−1xk+n‖)} ≤ µjφ −1{max1≤j,k≤n(ε+ ε+ (1− µ−1j )µj ×‖xi+n − xk+n‖} ≤ µjφ −1{max1≤j,k≤n(ε+ ε+ (1− µ−1j )µj ×(‖xi+n‖+ ‖xk+n‖}. eur. j. math. anal. 1 (2021) 52thus, ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖ ≤ µjφ−1(ε+ ε+ 2r(1− µ−1j )µj), (3.7) since xi+n and xk+n are both in c.also, (3.5), (3.6) and (3.7) imply that ‖t (pt )j−1yn − yn‖ ≤ µjφ−1(ε+ ε+ 2r(1− µ−1j )µj). (3.8) taking lim supn→∞ on both sides of (3.8) and noting that ε > 0 is arbitrary, we have that lim sup n→∞ ‖t (pt )j−1yn − yn‖ ≤ µjφ−1(2r(1− µ−1j )µj). (3.9) on the other hand, for any j ≥ 1, it follows from (3.1) that ‖t (pt )j−1ω − ω‖ ≤ ‖t (pt )j−1ω − t (pt )j−1yn‖+ ‖t (pt )j−1yn − yn‖+ ‖yn − ω‖ ≤ µj‖yn − ω‖+ ‖t (pt )j−1yn − yn‖+ ‖yn − ω‖. (3.10) taking lim supn→∞ on both sides of the above inequality and using (3.1) and (3.9), we have ‖t (pt )j−1ω − ω‖ ≤ µjφ−1(2r(1− µ−1j )µj). again, taking lim supj→∞ on both sides of the above inequality, we have lim sup j→∞ ‖t (pt )j−1ω − ω‖ ≤ φ−1(0) = 0, which implies that ‖t (pt )j−1ω − ω‖ → 0 as j →∞, and hence proving our claim. by continuityof tp, we have that lim j→∞ tp (t (pt )j−1ω) = tpω = tω = ω. this completes the proof. � lemma 3.2. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1, s2, s3 : k −→ k be three total asymptotically nonexpansive self mapping with sequences {k(1)n }, {k(2)n }, {k(3)n } ∈ [1,∞), {w (1)n }, {w (2)}n , {w (3)n } ∈ [1,∞) and t1, t2, t3 : k −→ e are three total asymptotically nonexpansive nonself mappings with sequences {µ(1)n }, {µ(2)n }, {µ(3)n } ∈ [1,∞), {ν(1)n }, {ν(2)n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = (f (ti) ∩ f (si)) 6= ∅. if the following conditions hold:i. ∑∞ n=1 k (1) n <∞, ∑∞ n=1 k (2) n <∞, ∑∞ n=1 k (3) n <∞, ∑∞ n=1 µ (1) n <∞, ∑∞ n=1 µ (2) n <∞,∑∞ n=1 µ (3) n <∞, ∑∞ n=1 ν (1) n <∞, ∑∞ n=1 ν (2) n <∞, ∑∞ n=1 ν (3) n <∞,ii. there exists a constant m > 0 such thatψ(t) = φ(t) ≤ mt, t ≤ 0. then, limn∞ ‖xn − q‖ and limn∞ d(xn − f )both exist for all q ∈ f . eur. j. math. anal. 1 (2021) 53 proof. set hn = max(k (1) n , k (2) n , k (3) n , µ (1) n , µ (2) n , µ (3) n ),m = max(m1,m2,m3,m4,m5,m6) and θn =max(ν (1) n , ν (2) n , ν (3) n , ω (1) n , ω (2) n , ω (3) n ). then, ∑∞ n=1 hn < ∞ and ∑∞ n=1 θn < ∞. for any q ∈ f , itfollows from (3.1) that ‖zn − q‖ = |p ((1− βn)sn3xn + βnt3(pt3) n−1xn)− p (q)‖ ≤ ‖(1− βn)sn3xn + βnt3(pt n−1 3 xn − q‖ = ‖(1− βn)sn3xn + βnq − q − βnq + βnt3(pt3) n−1xn‖ = ‖(1− βn)sn3xn − (1− βn)q + βn(t3(pt3) n−1xn − q)‖ = ‖(1− βn)(sn3xn − q) + βn(t3(pt3) n−1xn − q)‖ (3.11) ≤ (1− βn)‖sn3xn − q‖+ βn‖t3(pt3)n−1xn − q‖ ≤ (1− βn)[‖xn − q‖+ k (3) n ψ(‖xn − q‖) + ω (3) n ] + βn[‖xn − q‖+ µ (3) n φ(‖xn − q‖) +ν (3) n ] = (1− βn)‖xn − q‖+ (1− βn)hnψ(‖xn − q‖) + (1− βn)θn + βn‖xn − q‖ +βnhnφ(‖xn − q‖) + βnθn ≤ (1− βn)(1 + hnm5)‖xn − q‖+ βn(1 + hnm6)‖xn − q‖+ θn ≤ (1− βn)(1 + hnm)‖xn − q‖+ βn(1 + hnm)‖xn − q‖+ θn ≤ (1 + hnm)‖xn − q‖+ θn. (3.12) also, form (1.7), we get ‖yn − q‖ = |p ((1− βn)sn2xn + βnt2(pt2) n−1zn)− p (q)‖ ≤ ‖(1− βn)sn2xn + βnt2(pt2) n−1xn − q‖ = ‖(1− βn)sn2xn + βnq − q − βnq + βnt2(pt2) n−1zn‖ = ‖(1− βn)sn2xn − (1− βn)q + βn(t2(pt2) n−1zn − q)‖ (3.13) = ‖(1− βn)(sn2xn − q) + βn(t2(pt2) n−1zn − q)‖ ≤ (1− βn)‖sn2xn − q‖+ βn‖t2(pt2)n−1zn − q‖ ≤ (1− βn)[‖xn − q‖+ k (2) n ψ(‖xn − q‖) + ω (2) n ] + βn[‖zn − q‖+ µ (2) n φ(‖xn − q‖) +ν (2) n ] = (1− βn)‖xn − q‖+ (1− βn)hnψ(‖xn − q‖) + (1− βn)θn + βn‖xn − q‖ +βnhnφ(‖zn − q‖) + βnθn ≤ (1− βn)(1 + hnm3)‖xn − q‖+ βn(1 + hnm4)‖zn − q‖+ θn ≤ (1− βn)(1 + hnm)‖xn − q‖+ βn(1 + hnm)‖zn − q‖+ θn. (3.14) eur. j. math. anal. 1 (2021) 54putting (3.12) into (3.14), we have ‖yn − q‖ ≤ (1− βn)(1 + hnm)‖xn − q‖+ βn(1 + hnm)[(1 + hnm)‖xn − q‖+ θn] + θn = (1 + hnm)[(1− βn)‖xn − q‖+ βn((1 + hnm)‖xn − q‖+ θn)] + θn = (1 + hnm)[(1− βn + βn + βnhnm))‖xn − q‖+ θn)] + θn ≤ (1 + hnm)[1 + hnm))‖xn − q‖+ θn)] + θn = (1 + hnm)2‖xn − q‖+ (2 + hnm)θn. (3.15) again, using (1.7), we have ‖xn+1 − q‖ = |p ((1− αn)sn1xn + αnt1(pt1) n−1yn)− p (q)‖ ≤ ‖(1− αn)sn1xn + αnt1(pt1) n−1yn − q‖ = ‖(1− αn)sn1xn + αnq − q − αnq + αnt1(pt1) n−1yn‖ = ‖(1− αn)sn1xn − (1− αn)q + αn(t1(pt1) n−1yn − q)‖ = ‖(1− αn)(sn1xn − q) + αn(t1(pt1) n−1yn − q)‖ (3.16) ≤ (1− αn)‖sn1xn − q‖+ αn‖t1(pt1)n−1yn − q‖ ≤ (1− αn)[‖xn − q‖+ k (1) n ψ(‖xn − q‖) + ω (1) n ] + αn[‖yn − q‖ +µ (1) n φ(‖yn − q‖) + ν (1) n ] ≤ (1− αn)‖xn − q‖+ (1− αn)hnψ(‖xn − q‖) + (1− αn)θn + αn‖yn − q‖ +αnhnφ(‖yn − q‖) + αnθn ≤ (1− αn)(1 + hnm1)‖xn − q‖+ αn(1 + hnm2)‖yn − q‖+ θn ≤ (1− αn)(1 + hnm)‖xn − q‖+ αn(1 + hnm)‖yn − q‖+ θn. (3.17) putting (3.15) into (3.17), we obtain ‖xn+1 − q‖ ≤ (1− αn)(1 + hnm)‖xn − q‖+ αn(1 + hnm)[(1 + hnm)2‖xn − q‖ +(2 + hnm)θn] + θn] = (1 + hnm)‖xn − q‖ − αn(1 + hnm)‖xn − q‖+ αn(1 + hnm)3‖xn − q‖ +αn(1 + hnm)(2 + hnm)θn + θn ≤ [1 + (3 + 3hnm + h2nm 2)hnm]‖xn − q‖+ [1 + (1 + hnm)(2 + hnm]θn = (1 + δn)‖xn − q‖+ ρn. (3.18) where δn = 1+(3+3hnm+h2nm 2)hnm and ρn = [1+(1+hnm)(2+hnm]θn. since ∑∞ n=1 δn <∞and ∑∞ n=1 ρn <∞, it follows from lemma 2.1 that limn→∞ ‖xn − q‖ exists. eur. j. math. anal. 1 (2021) 55now taking the infimum over all q ∈ f in (3.18), we get d(xn+1, f ) ≤ (1 + δn)d(xn, f ) + ρn,∀n ∈ n. (3.19) again, since ∑∞ n=1 δn <∞ and ∑∞ n=1 ρn <∞, it follows from lemma 2.1 and (3.19) that limn→∞ d(xn, f ) exists. this completes the proof. � lemma 3.3. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1, s2, s3 : k −→ k be three total asymptotically nonexpansive self mapping with sequences {k(1)n }, {k(2)n }, {k(3)n } ∈ [1,∞), {w (1)n }, {w (2)}n , {w (3)n } ∈ [1,∞) and t1, t2, t3 : k −→ e are three total asymptotically nonexpansive nonself mappings with sequences {µ(1)n }, {µ(2)n }, {µ(3)n } ∈ [1,∞), {ν(1)n }, {ν(2)n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = (f (ti)∩f (si)) 6= ∅. if the following conditions hold: i. ∑∞ n=1 k (1) n <∞, ∑∞ n=1 k (2) n <∞, ∑∞ n=1 k (3) n <∞, ∑∞ n=1 µ (1) n <∞, ∑∞ n=1 µ (2) n <∞,∑∞ n=1 µ (3) n <∞, ∑∞ n=1 ν (1) n <∞, ∑∞ n=1 ν (2) n <∞, ∑∞ n=1 ν (3) n <∞,ii. ‖x−t1(pt1)n−1y‖ ≤ ‖sn1x−t1(pt1)n−1y‖, ‖x−t2(pt2)n−1y‖ ≤ ‖sn2x−t2(pt2)n−1y‖, ‖x − t3(pt3)n−1y‖ ≤ ‖sn3x − t3(pt3)n−1y‖iii. there exists a constant m1,m2 > 0 such that ψ(t) ≤ m1t, φ(t) ≤ m2t, t ≥ 0. then, limn∞ ‖xn − sixn‖ = 0 and limn∞ ‖xn − tixn‖ = 0, for i = 1, 2, , 3. proof. set hn = max(k (1) n , k (2) n , k (3) n , µ (1) n , µ (2) n , µ (3) n ),m = max(m1,m2,m3,m4,m5,m6) and θn =max(ν (1) n , ν (2) n , ν (3) n , ω (1) n , ω (2) n , ω (3) n ). then, ∑∞ n=1 hn <∞ and ∑∞ n=1 θn <∞. for any given q ∈ f , limn∞ ‖xn − q‖ exists by lemma 3.2. now, assume that limn∞ ‖xn − q‖ = c. it follows from (3.15),(3.16) and the fact that ∑∞ n=1 hn <∞ and ∑∞ n=1 θn <∞ that lim ‖(1− αn)(sn1xn − q) + αnt1(pt1) n−1yn − q)‖ = c. (3.20) also, we have ‖sn1xn − q‖ ≤ ‖xn − q‖+ k (1) n ψ(‖xn − q‖) + ω (1) n ≤ ‖xn − q‖+ k (1) n m‖xn − q‖) + ω (1) n ≤ (1 + k (1) n m)‖xn − q‖+ ω (1) n ≤ (1 + hnm)‖xn − q‖+ θn ⇒ lim sup ‖sn1xn − q‖ ≤ lim sup[(1 + hnm)‖xn − q‖+ θn] = c. (3.21) eur. j. math. anal. 1 (2021) 56furthermore, ‖t1(pt1)yn − q‖ ≤ ‖yn − q‖+ µ (1) n φ(‖yn − q‖) + ν (1) n ≤ ‖yn − q‖+ µ (1) n m‖yn − q‖) + ν (1) n ≤ (1 + µ (1) n m)‖yn − q‖+ ν (1) n ≤ (1 + hnm)‖yn − q‖+ θn taking limsup on both sides of (3.15), we obtain lim sup ‖yn−q‖ ≤ c and so lim sup ‖t1(pt1)yn−q‖ ≤ lim sup[(1 +hnm)‖yn−q‖+ θn] ≤ c . thus, lim sup ‖t1(pt1)yn − q‖ ≤ lim sup[(1 + hnm)‖yn − q‖+ θn] = c. (3.22) using lemma 2.2, we get lim n→∞ ‖sn1xn − t1(pt1)n−1yn‖ = 0. (3.23)by condition (ii), it follows that ‖xn − t1(pt1)n−1yn‖ ≤ ‖sn1xn − t1(pt1)n−1yn‖, and so from (3.23), we have lim n→∞ ‖xn − t1(pt1)n−1yn‖ = 0. (3.24)also, we have ‖sn2xn − q‖ ≤ ‖xn − q‖+ k (2) n ψ(‖xn − q‖) + ω (2) n ≤ ‖xn − q‖+ k (2) n m‖xn − q‖) + ω (2) n ≤ (1 + k (2) n m)‖xn − q‖+ ω (2) n ≤ (1 + hnm)‖xn − q‖+ θn ⇒ lim sup ‖sn2xn − q‖ ≤ lim sup[(1 + hnm)‖xn − q‖+ θn] = c. (3.25) furthermore, ‖t2(pt2)zn − q‖ ≤ ‖zn − q‖+ µ(2)n φ(‖zn − q‖) + ν(2)n ≤ ‖zn − q‖+ µ(2)n m‖zn − q‖) + ν(2)n ≤ (1 + µ(2)n m)‖zn − q‖+ ν(2)n ≤ (1 + hnm)‖zn − q‖+ θn eur. j. math. anal. 1 (2021) 57 taking lim sup on both sides of (3.12), we obtain lim supn→∞ ‖zn − q‖ ≤ c and so lim sup ‖t2(pt1)zn − q‖ ≤ lim sup[(1 + hnm)‖zn − q‖+ θn] ≤ c. (3.26) (3.13), (3.25), (3.26) and lemma 2.2 imply lim n→∞ ‖sn2xn − t2(pt2)n−1zn‖ = 0. (3.27) (3.27) and condition (ii) yields lim n→∞ ‖xn − t2(pt2)n−1zn‖ = 0. (3.28) from (3.11), using the same argument as was used in obtaining (3.27) above, we get lim n→∞ ‖sn3xn − t3(pt3)n−1xn‖ = 0. (3.29)now, we prove that lim n→∞ ‖xn − t1(pt1)n−1xn‖ = lim n→∞ ‖xn − t2(pt2)n−1xn‖ lim n→∞ ‖xn − t3(pt3)n−1xn‖ = 0.indeed, since ‖xn − t3(pt3)n−1xn‖ ≤ ‖sn3xn − t3(pt3)n−1xn‖, (by condition (ii)), it follows from(3.29) that lim n→∞ ‖xn − t3(pt3)n−1xn‖ = 0. (3.30)since, p (snxn) = snxn and p : e −→ k is a nonexpansive retraction of e onto k, we get ‖zn − sn3xn‖ = ‖p ((1− γn)sn3xn + γnt3(pt3) n−1xn)− sn3xn‖ ≤ ‖(1− γn)sn3xn + γnt3(pt3) n−1xn − sn3xn‖ = ‖ − γn(sn3xn − γnt3(pt3)n−1xn)‖ = γn‖(sn3xn − γnt3(pt3)n−1xn)‖, which by (3.29) gives lim n→∞ ‖zn − sn3xn‖ = 0. (3.31)observe that ‖zn − xn‖ = ‖zn − sn3xn + sn3xn − t3(pt3)n−1xn + t3(pt3) n−1xn − xn‖ ≤ ‖zn − sn3xn‖+ ‖sn3xn − t3(pt3)n−1xn‖ +‖t3(pt3)n−1xn − xn‖. (3.32) thus, it follows from (3.29), (3.30),(3.31) and (3.32) that lim n→∞ ‖zn − xn‖ = 0. (3.33) eur. j. math. anal. 1 (2021) 58again, observe that ‖sn2xn − t2(pt2)n−1xn‖ ≤ ‖sn2xn − t2(pt2)n−1zn‖+ ‖t2(pt2)n−1zn − t2(pt2)n−1xn‖ ≤ ‖sn2xn − t2(pt2)n−1zn‖+ (‖zn − xn‖+ k (2) n φ(‖zn − xn‖) + ν (2) n ≤ ‖sn2xn − t2(pt2)n−1zn‖+ ‖zn − xn‖+mhn(‖zn − xn‖) + θn = ‖sn2xn − t2(pt2)n−1zn‖+ (1 +mhn)‖zn − xn‖+ θn. (3.34) from (3.27),(3.33), (3.34) and the fact that ∑∞ n=1 θn <∞, we get lim n→∞ ‖sn2xn − t2(pt2)n−1xn‖ = 0. (3.35) since ‖xn−t2(pt2)n−1xn‖ ≤ ‖sn2xn−t2(pt2)n−1xn‖ (by condition (ii), it follows from (3.35) that lim n→∞ ‖xn − t2(pt2)n−1xn‖ = 0. (3.36) also, since p (snxn) = snxn and p : e −→ k is a nonexpansive retraction of e onto k, we get ‖yn − sn2xn‖ = ‖p ((1− βn)sn2xn + βnt2(pt2) n−1zn)− sn2xn‖ ≤ ‖(1− βn)sn2xn + βnt2(pt2) n−1zn − sn2xn‖ = ‖ − βn(sn2xn − βnt2(pt2)n−1zn)‖ = βn‖(sn2xn − βnt2(pt2)n−1zn)‖, which by (3.27) gives lim n→∞ ‖yn − sn2xn‖ = 0. (3.37)moreover, since ‖yn − xn‖ = ‖yn − sn2xn + sn2xn − t2(pt2)n−1zn + t2(pt2) n−1xn − zn‖ ≤ ‖yn − sn2xn‖+ ‖sn2xn − t2(pt2)n−1zn‖+ ‖t2(pt2)n−1zn − xn‖, it follows from (3.27), (3.28) and (3.37) that lim n→∞ ‖yn − xn‖ = 0. (3.38) observe that ‖sn1xn − t1(pt1)n−1xn‖ ≤ ‖sn1xn − t1(pt1)n−1yn‖+ ‖t1(pt1)n−1yn − t1(pt1)n−1xn‖ ≤ ‖sn1xn − t1(pt1)n−1yn‖+ (‖yn − xn‖+ k (1) n ψ(‖yn − xn‖) + ν (1) n ≤ ‖sn2xn − t2(pt2)n−1zn‖+ ‖yn − xn‖+mhn(‖zn − xn‖) + θn = ‖sn1xn − t1(pt1)n−1yn‖+ (1 +mhn)‖yn − xn‖+ θn. (3.39) from (3.23), (3.38), (3.39) and the fact that ∑∞ n=1 θn <∞ lim n→∞ ‖sn1xn − t1(pt1)n−1xn‖ = 0. (3.40) eur. j. math. anal. 1 (2021) 59now, since ‖xn−t1(pt1)n−1xn‖ ≤ ‖sn1xn−t1(pt1)n−1xn‖ (by condition (ii), it follows from (3.40)that lim n→∞ ‖xn − t1(pt1)n−1xn‖ = 0. (3.41) from ‖xn+1 − sn1xn‖ = ‖p [(1− αn)sn1xn + αnt1(pt1) n−1yn]− sn1xn‖ ≤ ‖(1− αn)sn1xn + αnt1(pt1) n−1yn − sn1xn‖ = ‖ − αn(sn1xn − t1(pt1)n−1yn])‖ = αn‖sn1xn − t1(pt1)n−1yn]‖ and (3.23), we obtain lim n→∞ ‖xn+1 − sn1xn‖ = 0. (3.42) from ‖xn+1 − t1(pt1)n−1yn‖ ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t1(pt1)n−1yn‖, (3.23) and (3.42), we get lim n→∞ ‖xn+1 − t1(pt1)n−1yn‖ = 0. (3.43) also, from (3.23), (3.24) and the inequality ‖sn1xn − xn‖ ≤ ‖sn1xn − t1(pt1)n−1yn‖+ ‖t1(pt1)n−1yn − xn‖, we have lim n→∞ ‖sn1xn − xn‖ = 0. (3.44) again, from (3.41), (3.44) and the inequality ‖sn1xn − t2(pt2)n−1xn‖ ≤ ‖sn1xn − xn‖+ ‖xn − t2(pt2)n−1xn‖, we have lim n→∞ ‖sn1xn − t2(pt2)n−1xn‖ = 0. (3.45) eur. j. math. anal. 1 (2021) 60since ‖xn+1 − t2(pt2)n−1yn‖ ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t2(pt2)n−1xn‖ +‖t2(pt2)n−1xn − t2(pt2)n−1yn‖ ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t2(pt2)n−1xn‖+ (‖xn − yn‖ +k (2) n φ(‖xn − yn‖) + ν (2) n ) ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t2(pt2)n−1xn‖+ ‖xn − yn‖ +mhn‖xn − yn‖) + θn = ‖xn+1 − sn1xn‖+ ‖sn1xn − t2(pt2)n−1xn‖ +(1 +mhn)‖xn − yn‖) + θn, it follows from (3.38), (3.42), (3.45) and the fact that ∑∞ n=1 θn <∞ that lim n→∞ ‖xn+1 − t2(pt2)n−1yn‖ = 0. (3.46) now, from (3.30), (3.41) and the inequality ‖sn1xn − t3(pt3)n−1xn‖ ≤ ‖sn1xn − xn‖+ ‖xn − t3(pt3)n−1xn‖, we obtain lim n→∞ ‖sn1xn − t3(pt3)n−1xn‖ = 0. (3.47) since ‖xn+1 − t3(pt3)n−1yn‖ ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t3(pt3)n−1xn‖ +‖t3(pt3)n−1xn − t3(pt3)n−1yn‖ ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t3(pt3)n−1xn‖+ (‖xn − yn‖ +k (3) n φ(‖xn − yn‖) + ν (3) n ) ≤ ‖xn+1 − sn1xn‖+ ‖sn1xn − t3(pt3)n−1xn‖+ ‖xn − yn‖ +mhn‖xn − yn‖) + θn = ‖xn+1 − sn1xn‖+ ‖sn1xn − t3(pt3)n−1xn‖ +(1 +mhn)‖xn − yn‖) + θn it follows from (3.38), (3.42), (3.47) and the fact that ∑∞ n=1 θn <∞ that lim n→∞ ‖xn+1 − t3(pt3)n−1yn‖ = 0. (3.48) eur. j. math. anal. 1 (2021) 61again, since (pt i)(pt i)n−2yn−1, xn ∈ k for i = 1, 2, 3 and t1, t2, t3 are three total asymptoti-cally nonexpansive nonself mappings, we have ‖ti(pti)n−1yn−1 − tixn‖ = ‖ti(pti)(pti) n−2yn−1 − ti(pxn)‖ ≤ ‖(pti)(pti) n−2yn−1 − p (xn)‖ +k (i) n φ(‖(pti)(pti) n−2yn−1 − p (xn)‖) + ν (i) n ≤ ‖(pti)(pti) n−2yn−1 − p (xn)‖ +mhn‖(pt i)(pt i)n−2yn−1 − p (xn)‖+ θn = (1 +mhn)‖(pti)(pti) n−2yn−1 − p (xn)‖+ θn = (1 +mhn)‖ti(pti)n−2yn−1 − xn‖+ θn. (3.49) for i = 1.2.3,, it follows from (3.43), (3.46) and (3.48) that lim n→∞ ‖t i(pt i)n−1yn−1 − t ixn‖ = 0. (3.50) observe that ‖xn+1 − yn‖ ≤ ‖xn+1 − t1(pt1)n−1yn‖+ ‖t1(pt1)n−1yn − xn‖+ ‖xn − yn‖, so that, by (3.24), (3.38) and (3.43), we get lim n→∞ ‖xn+1 − yn‖ = 0. (3.51) next,observe, for i = 1, 2, 3, that ‖xn − tixn‖ ≤ ‖xn − ti(pti)n−1xn‖+ ‖ti(pti)n−1xn − ti(pti)n−1yn−1‖ +‖ti(pti)n−1yn−1 − tixn‖ ≤ ‖xn − ti(pti)n−1xn‖+ [‖xn − yn−1‖+ k (i) n φ(‖xn − yn−1‖) +ν (i) n ] + ‖ti(pti)n−1yn−1 − tixn‖ ≤ ‖xn − ti(pti)n−1xn‖+ ‖xn − yn−1‖+ k (i) n m‖xn − yn−1‖ +ν (i) n + ‖ti(pt1)n−1yn−1 − tixn‖ = ‖xn − ti(pti)n−1xn‖+ (1 + k (i) n m)‖xn − yn−1‖+ ν (i) n ] +‖ti(pti)n−1yn−1 − tixn‖ ≤ ‖xn − ti(pti)n−1xn‖+max [supn≥1(1 + k (i) n m)]‖xn − yn−1‖ +max [supn≥1]ν (i) n ] + ‖ti(pti)n−1yn−1 − tixn‖ thus, it follows from (3.30), (3.36), (3.41), (3.50) and (3.51) that limn→∞ ‖xn − tixn‖ = 0, for i = 1, , 2, 3. eur. j. math. anal. 1 (2021) 62finally, we prove that limn→∞ ‖xn − sni xn‖ = 0, for i = 1, , 2, 3.infact, by condition (ii), we have for i = 1, 2, 3, that ‖xn − sni xn ≤ ‖xn − ti(pti)n−1xn‖+ ‖sni xn − ti(pti)n−1xn‖ thus, it follows from (3.29), (3.30), (3.36), (3.40), (3.41) and (3.45) that lim n→∞ ‖xn − sni xn‖ = 0, f or i = 1, 2, 3. (3.52) this completes the proof of lemma 3.3. � lemma 3.4. under the assumption of lemma 3.2, for all p1, p2 ∈ ∩3i1(f (si) ∩ f (ti)), the limit limn→∞ ‖xn + (1− t)p1 − p2‖ exists for all t ∈ [0, 1], where {xn} is the sequence defined by (1.7). proof. by lemma 3.2, limn→∞ ‖xn − q‖ exists for all q ∈ f and therefor {xn} is bounded. let an(t) = ‖xn + (1 − t)p1 − p2‖ exists for all t ∈ [0, 1]. then, limn→∞ a(0) = ‖p1 − p2‖ and limn→∞ a(1) = ‖xn− p2‖ exist by lemma 3.2. it remains therefor to prove lemma 3.4 for t ∈ (0, 1).for all x ∈ k, we define the mapping  rn(x) = p [(1− γn)sn3 + γt3(pt3) n−1xn]; wn(x) = p [(1− βn)sn2 + βt2(pt2) n−1xn]; vn(x) = p [(1− αn)sn1 + αt1(pt1) n−1xn], n ≥ 1. (3.53) then, it follows that xn+1 = vnxn, vnp = p,∀p ∈ f . now, from (3.12), (3.15) and (3.18) of lemma3.2, we see that  ‖rn(x) − rn(y)‖ ≤ (1 + hn)m‖x − y‖+ θn; ‖wn(x) −wn(y)‖ ≤ (1 + rn)m‖x − y‖+ δnθn; ‖vn(x) − vn(y)‖ ≤ (1 + en)m‖x − y‖+ θn = fn‖x − y‖+ gn, (3.54) where rn = 2hn+h2nm 2, δn = 2+hnm, en = 3hnm+3h2nm 2+h3nm 3 and gn = (1+hnm)(2+hnm)θnwith ∑∞ n=1 en <∞, ∑∞ n=1 gn <∞ and fn = 1 + en. since ∑∞ n=1 en <∞, it follows that fn → 1 as n →∞. set sn,m = vn+m−1vn+m−2 · · · vn, m ∈ n; bn,m = ‖sn,m(txn + (1− t)p1)− sn.m(txm + (1− t)p2‖. (3.55) eur. j. math. anal. 1 (2021) 63from (3.54) and (3.55), we have ‖sn,m(x)− sn,m(y)‖ = ‖vn+m−1vn+m−2 · · · vn(x)− vn+m−1vn+m−2 · · · vn(y)‖ ≤ fn+m−1‖vn+m−2vn+m−3 · · · vn(x)− vn+m−2vn+m−3 · · · vn(y)‖ +gn+m−1 ≤ (fn+m−1)(fn+m−2)‖vn+m−3vn+m−4 · · · vn(x) −vn+m−3vn+m−4 · · · vn(y)‖+ gn+m−1 + gn+m−2... ≤ ( n+m−1∏ i=n fi)‖x − y‖+ n+m−1∑ i=n gi = bn‖x − y‖+ n+m−1∑ i=n gi , (3.56) for all x, y ∈ k, where bn = ∏n+m−1 i=n fi , sn,mxn = xn and sn,mp = p for all p ∈ f . thus, an+m(t) = ‖txn + (1− t)p1 − p2‖ = ‖sn,m(txn + (1− t)p1 − p2‖ ≤ bn,m + ‖sn,m(txn + (1− t)p1 − p2‖. (3.57) by using theorem 2.3 in [5], we have bn,m ≤ ψ−1(‖(xn − u‖ − ‖xn+1 − sn,mu‖) = ψ−1(‖(xn − u‖ − ‖xn+1 − u + u − sn,mu‖) ≤ ψ−1(‖(xn − u‖ − (‖xn+1 − u‖+ ‖sn,mu − u‖)), (3.58) so that the sequence {bn.m} converges uniformly to 0, i.e, bn,m → 0 as n →∞. since limn→bn = 1and limn→∞ bn,m = 0, it follows from (3.57) that lim supn→∞ an(t) ≤ lim infb→∞ bn.m ≤ lim infn→∞ an(t).this shows that limn→∞ an(t) exists, i.e, limn→∞ ‖txn + (1 − t)p1 − p2‖ exists for all t ∈ [0, 1].this completes the proof lemma 3.4. � lemma 3.5. under the assumption of lemma 3.2, if e has frechet differentiable norm,then for all p1, jp2 ∈ f = ∩3i=1(f (ti) ∩ f (si)), the limn→∞(〈xn, j(p1 − p2)〉 exists, where {xn} is the sequence defined by (1.7). if ωω(xn) denotes the set of all weak subsequential limits of {xn}, then 〈q1 − q2, j(p1 − p2〉 = 0 for all p1, p2 ∈ f and for all q1, q2 ∈ ωω(xn). proof. suppose that x = p1 − p2 with p1 6= p2 and h = t(xn − p1) in(2.1). then, we have t(〈xn, j(p1 − p2)〉+ 1 2 ‖p1 − p2‖2 ≤ 1 2 ‖txn + (1− t)p1 − p2‖2 ≤ t(〈xn, j(p1 − p2)〉+ 1 2 ‖p1 − p2‖2 + b(t‖xn − p1‖) eur. j. math. anal. 1 (2021) 64since supn≥1 ‖xn − p‖ ≤ q for some q > 0, we have t lim n→∞ sup(〈xn, j(p1 − p2)〉+ 1 2 ‖p1 − p2‖2 ≤ 1 2 lim n→∞ sup‖txn + (1− t)p1 − p2‖2 ≤ t lim n→∞ inf (〈xn, j(p1 − p2)〉+ 1 2 ‖p1 − p2‖2 +b(tq) that is, t limn→∞ sup(〈xn, j(p1 − p2)〉 ≤ t lim infn→∞(〈xn, j(p1 − p2)〉 + b(tq). if t → 0, then limn→∞〈xn − p1, j(p1 − p2)〉 exists for all p1, p2 ∈ f and for all q2, q2 ∈ ωω(xn); in particular, (〈q1 − q2, j(p1 − p2)〉 = 0 for all q2, q2 ∈ ωω(xn). this completes the proof lemma 3.5. � theorem 3.6. under the assumption of lemma 3.2, if e has frechet differentiable norm, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1f (ti)∩f (si). proof. by lemma 3.5, (〈q1− q2, j(p1− p2)〉 = 0 for all q2, q2 ∈ ωω(xn). therefore, ‖q?− p ? ‖2 = 〈q? − p?, j(q? − p?)〉 = 0. this implies that p? = q?. consequently, {xn} converges to a commonfixed point of f = ∩3i=1f (ti) ∩ f (si). this completes the proof theorem 3.6. � theorem 3.7. under the assumption of lemma 3.2, if the dual space e? of e has the kadec klec (kk) property and the mappings i − si and i − ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1(f (ti) ∩ f (si)). proof. by lemma 3.2 {xn} is bounded and since e is reflexive, there exists a subsequence {xnk} of {xn} which converges weakly to some q? ∈ k. by lemma 3.3, we have limn→∞ ‖xnk − sixnk‖ = 0and limn→∞ ‖xnk − tixnk‖ = 0 for i = 1, 2, 3. since by hypothesis, the mappings i − si and i − tifor i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, siq? = q? and tiq ? = q? for i = 1, 2, 3.; which means q? ∈ f = ∩3i=1(f (ti) ∩ f (si)). now, we show that {xn}converges weakly to q?. suppose {xnj} is another subsequence of {xn} which converges weakly to p? ∈ k. by the same method as above, we have p? ∈ f and q? ∈ ωω(xn). by lemma 3.4, the limit limn→∞ ‖txn + (1 − t)q? − p?‖ exists for all t ∈ [0, 1] and so q? = p?. thus, the sequence {xn}converges weakly to q? ∈ f . this completes the proof. � theorem 3.8. under the assumption of lemma 3.2, if e satisfies opial’s condition and the mappings i−si and i−ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1(f (ti)∩ f (si)). proof. let q? ∈ f . from lemma 3.2, the squence {‖xn − p ? ‖} is convergent and hence bounded.since, e is uniformly convex , every bounded subset of e is weakly compact. thus, the existsa subsequence {xnk} of {xn} which converges weakly to some q? ∈ k. by lemma 3.3, we have eur. j. math. anal. 1 (2021) 65 limn→∞ ‖xnk − sixnk‖ = 0 and limn→∞ ‖xnk − tixnk‖ = 0 for i = 1, 2, 3. since by hypothesis, themappings i − si and i − ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosedat zero, siq? = q? and tiq? = q? for i = 1, 2, 3.; which means q? ∈ f = ∩3i=1(f (ti) ∩ f (si)).finally, we show that {xn} converges weakly to q?. suppose on the contrary that {xnj} is anothersubsequence of {xn} which converges weakly to p? ∈ k and q? 6= p? by lemma 3.2, limn→∞ ‖xn− q?‖ and limn→∞ ‖xn − p?‖ exist. by virtue of opial’s condition on e, we obtain lim n→∞ ‖xn − q?‖ = lim n→∞ ‖xnk − q ?‖ < lim n→∞ ‖xnk − p ?‖ = lim n→∞ ‖xn − p?‖ = lim n→∞ ‖xnj − p ?‖ < lim n→∞ ‖xnj − q ?‖ = lim n→∞ ‖xn − q?‖, (3.59) which is a contradiction, so q? = p? therefore, the sequence {xn} defined by (1.7) converges weaklyto q? ∈ f . this completes the proof. � corollary 3.9. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1, s2, s3 : k −→ k be three generalize asymptotically nonexpansive self mapping with sequences {k(1)n }, {k(2)n }, {k(3)n } ∈ [1,∞), {w (1)n }, {w (2)}n , {w (3)n } ∈ [1,∞) and t1, t2, t3 : k −→ e are three generalize asymptotically nonexpansive nonself mappings with sequences {µ(1)n }, {µ(2)n }, {µ(3)n } ∈ [1,∞), {ν(1)n }, {ν(2)n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1).. suppose f = ∩3i=1(f (ti) ∩ f (si)) 6= 0. if the following conditions hold:i. ∑∞ n=1 k (1) n <∞, ∑∞ n=1 k (2) n <∞, ∑∞ n=1 k (3) n <∞, ∑∞ n=1 µ (1) n <∞, ∑∞ n=1 µ (2) n <∞, ∑∞ n=1 µ (3) n < ∞, ∑∞ n=1 ν (1) n <∞, ∑∞ n=1 ν (2) n <∞, ∑∞ n=1 ν (3) n <∞,ii. there exists a constant m > 0 such that ψ(t) = φ(t) ≤ mt, t ≤ 0. then, limn∞ ‖xn − q‖ and limn∞ d(xn − f ) both exist for all q ∈ f . corollary 3.10. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1, s2, s3 : k −→ k be three generalize asymptotically nonexpansive self mapping with sequences {k(1)n }, {k(2)n }, {k(3)n } ∈ [1,∞), {w (1)n }, {w (2)}n , {w (3)n } ∈ [1,∞) and t1, t2, t3 : k −→ e are three generalize asymptotically nonexpansive nonself mappings with sequences {µ(1)n }, {µ(2)n }, {µ(3)n } ∈ [1,∞), {ν(1)n }, {ν(2)n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = ∩3i=1(f (ti) ∩ f (si)) 6= 0. if the following conditions hold: eur. j. math. anal. 1 (2021) 66 i. ∑∞ n=1 k (1) n <∞, ∑∞ n=1 k (2) n <∞, ∑∞ n=1 k (3) n <∞, ∑∞ n=1 µ (1) n <∞, ∑∞ n=1 µ (2) n <∞, ∑∞ n=1 µ (3) n < ∞, ∑∞ n=1 ν (1) n <∞, ∑∞ n=1 ν (2) n <∞, ∑∞ n=1 ν (3) n <∞,ii. ‖x−t1(pt1)n−1y‖ ≤ ‖sn1x−t1(pt1)n−1y‖, ‖x−t2(pt2)n−1y‖ ≤ ‖sn2x−t2(pt2)n−1y‖, ‖x − t3(pt3)n−1y‖ ≤ ‖sn3x − t3(pt3)n−1y‖iii. there exists a constant m1,m2 > 0 such that ψ(t) ≤ m1t, φ(t) ≤ m2t, t ≥ 0. then, limn∞ ‖xn − sixn‖ = 0 and limn∞ ‖xn − tixn‖ = 0, for i = 1, 2, , 3. abbreviations usednot applicable declaration: availability of data and materialnot applicable competing lnterestthe authors declare that there is no conflict of interest. fundingno specific funding received for this work authors contributionika and ncu wrote the paper while dii suggested the idea and did the analysis. the three authorsread and approved the final manuscript. acknowledgementthe authors thank the anonymous reviewers for their careful reading of this paper and approvedthe final manuscript. eur. j. math. anal. 1 (2021) 67references [1] ya.i. alber, c.e. chidume, h. zegeye, approximating fixed points of total asymptotically nonexpansive mappings,fixed point theory appl. 2006 (2006) 10673. https://doi.org/10.1155/fpta/2006/10673.[2] c.e. chidume, e.u. ofoedu, h. zegeye, strong and weak convergence theorems for asymptotically nonexpansivemappings, j. math. anal. appl. 280 (2003) 364?374. https://doi.org/10.1016/s0022-247x(03)00061-1.[3] c.e. chidume, n. shahzad, h. zegeye, convergence theorems for mappings which are asymptotically nonexpan-sive in the intermediate sense, numer. funct. anal. optim. 25 (2005) 239?257. https://doi.org/10.1081/ nfa-120039611.[4] c.e. chidume, e.u. ofoedu, approximation of common fixed points for finite families of total asymptotically 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approximating fixed point of nonexpansive mappings by the lshikawa iteration process, j. math.anal. appl. 178 (1993), 301-308.[17] l. wang, strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansivemappings, j. math. anal. appl. 323 (2006) 550?557. https://doi.org/10.1016/j.jmaa.2005.10.062.[18] e. yolacan, h. kiziltune, on convergence theorems for total asymptotically nonexpansive nonself mappings inbanach space, j. nonlinear sci. appl. 5 (2012), 389?402.[19] d.i. igbokwe, s.j. uko, weak and strong convergence theorems for approximating fixed points of nonexpansivemappings using composite hybrid iteration method, j. nig. math. soc. 33 (2014), 129-144.[20] d.i. igbokwe, s.j. uko, weak and strong convergence of hybrid iteration methods for fixed points of asymptoticallynonexpansive mappings, adv. fixed point theory. 5 (2015), 120-134.[21] p. wojtaszczyk, banach space for analyst, cambridge university press, 1991. https://doi.org/10.1155/fpta/2006/10673 https://doi.org/10.1016/s0022-247x(03)00061-1 https://doi.org/10.1081/nfa-120039611 https://doi.org/10.1081/nfa-120039611 https://doi.org/10.1016/j.jmaa.2006.09.023 https://doi.org/10.1016/s0362-546x(99)00200-x https://doi.org/10.1016/s0362-546x(99)00200-x https://doi.org/10.1090/s0002-9939-1972-0298500-3 https://doi.org/10.1016/j.aml.2011.06.022 https://doi.org/10.1186/1687-1812-2012-224 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/j.jmaa.2005.10.062 1. introduction 2. preliminary 3. main results references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 24doi: 10.28924/ada/ma.3.24 seventh order derivative-free methods for non-differentiable operator equations sunil kumar1, janak raj sharma2, ioannis k. argyros3,∗, samundra regmi4 1department of mathematics, university centre for research and development, chandigarh university, mohali-140413, india sfageria1988@gmail.com 2department of mathematics, sant longowal institute of engineering & technology, longowal, punjab 148106, india jrshira@yahoo.co.in 3department of computing and mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 4department of mathematics, university of houston, houston, tx, 77024, usa sregmi5@uh.edu ∗correspondence: iargyros@cameron.edu abstract. in nonlinear problems where function’s derivatives are difficult or expensive to compute,derivative-free iterative methods are good options to find the numerical solution. one of the importantparts in the development of such methods is to study their convergence properties. in this paper, wereview the concepts of local and semi-local convergence for a derivative-free method for nonlinearequations. in the earlier study of the considered method, the convergence analysis was carried outassuming the existence of higher order derivatives while no derivative is used in the method. suchassumptions certainly restrict its applicability. the present study further provides the estimate ofconvergence radius and bounds on the error for the given method. thus, the applicability of themethod clearly seems to be extended over the wider class of problems. we also review some of therecent developments in this area. the results presented in this paper can be useful for practitionersand researchers in developing and analyzing derivative-free numerical algorithms. 1. introduction there are several numerical methods such as newton’s method, broyden’s method, secant methodand steffensen’s method [3–11,13,14,17] that can be used to approximate x∗ of the equation f (x) = 0, (1.1) for f : ω ⊂ z → z, f is a continuous operator, acting between banach space z and itself.newton’s method is a popular iterative method used to find the roots of a nonlinear equation. received: 30 apr 2023. key words and phrases. divided difference; banach space; convergence; order of convergence.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 2iterative solution methods are commonly used when it is not possible to obtain the solution x∗ inclosed or analytical form. instead, these methods generate a sequence of approximate solutionsthat converge towards the true solution x∗.steffensen’s method [5, 9] defined for each n = 0, 1, 2, . . . by xn+1 = xn − b−1f (xn), (1.2) where b = bn = [un, xn;f ] and un = xn + f (xn), has been used extensively to generate such asequence converging quadratically to x∗.many iterative approaches have been developed to improve efficiency and order convergence(see [1, 2, 15,16]). an approach established in [16] that is defined for x0 ∈ ω by un = xn + f (xn), vn = xn − f (xn), d = dn = [un, vn;f ], yn = xn −d−1f (xn), zn = yn − (3i − 2d−1[yn, xn;f ])d−1f (yn), xn+1 = zn − (13 4 i −d−1[zn, yn;f ] (7 2 i − 5 4 d−1[zn, yn;f ] )) d−1f (zn), (1.3) has received significant attention in this paper. the convergence order seven is shown in [16],when z = rm etc. using assumption on f i , i = 1, 2, . . . , 8 not present in the method, significantlyreducing its applicability although it may converge.consider the function f (t) = { 7t3 log(t) + 5t5 − 5t4, t 6= 0 0, t = 0 (1.4) then, in any neighborhood of 0 and 1, say f ′′′ is unbounded. hence, the results in [16] cannotassure convergence to t∗ = 1. but the method converges.in this article we study convergence of the method (1.3) that includes mainly the local andsemi-local convergence (not provided in [16]).local convergence analysis uses information about the actual solution to determine the rateand radius of convergence of the method. this typically involves estimating the size of the regionaround the true solution where the method is guaranteed to converge. this type of analysis alsousually involves deriving upper bounds on the error norms, which provide an estimate of how closethe iterates of the method are to the true solution.in contrast, in semi-local convergence analysis, the convergence behavior of the method is studiedusing information from the initial point, typically by deriving sufficient conditions that guaranteeconvergence of the method. this analysis is usually carried out without any knowledge of theactual solution of the problem.generalized lipschitz-type conditions are often used in both semi-local and local convergenceanalysis. these conditions involve bounding the difference between the iterates of the method https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 3and the true solution using a lipschitz constant or a related quantity. these conditions can beused to derive sufficient conditions for convergence, as well as to estimate the rate and radius ofconvergence of the method.it is crucial to examine how technique (1.3) converges in both the local (section 2) and thesemi-local (section 3) cases. moreover, our approach gives a prior error estimates and isolationof the solution results not provided before and in banach space. this approach also enables acomparison of the convergence criteria of method. if the approach is examined separately, the newconvergence criteria may be weaker than those that have been provided. the numerical examplesare included in section 4, and the conclusions are discussed in section 5. 2. local convergence some real functions assist in the local analysis of the method. set t = [0,+∞). assume: (h1) there exist continuous as well as nondecreasing functions (cn) f1 : t → t, f2 : t → t,and w0 : t × t → r so that the equation w0(f1(t), f2(t))− 1 = 0 admits a smallest solution (ss) denoted by δ ∈ t − {0}. let t0 = [0, δ). (h2) there exist (cn) functions w : t0 → t, w1 : t0×t0×t0 → t, and w2 : t0×t0×t0 → tsuch that the equations hi(t)− 1 = 0, i = 1, 2, 3have (ss) solutions denoted by δi ∈ t0 − {0}, provided that h1(t) = w1(f1(t), f2(t), t) 1− w0(f1(t), f2(t)) , h2(t) = [w1(h1(t)t, f1(t), f2(t)) 1− w0(f1(t), f2(t)) + 2w2(t, h1(t)t, f1(t), f2(t))(1 + w(t)) (1− w0(f1(t), f2(t)))2 ] h1(t), h0(t) = 1 4 [ 5 (w2(h1(t)t, h2(t)t, f1(t), f2(t)) 1− w0(f1(t), f2(t)) )2 + 4 w2(h1(t)t, h2(t)t, f1(t), f2(t)) 1− w0(f1(t), f2(t)) ] , h3(t) = [w1(f1(t), f2(t), h2(t)t) 1− w0(f1(t), f2(t)) + h0(t)(1 + w(h2(t)t)) 1− w0(f1(t), f2(t)) ] h2(t). consider, the parameter δ∗ given as δ∗ = min{δi}. (2.5) let t1 = [0, δ∗). these definitions imply 0 ≤ w0(f1(t), f2(t)) < 1 (2.6) and 0 ≤ hi(t) < 1, (2.7) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 4for all t ∈ t1.let b(x̄ , r), b̄[x̄ , r ] abbreviate open and closed balls in s1, respectively so that thecenter is x̄ and the radius is some r > 0. the preceding real functions are associated tothe divided difference [., .;f ] as: (h3) there exists an invertible operator l ∈ l(z) so that for each x ∈ ω, u = x + f (x), v = x − f (x) ||l−1([u, v ;f ]− l)|| ≤ w0(||u − x∗||, ||v − x∗||), ||u − x∗|| ≤ f1(||x − x∗||), ||v − x∗|| ≤ f2(||x − x∗||). let b0 = b(x∗, δ). (h4) ||l−1([x, x∗;f ]− l)|| ≤ w(||x − x∗||), ||l−1([x, y ;f ]− [z, x∗;f ])|| ≤ w1(||x − x∗||, ||y − x∗||, ||z − x∗||) and ||l−1([u, v ;f ]− [y , x ;f ])|| ≤ w2(||x − x∗||, ||y − x∗||, ||u − x∗||, ||v − x∗||), for each x, y , z, u, v ∈ b0. (h5) b[x∗, δ∗] ⊂ ω.the local analysis is based on the conditions (h1)− (h5) under the preceding notations. theorem 2.1 assume the conditions (h1)− (h5) are validated. if x0 ∈ b(x∗, δ∗)− {x∗}, then the following items are valid {xn} ⊂ b(x∗, δ∗), (2.8) ‖yn − x∗‖ ≤ h1(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖ < δ∗, (2.9) ‖zn − x∗‖ ≤ h2(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖, (2.10) ‖xn+1 − x∗‖ ≤ h3(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖ (2.11) and the sequence {xn} is convergent to x∗. proof. these items are shown by mathematical induction. by the condition (h3), estimate (2.7)for u0 = x0 + f (x0), v0 = x0 − f (x0) it follows ‖l−1([u0, v0;f ]− l)‖ ≤ w0(‖u0 − x∗‖, ‖v0 − x∗‖) ≤ w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) ≤ w0(f1(u), f2(v)) < 1. (2.12) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 5then, the existence of [u0, v0;f ]−1 is assured by the estimate (2.12) and the perturbation lemmaon linear operators with inverses attributed to banach [6]. we also have ‖[u0, v0;f ]−1l‖ ≤ 1 1− w0(f1(‖u0 − x∗‖), f2(‖u0 − x∗‖) . (2.13) thus, the iterate y0 is well defined and y0 − x∗ = x0 − x∗ − [u0, v0;f ]−1f (x0). [u0, v0;f ]−1([u0, v0;f ]− [x0, x ∗;f ])(x0 − x∗). (2.14) then, the conditions (h3), (h4), (2.5), (2.7) (for i = 1), (2.13) and (2.14) give in turn ‖y0 − x∗‖ ≤ w1(‖u0 − x∗‖, ‖v0 − x∗‖, ‖x0 − x∗‖)‖x0 − x∗‖ 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) ≤h1(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < δ∗. (2.15) hence, the iterate y0 ∈ b(x∗, δ∗) and the item (2.9) is validated for n = 0.notice that the iterates z0 and x1 are also well defined by the second and the third substep ofthe method (1.3). in particular, we get z0 − x∗ = y0 − x∗ −d−1f (y0) + 2d−1(d − [yn, xn;f ])d−1f (yn) =d−1(d − [yn, x ∗;f ])(yn − x∗) + 2d−1([u0, v0;f ]− [y0, x0;f ])d−1f (y0). (2.16) but we can write by the first substep that f (y0) = f (y0)− f (x∗) = [y0, x ∗;f ](y0 − x∗), so by (h4) ‖l−1f (y0)‖ = ‖l−1([y0, x ∗;f ]− l+ l)(y0 − x∗)‖ ≤ (1 + w(‖y0 − x∗‖))‖y0 − x∗‖. (2.17) consequently, (2.5), (2.7) (for i = 2), (h4), (2.13), (2.16) and (2.17) imply ‖z0 − x∗‖ ≤ [ w1(‖y0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖) 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) + 2w2(‖x0 − x∗‖, ‖y0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖)(1 + w(‖y0 − x∗‖)) (1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)))2 ] ‖y0 − x∗‖, ≤ h2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖. (2.18) thus, the iterate z0 ∈ b(x∗, δ∗) and the item (2.10) is validated for n = 0. moreover, the thirdsubstep gives x1 − x∗ = z0 − x∗ −d−1f (z0)− ad−1f (z0) = d−1(d − [z0, x ∗;f ])(z0 − x∗)− ad−1f (z0), (2.19) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 6where a = − [9 4 i − 7 2 d−1[zn, yn;f ] + 5 4 (d−1[zn, yn;f ])2 ] = − 1 4 ( 5(d−1([zn, yn;f ]− [un, vn;f ]))2 − 4d−1([zn, yn;f ]− [un, vn;f ]) ) . therefore, ‖a‖ ≤ 1 4 [ 5 (w2(‖y0 − x∗‖, ‖z0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖) 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) )2 + 4 w2(‖y0 − x∗‖, ‖z0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖) 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) ] = h0. (2.20) then, by (2.5), (2.7) (for i = 3), (h4), (2.13) and (2.18)-(2.20) ‖x1 − x∗‖ ≤ [ w1(‖u0 − x∗‖, ‖v0 − x∗‖, ‖z0 − x∗‖) 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) + h0(1 + w(‖z0 − x∗‖)) 1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)) ] ‖z0 − x∗‖ ≤ h3(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖. (2.21) hence, the items (2.8) and (2.11) are validated for n = 0 and the iterate x1 ∈ b(x∗, δ∗). if thepreceding calculations are repeated with xm, ym, xm+1, replacing x0, y0, x1, respectively, theinduction for the items (2.8)-(2.11) is terminated. furthermore, from estimation ‖xm+1 − x∗‖ ≤ µ‖xm − x∗‖ < ‖xm − x∗‖, (2.22) where µ = h3(‖x0 − x∗‖) ∈ [0, 1), we conclude that limm→∞ xm = x∗ and the iterate xm+1 ∈ b(x∗, δ∗). � remark 2.2 the second and third hypotheses in (h3) are left as uncluttered as possible. somepossible choices for the functions f1 and f2 are specified. un − x∗ = xn − x∗ + f (xn) = (i + [xn, x ∗;f ])(xn − x∗) = (i + l+ ll−1([xn, x ∗;f ]− l))(xn − x∗), so ‖un − x∗‖ ≤ ( ‖i + l‖+ (‖l‖w(‖(xn − x∗)‖)) ) ‖(xn − x∗)‖.thus, we can choose f1(t) = ( ‖i + l‖+ ‖l‖w(t) ) t.notice also that we can set w(t) = w0(0, t).similarly, we define f2(t) = ( ‖i − l‖+ ‖l‖w(t) ) t.in view of the above the second and third conditions in (h3) can be dropped if (h5) is replaced by (h5) ′ b[x∗, δ̄] ⊂ ω, where δ̄ = max{δ∗, f1(δ∗)δ∗, f2(δ∗)δ∗}. https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 7possible choices for l are l = f ′(x∗) (the differentiable case) or l = [., .;f ] (the non-differentiable case). in practice l should be chosen to optimize the results.next, the point x∗ is shown to be the only solution of the equation f (x) = 0 in a certain set. preposition 2.3 assume: there exists a solution x∗1 ∈ b(x∗, δ4) of the equation f (x) = 0 for some δ4 > 0; the first assumption in (h3) is validated on the ball b(x∗, δ4) and there exists δ5 ≥ δ4 so that w(δ5) < 1. let b1 = ω ∩ b[x∗, δ5]. then, the only solution of the equation f (x) = 0 in the set b1 is x∗. proof. let q = [x∗, x∗1 ]. by the assumption it follows ‖l−1(q− l)‖ ≤ w(‖x∗1 − x∗‖) ≤ w(δ5) < 1, thus q−1 ∈ l(z) and consequently from the approximation x∗1 − x∗ = q−1(f (x∗1 )− f (x∗)) = q−1(0) = 0, it is concluded that x∗1 = x∗. �clearly, we can choose δ4 = δ∗. 3. semi-local analysis the role of x∗ is exchanged by x0. but there are some more differences.assume: (c1) there exist (cn) functions g1 : t0 → t, g2 : t0 → t and w0 : t0 × t0 → t so that theequation w0(g1(t), g2(t))− 1 = 0 has a (ss) denoted by r0 ∈ t0 − {0}. set t3 = [0, r0).define the scaler sequence {an} for a0 = 0, b0 ∈ [0, r0) and some (cn) functions g1 : t3 → t, g2 : t3 → t, w2 : t3 × t3 × t3 × t3 → t by cn = bn + [w2(an, bn, f1(an), f2(an)) 1− w0(g1(an), g2(an)) + 2w2(an, bn, f1(an), f2(an)) (1− w0(g1(an), g2(an)))2 ] (βn − an), (3.23) βn = 1 4 [ 5 (w2(an, bn, f1(an), f2(an)) 1− w0(g1(an), g2(an)) )2 + 4 (w2(an, bn, f1(an), f2(an)) 1− w0(g1(an), g2(an)) )2] , https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 8 γn = w2(an, bn, g1(an), g2(an))(bn − an), an+1 = cn + (1 + w0(bn, cn))(cn − bn)βn + γn 1− w0(g1(an), g2(an)) , δn+1 = (1 + v0(an, bn))(an+1 − an) + (1 + w0(g1(an), g2(an)))(bn − an) and bn+1 = an+1 + δn+1 1− w0(g1(an+1), g2(an+1)) . next, general convergence conditions are developed. lemma 3.1 assume there exists µ1 ∈ [0, r0) such that for each n = 0, 1, 2, . . ., (c2) w0(g1(an), g2(an)) < 1 and an ≤ µ1. then, the following items hold 0 ≤ an ≤ bn ≤ cn ≤ an+1 ≤ µ1 and there exists a∗ ∈ (0, µ1] such that limn→∞ an = a∗. proof. the conclusions follow immediately by the formula (3.23) and the condition (c2). �notice that the limit a∗ is unique, since it is the unique least upper bound of the sequence {an}. (c3) there exist an invertible operator l and a point x0 ∈ ω such that for each x, y ∈ ω, u = x + f (x), v = x − f (x) ‖l−1([u, v ;f ]− l)‖ ≤ w0(‖u − x0‖, ‖v − x0‖), ‖u − x0‖ ≤ g1(‖x − x0‖), ‖v − x0‖ ≤ g2(‖x − x0‖) and w0(g1(‖f (x0)‖), g2(‖f (x0)‖)) < 1.the existence of [u0, v0;f ]−1 is guaranteed, by the banach lemma and since l−1‖[u0, v0;f ]− l‖ ≤ w0(‖u0 − x0‖, ‖v0 − x0‖) < 1. (c4) ‖[u0, v0;f ]−1f (x0)‖ ≤ b0. let b2 = b(x0, µ0). (c5) ‖l−1([x, y ;f ]− [u, v ;f ])‖ ≤ w2(‖x−x0‖, ‖y−x0‖, ‖u−x0‖, ‖v−x0‖) for each x, y , u, v ∈ b2. (c6) b[x0, a ∗] ⊂ ω. next, the semi-local convergence is provided for the method (1.3). https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 9 theorem 3.2 assume the conditions (c1)− (c6) hold. then, the following items hold {xn} ⊂ b(x0, a ∗), (3.24) ‖yn − xn‖ ≤ bn − an, (3.25) ‖zn − yn‖ ≤ cn − bn, (3.26) ‖xn+1 − zn‖ ≤ an+1 − cn, (3.27) and there exists a solution x∗ of the equation f (x) = 0 such that ‖x∗ − xn‖ ≤ a∗ − an. (3.28) proof. the items (3.24)-(3.27) are shown by induction. notice that the iterates y0, z0, x1 existsby the invertibility of [u0, v0;f ] and the method (1.3). the estimate (3.25) is validated for n = 0,since by the condition (c4) ‖y0 − x0‖ = ‖[u0, v0;f ]−1f (x0)‖ ≤ b0 = b0 − a0 ≤ a∗ (3.29) and the iterate y0 ∈ b(x0, a ∗).then, as in the local convergence case but using x0, (c) instead of x∗, (h), we obtain from f (yn) = f (yn)− f (xn)−d(yn − xn) = ([yn, xn;f ]−d)(yn − xn), so ‖l−1f (yn)‖ ≤ w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖). (3.30) hence, by the second substep zn − yn = −d−1f (yn)− 2d−1(d − [yn, xn;f ])−1d−1f (yn), and ‖zn − yn‖ ≤ [w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖) 1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖)) + 2 (w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖) 1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖)) )2] ‖yn − xn‖ ≤ an − bn (3.31) and ‖zn − x0‖ ≤ ‖zn − yn‖+ ‖yn − x0‖ ≤ an − bn + bn − a0 = cn < a∗, thus the item (3.26) holds and the iterate z0 ∈ b(x0, a ∗).moreover, by the third substep xn+1 − zn = −bd−1f (zn), (3.32) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 10where b = − 1 4 ( 13i − 14d−1[zn, yn;f ] + 5(d−1[zn, yn;f ])2 ) = − 1 4 ( 5(d−1[zn, yn;f ]− i)2 − 4(d−1[zn, yn;f ]− i) + 4i ) , thus ‖b‖ ≤ 1 4 ( 5 (w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖) 1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖)) )2 + 4 (w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖) 1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖)) ) + 4 ) = β̄ < βn. consequently, we have ‖xn+1 − zn‖ ≤ β̄n(1 + w0(‖yn − x0‖, ‖zn − x0‖))‖zn − yn‖ 1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖)) ≤ βn(1 + w0(βn, cn))(cn − bn) 1− w0(f1(an), f2(an)) = an+1 − cn < βn (3.33) and ‖xn+1 − x0‖ ≤ ‖xn+1 − zn‖+ ‖zn − x0‖ ≤ an+1 − cn + cn − a0 = an+1 < a∗.hence, the item (3.27) is validated and iterate xn+1 ∈ b(x0, a ∗). � remark 3.3 as in the local case the functions g1 and g2 can be expressed in terms of the rest ofthe conditions.assume that there exists a cn function ϕ1 : t → r such that for each x ∈ ω ‖l−1([x, x0;f ]− l)‖ ≤ ϕ1(‖x − x0‖). then, from the estimate un − x0 = xn − x0 + f (xn)− f (x0) + f (x) =(i + l+ ll−1([xn, x0;f ]− l))(xn − x0) + f (x0), so we can choose g1(t) = (‖1 + l‖+ ‖l‖ϕ1(t))t + ‖f (x0)‖and similarly g2(t) = (‖1− l‖+ ‖l‖ϕ1(t))t + ‖f (x0)‖.under these choices of g1 and g2 ‖u − x0‖ = g1(‖x − x0‖), ‖v − x0‖ = g2(‖x − x0‖) and the second and third conditions in (h3) can be dropped.possible choices for l are l = f ′(x0) (the differentiable case) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 11or l = [., .;f ] (the non-differentiable case).the condition (c6) can be replaced by (c6) ′ b[x0, ā] ⊂ ω, where ā = max{a∗, g1(a∗), g2(a∗)}.a uniqueness of the solution set is specified. proposition 3.4 assume: there exists a solution d ∈ b(x0, δ6) of the equation f (x) = 0 for some δ6 > 0; the first condition in (h3) holds in the ball b(x0, δ6) and there exists δ7 ≥ δ6 such that w0(δ6, δ7) < 1. let b3 = b[x0, δ7] ∩ω. then, the point d is the only solution of the equation f (x) = 0 in the set b3. proof. let d1 ∈ b3 be such that f (d1) = 0. define the divided difference [d, d1;f ]. then, ‖l−1([d, d1;f ]− l)‖ ≤ w0(‖d − x0‖, ‖d1 − x0‖) ≤ w0(δ6, δ7) < 1, thus d1 = d. � remark 3.5 if all the conditions (c1)− (c6) hold, then set d = x∗ and δ6 = a∗. 4. numerical tests in order to validate the theoretical deductions, we take into account the following numericalexamples to estimate the real parameters defined in the preceding sections: example 1. let z = r× r× r and ω = b(ξ∗, 1) with ξ∗ = (0, 0, 0)t . define the mapping f for ξ = (ξ1, ξ2, ξ3) t , ξi ∈ r by f (ξ) = ( ξ1, e ξ2 − 1, (e − 1) 2 ξ23 + ξ3 )t . this definition gives that the f ′ of the mapping f is the jacobian matrix f ′(ξ) = 1 0 0 0 eξ2 0 0 0 (e − 1)ξ3 + 1  . notice that f (ξ∗) = o and f ′(ξ∗) = i. then, the conditions (h1)− (h4) are validated if w0(t1, t2) = 1 2 (e − 1)(t1 + t2), w(t) = 1 2 (e − 1)t, w1(t1, t2, t3) = 1 2 (e − 1)(t1 + t2 + t3), w2(t1, t2, t3, t4) = 1 2 (e − 1)(t1 + t2 + t3 + t4) https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 12and the functions f1 and f2 are given in remark 2.2. then, the radius δ∗ using (2.5) is δ1 = 0.20415, δ2 = 0.13109, δ3 = 0.11960 and δ∗ = 0.11960. figure 1. graph of radius of convergence of example 1. ∆1 ∆2 ∆3 -0.2 -0.1 0.0 0.1 0.2 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 t h i t h i h1 h2 h3 example 2. let z = c[0, 1] be the space of continuous functions defined in [0, 1] and ω = b̄(l∗, 1).consider the integral equation of the mixed hammerstein-type [6, 12] by l(d) = ∫ 1 0 t (d, ω) ( l(ω)3/2 + l(ω)2 2 ) dω, (4.34) t (d, ω) = { (1− d)ω, ω ≤ d, d(1− ω), d ≤ ω.notice that l∗(d) = 0. define h : ω ⊆ [0, 1]→ c[0, 1] as h(l)(d) = l(d)− ∫ 1 0 t (d, ω) ( l(ω)3/2 + l(ω)2 2 ) dω. the derivative h′ is given by h′(l)q(d) = q(d)− ∫ 1 0 t (d, ω) (3 2 l(ω)1/2 + l(ω) ) dω, since h′(l∗(d)) = 1, it follows ‖h′(α)−1(h′(l)−h′(q))‖ ≤ 5 16 ‖l − q‖. (4.35) in (4.35), switch q by l0 ‖h′(α)−1(h′(l)−h′(l0))‖ ≤ 5 16 ‖l − l0‖.thus, we take w0(t1, t2) = t1 + t2, w(t) = t, w1(t1, t2, t3) = t1 + t2 + t3, w2(t1, t2, t3, t4) = 5. https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 13hence, we obtain δ1 = 0.17539, δ2 = 0.02678, δ3 = 0.90819× 10−3 and δ∗ = 0.90819× 10−3. figure 2. graph of radius of convergence of example 2. ∆1 ∆2 ∆3 -0.6 -0.4 -0.2 0.0 0.2 -20 0 20 40 60 80 t h i t h i h1 h2 h3 example 3. let z = c[0, 1] be the space of continuous functions [12] defined on the interval [0, 1]and ω = b̄(0, 1). define the function h on ω by h(ϕ)(l) = ϕ(l)− 10 ∫ 1 0 lρϕ(ρ)3dρ. it follows that h′(ϕ(ξ))(l) = ξ(l)− 30 ∫ 1 0 lρϕ(ρ)2ξ(ρ)dρ, for each ξ ∈ ω. since l∗ = 0, so we can set w0(t1, t2) = 2(t1 + t2), w(t) = t 5 , w1(t1, t2, t3) = 2(t1 + t2 + t3), w2(t1, t2, t3, t4) = 2(t1 + t2 + t3 + t4). hence, we obtain δ1 = 0.98449× 10−1, δ2 = 0.62003× 10−1, δ3 = 0.55704× 10−1 and δ∗ = 0.55704× 10−1. example 4. lastly, a nondifferentiable nonlinear system on r × r is solved using the method(1.3), where the divided difference is defined by the 2 × 2 matrix given for t̄ = (t1, t2) ∈ r × r, t̃ = (t3, t4) ∈ r× r, and f = (f1, f2) by [t̄ , t̃;f ]i ,1 = fi(t3, t4)− fi(s1, s4) t3 − t1 t3 6= t1 and [t̄ , t̃;f ]i ,2 = fi(s1, s4)− fi(s1, s2) s4 − s2 . s4 6= s2. https://doi.org/10.28924/ada/ma.3.24 eur. j. math. anal. 10.28924/ada/ma.3.24 14 figure 3. graph of radius of convergence of example 3. ∆2 ∆1 ∆3 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 -1 0 1 2 t h i t h i h1 h2 h3 otherwise, we set [., .; .] = o. 7 let us consider the nonlinear and nondifferentiable system as 3t21 t2 + t22 − 1 + |t1 − 1| = 0, t41 + t1t 3 2 − 1 + |t2| = 0. then, we set f = (f1, f2), where f1(t1, t2) = f1 = 3t21 t2 + t22 − 1 + |t1 − 1| = 0 and f2(t1, t2) = f2 = t41 + t1t 3 2 − 1 + |t2| = 0. choose initial points (5, 5) and (1, 0). then, using the aforementioned divided difference andthe method (1.3) we obtain the solution x∗ = (x∗1 , x ∗ 2 ) after three iterations with x∗1 = 0.894655074966771 and x∗2 = 0.327826643198819. 5. conclusion the focus of this paper is to provide a comprehensive analysis of the local and semilocal con-vergence of a derivative-free seventh-order method in banach space. it is noteworthy that theconvergence has been investigated in earlier studies by assuming the existence of some higherorder derivatives, which in fact are not used in the iterative method. contrary to this, our approachonly considers the first-order divided differences that are actually present in the iterative process.this unique feature makes the method applicable to a wider range of functions, thereby expandingits utility. in the analysis, we present an error estimate and convergence ball that bounds the iter-ates, providing further benefits to the analysis of convergence. in addition, the sufficient conditionsare developed to show the uniqueness of solution in the given domain. to verify the theoreticalresults, we have 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https://doi.org/10.1007/s11075-015-9960-2 https://doi.org/10.1016/j.amc.2010.03.028 1. introduction 2. local convergence 3. semi-local analysis 4. numerical tests 5. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 6doi: 10.28924/ada/ma.4.6 the constants to measure the differences between isosceles and α− β orthogonalities qichuan ni, qi liu∗, yin zhou, qin qian school of mathematics and physics, anqing normal university, anqing 246133, p. r. china niqichuan111@163.com, liuq67@aqnu.edu.cn, zhouyin0330@163.com, 15212956918@163.com ∗correspondence: liuq67@aqnu.edu.cn abstract. in this paper, by combining the isosceles orthogonality and α−β orthogonality of banachspaces, we first introduce a new geometric constant. we demonstrate some basic properties aboutit, such as calculating its value in the common norm spaces. moreover, the necessary and sufficientconditions for the new constant to characterize hilbert spaces are given. finally, only consider thepoints on the unit sphere, we introduce another new geometric constant and some basic propertiesare also obtained. 1. introduction traditional orthogonality plays a key role in the geometry of banach spaces and is a geometricfeature of hilbert spaces. we repeat the definitions of the next three orthogonality types. in 1945,james [8] introduced isosceles orthogonality as follows: x ⊥i y if and only if ‖x + y‖ = ‖x − y‖. balestro [4] introduced the orthogonality of pythagoras as follows: x ⊥p y if and only if ‖x − y‖2 = ‖x‖2 + ‖y‖2. birkhoff defined the following birkhoff orthogonality [5] in linear metric spaces: x ⊥b y if and only if ‖x + αy‖ ≥ ‖x‖ for all α ∈ r. these three orthogonalities have been investigated in several papers (see [2], [9] and so on).over the years, many scholars have introduced the concept of extended orthogonality. the familyof orthogonalities introduced by carlson [6] in 1961, covering isosceles and pythagorean orthogo-nalities, is known as carlson orthogonality: x ⊥c y if and only if n∑ i=1 ai‖bix + ciy‖2 = 0. received: 22 jan 2024. key words and phrases. isosceles orthogonality; α− β orthogonality; geometric constant; hilbert spaces.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 2a further special case of carlson orthogonality was introduced by dimitini et al. [7] in 1983 asfollows: x ⊥α y if and only if (1 + α2)‖x − y‖2 = ‖x − αy‖2 + ‖y − αx‖2,where fixed α 6= 1. two years later, this orthogonality was generalized by the same author [3] as: x ⊥αβ y if and only if ‖x − y‖2 + ‖αx − βy‖2 = ‖x − βy‖2 + ‖y − αx‖2. it is well known that in the general banach spaces, these orthogonalities are not the same. forexample, isosceles orthogonality has symmetry, but birkhoff orthogonality does not have symmetry,which means that x ⊥b y cannot show that y ⊥b x . because of this difference, measuring thedifference between the two types of orthogonality is of great significance. many scholars havedefined and studied many novel orthogonality geometric constants and given a large number ofresults, including well-known constants (see [11], [12]): bi(x) = sup { ‖x + y‖ − ‖x − y‖ ‖x‖ : x, y ∈ sx , x, y 6= 0, x ⊥b y } and br(x) = sup α>0 { ‖x + αy‖ − ‖x − αy‖ α : x, y ∈ sx , x ⊥b y } . for more information, refer to references [1], [13-15]. throughout the article, we use x torepresent a real banach space with norm ‖ · ‖, the unit ball is denoted as bx = {x ∈ x : ‖x‖ ≤ 1}and the unit sphere is denoted as sx = {x ∈ x : ‖x‖ = 1}. let’s assume that dimx is greaterthan or equal to 2. 2. the constant lα,β(x) combining the extended isosceles and pythagorean orthogonalities, we define a new constantto describe the difference between these two orthogonalities. definition 2.1. let x be a banach space, the geometric constant of the isosceles orthogonality type is defined as lα,β(x) = sup { ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 : x ⊥i y , (x, y) 6= (0, 0) } , where α, β ≥ 0, α 6= 1, β 6= 1. in this section, we first give bounds on the constant lα,β(x) and its value in some particularspaces is obtained. we find that x is a hilbert space if and only if the constant lα,β(x) value is1. moreover, the relationship between lα,β(x) and uniformly non-square is given. now, we recallthe notion of the uniformly non-square. definition 2.2. ([10]) a banach space x is called uniformly non-square if there exists δ ∈ (0, 1) such that for any x, y ∈ sx , then ‖x + y‖ 2 ≤ 1− δ or ‖x − y‖ 2 ≤ 1− δ. https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 3 proposition 2.1. let x be a banach space, then 1 ≤ lα,β(x) ≤  2 1+β2 , 0 ≤ β ≤ α < 1; 2 1+α2 , 0 ≤ α ≤ β < 1; α2+β2 1+β2 , 1 < β ≤ α; α2+β2 1+α2 , 1 < α ≤ β. proof. let x0 = 0, y0 6= 0, satisfy x0 ⊥i y0, hence lα,β(x) ≥ ‖x0 − βy0‖2 + ‖αx0 − y0‖2 ‖x0 − y0‖2 + ‖αx0 − βy0‖2 = 1. on the other hand, using the triangle inequality, we get: ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≤ ( ∣∣∣1−β2 ∣∣∣ ‖x + y‖+ ∣∣∣1+β2 ∣∣∣ ‖x − y‖)2 + ( ∣∣α−12 ∣∣ ‖x + y‖+ ∣∣α+12 ∣∣ ‖x − y‖)2 ‖x − y‖2 + ( ∣∣∣α−β2 ∣∣∣ ‖x + y‖ − ∣∣∣α+β2 ∣∣∣ ‖x − y‖)2 . if we take the categorical approach, there will be the following four approaches. case 1 when 0 ≤ β ≤ α < 1, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≤ 2 1 + β2 . case 2 when 0 ≤ α ≤ β < 1, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≤ 2 1 + α2 . case 3 when 1 < β ≤ α, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≤ α2 + β2 1 + β2 . case 4 when 1 < α ≤ β, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≤ α2 + β2 1 + α2 . � next, we will show that there are points in some particular spaces where the value of the constant lα,β(x) is an upper bound. example 2.1. let 0 ≤ β < 1, α = β and x be the space r2 with l1 norm defined by ‖(x1, x2)‖ = |x1|+ |x2|, then lα,β(x) = 2 1+β2 . https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 4 let x = (1, 1), y = (1,−1), satisfy x ⊥i y . we get ‖αx − βy‖ = 2β, ‖x − y‖ = ‖x − βy‖ = ‖αx − y‖ = 2. thus, lα,β(x) = 2 1+β2 . example 2.2. let 0 ≤ β < 1, α = β and x be the space r2 with l∞ norm defined by ‖(x1, x2)‖ = max{|x1|, |x2|}, then lα,β(x) = 2 1+β2 . let x = (1, 0), y = (0,−1), satisfy x ⊥i y . we get ‖αx − βy‖ = β, ‖x − y‖ = ‖x − βy‖ = ‖αx − y‖ = 1. thus, lα,β(x) = 2 1+β2 . proposition 2.2. let x be a banach space, 0 ≤ β ≤ α < 1, then lα,β(x) = 1 if and only if x is a hilbert space. proof. since x is a hilbert space, combined with reference [3], we have lα,β(x) = 1. conversely,assuming that lα,β(x) = 1 and useing the homogeneity of x ⊥ y , we can prove by induction that ‖x − y‖2 + ‖αnx − βy‖2 ≥ ‖αnx − y‖2 + ‖x − βy‖2. since 0 ≤ α < 1, taking the limit n →∞, we get ‖x − y‖2 ≥ ( 1− β2 ) ‖y‖2 + ‖x − βy‖2. if 0 ≤ β < 1, a second induction shows that ‖x − y‖2 = (1− β2n) ‖y‖2+ ‖x − βny‖2. also takingthe limit n →∞, therefore, in this case we get ‖x − y‖2 ≥ ‖x‖2 + ‖y‖2. then, x ⊥i y implies ‖x + y‖2 + ‖x − y‖2 ≥ 2‖x‖2 + 2‖y‖2 for all x, y ∈ x , hence we can assert that x is a hilbert space. � among the many properties of banach spaces, we give below a sufficient condition that x is nota uniform non-square space. in the process of proving, we apply the lemma given by james. lemma 2.1. [8, lemma 4.1] let x be a banach space and x, y ∈ x . if x ⊥i y , then the following inequality holds. (i) ‖x + ky‖ ≤ |k |‖x ± y‖ and ‖x ± y‖ ≤ ‖x + ky‖, when |k | ≥ 1. (ii) ‖x + ky‖ ≤ ‖x ± y‖ and |k |‖x ± y‖ ≤ ‖x + ky‖, when |k | ≤ 1. proposition 2.3. let x be a finite dimensional banach space, if lα,β(x) = 2 1+β2 for some 0 ≤ β0 ≤ α0 < 1, then x is not uniformly non-square. proof. since lα,β(x) = 2 1+β2 , there exist xn ∈ sx , yn ∈ bx that satisfy xn ⊥i yn and lim n→∞ ‖xn − βyn‖2 + ‖αxn − yn‖2 ‖xn − yn‖2 + ‖αxn − βyn‖2 = 2 1 + β2 . https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 5at the same time, a banach space x is finite dimensional, then there exist x0, y0 ∈ bx that satisfy x0 ⊥i y0 and lim k→∞ ∥∥xnk∥∥ = ‖x0‖ , lim k→∞ ∥∥ynk∥∥ = ‖y0‖ . combine lemma 2.1, we have ‖xn − β0yn‖ ≤ ‖xn + yn‖ and ‖α0xn − yn‖ ≤ ‖xn + yn‖.thus, ‖xn + yn‖2 + ‖xn + yn‖2 ‖xn + yn‖2 + ( ∣∣∣α−β2 ∣∣∣− ∣∣∣α+β2 ∣∣∣)2‖xn + yn‖2 ≤ 2 1 + β2 , ‖xn + yn‖2 + ‖xn + yn‖2 (1 + β2) ‖xn + yn‖2 ≤ 2 1 + β2 . we can obtain ‖x0 − β0y0‖ = ‖x0 + y0‖ and ‖α0x0 − y0‖ = ‖x0 + y0‖. since ‖x0 − β0y0‖ ≤ (1− β0) ‖x0‖+β0 ‖x0 + y0‖, then ‖x0 + y0‖ ≤ ‖x0‖. moreover, we can prove that ‖x0 + y0‖ ≤ ‖y0‖,then max {‖x0 + y0‖ , ‖x0 − y0‖} = ‖x0 + y0‖ ≤ min {‖x0‖ , ‖y0‖} ≤ 1 < 1 + δ for any δ ∈ (0, 1), this means that x is not uniformly non-square. � 3. the constant l′α,β(x)in this section, if x and y satisfy the isosceles orthogonality condition and restrict x , y ∈ sx ,then we define the new constant: definition 3.1. let x be a banach space, another geometric constant of the isosceles orthogonal type is defined as l′α,β(x) = sup { ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 : x, y ∈ sx , x ⊥i y } , where α, β ≥ 0, α 6= 1, β 6= 1. we give bounds on the constant l′α,β(x) and calculate its value in `∞− `1 normed linear spacewhen α = β = 1 2 . proposition 3.1. let x be a banach space, then α2+β2 1+α2 α2+β2 1+β2 2 1+α2 2 1+β2 ≤ l′α,β(x) ≤  2 1+β2 , 0 ≤ β ≤ α < 1; 2 1+α2 , 0 ≤ α ≤ β < 1; α2+β2 1+β2 , 1 < β ≤ α; α2+β2 1+α2 , 1 < α ≤ β. https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 6 proof. combined with the idea of proposition 2.1, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≥ ( ∣∣∣1−β2 ∣∣∣ ‖x + y‖ − ∣∣∣1+β2 ∣∣∣ ‖x − y‖)2 + ( ∣∣α−12 ∣∣ ‖x + y‖ − ∣∣α+12 ∣∣ ‖x − y‖)2 ‖x − y‖2 + ( ∣∣∣α−β2 ∣∣∣ ‖x + y‖+ ∣∣∣α+β2 ∣∣∣ ‖x − y‖)2 . case 1 when 0 ≤ β ≤ α < 1, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≥ α2 + β2 1 + α2 . case 2 when 0 ≤ α ≤ β < 1, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≥ α2 + β2 1 + β2 . case 3 when 1 < β ≤ α, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≥ 2 1 + α2 . case 4 when 1 < α ≤ β, we have ‖x − βy‖2 + ‖αx − y‖2 ‖x − y‖2 + ‖αx − βy‖2 ≥ 2 1 + β2 . on the other hand, the upper bound on the constant l′α,β(x) is the same as the upper bound onthe constant lα,β(x). � example 3.1. let α = β = 1 2 and x be the space r2 with `∞ − `1 norm defined by ‖x‖ = { ‖x‖1, x1x2 ≤ 0, ‖x‖∞, x1x2 ≥ 0. then l′1 2 , 1 2 (x) = 0.91. proof. if x = (y1, 1+y1), y = (y2, 1+y2), where −1 ≤ y1 ≤ y2 ≤ 0; x = (y1, y1−1), y = (y2, y2−1),where 0 ≤ y1 ≤ y2 ≤ 1. the two cases above, which are determined by x ⊥i y , we have |y1 − y2| = 2, are contradictory. to estimate this constant value, it is only necessary to considerthe following two cases. case 1: assuming that x = (x1, 1) , y = (1, y2), 0 ≤ x1 ≤ y2 ≤ 1. since x ⊥i y , we have 1 + y2 = (1− x1) + (1− y2) , https://doi.org/10.28924/ada/ma.4.6 eur. j. math. anal. 10.28924/ada/ma.4.6 7hence x1 + 2y2 = 1, y2 ∈ [13 , 12]. then ‖αx − y‖ = 1, ‖x − βy‖ = 1 2 + 3 2y2, ‖αx − βy‖ = 1 2 + 1 2y2and ‖x − y‖ = 1 + y2.in fact, l′1 2 , 1 2 (x) = max 1 3 ≤y2≤ 12 9y22 + 6y2 + 5 5y22 + 10y2 + 5 . by simple calculation, we find that l′1 2 , 1 2 (x) = 0.91 is obtained at the point (1, 12). case 2: assuming that x = (x1, 1) , y = (y1, 1 + y1) satisfy −1 ≤ y1 ≤ 0 ≤ x1 ≤ 1. since x ⊥i y , we have ‖(x1 + y1, 2 + y1)‖ = ‖(x1 − y1,−y1)‖.if −x1 ≤ y1, then 2 + y1 = x1 − y1 is true, hence x1 − 2y1 = 2, y1 ∈ [−23 ,−12] . we have ‖αx − y‖ = 1, ‖x − βy‖ = 2 + 32y1, ‖αx − βy‖ = 1 + 12y1 and ‖x − y‖ = 2 + y1.in the same way, l′1 2 , 1 2 (x) = max − 2 3 ≤y1≤− 12 9y21 + 24y1 + 20 5y21 + 20y1 + 20 . by simple calculation, we find that l′1 2 , 1 2 (x) = 0.91 is obtained at the point (1,−12).similarly, if y1 ≤ −x1, such as case 2, prove omission.combined with all of the above, we get l′1 2 , 1 2 (x) = 0.91. � 4. funding statement this research work was funded by anhui province higher education science research project(natural science), 2023ah050487. references [1] j. alonso, c. benítez, orthogonality in normed linear spaces, a survey, ii, relations between main orthogonalities,extracta math. 4 (1989) 121–131.[2] j. alonso, h. martin, s. wu, on birkhoff orthogonality and isosceles orthogonality in normed linear spaces, aequat.math. 83 (2012) 153-189. https://doi.org/10.1007/s00010-011-0092-z.[3] e. z. andalafte, c. r. diminnie, r. w. freese, (α, β)−orthogonality and a characterization of inner product spaces,math. japon. 30 (1985) 341–349.[4] v. balestro, angles in normed spaces, aequat. math. 91 (2017) 201–236. 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constants of isosceles orthogonal type, (2022). http://arxiv.org/abs/ 2111.08392.[15] z. yang, y. li, a new geometric constant in banach spaces related to the isosceles orthogonality, kyungpookmath. j. 62 (2022) 271–287. https://doi.org/10.5666/kmj.2022.62.2.271. https://doi.org/10.28924/ada/ma.4.6 https://doi.org/10.1215/s0012-7094-45-01223-3 https://doi.org/10.1215/s0012-7094-45-01223-3 https://doi.org/10.2307/1990220 https://doi.org/10.2307/1970663 https://doi.org/10.2307/1970663 https://doi.org/10.1016/j.jmaa.2005.10.004 https://doi.org/10.2298/fil1610761m https://doi.org/10.1016/j.jmaa.2012.07.059 http://arxiv.org/abs/2111.08392 http://arxiv.org/abs/2111.08392 https://doi.org/10.5666/kmj.2022.62.2.271 1. introduction 2. the constant l,(x) 3. the constant l,(x) 4. funding statement references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 1doi: 10.28924/ada/ma.3.1 global analysis of a spatiotemporal cellular model for the transmission of hepatitis c virus with hattaf-yousfi functional response alexis nangue1,∗, bruno nde tchiffo2 1university of maroua, higher teachers’ training college, department of mathematics, p.o.box 55 maroua, cameroon alexnanga02@gmail.com 2university of maroua, faculty of science, department of mathematics and computer science, p.o. box 814 maroua, cameroon ndebruno2@gmail.com ∗correspondence: alexnanga02@gmail.com abstract. this paper carries out a mathematical analysis of the global dynamics of a partial differen-tial equation viral infection cellular model. we study the dynamics of a hepatitis c virus (hcv) model,under therapy, that considers both absorption phenomenon and diffusion of virions, infected and un-infected hepatocytes in the liver. firstly, we prove the boundedness of the potential solutions, globalexistence, uniqueness, and positivity of the obtained initial value and boundary problem solution.then, the dynamical behaviour of the model is entirely determined by a threshold parameter calledthe basic reproduction number denoted r0. we show that the uninfected spatially homogeneousequilibrium of the model is globally asymptotically stable if r0 ≤ 1 by using the direct lyapunovmethod. the latter means that the hcv infection is cleared, and the disease dies out. also, the globalasymptotical properties stability of the infected spatially homogeneous equilibrium of the model arestudied via a skilful construction of a suitable lyapunov functional. it means that the hcv infectionpersists in the host, and the infection becomes chronic. finally, numerical simulations are performedto support the obtained theoretical results. 1. introduction the dynamics of viruses, in particular the dynamics of the hepatitis c virus, remains a veryactive field of research in the world of sciences. moreover, the 2020 nobel prize in medicinewas awarded to three researchers, namely the british michael hougton and the americans harveyalter and charles rice. they were awarded this nobel prize for their very advanced research workon the hepatitis c virus. according to world health organization(who) [41], 71 million personswere living with chronic hepatitis c virus (hcv) infection worldwide and 399 000 persons had received: 29 apr 2022. key words and phrases. reaction-diffusion model; hcv infection; hattaf-yousfi functional response; semigroup; globalstability; variational method; cure rate. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 2died from cirrhosis or hepatocellular carcinoma following a survey done in 2015. aside from theburden of hcv infection secondary to liver-related sequelae, hcv causes an additional burdenthrough comorbidities among persons with hcv infection, including depression, diabetes mellitusand chronic renal disease. in may 2016, the world health assembly endorsed the global healthsector strategy for 2016-2021 on viral hepatitis (hbv and hcv infection), which proposes toeliminate viral hepatitis as a public health threat by 2030. elimination is defined as a 90%reduction in new chronic infections and a 65% reduction in mortality compared with the 2015baseline. mathematicians cannot stay aside from this disastrous situation decried by who. inview of the vital importance of the liver and the aforementioned facts, any contribution to a betterunderstanding of hcv infection process and strategy to eradicate this infection is of great interest.mathematical models have been developed to help understand and control the dynamics of hcvwithin an infected host such as in [6, 7, 14, 35]. the dynamics of viral infections such as theebola virus disease(evd), the human immunodeficiency virus (hiv) infection, the hepatitis b virus(hbv) infection, the hepatitis c virus (hcv) infection and, new corona virus infection have beenmodeled mathematically in a host. one of the earliest temporal models was the within-host basicviral infection model proposed in [31] to study hiv infection, and later adopted to hbv [8, 32].particularly, numerous mathematical models describing the temporal dynamics of hcv have beeninitially proposed by neumann and al [30] using the classical viral infection cellular model, andlater have been extended in [6, 10, 14, 35]. motivated by what has been done in [8, 30, 32], chongand al. [7] formulated the basic hcv temporal intra-host model with therapy as a system of threedifferential equations :  dh(t) dt = λ− dh(t)− (1− η)βh(t)v (t), di(t) dt = (1− η)βh(t)v (t)− αi(t), (1.1) dv (t) dt = (1− ε)ki(t)− µv (t), where the equations relate the dynamics relationship between, h as the uninfected target cells(hepatocytes), i as the infected cells and v as the viral load (amount of viruses present in theliver). in the system (1.1) the key assumption is that hepatocytes and viruses are well mixed, andneglects the mobility of hepatocytes c viruses, the infected and uninfected target cells. to studythe influences of spatial structures of virus dynamics, wang and wang in [39] assuming that themotion of virus follows fickian diffusion, that is to say, the population flux of virus is proportionalto the concentration gradient and the proportionality constant is taken to be negative [13]. more-over, in model (1.1), the rate of infection is assumed to be bilinear in the virus v and uninfectedhepatocytes t. it is shown in [29] that this bilinear rate of infection could be unrealistic. however,the actual incidence rate is probably not linear over the entire range of t and v. thus is reason-able to assume that the infection rate is given by a more general one, known as the hattaf-yousfi https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 3 functional response [18] of the form βhv α0+α1h+α2v +α3hv where α0 > 0, α1 ≥ 0, α2 ≥ 0, α3 ≥ 0are constants. the function βh α0+α1h+α2v +α3hv satisfies the hypotheses (h1), (h2) and (h3) ofgeneral incidence rate presented in [16,19–21]. the hattaf-yousfi type of functional response wasintroduced by hattaf and al. [18]. this functional response generalizes many functional responsesand it was used in [34] to describe the dynamics of labour market. thus, when α0 = 1, the hattaf-yousfi functional response is reduced to the specific functional response used by hattaf and alin [17]. furthermore, if α3 = α1α2 and α0 = 1, the hattaf-yousfi functional response is reducedto crowley-martin functional response [9] and was used in [43]. when α3 = 0 et α0 = 1 thehattaf-yousfi functional response is simplified to beddington-deangelis functional response [5,11],and was used in [25, 26, 38, 42]. when α1 > 0, α2 = α3 = 0 and α0 = 1, the hattaf-yousfifunctional response is reduced to holling type ii functional response [28]. and when α1 = α3 = 0, α2 > 0 and α0 = 1 it expresses a saturation response [36]. moreover, when α1 = α2 = α3 = 0,and α0 = 1 the hattaf-yousfi functional response is reduced to the mass action principle(or hollingtype i functional response). also ordinary differential system (1.1) don’t take into consideration thecure of infected hepatocytes. in this work, motivated by the breaches observed in the analysis andthe formulation of system (1.1), we construct and analyze a partial differential equation (pde)-cellular model system for hcv infection, which derives from system (1.1) by incorporating the space,hattaf-yousfi incidence rate, absorption effect and spontaneous cure. it is worth mentioning thatin [7] the authors used mass-action kinetics for viral infection, neglected the cure rate, ignored theabsorption effect and the diffusion of free virions, susceptible cells and infected cells. thus theobtained model is an extension of the one in the first part of the work done by chong et al. [7].the work is organized as follows. in section 2, we model the phenomenon described througha reaction-diffusion equations which leads to a initial value and boundary problem. section 3is devoted to the study of the existence and uniqueness of the global solution of our initial andboundary value problem, and of the properties of this solution, namely positivity and bounded-ness. section 4 deals with the stability and the analysis of spatially homogeneous equilibria andnumerical simulations in section 5. we conclude our work and provide a discussion in section 6. 2. formulation of the pde-cellular model let ω ⊂ r3 be a bounded connected domain representing the liver. let t ≥ 0 be a given timeand x = (x1, x2, x3) ∈ ω. denote respectively by h(x, t), i(x, t) and v (x, t) the concentrations ofhealthy hepatocytes, hcv infected hepatocytes, and free hcv virions at time t and location x . thedynamics of hcv infection intra-host is the result of the dynamics of each compartment h, i, andv, and the various interactions between them. we now describe the evolution of each compartment. 2.1. fluctuation of healthy hepatocytes. let ν be an elementary volume in ω. the variation ofthe quantity of healthy hepatocytes in ν is described under the following assumptions. healthy https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 4hepatocytes are produced at constant rate λ from the bone narrow and die at rate dh. virions infectthe healthy hepatocytes at the rate βhv α0+α1h+α2v +α3hv , where β is the rate of transmission of theinfection and αj , j = 0, 1, 2, 3 are positive constants. this generalized incidence function replacesthe mass-action function which has been shown to cause unrealistic conditions for successful chronichcv infection. ρi is the cure rate of infected hepatocytes either by noncytolytic mechanism orimmunity or treatment. in addition, the therapeutic effect of treatment in this model involved thereduction of new infections, which is described in a fraction as (1 − η). the spatial motion ofhealthy hepatocytes follows the fickian diffusion law. thus, the variation of healthy hepatocytesis expressed by the following equation: ∂h ∂t = d1∆h(x, t) + λ− dh(x, t)− (1− η)βh(x, t)v (x, t) α0 + α1h(x, t) + α2v (x, t) + α3h(x, t)v (x, t) + ρi(x, t), where d1 represents the healthy hepatocytes diffusion coefficient and ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 + ∂2 ∂x2 3is the usual laplacian operator in three-dimensional space. 2.2. fluctuation of hcv infected cells. the hcv infected cells die at rate α per day so that 1 αis the life-expectancy of hcv infected hepatocytes. healthy hepatocytes become infected at therate βhv α0+α1h+α2v +α3hv . the spatial motion of hcv infected cells follows the fickian diffusion law.thus, the variation of infected hepatocytes is expressed by the following equation ∂i ∂t = d2∆i(x, t) + (1− η)βh(x, t)v (x, t) α0 + α1h(x, t) + α2v (x, t) + α3h(x, t)v (x, t) − (α+ ρ)i(x, t), where d2 represents the hcv infected cells diffusion coefficient. 2.3. fluctuation of free hcv virions. the infected hepatocytes produce virus at rate ki , and virusis cleared at the rate µv . also, the population of virions decreases due to the infection at the rate u(1−η)βhv α0+α1h+α2v +α3hv due to absorption effect, where u ∈ {0, 1}. the spatial motion of virions followsthe fickian diffusion law. in addition, the therapeutic effect of treatment in this model involvedblocking virions production (referred to as drug effectiveness) which, is described in fraction (1−ε).thus, the variation of free virions is expressed by the following equation: ∂v ∂t = d3∆v (x, t)+(1−ε)ki(x, t)−µv (x, t)−u (1− η)βh(x, t)v (x, t) α0 + α1h(x, t) + α2v (x, t) + α3h(x, t)v (x, t) , where d3 represents the free hcv virions diffusion coefficient. 2.4. the initial boundary value problem associated to pde-cellular model. in this section, we usethe previous equations describing variables variations to set up a complete pde system modellingbiological dynamics for hcv infection. let t > 0 be a fixed time and define ωt = ω× (0, t ). https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 5therefore, in ωt the full system of pde governing the hcv infection becomes : ∂h ∂t = d1∆h(x, t) + λ− dh − (1− η)βhv α0 + α1h + α2v + α3hv + ρi, ∂i ∂t = d2∆i(x, t) + (1− η)βhv α0 + α1h + α2v + α3hv − (α+ ρ)i, (2.1) ∂v ∂t = d3∆v + (1− ε)ki − µv − u(1− η)βhv α0 + α1h + α2v + α3hv . we use the neumann homogeneous boundary conditions: ∂h ∂η = ∂i ∂η = ∂v ∂η = 0 on ∂ω× [0, t ], (2.2) where ∂ ∂η denotes the outward normal derivative on ∂ω. the initial conditions are the following : h(x, 0) = h0, i(x, 0) = i0, v (x, 0) = v0, x ∈ ω. (2.3) the boundary conditions in (2.2) imply that the healthy hepatocytes, the hcv infected cells and freehcv virions do not move across the boundary ∂ω. for an epidemiological significance, we assumethat the initial conditions are positive and hölder continuous, and satisfy ∂h0 ∂η = ∂i0 ∂η = ∂v0 ∂η = 0 on ∂ω. we then obtain the following initial boundary value problem, denoted ibvp associated to theprevious pde-cellular model: ∂h ∂t = d1∆h + λ− dh − (1− η)βhv α0 + α1h + α2v + α3hv + ρi in ωt , ∂i ∂t = d2∆i + (1− η)βhv α0 + α1h + α2v + α3hv − (α+ ρ)i in ωt , ∂v ∂t = d3∆v + (1− ε)ki − µv − u(1− η)βhv α0 + α1h + α2v + α3hv in ωt , (2.4) ∂h ∂η = ∂i ∂η = ∂v ∂η = 0 on ∂ω× [0, t ], h(x, 0) = h0, i(x, 0) = i0, v (x, 0) = v0, x ∈ ω,on which our study will focus on. 3. qualitative and quantitative analysis and some properties of the solutions for ibvp (2.4) in this section, we provide a thorough study of the dynamics of ibvp (2.4) which yields variousoutcomes. precisely, we prove existence, uniqueness, positivity and boundedness of solutions foribvp (2.4). this is done by combining variational method and semigroups techniques to someuseful functional analysis arguments. 3.1. local existence and uniqueness of solutions for the ibvp (2.4). set f (h, i, v ) = (f1(h, i, v ), f2(h, i, v ), f3(h, i, v ))t (3.1) where f1(h, i, v ) = λ− dh − (1− η)βhv α0 + α1h + α2v + α3hv + ρi, https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 6 f2(h, i, v ) = (1− η)βhv α0 + α1h + α2v + α3hv − (α+ ρ)i,and f3(h, i, v ) = (1− ε)ki − µv − u (1− η)βhv α0 + α1h + α2v + α3hv .we have the following result which guarantees that the right-hand side, without diffusion, of thepde-model system (2.4) is lipschitz. proposition 3.1. let t ∈ r∗+ and (h, i, v ) ∈ (c0 b(ω × [0, t ]))3, where c0 b(ω × [0, t ]) is the space of bounded and continuous functions on ω × [0, t ]. we suppose that f in (3.1) is defined on l2(ω× (0, t )). then f1, f2 and f3 are uniformly lipschitz continuous on l2(ω× (0, t )) with respect to h, i and v. proof. let t ∈ r∗+ and (h1, i1, v1), (h2, i2, v2) ∈ (c0 b(ω× [0, t ]))3. first, by direct computation,we have : ‖f1(h1, i1, v1)− f1(h2, i2, v2)‖2 ≤ k1 1‖h1 −h2‖2 +k1 2‖i1 − i2‖2 +k1 3‖v1 − v2‖2, (3.2) with k1 1 = d + (1− η)β ( 1 α2 + vm α0 ) , k1 2 = ρ, k1 3 = (1− η)β ( 1 α1 + hm α0 ) , (3.3)with hm and vm given below.then ‖f2(h1, i1, v1)− f2(h2, i2, v2)‖2 ≤ k2 1‖h1 −h2‖2 +k2 2‖i1 − i2‖2 +k2 3‖v1 − v2‖2, (3.4) with k2 1 = (1− η)β ( 1 α2 + vm α0 ) , k2 2 = (α+ ρ) and k2 3 = (1− η)β ( 1 α1 + hm α0 ) . (3.5) finally ‖f3(h1, i1, v1)− f3(h2, i2, v2)‖2 ≤ k3 1‖h1 −h2‖2 +k3 2‖i1 − i2‖2 +k3 3‖v1 − v2‖2, (3.6) with k3 1 = u(1− η)β ( 1 α2 + vm α0 ) , k3 2 = k(1− ε), and k3 3 = µ+ u(1− η)β ( 1 α1 + hm α0 ) . (3.7) this completes the proof of proposition 3.1. � now, consider the following ibvp ∂th −d1∆h = f (t, h, i, v ) in ω× (0, t ) ∂t i −d2∆i = g(t, h, i, v ) in ω× (0, t ) ∂tv −d3∆v = h(t, h, i, v ) in ω× (0, t ) ∂h ∂η = 0; ∂i ∂η = 0; ∂v ∂η = 0 on ∂ω× [0, t ] h = h0, i = i0, v = v0 on ω× {t = 0}. (3.8) in what follows, we will need the following definition and results. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 7 definition 3.1. (sectorial operator, [22]) let a be a linear operator in a banach space x andsuppose a is closed and densely defined. if there exist real numbers a, ω ∈ (0, π), m ≥ 1 suchthat ρ(a) ⊃ σ = {λ0 ∈ c : ω ≤ arg(λ0 − a) ≤ π, λ0 6= 0} (3.9)and ‖rλ0 (a)‖ ≤ m |λ0 − a| for all λ0 ∈ σ, (3.10)then we say that a is sectorial. remark 3.1. the neumann realization of the laplacian a = −∆, with domain d(a) = { ω ∈ h2(ω) : ∂ω ∂η = 0 } is a sectorial operator in l2(ω). but since c∞0 (ω) ⊂ d(a), it is densely defined in l2(ω). for β ≥ 0 large enough, we define the fractional powers of the helmholtz operator, h� = −∆ + βi ,with domain d(h�) equipped with graph norm ‖ . ‖d(h�) = ‖ . ‖2 + ‖h�.‖2. we have the following general results. lemma 3.2. [1] let 1 ≤ p <∞. then d(h�) ⊂ c∞0 (ω) with continuous injection for β > n 2p . lemma 3.3. [22] d(hβ) ⊂ c0 b(ω) with continuous injection for β > n 4 . theorem 3.4. [22] if a is sectorial, then −a is the infinitesimal generator of an analytic semigroup, g(t). if rλ0 > a, a ∈ r whenever λ0 ∈ σ, then for any t > 0, ‖g(t)‖ ≤ ce−at , ‖ag(t)‖ ≤ c t e−at and d dt g(t) = −ag(t) , t > 0. corollary 3.5. let g be the analytic semigroup generated by −a. the following properties hold for the semigroup g and the fractional powers of the helmholtz operator hβ :1) g(t) : l2(ω)→ d(h�) for all t > 0,2) ‖g(t)ω‖h� ≤ cβ,2t−β‖ω‖2 for all t > 0, ω ∈ l2,3) g(t)h�ω = h�g(t)ω for all t > 0, ω ∈ d(h�). remark 3.2. the following basic hypotheses are assumed to hold : (h1): d1 > 0, d2 > 0 and d3 > 0, (h2): h0 ≥ 0, i0 ≥ 0 and v0 ≥ 0 are continuous on ω, h0, i0, v0 ∈ c0 b(ω), (h3): f , g and h are continuously differentiable functions fromr4 + intor with f (t, 0, s, z) ≥ 0, g(t, r, 0, z) ≥ 0 and h(t, r, s, 0) ≥ 0 for all t , r , s , z ≥ 0. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 8for x ∈ ω, t ≥ 0, h, i , v ∈ (c0 b(ω))3, define f ,g and q on r+ × (c0 b(ω))3 by : [f(t, h, i, v )](x) = f (t, h(x), i(x), v (x)), [g(t, h, i, v )](x) = g(t, h(x), i(x), v (x)), [q(t, h, i, v )](x) = h(t, h(x), i(x), v (x)).in addition, we let g1, g2 and g3 be the analytical semigroup generated by a1 = d1 × ∆, a2 = d2 × ∆ and a3 = d3 × ∆ respectively. in the sequel, we will need the following results. lemma 3.6. [22] if h, i and v are continuous from [0, t ] to l2(ω), then the integrals : i1(t) = ∫ t 0 g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ , i2(t) = ∫ t 0 g2(t − τ)g(τ,h(τ), i(τ), v (τ))dτ , i3(t) = ∫ t 0 g3(t − τ)q(τ,h(τ), i(τ), v (τ))dτ , exist and i1(t), i2(t) and i3(t) are continues on [0, t [ with i1(t) ∈ d(a1), i2(t) ∈ d(a2), i3(t) ∈ d(a3) and i1(t)→ 0+ in l2 as t → 0+, i2(t)→ 0+ in l2 as t → 0+ and i3(t)→ 0+ in l2 as t → 0+. lemma 3.7. if the ibvp (3.8) has a classical solution, then h, i and v satisfy the following equalities : h(t) = g1(t)h0 + ∫ t 0 g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ, (3.11) i(t) = g2(t)i0 + ∫ t 0 g2(t − τ)g(τ,h(τ), i(τ), v (τ))dτ, (3.12) v (t) = g3(t)v0 + ∫ t 0 g3(t − τ)q(τ,h(τ), i(τ), v (τ))dτ. (3.13) proof. consider the l2−valued functions θj(τ) = gj(t − τ)ωj(τ), j = 1, 2, 3 with ω1 = h, ω2 = iand ω3 = v . then θj is differentiable since gj is analytic and ωj is differentiable. then bytheorem 3.4, we have dθ1 dτ = d dτ [ g1(t − τ) ] h(τ) + g1(t − τ)h ′ (τ), = −d1 × ∆ ( g1(t − τ) ) h(τ) + g1(t − τ) [ d1 × ∆h(τ) + f(τ,h(τ), i(τ), v (τ)) ] , = −d1 × ∆ ( g1(t − τ) ) h(τ) +d1 × g1(t − τ)× ∆h(τ) + g1(t − τ)f(τ,h(τ), i(τ), v (τ)), according to corollary 3.5 with β = 0, we have d1 × g1(t − τ)∆h(τ) = d1 × ∆g1(t − τ)h(τ). therefore dθ1 dτ = −d1 × ∆g1(t − τ)h(τ) +d1 × ∆g1(t − τ)h(τ) + g1(t − τ)f(τ,h(τ), i(τ), v (τ)), = g1(t − τ)f(τ,h(τ), i(τ), v (τ)). (3.14) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 9in the similar way, we also have : dθ2 dτ = d dτ [ g2(t − τ) ] i(τ) + g2(t − τ)i ′ (τ), = −d2 × ∆ ( g2(t − τ) ) i(τ) +d2 × g2(t − τ)× ∆i(τ) + g2(t − τ)g(τ,h(τ), i(τ), v (τ)), = g2(t − τ)g(τ,h(τ), i(τ), v (τ)). (3.15) dθ3 dτ = d dτ [ g3(t − τ) ] v (τ) + g3(t − τ)v ′ (τ), = −d × ∆g3(t − τ)v (τ) +d × g3(t − τ)∆v (τ) + g3(t − τ)q(τ,h(τ), i(τ), v (τ)), = g3(t − τ)q(τ,h(τ), i(τ), v (τ)). (3.16) integrating equations (3.14), (3.15) and (3.16) with respect to time, we obtain equations (3.11),(3.12) and (3.13) respectively. � remark 3.3. since in this work, n = 3, we take p = 2 so that β > 3 4 and therefore the domain d(h�) is continuously embedded in c∞0 (ω) by lemma 3.2. now, let h, i and v be continuousfunctions from [0, t ] to d(h�) ↪→ c0 b(ω) satisfying (3.11), (3.12) and (3.13) respectively. we canthen claim that h, i and v verify system (3.8). the continuity of h, i and v implies continuity of t 7→ f(t, h(t), i(t), v (t)), t 7→ g(t, h(t), i(t), v (t)) and t 7→ q(t, h(t), i(t), v (t)).one can then conclude that, the linear cauchy problem ∂ty1 −d1∆y1 = f(t, h(t), i(t), v (t)), ∂ty2 −d2∆y2 = g(t, h(t), i(t), v (t)), ∂ty3 −d∆y3 = q(t, h(t), i(t), v (t)), y1(0) = h0, y2(0) = i0, y3(0) = v0, has a unique solution, with y1, y2 and y3 given by (3.11), (3.12) and (3.13) respectively. following [2], [4], [3], [22], [24], we have the following main result for the local existence of (2.4),based on l2-theory. proposition 3.8. if hypotheses (h1), (h2) and (h3) are satisfied, then the initial value and boundary problem (3.8) admits a unique solution (h, i, v ) ∈ (c0 b(]0, t ], d(hβ)))3, with h(0) = h0 ∈ c0 b(ω), i(0) = i0 ∈ c0 b(ω) and v (0) = v0 ∈ c0 b(ω). the proof of this proposition is given in "appendix a ". 3.2. boundedness of the solutions for ibvp (2.4). proposition 3.9. let (h, i, v ) ∈ ( c0 ( ω× [0, t ) ) ∩ c2,1 b (ω× [0, t )) )3 be the solution of (2.4) with bounded initial conditions i.e. 0 < h0(x) < hm, 0 < i0(x) < hm, https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 10 0 < v0(x) < vm for all x ∈ ω, and satisfying the boundary condition ∂h0 ∂η = 0, ∂i0∂η = 0, ∂v0 ∂η = 0 on ∂ω. then, ∀(x, t) ∈ ω× [0, t ], h(x, t) ≤ hm, i(x, t) ≤ hm and v (x, t) ≤ vm with hm = max { λ δ2 ,max x∈ω̄ {h(x, 0) + i(x, 0)} } and vm = max { (1− ε)khm µ ,max x∈ω v0(x) } . proof. consider the function s defined for all (x, t) ∈ ω× [0, t ] by s(x, t) = h(x, t) + i(x, t). adding the first two equations in (2.4), yields ∂s(x, t) ∂t −d1∆h(x, t)−d2∆i(x, t) = λ− dh(x, t)− αi(x, t). it follows that ∂s(x, t) ∂t −max{d1, d2}∆ (h(x, t) + i(x, t)) ≤ λ−min{d, α} (h(x, t) + i(x, t)) , we have  ∂s(x,t) ∂t − δ1∆s(x, t) ≤ λ− δ2s(x, t), x ∈ ω, t ∈ [0, t ] ∂s(x,t) ∂η = 0, x ∈ ∂ω, t ∈ [0, t ] s(x, 0) = max x∈ω s0(x), (3.17) where s0(x) = {h(x, 0) + i(x, 0)}, δ1 = max{d1, d2} et δ2 = min{d, α}. by using the standardparabolic comparison of the scalar parabolic equations [33], one has s(x, t) ≤ s̄(t), where s̄(t) = λ δ2 ( 1− e−δ2t ) + max x∈ω̄ s0(x)e−δ2t is the solution of the problem ds̄(t) dt = λ− δ2s̄(t), s̄(0) = max x∈ω̄ s0(x), (3.18) which dominates system (3.17). the general solution of (3.18) is on the form s̄(t) = k(t)e−δ2t . bylagrange’s method, we have k(t) = λ δ2 eδ2t + c , c ∈ r. hence s̄(t) = ( λ δ2 eδ2t + c ) e−δ2t . initial condition yields c = max x∈ω s0(x)− λ δ2 .therefore s̄(t) = λ δ2 ( 1− e−δ2t ) + max x∈ω̄ s0(x)e−δ2t . https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 11then, it follows that : s(x, t) ≤ s̄(t) ≤ λ δ2 ( 1− e−δ2t ) + max x∈ω̄ s0(x)e−δ2t ≤ max { λ δ2 ,max x∈ω̄ s0(x) }( 1− e−δ2t ) + max { λ δ2 ,max x∈ω̄ s0(x) } e−δ2t ≤ max { λ δ2 ,max x∈ω̄ s0(x) } . thus, s(x, t) ≤ max { λ δ2 ,max x∈ω̄ {h(x, 0) + i(x, 0)} } . therefore s(x, t) ≤ hm = max { λ δ2 ,max x∈ω̄ {h(x, 0) + i(x, 0)} } ,∀(x, t) ∈ ω× [0, tmax), where tmax is the maximal time of existence of the solution of system (2.4), this implies that s isbounded.hence h and i are bounded since s is bounded. this prove that h and i are bounded.now, to show that v is bounded, from the third equation of ibvp (2.4), we have ∂v (x, t) ∂t −d3∆v (x, t) ≤ (1− ε)ki(x, t)− µv (x, t), x ∈ ω, t ∈ [0, t ] ∂v (x, t) ∂η = 0, x ∈ ∂ω, t ∈ [0, t ] v (x, 0) = max x∈ω v0(x). it follows from the previous system, inequality ∂v (x,t) ∂t −d∆v (x, t) ≤ (1− ε)khm − µv (x, t) ∂v (x,t) ∂η = 0 v (x, 0) = max x∈ω̄ v0(x), (3.19) by using the standard parabolic comparison of the scalar parabolic equations [33], one has v (x, t) ≤ v (t), where v (t) = (1−ε)khm µ (1− e−µt) + maxx∈ω̄ v0(x)e−µt is the solution of the problem dv (t) dt = (1− ε)khm − µv (t), v (0) = max x∈ω v0(x), (3.20) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 12which dominates system (3.19). indeed, the general solution of (3.20) is on the form v (t) = c(t)e−µt . by the lagrange’s method, we have c(t) = (1−ε)khm µ (eµt − 1) + c0, c0 ∈ r.thus, v (t) = [(1− ε)khm µ (eµt − 1) + c0 ] e−µt . initial condition yields max x∈ω̄ v0(x) = v (0) = c0. it follows from that v (t) = (1− ε)khm µ (1− e−µt) + max x∈ω̄ v0(x)e−µt . therefore v (x, t) ≤ v (t) ≤ max {(1− ε)khm µ ,max x∈ω̄ v0(x) } (1− e−µt) + max {(1− ε)khm µ ,max x∈ω̄ v0(x) } e−µt ≤ max {(1− ε)khm µ ,max x∈ω̄ v0(x) } . since v (x, t) ≤ v (t) ≤ max {(1− ε)khm µ ,max x∈ω v0(x) } , ∀(x, t) ∈ ω× [0, tmax); where tmax is the maximal time of existence of the solution of system (2.4), this implies that v isbounded.thus h(x, t), i(x, t) and v (x, t) are bounded on ω × [0, tmax). therefore, it follows from thestandard theory of semi-linear parabolic system in [23] that tmax = +∞. this completes the proofof proposition 3.9. � 3.3. global existence, uniqueness and positivity for the ibvp (2.4). we recast the ibvp (2.4) asfollows:  ∂w ∂t −d∆w + q(w)w = f (w) in ω× [0, t ), ∂w1 ∂η = 0, ∂w2 ∂η = 0, ∂w3 ∂η = 0 on ∂ω× [0, t ), (3.21) w(x, 0) = w0(x) in ω, https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 13where w = (w1, w2, w3)t = (h, i, v )t , d = diag(d1, d2, d3), q(w) = diag (q1(w), q2(w), q3(w)), f (w) = ( f1(w), f2(w), f3(w) )t , with q1(w) = d + (1− η)βw3 α0 + α1w1 + α2w3 + α3w1w3 , q2(w) = (α+ ρ), q3(w) = µ+ u(1− η)βw1 α0 + α1w1 + α2w3 + α3w1w3 , f1(w) = λ+ ρw2, f2(w) = (1− η)βw1w3 α0 + α1w1 + α2w3 + α3w1w3 , f3(w) = (1− ε)kw2. note that d1, d2, d3 > 0. denote h = l2(ω) and e = h1(ω) and define as in [12] the hilbertspace w (0, t, e, e′) = { u ∈ l2 ((0, t ), e) : ∂u ∂t ∈ l2 ( (0, t ), e′ )} , endowed with the norm ‖u‖2 w = ‖u‖2 l2((0,t ),e) + ∥∥∥∂u ∂t ∥∥∥2 l2((0,t ),e′)and the following hypothesis for initial conditions: w01 ∈ l∞(ω), w02, w03 ∈ h and w0i ≥ 0 for i ∈ {1, 2, 3}. (3.22) here, we apply theorem 2.7 of [12]. so, one approaches the solution by a sequence of solutions oflinear equations. for n = 0, w0 denotes the solution of ∂w0 ∂t −d∆w0 = 0 in ω× (0, t ), w0(0) = w0 in ω, ∂w0 i ∂η = 0. on ∂ω (3.23) this equation admits a strong solution and w0 ≥ 0.by induction, wn = ((wn1 , w n 1 , w n 1 )) denotes the solution of ∂wn ∂t −d∆wn + q(wn−1)wn = f (wn−1) in ω× (0, t ), wn(0) = w0 in ω, ∂wn ∂η = 0. on ∂ω. (3.24) since (3.24) is a linear equation, qi(wn−1) and fi(wn−1) can replace a0 and f (t) of corollary 2.10in [12]. suppose that there exists a unique nonnegative solution wn−1 i . assuming by induction that w ji ≥ 0 for 0 ≤ j ≤ n − 1 and that by proposition 3.9 w ji is bounded for 0 ≤ j ≤ n − 1, one has 0 ≤ u(1− η)βwn−1 1 α0 + α1w n−1 1 + α2w n−1 3 + α3w n−1 1 wn−1 3 ≤ u(1− η)β (3.25) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 14which implies that µ ≤ q3(wn−1) ≤ µ+ u(1− η)β. (3.26) since w ji are bounded , we have d ≤ q1(wn−1) ≤ d + (1− η)β. in addition, q2 is a constant.it then follows that q1(wn−1), q2(wn−1), q3(wn−1) ∈ l∞(ω× (0, t )). we also have f (wn−1) ≥ 0and f (wn−1) ∈ l2((0, t ), e′). then, by corollary 2.10 of [12], there exists a unique solution wn ∈ w (0, t, e, e′) with wn ≥ 0. since f1(w) = λ + ρw2, f2(w) = (1−η)βw1w3 α0+α1w1+α2w3+α3w1w3 ≤ (1− η)βw3 and f3(w) = (1− ε)kw2, then f1(wn−1) = λ+ ρwn−1 2 , f2(wn−1) ≤ (1− η)βwn−1 3 and f3(wn−1) = (1 − ε)kwn−1 2 remain bounded in l2(]0, t [, e). we deduce that wn2 and wn3 remainbounded in c0([0, t ],h) and l2((0, t ), e).now, we deduce that the sequence (wni )n≥0 (one can extract a subsequence (wmi )m≥0) convergesweakly to wi in l2((0, t ), e) and weakly star in l∞((0, t ),h) to wi . applying proposition 2.11in [12], it holds that for all n, wni (t) = gi(t)w0i + ∫ t 0 gi(t − s)gni (s)ds, (3.27) where gi(t) is the semigroup generated by the unbounded operator ai = −diah, and gni (s) = −qi(wn−1(s))wni (s) + fi(w n−1(s)). (3.28) then, gni ∈ l2((0, t ), e). since the sequence (wni )n≥0 is bounded in c0([0, t ],h), the se-quence (gni )n≥0 is bounded in c0([0, t ],h). now, consider the operator gi from c0((0, t ),h)into c0((0, t ),h) defined by gi(f ) = ∫ t 0 gi(t − s)f (s)ds. (3.29) let us prove that gi is a compact operator. considering the triple (l2(ω), h1(ω), a) with a(w, v) = 3∑ j=1 ∫ ω ∂w ∂xj ∂v ∂xj dx, (3.30) where ω is regular and bounded. as in [12], the unbounded variational operator ah associated to a is a positive symmetric operator with compact resolvent rλ(ah). it admits a sequence (λk)k ofpositive eigenvalues with lim k→+∞ λk = +∞ and a hilbert basis (ek)k ofh consisting of eigenvectorsof ah. if (g(t))t>0 is the semigroup generated by −ah, then for all w0 ∈ h, g(t)w0 = +∞∑ k=0 e−tλk (w0, ek)ek . (3.31) this proves that the operator is compact for all t > 0 since lim k→+∞ e−tλk = 0. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 15we have the same formula for gi(t), and it suffices to replace λk by diλk . setting gn(t)w = n∑ k=0 e−tλk (w, ek)ek , (3.32) one sees that gn(t) is an operator with finite rank which converges to g(t). the following theoremis relevant in the sequel. theorem 3.10. [12] let t 7→ g(t) be an application from [0,+∞) into l(h). one assumes that there exists a sequence of operators (gn(t))n≥0 on h verifying the following properties:1) for all n and all t > 0, gn(t) has finite rank independent of t ,2) t 7→ gn(t) is continuous from [0,+∞) into l(h) for all n ,3) for n → +∞, gn(t) converges to g(t) in l1(]0, t [,l(h)) for all t > 0. then the operator g is compact from c0([0, t ],h) to c0([0, t ],h) for all t > 0. we are now in the position to prove the global existence, uniqueness and positivity of the solutionto the ibvp (2.4). theorem 3.11. if the initial condition satisfies (3.22), then the ibvp (3.21) admits a unique nonnegative solution w ∈ (w (0, t, e, e′))3. the proof of theorem 3.11 is contained in "appendix b ". remark 3.4. it is worth noting that positivity of the solution may be proved by applying the maximumprinciple. moreover, from the above results and the boundedness of the solution, one has observedthat the solution of ibvp (2.4) enters the region: σ = { (h, i, v ) ∈ ω3 ×r3 + : 0 < h(x, t) ≤ hm, 0 < i(x, t) ≤ hm, 0 < v (x, t) ≤ vm } , where hm = max { λ δ2 ,max x∈ω {h(x, 0) + i(x, 0)} } et vm = max {(1− ε)khm µ ,max x∈ω v (x, 0) } . hence the region σ, of biological interest, is positively-invariant under the flow induced by ibvp(2.4). 4. stability analysis of the spatially homogeneous equilibria 4.1. hcv-spatial homogeneous uninfected equilibrium e0. the spatial homogeneous uninfectedequilibrium of the pde-model system (2.4) arises when there is no virus within a host i.e., v=0.easy calculations shows that the hcv-spatial homogeneous uninfected equilibrium for pde-modelsystem (2.4) is given by e0 = (λ, 0, 0) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 16where λ = λ d . 4.2. basic reproduction number r0. in order to define the basic reproduction number r0 forsystem (2.4), we first observe that system (2.4) has a spatially homogeneous uninfected equilibrium e0. it should be noted that one of the main tools in epidemic models is the basic reproductionnumber r0 which is an important threshold parameter to discuss the dynamic behaviour of theepidemic model. it quantifies the infection risk. it measures the expected average number ofnew infected hepatocytes generated by a single virion in a completely healthy hepatocyte. itshould be also noted that, while a huge number of works deals with the threshold dynamics forode-models, very few studies are devoted to pde-models. this is eventually due to the factthat the concept of basic reproduction number has just recently been extended to pde-modelssuch as reaction-diffusion and reaction-convection-diffusion epidemic models with mixed boundaryconditions [37,40]. the definition of r0 in this work follows the approach developed in [40]. in order to find the basic reproduction number r0 for the system (2.4), we obtain the followinglinear system at e0 for the infected classes:  ∂i ∂t = d2∆i − (α+ ρ)i + (1− η)βλ α0 + α1λ v in ωt , ∂v ∂t = d3∆v + (1− ε)ki − µv − u(1− η)βλ α0 + α1λ v in ωt , (4.1) ∂i ∂η = ∂v ∂η = 0 on ∂ω× [0, t ]. substituting i(x, t) = eλtψ2(x) and v (x, t) = eλtψ3(x) in (4.1), we obtain the following cooper-ative eigenvalue problem:  λψ2(x) = d2∆ψ2(x)− (α+ ρ)ψ2(x) + (1− η)βλ α0 + α1λ ψ3(x) in ω, λψ3(x) = d3∆ψ3(x) + (1− ε)kψ2(x)− µψ3(x)− u(1− η)βλ α0 + α1λ ψ3(x) in ω, (4.2) ∂ψ2(x) ∂η = ∂ψ3(x) ∂η = 0 on ∂ω. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 17as in [40], let t: c(ω̄,r2)→ c(ω̄,r2) be the solution semigroup of the following reaction-diffusionsystem:  ∂i ∂t = d2∆i − (α+ ρ)i in ωt , ∂v ∂t = d3∆v + (1− ε)ki − µv − u(1− η)βλ α0 + α1λ v in ωt , (4.3) i(x, 0) = ψ2(x), v (x, 0) = ψ3(x), in ωt ∂i ∂η = ∂v ∂η = 0 on ∂ω. thus, with initial infection ψ(x) = (ψ2, ψ3) the distribution of those infections members becomes t (t)ψ(x) as time evolves. therefore, the distribution of total new infections is∫ ∞ 0 f (x)t (t)ψ(x)dt, then, we define l(φ)(x) := ∫ ∞ 0 f (x)t (t)ψ(x)dt = f (x) ∫ ∞ 0 t (t)ψ(x)dt. l is a positive and continuous operator which maps the initial infection distribution to the distri-bution of the total infective members produced during the infection period. applying the idea ofnext generation operators [40], we define the spectral radius of l as the basic reproduction number r0 := ρ(l). the matrices f and v defined as f (x) =  0 (1−η)βλ α0+α1λ 0 0  , v (x) =  α+ ρ 0 −(1− ε)k [ µ+ u (1−η)βλ α0+α1λ ]  . then fv −1 = α0 + α1λ (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ]  (1−η)(1−ε)kβλ α0+α1λ (α+ρ)(1−η)βλ α0+α1λ 0 0  . by [40] (theorem 3.4), one has r0 = (1− η)(1− ε)kβλ (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] . (4.4) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 184.3. existence and uniqueness of hcv-spatial homogeneous infected equilibrium e∗. in thissection, we address the existence and uniqueness of infected spatial homogeneous equilibrium(2.4). the latter denoted as e∗ = (h∗, i∗, v ∗) with h∗ 6= 0, i∗ 6= 0 et v ∗ 6= 0 satisfying thefollowing algebraic system : λ− dh∗ − (1− η)l(h∗, i∗, v ∗)v ∗ + ρi∗ = 0, (1− η)l(h∗, i∗, v ∗)v ∗ − (α+ ρ)i∗ = 0, (1− ε)ki∗ − µv ∗ − u(1− η)l(h∗, i∗, v ∗)v ∗ = 0, (4.5) where l(h, i, v ) = βh α0 + α1h + α2v + α3hv .adding the first and second equation of (4.5), we have λ− dh∗ − αi∗ = 0 which yields i∗ = λ− dh∗ α . (4.6)as far as, using the second and third equation of (4.5), one has −u(α+ ρ)i∗ + (1− ε)ki∗ − µv ∗ = 0, i.e., v ∗ = (1− ε)k − u(α+ ρ) µ i∗, hence v ∗ = (1− ε)k − u(α+ ρ) µ λ− dh∗ α (4.7) according to (4.6). the substitution of (4.7) in (4.5) yields : (1− η)l ( h∗, λ− dh∗ α , (1− ε)k − u(α+ ρ) µ λ− dh∗ α ) (1− ε)k − u(α+ ρ) µ i∗ − (α+ ρ)i∗ = 0. thus, we have (1− η) ( (1− ε)k − u(α+ ρ) ) l ( h∗, λ− dh∗ α , (1− ε)k − u(α+ ρ) µ λ− dh∗ α ) = (α+ ρ)µ since i∗ 6= 0. furthermore, i∗ ≥ 0, gives λ−dh∗ α ≥ 0. thus h∗ ≤ λ d . hence there is not a biologicalequilibrium when h∗ > λ d .let us consider the function ψ defined on [0, λd ] by : ψ(x) = (1− η)γl ( x, λ− dx α , γ(λ− dx) µα ) − (α+ ρ)µ, where γ = (1− ε)k − u(α+ ρ). https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 19we have ψ(0) = −(α+ ρ)µ < 0 and ψ ( λ d ) = (1− η)γl ( λ d , 0, 0 ) − (α+ ρ)µ, = (1− η) [(1− ε)k − u(α+ ρ)] βλ α0 + α1λ − (α+ ρ)µ, = (1− η)(1− ε)kβλ α0 + α1λ − u(1− η)(α+ ρ)βλ α0 + α1λ − (α+ ρ)µ, = 1 α0 + α1λ [(1− η)(1− ε)kβλ− µ(α+ ρ)(α0 + α1λ)− u(1− η)(α+ ρ)βλ] , = 1 α0 + α1λ [ (1− η)(1− ε)kβλ− (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] ] , = (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] α0 + α1λ (r0 − 1), it follows that ψ (λ d ) = (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] α0 + α1λ (r0 − 1) > 0 if and only if r0 > 1. moreover, letting y = λ− d.x α and z = γ(λ− d.x) µα , we have ψ ′ (x) = (1− η)γ. d dx [ l ( x, λ− dx α , γ(λ− dx) µα ) − (α+ ρ)µ ] , = (1− η)γ ( ∂l ∂x − d α ∂l ∂y − γd µα ∂l ∂z ) , = (1− η)γ ( ∂l ∂x − d α ∂l ∂x ∂x ∂y − γd µα ∂l ∂x ∂x ∂z ) , = (1− η)γ ( ∂l ∂x − d α ∂l ∂x ( − α d ) − γd µα ∂l ∂x ( − µα γd )) , = 3(1− η)γ ∂l ∂x , = 3(1− η)γ βα0 + βα2v (α0 + (α1 + α3v )x + α2v )2 > 0 if γ > 0. therefore, if r0 > 1 there exists a unique spatially homogeneous infected equilibrium e∗ = (h∗, i∗, v ∗) with h∗ ∈ (0, λd ), i∗ > 0 and v ∗ > 0.the previous investigations can be summarized in the following theorem : theorem 4.1. 1) if r0 ≤ 1, then the pde-system (2.4) admits a unique spatially homogeneous uninfected equilibrium e0 = ( λ d , 0, 0 ) . 2) if r0 > 1 and γ > 0, then the pde system (2.4) admits a unique spatially homogeneous infected equilibrium e∗ = (h∗, i∗, v ∗) with h∗ ∈ ( 0, λd ) , i∗ > 0 and v ∗ > 0. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 20 remark 4.1. due to the spatial dependence of the state variables, spatially-inhomogeneous steadystates can exist.indeed, any spatially-inhomogeneous equilibrium point e = (h, i, v ) of the model (2.4) subjectto the homogeneous neumann boundary condition must solve the following system. d1∆h + λ− dh − (1− η)βhv α0 + α1h + α2v + α3hv + ρi = 0, d2∆i + (1− η)βhv α0 + α1h + α2v + α3hv − (α+ ρ)i = 0, d3∆v + (1− ε)ki − µv − u(1− η)βhv α0 + α1h + α2v + α3hv = 0, (4.8) ∂h ∂η = ∂i ∂η = ∂v ∂η = 0. investigation of the local stability of such spatially-inhomogeneous equilibria will be the concernof a forthcoming paper via an in-depth analysis of the above system. 4.4. local stability of hcv-uninfected equilibrium. the objective of this section is to discuss thelocal stability of the spatially homogeneous uninfected equilibrium for the pde system (2.4). weaddress local stability by analysing the characteristic equation. theorem 4.2. the spatially homogeneous uninfected equilibrium e0 of pde-model system (2.4) is locally asymptotically stable if r0 ≤ 1 and it is unstable if r0 > 1. proof. let {µl , ϕl} be an eigenpair of the laplace operator −∆ on ω with the homogeneous neu-mann boundary condition where 0 = µ1 < µ2 < µ3 < · · ·. let eµl be the eigenspace correspondingto µl in c1(ω) and {ϕl j , j = 1, 2, · · ·, dimeµl} be an orthogonal basis of eµl . let x = (c1(ω))3and xl j = {ϕl jc, / c ∈ r3}.consider the following direct sum x = ∞⊕ l=1 xl with xl = dimeµl⊕ j=1 xl j , where xl j is the eigenspace corresponding to µl . linearizing (2.4) at the spatially homogeneousuninfected equilibrium e0 we obtain the following linearized system : ∂w1 ∂t = d1∆w1 − dw1 + ρw2 − (1− η)βλ α0 + α1λ w3, ∂w2 ∂t = d2∆w2 − (α+ ρ)w2 + (1− η)βλ α0 + α1λ w3, (4.9) ∂w3 ∂t = d3∆w3 + (1− ε)kw2 − [ µ+ u (1− η)βλ α0 + α1λ ] w3, https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 21where w = (w1, w2, w3)t = (h, i, v )t .from the previous system (4.9), we obtain wt = lw = d∆w +k(e0)w where k(e0)w =  −dw1 + ρw2 − (1−η)βλ α0+α1λ w3 −(α+ ρ)w2 + (1−η)βλ α0+α1λ w3 (1− ε)kw2 − ( µ+ u (1−η)βλ α0+α1λ ) w3  . (4.10) for each l ≥ 1, xl is invariant under the operator l, and λ̃ is an eigenvalue of l if and only if it isan eigenvalue of the matrix −µld + k(e0) for some l ≥ 1, in which case, there is an eigenvectorin xl . so, one has det ( −µld +k(e0)− λ̃id ) = ∣∣∣∣∣∣∣∣∣∣∣∣∣ − ( µld1 + d + λ̃ ) ρ − (1−η)βλ α0+α1λ 0 − ( µld2 + (α+ ρ) + λ̃ ) (1−η)βλ α0+α1λ 0 (1− ε)k − ( µld3 + µ+ u (1−η)βλ α0+α1λ ) − λ̃ ∣∣∣∣∣∣∣∣∣∣∣∣∣ the characteristic equation of −µld +k(e0) is − (µld1 + d + λ̃) [[ µld2 + (α+ ρ) + λ̃ ] [( µld3 + µ+ u (1− η)βλ α0 + α1λ ) + λ̃ ] − (1− ε)(1− η)kβλ α0 + α1λ ] = 0, (4.11) from (4.11), we get λ̃0 = −µld1 − d < 0, and another characteristic eigenvalues are the roots of the following equation : λ̃2 + bλ̃+ (µld2 + α+ ρ) ( µld3 + µ+ u (1− η)βλ α0 + α1λ ) − (1− ε)(1− η)kβλ α0 + α1λ = 0, (4.12) where b = [( µld3 + µ+ u (1− η)βλ α0 + α1λ ) + µld2 + (α+ ρ) ] . let c = (µld2 + α+ ρ) ( µld3 + µ+ u (1− η)βλ α0 + α1λ ) − (1− ε)(1− η)kβλ α0 + α1λ . one has, c = µld2 ( µld3 + µ+ u (1− η)βλ α0 + α1λ ) + (α+ ρ)µld3 + (α+ ρ) ( µ+ u (1− η)βλ α0 + α1λ ) − (1− ε)(1− η)kβλ α0 + α1λ , = µld2 ( µld3 + µ+ u (1− η)βλ α0 + α1λ ) + (α+ ρ)µld3 + 1 α0 + α1λ [ (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ]− (1− ε)(1− η)kβλ ] , https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 22 = µld2 ( µld3 + µ+ u (1− η)βλ α0 + α1λ ) + (α+ ρ)µld3 + (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] α0 + α1λ (1−r0). since b > 0, if r0 ≤ 1 then c is also positive. hence by virtue of the routh-hurwitz criterion,equation (4.12) does not admit solution with positive real part. thus none characteristic eigenvaluehave positive real part. therefore if r0 ≤ 1, the spatially homogeneous uninfected equilibrium e0 = ( λ d , 0, 0 ) of (2.4) is locally asymptotically stable.otherwise if r0 > 1, then for l = 1, (in this case µ1 = 0) one has, c = (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] α0 + α1λ (1−r0) < 0. hence there is a complex root of equation (4.12) with positive real part in the spectrum of kaccording to routh-hurwitz criterion. therefore the uninfected equilibrium e0 = ( λ d , 0, 0 ) of (2.4)is unstable. this completes the proof of theorem 4.2. � 4.5. global stability of hcv-uninfected equilibrium. the objective of this section is to discuss theglobal stability of the spatially homogeneous uninfected equilibrium for the pde system (2.4). weaddress global stability by using the construction of lyapunov functional method. this lyapunovfunctional is obtained from those of differential equations by applying the method presented in [15].for this purpose, we start by letting τ0 = (1− ε)k(1− η)βλ µα0(α+ ρ) . then, it is easy to see that (1− ε)(1− η)kβλ (α+ ρ) [µ(α0 + α1λ) + u(1− η)βλ] ≤ (1− ε)k(1− η)βλ µα0(α+ ρ) , i.e., r0 ≤ τ0.we state the following result on global stability at e0 as follows : theorem 4.3. the spatially homogeneous uninfected equilibrium e0 of pde-model system (2.4) is globally asymptotically stable in the positively-invariant region σ if τ0 < 1. proof. let us consider the following function g1(t) = (1− ε)k α+ ρ i(t) + v (t). then, the differentiation of g1 with respect to t gives dg1 dt = ( (1− ε)k − u(α+ ρ) µ(α+ ρ)(α0 + α1h + α2v + α3hv ) (1− η)βh − 1 ) µv. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 23since h ≤ λ d = λ in the positively-invariant region σ, one has dg1 dt ≤ [[ (1− ε)k − u(α+ ρ) ] (1− η)βλ µα0(α+ ρ) − 1 ] µv, ≤ [(1− ε)(1− η)kβλ µα0(α+ ρ) − 1 ] µv, ≤ (τ0 − 1)µv. now, we define the lyapunov function as follows l1 = ∫ ω g1dx. the computation of the time derivative of l1 along the positive solutions of the pde-model system(2.4) yields dl1 dt = d dt [ ∫ ω g1dx ] , = ∫ ω dg1 dt dx, ≤ ∫ ω [ (τ0 − 1)µv ] dx. it is clear that the condition τ0 ≤ 1 gives dl1 dt ≤ 0 for all h, i, v > 0. we note that the solutionsof system (2.4) are limited by υ, the greatest invariant subset of e={(h, i, v ) ∈ σ|dl1 dt = 0 }.we realize that dl1 dt = 0 if and only if v = 0 and i = 0. each element of υ satisfies v = 0 andconsequently i=0. by lyapunov-lasalle invariance principle [27], e0 is globally asymptoticallystable if τ0 < 1. so, we obtain a sufficient condition r0 ≤ τ0 which ensures that the hcv spatiallyhomogeneous equilibrium e0 of pde-model system (2.4) is globally asymptotically stable if τ0 < 1.this completes the proof of theorem 4.3. � 4.6. local stability of hcv spatially homogeneous infected equilibrium. let us study the localstability of the unique infected spatially homogeneous equilibrium e∗ of our pde-model system.consider the laplace operator −∆ and let 0 = µ1 < µ2 < µ3 < · · · be its eigenvalues on ω withthe homogeneous neumann boundary condition, and eµl be the eigenspace corresponding to µlin c1(ω). let also x = (c1(ω))3, {ϕl j , j = 1, 2, · · ·, d imeµl} be an orthogonal basis of eµl and xl j = {ϕl jc / c ∈ r3}.then, x = ∞⊕ l=1 xl with xl = dimeµl⊕ j=1 xl j . now, let set w1 = h, w2 = i , w3 = v . further we use the vector notation w = (w1, w2, w3)t = (h, i, v )t . then the linearization of the pde system at e∗ is of the form wt = lw = d∆w +k(e∗)w, https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 24where k(e∗)w =  − (d + a)w1 + ρw2 − bw3 aw1 − (α+ ρ)w2 + bw3 −uaw1 + (1− ε)kw2 − (µ+ ub)w3  , (4.13) with a = (1− η)(α0 + α2v ∗)βv ∗ (α0 + α1h∗ + α2v ∗ + α3h∗v ∗)2and b = (1− η)(α0 + α1h ∗)βh∗ (α0 + α1h∗ + α2v ∗ + α3h∗v ∗)2 . for each l ≥ 1, xl is invariant under the operator l, and λ̃ is an eigenvalue l if and only if it isan eigenvalue of the matrix −µld + k(e∗) for some l ≥ 1, in which case, there is an eigenvectorin xl . therefore we get: det ( − µld+k(e∗)− λ̃id ) = ∣∣∣∣∣∣∣∣ −(µld1 + d + a)− λ̃ ρ −b a −(µld2 + α+ ρ)− λ̃ b −ua (1− ε)k − (µld3 + µ+ ub)− λ̃ ∣∣∣∣∣∣∣∣ . the characteristic equation of −µld +k(e∗) is on the form λ̃3 + a2λ̃ 2 + a1λ̃+ a0 = 0 (4.14) where a2 = (µld1 + d + a+ µld2 + α+ ρ+ µld3 + µ+ ub) > 0, a1 = (µld1 +d+a)(µld2 +α+ρ+µld3 +µ+ub) + (µld2 +α+ρ)(µld3 +µ+ub)−(1−ε)kb, a0 = (µld1 + d + a)(µld2 + α+ ρ)(µld3 + µ+ ub)− (µld1 + d + a)(1− ε)kb.if a1 > 0 and a1a2 > a0 from the above investigations, it then follows from routh-hurwitz criterionthat all roots of (4.14) have negative real parts and therefore we have the following result. theorem 4.4. if a1 > 0 and a1a2 > a0, then the spatially homogeneous infected equilibrium e∗ = (h∗, i∗, v ∗) of the pde-model system (2.4) is locally asymptotically stable when it exists. 4.7. global stability of hcv-spatially homogeneous infected equilibrium. the objective of thissection is to discuss the global stability of the spatially homogeneous infected equilibrium e∗ forthe pde system (2.4). we address global stability by using the method of construction of lyapunovfunctionals. these lyapunov functional is obtained from those for differential equations by applyingthe method of hattaf and yousfi presented in [15]. we address this study with certain assumptionsnamely : u = 0 (i.e., there is no absorption effect), α0 = 1 et α3 = α1α2. thus we have thefollowing results. theorem 4.5. the spatially homogeneous infected equilibrium e∗ of pde-model system (2.4) is globally asymptotically stable when it exists. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 25 proof. we first define the function g2(h, i, v ) = h −h∗ − ∫ h h∗ (α+ ρ)i∗ (1−η)βτv ∗ (1+α1τ)(1+α2v ∗) dτ + i − i∗ − i∗ ln( i i∗ ) + α+ ρ (1− ε)k (α0 + α2v ∗) v − v ∗ − ∫ v v ∗ (α+ ρ)i∗ (1−η)βτh∗ (1+α1h∗)(1+α2τ) dτ  . then, the computation of the derivative of g2 with respect to t yields : dg2 dt = [ λ− dh − αi − (α+ ρ)µ (1− ε)k v ] − (α+ ρ)i∗ (1 + α1h)(1 + α2v ∗) (1− η)βhv ∗ [ λ− dh − (1− η)βhv (1 + α1h)(1 + α2v ) + ρi ] − i∗ i [ (1− η)βhv (1 + α1h)(1 + α2v ) − (α+ ρ)i ] − α+ ρ (1− ε)k v ∗ v [(1− ε)ki − µv ] . since (1− η)βh∗v ∗ (1 + α1h∗)(1 + α2v ∗) = (α+ ρ)i∗, λ = dh∗ + αi∗, and (α+ ρ)µ (1− ε)k = (α+ ρ)i∗ v ∗ , we have dg2 dt = [ dh∗ + αi∗ − dh − αi − (α+ ρ)i∗ v v ∗ ] −(α+ ρ)i∗ (1 + α1h)(1 + α2v ∗) (1− η)βhv ∗ [ dh∗ + αi∗ − dh − (1− η)βhv (1 + α1h)(1 + α2v ) + ρi ] − i∗ i (1− η) (1+α1h ∗)(1+α2v ∗)(α+ρ)i∗ (1−η)h∗v ∗ hv (1 + α1h)(1 + α2v ) − (α+ ρ)i − (α+ ρ)i v ∗ v + (α+ ρ)µ (1− ε)k v ∗, = [ dh∗ + (α+ ρ)i∗ − ρi∗ − dh − αi − (α+ ρ)i∗ v v ∗ ] − [h∗ h 1 + α1h 1 + α1h∗ dh∗ + h∗ h 1 + α1h 1 + α1h∗ αi∗ − 1 + α1h 1 + α1h∗ dh∗ − v v ∗ 1 + α2v ∗ 1 + α2v (α+ ρ)i∗ + h∗ h 1 + α1h 1 + α1h∗ ρi ] + (α+ ρ)i∗ [ 1− hi∗v (1 + α1h ∗)(1 + α2v ∗) h∗iv ∗(1 + α1h)(1 + α2v ) ] + (α+ ρ)i∗ ( 1− i i∗ v ∗ v ) , https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 26 =dh∗ [ 1− h h∗ − h∗ h 1 + α1h 1 + α1h∗ + 1 + α1h 1 + α1h∗ ] + (α+ ρ)i∗ [ 1− hi∗v (1 + α1h ∗)(1 + α2v ∗) h∗iv ∗(1 + α1h)(1 + α2v ) + v v ∗ 1 + α2v ∗ 1 + α2v ] + (α+ ρ)i∗ ( 2− v v ∗ − i i∗ v ∗ v ) − αi∗ ( ρ α + i i∗ ) − h∗ h 1 + α1h 1 + α1h∗ αi∗ − h∗ h 1 + α1h 1 + α1h∗ ρi, =− d(h −h∗)2 h(1 + α1h∗) + (α+ ρ)i∗ [ −1− v v ∗ + v v ∗ 1 + α2v ∗ 1 + α2v + 1 + α2v 1 + α2v ∗ ] + (α+ ρ)i∗ [ 4− h∗ h 1 + α1h 1 + α1h∗ − hi∗v (1 + α1h ∗)(1 + α2v ∗) h∗iv ∗(1 + α1h)(1 + α2v ) − i i∗ v ∗ v − 1 + α2v 1 + α2v ∗ ] − αi∗ ( ρ α + i i∗ ) − h∗ h 1 + α1h 1 + α1h∗ αi∗ − h∗ h 1 + α1h 1 + α1h∗ ρi + (α+ ρ)i∗ h∗ h 1 + α1h 1 + α1h∗ . therefore dg2 dt =− d(h −h∗)2 h(1 + α1h∗) − α2(α+ ρ)i∗(v − v ∗)2 v ∗(1 + α2v ∗)(1 + α2v ) − αi∗ ( ρ α + i i∗ ) + (α+ ρ)i∗ [ 4− h∗ h 1 + α1h 1 + α1h∗ − hi∗v (1 + α1h ∗)(1 + α2v ∗) h∗iv ∗(1 + α1h)(1 + α2v ) − i i∗ v ∗ v − 1 + α2v 1 + α2v ∗ ] . we have : (α+ ρ)i∗ [ 4− h∗ h 1 + α1h 1 + α1h∗ − hi∗v (1 + α1h ∗)(1 + α2v ∗) h∗iv ∗(1 + α1h)(1 + α2v ) − i i∗ v ∗ v − 1 + α2v 1 + α2v ∗ ] ≤ 0 since the left side of the latter inequality is the difference between the geometric mean and thearithmetic mean. that is dg2 dt ≤ 0. otherwise dg2 dt = 0 if and only if h = h∗, i = i∗ et v = v ∗.thus g2 is a lyapunov functional of the differential equation associated to the pde-model system(2.4). therefore using lyapunov-lasalle invariance principle [27] combined to the method presentedin [15], the functional defined by l2(t) = ∫ ω g2(t)dxis a lyapunov functional of the pde-model system (2.4) at the spatially homogeneous infectedequilibrium e∗. therefore e∗ is globally asymptotically stable. this completes the proof of theo-rem 4.5. � 5. numerical simulations in this section, we present the numerical simulations to illustrate our theoretical results. tosimplify, we consider ibvp (2.4) with ω = (1) under neumann boundary condition ∂h ∂ν = 0, ∂i ∂ν = 0, ∂v ∂ν = 0 t > 0, x = 1 (5.1) and, following initial conditions h(x, 0) = 5, i(x, 0) = 5, v (x, 0) = 5, (5.2) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 27and h(x, 0) = 15, i(x, 0) = 5, v (x, 0) = 5. (5.3)now we choose the numerical values of the parameters for the pde-cellular model system (2.4)as follows: λ = 50; d = 5; ρ = 0, 01; α = 0, 05; d1 = d2 = d3 = 0, 1; η = 0, 00004; α3 = 0, 03; ε = 0, 5; α2 = 0, 02; k = 2; α1 = 0, 1; µ = 20; α0 = 1; β = 0, 24 et u = 1. by calculation wehave r0 = 0.943361. in this case, pde-cellular model system (2.4) has a spatially homogeneousequilibrium e0 = (10, 0, 0). hence by theorem 4.3 e0 is globally asymptotically stable. numericalsimulation illustrates our result (see figure 1). otherwise we choose the numerical values of (a) (b) (c) figure 1. simulations of ibvp (2.4) under neumann boundary conditions (5.1) andinitial condition (5.2) the parameters for the pde-cellular model system (2.4) as follows : λ = 50; d = 5; ρ = 0, 01; α = 0, 05; d1 = d2 = d3 = 0, 1; η = 0, 00004; ε = 0, 5; α0 = 1; α1 = 0, 1; α2 = 0, 02; α3 = 0, 03; k = 2; µ = 2; β = 0, 24 et u = 1. by calculation we have r0 = 6.25009. in this case,pde-cellular model system (2.4) has a spatially homogeneous equilibrium e∗ = (5; 500; 235). https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 28hence by theorem 4.5 e∗ is globally asymptotically stable. numerical simulation illustrates ourresult (see figure 2). (a) (b) (c) figure 2. simulations of ibvp (2.4) under neumann boundary conditions (5.1) andinitial condition (5.3) 6. conclusion in this work, we addressed the dynamics of a reaction diffusion hcv intra-host infection modelwith the hattaf-yousfi incidence rate, which is a generalized non-linear incidence rate. the object ofthis work was to make a mathematical analysis of a cellular model of hcv infection which assumesthat virions diffuse into the liver, which uses the hattaf-yousfi functional response generalizingmost of the functional responses that exist. our model also takes into account the absorption effectwhich is much neglected in the literature. we first showed that the initial value and boundaryproblem (2.4) admits a unique global solution in time. and secondly, we have shown that thisunique solution is positive and uniformly bounded. then, we determined the expression of basic https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 29reproduction number r0 which is the parameter from which we studied the dynamics of our modelat equilibria whose existence and uniqueness of these have been previously proven. more precisely,we have shown that, if r0 < 1, the unique uninfected equilibrium point is locally and globallyasymptotically stable. this means that, under this condition,infection disappears. otherwise, theuninfected equilibrium point is unstable; and in this case the infection persists in the host. it hasalso been shown under the hypothesis r0 > 1, that the infected equilibrium point is locally andglobally asymptotically stable. this edifying work ended with numerical simulations, carried outon the mathematica software, which confirmed our theoretical results. in order to get as close aspossible to complex reality of biological phenomena, we envisage in the future, the mathematicalanalysis of models taking into account cell proliferation, the delays and more generalization of theincident rate function. appendix a. proof of proposition 3.8 the proof is established by using banach’s fixed point theorem.choose β such that 3 4 < β < 1, then the injection i : d(hβ) → c0 b is continuous by lemma 3.3.for r0 > 0 and t > 0, the following closed ball is considered : br0 (h0, i0, v0) = { (h, i, v ) ∈ (c0 b((0, t ], d(hβ)))3 : ‖h −h0‖β, ‖i − i0‖β, ‖v − v0‖β ≤ r0 } .(a.1)by proposition 3.1, we have the local lipschitz properties: ‖f(t, h1, i1, v1)−f(t, h2, i2, v2)‖2 ≤ k1 1‖h1 −h2‖d(hβ) +k1 2‖i1 − i2‖d(hβ) +k1 3‖v1 − v2‖d(hβ), ‖g(t, h1, i1, v1)− g(t, h2, i2, v2)‖2 ≤ k2 1‖h1 −h2‖d(hβ) +k2 2‖i1 − i2‖d(hβ) +k2 3‖v1 − v2‖d(hβ), ‖q(t, h1, i1, v1)−q(t, h2, i2, v2)‖2 ≤ k3 1‖h1 −h2‖d(hβ) +k3 2‖i1 − i2‖d(hβ) +k3 3‖v1 − v2‖d(hβ),(a.2) for t ∈ [0, t ], (h1, i1, v1), (h2, i2, v2) ∈ br0 (h0, i0, v0) and lipschitz-constants k ij > 0, i , j = 1, 2, 3. in addition (y1, y2, y3) ∈ br0 (h0, i0, v0), define p : [0, t ] → l2(ω), q : [0, t ] → l2(ω)and r : [0, t ]→ l2(ω) as follows py1(t) = g1(t)h0 + ∫ t 0 g1(t − τ)f(t, y1(τ), y2(τ), y3(τ))dτ, qy2(t) = g2(t)i0 + ∫ t 0 g2(t − τ)g(t, y1(τ), y2(τ), y3(τ))dτ, ry3(t) = g3(t)v0 + ∫ t 0 g3(t − τ)q(t, y1(τ), y2(τ), y3(τ))dτ. finally, set m1 = sup t∈[0,t ] ‖f(t, y1(0), y2(0), y3(0))‖2, m2 = sup t∈[0,t ] ‖g(t, y1(0), y2(0), y3(0))‖2, m3 = sup t∈[0,t ] ‖q(t, y1(0), y2(0), y3(0))‖2 and choose t so that: ‖g1(h)h0 −h0‖β, ‖g2(h)i0 − i0‖β, ‖g3(h)v0 − v0‖β ≤ 3r0 4 , 0 ≤ h ≤ t, (a.3) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 30 c iβ,2(mi + r0k i 1 + r0k i 2 + r0k i 3) ∫ t 0 s−βds ≤ r0 4 , i = 1, 2, 3, (a.4) where c iβ,2 is the constant in property 2) of corollary 3.5 for operator a (a is zero for i = 1, 2).note that such t exists since g1(h), g2(h) and g3(h) converge to id as h tends to 0+ by definitionof an analytic semigroup and∫ t 0 s−βds = 1 1− βh 1−β → 0, h → 0+ f or β < 1. then the proof will continue according to the following two points : (a): it is shown that (p,q,r) maps br0 (h0, i0, v0) into itself, (b): it is shown that (p,q,r) is a strict contraction on br0 (h0, i0, v0), allowing the use ofbanach’s fixed point theorem to get the existence of a unique fixed point in br0 (h0, i0, v0).let (h, i, v ) ∈ br0 (h0, i0, v0). then, using (a.3) and property 2) of corollary 3.5, we have : ‖ph(t)−h0‖d(hβ) = ∥∥∥g1(t)h0 −h0 + ∫ t 0 g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ ∥∥∥ d(hβ) , ≤ ‖g1(t)h0 −h0‖d(hβ) + ∫ t 0 ‖g1(t − τ)f(τ,h(τ), i(τ), v (τ))‖d(hβ)dτ, ≤ 3r0 4 + ∫ t 0 c1 β,2(t − s)−β ∥∥∥f(τ,h(τ), i(τ), v (τ)) ∥∥∥ 2 dτ, ≤ 3r0 4 + ∫ t 0 c1 β,2(t − s)−β ∥∥∥f(τ,h(τ), i(τ), v (τ))−f(τ,h0, i0, v0) +f(τ,h0, i0, v0) ∥∥∥ 2 dτ, ≤ 3r0 4 + ∫ t 0 c1 β,2(t − s)−β (∥∥∥f(τ,h(τ), i(τ), v (τ))−f(τ,h0, i0, v0) ∥∥∥ 2 + ∥∥∥f(τ,h0, i0, v0) ∥∥∥ 2 ) dτ, ‖ph(t)−h0‖d(hβ) ≤ 3r0 4 + ∫ t 0 c1 β,2(t − s)−β ( k1 1 r0 +k1 2 r0 +k1 3 r0 +m1 ) dτ, ≤ 3r0 4 + c1 β,2 ( k1 1 r0 +k1 2 r0 +k1 3 r0 +m1 ) ∫ t 0 (t − s)−βdτ, ≤ 3r0 4 + 0, ≤ r0 for 0 ≤ t ≤ t, and similarly ‖qi(t)− i0‖d(hβ) ≤ r0 and < ‖rv (t)− v0‖d(hβ) ≤ r0.showing that (p,q,r) maps br0 (h0, i0, v0) into itself. furthermore, from property 1) in corol-lary 3.5 and lemma 3.6 we compute: ‖ph(t + h)− ph(t)‖d(hβ) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 31 = ∥∥∥g1(t + h)h0 − g1(t)h0 + ∫ t+h 0 g1(t + h − τ)f(τ,h(τ), i(τ), v (τ))dτ − ∫ t 0 g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ ∥∥∥ d(hβ) , = ∥∥∥g1(t)g1(h)h0 − g1(t)h0 + ∫ t 0 g1(h)g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ + ∫ t+h t g1(h)g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ − ∫ t 0 g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ ∥∥∥ d(hβ) , = ∥∥∥g1(t)(g1(h)− id)h0 + ∫ t 0 (g1(h)− id)g1(t − τ)f(τ,h(τ), i(τ), v (τ))dτ + ∫ t+h t g1(t + h − τ)f(τ,h(τ), i(τ), v (τ))dτ ∥∥∥ d(hβ) . the strong continuity of the semigroup and lemma 3.6 yields ‖ph(t + h)− ph(t)‖d(hβ) → 0 as h → 0+. therefore p is continuous from [0,t] into d(hβ).a similar calculation shows that q and r have the same properties and item (a) is proved.presently let us show that (p,q,r) is a strict contraction on br0 (h0, i0, v0).let (y1, y2, y3), (z1, z2, z3) ∈ br0 (h0, i0, v0). then ‖py1(t)− pz1(t)‖d(hβ) = ∥∥∥∫ t 0 g1(t − τ) ( f(τ, y1(τ), y2(τ), y3(τ))−f(τ, z1(τ), z2(τ), z3(τ)) ) dτ ∥∥∥ d(hβ) , ≤ ∫ t 0 ∥∥∥g1(t − τ) ( f(τ, y1(τ), y2(τ), y3(τ))−f(τ, z1(τ), z2(τ), z3(τ)) )∥∥∥ d(hβ) dτ, ≤ ∫ t 0 c1 β,2(t − τ)−β ∥∥∥f(τ, y1(τ), y2(τ), y3(τ))−f(τ, z1(τ), z2(τ), z3(τ)) ∥∥∥ 2 dτ, ≤ ∫ t 0 c1 β,2(t − τ)−β ( k1 1‖y1 − z1‖d(hβ) +k1 2‖y2 − z2‖d(hβ) +k1 3‖y3 − z3‖d(hβ) ) dτ, ≤ c1 β,2 ( k1 1 sup τ∈[0,t] ‖y1 − z1‖d(hβ) +k1 2 sup τ∈[0,t] ‖y2 − z2‖d(hβ) + k1 3 sup τ∈[0,t] ‖y3 − z3‖d(hβ) )∫ t 0 (t − τ)−βdτ. that is : ‖py1(t)− pz1(t)‖d(hβ) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 32 ≤ c1 β,2 ( k1 1 +k1 2 +k1 3 + m1 r0 )( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) )∫ t 0 (t − τ)−βdτ, ≤ ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) ×c1 β,2 ( k1 1 +k1 2 +k1 3 + m1 r0 )∫ t 0 s−βds, ≤ ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) × 1 r0 c1 β,2 ( r0k 1 1 + r0k 1 2 + r0k 1 3 +m1 ) ∫ t 0 s−βds, ≤ ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) 1 r0 × r0 4 , ≤ 1 4 ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) , for every t ∈ [0, t ]. hence sup t∈[0,t ] ‖py1(t)− pz1(t)‖d(hβ) ≤ 1 4 ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) . similarly sup t∈[0,t ] ‖qy1(t)−qz1(t)‖d(hβ) ≤ 1 4 ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) and sup t∈[0,t ] ‖ry1(t)− rz1(t)‖d(hβ) ≤ 1 4 ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) . thus sup t∈[0,t ] ‖(p,q,r)(y1(t), y2(t), y3(t))− (p,q,r)(z1(t), z2(t), z3(t))‖d(hβ)3 ≤ sup τ∈[0,t] ( ‖py1 − pz1‖d(hβ) + ‖qy2 −qz2‖d(hβ) + ‖ry3 − rz3‖d(hβ) ) , ≤ sup τ∈[0,t] ‖py1 − pz1‖d(hβ) + sup τ∈[0,t] ‖qy2 −qz2‖d(hβ) + sup τ∈[0,t] ‖ry3 − rz3‖d(hβ), ≤ 3 4 ( sup τ∈[0,t] ‖y1 − z1‖d(hβ) + sup τ∈[0,t] ‖y2 − z2‖d(hβ) + sup τ∈[0,t] ‖y3 − z3‖d(hβ) ) , ≤ 3 4 sup t∈[0,t ] ‖(y1(t), y2(t), y3(t))− (z1(t), z2(t), z3(t))‖d(hβ)3 . hence (p,q,r) is a strict contraction on br0 (h0, i0, v0) and this proves part b). according tobanach’s fixed point theorem, (p,q,r) has a unique fixed point in br0 (h0, i0, v0). this is thesolution of (2.4) on [0,t] with initial value (h(0), i(0), v (0)) = (h0, i0, v0) in (d(hβ))3. thiscompletes the proof of proposition 3.8. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 33appendix b. proof of theorem 3.11 we first prove the existence and positivity of the solution. from theorem 3.10, there exist asequence (wmi ) and a function wi such that wmi → wi in c0([0, t ],h), with wi ≥ 0 and wi(0) = w0i .we check that qi(w m−1)→ qi(w) and qi(w m−1)wmi → qi(w)wi in c0([0, t ],h). we also have fi(w m−1)→ fi(w) in c0([0, t ],h) ⋂ l2((0, t ), e′).but, wmi is solution of〈∂wmi ∂t , vi 〉 + 〈 aiw m i , vi 〉 + ( qi(w m−1)wmi , vi ) = 〈 fi(w m−1), vi 〉 , ∀vi ∈ e. (b.1) we take φ ∈ d((0, t )), such that φvi ∈ l2((0, t ), e),∫ t 0 〈∂wmi ∂t , φvi 〉 dt+ ∫ t 0 〈 aiw m i , φvi 〉 dt+ ∫ t 0 ( qi(w m−1)wmi , φvi ) dt = ∫ t 0 〈 fi(w m−1), φvi 〉 dt.(b.2)the second term in the left side and the right side of the equality (b.2) converges due to the weakconvergence in l2((0, t ), e′). the third term in the left-hand side of (b.2) also converges, due tothe convergence in c([0, t ],h). we deduce that ∂wmi ∂t converges weakly in l2((0, t ), e′).but we have wmi → wi in c0([0, t ],h).then ∂wmi ∂t → ∂wi ∂t in d′((0, t ),h)therefore, we obtain ∂wmi ∂t → ∂wi ∂t weakly in l2((0, t ), e′),and∫ t 0 〈∂wi ∂t , φvi 〉 dt + ∫ t 0 〈 aiwi , φvi 〉 dt + ∫ t 0 ( qi(w)wi , φvi ) h dt = ∫ t 0 〈 fi(w), φvi 〉 dt. (b.3) this being true for all φ, one has〈∂wi ∂t , vi 〉 + 〈 aiwi , vi 〉 + ( qi(w)wi , vi ) h = 〈 fi(w), vi 〉 , ∀vi ∈ e. that is to say, d dt (wi , vi)h + a(wi , vi) + ( qi(w)wi , vi ) h = 〈 fi(w), vi 〉 , ∀vi ∈ e, (b.4) ∂wi ∂t = fi(w)− aiwi − qi(w)wi in l2((0, t ), e′). (b.5) https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 34according to (3.27) and (3.28) we have wmi (t) = gi(t)w0i + ∫ t 0 gi(t − s)(−qi(wm−1)wmi + fi(w m−1))(s)ds, (b.6) and in addition, as qi(wm−1)wmi and fi(w m−1) converge in c0([0, t ],h) and the operator gi ,defined by the relation (3.29), is compact, using the limit in (b.6) one has, wi(t) = gi(t)w0i + ∫ t 0 gi(t − s)(−qi(w)wi + fi(w))(s)ds. (b.7) it remains to prove uniqueness.let v be another solution of ibvp (3.21). then vi ∈ w (0, t, e, e′)⇒ vi ∈ c0([0, t ],h) and vi ≥ 0. consequently we obtain qi(v)vi + fi(v) ∈ l2((0, t ), e′). thus, by proposition 2.11 of [12], one has vi(t) = gi(t)w0i + ∫ t 0 gi(t − s)(−qi(v)vi + fi(v))(s)ds. subtracting, we have wi(t)− vi(t) = ∫ t 0 gi(t − s) ( − (qi(w)wi − qi(v)vi) + (fi(w)− fi(v)) ) (s)ds, (b.8) with qi(w)wi − qi(v)vi = qi(w)wi − qi(w)vi + qi(w)vi − qi(v)vi , = qi(w)(wi − vi) + (qi(w)− qi(v))vi . since wi is positive, one has∥∥∥∥∥ wj α0 + α1wk + α2wj + α3wkwj ∥∥∥∥∥ ≤ 1 k ‖wj‖∞ where ‖wj‖∞ = ‖wj‖l∞((0,t ),h). if we define ‖w‖∞ = 3∑ j=1 ‖wj‖∞, there is m1 > 0 such that ‖q(w)‖∞ ≤ m1‖wj‖∞. https://doi.org/10.28924/ada/ma.3.1 eur. j. math. anal. 10.28924/ada/ma.3.1 35so, for r = 1, 2, 3, the numerator of qr (w) − qr (v) is the sum of terms of the form (wk − vk)vj or (wj − vj)wk , and we can find m2 > 0 such that ∣∣qr (w)− qr (v) ∣∣ h(s) ≤ m2 ( 3∑ j=1 |wj(s)− vj(s)|h ) . also we can find m3 > 0 such that∣∣fr (w)− fr (v) ∣∣ h(s) ≤ m3 ( 3∑ j=1 |wj(s)− vj(s)|h ) . summing up |wj(s) − vj(s)|h and noting that ‖gj(t − s)‖ ≤ njeθjt with nj , θj > 0, we can find m > 0 such that 3∑ j=1 |wj(s)− vj(s)|h ≤ m‖w − v‖∞. replacing in (b.8), we obtain 3∑ j=1 |wj(s)− vj(s)|h ≤ m2‖w − v‖∞ ∫ t 0 sds = m2 t 2 2 ‖w − v‖∞. by induction, we have 3∑ j=1 |wj(s)− vj(s)|h ≤ mn n! t n‖w − v‖∞, with lim n→+∞ mn n! t n‖w − v‖∞ = 0. therefore w = v . this ends the proof of theorem 3.11. acknowledgmenta.nangue acknowledges support from the faculty of sciences of the university of maroua, wherethis work was initiated. conflict of intereststhe authors declare that they have no conflict of interests regarding the publication of this paper. authors’ contributiona. nangue provided the subject, wrote the introduction, the conclusion, checked the proofs andverified the calculation. he also managed the wellposedness of the initial and boundary valueproblem. b. nde tchiffo conceived the study and computed 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https://doi.org/10.4134/bkms.2011.48.3.555. https://doi.org/10.28924/ada/ma.3.1 https://doi.org/10.1137/0518057 https://doi.org/10.1016/j.aml.2009.06.004 https://doi.org/10.1016/j.aml.2011.02.007 https://doi.org/10.1016/j.jmaa.2007.02.006 https://www.jstor.org/stable/44239519 https://doi.org/10.1126/science.282.5386.103 https://doi.org/10.1126/science.272.5258.74 https://doi.org/10.1073/pnas.93.9.4398 https://doi.org/10.9734/bjmcs/2016/28640 https://doi.org/10.1016/j.mbs.2013.04.012 https://doi.org/10.1016/j.jmaa.2006.06.064 https://doi.org/10.1137/080732870 https://doi.org/10.1007/s11071-011-9954-0 https://doi.org/10.1016/j.mbs.2007.05.004 https://doi.org/10.1016/j.mbs.2007.05.004 https://doi.org/10.1137/120872942 http://apps.who.int/iris/bitstream/10665/255016/1/9789241565455eng.pdf http://apps.who.int/iris/bitstream/10665/255016/1/9789241565455eng.pdf https://doi.org/10.1016/j.nonrwa.2013.06.005 https://doi.org/10.4134/bkms.2011.48.3.555 1. introduction 2. formulation of the pde-cellular model 2.1. fluctuation of healthy hepatocytes 2.2. fluctuation of hcv infected cells 2.3. fluctuation of free hcv virions 2.4. the initial boundary value problem associated to pde-cellular model 3. qualitative and quantitative analysis and some properties of the solutions for ibvp (2.4) 3.1. local existence and uniqueness of solutions for the ibvp (2.4) 3.2. boundedness of the solutions for ibvp (2.4) 3.3. global existence, uniqueness and positivity for the ibvp (2.4) 4. stability analysis of the spatially homogeneous equilibria 4.1. hcv-spatial homogeneous uninfected equilibrium e0 4.2. basic reproduction number r0 4.3. existence and uniqueness of hcv-spatial homogeneous infected equilibrium e* 4.4. local stability of hcv-uninfected equilibrium 4.5. global stability of hcv-uninfected equilibrium 4.6. local stability of hcv spatially homogeneous infected equilibrium 4.7. global stability of hcv-spatially homogeneous infected equilibrium 5. numerical simulations 6. conclusion appendix a. proof of proposition 3.8 appendix b. proof of theorem 3.11 references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 13doi: 10.28924/ada/ma.2.13 on the semi-local convergence of a third order scheme for solving nonlinear equations samundra regmi1, ioannis k. argyros2,∗, santhosh george3, christopher argyros4 1learning commons, university of north texas at dallas, dallas, tx, usa samundra.regmi@untdallas.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. the semi-local convergence analysis of a third order scheme for solving nonlinear equationin banach space has not been given under lipschitz continuity or other conditions. our goal isto extend the applicability of the cordero-torregrosa scheme in the semi-local convergence underconditions on the first fréchet derivative of the operator involved. majorizing sequences are used forproving our results. numerical experiments testing the convergence criteria are given in this study. 1. introduction cordero and torregrosa in [10] considered the third order scheme, defined for n = 0, 1, 2, . . . , by yn = xn − f ′(xn)−1f (xn) xn+1 = xn − 3m−1n f (xn), (1.1) for solving the nonlinear equation f (x) = 0, (1.2) where mn = 2f ′ ( 3xn+yn 4 ) − f ′ ( xn+yn 2 ) + 2f ′ ( xn+3yn 4 ) . here f : d ⊂ e −→ e1 is an operatoracting between banach spaces e and e1 with d 6= ∅. in general a closed form solution for (1.2) isnot possible, so iterative schemes are used for approximating a solution x∗ of (1.2) (see [1–27]). received: 14 feb 2022. key words and phrases. semi-local convergence; cordero-torregrosa scheme; iterative schemes; banach space; con-vergence criterion. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 2the local convergence of the this scheme in the special case when e = e1 = r was shown to beof order three using taylor expansion and assumptions on the fourth order derivative of f, whichis not on these schemes [10]. so, the assumptions on the fourth derivative reduce the applicabilityof these schemes [1–27].for example: let e = e1 = r, d = [−0.5, 1.5]. define λ on d by λ(t) = { t3 log t2 + t5 − t4 i f t 6= 0 0 i f t = 0. then, we get f (1) = 0, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. obviously λ′′′(t) is not bounded on d. so, the convergence of scheme (1.1) is not guaranteed bythe previous analyses in [1–27].in this study we introduce a majorant sequence and use general continuity conditions to extendthe applicability of scheme (1.1). our analysis includes error bounds and results on uniqueness of x∗ based on computable lipschitz constants not given before in [1–27] and in other similar studiesusing taylor series. our idea is very general. so, it applies on other schemes too.the rest of the study is set up as follows: in section 2 we present results on majorizing sequences.sections 3,4 contain the semi-local and local convergence, respectively, where in section 4 thenumerical experiments are presented. concluding remarks are given in the last section 5. 2. majorizing sequences scalar sequences are developed that majorize scheme (1.1). let k0 > 0, k > 0 and η > 0 begiven constants. define sequences {tn}, {sn} by t0 = 0, s0 = η tn+1 = sn + 2k(sn − tn)(tn+1 − tn) 9(1−k0tn)(1− pn) , sn+1 = tn+1 + k(tn+1 − tn + sn − tn)(tn+1 − tn) 2(1−k0tn+1) , (2.1) where pn = 5k0 6 (sn + tn). notice that tn+1 is given implicitly in the first substep of sequence (2.1).it we solve for tn+1, we get its explicit form tn+1 = 9sn(1−k0tn)(1− pn)− 2tnk(sn − tn) 9(1−k0tn)(1− pn)− 2k(sn − tn) . but for the convergence analysis in theorem 3.1 we prefer tn+1 in its implicit form.next, we present sufficient conditions for the convergence scheme (1.1). https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 3 lemma 2.1. suppose that 5(tn + sn) < 6 k0 . (2.2) for all n = 0, 1, 2, . . . . then, sequences {tn} is nondecreasing and bounded from above by t ∗ = 3 5k0 and as such it converge to its unique least upper t ∈ [0, t ∗]. proof. it follows from the definition (2.1) of sequences {tn} and (2.2) that this sequence isnondecreasing and bounded from above by t ∗, and as such it converges to t. �the next result shows the convergence of sequence {tn}, under stronger but easier to verifyconditions than (2.2). but first we need to introduce some functions and parameters. definefunctions g1 and g2 on the interval (0, 1) by g1(t) = 4k(1 + t)t − 4k(1 + t) + 9k0t, and g2(t) = k(2 + t)(1 + t)t −k(2 + t)(1 + t) + 2k0t 3.then, we get g1(0) = −4k, g1(1) = 9k0, g2(0) = −2k and g2(1) = 2k0.hence, functions g1 and g2 have roots in (0, 1). denote the minimal such roots by α1 and α2, re-spectively. set a = 2k(t1−t0) 9(1−k0t)(1−p0) , b = k(t1−t0+s0−t0)(t1−t0) 2η(1−k0t1) , c̄ = min{a, b}, c = max{a, b}, α3 = min{α1, α2} and α = max{α1, α2}.then, we can show the second result on majorizing sequences for method (1.2). lemma 2.2. suppose 0 < c̄ ≤ c ≤ α3 ≤ α ≤ 1− 10 3 k0η. (2.3) then, sequence {tn} is nondecreasing, bounded from above by t = η 1−α and as such it converges to its unique least upper bound t∗ ∈ [0, t ]. proof. items 0 ≤ 2k(tk+1 − tk) 9(1−k0tk)(1− pk) ≤ α, (2.4) 0 ≤ k(tk+1 − tk + sk − tk)(tk+1 − tk) 2(1−k0tk+1) ≤ α(sk − tk), (2.5) 0 ≤ 1 1− pk ≤ 2 (2.6)and tk ≤ sk ≤ tk+1 (2.7)are shown using induction on k. these estimates are true for k = 0 by (2.3). suppose thesehold for all k smaller than n − 1. by induction hypotheses and (1.2), we have 0 ≤ sk − tk ≤ α(sk−1 − tk−1) ≤ . . . ≤ αkη, tk+1 − tk = (tk+1 − sk) + (sk − tk) ≤ (1 + α)(sk − tk) https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 4and tk+1 ≤ (1− αk+2)η 1− α < t. evidently, (2.4) holds if 4k(1 + α)αk−1η 9(1−k0 1−α k+1 1−α η ≤ α, (2.8) where we used (2.6). define recurrent polynomials f (1)k on the interval (0, 1) by f (1) n (t) = 4k(1 + t)tk−1η + 9k0(1 + t + . . .+ tk−1)η − 9. (2.9) then, estimate (2.8) holds if f (1) n (t) ≤ 0 at t = α1. (2.10) we need a relationship between two consecutive polynomials f (1)k : f (1) k+1(t) = 4k(1 + t)tkη + 3k0(1 + t + . . .+ tk)η − 9 + f (1) k (t) −4k(1 + t)tk−1 + 3k0(1 + t + . . .+ tk−1)η + 9 = f (1) k (t) + g1(t)t k−1η. (2.11) in particular, one gets f (1)k+1(α1) = f (1) k (α1) since by the definition of α1 and g1, g1(α1) = 0.define function f (1)∞ (t) = lim k−→∞ f (1) k (t). (2.12) then, (2.10) holds if f (1)∞ (t) ≤ 0 at t = α1. (2.13) but by (2.9) and (2.12) one gets f (1)∞ (t) = 9k0η 1− t − 9, (2.14) so (2.13) holds if f (1)∞ (t) ≤ 0 at t = α1 which is true by (2.3).similarly, (2.5) holds if k(2 + α)(1 + α)αkη 2(1−k0 1−α k+2 1−α η) ≤ α. (2.15) define polynomials f (2)k (t) on the interval (0, 1) by f (2) k (t) = k(2 + t)(1 + t)tk−1η + 2k0(1 + t + . . .+ tk+1)η − 2. (2.16) then, (2.15) holds if f (2) k (t) ≤ 0 at t = α2. (2.17) https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 5we get f (2) k+1(t) = k(2 + t)(1 + t)tkη + 2k0(1 + t + . . .+ tk+2)η − 2 + f (2) k (t) −k(2 + t)(1 + t)tk−1η − 2k0(1 + t + . . .+ tk+1)η + 2 = f (2) k (t) + g2(t)t k−1η, (2.18) and f (2) k+1(α2) = f (2) k (α2). (2.19) define function f (2)∞ (t) = lim k−→∞ f (2) k (t). (2.20) then, (2.17) holds if f (2)∞ (t) ≤ 0 at t = α2. (2.21) by (2.16) and (2.20), we get f (2)∞ (t) = k0η 1− t − 1, so (2.21) holds by (2.3). moreover, estimate (2.6) certainly holds if 2pk = 5k0 3 (sk+tk) < 5k0 3 ( η 1−α+ η 1−α) = 10k0η 3(1−α) < 1, which is true by (2.3). furthermore, estimate (2.7) holds by (2.4)-(2.6) and thedefinition of sequence {tk}. hence the induction for estimates (2.4)-(2.7) is completed. it followsthat sequence {tk} is nondecreasing and bounded from above by t ∗, and such it converges to t. �if one desires iterates to be given explicitly in (2.1), then define instead sequence {tn} as follows t0 = 0, s0 = η tn+1 = sn + 2k(1 +k0tn)(sn − tn)2 3(1−k0tn)(1− pn) (2.22) sn+1 = tn+1 + 2k(tn+1 − tn + sn − tn)(tn+1 − tn) 2(1−k0tn+1) . moreover, define recurrent polynomial on the interval [0, 1) by f (1) n (t) = 4k 3 tn−1η + 4kk0 3 tn−1(1 + t + . . .+ tn)η2 +k0(1 + t + . . .+ tn)η − 1. this time we have f (1) n+1(t) = f (1) n (t) + g (1) n (t)tn−1η, (2.23) where g (1) n (t) = 4kk0 3 tn+2η + 4kk0 3 tn+1η + 4 3 kt − 4k 3 (1−k0η). https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 6 we get g(1)n (0) = −4k3 (1 − k0η) < 0 for k0η < 1, and g(1)1 (1) = 4kk0η > 0. denote by rn thesmallest solution of g(1)n (t),respectively. notice that these solutions are increasing as n increases,since g(1)n (t) ≤ g(1)n−1(t). hence, it follows by (2.23) that f (1) n+1(t) ≤ f (1) n (t) + g (1) 1 (t)tn−1η. in particular for α1 = r1, we get f (1) n+1(t) ≤ f (1) n (t) at t = α1. hence, f (1) n (t) ≤ 0 holds if f (1) 1 (t) ≤ 0 at t = α1.but f (1) 1 (t) = 4k 3 η + 4 3 kk0η 2 +k0η − 1. define b = 2k(s0−t0) 3 . then, we arrive at the following convergence results for majorizing sequence(2.2). lemma 2.3. suppose 5(tn + sn) < 6 k0 , where {tn} is the sequence defined by (2.22). then, the conclusions of lemma 2.2 hold for this sequence. lemma 2.4. suppose 0 < c̄ ≤ c ≤ α3 ≤ α ≤ 1− 10k0 3 η (2.24) and ( 4k 3 + 4 3 k0kη +k0 ) η ≤ 1. (2.25) then, the conclusions of lemma 2.2 hold for sequence {tn} given by (2.22). remark 2.5. the solutions α1 and α2 in lemma 2.2 depend only on k0 and k. similarly α2 in lemma 2.4 depends on k0 and k1. but α1 depends k0, k and η. to avoid this dependence pick any γ ∈ (0, 1] and set γ = k0η. define functions ḡ(1)n (t) on [0, 1) by ḡ (1) n (t) = 4kγ 3 tn+1 + 4kγ 3 tn=1 + 4 3 kt − 4k 3 (1− γ). then, according to the proof of lemma 2.2 we can set α1 = r̄1, where r̄1 is the smallest solution in (0, 1) of equation ḡ(1)1 (t) = 0 assured also to exist. finally, notice that the first condition shows implicitly and the second explicitly the smallness of η. https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 73. semi-local convergence the following sufficient convergence criteria (a) are used. suppose:(a1) there exist x0 ∈ d and η > 0 such that f ′(x0)−1 exists and ‖f ′(x0)−1f (x0)‖ ≤ η. (a2) ‖f ′(x0)−1(f ′(w)− f ′(x0))‖ ≤ k0‖w − x0‖for all w ∈ d. set d0 = d ∩ u(x0, 1 k0 ).(a3) ‖f ′(x0)−1(f ′(w)− f ′(v)‖ ≤ k‖w − v‖for all v ∈ d0 and w = v − f ′(v)−1f (v). denote by l the constant, if (a3) holds for all u, v ∈ d0, and by l1 the constant for all u, v ∈ d. it follows that k ≤ l ≤ l1. in practicewe shall use whichever of k or l is easier to compute (see also the numerical section).(a4) hypotheses of lemma 2.1 or lemma 2.2 hold.and(a5) u[x0, t ∗] ⊂ d (or u[x0, t ] ⊆ d).next, the semi-local convergence of scheme (1.1) is developed based on conditions (a) and theaforementioned notation. theorem 3.1. suppose conditions (a) hold. then, the following items hold {xn} ∈ u(x0, t ∗) (3.1) and ‖x∗ − xn‖ ≤ t∗ − tn, (3.2) where x∗ = limn−→∞ xn ∈ u[x0, t ∗] and f (x∗) = 0. proof. mathematical induction is used to show ‖yk − xk‖ ≤ sk − tk (3.3) and ‖xk+1 − yk‖ ≤ tk+1 − sk . (3.4)it follows from (a1) and (1.1) that ‖y0 − x0‖ = ‖f ′(x0)−1f (x0) ≤ η =≤ s0 − t0 = η ≤ t, (3.5) so y0 ∈ u(x0, t ∗) and (3.3) hold for k = 0. let z ∈ u(x0, t ∗). in view of (a2), one has ‖f ′(x0)−1(f ′(z)− f ′(x0)) ≤ k0‖z − x0‖ ≤ k0t∗ < 1, so f ′(z)−1 ∈ l(e1, e) and https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 8 ‖f ′(z)−1f ′(x0)‖ ≤ 1 1−k0‖z − x0‖ . (3.6) by a result due to banach [14] on linear invertible operators. operator mk can be shown to beinvertible. indeed, by the definition of operator mk , (2.2) and (a2) we obtain ‖(3f ′(x0))−1(mk − 3f ′(x0))‖ ≤ 1 3 [2‖f ′(x0)−1 ( f ′ ( 3xk + yk 4 ) − f ′(x0))‖ +‖f ′(x0)−1 ( f ′ ( xk + yk 2 ) − f ′(x0) ) ‖ +2‖f ′(x0)−1 ( f ′ ( xk + 3yk 4 ) − f ′(x0) ) ≤ 1 3 (2k0‖ 3xk + yk 4 − x0‖+k0‖ xk + yk 2 − x0‖ +2k0‖ xk + 3yk 4 − x0‖) ≤ 1 3 (2k0 3tk + sk 4 +k0 sk + tk 2 + 2k0 tk + 3sk 4 ) = 5k0 6 (tk + sk) = pk < 1, so mk is invertible and ‖m−1k f ′(x0)‖ ≤ 1 3(1− pk) , (3.7) and xk+1 is well defined by the second substep of method (1.1). then, we can write by method(1.1) that xk+1 = xk − f ′(xk)−1f (xk) + (f ′(xk)−1 − 3m−1k )f (xk) = yk − 1 3 f ′(xk)−1(mk − 3f ′(xk))m−1k (xk+1 − xk). (3.8) we need the estimate, mk − 3f ′(xk) = 2f ′ ( 3xk + yk 4 ) − f ′ ( xk + yk 2 ) +2f ′ ( xk + 3yk 4 ) − 3f ′(xk) = ( f ′ ( 3xk + yk 4 ) − f ′ ( xk + yk 2 )) + ( f ′ ( 3xk + yk 4 ) − f ′(xk) ) + 2 ( f ′ ( xk + 3yk 4 ) − f ′(xk) ) , https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 9so by (a3) ‖f ′(x0)−1(mk − 3f ′(xk))‖ ≤ k‖ 3xk + yk 4 − 2xk + 2yk 4 ‖ k‖ 3xk + yk 4 − 4xk 4 ‖+ 2k‖ xk + 3yk 4 − 4xk 4 ‖ = 2k‖yk − xk‖ ≤ 2k(sk − tk). (3.9) using (1.1), (3.6) (for z = xk ) and (3.7)-(3.9) ‖xk+1 − yk‖ ≤ 2k(sk − tk)(tk+1 − tk) 9(1−k0tk)(1− pk) = tk+1 − sk . (3.10) we also have ‖xk+1 − x0‖ ≤ ‖xk+1 − yk‖+ ‖yk − x0‖ ≤ tk+1 − sk + sk − t0 = tk+1 ≤ t∗, (3.11) so xk+1 ∈ u(x0, t ∗). we can write by method (1.1) f (xk+1) = f (xk+1)− f (xk)− 1 3 mk(xk+1 − xk) = ∫ 1 0 (f ′(xk + θ(xk+1 − xk))dθ − 1 3 mk)(xk+1 − xk). (3.12) one can obtain the estimate∫ 1 0 (f ′(xk + θ(xk+1 − xk))dθ − 2 3 f ′ ( 3xk + yk 4 ) + 1 3 f ′ ( xk + yk 2 ) − 2 3 f ′ ( xk + 4yk 4 ) = ∫ 1 0 f ′(xk + θ(xk+1 − xk))dθ − f ′(xk)) + 2 3 (f ′(xk)− f ′ ( 3xk + yk 4 ) ) + 1 3 (f ′(xk)− f ′ ( xk + 3yk 4 ) + 1 3 (f ′ ( xk + yk 2 ) − f ′ ( xk + 3yk 4 ) ), (3.13) so ‖f ′(x0)−1 ∫ 1 0 (f ′(xk + θ(xk+1 − xk))dθ − 1 3 mk)‖ ≤ k [ ‖xk+1 − xk‖ 2 + ‖yk − xk‖ 6 + ‖yk − xk‖ 4 + ‖yk − xk‖ 12 ] ≤ k( tk+1 − tk 2 + sk − tk 6 + sk − tk 4 + sk − tk 12 ) = k 2 (tk+1 − tk + sk − tk). (3.14) it follows from method (1.1), (3.6) (for z = xk+1), (3.11) and (2.10) that https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 10 ‖yk+1 − xk+1‖ ≤ ‖(f ′(xk+1)−1f ′(x0)f ′(x0)−1f (xk+1)‖ ≤ k(tk+1 − tk + sk − tk)(tk+1 − tk) 2(1−k0tk+1) = sk+1 − tk+1, (3.15) showing (3.3). moreover, we get ‖yk+1 − x0‖ ≤ ‖yk+1 − xk+1‖+ ‖xk+1 − x0‖ ≤ sk+1 − tk+1 + tk+1 − t0 = sk+1 ≤ t∗, (3.16) so yk+1 ∈ u(x0, t ∗). the induction for (3.3) and (3.6) is completed. it follows from(3.3), (3.6), (3.10)and (3.16) that sequence {xn} is fundamental in banach space e, and as such it converges to x∗ ∈ u[x0, t ∗]. using (3.9) and letting k −→ ∞ in ‖f ′(x0)−1f (xk+1)‖ ≤ k 2 (tk+1 − tk + sk − tk),we obtain f (x∗) = 0. �next, a uniqueness of the solution x∗ result is presented. proposition 3.2. suppose: (1) the element x∗ ∈ u(x∗, s ∗) is a simple solution of (1.2), and (a2) holds. (2) there exists δ ≥ s∗ so that k0(s ∗ + δ) < 2. (3.17) set d1 = d ∩ u[x∗, δ]. then, x∗ is the unique solution of equation (1.2) in the domain d1. proof. let q ∈ d1 with f (q) = 0. define s = ∫ 1 0 f ′(q + θ(x∗ − q))dθ. using (h2) and (3.17)one obtains ‖f ′(x0)−1(s − f ′(x0))‖ ≤ k0 ∫ 1 0 ((1− θ)‖q − x0‖+ θ‖x∗ − x0‖)dθ ≤ k0 2 (s∗ + δ) < 1, so q = x∗, follows from the invertability of s and the identity s(q−x∗) = f (q)−f (x∗) = 0−0 = 0. � remark 3.3. (i) point t given in closed form can repalce t∗ in theorem 3.1. (ii) we used majorizing sequence {tn} given by (2.1) and lemma 2.2 to prove theorem 3.1. but we can also use majorizing sequence {tn} given by (2.22) and lemma 2.3 to arrive at the conclusions of the theorem 3.1. simply notice that in the proof of this theorem we got using the second substep of https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 11 scheme (1.1), (3.8) and (3.9) estimate (3.10) leading to the definition of the first substep of sequence (2.1). but we can use the first substep of scheme (1.1) to write instead of (3.8) that xk+1 = yk − f ′(xk)−1(mk − 3f ′(xk))m−1k f (xk)(yk − xk) leading to ‖xk+1 − yk‖ ≤ 2k(1 +k0tk)(sk − tk)2 3(1−k0tk)(1− pk) = tk+1 − sk , where, we also used ‖f ′(x0)−1f (xk)‖ = ‖f ′(x0)−1((f ′(xk)− f (x0)) + f ′(x0))‖ ≤ 1 +k0‖xk − x0‖ ≤ 1 +k0tk . hence, we arrive at the second semi-local convergence rsult for scheme (1.1). theorem 3.4. suppose:conditions (a) hold with (a4) replaced by (a4)’ hypotheses of lemma 2.3 or lemma 2.4 hold. then, the conclusions of theorem 3.1 hold with (2.22) replacing (2.1). in practice we shall use the theorem providing the best results. 4. numerical experiments lipschitz parameters are determinded and convegence criteria are tested for some numericalexperiments. example 4.1. define scalar function ζ(t) = ξ0t + ξ1 + ξ2 sin ξ3t, x0 = 0, where ξj , j = 0, 1, 2, 3 are parameters. then, clearly for ξ3 large and ξ2 small, k0l1 can be small (arbitrarily). in particular, notice that k l1 −→ 0. example 4.2. let e = e1 = c[0, 1] and d = u[0, 1]. it is well known that the boundary value problem [12]. ς(0) = 0, ς(1) = 1, ς ′′ = −ς − σς2 can be given as a hammerstein-like nonlinear integral equation ς(s) = s + ∫ 1 0 q(s, t)(ς3(t) + σς2(t))dt where σ is a parameter. then, define f : d −→ e1 by [f (x)](s) = x(s)− s − ∫ 1 0 q(s, t)(x3(t) + σx2(t))dt. https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 12 choose ς0(s) = s and d = u(ς0, ρ0). then, clearly u(ς0, ρ0) ⊂ u(0, ρ0 + 1), since ‖ς0‖ = 1. suppose 2σ < 5. then, conditions (a) are satisfied for k0 = 2σ + 3ρ0 + 6 8 , l = σ + 6ρ0 + 3 4 , and η = 1+σ 5−2σ . notice that k0 < l. example 4.3. let us consider a scalar function ψ defined on the set d = u[x0, 1 − q] for q ∈ (0, 12), by ψ(x) = x3 − q. choose x0 = 1. then, we obtain the estiamtes |ψ′(x0)−1(ψ′(x)− ψ′(x0))| = |x2 − x20 | ≤ |x + x0||x − x0| ≤ (|x − x0|+ 2|x0|)|x − x0| = (1− q + 2)|x − x0| = (3− q)|x − x0|, for all x ∈ d, so k0 = 3− q, d0 = u(x0, 1 k0 ) ∩d = u(x0, 1 k0 ), |ψ′(x0)−1(ψ′(y)− ψ′(x)| = |y2 − x2| ≤ |y + x ||y − x | ≤ (|y − x0 + x − x0 + 2x0)||y − x | = (|y − x0|+ |x − x0|+ 2|x0|)|y − x | ≤ ( 1 k0 + 1 k0 + 2)|y − x | = 2(1 + 1 k0 )|y − x |, for all x, y ∈ d0, so l = 2(1 + 1 k0 ), |ψ′(x0)−1(ψ′(y)− ψ′(x)| = (|y − x0|+ |x − x0|+ 2|x0|)|y − x | ≤ (1− q + 1− q + 2)|y − x | = 2(2− q)|y − x |, for all x, y ∈ d and l1 = 2(2− q). notice that for all q ∈ (0, 12) k0 < l < l1. next, set y = x − ψ′(x)−1ψ(x), x ∈ d. then, we have y + x = x − ψ′(x)−1ψ(x) + x = 5x3 + q 3x2 . define fundtion ψ̄ on the interval d = [q, 2− q] by ψ̄(x) = 5x3 + q 3x2 . https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 13 then, we get by this definition that ψ̄′(x) = 15x4 − 6xq 9x4 = 5(x − q)(x2 + xq + q2) 3x3 , where p = 3 √ 2q 5 is the critical point of function ψ̄. notice that q < p < 2 − q. it follows that this function is decreasing on the interval (q, p) and increasing on the interval (q, 2 − q), since x2 + xq + q2 > 0 and x3 > 0. so, we can set k1 = 5(2− q)2 + q 9(2− q)2 , η = 1− q 3 and k1 < k0. but if x ∈ d0 = [1− 1 k0 , 1 + 1 k0 ], then k = 5%3 + q 9%2 , where % = 4−q 3−q and k < k1 for all q ∈ (0, 12). next, we verify conditions (2.2), (2.3), (2.24) and (2.25). then for q = 0.95, 6k0 = 2.9268 and n 1 2 3 4 5 tn 0.1683 0.1694 0.1694 0.1694 0.1694 α1 = 0.1643 = α3, α2 = 0.6588 = α, a = 0.0030 = c̄ , b = 0.0136 = c, 1 − 10k0η3 = 0.8861, and (4k3 + 4 3k0kη + k0)η = 0.0521 < 1. hence, conditions (2.2),(2.3), (2.24) and (2.25) hold. 5. conclusion the semi-local convergence of scheme (1.1) with order three is extended using general conditionson f ′ and recurrent majorizing sequences. references [1] i.k. argyros, on the newton kantorovich hypothesis for solving equations, j. comput. math. 169 (2004) 315-332. https://doi.org/10.1016/j.cam.2004.01.029[2] i.k. argyros, computational theory of iterative schemes. series: studies in computational mathematics, 15, editors:c.k.chui and l. wuytack, elsevier publ. co. new york, u.s.a, 2007.[3] i.k. argyros, convergence and applications of newton-type iterations, springer verlag, berlin, germany, (2008).[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s scheme, j. complex. 28 (2012) 364–387. https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s scheme, appl. math. comput. 225 (2013)372-386. https://doi.org/10.1016/j.amc.2013.09.049 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(1984).[20] p.d. proinov, general local convergence theory for a class of iterative processes and its applications to newton’sscheme, j. complex. 25 (2009) 38-62. https://doi.org/10.1016/j.jco.2008.05.006[21] p.d. proinov, new general convergence theory for iterative processes and its applications to newton-kantorovichtype theorems, j. complex. 26 (2010) 3-42. https://doi.org/10.1016/j.jco.2009.05.001[22] w.c. rheinboldt, an adaptive continuation process of solving systems of nonlinear equations, banach center publ.3 (1978) 129-142.[23] s.m. shakhno, o.p. gnatyshyn, on an iterative algorithm of order 1.839. . . for solving the nonlinear least squaresproblems, appl. math. comput. 161 (2005) 253–264. https://doi.org/10.1016/j.amc.2003.12.025.[24] s.m. shakhno, r.p. iakymchuk, h.p. yarmola, convergence analysis of a two step scheme for the nonlinear squaresproblem with decomposition of operator, j. numer. appl. math. 128 (2018) 82-95.[25] j.r. sharma, r.k. guha, r. sharma, an efficient fourth order weighted newton scheme for systems of nonlinearequations, numer. algorithms, 62 (2013) 307–323, https://doi.org/10.1007/s11075-012-9585-7.[26] j.f. traub, iterative schemes for the solution of equations, prentice hall, new jersey, u.s.a. (1964).[27] r. verma, new trends in fractional programming, nova science publisher, new york, usa, (2019). https://doi.org/10.28924/ada/ma.2.13 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1090/s0025-5718-04-01646-1 https://doi.org/10.1016/j.amc.2007.01.062 https://doi.org/10.1016/j.amc.2011.08.011 https://doi.org/10.1007/ s10910-018-0856-y https://doi.org/10.1007/ s10910-018-0856-y https://doi.org/10.1016/j.cam.2013.11.019 https://doi.org/10.1016/j.jco.2008.05.006 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/j.amc.2003.12.025 https://doi.org/10.1007/s11075-012-9585-7 1. introduction 2. majorizing sequences 3. semi-local convergence 4. numerical experiments 5. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 7doi: 10.28924/ada/ma.3.7 a note on the stability of functional equations via a celebrated direct method dongwen zhang1, john michael rassias2, qi liu3, yongjin li4,∗ 1school of mathematics (zhuhai), sun yat-sen university, zhuhai 519082, p.r. china zhangdw25@mail2.sysu.edu.cn 2national and kapodistrian university of athens, department of mathematics and informatics, attikis 15342, greece jrassias@primedu.uoa.gr 3school of mathematics and physics, anqing normal university, anqing 246133, p.r. china liuq325@mail2.sysu.edu.cn 4department of mathematics, sun yat-sen university, guangzhou, 510275, p.r. china stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. more than ten years after justyna sikorska [8] attempted to solve the heyers-ulam sta-bility of a single variable equation by using direct method. in this paper, we will improve the resultsof justyna sikorska by using a more efficient approach. relations between the generalized functionalequation, the dependence of their different parameters and several properties are also further ex-plored. to achieve the problem, we try to develop some new techniques to overcome the fundamentaldifficulties caused by the different properties of the function and the presence of several variables inthe equation. furthermore, we continue to construct and study a couple of functional equations bymaking a new direct method. 1. introduction the core idea of the hyers-ulam stability for functional equations has been dated back to awell-known problem concerning about group homomorphisms solved by s.m. ulam and d.h. hyers(see [1–3]). in the last decades, a great number of papers treating the stability problem aboutfunctional equations has already been achieved and a great deal of important problems about thisfield has been studied ( [4–7]). it follows that the most efficient methods have been stated in manypapers ( [10, 18–24,27]) such as the direct approach, the shadowing approach, and invariant meanapproach and so on. in particular, the direct method is always the main studying tool on theinvestigation of functional equations of different types. received: 15 sep. 2022. key words and phrases. stability; several functional equations; approximations; odd function; even function.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 2the stability problems for an appropriate simple variable functional equations have earlier beeninvestigated by direct method. the direct method is familiar with many readers to derive thesolutions of the equations. the author in [8] have made full use of quite a general way to solve thehyers-ulam stability problems on the functional equations under which many excellent outcomeshave been achieved without reduplicating the similar procedure in the whole process of computation.however, her results can only be used to derive the solutions of the equation where the mediatefunction is odd. this is exactly our contribution to the paper. in fact, a straightforward observationis that the inequality ‖f (x)− uf (e(x))− vf (−e(x))‖ 6 δ(x) can be solved if the function h is even. next, the present studying approach calls us to investigatethe following functional inequality, by using a direct method, under which the result can not becovered by earlier works ‖f (x + y + z) + f (x) + f (y) + f (z)− f (x + y)− f (z + y)− f (x + z)‖ 6 k (‖x‖r + ‖y‖r + ‖z‖r ) . (1.1) in fact, the functional inequality (1.1) comes from some equivalent characterizations of hilbert spacein [15]. the investigator described several properties of an inner product space and applies theseresults to solve many interesting functional inequalities such as: zarantone’s inequality, hayashi’sinequality and so on. however, the more far reaching work can be done m. fréchet in [16] underwhich he ascertained that the corresponding equation is a necessary prerequisite condition whencomplex or real normed completed spaces become hilbert spaces. investigator in [17] studied thestability of fréchet functional equation from which a characterization of inner product spaces hadbeen achieved by using a stationary point theorem in banach spaces. compared with the beforestudying approaches, we further explored solutions of the equation (1.1) in this literature. of course,to the best of our knowledge, it has also already been solved by [8] under which a direct methodwas to derive solutions of the equation (1.1) and to look for some improvement approximations.however, in this literature we make a new direct method to achieve the solution of equation (1.1)must be close to the approximate solution, approximately satisfying the corresponding equation.besides this, we will consider that the functions on the functional equation of different typeshave been defined in a more general domain. for instance, the papers [11, 12] have defined anadditive ρ-functional inequalities in nonarchimedean normed spaces and banach spaces. however,this phenomenon can not attract enough attention to the study of functional equations in themore general and complex nonlinear structure of f-spaces (see the definition in [13, 14]). but,the nonlinear structure of space has always stood in a very important position of leadership infunctional analysis. based on the above analysis, it is of great significance that the functionalinequality is considered in β-homogeneous f-space. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 3in section 2, the counterpart of theorem 2.1 from [8] where the mediate function is odd willbe considered. in the subsequent part, a new direct method for solving (1.1) in f-space will bedescribed and some new extended results of theorem 2.1 from [8] will be presented. with it, twonew different applications of the results will be described in the final part. 2. a simple variable of abstract equation in theorem 2.1 from [8], sikorska solved the equation (2.1) where the mediate function e is oddand the related parameters u, v are restricted on the real field. for simplicity in notation, weprovide traditionally our first result with the studying mapping defined in banach space. by makinguse of small conjectures the more general form of the results will be provided in β-homogeneous f -space in section 3. therefore, our first result is simply considered in banach space. theorem 2.1 let (x,+) be a group, and (y, ‖ · ‖) be a banach space, and assume the mapping f : x → y satisfying the inequality ‖f (x)− uf (e(x))− vf (−e(x))‖ 6 δ(x), x ∈ x, (2.1) where u, v ∈ (−∞,+∞), and the mappings e : x → x, δ : x → [0,∞) satisfy that e is even ( i.e., e(−x) = e(x) for every x ∈ x). let the infinite progression ∑∞n=0 [|un| δ (en(x)) + |vn|δ (−en(x))]with u0 := 1, un := [ u(u + v)n−1 ] , v0 := 0, vn := [ v(u + v)n−1 ] , n ∈ n(em states the m-th composition of function e ), be assumed convergence for every x ∈ x . thenthere has a unique even mapping g : x → y satisfying g(x) = ung(en(x)) + vng(−en(x)), and ‖f (x)− g(x)‖ 6 ∞∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] , x ∈ x and n ∈ n. (2.2) proof. we will prove that ‖f (x)− unf (en(x))− vnf (−en(x))‖ 6 γn(x), x ∈ x, (2.3) where γn(x) := n−1∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] , x ∈ x, n ∈ n. first of all, consider with every m, n ∈ n and it is easy to observe that un+1 = uun + uvn, vn+1 = vvn + vun, and un+m = umun + vmun, vn+m = umvn + vmvn. (2.4) https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 4from the definition of sequences (un) and (vn) we also have uvn = vun, vmun = umvn. first, (2.1) gives (2.3) with setting n = 1, and by mathematical induction, later we suppose that (2.3) establishes for some n ∈ n . we prove that in the case for n + 1 by virtue of (2.1) ‖f (x)− un+1f ( en+1(x) ) − vn+1f ( −en+1(x) ) ‖ 6 ‖f (x)− unf (en(x))− vnf (−en(x))‖ + |un| ∥∥f (en(x))− uf ( en+1(x) ) − vf ( −en+1(x) )∥∥ + |vn| ∥∥f (−en(x))− uf ( en+1(x) ) − vf ( −en+1(x) )∥∥ 6 n−1∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] + |un| δ (en(x)) + |vn| δ (−en(x)) = n∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] . since the series ∑∞i=0 [|ui | δ (e i(x) ) + |vi | δ ( −e i(x) )] is convergent for every x ∈ x , combinedwith (2.3) and by virtue of the completeness of y , the mapping can be well defined as in thefollowing: g(x) := lim n→∞ [unf (en(x)) + vnf (−en(x))] , x ∈ x, (2.5) and we prove the following properties of the function g.an easy computation is to prove that ug(e(x)) + vg(−e(x)) = u lim n→∞ [ unf ( en+1(x) ) + vnf ( −en+1(x) )] + v lim n→∞ [ unf ( en+1(x) ) + vnf ( −en+1(x) )] = lim n→∞ [ (uun + vun) f ( en+1(x) ) + (uvn + vvn) f ( −en+1(x) )] = g(x). furthermore, we will prove the more general property of g g(x) = ung (en(x)) + vng (−en(x)) , f or al l x ∈ x and n ∈ n. (2.6) by induction, we assume that the equation is true for all natural number k with k ≤ n for some n ∈ n. let us calculate with k = n + 1 g(x) = ung (en(x)) + vng (−en(x)) = un(ug ( en+1(x) ) + vg ( −en+1(x) ) ) + vn(ug ( en+1(x) ) + vg ( −en+1(x) ) ) = un+1g ( en+1(x) ) + vn+1g ( −en+1(x) ) ). in particular, we also have that the function g is even. an easy computation is to state that g(−x) = ung (en(x)) + vng (−en(x)) = g(x), f or every x ∈ x and n ∈ n. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 5in order to achieve the uniqueness of g, suppose further that ḡ : x → y is the another mappingsuch that (2.2) and (2.6) hold. then ‖g(x)− ḡ(x)‖ 6 2 ∞∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] , x ∈ x. moreover, we have g(x)− ḡ(x) = un [g (en(x))− ḡ (en(x))] + vn [g (−en(x))− ḡ (−en(x))] , x ∈ x, and on account of (2.4) and (2.6) we can rewrite ‖g(x)− ḡ(x)‖ 6 |un + vn| ‖g (en(x))− ḡ (en(x))‖ 6 2|un + vn| ∞∑ i=0 [ |ui | δ ( e i+n(x) ) + |vi | δ ( −e i+n(x) )] = 2 ∞∑ i=0 [(|ui(un + vn)|) δ ( e i+n(x) ) + |vi(un + vn)|δ ( −e i+n(x) ) ] = 2 ∞∑ i=0 [ |ui+n| δ ( e i+n(x) ) + |vi+n| δ ( −e i+n(x) )] = 2 ∞∑ j=n [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] for every x ∈ x and n ∈ n, where it states that g = ḡ as n →∞. this proves the theorem. � the purpose of stating and proving this results is of particular interest and give out a solutionof a simple variable functional equation (2.1) at least. in section 3, we will extend the results oftheorem 2.1 form [8] to a more general setting. in particular, the related parameters u, v can beextended to complex numbers.according to the above analysis, we give out a corollary of theorem 2.1 (still quite general).first of all, we must state that the absolute of an element x ∈ x can be given out in the real fieldconsidering that the function h is even for the meaningful of the results, for example h(x) = a|x |.as a matter of fact, we can also present the absolute of x = (x1, x2, · · · , xn) ∈ rn by |x | = (|x1|, |x2|, · · · , |xn|). thus, it worth stating the results. in particular, we can present the followingresults in the euclidean space if the more general setting can not be judged. corollary 2.1 assume that (x,+) is a real or complex normed linear space and set (y, ‖ · ‖) isa banach space. suppose further that the mapping f : x → y fulfils the inequality∥∥∥∥f (x)− a + 1 2a2 f (a|x |) + a − 1 2a2 f (−a|x |) ∥∥∥∥ 6 δ(x), x ∈ x, (2.7) where a ∈ r with a > 1 and mappings e : x → x, δ : x → [0,∞) make that e is an even function ( i.e., e(−x) = e(x) for every x ∈ x). the infinite progression ∑∞i=0 1ai δ (ai |x |) is convergence for https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 6every x ∈ x . then there has a unique mapping g : x → y fulfilling the following equations for all x ∈ x g(x) = a + 1 2a2 g(a|x |)− a − 1 2a2 g(−a|x |), and ‖f (x)− g(x)‖ 6 ∆(x) + λ(x), where ∆(x) := 1 2 ∑∞ i=0 1 ai [ δ ( ai |x | ) + δ ( −ai |x | )] , λ(x) := 1 2 ∑∞ i=0 1 a2i [ δ ( ai |x | ) − δ ( −ai |x | )] , x ∈ x.furthermore, g can be obtained in the following limiting equality g(x) := lim n→∞ ( an + 1 2a2n f (an|x |)− an − 1 2a2n f (−an|x |) ) , x ∈ x. proof. by using the results of theorem 2.1, u := 1+a 2a2 , v := 1−a 2a2 and together e(x) := a|x |, for all x ∈ x , a computation is to prove that un := 1 + a 2a2n , vn := 1− a 2a2n , n ∈ n. for the convergent series ∑∞i=0 1ai δ (ai |x |) with x ∈ x , therefore ∑∞i=0 1a2i δ (ai |x |) is convergence.applying theorem 2.1, there has a unique limiting function g : x → y fulfilling ‖f (x)− g(x)‖ 6 ∞∑ i=0 [∣∣∣∣1 + ai 2a2i ∣∣∣∣ δ (ai |x |)+ ∣∣∣∣1− ai2a2i ∣∣∣∣ δ (−ai |x |)] = ∞∑ i=0 1 a2i [ δ ( ai |x | ) − δ ( −ai |x | )] + ∞∑ i=0 1 ai [ δ ( ai |x | ) + δ ( −ai |x | )] = λ(x) + ∆(x). function g has been dated back to derived from (2.5). we complete the proof. � remark 2.2 if u = 1, v = 0 and e(x) = |x |, the above results may be trivial and meaningless.in the above results, suppose that a ∈ (−∞,∞) which is not equal to −1, 0, 1. exchanging a with −a, this transformation may not be different from primary inequality (2.7). this is a basic factleaving to the reader to check it. assume that the convergent series ∑∞i=0 1|a|i δ (|a|i |x |) establishesfor every x ∈ x . in fact, the assertions with |a| exchanging for a has also been achieved by asimilar way. corollary 2.2 assume that (x,+) is a group and set (y, ‖ · ‖) is a banach space. supposefurther that the mapping f : x → y fulfils the inequality∥∥∥∥f (x)− a + 1 2a2 f (a|x |) + a − 1 2a2 f (−a|x |) ∥∥∥∥ 6 δ, x ∈ x, https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 7where a ∈ (−∞,∞) with |a| > 1 and δ > 0 is constant. then there is a unique limiting evenfunction g : x → y fulfilling ‖f (x)− g(x)‖ 6 |a|δ |a| − 1 . proof. since the function δ is a positive constant, thus ∆(x) = |a| |a|−1δ and λ(x) = 0 for every x ∈ x . the mapping g has been stated in the following shape: g(x) := lim n→∞ ( |a|n + 1 2a2n f (|a|n|x |)− |a|n − 1 2a2n f (−|a|n|x |) ) , x ∈ x. this proves the proof. � remark 2.3 the above corollaries 2.1 and 2.2 will still establish in β-homogeneous f -spacewith a ∈ (−∞,∞) and |a| > 1. if we exchange a for 1a in the equation f (x)− a + 1 2a2 f (a|x |) + a − 1 2a2 f (−a|x |) from (2.7), the second group of results will also be obtained with a is a positive constant stated inthe following results. corollary 2.3 assume that (x,+) is a group divisible by a with a ∈ (−∞,∞) and |a| > 1 andset (y, ‖ · ‖) is a banach space. suppose further that the mapping f : x → y fulfils the inequality∥∥∥∥f (x)− a2 + a 2 f ( 1 a |x | ) − a2 − a 2 f ( − 1 a |x | )∥∥∥∥ 6 δ(x), x ∈ x, with δ : x → [0,∞) is such that the convergent series ∑∞i=0 a2iδ ( 1ai |x |) holds for every x ∈ x .then there has a unique even limiting mapping g : x → y fulfilling for every x ∈ x , g(x) = a2 + a 2 g ( 1 a |x | ) + a2 − a 2 g ( − 1 a |x | ) , and ‖f (x)− g(x)‖ 6 ∆̃(x) + λ̃(x). furthermore, the mapping g can be stated in the following shape: g(x) := lim n→∞ [ a2n + an 2 f ( 1 an |x | ) + a2n − an 2 f ( − 1 an |x | )] , x ∈ x. proof. applying for theorem 2.1 for u := a2+a 2 , v := a2−a 2 and e(x) := 1 a |x |, for all x ∈ x , an easycomputation is to show that un := a2n + a2n−1 2 , vn := a2n − a2n−1 2 , n ∈ n. according to the convergent series ∑∞ i=0 a 2iδ ( 1 ai |x | ) for all x ∈ x , hence the series∑∞ i=0 a 2i−1δ ( 1 ai |x | ) is convergence, and there has a unique even limiting mapping g : x → y https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 8fulfilling ‖f (x)− g(x)‖ 6 ∞∑ i=0 [∣∣∣∣a2i + ai 2 ∣∣∣∣ δ( 1 ai |x | ) + ∣∣∣∣a2i − ai2 ∣∣∣∣ δ(− 1 ai |x | )] = ∞∑ i=0 a2i 2 [ δ ( 1 ai |x | ) + δ ( − 1 ai |x | )] + ∞∑ i=0 ai 2 [ δ ( 1 ai |x | ) − δ ( − 1 ai |x | )] = ∆̃(x) + λ̃(x). the definition of g is derived from (2.5). we complete the proof. � remark 2.4 corollary 2.3 can be used to investigate the function from which it could be splitinto even and odd parts. there is a good point of the approach achieved here where the functionssplit into two two parts of odd and even functions can give more concise approximations than thebefore approximations in theorem 2.1. the above results is the counterpart of the correspondingresults of sikorska’s paper. however, it is not copied word by word. it is the counterpart of evenfunction. 3. the stability of functional equations in f-space an f -space is called β-homogeneous if it satisfies ‖tx‖ = |t|β‖x‖ for every x ∈ x , t ∈ c. inthis section of the first two theorems, β1, β2 are to be 0 < β1 ≤ 1 and 0 < β2 ≤ 1. furthermore,we suppose x is β1-homogeneous f-space and y is β2-homogeneous f-space. before applyingtheorem 2.1 we would like to make an answer that all roads lead to rome. therefore anotherapproach to prove the following functional inequality has been stated in the following. in fact,there is also a similar solution about functional equation being stated in [8]. theorem 3.1 assume the mapping f : x → y fulfilling for some k ≥ 0 and r < β2 β1 ‖f (x + y + z) + f (x) + f (y) + f (z)− f (x + y)− f (z + y)− f (x + z)‖ 6 k (‖x‖r + ‖y‖r + ‖z‖r ) (3.1) for x, y , z ∈ x . then there has a unique limiting mapping ψ1 : x → y such that ‖f (x)− ψ1(x)‖ 6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for x ∈ x . moreover, ψ1 satisfying the above inequality is also satisfying the following equation ψ1(x + y + z) + ψ1(x) + ψ1(z) + ψ1(y) = ψ1(x + y) + ψ1(z + y) + ψ1(x + z) (3.2) for all x, y , z ∈ x . proof. from (x, x, x) in place of (x, y , z) in (3.1) we have ‖f (3x) + 3f (x)− 3f (2x)‖ 6 3k (‖x‖r ) . https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 9hence ‖2f (3x) + 6f (x)− 6f (2x)‖ 6 3 · 2β2k (‖x‖r ) . substitute (x, x, 2x) in place of (x, y , z) in (3.1), yielding that ‖f (4x) + 2f (x)− 2f (3x)‖ 6 (2 + 2rβ1)k (‖x‖r ) . and combining the above two inequalities, we get ‖f (4x) + 8f (x)− 6f (2x)‖ 6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) . (3.3) let us define g(x) = f (2x)− 4f (x) for all x ∈ x . hence ‖g(2x)/2− g(x)‖ 6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) /2β2 (3.4) for all x ∈ x . therefore ‖g(2nx)/2n − g(2mx)/2m‖ 6 n−1∑ j=m (2 + 2rβ1 + 3 · 2β2)k 2jβ1r 2β22jβ2 (‖x‖r ) (3.5) for m, n ∈ n with n > m and all x ∈ x . since the sequence {g(2nx)/2n} is a cauchy sequencein y for all x ∈ x and y is complete, the mapping can be well defined as: φ(x) = lim n→∞ g(2nx)/2n for all x ∈ x . in particular, letting m = 0 and setting n →∞ in (3.4), we have ‖φ(x)− g(x)‖ 6 (2+2rβ1+3·2β2)k 2β1r−2β2 (‖x‖r ) . (3.6) now, we prove the mapping φ is additive and is unique. from (x, y , y + x) in (3.1) yields that ‖f (2x + 2y) + f (x) + f (y)− f (x + 2y)− f (2x + y)‖ 6 k (‖x‖r + ‖y‖r + ‖x + y‖r ) . from (x, x, y) in equation (3.1) yields that ‖f (2x + y) + 2f (x) + f (y)− f (2x)− 2f (x + y)‖ 6 k (2‖x‖r + ‖y‖r ) . from (x, y , y) in (3.1) we have ‖f (x + 2y) + f (x) + 2f (y)− f (2y)− 2f (x + y)‖ 6 k (‖x‖r + 2‖y‖r ) . https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 10combining the above three inequalities, we have that for x, y , z ∈ x ‖φ(x + y)− φ(x)− φ(y)‖ = lim n→∞ 1 2β2n ‖f (2n+1x + 2n+1y) + 4f (2nx) + 4f (2ny)− f (2n+1x)− f (2n+1y)− 4f (2nx + 2ny)‖ 6 lim n→∞ 1 2β2n ∥∥f (2n+1x + 2n+1y) + f (2nx) + f (2ny)− f (2n+1x + 2ny)− f (2n+1y + 2nx) ∥∥ + lim n→∞ 1 2β2n ∥∥f (2n+1x + 2ny) + 2f (2nx) + f (2ny)− f (2n+1x)− 2f (2nx + 2ny) ∥∥ + lim n→∞ 1 2β2n ∥∥f (2nx + 2n+1y) + f (2nx) + 2f (2ny)− f (2n+1y)− 2f (2nx + 2ny) ∥∥ 6 lim n→∞ 2β1rn 2β2n k (4‖x‖r + 4‖y‖r + ‖x + y‖r ) . so we have φ(x + y) = φ(x) + φ(y) for all x, y ∈ x .next, the uniqueness of the mapping φ will be proved. let u(x) be another additive mappingsuch that for some k2 ≥ 0 and r < β2 β1 , ‖g(x)− u(x)‖ 6 k2‖x‖r2 . hence ‖φ(x)− u(x)‖ =‖φ(nx)− u(nx)‖/nβ2 6‖φ(nx)− g(nx)‖/nβ2 + ‖g(nx)− u(nx)‖/nβ2 6 (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 ‖x‖rnrβ1−β2 +k2‖x‖r2nr2β1−β2 for all x ∈ x. therefore φ(x) = u(x) for all x ∈ x. by the condition r < β2 β1 . so there has a uniqueadditive limiting mapping φ fulfilling ‖(f (x)− 1 2 φ(x))− (f (2x)− 1 2 φ(2x))/4‖ ≤ (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 ‖x‖r/22β2 . hence ‖(f (x)− 1 2 φ(x))− (f (2nx)− 1 2 φ(2nx))/4n‖ ≤ n−1∑ j=0 2β1r j 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r . then the mapping can be well defined as ψ(x) = lim n→∞ (f (2nx)− 1 2 φ(2nx))/4n for all x ∈ x , by the completeness of the space y . thus ‖ψ(x)− f (x) + φ(x)/2‖ ≤ (2 + 2rβ1 + 3 · 2β2)k (2β1r − 2β2)(2β1r − 22β2) ‖x‖r . let u(x) be another limiting mapping which has the same property to the function ψ(x) such that, ‖u(x)− φ(x)‖ 6 ‖u(x)− (f (2nx)− 1 2 φ(2nx))/4n‖+ ‖(f (2nx)− 1 2 φ(2nx))/4n − φ(x)‖ 6 2 ∞∑ j=n 2β1r j 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 11which shows that the approximation function φ(x) is unique. finally, it remains to prove that φ(x)satisfies (3.2) and we obtain 1 4n ‖f (2nx + 2ny + 2nz) + f (2nx) + f (2ny) + f (2nz)− f (2nx + 2ny)− f (2nz + 2ny)− f (2nx + 2nz)‖ 6 2β1n 4n k (‖x‖r + ‖y‖r + ‖z‖r ) . (3.7) letting n →∞, and we get our assertion by using the additivity of φ(x). � in another direction, we will describe the similar stability results of the above theorem 3.1. theorem 3.2 let r > β2 β1 and assume that f : x → y is a mapping satisfying the equation (3.1).then there has a unique limiting mapping ψ1 : x → y satisfying ‖f (x)− ψ1(x)‖ 6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for all x ∈ x . moreover, ψ1 solves also the following equation ψ1(x + y + z) + ψ1(x) + ψ1(z) + ψ1(y) = ψ1(x + y) + ψ1(z + y) + ψ1(x + z) (3.8) for all x, y , z ∈ x . proof. according to the equation (3.3), we obtain ‖g(x)− 2g( x 2 )‖ 6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) /2β1r . therefore ‖2ng( x 2n )− 2mg( x 2m )‖ 6 n−1∑ j=m (2 + 2rβ1 + 3 · 2β2)k 2jβ2 2jβ1r2β1r (‖x‖r ) for m, n ∈ n with n > m and x ∈ x . since the sequence {2ng( x2n )} is a cauchy sequence in yfor all x ∈ x and y is complete, the mapping can be well defined as: φ(x) = lim n→∞ 2ng( x 2n ) for all x ∈ x . using a similar manner, we can complete the rest part. � if f (x) is odd, then (x, y ,−x − y) in (3.2) can give a precise condition to ascertain the additiveproperty of the function f (x) (see [9]). obviously, the additive property is stronger than theproperty of the equation (3.2), but vice versa is not true. in contrast with the subadditive property,we can not get obvious strong or weak property temporarily. by using another approach to solvethe theorem 3.1, according to (3.6), we have ‖f (2nx)/22n − f (x)− n−1∑ j=0 φ(2jx)/22(j+1)‖ ≤ n−1∑ j=0 2β1r j 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r . then the mapping can be well defined as ψ(x) = lim n→∞ f (2nx)/22n https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 12for all x ∈ x , by the completeness of the space y . thus ‖ψ(x)− f (x)− φ(x)/2‖ ≤ (2 + 2rβ1 + 3 · 2β2)k (2β1r − 2β2)(2β1r − 22β2) ‖x‖r . in a similar way, we can use two steps to prove that the mapping ψ(x) is unique. the first stepwe show that the mapping satisfies the property: ψ(kx) = k2ψ(x) for all k ∈ n , x ∈ x . we provethis by mathematical induction, for a fixed element x ∈ x. we will prove that the property is truefor k = 2. from (x,−x, x) in equation (3.1), we can get that ‖3f (x) + f (−x)− f (2x)‖ 6 3k (‖x‖r ) for all x ∈ x.thus ‖f (−x)− f (x)− φ(x)‖ ≤ ‖f (2x)− 4f (x)− φ(x)‖+ ‖3f (x) + f (−x)− f (2x)‖ 6 ( (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 + 3k) (‖x‖r ) for all x ∈ x. using the similar above argumentation together the above inequality and equation (3.1), yields ψ(−x) = ψ(x) + lim n→∞ φ(x) 2nand ψ(x + y + z) + ψ(x) + ψ(z) + ψ(y) = ψ(x + y) + ψ(z + y) + ψ(x + z) (3.9) for all x, y , z ∈ x. from (x,−x, x) in equation (3.7), we achieve ψ(2x) = 3ψ(x) + ψ(−x) = 4ψ(x). fixed x ∈ x, we prove this by induction. we have already proved that the property is true for n = 2. supposing that ψ(nx) = n2ψ(x) for all natural n ≤ 2k , with k ≥ 1, let us calculate ψ((2k + 1)x). from (kx, kx, x) in (3.7), we know ψ((2k + 1)x) = ψ(2kx) + 2ψ((k + 1)x)− 2ψ(kx)− ψ(x) = (4k2 + 2(k + 1)2 − 2k2 − 1)ψ(x) = (2k + 1)2ψ(x). now, we show the mapping ψ satisfies the property ψ(kx) = k2ψ(x) for all k ∈ n , x ∈ x . thesecond step, we claim that the mapping φ is unique. let u(x) be another limiting mapping suchthat for some k2 ≥ 0 and r < β2 β1 , ‖u(x)− f (x)− φ(x)/2‖ 6 k2‖x‖r2 https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 13which satisfies the property u(kx) = k2u(x) for all k ∈ n and x ∈ x . therefore ‖ψ(x)− u(x)‖ =‖ψ1(kx)− u(kx)‖/k2β2 6‖u(xk)− f (kx)− φ(kx)/2‖/k2β2 + |ψ(xk)− f (kx)− φ(kx)/2‖/k2β2 6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖rk rβ1−2β2 + |k2‖x‖r2k r2β1−2β2 . hence φ(x) = u(x) for all x ∈ x . this shows that ψ is unique. let ψ1(x) = ψ(x)− φ(x)/2. thiscompletes the uniqueness of ψ1(x). we have ‖ψ1(x)− f (x)‖ 6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for all x ∈ x and also the equation (3.2) holds by using the additive property of φ and equation (3.7). we complete the proof. we may also assume that limn→∞ φ(x) 2n = limn→∞ g(2nx)/2n 2n = 0.otherwise, this limit may not be convergence to zero. conversely, we may add some similar smalladditional assumptions to guarantee the convergence in theorem 3.2. 4. the stability of functional equations in banach space in this section, we will prove the counterpart of the results of theorem 2.1 from [8] to moregeneral case. we generalize the results of sikorska in 2010. in particular, the related parameters u, v can be extended to complex numbers by using a more efficient approach. beyond that, westate that the first results in section 2 are presented and combined the first results in [8]. ourcontribution to the parameters u, v are complex numbers. the results is stated in this section inmore detail. theorem 4.1 suppose that (x,+) is a group, and (y, ‖ · ‖) is a banach space, and let themapping f : x → y satisfy the inequality ‖f (x)− uf (e(x))− vf (−e(x))‖ 6 δ(x), x ∈ x, where u, v ∈ c (c denotes the complex field.), and e : x → x, δ : x → [0,∞) are arbitrary givenfunctions.(1): if e is a even function ( i.e., e(−x) = e(x) for x ∈ x) and the convergent series ∞∑ n=0 [|un| δ (en(x)) + |vn|δ (−en(x))] with u0 := 1, un := [ u(u + v)n−1 ] , n ∈ n, v0 := 0, vn := [ v(u + v)n−1 ] , n ∈ n https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 14(and where en states the n-th composition of the function e ), establishes for every x ∈ x . thenthere has a unique even limiting function g : x → y fulfilling g(x) = ung(en(x)) + vng(−en(x)), x ∈ x and n ∈ n, (4.1) and ‖f (x)− g(x)‖ 6 ∞∑ i=0 [ |ui | δ ( e i(x) ) + |vi | δ ( −e i(x) )] , x ∈ x. (4.2) (2): if e is odd ( i·e., e(−x) = −e(x) for al l x ∈ x) and the convergent series ∞∑ n=0 [|un| δ (en(x)) + |vn| δ (−en(x))] . with u0 := 1, un := 1 2 [(u + v)n + (u − v)n] , n ∈ n, v0 := 0, vn := 1 2 [(u + v)n − (u − v)n] , n ∈ nestablishes for all x ∈ x . then there has a unique limiting mapping g : x → y fulfilling (3.10)and (3.11). proof. we only need to prove the uniqueness of the approximation function. (1): let us supposethat g̃ : x → y is another approximating mapping. so let’s first prove the inequality together withthe equation (2.5) and g(−x) = g(x) ‖f (em(x))− um(unf ( en+m(x) ) + vnf ( −em+n(x) ) ) − vm(unf ( en+m(x) ) + vnf ( −em+n(x) ) )‖ =‖f (em(x))− un+mf ( en+m(x) ) − vn+mf ( −en+m(x) ) ‖ 6 n+m−1∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] , and letting n →∞ we have for any m ∈ n ‖f (em(x))− umg (em(x))− vmg (−em(x)) ‖ 6 ∞∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] , and we can rewrite ‖g(x)− g̃(x)‖ 6‖f (km(x))− umg (em(x))− vmg (−em(x)) ‖ + ‖f (km(x))− umg̃ (em(x))− vmg̃ (em(x)) ‖ 62 ∞∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] for any x ∈ x and m ∈ n, which yields g = g̃ in x as m →∞.(2): combined with the results of theorem 2.1 from [8] where e is odd, we only need to prove theuniqueness of the approximation function. let us suppose that g̃ : x → y is another approximating https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 15mapping. so let’s first prove the inequality together with the equation the results in theorem 2.1in [8] ‖f (em(x))− um(unf ( en+m(x) ) + vnf ( −em+n(x) ) ) − vm(unf ( −en+m(x) ) + vnf ( em+n(x) ) )‖ =‖f (em(x))− un+mf ( en+m(x) ) − vn+mf ( −en+m(x) ) ‖ 6 n+m−1∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] , and letting n →∞ we have for any m ∈ n ‖f (em(x))− umg (em(x))− vmg (−em(x)) ‖ 6 ∞∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] , and we can rewrite ‖g(x)− g̃(x)‖ 6‖f (km(x))− umg (em(x))− vmg (−em(x)) ‖ + ‖f (km(x))− umg̃ (em(x))− vmg̃ (em(x)) ‖ 62 ∞∑ j=m [∣∣uj ∣∣ δ (e j(x) ) + ∣∣vj ∣∣ δ (−e j(x) )] for any x ∈ x and m ∈ n, which yields g = g̃ in x as m →∞. this completes the proof. � for the euler-lagrange equation, we provide another method to solve it in contrast with [10]. theorem 4.2 suppose that (x,+) is a group, and (y, ‖ · ‖) is a banach space and let themapping f : x → y satisfy the inequality for all x, y , z ∈ x and some ε > 0 ‖f (x + y + z) + f (x − y + z) + f (x + y − z) + f (x − y − z)− 4f (x)− 4f (y)− 4f (z)‖ 6 ε. (4.3) then there has a unique limiting function g : x → y such that g(x) = 2 9 g(3x)− 1 9 g(−3x), x ∈ x and ‖f (x)− g(x)‖ 6 3ε 8 x ∈ x.in particular, if x is abelian, then g is a solution of the equation in the following f (x + y + z) + f (x − y + z) + f (x + y − z) + f (x − y − z) = 4f (x) + 4f (y) + 4f (y), (4.4) for all x, y ∈ x . proof. from (x, x,−x) in (4.3), we obtain ‖6f (x) + 3f (−x)− f (3x)‖ 6 ε, x ∈ x. replacing x by −x in the above inequality we obtain ‖6f (−x) + 3f (x)− f (−3x)‖ 6 ε, x ∈ x. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 16consequently, combining the above two inequalities yield that ‖9f (x) + f (−3x)− 2f (3x)‖ 6 3ε, x ∈ x. by using the results second part of theorem 4.1, a computation is to prove that un := 3n + 1 2 · 9n , vn := 1− 3n 2 · 9n , n ∈ n. and the convergent series can be described as ∞∑ n=0 [|un| δ (en(x)) + |vn|δ (−en(x))] = 3ε 8 . and we show that if x is commutative, by using (x, y , z) = (3nx, 3ny , 3nz), then ‖un[f (3n(x + y + z)) + f (3n(x − y + z)) + f (3n(x + y − z)) + f (3n(x − y − z)) − 4f (3nx)− 4f (3ny)− 4f (3ny)] + vn[f (3n(x + y + z)) + f (3n(x − y + z)) + f (3n(x + y − z)) + f (3n(x − y − z))− 4f (3nx)− 4f (3ny)− 4f (3ny)]‖ 6 ε 9nwhich we achieve our result (3.13) by letting n →∞. � theorem 4.3 suppose that x is a group, and (y, ‖ · ‖) is a banach space and let the mapping f : x → y satisfy the inequality for all x, y ∈ x and some ε > 0 ‖f (x + y) + f (x − y)− 2f (x)− f (y)− f (−y)‖ 6 ε, x, y ∈ x. (4.5) then there has a unique limiting function g : x → y fulfilling g(x) = 3 8 g(2x)− 1 8 g(−2x), x ∈ x and ‖f (x)− g(x)‖ 6 2ε 3 x ∈ x.in particular, if x is commutative, then g also fulfils g(x + y) + g(x − y) = 2g(x) + g(y) + g(−y), x, y ∈ x. proof. substituting in the sequel (x, x) in (4.5), we obtain ‖f (2x) + f (0)− 3f (x)− f (−x)‖ 6 ε, x ∈ x. (4.6) replacing x by −x in (4.6) we have ‖f (−2x) + f (0)− 3f (−x)− f (x)‖ 6 ε, x ∈ x. (4.7) consequently, (4.6) and (4.7) yield that ‖8f (x) + f (−2x)− 3f (2x)‖ 6 4ε, x ∈ x. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 17by using the results of theorem 3.3, a computation is to prove that un := 2n + 1 2 · 4n , vn := 1− 2n 2 · 4n , n ∈ n. and the convergent series ∞∑ n=0 [|un| δ (en(x)) + |vn|δ (−en(x))] = 2ε 3 . and we show that if x is commutative, by using (x, y) = (2nx, 2ny), then ‖un[f (2nx + 2ny) + f (2nx − 2ny)− 2f (2nx)− f (2ny)− f (−2ny)] + vn[f (2nx + 2ny) + f (2nx − 2ny)− 2f (2nx)− f (2ny)− f (−2ny)]‖ 6 ε 4nwhich we achieve our result (∗) by letting n →∞. � if we can not set f (0) = 0, then the approximate constat is 56ε. acknowledgments the authors express their gratitude to the anonymous reviewers and editor for their carefulreading the manuscript and for many valuable remarks and suggestions. conflict of interest the author(s) declare(s) that there is no conflict of interest regarding this manuscript. data availability data 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functional equations with a new direct method, mathematics.10 (2022) 1188. https://doi.org/10.3390/math10071188. https://doi.org/10.28924/ada/ma.3.7 https://doi.org/10.4134/bkms.2005.42.1.057 https://doi.org/10.3390/math10071188 1. introduction 2. a simple variable of abstract equation 3. the stability of functional equations in f-space 4. the stability of functional equations in banach space acknowledgments conflict of interest data availability funding statement references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 9doi: 10.28924/ada/ma.4.9 global stability analysis of onchocerciasis transmission dynamics with vigilant compartment in two interacting populations k. m. adeyemo department of mathematics, hallmark university ijebu-itele, ogun state, nigeria mikyade2019@gmail.com abstract. a deterministic compartmental model for the transmission dynamics of onchocerciasis withvigilant compartment in two interacting populations is studied. the model is qualitatively analyzedto investigate its global asymptotic behavior with respect to disease-free and endemic equilibria. itis shown, using a linear lyapunov function, that the disease-free equilibrium is globally asymptoti-cally stable when the associated basic reproduction number, r0 < 1. when the basic reproductionnumber r0 > 1, under some certain conditions on the model parameters, we prove that the endemicequilibrium is globally asymptotically stable with the aid of a suitable nonlinear lyapunov function. 1. introduction onchocerciasis is one of the neglected tropical diseases caused by the parasite onchocercavolvulus, a filarial nematode [3]. the disease is transmitted from one person to another by re-peated bites of black flies. the disease is endemic in sub-saharan africa. many researchers haveworked on many ways to reduce the spread of the disease. for instance, remme et al. [14] usedskin snip survey in west africa to investigate the impact of controlling black flies by larviciding.plaisier et al. [13] used micro simulation model to determine the period required for combiningannual ivermectin treatment and vector control in the onchocerciasis control programme in westafrica. alley et al. [3] used a computer simulation model to study prevention of onchocerciasis byusing macrofilaricide which kills the adult worms. asha hassan & nyimvua shaban [5] investigatedthe effects of four control strategies on the spread of the disease.in this paper, we consider global stability analysis of onchocerciasis transmission dynamics withvigilant compartment. the human population is sub-divided into four compartments and the vec-tor population is sub-divided into three compartments. we show global asymptotic behaviour indisease-free and endemic equilibria. this is an extension of the work done in [1] where the authorworked on the local stability of the model without the vigilant compartment. received: 18 jan 2024. key words and phrases. onchocerciasis epidemic model; vigilant compartment; global dynamics; lyapunov function.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 2the case of onchocerciasis model presented in this paper incorporates a new class of humancompartment called vigilant individuals denoted by vh(t, xi). the individuals in the compartmentare assumed to be tired of onchocerciasis and guide against it by strictly adhering to the vectorcontrol measures such as: regular indoor residual spraying (irs), insecticide-treated bed-nets(itns), clearing of stagnant water bodies and drainages and the use of head-nets in the outdoor.the rest of the paper is organized as follows: the description of the model and theorems on positivityof solutions and reproduction number are given in section 2 while section 3, we explored the globalasymptotic stability of the disease-free equilibrium and endemic equilibrium with a concludingremark. 2. model description two interacting populations are considered; the humans and the black-flies populations. thehuman population is partitioned into four compartments: the susceptible human compartment; sh,the exposed compartment; eh, the infectious human compartment; ih and the vigilant compartment; vh. the black-fly population is partitioned into three compartments: susceptible vector; sv , theexposed vector compartment; ev and the infective vector compartment. the total human and vectorpopulations at any given time, t, are respectively given by; n = sh(t) +eh(t) + ih(t) +vh(t) and v e = sv (t)+ev (t)+iv (t). we assume that the transmission of onchocerciaisis in susceptible hostsis only through contact with infectious vector. we also assume that susceptible vector becomesinfectious as a result of contact with infectious hosts during blood meal. the population understudy is assumed to be large enough to be modelled deterministically. the following systemof non-linear ordinary differential equations, with non-negative initial conditions, describes thedynamics of onchocerciaisis epidemics. dsh(t,xi ) dt = ∑l i=0(1− τ)ψh(xi)− δλh(xi )σh(t,xi )iv (t) nh(t,xi ) − µh(xi)sh(t, xi) deh(t,xi ) dt = ∑l i=0 δλh(xi )σsh(t,xi )iv (t) nh(t,xi ) − (αh(xi) + µh(xi))eh(t, xi) d ih(t,xi ) dt = ∑l i=0(1− θ)αh(xi)eh − (γ(xi) + µh(xi))ih(t, xi) dvh(t,xi ) dt = τψh(xi)n(t, xi) + θαh(xi)eh(t, xi) + γ(xi)ih(t, xi)− µh(xi)vh(t, xi) dsv dt = ψv − δλv (xi )sv (t)ih(t,xi ) nh(t,xi ) − µvsv (t) dev dt = δλv (xi )sv (t)ih(t,xi ) n(t,xi ) − (αv + µv )ev (t) d iv dt = αvev (t)− µv iv (t)  (2.1) subject to the following initial conditions: sh(0, xi) = s0h(xi), eh(0, xi) = e0h(xi), ih(0, xi) = i0h(xi), vh(0, xi) = v0h(xi) (2.2) sv (0) = s0v , ev (0) = e0v , iv (0) = i0v https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 3 symbols definitionss sh(t, xi) number of susceptible humans at time t and discrete age xi eh(t, xi) number of exposed humans at time t and discrete age xi ih(t, xi) number of infectious humans at time t and discrete age xi vh(t, ai) number of vigilant host humans at time t and discrete age xi sv (t) number of susceptible black-flies at time t ev (t) number of exposed black-flies at time t iv (t) number of infectious black-flies at time t ψh(xi) recruitment term of the susceptible humans at discrete age xi ψv recruitment term of the susceptible vectors δ biting rate of the vector λh(xi) probability that a bite by an infectious vector results in transmissionof disease to humanat discrete age xi λv probability that a bite results in transmission of parasiteto a susceptible vector µh(xi) per capita death rate of humans at discrete age xi µv per capita death rate of vector γh(xi) disease-induced death rate of humans at discrete age xi γv disease-induced death rate of vectors αh(xi) per capita rate of progression of humans from the exposed state to theinfectious stateat discrete age xi αv per capita rate of progression of vectors from the exposed state to theinfectious state νh(xi) humans disease-inhibiting factor at discrete age xi νv vectors disease-inhibiting factor τ(xi) proportion of human population that is born vigilant at discrete age xi θ(xi) proportion of exposed humans that becomes vigilant at discrete age xi γ(xi) per capita recovery rate of infectious humans to the vigilant state at discrete age xi model assumptionsthe formulation of the compartmental model is based on the following assumptions: 1. that only humans are vigilant.2. that humans are born either susceptible or vigilant.3. that exposed humans progress to either become infectious or vigilant. the assumption thatexposed humans can become vigilant is motivated by the possibility of treating plasmodiumvivax infection which is at the dormant liver stage4. that all infectious humans become vigilant upon recovery due to treatment5. that strict adherence to vector control measures by the vigilant humans does not result intore-infection. https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 46. all black-flies are born susceptible.7. that the susceptible black-flies, when infected, becomes exposed black-flies who are notyet infectious.8. that the exposed black-flies progress to become infectious only.9. that the infectious black-flies remain infectious for life. that is, there is no recovered classfor black-fly population.10. that a proportion of susceptible humans is infected by infectious mosquitoes and thatsusceptible mosquitoes become infected when in contact with a proportion of infectioushumansto carry out the analysis of the formulated model (2.1), it is convenient to rescale the variablesby dividing the number of the individuals in the subpopulations by their respective total numberof populations nh(t, xi) and nv (t). this process is achieved by making the following change ofvariables: s̄h(t, xi) = sh(t,xi ) nh(t,xi ) , ēh(t, xi) = eh(t,xi ) nh(t,xi ) , īh(t, xi) = ih(t,xi ) nh(t,xi ) , v̄h(t, xi) = vh(t,xi ) nh(t,xi ) , s̄v (t, xi) = sv (t,xi ) nv (t,xi ) , ēv (t, xi) = ev (t,xi ) nv (t,xi ) , īv (t, xi) = iv (t,xi ) nv (t,xi )so that̄ sh(t, xi) + ēh(t, xi) + īh(t, xi) + v̄h(t, xi) = 1 and s̄v (t, xi) + ēv (t, xi) + īv (t, xi) = 1 the consequence of this, we have ψh(xi) = µh(xi), ψv (xi) = µv and σ = nv (t) nh(t,xi ) . after droppingof bars (̄), model (2.1) gives rise to the following system of equations: dsh(t,xi ) dt = (1− τ)ψh(xi)− ∑l i=0 δλh(xi)σh(t, xi)iv (t)− µh(xi)sh(t, xi) deh(t,xi ) dt = ∑l i=0 δλh(xi)σsh(t, xi)iv (t)− (αh(xi) + µh(xi))eh(t, xi) d ih(t,xi ) dt = ∑l i=0(1− θ)αh(xi)eh − (γ(xi) + µh(xi))ih(t, xi) dvh(t,xi ) dt = τψh(xi) + θαh(xi)eh(t, xi) + γ(xi)ih(t, xi)− µh(xi)vh(t, xi) dsv dt = ψv − δλv (xi)sv (t)ih(t, xi)− µvsv (t) dev dt = δλv (xi)sv (t)ih(t, xi)− (αv + µv )ev (t) d iv dt = αvev (t)− µv iv (t)  (2.3) subject to the following initial conditions: sh(0, xi) = s0h(xi), eh(0, xi) = e0h(xi), ih(0, xi) = i0h(xi), vh(0, xi) = v0h(xi) (2.4) sv (0) = s0v , ev (0) = e0v , iv (0) = i0v 3. global stability analysis here, we explore the global asymptotic stability of the dfe and ee for the special case with noloss of immunity acquired by the recovered individuals. we use the concept of lyapunov functionsto analyze the global stability https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 53.1. global stability of disease-free equilibrium. the following result establishes the globalasymptotic behavior of system (2.1) around e0 which is determined by the basic reproductionnumber r0. theorem 3:the disease-free equilibrium (2.12) of model (2.1) is globally asymptotically stable in ω whenever r0 ≤ 1 proof:consider the linear lyapunov function of the form m = d1eh(t, xi) + d2ih(t, xi) + d3ev (t) + d4iv (t) (3.1) where d1 = αh(xi)(1− θ) (αh(xi) + µh(xi))(γ(xi) + µh(xi)) d2 = 1 (γ(xi) + µh(xi)) d3 = 1 δλv d4 = αv + µv δλvαvin what follows, the time derivative of m given by (3.1) along the solutions of the model (2.3)yields ṁ = αh(xi)(1− θ)[δλh(xi)σh(t, xi)iv − (αh(xi)iv + µh(xi))eh(t, xi)] (αh(xi) + µh(xi))(γ(xi) + µh(xi)) + l∑ i=0 (γ(xi) + µh(xi))[(1− θ)αh(xi)eh(t, xi)− (γ(xi) + µh(xi))ih(t, xi)] + 1 δλv [δλvsv ih(t, xi)− (αv + µv )ev ] + αv + µv δλvαv [αvev − µv iv ] = l∑ i=0 δλh(xi)σαh(xi)(1− θ)sh(t, xi)iv (αh(xi))(γ(xi) + µh(xi)) − αh(xi)(1− θ)eh(t, xi) γ(xi) + µh(xi) + l∑ i=0 (1− θ)eh(t, xi) (γ(xi) + µh(xi)) − ih(t, xi) + sv ih(t, xi)− (αv + µv )µv iv δλvαv ≤ l∑ i=0 δλh(xi)σαh(xi)(1− θ)(1− τ)iv (αh(xi) + µh(xi))(γ(xi) + µh(xi)) − (αv + µv )µv iv δλvαv = [ l∑ i=0 δλh(xi)σαh(xi)(1− θ)(1− τ) (αh(xi) + µh(xi))(γ(xi) + µh(xi)) − (αv + µv )µv δλvαv ] iv = (αv + µv )µv δλvαv [r20 − 1]iv https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 6we have that ṁ ≤ 0 whenever r0 ≤ 1 with ṁ = 0 if and only if iv = 0. we also see that (sh(t, xi), eh(t, xi), ih(t, xi), vh(t, xi), sv (t), ev (t)) tends to ((1− τ), 0, 0, 0, 1, 0) as t →∞ since iv (t) → 0 as t → ∞. by lasalle’s principle [7], one concludes that every solution of the model(2.3) in ω approaches the disease-free equilibrium, e0, as t →∞.we have that ṁ ≤ 0 whenever r0 ≤ 1 with ṁ = 0 if and only if iv = 0. we also see that (sh(t, xi)), eh(t, xi), ih(t, xi), vh(t, xi), sv (t), ev ()) hence e0 is globally asymptotically stable in ω if r0 ≤ 1 2the global asymptotic stability analysis of the endemic equilibrium is considered next forthe special case with τ = θ = 0. the disease-present (endemic) equilibrium of the model(2.3) is referred to the steady-state solution where at least one of the infected compartmentsis nonzero. let the arbitrary endemic equilibrium of the model (2.1) be represented by ee = (s∗∗h (xi), e ∗∗ h (xi), i ∗∗ h (xi), v ∗∗ h (xi), s ∗∗ m , e ∗∗ m , i ∗∗ m ) in order to do this, nonlinear lyapunov function isused of goh-volterra type [6, 15]. theorem 4: the unique endemic equilibrium,ee , of the model (2.3) is globally asymptoticallystable if r0 > 1. proof: let r0 > 1 so that there exists a unique endemic equilibrium and consider the nonlinearlyapunov function defined by m = ( sh(t, xi)− s∗∗h (xi)− s∗∗h (xi) ln sh(t, xi) s∗∗h (xi) ) + ( eh(t, xi)− e∗∗h (xi)− e∗∗h (xi) ln eh(t, xi) e∗∗h (xi) ) + l∑ i=0 αh(xi) + µh(xi) αh(xi) [ ih(t, xi)− i∗∗h (xi)− i∗∗h (xi) ln ih(t, xi) i∗∗h (xi) ] + ( sv − s∗∗v − s∗∗v ln sv s∗∗v ) + ( ev − e∗∗v − e∗∗v ln ev e∗∗v ) + αv + µv αv [ iv − i∗∗v − i∗∗v ln iv i∗∗v ] with lyapunov time-derivative given as ṁ = ṡh(t, xi)− s∗∗h (xi) sh(xi) ṡh(t, xi) + ėh(t, xi)− e∗∗h (xi) eh(xi) ėh(t, xi) + l∑ i=0 αh(xi) + µh(xi) αh(xi) ( i̇h(t, xi)− i∗∗h (xi) ih(xi) i̇h(t, xi) ) + ṡv − s∗∗v sv ṡv + ėv − e∗∗v ev ėv + αv + µv αm ( i̇v − i∗∗v iv i̇v ) (3.2) https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 7using equations of the model (2.3) in (3.3) we obtain ṁ = (1− τ)ψh(xi)− l∑ i=0 δλh(xi)σsh(t, xi)iv − µh(xi)sh(t, xi) (3.3) − l∑ i=0 s∗∗h (xi) sh(t, xi) (ψh(xi)− δλh(xi)σsh(t, xi)iv − µh(xi)sh(t, xi)) + l∑ i=0 δλh(xi)sh(t, xi)iv + [αh + µh]eh(t, xi)− l∑ i=0 e∗∗h (xi) eh(t, xi) (δλh(xi)σsh(t, xi)iv + [αh + µh]eh(t, xi)) + l∑ i=0 αh(xi) + µh(xi) αh(xi) × ((1− θ)αh(xi)eh(t, xi)− [r(xi) + µh(xi) + γh(xi)])ih(t, xi) − l∑ i=0 i∗∗h (xi)(αh(xi) + µh(xi)) ih(t, xi)αh(xi) × ((1− θ)αh(xi)eh(t, xi)− [r(xi) + µh(xi) + γh(xi)])ih(t, xi) + ψv − δλvsv ih(t, xi)− µvsv − s∗∗v sv (ψv − δλvsv ih(t, xi)− µvsv ) + δλvsv ih(t, xi) + [αv + µv ]ev − e∗∗v eh(t, xi) (δλh(xi)sh(t, xi)iv + [αh + µh]ev ) + αv + µv αv [ αvev − [µv + γv ]iv − i∗∗v iv (αvev − [µv + αv ]iv ) ] simplifying ṁ gives ṁ = l∑ i=0 ψh(xi) ( 1− s∗∗h (xi) sh(t, xi) ) − l∑ i=0 µh(xi)sh(t, xi) ( 1− s∗∗h (xi) sh(t, xi) ) + l∑ i=0 δλh(xi)s ∗∗ h (xi)iv (3.4) − l∑ i=0 e∗∗h (xi)δλh(xi)sh(t, xi)iv eh(t, xi) + l∑ i=0 (αh(xi) + µh(xi))e∗∗h (xi) (3.5) − l∑ i=0 (αh(xi) + µh(xi)) αh(xi) (r(xi) + µh(xi)γh(xi))ih(t, xi) (3.6) − l∑ i=0 (αh(xi) + µh(xi))i∗∗h (xi)eh(t, xi) ih(t, xi) + l∑ i=0 αh(xi) + µh(xi) αh(xi) (r(xi) + µh(xi) + γh(xi))i∗∗h (xi) (3.7) + µv ( 1− s∗∗v sv ) − µvsv ( 1− s∗∗v sv ) + δλvs ∗∗ v (xi)ih − e∗∗v δλvsv ih ev + (αv + µv )e∗∗v (3.8) − (αv + µv )(µv + γv )iv αv − (αv + µv )i∗∗v ev iv + (αv + µv )(µv + γv )iv αv (3.9) at the endemic equilibrium ee , we get from model (2.4) that ψh(xi) = ∑l i=0 δλh(xi)σ ∗ h(xi)i ∗ v + ∑l i=0 µh(xi)s ∗ h(xi) αh(xi) + µh(xi) = ∑l i=0 δλh(xi )σ ∗ hi ∗ v e∗h(xi ) µh(xi) + γh(xi) = ∑l i=0 αh(xi )e ∗ h(xi ) i∗h(xi ) ψv = δλvs ∗ v i ∗ h + µvs ∗ v αv + µv = δλvs∗v i ∗ h e∗v µv + γv = αve∗v i∗v  (3.10) https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 8using (3.6) in (3.5), we have ṁ = l∑ i=0 µh(xi)σ ∗ h ( 2− s∗h(xi) sh(t, xi) − sh(t, xi) s∗h(xi) ) + l∑ i=0 δλh(xi)σs ∗ h(xi)i ∗ v (3.11) − l∑ i=0 δλh(xi)(s∗h)2i∗v sh(xi) + δλh(xi)σs ∗ h(xi)iv − l∑ i=0 e∗h(xi)δλh(xi)σsh(t, xi)iv eh(t, xi) + δλh(xi)σs ∗ h(xi)iv − l∑ i=0 δλh(xi)σs ∗ h(xi)ih(t, xi)i ∗ v i∗h(xi) − l∑ i=0 δλh(xi)i ∗ h(xi)eh(t, xi)i ∗ v e∗h(xi)ih(t, xi) + l∑ i=0 δλh(xi)σs ∗ hi ∗ v + µvs ∗ v ( 2− s∗∗v sv − sv s∗v ) − δλvs∗v i∗h − δλv (s∗v )2i∗h sv + δλvs ∗ v ih − e∗v δλvsv ih ev + δλvs ∗ v ih − δλvs ∗ v iv i ∗ h i∗v − δλv i ∗ vev i ∗ h e∗v iv + δλvs ∗ v i ∗ h simplifying further, we have ṁ = l∑ i=0 µh(xi)s ∗ h ( 2− s∗h(xi) sh(t, xi) − sh(t, xi) s∗h(xi) ) + l∑ i=0 δλh(xi)σs ∗∗ h i ∗ v (3.12) × [ 4− s∗h(xi) sh(t, xi) − e∗h(xi)σsh(t, xi)iv eh(t, xi)σs ∗ hi ∗ v − i∗h(xi)eh(t, xi) ih(t, xi)e ∗ h(xi) − ih(t, xi)i ∗ v i∗h(xi)iv ] + l∑ i=0 δλh(xi)σs ∗ hi ∗ v − δλ(xi)σs ∗ h(xi)ih(t, xi)i ∗ v i∗h(xi + l∑ i=0 δλh(xi)σs ∗ h(xi)ih(t, xi)(i∗v i∗h(xi)iv − δλh(xi)σs ∗ hi ∗∗ v + µvs ∗∗ v ( 2− s∗∗v sv − sv s∗∗v ) + δλvs ∗ v i ∗ h × [ 4− s∗v sv − e∗vsh(t, xi)g(ih) evs∗∗v g(i∗∗h ) − i∗∗v ev ive∗∗v − ivg(i∗∗h ) i∗∗v g(ih) ] + δλvs ∗ v i ∗ h)− δλvs ∗ v iv i ∗ h) i∗v + δλvs ∗ v iv (i∗h(xi) ) 2 i∗v ih − δλvs∗v i∗h further simplification yields ṁ = −ṁ1 − ṁ2 − l∑ i=0 δλh(xi)s ∗ h(xi)i ∗ v [ 1− iv i∗v + ih(t, xi) i∗h(xi) + ih(t, xi)i ∗ v i∗h(xi)iv ] − ṁ3 − ṁ4 − l∑ i=0 δλvs ∗∗ v i ∗ h [ 1− ih i∗∗h + iv i∗v + iv i ∗ h i∗v ih ] (3.13) where m1 = l∑ i=0 µh(xi)s ∗ h(xi) ( s∗h(xi) sh(t, xi) + sh(t, xi) s∗h(xi) − 2 ) , m2 = l∑ i=0 δλh(xi)s ∗ h(xi)iv × [ s∗h(xi) sh(t, xi) + e∗h(xi)sh(t, xi)iv eh(t, xi)s ∗ hi ∗ v + i∗h(xi)eh(t, xi) ih(t, xi)e ∗ h(xi) + ih(t, xi)i ∗ v i∗h(xi)iv − 4 ] , m3 = µvs ∗ v ( s∗v sv + sv s∗v − 2 ) m4 = δλvs ∗ [ s∗v sv + e∗vsv (t, xi)g(ih) evs∗v i ∗ h + i∗vev ive∗v + iv i ∗ h i∗v ih − 4 ] https://doi.org/10.28924/ada/ma.4.9 eur. j. math. anal. 10.28924/ada/ma.4.9 9 conclusion: in this article, a onchocerciasis transmission dynamics with vigilant compartmentgoverned by system of differential equations has been theoretically analyzed. the analysis iscentered on the global asymptotic behavior of solutions of the system (2.3) around the disease-free and endemic equilibria using lyapunov functions. the system has a globally asymptoticallystable disease-free equilibrium whenever the basic reproduction r0 < 1. moreover, the endemicequilibrium of the system, when it exists, is shown to be globally asymptotically stable wheneverthe associated basic reproduction number r0 > 1. references [1] k.m. adeyemo, local stability of onchocerciasis transmission dynamics with nonlinear incidence functions in twointeracting populations, eur. j. math. anal. 3 (2023) 22.[2] w.s. alley, b.a.b. boatin, n.j.d.n. nagelkerke, macrofilaricides and onchocerciasis control, mathematical modellingof the prospects for elimination, bmc public health 1 (2001) 12.[3] u. amazigo, m. noma, j. bump, b. bentin, b. liese, l. yameogo, h. zouré, and a. seketeli, onchocerciasis diseaseand mortality in sub saharan africa, chapter 15, world bank, washington, dc, 2006.[4] a. hassan, n. shaban, onchocerciasis dynamics: modelling the effects of treatment, education and vector control,j. biol. dyn. 14 (2020) 245-268.[5] c. castillo-chavez, b. song, dynamical models of tuberculosis and their applications, math. biosci. eng. 1 (2004)361-404.[6] p. georgescu, h. zhang, a lyapunov functional for a siri model with nonlinear incidence of infection and relapse,appl. math. comp. 219 (2013) 8496-8507.[7] j.p. lasalle, the stability of dynamical systems, siam, philadelphia, 1976.[8] e.m. poolman, a.p. galvani, modeling targeted ivermectin treatment for controlling river blindness, amer. j. trop.med. hyg. 75 (2006) 921–927.[9] j.p. mopecha, h.r. thieme, competitive dynamics in a model for onchocerciasis with cross-immunity, canad. appl.math. q. 11 (2003) 339–376.[10] m.g. basáñez, m. boussinesq, population biology of human onchocerciasis, phil. trans. r. soc. lond. b: biol. sci.354 (1999) 809–826.[11] m.g. basáñez, j. ricárdez-esquinca, models for the population biology and control of human onchocerciasis, trendsparasitol. 17 (2001) 430–438.[12] j.d. murray, mathematical biology i., an introduction. 3rd ed. springer-verlag, berlin, 2002.[13] a.p. plaisier, e.s. alley, g.j. van oortmarssen, b.a. boatin, j.d.f habbema, required duration of combined annualivermectin treatment and vector control program in west africa, bull. world health organ. 75 (1997) 237.[14] j. remme, g. de sole, g.j. van oortmarssen, the predicted and observed decline in onchocerciasis infection during14 years of successful control of black flies in west africa, bull. world health organ. 68 (1990) 331–339.[15] m.a. safi, s.m. garba, global stability analysis of seir model with holling type ii incidence function, comp. math.meth. med. 2012 (2012) 826052.[16] s.i. omade, a.t. omotunde, a.s. gbenga, mathematical modeling of river blindness disease with demography usingeuler method, math. theory model. 5 (2015) 75–85.[17] world health organization, african programme for onchocerciasis control: meeting of national onchocerciasis taskforces, september 2012, weekly epidemiol. record 87 (2012) 494–502. https://doi.org/10.28924/ada/ma.4.9 1. introduction 2. model description 3. global stability analysis 3.1. global stability of disease-free equilibrium references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 20doi: 10.28924/ada/ma.3.20 on degree-based topological indices of petersen subdivision graph mukhtar ahmad1, saddam hussain2, ulfat parveen3, iqra zahid3, muhammad sultan3,ather qayyum3,∗ 1department of mathematics, khawaja fareed university of engineering and information technology rahim yar khan, pakistan itxmemuktar@gmail.com 2department of statistics, university of mian wali, pakistan saddamhussain.stat885@gmail.com 3department of mathematics, institute of southern punjab multan, pakistan uulfat05@gmail.com, iqraimran57@gmail.com, sultan.sadeeq7866127@gmail.com, atherqayyum@isp.edu.pk ∗correspondence: atherqayyum@isp.edu.pk abstract. in this paper, we adequately describe the generalised petersen graph, expanding to thecategories of graphs. we created a petersen graph, which is cyclic and has vertices that are arrangedin the centre and nine gons plus one vertex, leading to the factorization of regular graphs. petersengraph is still shown in graph theory literature, nevertheless. 1. introduction named after julius petersen, a danish mathematician, the graph of petersen is(from 1839 to1910). petersen researched factorizations of normal factorizations during the 1890s. in 1891, asignificant paper of graphs was published which is commemorated in that volume. petersen provedthat any graph of 3-regular with at a i-factor includes much of the two bridges. tait had writtena few years ago that he had shown i-factorable for each 3-regular graph,but that this outcomeit was not valid without restriction. but tait’s comment in 1898 was interpreted by petersen toimply that each 3-regular bridge less graph is l-factorable. if this outcome were valid, then itwould have been stronger than theorem for petersen. the key characteristics of the petersengraph were examined in detail in 1985. the graph of petersen continously to express in the entiregraph-theory education. we update our previous analysis in the present article by denoting extrarecently findings concerning the petersen graph.julius petersen’s ’die theorie der regulken graphs’ is an exceptional paper that developed a new received: 15 apr 2023. key words and phrases. atom-bond connectivity index; reduced zagreb; randic indices; general connectivity index;petersen graph. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 2theory in graph theory, based on the exchange property of trees spanning and the cyclomaticnumber of trees spanningresently resently zaib hassan niazi et.al[15]. 1.1. the graph of petersen. every petersen graph is cyclic graph and the graph g′ in general formconsists v having set of vertex and e having set of edge, if the natural number, there exist n thegraph with vertices are v (g′) = 4n, edges are e(g′) = 6n, and the specific of this graph is thatabout degree of every each vertex is p(k,t) = [d(x1), d(x2)] = 3. then this graphic which is said tobe petersen graphic. then petersen graphic is denoted by p[v (g′),e(g′)] = (4n, 6n) petersen mapexplored by 1985, updated by new analysis.sylvester’s association to graphs of invariants and covariants requires interpretation of principleof invariants in 1880s. 1.2. graphical idea of petersen graph. if the set of natural number is tn = {1, 2, 3, ...}, if thereexists n then graph with vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form ofpetersen expressed by p[v (g′),e(g′)] = (4n, 6n). this graph having a specification, that degree ofevery each vertex is p(k, t) = [d(x1), d(x2)] = 3.now we write; tn = 1, 2, 3, .... v (g′) = 4n → [1] e(g′) = 6n → [2] k = d(x1) = 3 t = d(x2) = 3 then p(k,t) = [d(x1), d(x2)] = 3 p[v (g′),e(g′)] = (4n, 6n). next we discuss the topological indiceszagreb indices of the group were recognized in the early 1980s and are now known as thefirst and second zagreb indices. they are important molecular descriptors and have been closelycorrelated with chemical properties.[degree based topological indices]the first zagreb index m1(g) is equal to the sum of the squares of the degrees of the vertices for https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 3the (molecular) graph g[1]. it can also be considered as the sum over the edges of g, and m1(g)is defined as:[the first and second zagreb indices of some graph operations] m1(g, x) = ∑ [x1,x2∈e(g)] [d(x1) + d(x2)] (1) the second zagreb index m2(g) is equal to the sum of the products of the degrees of the adjacentvertices for the pair of vertices for the (molecular) graph g, and m2(g) is defined as:[the first andsecond zagreb indices of some graph operations] m2(g, x) = ∑ [x1,x2∈e(g)] [d(x1)d(x2)] (2) in 1972, the first zagreb index, a very old topological index, was launched and several variants ofthe zagreb index were subsequently proposed, e.g. shirdel et al. described a novel index in 2013under the title of ’hyper-zagreb index’ and then it was identified as[2]: [a note on hyper-zagrebindex of graph operations] hm1(g) = ∑ [x1,x2∈e(g)] [d(x1) + d(x2)] 2 (3) e. deutshi and s. klavzar,in 2015, defined a new polynomial, m-polynomial in the followingway, based on the degree of the vertex[3]:[computing hyper zagreb index and m-polynomials] m1(g, y , z) = ∑ [x1,x2∈e(g)] y [d(x1)]z [d(x2)] (4) in shuxian defined two polynomials related to the first zagreb index as in the form: m∗1(g, x) = ∑ [xi∈v (g)] [d(xi)][x (xi )] (5) m0(g, x) = ∑ [xi∈v (g)] (x)[d(xi )] (6) two zagreb type polynomials are defined as follow: ma,b(g, x) = ∑ [xi ,xj∈e(g)] (x)[a{d(xi )}+b{d(xi )}] (7) m ′a,b(g, x) = ∑ [xi ,xj∈e(g)] (x)([a+{d(xi )}][b+{d(xi )}]) (8) todeshine et al. introduced two updated models of the zagreb index for moleculargraphs[4]:[multiplicative zagreb indices of trees] first multiplicative zagreb index formolecular graph g defined as follows: pm1(g) = ∏ [x1,x2∈e(g)] [d(x1) + d(x2)] (9) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 4second multiplicative zagreb index for molecular graph g defined as follows: pm2(g) = ∏ [x1,x2∈e(g)] [d(x1)× d(x2)] (10) first multiplicative zagreb polynomial for molecular graph g defined as follows: pm1(g, x) = ∏ [x1,x2∈e(g)] x[d(x1)+d(x2)] (11) second multiplicative zagreb polynomial for molecular graph g defined as follows: pm2(g, x) = ∏ [x1,x2∈e(g)] x[d(x1)d(x2)] (12) the first degree-based topological index was proposed by milan randic in 1975[5]:[degree-basedtopological indices] r1(α)(g) = ∑ [x1,x2∈e(g)] [d(x1) + d(x2)] α (13) atom-bond connectivity index (abc) is a topological index used in chemistry, environmental sci-ences and pharmacology[6]: [estrada, torres, rodriguez, and gutman, 1998b] abc(g) = ∑ [x1,x2∈e(g)] √ [d(x1) + d(x2)]− 2 d(x1)× d(x2) (14) first, second and third reduced zagreb indices[7] are described as follow: mr1(g) = ∑ [x1,x2∈e(g)] |(d(x1)− 1) + (d(x2)− 1)| (15) mr2(g) = ∑ [x1,x2∈e(g)] [(d(x1)− 1)(d(x2)− 1)] (16) mr3(g) = ∑ [x1,x2∈e(g)] |(d(x1)− 1)− (d(x2)− 1)| (17) rr(g) = ∑ [x1,x2∈e(g)] √ d(x1)× d(x2) (18) the reduced reciprocal randic index is defined as[8]: rrr(g) = ∑ [x1,x2∈e(g)] √ [d(x1)− 1]× [d(x2)− 1] (19) recently in 2015 furtula and gutman [8] introduced another topological index known as forgottenindex or f − index . for more detail on the f − index , we refer to the articles [9].the forgottenindex of a graph g is defined as[10, 11, 12]. f (g) = ∑ [x1,x2∈e(g)] [(dx1) 2 + (dx2) 2] (20) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 5the forgotten polynomial of a graph g is defined as: f (g, x) = ∑ [x1,x2∈e(g)] (x)[(dx1) 2+(dx2) 2] (21) the symmetric division degree index of a connected graph g is defined as: sdd(g) = ∑ [x1,x2∈e(g)] mini(d(x1), d(x2)) max(d(x1), d(x2)) + maxi(d(x1), d(x2)) mini(d(x1), d(x2)) (22) there are two types of general connectivity index. the general randic index (or product-connectivityindex) was proposed by bolloba and erdos and is defined as follows: m1(g) = ∑ [x2∈v (g)] [dg(x2)] 2 (23) where α is a real number. if α = −12 , then it becomes the randic index and if α = 1 then it becomesthe second zagreb index. zhou and trinajstic developed the general sum-connectivity index: [onthe general sum-connectivity index of trees] m1(g) = ∑ [x1,x2∈e(g)] [d(x1) + d(x2)] α (24) where α is a real number. if α = 1, then the general sum connectivity index becomes the firstzagreb index resently asghar et.al[14]. 2. main results in this section, we established some results on degree based topological indices of petersengraph. theorem 2.1 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, first zagrebpolynomials indices are, m1(g, x) =[|6n|](x)(6) proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now first zagreb polynomials indices are i.e. , ⇒ ga(r) = ∑ y1,y2∈e(r) 2 √ dy1dy2 dy1+dy2now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in first zagrebtopological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 6 ⇒ m1(g, x) =[|e(g)|](x)[(3)+(3)] ⇒ m1(g, x) =(6n)(x)(6) ⇒ m1(g, x) = (6n)(x)6. m1(g, x) = (x)6× [general edges of petersen graph] theorem 2.2 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, second zagrebpolynomials indices are, m2(g, x) = (6n)(x)9 proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now second zagreb polynomials indices are i.e. , m2(g, x) = ∑ [x1,x2∈e(g)](x) [d(x1)×d(x2)] → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in second zagrebtopological index of the general form, ⇒ m2(g, x) =∑ x1,x2∈e(g)(x) [(3)(3)] ⇒ m2(g, x) =[|e(g)|](x)9 ⇒ m2(g, x) = (6n)(x)9. m2(g, x) = (x)9× [general edges of petersen graph] theorem 2.3 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, randic indicesare, r1(α)(g) = (6n)[6]α proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now randic indices are i.e. , r1(α)(g) = ∑ [x1,x2∈e(g)][d(x1) + d(x2)] α now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in randic indices https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 7topological index of the general form, r1(α)(g) = ∑ [x1,x2∈e(g)][d(x1) + d(x2)] α in general form of topological index becomes; ⇒ r1(α)(g) = ∑ [x1,x2∈e(g)][d(x1) + d(x2)] αnow putting values in above equation, ⇒ r1(α)(g) =[|e(g)|][(3) + (3)]α ⇒ r1(α)(g) =[|e(g)|][6]α ⇒ r1(α)(g) =[|6n|][6]α ⇒ r1(α)(g) = (6n)[6]α. r1(α)(g) = [6α]×[ the general edges of petersen graph] theorem 2.4 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, reducedreciprocal randic are, rrr(g) = 12n. proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now reduced reciprocal randic are i.e. , rr(g) = ∑ [x1,x2∈e(g)] √ d(x1)× d(x2) now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in reducedreciprocal randic topological index of the general form, ⇒ rrr(g) = ∑ [x1,x2∈e(g)] √ [d(x1)− 1]× [d(x2)− 1]now puttings the values then; ⇒ rrr(g) = ∑ [x1,x2∈e(g)] √ (3− 1)× (3− 1) ⇒ rrr(g) = [|e(g)|]√(4) ⇒ rrr(g) = [|6n|]√(4) ⇒ rrr(g) = (6n)√(4) ⇒ rrr(g) = 6n(2) ⇒ rrr(g) = 12n. https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 8 theorem 2.5 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, hyper zagrebindex are, hm1(g) = 216n proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now hyper zagreb index are i.e. , hm1(g)= ∑ [x1,x2∈e(g)][d(x1) + d(x2)] 2 → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in hyper zagrebindex topological index of the general form, ⇒ hm1(g) =[|e(g)|][(3 + 3)]2 ⇒ hm1(g) =[|6n|](6)2 ⇒ hm1(g) = (6n)(6)2 ⇒ hm1(g) = (6n)(36) ⇒ hm1(g) = 216n. hm1(g) = thirty six times to general edges of petersen graph. theorem 2.6 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, two polynomialrelated to the first zagreb index are, m∗1(g, x) =(12n)x4n m0(g, x) =4nx3 proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersengraph tn = {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. thedegree of each vertex in p(k,t) is 3 and now two polynomial related to the first zagreb index are i.e. , m∗1(g, x) =∑ [xi∈v (g)][d(xi)][x [xi ]] m0(g, x) =∑ [xi∈v (g)](x) [d(xi )] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in two polynomialrelated to the first zagreb index topological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 9 ⇒ m∗1(g, x) =∑ [xi∈v (g)](3)[x [xi ]] ⇒ m∗1(g, x) =∑ [xi∈v (g)](3)x [4n] ⇒ m∗1(g, x) =[|v (g)|](3)x4n ⇒ m∗1(g, x) =4n(3)x4n m∗1(g, x) =(12n)x4n. m∗1(g, x) =[3x4n]× [general vertices of petersen graph]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in two polynomialrelated to the first zagreb index topological index of the general form, ⇒ m0(g, x) =∑ [xi∈v (g)](x) 3 ⇒ m0(g, x) =[|v (g)|](x)3 m0(g, x) =4nx3. m0(g, x) =[x3]× [general vertices of petersen graph] theorem 2.7 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, zagreb typepolynomials are, ma,b(g, x) =6nx [3(a+b)] m ′a,b(g, x) = (6n)(x)[(a+3)(b+3)] proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now zagreb type polynomials are i.e. , ma,b(g, x) = ∑ [xi ,xj∈e(g)](x) [a{d(xi )}+b{d(xi )}] m ′a,b(g, x) = ∑ [xi ,xj∈e(g)](x) ([a+{d(xi )}][b+{d(xi )}])now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in zagreb typepolynomials topological index of the general form, ⇒ ma,b(g, x) =∑ [xi ,xj∈e(g)](x) [a(3)+b(3)] ⇒ ma,b(g, x) =∑ [xi ,yj∈e(g)](x) [3(a+b)] ⇒ ma,b(g, x) =[|e(g)|](x)[3(a+b)] ma,b(g, x) =6nx [3(a+b)].now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in zagreb typepolynomials topological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 10 ⇒ m ′a,b(g, x) =∑ [xi ,xj∈e(g)](x) ([(a+(3))][(b+(3))]) ⇒ m ′a,b(g, x) =[|e(g)|](x)[(a+(3))(b+(3))] ⇒ m ′a,b(g, x) =[|6n|](x)[(a+3)(b+3)] m ′a,b(g, x) =(6n)(x)[(a+3)(b+3)]. m ′a,b(g, x) =[(x)[(a+3)(b+3)]]× [general edges of petersen graph] theorem 2.8 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, atomic-bond-connectivity (abc) index are, abc(g) = (4n). proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now atomic-bond-connectivity (abc) index are i.e. , abc(g) = ∑ [x1,x2∈e(g)] √ [d(x1)+d(x2)]−2 d(x1)×d(x2) → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values inatomic-bond-connectivity (abc) index topological index of the general form, ⇒ abc(g) = ∑ [x1,x2∈e(g)] √ [d(x1)+d(x2)]−2 d(x1)×d(x2)putting values in above equation; ⇒ abc(g) =[|e(g)|]√ [(3)+(3)−2] (3)×(3) ⇒ abc(g) =[|6n|]√ (4) (3)2 ⇒ abc(g) =(6n)√4(3) ⇒ abc(g) = (4n). abc(g) = general vertices of petersen graph theorem 2.9 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, geometricarithmetic(ga) index are, ga(g) =6n proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now geometric arithmetic(ga) index are i.e. , ga(g) = ∑ [x1,x2∈e(g)] 2 √ d(x1)×d(x2) d(x1)+d(x2) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 11now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in geometricarithmetic(ga) index topological index of the general form, ⇒ ga(g) =[|e(g)|]2√(3)×(3)(3)+(3) ⇒ ga(g) =[|6n|]2√(3)2(6) ⇒ ga(g) =(6n)2√(3)26 ⇒ ga(g) =(6n)2(3)(6) ⇒ ga(g) =6n. ga(r) = general edges of petersen graph. theorem 2.10 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, first multiplezagreb index are, pm1(g) = (6)6n proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now first multiple zagreb index are i.e. , pm1(g) = ∏ [x1,x2∈e(g)][d(x1) + d(x2)]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in first multiplezagreb index topological index of the general form, ⇒ pm1(g) = ∏ [x1,x2∈e(g)][(3 + 3)] ⇒ pm1(g) = (6)[|e(g)|] ⇒ pm1(g) = (6)[|6n|] ⇒ pm1(g) = (6)6n. pm1(g) = general edges of petersen graph to the power of six. theorem 2.11 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, secondmultiple zagreb index are, pm2(g) = (9)[6n] proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now second multiple zagreb index are i.e. , pm2(g) = ∏ [x1,x2∈e(g)][d(x1)× d(x2)] https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 12now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in second multiplezagreb index topological index of the general form, ⇒ pm2(g) = ∏ [x1,x2∈e(r)][(3)× (3)] ⇒ pm2(g) = (9)[|e(r)|] ⇒ pm2(g) = (9)[6n]. pm2(g) = (9)6n pm2(g) = general edges of petersen graph to the power of nine. theorem 2.12 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, forgottenpolynomial are, f (r) = (6n)x18 proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now forgotten polynomial are i.e. , f (g, x) =∑ [x1,x2∈e(g)](x) [(dx1) 2+(dx2) 2]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in forgottenpolynomial topological index of the general form, ⇒ f (g, x) =∑ [x1,x2∈e(g)](x) [(3)2+(3)2] ⇒ f (g, x) =[|e(g)|](x)[9+9] ⇒ f (g) =[6n](x)18 ⇒ f (r) =(6n)x18. f (r) =[x18]× [general edges of petersen graph] theorem 2.13 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, symmetricdivision deg. index are, sdd(g) = 12n proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now symmetric division deg. index are i.e. , sdd(g) = ∑ [x1,x2∈e(g)][ d(x1) 2+d(x2) 2 d(x1)d(x2) ]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in symmetric https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 13division deg. index topological index of the general form, ⇒ sdd(g) = ∑ [x1,x2∈e(g)] mini(3,3) max(3,3) + maxi(3,3) mini(3,3) ⇒ sdd(g) =[|e(g)|][33 + 33 ] ⇒ sdd(g) =[|e(g)|][ (3)+(3)3 ] ⇒ sdd(g) =[|6n|][ (6)3 ] ⇒ sdd(g) =(6n)[2] ⇒ sdd(g) =12n. sdd(g) = two times of general edges of petersen graph. theorem 2.14 let p(k,t) be petersen subdivision graph. then, for tn = {1, 2, 3, ...}, generalconnectivity index are, sdd(g) = 12n proof: the petersen graph tn = {1, 2, 3, ...} appears in figure(graph). the petersen graph tn= {1, 2, 3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now general connectivity index are i.e. , m1(g) = ∑ [x2∈v (g)][dg(x2)] 2 m2(g) = ∑ [x1,x2∈e(g)][(dg(x1))× (dg(x2))] → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1), d(x2)] = 3. now putting the values in generalconnectivity index topological index of the general form, ⇒ m1(g) = ∑ [x2∈v (g)] [(3) 2] ⇒ m1(g) = [|v (g)|](3)2 ⇒ m1(g) = (4n)(9) ⇒ m1(g) = 36n. m1(g) = nine times to vertices of petersen graph. in general form of real number index is, in equation [1] becomes; ⇒ m2(g) = ∑ [x1,x2∈e(g)][dg(x1)× dg(x2)]now putting values in above equation. ⇒ m2(g) = ∑ [x1,x2∈e(g)](3)(3) ⇒ m2(g) = [|e(g)|](3)(3) ⇒ m2(g) = (6n)(9) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 14 ⇒ m2(g) = 54n. m2(g) = nine times to edges of petersen graph. 3. numerical examples every petersen graph is cyclic graph and the graph g′ in general form consists v having set ofvertex and e having set of edge, if the natural number, there exist n the graph with vertices are v (g′) = 4n, edges are e(g′) = 6n, and the specific of this graph is that about degree of everyeach vertex is p(k,t) = [d(x1), d(x2)] = 3. then this graphic which is said to be petersen graphic.then petersen graphic is denoted by p[v (g′),e(g′)] = (4n, 6n)the core features of petersen map explored by length in 1985. however, the petersen line contin-uously arise in literature of the theoretical graphing. by this article, we update previous analysisto introduce additionally fresh findings on the petersen mapping.sylvester’s association to graphs of invariants and covariants requires interpretation of principleof invariants in 1880s. example 3.1. if there exists n is positive natural number then tn = {1, 2, 3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] = (4n, 6n). this graph having a specification, that degree of every each vertex is p(k, t) = [d(x1), d(x2)] = 3.now we write; tn = 1, 2, 3, .... v (g′) = 4n .........(1) e(g′) = 6n .........(2) k = d(x1) = 3 t = d(x2) = 3put n = 3 in equations (1) and (2) and these equations become; t3 = 3 v (g′) = 12 e(g′) = 18 p(k,t) = [d(x1), d(x2)] = 3 p[v (g′),e(g′)] = (12, 18)now figure is; example 3.2. if there exists n is positive natural number then tn = {1, 2, 3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] = (4n, 6n). this graph having a specification, that degree of every each vertex is p(k, t) = [d(x1), d(x2)] = 3. https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 15 figure 1. petersen graph now we write; tn = 1, 2, 3, .... v (g′) = 4n .........(1) e(g′) = 6n .........(2) k = d(x1) = 3 t = d(x2) = 3put n = 4 in equations (1) and (2) and these equations become; t4 = 4 v (g′) = 16 e(g′) = 24 p(k,t) = [d(x1), d(x2)] = 3 p[v (g′),e(g′)] = (16, 24)now figure is; figure 2. petersen graph example 3.3. if there exists n is positive natural number then tn = {1, 2, 3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] = (4n, 6n). this graph having a specification, that degree of every each vertex is p(k, t) = [d(x1), d(x2)] = 3.now we write; https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 16 tn = 1, 2, 3, .... v (g′) = 4n .........(1) e(g′) = 6n .........(2) k = d(x1) = 3 t = d(x2) = 3put n = 5 in equations (1) and (2) and these equations become; t5 = 5 v (g′) = 20 e(g′) = 30 p(k,t) = [d(x1), d(x2)] = 3 p[v (g′),e(g′)] = (20, 30)now figure is; figure 3. petersen graph example 3.4. if there exists n is positive natural number then tn = {1, 2, 3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] = (4n, 6n). this graph having a specification, that degree of every each vertex is p(k, t) = [d(x1), d(x2)] = 3.now we write; tn = 1, 2, 3, .... v (g′) = 4n .........(1) e(g′) = 6n .........(2) k = d(x1) = 3 t = d(x2) = 3put n = 10 in equations (1) and (2) and these equations become; t10 = 10 v (g′) = 40 https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 17 e(g′) = 60 p(k,t) = [d(x1), d(x2)] = 3 p[v (g′),e(g′)] = (40, 60)now figures are; figure 4. petersen graph 4. conclusion and future studies frequently, graph theory is refuted using the petersen graph. in this paper, the general petersengraph was constructed, and the exact expressions of the first and second zagreb indices, the forgottentopological index, the hyper zagreb index, the reduced second zagreb index and the petersen graphin terms of cyclic graph were then examined. the future work will concentrate on topologicalindeces, then generalised petersen via graph operations. references [1] m.h. khalifeh, h. yousefi-azari, a.r. ashrafi, the first and second zagreb indices of some graph operations, discr.appl. math. 157 (2009) 804-811. https://doi.org/10.1016/j.dam.2008.06.015.[2] v. anandkumar, r.r. iyer, on the hyper-zagreb index of some operations on graphs, int. j. pure appl. math. 112(2017) 213-220. https://doi.org/10.12732/ijpam.v112i2.2.[3] s.m. sankarraman, a computational approach on acetaminophen drug using degree-based topological indices andm-polynomials, biointerface res. appl. chem. 12 (2021) 7249-7266. https://doi.org/10.33263/briac126. 72497266.[4] i. gutman, multiplicative zagreb indices of trees, bull. soc. math. banja luka. 18 (2011) 17-23.[5] e. estrada, l. torres, l. rodriguez, i. gutman, an atom-bond connectivity index: modelling the enthalpy of formationof alkanes, indian j. chem. 37 (1998) 849-855.[6] x. ren, x. hu, b. zhao, proving a conjecture concerning trees with maximal reduced reciprocal randic index, matchcommun. math. comput. chem. 76 (2016) 171-184.[7] a.r. bindusree, n. cangul i., v. lokesha, s. cevik a., zagreb polynomials of three graph operators, filomat. 30(2016) 1979-1986. https://doi.org/10.2298/fil1607979b. https://doi.org/10.28924/ada/ma.3.20 https://doi.org/10.1016/j.dam.2008.06.015 https://doi.org/10.12732/ijpam.v112i2.2 https://doi.org/10.33263/briac126.72497266 https://doi.org/10.33263/briac126.72497266 https://doi.org/10.2298/fil1607979b eur. j. math. anal. 10.28924/ada/ma.3.20 18 [8] b. furtula, i. gutman, a forgotten topological index, j. math. chem. 53 (2015) 1184-1190. https://doi.org/10. 1007/s10910-015-0480-z.[9] b. bollabas, p. erd, graphs of extremal weights, ars comb. 50 (1998) 225-233.[10] a.r. ashrafi, m. mirzargar, pi, szeged and edge szeged of an infinite family of nanostardendrimers, indian j. chem.47 (2008) 1656-1660.[11] z. chen, m. dehmer, f. emmert-streib, y. shi, entropy boundsfor dendrimers, appl. math. comput. 242 (2014)462-472.[12] m.v. diudea, a.e. vizitiu, m. mirzagar, a.r. ashrafi, sadhana polynomial in nano-dendrimers, carpathian j. math.26 (2010) 59-66.[13] b. zhou, n. trinajstic, on general sum-connectivity index, j. math. chem. 47 (2010) 210-218.[14] a. asghar, a. qayyum, n. muhammad, different types of topological structures by graphs, eur. j. math. anal. 3(2022) 3. https://doi.org/10.28924/ada/ma.3.3.[15] z.h. niazi, m.a.t. bhatti, m. aslam, y. qayyum, m. ibrahim, a. qayyum, d-lucky labelling of some special graphs,amer. j. math. anal. 10 (2022) 3-11. https://doi.org/10.12691/ajma-10-1-2. https://doi.org/10.28924/ada/ma.3.20 https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.12691/ajma-10-1-2 1. introduction 1.1. the graph of petersen 1.2. graphical idea of petersen graph 2. main results 3. numerical examples 4. conclusion and future studies references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 15doi: 10.28924/ada/ma.3.15 developments on the convergence analysis of newton-kantorovich method for solving nonlinear equations samundra regmi1, ioannis k. argyros2,∗, santhosh george3, michael i. argyros4 1department of mathematics, university of houston, houston, tx 77204, usa sregmi5@uh.edu 2department of computing and mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computer science, university of oklahoma, norman, 73019, ok, usa michael.i.argyros-1@ou.edu ∗correspondence: iargyros@cameron.edu abstract. developments are presented for the semi-local convergence of newton’s method to solvebanach space-valued nonlinear equations. by utilizing a new methodology, we provide a finer con-vergence analysis with no additional conditions than in earlier results. in particular, this is done byintroducing the center-lipschitz condition by which we construct a stricter domain than the originaldomain of the operator. then, the lipschitz constants in the new domain are at least as small asthe original constants leading to weaker sufficient convergence criteria, tighter error bounds on theerror distances involved, and a piece of better information on the location of the solution. thesebenefits are obtained under the same computational cost since in practice the computation of theoriginal constants requires the computation of the new constants as special cases. the same benefitsare obtained if the lipschitz conditions are replaced by hölder conditions or even more general ω−continuity conditions. this methodology can be applied to other methods using such as the secant,stirling’s newton-like, and other methods along the same lines. numerical examples indicate thatthe new results can be utilized to solve nonlinear equations, but not earlier ones. 1. introduction consider the problem of finding a solution x∗ ∈ ω of the equation f (x) = 0, (1.1) received: 26 jan 2023. key words and phrases. newton-kantorovich method; convergence; banach space.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 2where f : ω −→ e2 is a continuously differentiable operator in the fréchet-sense, e1, e2 arebanach spaces and ω ⊂ e1 is an open set.the solution x∗ in closed form is desirable. but this is possible only in special cases. so,most solution methods for (1.1) are iterative methods. the convergence regions for these methodsare small in general, so their applicability is reduced. the error bounds are also pessimistic (ingeneral).among the iterative methods, the most famous one is newton’s method (nm) defined for n = 0, 1, 2, . . . by xn+1 = xn − f ′(xn)−1f (xn) (1.2)kantorovich provided the semi-local convergence analysis of nm utilizing the contraction map-ping principle attributed to banach. in particular, he presented two different proofs using majorantfunctions or recurrence relations [15]. his so-called newton-kantorovich theorem is that no as-sumption on the solution is made and at the same time, the existence of the solution x∗ is established.numerous researchers used this theorem in applications and also as a theoretical tool [1–16]. butthe convergence criteria may not hold although nm may converge. motivated by these concernsand optimization considerations we present new results that not only extend the convergence regionbut also provide more precise error estimates and better knowledge of the location of the solution.the novelty of the article is that these benefits require no additional conditions. this is how theusage of nm is extended. the technique used can be applied to extend other iterative methodsalong the same lines. 2. convergence analysis let α > 0, λ ≥ 0 and x0 ∈ ω be such that ‖f ′(x0)−1‖ ≤ α, ‖f ′(x0)−1f (x0)‖ ≤ λ and f ′(x0) −1 ∈ l(e2, e1), the space of bounded linear operators from e2 to e1. by b(x, b), b[x, b] wedenote the open and closed balls in e1, respectively with center x ∈ e1 and of radius b > 0.some lipschitz-type conditions are needed. definition 2.1. operator f ′ is center-lipschitz continuous about x0 on ω if there exists l0 > 0 such that for all u ∈ ω ‖f ′(u)− f ′(x0)‖ ≤ l0‖u − x0‖. (2.1) set ω0 = b(x0, 1 αl0 ) ∩ω. (2.2) definition 2.2. operator f ′ is 1−restricted lipschitz continuous on ω0 if there exists l > 0 such that ‖f ′(u)− f ′(v)‖ ≤ l‖u − v‖ (2.3) for all u ∈ ω0, v = u − f ′(u)−1f (u) ∈ ω0. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 3 definition 2.3. operator f ′ is 2−restricted lipschitz continuous on ω0 if there exists l1 > 0 such that for all u, v ∈ ω0 ‖f ′(u)− f ′(v)‖ ≤ l1‖u − v‖. (2.4) definition 2.4. operator f ′ is lipschitz continuous on ω if there exists l2 > 0 such that for all u, v ∈ ω ‖f ′(u)− f ′(v)‖ ≤ l2‖u − v‖. (2.5) definition 2.5. assume: λαl0 < 1 (2.6) and ω1 = b(x1, 1 αl0 − ‖x1 − x0‖) ⊂ ω (2.7) then, operator is 3− restricted lipschitz continuous on ω1 if there exists a constant k > 0 such that for all u ∈ ω1 ‖f ′(u)− f ′(v)‖ ≤ k‖u − v‖ (2.8) for v = u − f ′(u)−1f (u) ∈ ω1. remark 2.6. by the definition of sets ω0 and ω1, we get ω0 ⊆ ω, (2.9) and ω1 ⊆ ω0. (2.10) indeed, if y ∈ ω1, then we obtain ‖y − x1‖ ≤ 1 αl0 − ‖x1 − x0‖ ⇒ ‖y − x1‖+ ‖x1 − x0‖ ≤ 1 αl0 ⇒ ‖y − x0‖ ≤ 1 αl0 ⇒ y ∈ ω0 ⇒ ω1 ⊆ ω0. it follows by these definitions, (2.9) and (2.10) that if the best constants are chosen in the definitions 2.1-2.5, then l ≤ l1 ≤ l2, (2.11) l0 ≤ l2, (2.12) and k ≤ l. (2.13) hence, parameter k can replace results on newton’s using the constants l, l1 and l2. notice also that l0 = l0(f ′,ω), l = l(f ′,ω0), l1 = l1(f ′,ω0), l2 = l2(f ′,ω) and k = k(f,ω0,ω1). examples, where (2.9)-(2.13) are strict can be found in the numerical section. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 4 notice that the computation of the constant l2 requires the computation of the other constants as special cases. hence, no additional effort is needed to compute them. moreover, they all depend on the initial data (x0, f,ω). it is also worth noticing that under (2.1) we obtain ‖f ′(u)−1‖ ≤ α 1− αl0‖u − x0‖ . (2.14) this is a tighter estimate than using the stronger (2.5) to get ‖f ′(u)−1‖ ≤ α 1− αl2‖u − x0‖ . (2.15) we assume from now on that l0 ≤ k. (2.16) but if k < l0 then, the following results hold with l0 replacing k. based on the above we present two extended theorems on newton’s method. an important role is played in the convergence of nm by the majorizing sequence {sn} definedby s0 = 0, sn+1 − sn = − p(sn) p′0(sn) = αk(sn − sn−1)2 1− l0αsn , p(s) = k 2 s2 − s α + λ α , p0(s) = l0 2 s2 − s α λ α . theorem 2.7. (extended newton-kantorovich theorem [1,2,10,12,13,15,16]) under conditions (2.1), (2.6)-(2.8) further suppose b(x0, s∗) ⊂ ω, h = kαλ ≤ 1 2 . (2.17) then, newton’s method (1.2) initiated at x0 ∈ ω generates a sequence {xn} such that:{xn} ⊆ b(x0, s∗), limn−→∞ xn = x∗ ∈ b[x0, s∗]. ‖xn+1 − xn‖ ≤ sn+1 − sn (2.18) ‖x∗ − xn‖ ≤ s∗ − sn, (2.19) where, limn−→∞ sn = s∗ = 1− √ 1−2h kα and s∗∗ = 1+ √ 1−2h kα . moreover, the following items hold for τ = s∗ s∗∗ s∗ − sn = { (s∗∗−s∗)τ2 n 1−τ2n , if s∗ < s∗∗ 1 2n s∗, if s∗ = s∗∗. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 5 furthermore, the element x∗ is the unique solution of equation f (x) = 0 in b[x0, s̄], where s̄ = 2 l0α − s∗ if l0αs∗ < 2. proof. simply replace l2 by k and use (2.14) instead of (2.15) in the proof of the version ofnewton-kantorovich theorem given in [10] (see also [3–9,14–16]. � remark 2.8. (i)if k = l2, the result of theorem 2.7 reduces to one in the newton-kantorovich theorem where hk = l2αλ ≤ 1 2 , (2.20) t0 = 0, tn+1 − tn = − p̄(tn) p̄′(tn) = αl2(tn − tn−1)2 1− l2αtn , p̄(s) = l2 2 s2 − s α + λ α , and limn−→∞ tn = t∗ = 1− √ 1−2kk l2α and t∗∗ = 1+ √ 1−2kk l2α , ¯̄s = 2 l2α − t∗, µ = t∗ t∗∗ , t∗ − tn =  (t∗∗−t∗)µ2 n 1−µ2n , if t∗ < t∗∗ 1 2n t∗, if t∗ = t∗∗. then, in view of estimates (2.11)-(2.13) we have hk ≤ 1 2 ⇒ h ≤ 1 2 , (2.21) s∗ ≤ t∗, ¯̄s ≤ s̄ , (2.22) 0 ≤ sn+1 − sn ≤ tn+1 − tn (2.23) and 0 ≤ s∗ − sn ≤ t∗ − tn. (2.24) estimates (2.21)-(2.24) justify the advantages (a) as stated in the introduction. (ii)a more careful look at the proof shows that tighter sequence {rn} defined by r0 = 0, r1 = λ, r2 = r1 + αl0(r1 − r0)2 2(1− l0αr1) , rn+2 = rn+1 + kα(rn+1 − rn)2 2(1− l0αrn+1) , also majorizes sequence {xn}. the sufficient convergence criterion for this sequence is given by ha = k̄αλ ≤ 1 2 , (2.25) https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 6 where k̄ = 1 8(4l0 + √ kl0 + 8l20 + √ l0k). this criterion was given by us in [4] for k = l− 2. notice that h ≤ 1 2 ⇒ ha ≤ 1 2 . (2.26) hence, if (2.25) and {rn} replace (2.17) and {sn} the conclusions of theorem 2.7 hold with these changes too. (iii)suppose that there exist a > 0, b > 0 such that ‖f ′(x0 + θ(x1 − x0))− f ′(x0)‖ ≤ τa‖x1 − x0‖ (2.27) and ‖f ′(x1)− f ′(x0)‖ ≤ b‖x1 − x0‖ (2.28) for all τ ∈ [0, 1]. then, it was shown in [5] that sequence {qn} defined by q0 = 0, q1 = λ, q2 = q1 + αa(q1 − q0)2 2(1− bαq1) , qn+2 = qn+1 + kα(qn+1 − qn)2 2(1− l0αqn+1) is also majorizing for sequence {xn}. the convergence criterion for sequence {qn} is given by haa = λ 2c ≤ 1 2 , (2.29) where p1(s) = (ka + 2dl0(a − 2b))s2 + 4p(l0 + b)s − 4d, d = 2k k + √ k2 + 8l0k , and c =  1 l0+b , ka + 2dl0(a − 2b) = 0positive root of p1, ka + 2dl0(a − 2b) > 0smaller positive root of p1, ka + 2dl0(a − 2b) < 0. notice that b ≤ a ≤ l0. hence, {qn} is a tighter majorizing sequence than {rn}. criterion (2.29) was given by us in [4] for k = l2. therefore (2.29) and {qn} can also replace (2.17) and {sn} in theorem 2.7. (iv) it follows from the definition of sequence {sn} that if l0αsn < 1. (2.30) then, sequence {sn} is such that 0 ≤ sn ≤ sn+1 and limn−→∞ sn = s∗ ≤ 1 l0α . hence, weaker than all conditions (2.30) can be used in theorem 2.7. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 73. examples we test the convergence criteria. example 3.1. defined the real function f on ω = b[x0, 1− δ], x0 = 1, δ ∈ (0, 12) by f (s) = s3 − δ. then, the definitions are satisfied for λ = 1−δ 3 , α = 1 3 , l0 = 3(3 − δ), l2 = 6(2 − δ), l1 = 6(1 + 1 3−δ ), x1 = 2+δ 3 , l = 5( 4−δ 3−δ ) 3+δ 3( 4−δ 3−δ ) 2 , a = b = δ + 5, k = 5h3+δ 3h2 , and h = δ+2 3 + 3−(1−δ)(3−δ) 3(1−δ) . denote by m1,m2,m3,m4 the set of values δ ∈ (0, 12) for which (2.20), (2.17), (2.25) and (2.29) are satisfied, respectively. then, by solving these inequalities for δ, we get m1 = ∅, m2 = (0.0751, 0.5), m3 = (0.1320, 0.5) and m4 = (0.3967, 0.5). notice in particular that the newton-kantorovich criterion (2.20) [1, 9–15] cannot assure convergence of nm since m1 = ∅. a second example is provided to show that our conditions can be used to solve equations incases where the ones in [1, 2, 10,12,13] cannot. example 3.2. consider e1 = e2 = c[0, 1] with the norm-max. set ω = b(x0, 3). define, hammerstein-type integral operator m on ω by m(z)(w) = z(w)− y(w)− ∫ 1 0 t (w, t)v3(t)dt, (3.1) w ∈ [0, 1], z ∈ c[0, 1], where y ∈ c[0, 1] is fixed and t is a green’s kernel defined by t (w, u) = { (1− w)u, i f u ≤ w w(1− u), i f w ≤ u. (3.2) then, the derivative m ′ according to fréchet is defined by [m ′(v)(z)](w) = z(w)− 3 ∫ 1 0 t (w, u)v2(t)z(t)dt, (3.3) w ∈ [0, 1], z ∈ c[0, 1]. let y(w) = x0(w) = 1. then, using (3.1)-(3.3), we obtain m ′(x0)−1 ∈ l(e2, e1), ‖i − m ′(x0)‖ < 3 8 , ‖m ′(x0) −1‖ ≤ 8 5 := α, λ = 1 5 , l0 = 12 5 , l2 = 18 5 , and ω0 = b(1, 3) ∩ b(1, 512) = b(1, 512), so l1 = 3 2 , and l0 < l2, l1 < l2. set k = l = l1. then, the old sufficient convergence criterion is not satisfied, since αλl2 = 1 5 8 5 18 5 = 144 125 > 1 2 holds. therefore, there is no guarantee that newton’s method (1.2) converges to x∗ under the conditions of the aforementioned references. but our condition hold, since dba = 1 5 8 5 3 2 = 24 50 < 1 2 . therefore, the conclusions of our theorem 2.7 follow. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 84. conclusion the technique of recurrent functions has been utilized to extend the sufficient conditions forconvergence of nm for solving nonlinear equations. the new results are finer than the earlierones. so, they can replace them. no additional conditions have been used. the technique is verygeneral rendering useful to extend the usage of other iterative methods. declarations the authors declare that there are no competing interests and that all authors contributedequally in conceptualization, methodology, formal analysis, and investigation. the original draftwas prepared by i. k. argyros and review and editing was done by s. regmi, s. george, and m.argyros. references [1] s. adly, h.v. ngai, v.v. nguyen, newton’s methods for solving generalized equations: kantorovich’s and smale’sapproaches, j. math. anal. appl. 439 (2016) 396-418. https://doi.org/10.1016/j.jmaa.2016.02.047.[2] s. adly, r. cibulka, h.v. ngai, newton’s method for solving inclusions using set-valued approximations, siam j.optim. 25 (2015) 159-184. https://doi.org/10.1137/130926730.[3] i.k. argyros, unified convergence criteria for iterative banach space valued methods with applications, mathematics,9 (2021) 1942. https://doi.org/10.3390/math9161942.[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s method, j. complex. 28 (2012) 364-387. https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s method, appl. math. comp. 225 (2013)372-386; https://doi.org/10.1016/j.amc.2013.09.049.[6] i.k. argyros, a.a. magréñan, a contemporary study of iterative procedures, elsevier (academic press), new york,2018. https://doi.org/10.1016/c2015-0-04301-5.[7] i.k. argyros, s. george, mathematical modeling for the solution of equations and systems of equations with appli-cations, volume-iv, nova publisher, ny, 2021.[8] r. behl, p. maroju, e. martinez, s. singh, a study of the local convergence of a fifth order iterative procedure,indian j. pure appl. math. 51 (2020) 439-455. https://doi.org/10.1007/s13226-020-0409-5.[9] p.g. ciarlet, c. madare, on the newton-kantorovich theorem, anal. appl. 10 (2012) 249-269. https://doi.org/ 10.1142/s0219530512500121.[10] r. cibulka, a.l. dontchev, j. preininger, v. veliov, t. roubai, kantorovich-type theorems for generalized equations,j. convex anal. 25 (2018) 459-486. http://hdl.handle.net/20.500.12708/144705.[11] j.a. ezquerro, m.a. hernandez, newton’s procedure: an updated approach of kantorovich’s theory, cham switzer-land, (2018). https://www.booksandcranniesva.com/book/9783319559759.[12] l.v. kantorovich, g.p. akilov, functional analysis in normed spaces, the macmillan co, new york, (1964).[13] a.a. magréñan, j.m. gutiérrez, real dynamics for damped newton’s procedure applied to cubic polynomials, j.comp. appl. math. 275 (2015) 527–538; https://doi.org/10.1016/j.cam.2013.11.019.[14] f.a. potra, v. pták, nondiscrete induction and iterative processes, research notes in mathematics 103, pitman,boston (1984). https://archive.org/details/nondiscreteinduc0000potr.[15] p.d. proinov, new general convergence theory for iterative processes and its applications to newton-kantorovichtype theorems, j. complex. 26 (2010) 3-42. https://doi.org/10.1016/j.jco.2009.05.001. https://doi.org/10.28924/ada/ma.3.15 https://doi.org/10.1016/j.jmaa.2016.02.047 https://doi.org/10.1137/130926730 https://doi. org/10.3390/math9161942 https://doi.org/10.1016/j.jco.2011.12.003 https://doi.org/10.1016/j.amc.2013.09.049 https://doi.org/10.1016/c2015-0-04301-5 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1142/s0219530512500121 https://doi.org/10.1142/s0219530512500121 http://hdl.handle.net/20.500.12708/144705 https://www.booksandcranniesva.com/book/9783319559759 https://doi.org/10.1016/j.cam.2013.11.019 https://archive.org/details/nondiscreteinduc0000potr https://doi.org/10.1016/j.jco.2009.05.001 eur. j. math. anal. 10.28924/ada/ma.3.15 9 [16] r. verma, new trends in fractional programming, nova science publisher, new york, usa, (2019). https:// novapublishers.com/shop/new-trends-in-fractional-programming/. https://doi.org/10.28924/ada/ma.3.15 https://novapublishers.com/shop/new-trends-in-fractional-programming/ https://novapublishers.com/shop/new-trends-in-fractional-programming/ 1. introduction 2. convergence analysis 3. examples 4. conclusion declarations references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 15doi: 10.28924/ada/ma.4.15 on the dirichlet boundary value problem for the cauchy–riemann equations in the half disc ali darya1,∗ , nasir tagizadeh2 1faculty of mathematics sciences, university of guilan, rasht, 19141, iran alidarya@phd.guilan.ac.ir 2faculty of mathematics sciences, university of guilan, rasht, 19141, iran taghizadeh@guilan.ac.ir ∗correspondence: alidarya@phd.guilan.ac.ir abstract. in this article, we investigate the dirichlet boundary value problem for the cauchy–riemann equations in the half disc. first, using the technique of parqueting–reflection and thecauchy–pompeiu representation formula for a half disc, we obtain an integral representation formulain the half disc. in other words, we construct a unique solution for the dirichlet boundary valueproblem. finally, we solve the dirichlet boundary value problem for both the homogeneous andthe inhomogeneous cauchy–riemann equations. in particular, the boundary behaviors at the cornerpoints are considered. 1. introduction and preliminaries boundary value problems are an essential concept in the field of mathematical analysis andpartial differential equations. they arise when seeking solutions partial differential equationssubject to specific conditions on different parts of the boundary of the domain. the dirichletboundary value problem is a fundamental concept in mathematical analysis, particularly in thefield of partial differential equations. it deals with finding a solution to a partial differentialequation that satisfies certain prescribed conditions on the boundary of a given domain.one of the most powerful tools for constructing solutions to the dirichlet problem is the integralrepresentation formula. it provides a way to express the solution in term of an integral over theboundary of domain, which can often simplify the problem and lead to explicit solutions. this formulaallows for the efficient and accurate computation of solutions wide range of partial differentialequations, making it an essential tool in the field of partial differential equations.the parqueting–reflection principle is a technique used in constructing integral representationformulas for the dirichlet boundary value problem. for specific regions of complex plane whoseboundary consists of sub-arcs of circles or straight lines, the parqueting–reflection method for received: 13 feb 2024. key words and phrases. cauchy–riemann equation, dirichlet problem, integral representation formula, half disc.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.15 https://orcid.org/0009-0009-7723-9718 https://orcid.org/0000-0002-3865-7943 eur. j. math. anal. 10.28924/ada/ma.4.15 2constructing integral representation formula to solve the dirichlet boundary value problem for thecauchy–riemann equation is used. if the boundary of the region has the mentioned characteristics,it is possible to create a new region by reflecting the main region with respect to its boundaryand by reflecting this new region with respect to its boundary, another region is obtained. bycontinuing this process, we reach pieces of the plane that the union of these pieces provides acover for the complex plane. this cover can be achieved through a single reflection or multipleconsecutive reflections or infinite repetitive reflections. therefore, the reflection of the main regionat its boundary is repeated to achieve a cover for complex plane.many results have been obtained for boundary value problems of complex partial differentialequations in some particular domains, see, e.g. [1–16]. in the year 2009, harmonic boundary valueproblem for the poisson equation in a half disc was presented by h. begehr and t. vaitekhovich [2].in 2012, y. wang introduced schwarz-type boundary value problems for the polyanalytic equationin the half unit disc [16].in this article, in addition to introducing the domain of half disc, we want to express the reflections,covering and points that we obtain at each stage. we also construct the integral representationformula using the parqueting–reflection method and investigate the dirichlet problem. in particular,we study the explicit solvability of the dirichlet boundary value problem for both the homogeneousand the inhomogeneous cauchy–riemann equations in the half disc.in this article, let m be the half disc domain in the complex plane c defined by m = {z ∈ c : |z | < 1, imz > 0} where d = {z ∈ c : |z | = 1} and the boundary of m is denoted by ∂m . it is formed by arc of thecircle d and a line segment on the real axis from point −1 to 1. see figure 1. figure 1. half disc m https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 3now, we introduce some important definitions and properties of complex analytic functions, re-sults which will be required in subsequent sections. defining the complex partial differential operators ∂ ∂z and ∂ ∂z̄ by ∂ ∂z = 1 2 ( ∂ ∂x − i ∂ ∂y ), ∂ ∂z̄ = 1 2 ( ∂ ∂x + i ∂ ∂y ). let the complex-valued function ω be defined in m and let u and v denote its real and imaginaryparts: ω = u+ iv , where u(x, y) and v(x, y) are real-valued functions. the two partial differentialequations ∂u ∂x = ∂v ∂y , (1.1) ∂u ∂y = − ∂v ∂x , (1.2) are called the cauchy–riemann equations for the pair of functions u and v . multiplying the bothsides of the equality (1.2) by i , i( ∂u ∂y + ∂v ∂x ) = 0. (1.3)adding (1.1) and (1.3) leads to ∂ ∂x (u + iv) + i ∂ ∂y (u − 1 i v) = 0. thus 1 2 ( ∂w ∂x + i ∂w ∂y ) = 0. the cauchy–riemann equation can be written as ∂ω ∂z̄ = ωz̄ = 0 and this is the condition for ωto be analytic function. recall the definition of the pompeiu integral operator t f (z) = − 1 π ∫ m f (t) t − z dξdη, t f (z) is weakly differentiable with ωz̄ = f when f ∈ lp(m;c), p > 2 and t = ξ + iη, [14]. 2. an integral representation formula for m in this section, the integral representation formula for the half disc domain is constructed. a con-venient technique for constructing the integral representation formula for domain with boundariesconsisting of arcs and straight lines is given by the parqueting–reflection method. now using theparqueting–reflection technique for the introduced domain, we obtain a cover for the entire complexplane. this coverage is obtained from three consecutive reflections. reflecting any z at circle gives |z − a| = r ⇒ (z − a) (z̄ − a) = r2 ⇒ zr = az̄ − aa + r2 z̄ − a . https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 4reflecting any z ∈ m at the d, gives |z | = 1⇒ zz̄ = 1⇒ z∗ = 1 z̄ . reflecting any z at the real axis gives z̄ . therefore, we can obtain the following points z∗1 = 1 z̄ , z∗2 = 1 z , z∗3 = z̄ . those reflections produce a parqueting of the entire complex plane. to solve the dirichletboundary value problems for analytic functions the integral representation formula is important.now, using the cauchy–pompeiu representation formula, we construct the integral representationformula for the half disc domain. theorem 2.1. any ω ∈ c1(m;c) ⋂ c(m;c) can be represented as ω(z) = 1 2πi ∫ ∂m ω(t) [ 1 t − z + z tz − 1 ] dt − 1 π ∫ m ωt̄(t) [ 1 t − z + z tz − 1 ] dξdη, (2.1) where t = ξ + iη. proof. applying the cauchy–pompeiu formula [3] 1 2πi ∫ ∂m ω(t) dt t − z − 1 π ∫ m ωt̄(t) dξdη t − z = { ω(z) z ∈ m, 0 z /∈ m. (2.2) for z ∈ m and z∗2 /∈ m. substitute the points into the cauchy–pompeiu formula (2.2) ω(z) = 1 2πi ∫ ∂m ω(t) dt t − z − 1 π ∫ m ωt̄(t) dξdη t − z . (2.3) since point z∗2 is outside the domain m , therefore, the cauchy–pompeiu formula is equal to zeroat this point. 0 = 1 2πi ∫ ∂m ω(t) zdt tz − 1 − 1 π ∫ m ωt̄(t) zdξdη tz − 1 , (2.4) by combining the obtained equations, the integral representation formula is obtained. � the integral representation formula (2.1), serves to solve the related the dirichlet problem forthe cauchy–riemann equations in m . 3. dirichlet problem for the cauchy–riemann equation in m in this section, we study the dirichlet boundary value problem for the homogeneous and the in-homogeneous cauchy–riemann equations. in the following, we solve the dirichlet boundary valueproblem for the homogeneous cauchy–riemann equation. https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 5 theorem 3.1. the dirichlet problem for the homogeneous cauchy–riemann equation ωz̄ = 0, in m, ω = γ, on ∂m, γ ∈ c(∂m;c) (3.1) with given γ ∈ c(∂m;c), γ(±1) = 0, is solvable, if and only if 1 2πi ∫ ∂m γ(t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 0, (3.2) and the unique solution can be presented as ω(z) = 1 2πi ∫ ∂m γ(t) [ 1 t − z + z tz − 1 ] dt, z ∈ m. (3.3) proof. let ω defined by (3.3) be a solution to the dirichlet problem. then the equality ω(z) = γ(t), t ∈ ∂m, (3.4) holds. we consider the following function h(z) = 1 2πi ∫ ∂m γ(t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt, (3.5) and take the difference ω(z)− h(z) = 1 2πi ∫ ∂m γ(t) [ 1 ζ − z + z tz − 1 ] dt − 1 2πi ∫ ∂m γ(t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 1 2πi ∫ ∂m γ(t) [ 1 t − z − 1 t − z̄ + z tz − 1 − z̄ tz̄ − 1 ] dt = 1 2πi ∫ ∂m∩d γ(t) [ t t − z + t̄ t̄ − z̄ − t̄ t̄ − z − t t − z̄ ] dt t + 1 2πi ∫ 1 −1 γ(s) [ 1 s − z − 1 s − z̄ + z sz − 1 − z̄ s z̄ − 1 ] ds = 1 2πi ∫ ∂m∩d γ(t) [ 1− |z |2 |t − z |2 − 1− |z |2 |t̄ − z |2 ] dt t + 1 2πi ∫ 1 −1 γ(s) [ z − z̄ |s − z |2 − z − z̄ |1− zs|2 ] ds. studying the boundary behavior of the boundary integral implies computations on the differentparts of the boundary ∂m . for |t0| = 1, imt0 > 0. as |t̄ − t0|2 6= 0, 1− |t0|2 = 0 and |s − t0|2 = |1− t0s|2 . thus, lim z→t (ω(z)− h(z)) = γ(t).for |t0| < 1, imt0 = 0. since |t − t0| = |t̄ − t0| , |1− t0s|2 6= 0, t − t̄0 = 0. thus, lim z→t (ω(z)− h(z)) = γ(t). now, we consider the boundary behavior at the corner points. let ω(z)− h(z) = 1 2πi ∫ ∂m∩d γ(t) [ t + z t − z − t̄ + z t̄ − z ] dt t https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 6we can write ω(z)− h(z) = 1 2πi ∫ ∂m∩d γ(t) t + z t − z dt t + 1 2πi ∫ ∂m∩d γ(t) t̄ + z t̄ − z dt t̄ = 1 2πi ∫ ∂m∩d γ(t) t + z t − z dt t − 1 2πi ∫ ∂m∩d γ(t̄) t + z t − z dt t = 1 2πi ∫ ∂m∩d υ(t) t + z t − z dt twhere υ(t) = { γ(t), imz ≥ 0, −γ(t̄), imz < 0.from the properties of the poisson kernel for unit disc [2,3], we have lim z→t (ω(z)− h(z)) = υ(t). in particular lim z→±1 (ω(z)− h(z)) = γ(±1) = 0, is seen because of the continuity of υ at ±1. similar to what was done above, from the properties of the poisson kernel for half plane [2], wehave lim z→t (ω(z)− h(z)) = γ(t). (3.6) by (3.4) and (3.6), we have lim z→t h(z) = 0, t ∈ ∂m. then, from the maximum principle for analytic functions h(z) = 0 for z ∈ m, which is given ascondition (3.2). conversely, if the condition (3.2) is is satisfied, then, the analytic function ω can be expressedas ω(z) = ω(z)− h(z) = 1 2πi ∫ ∂m γ(t) [ 1 t − z + z tz − 1 ] dt − 1 2πi ∫ ∂m γ(t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 1 2πi ∫ ∂m γ(t) [ 1 t − z − 1 t − z̄ + z tz − 1 − z̄ tz̄ − 1 ] dt. hence, lim z→t ω(z) = γ(t), t ∈ ∂m. (3.7) follows again from the properties of the poisson kernel. � https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 7in the next stage, we investigate the dirichlet problem for the inhomogeneous cauchy–riemannequation. to solve this problem, we reduce the inhomogeneous problem into a homogeneous one,using definition and properties of the pompeiu operator, and then find a solution for it using theprevious theorem. theorem 3.2. the dirichlet boundary value problem for the inhomogeneous cauchy–riemann equation ωz̄ = f (z), z ∈ m, f ∈ lp(m;c), p > 2, ω = γ, on ∂m, γ ∈ c(∂m;c), (3.8) is solvable if and only if for z ∈ m, 1 2πi ∫ ∂m γ(t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 1 π ∫ m f (t) [ 1 ζ − z̄ + z̄ tz̄ − 1 ] dξdη, (3.9) and its solution can be uniquely expressed as ω(z) = 1 2πi ∫ ∂m γ(t) [ 1 t − z + z tz − 1 ] dt − 1 π ∫ m f (t) [ 1 t − z + z tz − 1 ] dξdη. (3.10) where t = ξ + iη. proof. by the theorem 2.1, if the dirichlet problem (3.8) is solvable, its can be expressed in theform of (3.10). let ϕ(z) = ω(z)− t f (z), by applying the ∂z̄ operator to the function ’ we have, ∂z̄ϕ = ∂z̄ω − ∂z̄t f ⇒ ∂z̄ϕ = f − f = 0, ϕ = ω − t f ⇒ ϕ = γ − t f . then consider the homogeneous dirichlet problem ϕz̄ = 0, in z ∈ m, ϕ = γ − t f , on ∂m. (3.11) which is equivalent to equation (3.11) by the theorem 3.2 the solvability condition for equation(3.14) is 1 2πi ∫ ∂m (γ(t)− t f (t)) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 0, using the properties of the integral operator 1 2πi ∫ ∂m t f (t) [ 1 t − z̄ + z̄ tz̄ − 1 ] dt = 1 π ∫ m f (t̃) 1 2πi ∫ ∂m [ 1 t − z̄ + z̄ tz̄ − 1 ] dt t − t̃ d ξ̃dη̃ = 1 π ∫ m f (t̃) [ 1 t̃ − z̄ + z̄ t̃ z̄ − 1 ] dξ̃dη̃. which is just condition (3.9). https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 8on the other hand, if the condition of solvability (3.9) is satisfied, then (3.10) can be expressedas follows. ω(z) = 1 2πi ∫ ∂m γ(z) [ 1 t − z + z tz − 1 − 1 t − z̄ − z̄ tz̄ − 1 ] dt. − 1 π ∫ m f (z) [ 1 t − z + z tz − 1 − 1 t − z̄ − z̄ tz̄ − 1 ] dξdη. (3.12) since the area integral tends to 0 as z → t ∈ ∂m, by the proof of theorem (3.2), (3.12) impliesthat lim z→t ω(z) = γ(t), t ∈ ∂m.now, we are going to investigate the uniqueness of the dirichlet problem solution. assume that ωand w are two solutions to the dirichlet problem, therefore we have ωz̄ = f , in z ∈ m, ω = γ on ∂m, wz̄ = f , in z ∈ m, w = γ, on ∂m.by subtracting the above two relations, we conclude that (ω − w)z̄ = 0, in m ω − w = 0. in m,this completes the proof. � acknowledgments the authors would like to express their sincere gratitude to the editor in chief, associate editorand referees for their valuable comments that led to considerable improvement of the article. declarations there is no conflict of interest related to the present research. the work has not been publishedbefore and is not under consideration elsewhere. this research received no particular grant fromany funding agency. references [1] m. akel, s. mondal, dirichlet problems in lens and lune, bull. malays. math. sci. soc. 41 (2018) 1029–1043.[2] h. begehr, t. vaitekhovich, harmonic boundary value problems in half disc and half ring, funct. approx. 40 (2009)251–282.[3] h. begehr, t. vaitekhovich, schwarz problem in lens and lune, complex var. epllitic equ. 59 (2014) 76–84.[4] v.p. burskii, e.v. lesina, on boundary value problems for an improperly elliptic equation in a circle, comput. math.math. phys. 60 (2020) 1306–1321.[5] z. du, y. wang, m. ku, schwarz boundary value problems for polyanalytic equation in a sector ring, complex anal.oper. theory, 17 (2023) 33. https://doi.org/10.28924/ada/ma.4.15 eur. j. math. anal. 10.28924/ada/ma.4.15 9 [6] h. emkanpour, n. taghizadeh, three boundary value problems of the cauchy–riemann equation in eclipse domain,complex var. epllitic equ. 67 (2022) 510–529.[7] y. gao, y. zhao, b. zhao, boundary value problems of holomorphic vector functions in 1d qcs, physica b. 394(2007) 56–61.[8] v.v. karachik, class of neumann-type problems for the polyharmonic equation in a ball, comput. math. math. phys.60 (2020) 144–162.[9] v.g. nikolaev, schwarz problem for j-analytic functions in an ellipse, comput. math. math. phys. 62 (2022) 1089–1111.[10] a.v. petukhov, a.o. savchenko, solution of the exterior boundary value problem for the helmholtz equation usingoverlapping domain decomposition, comput. math. math. phys. 62 (2020) 784–796.[11] k. ravikumar, k. ramkumar, d. chalishajar, existence and stability results for second-order neutral stochasticdifferential equations with random impulses and poisson jumps, eur. j. math. anal. 1 (2021) 1.[12] n. taghizadeh, m. mirzazadeh, f. farahrooz, exact solutions of the nonlinear schrödinger equation by the firstintegral method, j. math. anal. appl. 374 (2011) 549–553.[13] n. taghizadeh, v.s. mohammadi, some boundary value problems for the cauchy–riemann equation in half lens,eurasian math j. 9 (2018) 73–84.[14] i.n. vekua, generalized analytic functions, pergamon press, oxford, 1962.[15] y. wang, x. zhao, schwarz boundary value problem for the cauchy–riemann equation in a rectangle, bound. valueprobl. 2016 (2016) 7.[16] y. wang, schwarz-type boundary value problems for the polyanalytic equation in the half unit disc, complex var.epllitic equ. 57 (2012) 983–993. https://doi.org/10.28924/ada/ma.4.15 1. introduction and preliminaries 2. an integral representation formula for m 3. dirichlet problem for the cauchy–riemann equation in m acknowledgments declarations references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 11doi: 10.28924/ada/ma.3.11 woven k − g−fusion frames in hilbert c∗−modules fakhr-dine nhari1, mohamed rossafi2,∗ 1laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, p. o. box 133 kenitra, morocco nharidoc@gmail.com 2lasma laboratory, department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, p. o. box 1796 fez atlas, morocco rossafimohamed@gmail.com ∗correspondence: rossafimohamed@gmail.com abstract. in this paper, we introduced the notion of woven k − g−fusion frames in hilbert c∗−modules. we present necessary and sufficient conditions for these woven and also constructthem by linear bounded operator. finally we study perturbation of weaving k − g−fusion frames. 1. introduction basis is one of the most important concepts in vector spaces study. however, frames generaliseorthonormal bases and were introduced by duffin and schaefer [3] in 1952 to analyse some deepproblems in nonharmonic fourier series by abstracting the fundamental notion of gabor [5] for signalprocessing. in 2000, frank-larson [4] introduced the concept of frames in hilbet c∗−modulesas a generalization of frames in hilbert spaces. the basic idea was to consider modules over c∗−algebras of linear spaces and to allow the inner product to take values in the c∗−algebras [6].many generalizations of the concept of frame have been defined in hilbert c∗-modules [7,9,11–16].throughout this paper, h is considered to be a countably generated hilbert c∗−module. let {hj}j∈j are the collection of hilbert c∗−module and {wj}j∈j is a collection of closed orthogonallycomplemented submodules of h, where j be finite or countable index set. end∗a(h,hj) is a setof all adjointable operator from h to hj . in particular end∗a(h) denote the set of all adjointableoperators on h. pwj denote the orthogonal projection onto the closed submodule orthogonally received: 31 jul 2022.2010 mathematics subject classification. primary 41a58; secondary 42c15. key words and phrases. fusion frames; k − g−fusion frames; woven k − g−fusion frames; c∗-algebra; hilbert c∗-modules. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 2complemented wj of h. define the module l2({hj}j∈j) = {{fj}j∈j : fj ∈ hj , ‖ ∑ j∈j 〈fj , fj〉‖ <∞} with a−valued inner product 〈f , g〉 = ∑ j∈j〈fj , gj〉, where f = {fj}j∈j and g = {gj}j∈j, clearly l2({hj}j∈j) is a hilbert a−module. definition 1.1. [8] let a be a unital c∗-algebra and h be a left a-module, such that the linearstructures of a and h are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued inner product 〈., .〉 : h×h → a, such that is sesquilinear, positive definite and respectsthe module action. in the other words,(i) 〈f , f 〉 ≥ 0 for all f ∈ h and 〈f , f 〉 = 0 if and only if f = 0.(ii) 〈af + g, h〉 = a〈f , h〉+ 〈g, h〉 for all a ∈ a and f , g, h ∈ h.(iii) 〈f , g〉 = 〈g, f 〉∗ for all f , g ∈ h.for f ∈ h, we define ||f || = ||〈f , f 〉|| 1 2 . if h is complete with ||.||, it is called a hilbert a-moduleor a hilbert c∗-module over a. for every a in a c∗-algebra a, we have |a| = (a∗a) 1 2 and the a-valued norm on h is defined by |f | = 〈f , f 〉 1 2 for f ∈ h. lemma 1.2. [10] let {wj}j∈j be a sequence of orthogonally complemented closed submodules of h and t ∈ end∗a(h) invertible, if t ∗twj ⊂ wj for each j ∈ j , then {twj}j∈j is a sequence of orthogonally complemented closed submodules and pwj t ∗ = pwj t ∗ptwj . lemma 1.3. [2]. let h and k two hilbert a-modules and t ∈ end∗a(h,k). then the following statements are equivalent:(i) t is surjective.(ii) t ∗ is bounded below with respect to norm, i.e., there is m > 0 such that ‖t ∗x‖ ≥ m‖x‖ for all x ∈ k.(iii) t ∗ is bounded below with respect to the inner product, i.e., there is m′ > 0 such that 〈t ∗x, t ∗x〉 ≥ m′〈x, x〉 for all x ∈ k. lemma 1.4. [1]. let u and h two hilbert a-modules and t ∈ end∗a(u,h). then:(i) if t is injective and t has closed range, then the adjointable map t ∗t is invertible and ‖(t ∗t )−1‖−1 ≤ t ∗t ≤ ‖t‖2. (ii) if t is surjective, then the adjointable map tt ∗ is invertible and ‖(tt ∗)−1‖−1 ≤ tt ∗ ≤ ‖t‖2. definition 1.5. [10] let {wi}i∈i be a sequence of closed orthogonally complemented submodulesof h, {vi}i∈i be a familly of positive weights in a, i.e., each vi is a positive invertible element from https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 3the center of the c∗−algebra a and λi ∈ end∗a(h,hi) for all i ∈ i . we say that λ = {wi ,λi , vi}i∈iis a g−fusion frame for h if and only if there exists two constants 0 < a ≤ b <∞ such that a〈x, x〉 ≤ ∑ i∈i v2 i 〈λipwi x,λipwi x〉 ≤ b〈x, x〉, ∀x ∈ h. (1.1) the constants a and b are called the lower and upper bounds of g−fusion frame, respectively. if a = b then λ is called tight g-fusion frame and if a = b = 1 then we say λ is a parseval g−fusionframe. if λ satisfies the inequality∑ i∈i v2 i 〈λipwi x,λipwi x〉 ≤ b〈x, x〉, ∀x ∈ h. then it is called a g−fusion bessel sequence with bound b in h. definition 1.6. [10]let λ = {wj ,λj , vj}j∈j be a g−fusion bessel sequence for h. then the operator tλ : l2({hj}j∈j)→ h defined by tλ({fj}j∈j) = ∑ j∈j vjpwj λ∗j fj , ∀{fj}j∈j ∈ l2({hj}j∈j). is called synthesis operator. we say the adjoint uλ of the synthesis operator the analysis operatorand it is defined by uλ : h → l2({hj}j∈j) such that uλ(f ) = {vjλjpwj (f )}j∈j, ∀f ∈ h. the operator sλ : h → h defined by sλf = tλuλf = ∑ j∈j v2 j pwj λ∗j λjpwj (f ), ∀f ∈ h. is called g−fusion frame operator. it can be easily verify that 〈sλf , f 〉 = ∑ j∈j v2 j 〈λjpwj (f ),λjpwj (f )〉, ∀f ∈ h. (1.2) furthermore, if λ is a g−fusion frame with bounds a and b, then a〈f , f 〉 ≤ 〈sλf , f 〉 ≤ b〈f , f 〉, ∀f ∈ h. it easy to see that the operator sλ is bounded, self-adjoint, positive, now we proof the inversibilityof sλ. let f ∈ h we have ||uλ(f )|| = ||{vjλjpwj (f )}j∈i || = || ∑ j∈j v2 j 〈λjpwj (f ),λjpwj (f )〉|| 1 2 . since λ is g−fusion frame then √ a||〈f , f 〉|| 1 2 ≤ ||uλf ||.then √ a||f || ≤ ||uλf ||. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 4frome lemma 1.3, tλ is surjective and by lemma 1.4, tλuλ = sλ is invertible. we now, aih ≤ sλ ≤ bih and this gives b−1ih ≤ s−1 λ ≤ a−1ih . 2. woven k − g−fusion frames in hilbert c∗−modules throughout this paper, [m] = {1, 2, ..., m} for each m > 1, {wi j}j∈j,i∈[m] is a collection of closedorthogonally complemented submodules of h, {vi j}j∈j,i∈[m] is a family of weights, k ∈ end∗a(h)and {λi j}j∈j,i∈[m] ∈ end∗a(h,hi j) where hi j are hilbert a−modules. definition 2.1. a family of g−fusion frames {wi j ,λi j , vi j}j∈j,i∈[m] for h is said to be k− g−fusionwoven if there exist universal positive constants 0 < a ≤ b such that for each partition {σi}i∈[m]of j, the family {wi j ,λi j , vi j}j∈σi ,i∈[m] is a k − g−fusion frame for h with bounds a and b. in next theorem, we provide a necessary and sufficient condition for weaving k−g−fusion frames. theorem 2.2. assume that {wj ,λj , vj}j∈j and {vj , θj , µj}j∈j are two k − g−fusion frames for h where λj ∈ end∗a(h,hj) and θj ∈ end∗a(h,hj) for any j ∈ j, the following assertions are equivalent.(1) {wj ,λj , vj}j∈j and {vj , θj , µj}j∈j are k − g−fusion woven.(2) there exists α > 0 such that for each σ ⊂ j there exists a bounded linear operator ψσ : lσ2 ({hj}j∈j)→ h, ψσ{xj}j∈j = ∑ j∈σ vjpwj λ∗j xj + ∑ j∈σc µjpvj θ ∗ j xj , such that αkk∗ ≤ ψσψ∗σ , where lσ2 ({hj}j∈j) = {{xj}j∈j = {fj}j∈σ ∪ {gj}j∈σc : fj ∈ hj , gj ∈ hj , ‖ ∑ j∈j 〈xj , xj〉‖ <∞}. proof. (1) =⇒ (2): suppose that a is an universal lower frame bound for {wj ,λj , vj}j∈j and {vj , θj , µj}j∈j. choose α = a and ψσ = tσ for every σ ⊂ j, where tσ is the synthesis operator of {wj ,λj , vj}j∈σ ∪ {vj , θj , µj}j∈σc . then, for any {xj}j∈j ∈ lσ2 ({hj}j∈j) we have ψσ{xj}j∈j = tσ{xj}j∈j = ∑ j∈σ vjpwj λ∗j xj + ∑ j∈σc µjpvj θ ∗ j xj , and also, for each f ∈ h, a〈k∗f , k∗f 〉 ≤ 〈t ∗σ f , t ∗σ f 〉 = 〈ψ∗σf , ψ∗σf 〉.thus, αkk∗ ≤ ψσψ∗σ . (2) =⇒ (1): let σ ⊂ j and f ∈ h, so it is easy to check that ψ∗σf = {vjλjpwj f }j∈σ ∪ {µjθjpvj f }j∈σc . https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 5therefore, α〈k∗f , k∗f 〉 = 〈αkk∗f , f 〉 ≤ 〈ψσψ∗σf , f 〉 = 〈ψ∗σf , ψ∗σf 〉 = ∑ j∈σ v2 j 〈λjpwj f ,λjpwj f 〉+ ∑ j∈σc µ2 j 〈θjpvj f , θjpvj f 〉. this gives that α is an universal lower frame bound of {wj ,λj , vj}j∈j and {vj , θj , µj}j∈j. � in next results, we construct a k − g−fusion woven by using a bounded linear operator. theorem 2.3. let {wi j ,λi j , vi j}j∈j,i∈[m] be a k−g−fusion woven for h with common frame bounds a,b and assume that u ∈ end∗a(h) has closed range so that r(k∗) ⊂ r(u) and ku = uk. then {uwi j ,λi jpwi j u∗, vi j}j∈j,i∈[m] is also k − g−fusion woven for r(u). proof. by the open mapping theorem, uwi j is closed for any j ∈ j and i ∈ [m]. using lemme(refk-g-fusion ), we can write for each f ∈ r(u), a〈k∗f , k∗f 〉 = a〈(u+)∗u∗k∗f , (u+)∗u∗k∗f 〉 ≤ a‖u+‖2〈k∗u∗f , k∗u∗f 〉 ≤ ‖u+‖2 ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗f ,λi jpwi j u∗f 〉 = ‖u+‖2 ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗puwi j f ,λi jpwi j u∗puwi j f 〉. the upper bound is obvious. � theorem 2.4. let k have closed range, {wi j ,λi j , vi j}j∈j,i∈[m] be a k− g−fusion woven for h with the universal bounds a,b and u ∈ end∗a(h) has closed range so that r(u∗) ⊂ r(k). then {uwi j ,λi jpwi j u∗, vi j}j∈j,i∈[m] is a k − g−fusion woven for h if and only if there exists a δ > 0 such that for every f ∈ h, 〈u∗f , u∗f 〉 ≥ δ〈k∗f , k∗f 〉. proof. let f ∈ h and {uwi j ,λi jpwi j u∗, vi j}j∈j,i∈[m] is a k − g−fusion woven for h with lowerbound c, we get c〈k∗f , k∗f 〉 ≤ ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗puwi j f ,λi jpwi j u∗puwi j f 〉 = ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗f ,λi jpwi j u∗f 〉 ≤ b〈u∗f , u∗f 〉. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 6 therefore, 〈u∗f , u∗f 〉 ≥√c b 〈k ∗f , k∗f 〉. for the opposite implication, we can write for all f ∈ h, 〈u∗f , u∗f 〉 = 〈(k+)∗k∗u∗f , (k+)∗k∗u∗f 〉 ≤ ‖k+‖2〈k∗u∗f , k∗u∗f 〉. hence, we have aδ‖k+‖−2〈k∗f , k∗f 〉 ≤ a‖k+‖−2〈u∗f , u∗f 〉 ≤ a〈k∗u∗f , k∗u∗f 〉 ≤ ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗f ,λi jpwi j u∗f 〉 = ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j u∗puwi j f ,λi jpwi j u∗puwi j f 〉 ≤ b‖u‖2〈f , f 〉. so, {uwi j ,λi jpwi j u∗, vi j}j∈j,i∈[m] is a k − g−fusion woven for h with frame bounds aδ‖k+‖−2and b‖u‖2. � theorem 2.5. let {wi j ,λi j , vi j}j∈j,i∈[m] be a k − g−fusion woven for h with common frame bounds a and b. suppose that 0 ≤ c ≤ |w (i) j | 2 ≤ d < ∞ for any i ∈ [m] and j ∈ j, then {wi j , w (i) j λi j , vi j}j∈j,i∈[m] is a k − g−fusion woven for h with frame bounds ac and bd. proof. for any partition {σi}i∈[m] of j and f ∈ h, we get ac〈k∗f , k∗f 〉 = min i∈[m] |w (i) j | 2a〈k∗f , k∗f 〉 ≤ ∑ i∈[m] ∑ j∈σi v2 i j 〈w (i) j λi jpwi j f , w (i) j λi jpwi j f 〉 ≤ max i∈[m] |w (i) j | 2b〈f , f 〉 = bd〈f , f 〉. � theorem 2.6. let i ⊂ j be arbitrary and {wi j ,λi j , vi j}j∈i,i∈[m] be a k − g−fusion woven for h. then {wi j ,λi j , vi j}j∈j,i∈[m] is a k − g−fusion woven. proof. assume that σi ⊂ j, so σi ∩ i ⊂ i and a is the lower bound of {wi j ,λi j , vi j}j∈σi∩i,i∈[m], thenfor every f ∈ h we have a〈k∗f , k∗f 〉 ≤ ∑ i∈[m] ∑ j∈σi∩i v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ≤ ∑ i∈[m] ∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉. this implies the statement. � next theorem is shows that even if one subspace is deleted, it dose not still remain a k−g−fusionwoven. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 7 theorem 2.7. let k has closed range, i ⊂ j and {wi j ,λi j , vi j}j∈j,i∈[m] be a k − g−fusion woven for h with the bounds a,b. if c = ∑ i∈[m] ∑ j∈i v2 i j‖λi jpwi j ‖2 < a‖k+‖2, then {wi j ,λi j , vi j}j∈j−i,i∈[m] is a k − g−fusion woven for r(k). proof. the upper bound is obvious. suppose that σi i∈[m] ⊂ j− i and f ∈ r(k), so we get∑ i∈[m] ∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 = ∑ i∈[m] ∑ j∈σi∪i v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 − ∑ i∈[m] ∑ j∈i v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ≥ a〈k∗f , k∗f 〉 − ∑ i∈[m] ∑ j∈i v2 i j‖λi jpwi j ‖2〈f , f 〉 ≥ (a− c‖k+‖2)〈k∗f , k∗f 〉. � theorem 2.8. let {wi j ,λi j , vi j}j∈j,i∈[m] be a k − g−fusion woven for h with bounds a,b. for each i ∈ [m],j ∈ j and a index set ii j , suppose that {f (k) i j }k∈ii j ∈ λi j(wi j) is a parseval frame for hi j such that for every finite subset ki j ⊂ ii j , the set {f kij }k∈ii j−ki j is a frame with the lower bound ci j . let w̃i j = span{λ∗i j f (k) i j }k∈ii j−ki j for any i ∈ [m] and j ∈ j, then {w̃i j ,λi j , vi j}j∈j,i∈[m] is a k − g−fusion woven for h with the bounds (mini∈[m],j∈j ci j)a and b. proof. obviously, b is the upper bound of {w̃i j ,λi j , vi j}j∈j,i∈[m]. assume that f ∈ h and {σi}i∈[m] ∈ j, so ∑ i∈[m] ∑ j∈σi v2 i j 〈λi jpw̃i j f ,λi jpw̃i j f 〉 = ∑ i∈[m] ∑ j∈σi v2 i j ∑ k∈ii j 〈λi jpw̃i j f , f (k) i j 〉〈f (k) i j ,λi jpw̃i j f 〉 ≥ ∑ i∈[m] ∑ j∈σi v2 i j ∑ k∈ii j−ki j 〈λi jpw̃i j f , f (k) i j 〉〈f (k) i j ,λi jpw̃i j f 〉 ≥ ∑ i∈[m] ∑ j∈σi v2 i jci j〈λi jpwi j f ,λi jpwi j f 〉 ≥ ( min i∈[m],j∈j ci j) ∑ i∈[m] ∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ≥ ( min i∈[m],j∈j ci j)a〈k∗f , k∗f 〉. � theorem 2.9. let {wi j ,λi j , vi j}j∈j is a k−g−fusion frame for h for each i ∈ [m]. suppose that for a partition collection of disjoint finite sets {δi}i∈[m] of j and for any ε > 0 there exists a partition {σi}i∈[m] of the set j − ∪i∈[m]δi such that {wi j ,λi j , vi j}j∈(σi∪δi ),i∈[m] has a lower k − g−fusion frame bound less than ε. then {wi j ,λi j , vi j}j∈j,i∈[m] is not a woven. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 8 proof. we can write j = ∪j∈njj , where jj are disjoint index sets. assume that δ1j = ∅ for all i ∈ [m] and ε = 1. then, there exists a partition σi1i∈[m] of j such that {wi j ,λi j , vi j}j∈(σi1∪δi1),i∈[m]has a lower bound (also, optimal lower bound) less than 1. thus, there is a f1 ∈ h such that∑ i∈[m] ∑ j∈(σi1∪δi1) v2 i j 〈λi jpwi j f1,λi jpwi j f1〉 < 〈k∗f1, k∗f1〉. since ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j f1,λi jpwi j f1〉 <∞, so, there is a k1 ∈ n such that∑ i∈[m] ∑ j∈k1 v2 i j 〈λi jpwi j f1,λi jpwi j f1〉 < 〈k∗f1, k∗f1〉, where, k1 = ∪i≥k1+1jj . continuing this way, for ε = 1 n and a partition {δni}i∈[m] of j1 ∪ ...∪ jkn−1such that δni = δ(n−1)i ∪ (σ(n−1)i ∩ (j1 ∪ ... ∪ jkn−1)) for all i ∈ [m], there exists a partition {σni}i∈[m] of j − (j1 ∪ ... ∪ jkn−1) such that {wi j ,λi j , vi j}j∈(σni∪δni ),i∈[m] has a lower bound less than 1 n . therefore, there is a fn ∈ h and kn ∈ n such that kn > kn−1 and∑ i∈[m] ∑ j∈kn v2 i j 〈λi jpwi j fn,λi jpwi j fn〉 < 1 n 〈k∗fn, k∗f1〉, where, kn = ∪i≥kn+1jj . choose a partition {αi}i∈[m] of j, where αi = ∪j∈n{δj i} = δ(n+1)i ∪ (αi ∩ j − (j1 ∪ ... ∪ jn)). assume that {wi j ,λi j , vi j}j∈αi ,i∈[m] is a k − g−fusion frame for h with theoptimal lower bound a. then, by the archimedean property, there exits a r ∈ n such that r > 2 a .now, there exists a fr ∈ h such that∑ i∈[m] ∑ j∈αi v2 i j 〈λi jpwi j fr ,λi jpwi j fr 〉 = ∑ i∈[m] ∑ j∈δ(r+1)i v2 i j 〈λi jpwi j fr ,λi jpwi j fr 〉 + ∑ i∈[m] ∑ j∈αi∩j−(j1∪...∪jr ) v2 i j 〈λi jpwi j fr ,λi jpwi j fr 〉 ≤ ∑ i∈[m] ∑ j∈(σr i∪δr i ) v2 i j 〈λi jpwi j fr ,λi jpwi j fr 〉 + ∑ i∈[m] ∑ j∈∪k≥r+1jk v2 i j 〈λi jpwi j fr ,λi jpwi j fr 〉 < 1 r 〈k∗fr , k∗fr 〉+ 1 r 〈k∗fr , k∗fr 〉 < a〈k∗fr , k∗fr 〉 and this is a contradiction with the lower bound of a. � https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 9 corollary 2.10. let {wi j ,λi j , vi j}j∈j,i∈[m] be a k − g−fusion woven for h. then there exists a collection of disjoint finite subsets {δi}i∈[m] of j and a > 0 such that for each partition {σi}i∈[m] of the set j − ∪i∈[m]δi , some the family {wi j ,λi j , vi j}j∈(σi∪δi ),i∈[m] is a k − g−fusion frame for h with the lower frame bound a. theorem 2.11. let {wi j ,λi j , vi j}j∈j be a k−g−fusion frame for h with bounds ai and bi for each i ∈ [m]. suppose that there exists n > 0 such that for all i , k ∈ [m] with i 6= k , i ⊂ j and f ∈ h,∑ j∈i 〈(vi jλi jpwi j − vkjλkjpwkj )f , (vi jλi jpwi j − vkjλkjpwkj )f 〉 ≤ n min{ ∑ j∈i v2 i j 〈λi jpwi j f ,λi jpwi j f 〉,∑ j∈i v2 kj〈λkjpwkj f ,λkjpwkj f 〉}. then the family {wi j ,λi j , vi j}j∈j,i∈[m] is woven with universal bounds a (m − 1)(n + 1) + 1 and b, where a = ∑ i∈[m] ai and b = ∑ i∈[m]bi . proof. let {σi}i∈[m] be a partition of j and f ∈ h. therefore,∑ i∈[m] ai〈k∗f , k∗f 〉 ∑ i∈[m] ∑ j∈j v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 = ∑ i∈[m] ∑ k∈[m] ∑ j∈σk v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ≤ ∑ i∈[m] (∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉+ ∑ k∈[m],k 6=i ∑ j∈σk {v2 kj〈λkjpwkj f ,λkjpwkj f 〉 + 〈(vi jλi jpwi j − vkjλkjpwkj )f , (vi jλi jpwi j − vkjλkjpwkj )f 〉} ) ≤ ∑ i∈[m] (∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 + ∑ k∈[m],k 6=i ∑ j∈σk (n + 1)v2 kj〈λkjpwkj f ,λkjpwkj f 〉 ) = {(m − 1)(n + 1) + 1} ∑ i∈[m] (∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ) . thus, we get a (m − 1)(n + 1) + 1 〈k∗f , k∗f 〉 ≤ ∑ i∈[m] (∑ j∈σi v2 i j 〈λi jpwi j f ,λi jpwi j f 〉 ) ≤ b〈f , f 〉. � in next theorem we study a paley-wiener type perturbation for weaving k − g−fusion frames. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 10 theorem 2.12. let {wj ,λj , wj}j∈j and {vj , θj , vj}j∈j be two k−g−fusion frames for h with frame bounds a1, b1 and a2, b2, respectively. suppose that there exist non-negative scalers µ and 0 ≤ λ < 1 2 such that ( 1 2 − λ)a1 > µ and for each f ∈ h,∑ j∈j 〈(wjλjpwj − vjθjpvj )f , (wjλjpwj − vjθjpvj )f 〉 ≤ λ ∑ j∈j 〈wjλjpwj f , wjλjpwj f 〉+ µ〈k∗f , k∗f 〉. then, {wj ,λj , wj}j∈j and {vj , θj , vj}j∈j are k−g−fusion woven for h with universal frame bounds ( 1 2 − λ)a1 − µ and b1 + b2. proof. the upper frame bound is clear. for the lower frame bound, assume that σ ⊂ j and we get,by the arithmetic-quadratic mean, for any f ∈ h∑ j∈σ w2 j 〈λjpwj f ,λjpwj f 〉+ ∑ j∈σc v2 j 〈θjpvj f , θjpvj f 〉 = ∑ j∈σ w2 j 〈λjpwj f ,λjpwj f 〉 + ∑ j∈σc 〈wjλjpwj f − (wjλjpwj − vjθjpvj )f , wjλjpwj f − (wjλjpwj − vjθjpvj )f 〉 ≥ ∑ j∈σ w2 j 〈λjpwj f ,λjpwj f 〉+ 1 2 ∑ j∈σc w2 j 〈λjpwj f ,λjpwj f 〉 − ∑ j∈σc 〈(wjλjpwj − vjθjpvj )f , (wjλjpwj − vjθjpvj )f 〉 = 1 2 ∑ j∈j w2 j 〈λjpwj f ,λjpwj f 〉+ 1 2 ∑ j∈σ w2 j 〈λjpwj f ,λjpwj f 〉 − ∑ j∈σc 〈(wjλjpwj − vjθjpvj )f , (wjλjpwj − vjθjpvj )f 〉 ≥ 1 2 ∑ j∈j w2 j 〈λjpwj f ,λjpwj f 〉 − ∑ j∈σc 〈(wjλjpwj − vjθjpvj )f , (wjλjpwj − vjθjpvj )f 〉 ≥ 1 2 ∑ j∈j w2 j 〈λjpwj f ,λjpwj f 〉 − λ ∑ j∈j w2 j 〈λjpwj f ,λjpwj f 〉 − µ〈k∗f , k∗f 〉 ≥ ( ( 1 2− λ)a1 − µ ) 〈k∗f , k∗f 〉. this completes the proof. � declarations availablity of data and materialsnot applicable. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 11 human and animal rightswe would like to mention that this article does not contain any studies with animals and does notinvolve any studies over human being. competing intereston behalf of all authors, the corresponding author states that there is no conflict of interest. fundingsauthors declare that there is no funding available for this article. authors’ contributionsthe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript. references [1] a. alijani, m. dehghan, ∗-frames in hilbert c∗modules, u.p.b. sci. bull., ser. a, 73 (2011), 89-106.[2] lj. arambašić , on frames for countably generated hilbert c∗-modules, proc. amer. math. soc. 135 (2007) 469-478. https://doi.org/10.1090/s0002-9939-06-08498-x.[3] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341–366. https://doi.org/10.1090/s0002-9947-1952-0047179-6.[4] m. frank, d.r. larson, a-module frame concept for hilbert c∗-modules, funct. harm. anal. wavel. contempt. math.247 (2000) 207-233.[5] d. gabor, theory of communication. part 1: the analysis of information, j. inst. electric. eng. 93 (1946) 429–441. https://doi.org/10.1049/ji-3-2.1946.0074.[6] e.c. lance, hilbert c∗−modules: a toolkit for operator algebraist, london math. soc. lecture note ser. cambridgeuniv. press, cambridge, 1995.[7] s. 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paedagog. crac. stud. math.17 (2018) 17-25. https://doi.org/10.2478/aupcsm-2018-0002.[15] m. rossafi, s. kabbaj, generalized frames for b(h,k), iran. j. math. sci. inf. 17 (2022) 01-09. https://doi.org/ 10.52547/ijmsi.17.1.1.[16] m. rossafi, f.d. nhari, c. park, s. kabbaj, continuous g-frames with c∗-valued bounds and their properties,complex anal. oper. theory 16 (2022) 44. https://doi.org/10.1007/s11785-022-01229-4. https://doi.org/10.28924/ada/ma.3.11 https://doi.org/10.1090/s0002-9939-06-08498-x https://doi.org/10.1090/s0002-9947-1952-0047179-6 https://doi.org/10.1049/ji-3-2.1946.0074 https://doi.org/10.2307/2372552 https://doi.org/10.2307/2372552 https://doi.org/10.1142/s0219691308002458 https://doi.org/10.28924/2291-8639-19-2021-836 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.2478/aupcsm-2018-0002 https://doi.org/10.52547/ijmsi.17.1.1 https://doi.org/10.52547/ijmsi.17.1.1 https://doi.org/10.1007/s11785-022-01229-4 1. introduction 2. woven k-g-fusion frames in hilbert c-modules declarations references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 25doi: 10.28924/ada/ma.3.25 on the kolmogorov distance for the maximum likelihood estimator in the explosive ornstein-uhlenbeck process jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte, 376 fretwell bldg,9201 university city blvd. charlotte, nc 28223-0001, usacorrespondence: j.bishwal@uncc.edu abstract. the paper estimates the kolmogorov distance between the distribution of the normalizedmaximum likelihood estimator of the positive drift parameter in the nonergodic ornstein-uhlenbeckprocess and the standard cauchy distribution and shows exponential error rate for large time limit. 1. introduction estimating the rate in the kolmogorov distance between two distributions has a long history inprobability and statistics. the estimate could be useful in finding confidence interval and inhypothesis testing, see bishwal [8, 11]. in the i.i.d. case, the berry-esseen bound for minimumcontrast estimators was obtained in pfanzagl [28] improving that from michel and pfanzagl [24].borokov [14] obtained the rate of convergence for the invariance principle in the i.i.d. case. hall andheyde [20] obtained rate of convergence in the central limit theorem for martingales using skorohodembedding. uniform rate of weak convergence for the minimum contrast estimator in the ornstein-uhlenbeck (o-u) process was studied in bishwal [5]. the rates of convergence of the conditionalleast squares estimator and an approximate maximum likelihood estimator when the o-u processis observed at discrete time points in [0, t ] has been studied in bishwal and bose [13](2001) inthe ergodic case. in a bayesian framework, the rates of convergence of the posterior distributionsand the bayes estimators has been studied in bishwal [6] and bishwal [10] for the continuousobservation and discrete observations respectively in the ergodic case. in finance, asset price maybehave in nonergodic manner, i.e., efficient market hypotheses may not hold, possibly be due tosocial interaction among consumers among other reasons, see horst and wenzelburger [21]. westudy the nonergodic ornstein-uhlenbeck process in this paper and focus on the rate of convergenceof the kolmogorov distance. received: 5 jul 2023. key words and phrases. itô stochastic differential equation, explosive ornstein-uhlenbeck process, maximum likeli-hood estimator, kolmogorov distance, inefficient market. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 2let (ω,f , {ft}t≥0, p ) be a stochastic basis on which is defined the ornstein-uhlenbeck process {xt} satisfying the itô stochastic differential equation dxt = θxtdt + dwt , t ≥ 0, x0 = 0 (1.1) where {wt}t≥0 is a standard wiener process with respect to the filtration {ft}t≥0 and θ > 0is the unknown parameter to be estimated on the basis of continuous observation of the process {xt}t≥0 on the time interval [0, t ].let us denote the realization {xt , 0 ≤ t ≤ t} by xt0 . let p tθ be the measure generated on thespace (ct , bt ) of continuous functions on [0, t ] with the associated borel σ-algebra bt generatedunder the supremum norm by the process xt0 and p t0 be the standard wiener measure. it is wellknown that when θ is the true value of the parameter p tθ is absolutely continuous with respect to p t0 and the radon-nikodym derivative (likelihood) of p tθ with respect to p t0 based on xt0 is givenby lt (θ) := dp tθ dp t0 (xt0 ) = exp { θ ∫ t 0 xtdxt − θ2 2 ∫ t 0 x2 t dt } . (1.2) maximizing the log-likelihood with respect to θ provides the maximum likelihood estimate (mle) θt := ∫ t 0 xtdxt∫ t 0 x2 t dt . (1.3) in this transient case, we show that this estimator converges to the cauchy distribution withan error rate o(e−θt ). note that in the transient case, with random norming, specifically if onenormalizes the mle by the square root of the observed fisher information, then the mle convergesto the normal distribution, see feigin [16]. maximum likelihood estimation in non-recurrent casewas studied in dietz and kutoyants [15]. local asymptotic mixed normality for discretely observednon-recurrent ornstein-uhlenbeck processes was studied in shimizu [30]. θt − θ := ∫ t 0 xtdwt∫ t 0 x2 t dt = zt it (1.4) where zt := ∫ t 0 xtdwt and it := ∫ t 0 x2 t dt. (1.5) hence eθt 2θ (θt − θ) = e−θt 2θzt e−2θt 4θ2it = ( e−2θt 4θ2 )1/2 zt e−2θt 4θ2it (1.6)in (1.6), the numerator of the normalized mle is a normalized martingale which converges tothe standard normal variable and the denominator is its corresponding increasing process whichconverges to a chi-square random variable as t → ∞ which is independent of the numerator.hence the ratio converges to a cauchy distribution with parameters (0, 1).let us introduce two wiener integrals: ξt := ∫ t 0 e−θsdws , and ηt := ∫ t 0 eθsdws , t ≥ 0. (1.7) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 3 and the respective limits ξ = limt→∞ ∫ t 0 e−θsdws := ∫∞ 0 e−θsdws which has n (0, 1 2θ ) distribu-tion and η := limt→∞ e −θtηt = limt→∞ e −θt ∫ t 0 eθsdws = ∫∞ 0 e−θsdws which has n (0, 1 2θ )distribution.with these notations zt := ∫ t 0 xtdwt = ∫ t 0 eθtξtdwt and it := ∫ t 0 x2 t dt = ∫ t 0 e2θtξ2 t dt, θt − θ = ∫ t 0 eθtξtdwt∫ t 0 e2θtξ2 t dt , (1.8) eθt 2θ (θt − θ) = ( e−2θt 4θ2 )1/2 ∫ t 0 eθtξtdwt e−2θt 4θ2 ∫ t 0 e2θtξ2 t dt = ( e−2θt 4θ2 )1/2 ∫ t 0 eθt( ∫ t 0 e −θsdws)dwt e−2θt 4θ2 ∫ t 0 e2θt( ∫ t 0 e −θsdws)2dt = ξt ξ 2θe−2θt ∫ t 0 e2θtξ2 t dt × e−θt ∫ t 0 eθtdwt ξ = ξt ξ 2θe−2θt it × e−θtηt ξ =: aθt × bθt . (1.9) we have aθt → 1 almost surely as t →∞, (1.10) bθt d→ n√ 2θξ as t →∞ (1.11) where √2θξ = n1, and n1 and n are independent standard normal random variables. since n√ 2θξ d = c(1) as t →∞ (1.12) where c(1) is the standard cauchy distribution, by slutsky’s theorem, we have aθt × bθt d→c(1) as t →∞. (1.13) note that ξt d→ ξ as t →∞. (1.14)using borel-cantelli lemma and stochastic fubini theorem, it can be shown that ξt → ξ almostsurely and in l2(ω) as t →∞. by integration by parts we have e−2θt it = e−2θt ∫ t 0 x2 s ds = e−2θt ∫ t 0 e2θsξ2 s ds = ξ2 t 2θ − e−2θt θ ∫ t 0 e2θsξsdξs − te−2θt θ = ξ2 t 2θ − e−2θt θ ∫ t 0 eθsξsdws − te−2θt θ . (1.15) this equality together with e (∫ t 0 eθsξsdws )2 = ∫ t 0 e2θse(ξ2 s )ds = 1 2θ ∫ t 0 e2θs(1− e−2θs)ds = e2θt − 1− 2θt 4θ2 (1.16) by the clt for stochastic integrals provides e−2θt ∫ t 0 x2 s ds d→ ξ2 2θ as t →∞ (1.17) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 4 i.e., e−2θt it d→ ξ2 2θ as t →∞.it can be shown that e−2θt it → ξ2 2θ almost surely as t →∞. (1.18)by itô formula, we have zt = ∫ t 0 xsdws = ∫ t 0 eθsξsdws = ∫ t 0 ξsdηs = ξtηt − t − ∫ t 0 ηsdξs = ξtηt − t − ∫ t 0 ηse −θsdws . (1.19) hence e−θtzt = e−θt ξtηt − e−θtt − e−θt ∫ t 0 ηse −θsdws . (1.20) direct calculation gives e−θtηt d→ η ∼ n ( 0, 1 2θ ) as t →∞ (1.21) and e(ξη) = lim t→∞ e(ξtηt e −θt ) = lim t→∞ te−θt = 0. (1.22)hence e−θtzt d→ ξη as t →∞. (1.23)hence the limit distribution of the pair (ξt , e −θtηt ) is a gaussian distribution of two indepen-dent variables. thus eθt ξt ηt d→ ζ as t →∞ (1.24) where ζ is the standard cauchy variable with probability density function f (x) = 1 π(1 + x2) , x ∈ r. (1.25) and cdf c(x) = 1 2 + 1 π arctan x, x ∈ r (1.26)and characteristic function ∫ ∞ −∞ e iλxdc(x) = e−|λ|. (1.27) hence eθt 2θ (θt − θ) d→ ζ. (1.28)we estimate the rate of convergence in this phenomenon. we need the following lemma in thesequel. lemma 1.1 (esseen’s smoothing lemma) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 5let f be a non-decreasing function and h be a differentiable function of bounded variation on thereal line with f (±∞) = g(±∞). denote the corresponding fourier-stieltjes transforms by f̂ and ĝ, respectively. then for all λ > 0, sup x∈r |f (x)− g(x)| ≤ 1 π ∫ λ −λ ∣∣∣f̂ (λ)− ĝ(λ) ∣∣∣ |λ| dλ+ 24 πλ sup x∈r |g′(x)|. proof: see petrov [27] or feller [18]. let φ(·) denote the standard normal distribution function and c(·) denotes the standard cauchydistribution function. throughout the paper c denotes a generic constant (perhaps depending on θ, but not on anything else).we need the following well known inequality. lemma 1.2 1√ 2π exp( −x2 2 )( 1 x − 1 x3 ) ≤ 1−φ(x) ≤ 1√ 2πx exp( −x2 2 )for x > 0. as x →∞, 1−φ(x)∼ 1√ 2πx exp( −x2 2 ). proof: see feller ( [17], p.166). https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 6 2. main results we start with the dambis-dubins-schwarz (dds) theorem, see protter [29]. since zt is a contin-uous time martingale, due to time change (skorohod embedding), zt = bit where b is a brownianmotion independent of w, we have e(exp(iuzt )) = e exp(− u2 2 it ), (2.1) e(exp(iue−θt √ 2θzt )) = e exp(− u2 2 e−2θt 2θit ), (2.2) θt − θ = bit it , (2.3) eθt 2θ (θt − θ) = e−θtbit e−2θt 2θit = ( e−2θt 4θ2 )1/2 bit e−2θt 4θ2it =: yt (2.4) where yt = e−θtbit e−2θt 2θit . (2.5) our main claim in the paper is to show that |e(e iuyt )− e−|u|| ≤ c|u|e−|u|/2e−θt . (2.6) this is done through several lemmas. once it is shown, let f (x) = p (yt ≤ x), (2.7) c(x) = 1 2 + 1 π arctan x, x ∈ r, (2.8) take λ = eθt . then sup x∈r |f (x)− c(x)| ≤ 1 π j + 24 πeθt sup c′(x) (2.9) where j := 1 π ∫ |λ|≤eθt ∣∣∣f̂ (λ)− ĉ(λ) ∣∣∣ |λ| dλ. (2.10) clearly sup c′(x) <∞and j ≤ c eθt ∫ |λ|≤eθt e−|λ|/2dλ ≤ c eθt ∫ ∞ −∞ e−|λ|/2dλ = o(e−θt ). (2.11) which would ultimately give sup x∈r |f (x)− c(x)| = o(e−θt ). (2.12) first we start with kolmogorov distance for wiener chaos and its relative: https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 7 lemma 2.1 we have the following rate of convergence for the double-stochastic integral or thesecond wiener chaos ∫ t0 eθt (∫ t 0 e −θsdws ) dwt : (a) sup x∈r ∣∣∣∣p {eθt (2θe−2θt ∫ t 0 eθt (∫ t 0 e−θsdws ) dwt ) ≤ x } −φ(x) ∣∣∣∣ ≤ ce−θt . (b) sup x∈r ∣∣∣∣p {eθt (2θe−2θt ∫ t 0 e2θtξ2 t dt − ξ2 ) ≤ x } −φ(x) ∣∣∣∣ ≤ ce−θt . proof. observe that zt = ∫ t 0 xtdwt = ∫ t 0 eθt (∫ t 0 e−θsdws ) dwt the integral ∫ t 0 eθt (∫ t 0 e−θsdws ) dwtis second wiener chaos. one can use the stein-malliavin method (see nourdin and peccati( [25], [26]) and estimate the kolmogorov distance for zt . however, part (a) follows as a conse-quence of lemma 2.4(c) below along with lemma 1.1 above. part (b) follows as a consequence oflemma 2.2 below along with lemma 1.1 above. note that ξ2 ∼ χ2 1. the next theorem gives an exponential estimate on the rate of convergenceto the chi-square distribution for energy it of the o-u process. theorem 2.1 sup x∈r ∣∣p {e−2θt 2θit ≤ x } − p { ξ2 ≤ x }∣∣ = o(e−θt ). the above theorem is a consequence of the following lemma and the esseen’s smoothing lemma 1.1 lemma 2.2 for |u| ≤ eθt ε, ε sufficiently small, we have∣∣∣∣∣e exp ( iue−2θt 2θit ) − 1 (1− 2iu) 1 2 ∣∣∣∣∣ ≤ c(|u|+ |u|3)e−θt . proof. from liptser and shiryayev [23], we have e exp ( iue−2θt 2θit ) = exp ( θt 2 )[ 2γ (γ − θ)e−γt + (γ + θ)eγt ]1/2 (2.12) where γ := ( θ2 − 2iue−2θt 2θ )1/2 . (2.13)the lemma is an easy consequence of this result. lemma 2.3 for every δ > 0, p {∣∣e−2θt 2θit − ξ2 ∣∣ ≥ δ} ≤ ce−2θt δ−2. https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 8 proof : it is clear that xt = ∫ t 0 e−θ(t−s)dws . (2.14) further, itô formula (see friedman [19]), we have∫ t 0 eθ(t−s)dws = wt − θ ∫ t 0 eθ(t−s)wsds, ξt = ∫ t 0 e−θsdws = e−θtwt − θ ∫ t 0 e−θsdws , ηt = ∫ t 0 eθsdws = eθtwt + θ ∫ t 0 eθsdws .note that e(x2 t ) = 1− e−2θt 2θ , e(x4 t ) = 3(1− e−2θt )2 4θ and e(it ) = 2θt − 1 + e−2θt 4θ2 . (2.15) by itô formula, we have it = x2 t 2θ − t 2θ − zt θ . (2.16)by chebyshev inequality, we have p {∣∣e−2θt 2θit − ξ2 ∣∣ ≥ δ} ≤ 1 δ2 e ∣∣e−2θt 2θit − ξ2 ∣∣2 = 1 δ2 e ∣∣∣∣e−2θt 2θ ∫ t 0 e2θtξ2 t dt − ξ2 ∣∣∣∣2 = 1 δ2 e ∣∣∣∣e−2θt 2θ ∫ t 0 e2θtξ2 t dt − ξ2 t + ξ2 t − ξ2 ∣∣∣∣2 ≤ 2 δ2 [ e|e−2θt 2θ ∫ t 0 e2θtξ2 t dt − ξ2 t |2 + e|ξ2 t − ξ2|2 ] ≤ 2 δ2 [ e|e−2θt 2θ ∫ t 0 e2θtξ2 t dt − ξ2 t |2 + e|ξt − ξ|2e|ξt + ξ|2 ] ≤ 2 δ2 [ e|e−2θt 2θ ∫ t 0 e2θtξ2 t dt − ξ2 t |2 + e−2θt √ 2θ ] ≤ ce−2θt δ−2 (2.17) since e|ξt + ξ|2 ≤ 2e|ξt |2 + 2e|ξ|2 <∞.since e(ξt − ξ)2 = ∫ ∞ t ∫ ∞ t e−θre−θs |r − s|−1drds = e−2θt √ 2θ (2.18) hence e(ξt − ξ)2 = e−2θt √ 2θ (2.19) gives the l2 convergence rate. recall that it = ∫ t 0 e2θtξ2 t dt, (2.20) e(ξt − ξ)2 → 0 as t →∞, (2.21) e(ξt − ξs)2 ≤ c(t − s). (2.22)we have e ( 2θe−2θt it − ξ2 )2 = e ( 2θe−2θt ∫ t 0 e2θtξ2 t dt − ξ2 )2 = e ( 2θ e2θt ∫ t 0 e2θtξ2 t dt − ξ2 )2 . (2.23) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 9further, by toeplitz’s lemma lim t→∞ 2θe−2θt ∫ t 0 e2θtξ2 t dt = lim t→∞ ξ2 t = ξ2 almost surely. (2.24) e(ξ2) <∞ which implies that p (ξ = 0) = 0. we have lim t→∞ [ 2θe−2θt ∫ t 0 e2θtξ2 t dt − ξ2 t ] = 0 almost surely. (2.25) because of the continuity of ξt , for every t ≥ 0,∫ t 0 e2θtξ2 t dt ≥ ∫ t t 2 e2θtξ2 t dt ≥ t 2 eθt ( inf t 2 <t<t ξ2 t ) almost surely. (2.26) furthermore the continuity of ξt , gives lim t→∞ ( inf t 2 <t<t ξ2 t ) = ξ2 almost surely. (2.27) lim t→∞ ∫ t 0 e2θtξ2 t dt =∞ almost surely. (2.28) by l’hopital rule, lim t→∞ ∫ t 0 e2θtξ2 t dt e2θt = lim t→∞ ξ2 t 2θ = ξ2 2θ almost surely. (2.29) θt − θ = ∫ t 0 eθtξtdwt∫ t 0 e2θtξ2 t dt = ξ2 t 2e−2θt ∫ t 0 e2θtξ2 t dt − θ. (2.30) θt − θ → 0 almost surely. (2.31) θ̂t − θ = ξ2 t − 2θe−2θt ∫ t 0 e2θtξ2 t dt 2e−2θt ∫ t 0 e2θtξ2 t dt . (2.32) it is easy to verify that e [ ξ2 t − 2θe−2θt ∫ t 0 e2θtξ2 t dt ]2 ≤ ce−2θt . (2.33) this completes the proof of the lemma. the following lemma (cameron-martin type theorem) gives the bound on the joint characteristicfunctions of the sufficient statistics defining the mle: lemma 2.4 (a) let φt (z1, z2) := e exp(z1it + z2x 2 t ), z1, z2 ∈ c. then φt (z1, z2) exists for |zi | ≤ δ, 1 = 1,2 for some δ > 0 and is given by φt (z1, z2) = exp ( θt 2 )[ 2γ (γ − θ + 2z2)e−γt + (γ + θ − 2z2)eγt ]1/2 where γ = (θ2 − 2z1)1/2 and we choose the principal branch of the square root. https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 10 (b) let ht,x := ( e−2θt 4θ2 )1/2 zt − ( e−2θt 4θ2it − ξ2 ) x. then for |x | ≤ 2(log e2θt )1/2 and for |u| ≤ εeθt , where ε is sufficiently small∣∣∣∣e exp(iuht,x)− exp(− u2 2 ) ∣∣∣∣ ≤ c exp( −|u| 2 )(|u|+ |u|3)e−θt . (c) for |u| ≤ ε1e θt , where ε1 is sufficiently small, we have as t →∞,∣∣∣∣e exp { iu ( e−θt 2θ ) zt } − exp(− u2 2 ) ∣∣∣∣ ≤ c exp(− |u| 2 )(|u|+ |u|3)e−θt . part (a) is from bishwal [5].we shall prove part (b) in details. proof of part (c) is very similar to part (b) and will be omitted. proof : by itô formula, zt = θit + x2 t 2 − t 2 .note that e exp(iuht,x) = e exp [ −iu ( e−2θt 4θ2 )1/2 zt − iu (( e−2θt 4θ2 ) it − ξ2 ) x ] = e exp [ −iu ( e−2θt 4θ2 )1/2 { θit + x2 t 2 − t 2 } − i t (( e−2θt 4θ2 ) it − ξ2 ) x ] = e exp(z1it + z2x 2 t + z3) = exp(z3)φt (z1, z2) where z1 = −iuθδt,x , z2 = − iu 2 ( e−2θt 4θ2 )1/2 , z3 = iut 2 δt,x , δt,x = ( e−2θt 4θ2 )1/2 + 2x t . note that (z1, z2) satisfies the conditions of (a) by choosing ε sufficiently small. let α1,t (u), α2,t (u), α3,t (u) and α4,t (u) be functions which are of the orders o(|u|e−θt/2), o(|u|2e−θt/2), o(|u|3e−3θt/2) and o(|u|3e−θt/2) respectively. note that for the given range of values of xand u, the conditions on zi for part (a) of lemma are satisfied. note also that z2 = α1,t (u).further, with βt (t) = 1 + iu δt,x θ + u2δ2 t,x 2θ2 , γ = (θ2 − 2z1)1/2 = θ [ 1− z1 θ2 − z2 1 2θ4 + z3 1 2θ8 + · · · ] = θ [ 1 + iu δt,x θ + u2δ2 t,x 2θ2 + iu3δ3 t,x 2θ3 + · · · ] = θ[1 + α1,t (u) + α2,t (u) + α3,t (u)] = θβt (u) + α3,t (u) = θ[1 + α1,t (u)]. thus γ − θ = α1,t , γ + θ = 2θ + α1,t . hence the above expectation equals exp ( z3 + θt 2 )[ 2θβt (u) + α3,t (u) α1,t exp{−θtβt (u) + α4,t (u)}+ (2θ + α1,t (u)) exp{θtβt (u) + α4,t (u)} ]1/2 = [ 1 + α1,t (u) α1,t exp(χt (u)) + (1 + α1,t (u)) exp(ψt (u)) ]1/2 https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 11where χt (u) := −θtβt (u) + α4,t (u)− 2z3 − θt = −2θt + α1,t (u) + t2α1,t (u), ψt (u) := θtβt (u) + α4,t (u)− 2z3 − θeθt = θt [ 1 + iu δt,x θ + u2δ2 t,x 2θ2 ] + α4,t (u)− i teθt δt,x − θeθt = u2eθt 2θ [( 4θ2 e2θt )1/2 + 2x eθt ]2 = u2 + u2α1,t (u). hence, for the given range of values of u, χt (u)−ψt (u) ≤ −θeθt . hence the above expectationequals exp(− t2 2 )(1 + α1,t )1/2 [ α1,t exp{−2θeθt + α1,t + u2α1,t }+ (1 + α1,t (u)) exp{t2α1,t (u)} ]−1/2 = exp(− u2 2 ) [ 1 + α1,t )(1 + α1,t (1 + α1,t ) exp{−θeθt + α1,t + t2α1,t } ] exp(u2α1,t (u)). lemma 2.4 (c) and lemma 2.2 respectively give the berry-esseen rate for zt and it immediatelyby using the esseen’s lemma 1.1. corollary 2.1 (a) sup xεr ∣∣∣∣∣p {( 4θ2 e2θt )1/2 zt ≤ x } −φ(x) ∣∣∣∣∣ ≤ ce−θt . (b) sup x∈r ∣∣∣∣∣p {( 4θ2 e2θt )1/2( θit − ξ2 e θt 2 ) ≤ x } −φ(x) ∣∣∣∣∣ ≤ ce−θt . remark though this was basically shown in lemma 2.1, here we obtain kolmogorov distance fora martingale and kolmogorov distance for its quadratic variation through cameron-matin typeresults which are generalization of levy area formula. in lemma 2.1, one could go directly to thestein-malliavin way through wiener chaos expansion which does not depend on any martingalecharacteristics.before we prove the results on the berry-esseen bound on the kolmogorov distance for themle with random norming we need the following large deviation result for the mle. this canbe obtained as a consequence of lemma 3.1 of bercu et al. [3] or bercu and richou [4] who usethe gartner-ellis’s theorem and the contraction principle. however we give a direct proof usingfeller’s approach. lemma 2.5 p {( e2θt 4θ2 )1/2 |θt − θ| ≥ 2(2θt )1/2 } ≤ ce−θt . proof : observe that p {( e2θt 4θ2 )1/2 |θt − θ| ≥ 2(2θt )1/2 } = p  ∣∣∣∣∣∣∣ ( 4θ2 e2θt )1/2 zt ( 2θ e2θt )it ∣∣∣∣∣∣∣ ≥ 2(2θt )1/2  https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 12 ≤ p {∣∣∣∣∣ ( 4θ2 e2θt )1/2 zt ∣∣∣∣∣ ≥ (θt )1/2 } + p {∣∣∣∣ 2θ e2θt it ∣∣∣∣ ≤ 1 2 } ≤ ∣∣∣∣∣p {( 4θ2 e2θt )1/2 |zt | ≥ (θt )1/2 } − 2φ(−(2θt )1/2) ∣∣∣∣∣+ 2φ(−(2θt )1/2) + p {∣∣∣∣ 2θ e2θt it − ξ2 ∣∣∣∣ ≥ 1 2 } ≤ sup x∈r ∣∣∣∣∣p {( 4θ2 e2θt )1/2 |zt | ≥ x } − 2φ(−x) ∣∣∣∣∣+ 2φ(−(2θt )1/2) + p {∣∣∣∣( 4θ2 e2θt ) it − ξ2 ∣∣∣∣ ≥ 1 2 } ≤ sup x∈r ∣∣∣∣∣p {( 4θ2 e2θt )1/2 |zt | ≥ x } − 2φ(−x) ∣∣∣∣∣+ 2φ(−(2θt )1/2) + p {∣∣∣∣( 4θ2 e2θt ) it − ξ2 ∣∣∣∣ ≥ 1 2 } ≤ ce−θt + c(e2θt 2θt )−1/2 + c(e2θt )−1 ≤ ce−θt . the bounds for the first and the third terms come from corollary 2.1 (a) and lemma 2.3 respectivelyand that for the middle term comes from feller ( [17], p. 166). we are now in a position to obtain the berry-esseen bound of the order o(e−θt ) on thekolmogorov distance for the mle. theorem 2.2 sup x∈r ∣∣∣∣p {(eθt2θ ) (θt − θ) ≤ x } − c(x) ∣∣∣∣ = o(e−θt ). proof : we shall consider two possibilities: (i) |x | > 2(θt )1/2 and (ii) |x | ≤ 2(θt )1/2.(i) we shall give a proof for the case x > 2(θt )1/2. the proof for the case x < −2(θt )1/2 runssimilarly. note that∣∣∣∣p {(eθt2θ ) (θt − θ) ≤ x } − c(x) ∣∣∣∣ ≤ p {(eθt2θ ) (θt − θ) ≥ x } + c(−x) but c(−x) ≤ c(−2(θt )1/2) ≤ ce−2θt . moreover by lemma 2.5, we have p {( eθt 2θ ) (θt − θ) ≥ 2(θt )1/2 } ≤ ce−θt/2. hence ∣∣∣∣∣p {( eθt 2θ )1/2 (θt − θ) ≤ x } − c(x) ∣∣∣∣∣ ≤ ce−θt/2. (ii) let at := {( eθt 2θ ) |θt − θ| ≤ 2(θt )1/2 } and bt := { it eθt > c0 } where 0 < c0 < 1 2θ . by lemma 2.5, we have p (act ) ≤ ce−θt . (2.34) by lemma 2.3, we have p (bct ) = p { 2θ eθt it − ξ2 < 2θc0 − ξ2 } < p {∣∣∣∣ 2θ eθt it − ξ2 ∣∣∣∣ > ξ2 − 2θc0 } ≤ ce−θt . (2.35) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 13let b0 be some positive number. for ω ∈ at ∩bt and for all t > t0 with 4b0(2θt0)1/2( 2θ eθt0 )1/2 ≤ c0, we have ( eθt 2θ ) (θt − θ) ≤ x ⇒ it + b0e θt (θt − θ) < it + ( eθt 2θ ) 2b0θx ⇒ ( eθt 2θ ) (θt − θ)[it + b0e θt (θt − θ)] < x [it + ( eθt 2θ ) 2b0θx ] ⇒ (θt − θ)it + b0t (θt − θ)2 < ( 2θ eθt ) it x + 2b0θx 2 ⇒ zt + (θt − θ)it + b0e θt (θt − θ)2 < zt + ( 2θ eθt ) it x + 2b0θx 2 ⇒ 0 < zt + ( 2θ eθt ) it x + 2b0θx 2 since it + b0e θt (θt − θ) > eθt c0 + b0e θt (θt − θ) > 4b0(θt )1/2 ( 2θ eθt ) − 2b0(θt )1/2( 2θ eθt ) = 2b0(θt )1/2 ( 2θ eθt ) > 0. hence, for ω ∈ at ∩ bt ,( eθt 2θ ) (θt − θ) ≤ x ⇒ zt + ( 2θ eθt ) it x + 2b0θx 2 > 0. on the other hand, for ω ∈ at ∩ bt and for all t > t0 with 4b0(2θt0)1/2( 2θ eθt0 ) ≤ c0, we have ( eθt 2θ ) (θt − θ) > x ⇒ it − b0e θt (θt − θ) < it − ( eθt 2θ ) 2b0θx ⇒ ( eθt 2θ ) (θt − θ)[it − b0e θt (θt − θ)] > x [it − ( eθt 2θ ) 2b0θx ] ⇒ (θt − θ)it − b0e θt (θt − θ)2 > ( 2θ eθt ) it x − 2b0θx 2 ⇒ zt + (θt − θ)it − b0e θt (θt − θ)2 > zt + ( 2θ eθt ) it x − 2b0θx 2 ⇒ 0 > zt + ( 2θ eθt ) it x − 2b0θx 2 since it − b0e θt (θt − θ) > eθt c0 − b0e θt (θt − θ) > 4b0(θt )1/2 ( 2θ eθt ) − 2b0(θt )1/2 ( 2θ eθt ) = 2b0(θt )1/2 ( 2θ eθt ) > 0. https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 14hence, for ω ∈ at ∩ bt , 0 < zt + ( 2θ eθt ) it x − 2b0θx 2 ⇒ ( eθt 2θ ) (θt − θ) ≤ x. we use the squeezing method developed in pfanzagl [28] for the i.i.d. case instead of the splittingmethod of michel and pfanzagl [28]. let us introduce the piecewise quadratic random functionsinvolving the martingale and quadratic variation part of θt − θ: g±(x) := zt + ( 2θ eθt )it x ± 2b0θx 2. let us introduce the events d±t,x := { zt + ( 2θ eθt )it x ± 2b0θx 2 > 0 } . thus we have d−t,x ∩ at ∩ bt ⊆ at ∩ bt ∩ {( eθt 2θ ) (θt − θ) ≤ x } ⊆ d+ t,x ∩ at ∩ bt . (2.36) this gives p (d−t,x ∩ at ∩ bt ) ≤ p ( at ∩ bt ∩ {( eθt 2θ ) (θt − θ) ≤ x }) ≤ p (d+ t,x ∩ at ∩ bt ) so that ∣∣∣∣p (at ∩ bt ∩{(eθt2θ ) (θt − θ) ≤ x }) − c(x) ∣∣∣∣ ≤ max { |p (d−t,x ∩ at ∩ bt )− c(x)|, |p (d+ t,x ∩ at ∩ bt )− c(x)| } ≤ max { |p (d−t,x)− c(x)|, |p (d+ t,x)− c(x)| } + p (at ∩ bt )c .from (2.34) and (2.35), p (at ∩ bt )c ≤ ce−θtfor all t > t0 and |x | ≤ 2(θt )1/2. if it is shown that∣∣p {d±t,x}− c(x) ∣∣ ≤ ce−θt (2.37) for all t > t0 and |x | ≤ 2(θt )1/2, then the theorem would follow from (2.34) – (2.37).we shall prove (2.37) for d+ t,x . the proof for d−t,x is analogous.note that ∣∣p {d+ t,x } − c(x) ∣∣ = ∣∣∣∣p {−( 2θ eθt )zt − ( 2θ eθt it − ξ2 ) x < x + 2 ( 2θ eθt ) b0θx 2 } − c(x) ∣∣∣∣ ≤ sup y∈r ∣∣∣∣p {−( 2θ eθt ) zt − ( 2θ eθt it − ξ2 ) x ≤ y } − c(y) ∣∣∣∣+ ∣∣∣∣c (x + ( 2θ eθt ) b0θx 2 ) − c(x) ∣∣∣∣ =: ∆1 + ∆2. (2.38)lemma 2.4 (b) and esseen’s smoothing lemma 1.1 immediately yield ∆1 ≤ ce−θt . (2.39) https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 15on the other hand, for all t > t0, ∆2 ≤ 2( 2θ eθt )b0θx 2(2π)−1/2 exp(−x2/2) where |x − x | ≤ 2 ( 2θ eθt ) b0θx 2. since |x | ≤ 2(θt )1/2, it follows that |x̄ | > |x |/2 for all t > t0 and consequently ∆2 ≤ 2 ( 2θ eθt ) b0θx 2(2π)−1/2x2 exp(−x2/8) ≤ ce−θt . (2.40) from (2.38) (2.40), we obtain ∣∣p {d+ t,x } − c(x) ∣∣ ≤ ce−θt . this completes the proof of the theorem. concluding remarks (1) the bound in theorem 2.2 is uniform over compact subsets of the parameter space θ.(2) the bound in theorem 2.2 is optimal and cannot be improved further.(3) note that in the critical case, i.e., when θ = 0, the mle has a distribution concentratedon a half line, precisely the distribution of the ratio of a noncentral chisquare to the to the sumof chisquares. note that the behaviour of the o-u process depends on both the initial condition x0 = x0 and the parameter space. classically it has been assumed that x0 is either has a normaldistribution or a nonzero constant and θ < 0 which makes the process stationary with gaussianinvariant distribution. if x0 = 0 is with θ < 0, then the process is asymptotically stationary andergodic. in above two cases the model satisfies the lan (local asymptotic normality) property.with x0 a nonzero constant and θ > 0 the process is transient and satisfies the lamn (localasymptotic mixed normality) property. with θ = 0, the process is nonstationary and satisfies thelabf (local asymptotic brownian functional) property. for all θ ∈ r, the model satisfies the labfproperty, see bishwal [9] for the definitions of these lan, lamn and labf properties. bishwal [9]has shown that sequential sampling based on a stopping rule unifies the three properties andmakes them lan.(4) it remains to study the kolmogorov distnace for bayes estimator from both continuous andsiscrete observations and approximate maximum likelihood estimator from discrete observations inthe nonergodic case.(5) extension to multidimensional process and to multiparameter case remains to be investigated.(6) it remains to investigate the nonuniform rates of convergence to cauchy distribution whichare more useful. https://doi.org/10.28924/ada/ma.3.25 eur. j. math. anal. 10.28924/ada/ma.3.25 16references [1] b. bercu, on large deviations in the 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https://doi.org/10.28924/ada/ma.3.25 references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 17doi: 10.28924/ada/ma.3.17 on certain properties of a degenerate sigmoid function thomas awinba akugre1,∗ , kwara nantomah2 , mohammed muniru iddrisu3 1department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana takugre.stu@cktutas.edu.gh 2department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana knantomah@cktutas.edu.gh 3department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana middrisu@cktutas.edu.gh ∗correspondence: takugre.stu@cktutas.edu.gh abstract. in this paper, we introduce a degenerate sigmoid function. by employing analytical tech-niques, we present some properties such as logarithmic concavity, monotonicity and inequalities ofthe new function. 1. introduction it is known that, what is currently referred to as the logistic equation or the s-shaped curve wasfirst introduced by verhulst (see [17]). it maps a very large input domain to a small range of outputof real numbers between 0 and 1. it is a one toone functioon and increases monotonically(see [8]). the sigmoid function, also known in the literature as the sigmoidal curve or standardlogistic function is defined as (see [13]), s (t) = et 1 + et = 1 1 + e−t , t ∈ (−∞,∞) , (1) = 1 2 + 1 2 tanh ( t 2 ) , t ∈ (−∞,∞) . (2) it has the following as its first and second derivatives received: 27 feb 2023. key words and phrases. degenerate sigmoid function; logarithmically concave; inequality.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.17 https://orcid.org/0009-0005-1387-377x https://orcid.org/0000-0003-0911-9537 https://orcid.org/0000-0001-7628-8168 eur. j. math. anal. 10.28924/ada/ma.3.17 2 s ′ (t) = et (1 + et)2 = s (t) (1− s (t)) , (3) s ′′ (t) = et ( 1− et ) (1 + et)3 = s (t) (1− s (t)) (1− 2s (t)) , (4) for all t ∈ (−∞,∞) .the sigmoid function is used in a wide range of scientific disciplines, including probability andstatistics, biology, demography, machine learning, population dynamics, ecology, and mathematicalpsychology(see [7], [16]). in the business sector, the sigmoid function has been utilized to analyzeperformance growth in manufacturing and service management (see [10]). at each neuron’s output,the function serves as an activation function in artificial neural networks (see [12], [18], [15]) andthe references therein.in addition, the function is used in medicine to research pharmacokinetic responses and mimictumor development (see [11]). in [5], the site index of unmanaged loblolly and slash pine plantationsin east texas is predicted using a generic variant of the sigmoid function. it is also used incomputer graphics and image processing to improve picture contrast (see [4], [9], [6]). it is clearfrom the above applications of the sigmoid function that, further research needs to be conductedon this very important function to unearth more of its properties and potential applications. recently, in [13], the author studied properties such as super multiplicativity, subadditivity,convexity and inequalities of the sigmoid function. in this paper, a degenerate sigmoid function is introduced and properties such as logarithmicconcavity, monotonicity and inequlities involving the function are provided. we start with thefollowing definitions and lemmas. 2. some definitions and lemmas definition 2.1. [1] a function m : (0,∞)×(0,∞)→ (0,∞) is called a mean function if it satisfiesthe following.(1) m (r, t) = m (t, r) ,(2) m (t, t) = t,(3) r < m (r, t) < t, for r < t,(4) m (ηr, ηt) = ηm (r, t) , for η > 0. there are many well-known mean functions in the literature. amongst them are the following.(1) arithmetic mean: a (r, t) = r+t 2 ,(2) geometric mean: g (r, t) = √ r t, https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 3(3) harmonic mean: h (r, t) = 1 a( 1r , 1 t ) = 2r t r+t ,(4) logarithmic mean: l (r, t) = r−t ln r−ln t , for r 6= t and l (t, t) = t,(5) identric mean: i (r, t) = 1 e ( r r tt ) 1 r−t , for r 6= t and i (t, t) = t. definition 2.2. [1] let g : i ⊆ (0,∞)→ (0,∞) be a continuous function and u and v be any twomean functions. then, g is said to be uv −convex (uv −concave) if g (u (r, t)) ≤ (≥) v (g (r) , g (t)) , for all r, t ∈ i. lemma 2.3. [1] let f : i ⊆ (0,∞)→ (0,∞) be a differentiable function. then(1) f is ag-convex(or concave) if and only if f ′ (t) f (t) is increasing(or decreasing) for all t ∈ i .(2) f is ah-convex( or concave) if and only if f ′ (t) f (t)2 is increasing(or decreasing) for all t ∈ i . lemma 2.4. [2] let f : i ⊆ (b,∞)→ (−∞,∞) with b ≥ 0. if the function defined by g (t) = f (t)−1 t is increasing on (b,∞) , then the function h (t) = f ( t2 ) satisfies the grumbaum-type inequality 1 + h ( z2 ) ≥ h ( r2 ) + h ( t2 ) , (5) where r, t ≥ b and z2 = r2 + t2. if g is decreasing, then the inequality (5) is reversed. 3. main results definition 3.1. the degenerate sigmoid function is defined for λ ∈ (0,∞) and t ∈ (−∞,∞) as sλ (t) = (1 + λt) 1 λ 1 + (1 + λt) 1 λ (6) = 1 1 + (1 + λt)− 1 λ (7) = 1 2 + 1 2 tanhλ ( t 2 ) . (8) it is clear that, taking the limit of sλ (t) as λ→ 0, then sλ (t)→ s (t) .the first derivative of the degenerate sigmoid function is given as s ′ λ (t) = (1 + λt) 1 λ −1[ 1 + (1 + λt) 1 λ ]2 > 0, (9) for all t ∈ (−∞,∞) and λ ∈ (0,∞) . the degenerate sigmoid function satisfies the following identities. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 4 sλ (t) + sλ (−t) = 1, (10) sλ (t)sλ (−t) = (1 + λt)s ′ λ (t) , (11) s ′ λ (t) = s ′ λ (−t) , (12) lim t→∞ sλ (t) = 1, (13) lim t→0 sλ (t) = 1 2 , (14) lim t→0 s ′ λ (t) = 1 4 , (15) lim t→∞ s ′ λ (t) = 0. (16) theorem 3.2. the function sλ (t) is ag-concave on (0,∞). in other words, for all r, t, λ ∈ (0,∞) , the inequality sλ ( r + t 2 ) ≥ [sλ (r)sλ (t)] 1 2 (17) is satisfied. proof. we have s ′ λ (t) sλ (t) =  (1 + λt) 1 λ −1[ 1 + (1 + λt) 1 λ ]2 (1 + (1 + λt) 1 λ (1 + λt) 1 λ ) = 1 (1 + λt) + (1 + λt) 1 λ +1 and ( s ′ λ (t) sλ (t) )′ = − λ+ (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, (18) which imlplies that s ′ λ(t) sλ(t) is decreasing on (0,∞). hence, by lemma 2.3(1), we obtain the desiredresult (17). � theorem 3.3. the function sλ (t) is ah-concave on (0,∞). in other words, for all r, t, λ ∈ (0,∞) , the inequality sλ ( r + t 2 ) ≥ 2sλ (r)sλ (t) sλ (r) + sλ (t) (19) is valid. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 5 proof. now we have s ′ λ (t) sλ (t)2 =  (1 + λt) 1 λ −1[ 1 + (1 + λt) 1 λ ]2   [ 1 + (1 + λt) 1 λ ]2 (1 + λt) 2 λ  = 1 (1 + λt) (1 + λt) 1 λ = 1 (1 + λt) 1 λ +1 and ( s ′ λ (t) sλ (t)2 )′ = − (1 + λ) (1 + λt) 1 λ (1 + λt) 2 λ +2 < 0. by lemma 2.3(2), we conclude that sλ (t) is ah-concave on (0,∞). this implies inequality(19). � theorem 3.4. the function sλ (t), for r, t, λ ∈ (0,∞) and z2 = r2+t2, satisfies the grunbaum-type inequality 1 + sλ ( z2 ) ≥ sλ ( r2 ) + sλ ( t2 ) . (20) proof. let h (t) be defined for t, λ ∈ (0,∞) as h (t) = sλ(t)−1 t . this implies h (t) = (1+λt) 1 λ 1+(1+λt) 1 λ − 1 t =− 1 t + t (1 + λt) 1 λ . differentiating h (t), we have h ′ (t) = 1 + (1 + λt) 1 λ + t (1 + λt) 1 λ −1[ t + t (1 + λt) 1 λ ]2 > 0, which implies that h (t) is increasing. by applying lemma 2.4, we obtain the desired result (20). � theorem 3.5. for λ ∈ (0,∞) , the function sλ (t) satisfies the inequalities s2λ (r + t) ≥ sλ (r)sλ (t) , r, t ∈ [0,∞) (21) and s2λ (r + t) ≤ sλ (r)sλ (t) , r, t ∈ (−∞, 0] . (22) equality holds if r = t = 0. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 6 proof. let r, t ∈ [0,∞) and λ ∈ (0,∞). recall that sλ (t) is increasing. thus we have sλ (r + t) ≥ sλ (r) > 0, (23) sλ (r + t) ≥ sλ (t) > 0, (24) since r + t ≥ r and r + t ≥ t. now by multiplying (23)and (24), we obtain the desired result (21).next, let r, t ∈ (−∞, 0] and λ ∈ (0,∞), we have 0 < sλ (r + t) ≤ sλ (r) , (25) 0 < sλ (r + t) ≤ sλ (t) , (26) since r + t ≤ r and r + t ≤ t. by multiplying the inequalities (25) and (26), we have the desiredresult. � theorem 3.6. the function sλ (t) , for λ ∈ (0,∞) , satisfies the inequalities s2λ (r t) ≤ sλ (r)sλ (t) , r, t ∈ [0, 1] (27) and s2λ (r t) ≥ sλ (r)sλ (t) , r, t ∈ [1,∞) . (28) equality holds if r = t = 1. proof. let r, t ∈ [0, 1] and λ ∈ (0,∞). recall that sλ (t) is increasing. thus we have 0 < sλ (r t) ≤ sλ (r) , (29) 0 < sλ (r t) ≤ sλ (t) , (30) since r t ≤ r and r t ≤ t. now by multiplying (29)and (30), we obtain the result (27).next, let r, t ∈ [1,∞, ) and λ ∈ (0,∞), we have sλ (r t) ≥ sλ (r) > 0, (31) sλ (r t) ≥ sλ (t) > 0, (32) since r t ≥ r and r t ≥ t. by multiplying the inequalities (31) and (32), the desired result isobtained (28). � theorem 3.7. for r, t ∈ (−∞,∞) and λ ∈ (0,∞), the function sλ (t) is logarithmically concave. in other words, the inequality sλ ( r a + t b ) ≥ [sλ (r)] 1 a [sλ (t)] 1 b (33) is satisfied. where a > 1 and 1a + 1 b = 1. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 7 proof. let q (t) = lnsλ (t) . then, q ′ (t) = s ′ λ (t) sλ (t) = (1+λt) 1 λ −1[ 1+(1+λt) 1 λ ]2 (1+λt) 1 λ 1+(1+λt) 1 λ =  (1 + λt) 1 λ (1 + λt) [ 1 + (1 + λt) 1 λ ]2 (1 + (1 + λt) 1 λ (1 + λt) 1 λ ) = 1 (1 + λt) + (1 + λt) 1 λ +1 . taking the second derivative of q (t) , we have q ′′ (t) =− λ+ (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, and this completes the proof. � corollary 3.8. for λ ∈ (0,∞) and t ∈ (−∞,∞) , the inequalities s ′′ λ (t)sλ (t) ≤ [ s ′ λ (t) ]2 (34) and sλ (1 + u)sλ (1− u) ≤ [ (1 + λ) 1 λ 1 + (1 + λ) 1 λ ]2 (35) are valid. proof. since sλ (t) is logarithmically concave, then [ln (sλ (t))] ′′ ≤ 0, for all t ∈ (−∞,∞) and λ ∈ (0,∞) . this implies that, [ln (sλ (t))] ′′ = [ s ′ λ (t) sλ (t) ]′ = s ′′ λ (t)sλ (t)− s′λ (t)s ′ λ (t) [sλ (t)]2 = s ′′ λ (t)sλ (t)− [ s ′ λ (t) ]2 [sλ (t)]2 ≤ 0. hence, s′′λ (t)sλ (t)− [ s ′ λ (t) ]2 ≤ 0, which yields equation (34). https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 8next, let a = b = 2, t = 1 + u and r = 1− u in equation (33). we have sλ ( 1 + u 2 + 1− u 2 ) ≥ [sλ (1 + u)] 1 2 [sλ (1− u)] 1 2 sλ (1) ≥ ([sλ (1 + u)] [sλ (1− u)]) 1 2[ (1 + λ) 1 λ 1 + (1 + λ) 1 λ ]2 ≥ sλ (1 + u)sλ (1− u) , resulting in equation (35). this concludes the proof. � theorem 3.9. for t, λ ∈ (0,∞) , the function sλ (t) satisfies the inequality 1 < sλ (t + 1) sλ (t) < 2 (1 + λ) 1 λ 1 + (1 + λ) 1 λ . (36) proof. recall from equation (18), that( s ′ λ (t) sλ (t) )′ = − λ+ (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, for all t, λ ∈ (0,∞) . this implies, the function s ′ λ(t) sλ(t) is decreasing on the given interval.now, let p (t) = sλ (t + 1) sλ (t) = ( [1 + λ (t + 1)] 1 λ 1 + [1 + λ (t + 1)] 1 λ )( 1 + (1 + λt) 1 λ (1 + λt) 1 λ ) = [1 + λ (t + 1)] 1 λ + (1 + λt) 1 λ [1 + λ (t + 1)] 1 λ (1 + λt) 1 λ + (1 + λt) 1 λ [1 + λ (t + 1)] 1 λ and ω (t) = lnp (t) = lnsλ (t + 1)− lnsλ (t) . then, ω ′ (t) = s ′ λ (t + 1) sλ (t + 1) − s ′ λ (t) sλ (t) < 0, since s ′ λ(t) sλ(t) is decreasing. this implies ω (t) and consequently p (t) are decreasing. hence, forall t, λ ∈ (0,∞) , we have 1 = lim t→∞ p (t) < p (t) < lim t→0 p (t) = 2 (1 + λ) 1 λ 1 + (1 + λ) 1 λ , which yields the desired result (36). � https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 94. conclusion we have introduced a degenerate sigmoid function. properties such as concavity, monotonicity 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https://doi.org/10.1007/bf02309004 https://doi.org/10.1007/bf02309004 https://doi.org/10.5772/intechopen.80416 https://doi.org/10.5772/intechopen.80416 1. introduction 2. some definitions and lemmas 3. main results 4. conclusion 5. conflicts of interest references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 68-85doi: 10.28924/ada/ma.1.68 unified convergence analysis of two-step iterative methods for solving equations ioannis k. argyros department of mathematical sciences, cameron university, lawton, ok 73505, usa correspondence: iargyros@cameron.edu abstract. in this paper we consider unified convergence analysis of two-step iterative methods forsolving equations in the banach space setting. the convergence order four was shown using taylorexpansions requiring the existence of the fifth derivative not on this method. but these hypotheseslimit the utilization of it to functions which are at least five times differentiable although the methodmay converge. as far as we know no semi-local convergence has been given in this setting. ourgoal is to extend the applicability of this method in both the local and semi-local convergence caseand in the more general setting of banach space valued operators. moreover, we use our idea ofrecurrent functions and conditions only on the first derivative and divided differences which appearon the method. this idea can be used to extend other high convergence multipoint and multistepmethods. numerical experiments testing the convergence criteria complement this study. 1. introduction we consider the problem of approximating a solution x∗ of equation f (x) = 0, (1.1) where f : ω ⊂ b −→ b1 is a continuous operator acting between banach spaces b and b1 with ω 6= ∅. since a closed form solution is not possible in general, iterative methods are used forsolving (1.1). many iterative methods are studied for approximating x∗. in this paper, we considerthe iterative methods, defined for n = 0, 1, 2, . . . , by yn = xn − f ′(xn)−1f (xn) xn+1 = yn − anf ′(xn)−1f (yn), (1.2) an = a(xn, yn), a : ω×ω −→ l(b,b1), where a−1 ∈ l(b1, b). many methods are special casesof (1.2). for example: received: 31 aug 2021. key words and phrases. iterative methods; banach space; convergence criterion; continuous functions.68 https://adac.ee https://doi.org/10.28924/ada/ma.1.68 eur. j. math. anal. 1 (2021) 69 traub [35] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − f ′(xn)−1f (yn), (1.3) newton [6] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − f ′(yn)−1f (yn), (1.4) ostrowski [25] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − (2[xn, yn;f ]− f ′(xn))−1f (yn), (1.5) kung-traub [35–37] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − [xn, yn;f ]−1f ′(xn)[xn, yn;f ]−1f (yn), (1.6) ostrowski-type [25] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − (2[xn, yn;f ]−1 − f ′(xn)−1)f (yn), (1.7) sharma type [32] yn = xn − f ′(xn)−1f (xn) xn+1 = yn − p (xn, yn)f ′(xn)−1f (yn). (1.8) to obtain all these special cases choose, an = i, an = f ′(yn)−1f ′(xn), an = (2[xn, yn;f ] − f ′(xn))f ′(xn), an = [xn, yn;f ]−1f ′(xn)[xn, yn;f ]−1f ′(xn), an = (2[xn, yn;f ]−1−f ′(xn)−1)f ′(xn), an = p (xn, yn), respectively, where [., .;f ] : ω × ω −→ l(b,b1) is a divided difference of orderone and p : ω×ω −→ l(b,b1) is weight operator [32] (see also [15,28,40] and reference therein).these special methods were shown to be of order four using taylor expansion and assumptions onthe fifth order derivative of f, which is not on these methods . so, the assumptions on the fifthderivative reduce the applicability of these methods [1–41].for example: let b = b1 = r, ω = [−0.5, 1.5]. define λ on ω by λ(t) = { t3 log t2 + t5 − t4 i f t 6= 0 0 i f t = 0.then, we get t∗ = 1, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. eur. j. math. anal. 1 (2021) 70obviously λ′′′(t) is not bounded on ω. so, the convergence of method (1.2) is not guaranteed bythe previous analyses in [1–41].in this paper we introduce a majorant sequence and use our idea of recurrent functions to extendthe applicability of method (1.2). our analysis includes error bounds and results on uniqueness of x∗ based on computable lipschitz constants not given before in [1–41] and in other similar studiesusing taylor series. our idea is very general. so, it applies on other methods too.the rest of the paper is set up as follows: in section 2 we present results on majorizing sequences.sections 3,4 contain the semi-local and local convergence, respectively, where in section 4 thenumerical experiments are presented. concluding remarks are given in the last section 5. 2. results on majorizing sequences we recall the definition followed by convergence results. definition 2.1. let {w̄n} be a sequence in a banach space. then, a nondecreasing scalar sequence {wn} is called majorizing for {w̄n} if ‖w̄n+1 − w̄n‖ ≤ wn+1 − wn for each n = 0, 1, 2, . . . . (2.1) sequence {wn} is used instead to study the convergence of {w̄n} [23–25]. set m = [0,∞).let η > 0, p0 : m −→ r, p : m −→ r, a : m × m × m −→ r, ā : m × m × m −→ rand b : m ×m ×m ×m −→ r be continuous and nondecreasing functions. set an = a(n) and ξn = b(n). define scalar sequences {sn}, {tn} for each n = 0, 1, 2, . . . by t0 = 0, s0 = η, tn+1 = sn + ᾱn(sn − tn) sn+1 = tn+1 + βn(tn+1 − sn), (2.2) where ᾱn = ān ∫ 1 0 p̄ ((1− θ)(sn − tn))dθ and βn = ξn 1− p0(tn+1) , ān = { ā, i f n = 0 a, i f n = 1, 2, . . . , p̄ = { p0, i f n = 0 p, i f n = 1, 2, . . .next, we present results on the convergence of sequence {sn}, {tn}. lemma 2.2. suppose that there exists µ > 0 such that for each n = 0, 1, 2, . . . , tn ≤ µ (2.3) and p0(µ) < 1. (2.4) eur. j. math. anal. 1 (2021) 71 then, sequences {sn}, {tn} converge to their unique least upper bound t∗ ∈ [η, µ] and tn ≤ sn ≤ tn+1. proof. it follows from (2.2)-(2.4) that these sequences are nondecreasing, bounded from aboveby µ, and as such they converge to t∗. � lemma 2.3. if function p0 is increasing then conditions (2.3) and (2.4) can be replaced by tn ≤ p−10 (1). (2.5) proof. set µ = p−10 (1) in lemma 2.2. � remark 2.4. conditions (2.3)-(2.5) are very general and can be verified only in special cases. that is why we present stronger conditions that are easier to verify. define functions f and g on the interval [0, 1) by f (t) = a( η 1− t , η 1− t , t 2η) ∫ 1 0 p ((1− θ)t2η)dθ − t and g(t) = b( η 1− t , η 1− t , t 2η, t3η) + tp0( η 1− t )− t. suppose that these functions have minimal zeros λf and λg in (0, 1), respectively. set λ = min{λf , λg} and λ0 = max{α0, β0}. then, we can show the third result on majorizing sequencefor method (1.2). lemma 2.5. suppose that µ0 ≤ λ0 ≤ λ. (2.6) then, sequences {sn}, {tn} are nondecreasing, bounded from above by t∗∗ = η 1−λ , and converge to t∗ ∈ [0, t∗∗]. moreover, the following estimates hold for each n = 1, 2, . . . 0 ≤ sn − tn ≤ λ(tn − sn−1) ≤ λ2nη, (2.7) 0 ≤ tn+1 − sn ≤ λ(sn − tn) ≤ λ2n+1η, (2.8) 0 ≤ sn ≤ 1− λ2n+1 1− λ η (2.9) and 0 ≤ tn+1 ≤ 1− λ2n+1 1− λ η. (2.10) eur. j. math. anal. 1 (2021) 72 proof. estimates (2.7)-(2.10) hold if 0 ≤ αm ≤ λ, (2.11) 0 ≤ βm ≤ λ, (2.12)and tm ≤ sm ≤ tm+1, (2.13)are true for m = 0, 1, 2, . . . . these estimates hold for m = 0 by (2.6). we suppose that (2.11)-(2.13) are true for m = 1, 2, . . . n. by induction hypotheses, (2.7) and (2.8), we have sm ≤ tm + λ2mη ≤ sm−1 + λ2m−1η + λ2mη ≤ η + λη + . . .+ λ2mη = 1− λ2m+1 1− λ η < η 1− λ = t∗∗, and tm+1 ≤ sm + λ2m+1η ≤ tm + λ2mη + λ2m+1η ≤ η + λη + . . .+ λ2m+1η = 1− λ2m+2 1− λ η < η 1− λ = t∗∗. therefore, by (2.13) and the induction hypotheses, we see that sequences {sm} and {tm} arenondecreasing. then, (2.11) shall be true if a(tm, sm, sm − tm) ∫ 1 0 ψ((1− θ)(sm − tm))dθ ≤ λ or a( 1− λ2m 1− λ η, 1− λ2m+1 1− λ η, λ2mη) ∫ 1 0 p ((1− θ)λ2mη)dθ ≤ λor a( η 1− λ, η 1− λ, λ 2η) ∫ 1 0 ψ((1− θ)λ2η)dθ ≤ λor f (λ) ≤ 0,which is true by the definition of λf and λ. similarly, (2.12) shall be true if b( 1− λ2m 1− λ η, 1− λ2m 1− λ η, λ2mη, λ2m+1η) +λp0( 1− λ2m+2 1− λ η) ≤ λ, or b( η 1− λ, η 1− λ, λ 2η, λ3η) + λp0( η 1− λ) ≤ λ eur. j. math. anal. 1 (2021) 73or g(λ) ≤ 0, which is also true by the definition of λg and λ. hence, we conclude (2.13) holds and limm−→∞ sm = limm−→∞ tm = t∗. � 3. semi-local convergence let u(x0, r) = {x ∈ b : ‖x − x0‖ < r, r > 0} and u[x0, r ] = {x ∈ b : ‖x − x0‖ ≤ r, r > 0}.we use some parameters and functions. consider m = [0,∞). suppose that there exists function p0 : m −→ m which is continuous and nondecreasing such that functions p0(t) − 1 = 0 has aminimal zero s ∈ (0,∞). set m0 = [0, s). suppose function p0 : m0 −→ m is continuous andnondecreasing. the following conditions (c) are needed:(c1) there exists x0 ∈ ω and η > 0 such that f ′(x0)−1 ∈ l(b1, b) and ‖f ′(x0)−1f (x0)‖ ≤ η. (c2) for each x ∈ ω ‖f ′(x0)−1(f ′(u)− f ′(x0))‖ ≤ p0(‖u − x0‖). set s0 = u(x0, s) ∩ω.(c3) for each x, y ∈ s0 the following hold ‖f ′(x0)−1(f ′(y)− f ′(x))‖ ≤ p (‖y − x‖) (c4) for each n = 0, 1, 2, . . . ‖anf ′(xn)−1f ′(x0)‖ ≤ an f ′(x0) −1([y , x ;f ]− f ′(x))‖ ≤ l2‖y − x‖and ‖f ′(x0)−1hn‖ ≤ ξn,where hn = f ′(x0) −1 ∫ 1 0 (f ′(yn + θ(xn+1 − yn))− f ′(xn)a−1n )dθ. (c5) conditions of lemma 2.2 or lemma 2.3 or lemma 2.5 hold.and(c6) u[x0, t ∗] ⊂ ω.then, we can show the semi-local convergence of method (1.2) using the conditions (c) and thepreceding notation. eur. j. math. anal. 1 (2021) 74 theorem 3.1. under the conditions (c), sequences {yn}, {xn} generated by method (1.2) are well defined in u[x0, t ∗], remain in u[x0, t ∗] for each n = 0, 1, 2, . . . and converge to a solution x∗ ∈ u[x0, t ∗] of equation f (x) = 0. moreover, the following error estimates hold for each n = 0, 1, 2, . . . ‖x∗ − xn‖ ≤ t∗ − tn. proof. we shall show items(pm) ‖ym − xm‖ ≤ sm − tm(qm) ‖xm+1 − ym‖ ≤ tm+1 − smusing mathematical induction on integer m. by the first substep of method (1.2) for n = 0 and (c1),we have ‖y0 − x0‖ = ‖f ′(x0)−1f (x0)‖ ≤ η = s0 − t0 = s0 ≤ t∗,so y0 ∈ u[x0, t ∗] and (p0) holds. we can write by the first sustep of method (1.2) that f (y0) = f (y0)− f (x0)− f ′(x0)(y0 − x0) = ∫ 1 0 (f ′(x0 + θ(y0 − x0))− f ′(x0))(y0 − x0)dθ, leading by (c2) and (p0) to ‖f ′(x0)−1f (y0)‖ ≤ ∫ 1 0 p0(θ‖y0 − x0‖)dθ‖y0 − x0‖ ≤ ∫ 1 0 p̄ (θ(s0 − t0))dθ(s0 − t0). (3.1) let z ∈ u(x0, t ∗). in view of (c2), we get ‖f ′(x0)−1(f ′(z)− f ′(x0))‖ ≤ p0(‖z − x0‖) ≤ p0(t ∗) < 1, (3.2) so ‖f ′(z)−1f ′(x0)‖ ≤ 1 1− p0(‖z − x0‖) (3.3) holds by a lemma on invertible linear operators due to banach [24] and (3.2). therefore, iterate x1is well defined and we can write in turn by (c3) and (3.3) (for z = x0, y0) ‖x1 − y0‖ = ‖a0f ′(x0)−1f (y0)‖ ≤ ‖a0f ′(x0)−1f ′(x0)‖‖ ∫ 1 0 f ′(x0) −1(f ′(x0 + θ(y0 − x0))− f ′(x0))dθ(y0 − x0)‖ ≤ a0 ∫ 1 0 p̄ ((1− θ)‖y0 − x0‖)dθ‖y0 − x0‖ 1− p0(‖x0 − x0‖) ≤ a0 ∫ 1 0 p̄ ((1− θ)(s0 − t0))dθ 1− p0(0) (s0 − t0) = t1 − s0, (3.4) eur. j. math. anal. 1 (2021) 75showing (q0). then, we have ‖x1 − x0‖ ≤ ‖x0 − y0‖+ ‖y0 − x0‖ ≤ t1 − s0 + s0 − t0 = t1 ≤ t∗, so x1 ∈ u[x0, t ∗]. moreover, we can write f (x1) = f (x1)− f (y0) + f (y0) = f (x1)− f (y0)− f ′(x0)a−10 (x1 − y0) = ∫ 1 0 (f ′(y0 + θ(x1 − x0))− f ′(x0)a−10 )dθ(x1 − y0) = h0(x1 − y0), (3.5) since by the second substep of method (1.2), we have f (y0) = −f ′(x0)a−10 (x1− y0). by (c3), (3.4)and (3.5), we obtain ‖f ′(x0)−1f (x1)‖ ≤ ‖f ′(x0)−1h0‖‖x1 − y0‖ ≤ ξ0(t1 − s0), (3.6) so ‖y1 − x1‖ ≤ ‖f ′(x1)−1f ′(x0)‖‖f ′(x0)−1f (x1)‖ ≤ ξ0(t1 − s0) 1− p0(t1) = s1 − t1, (3.7) showing (p1) for m = 1. suppose (pm), (qm) hold ym and xm+1 ∈ u[x0, t ∗]. then, by repeatingthese computations with xm, ym, xm+1 replacing x0, y0, x1, respectively, we complete the induction.moreover, sequence {xm} is complete in a banach space, so it converges to some x∗ ∈ u[x0, t ∗].finally, by letting m −→∞ in the estimation ‖f ′(x0)−1f (xm+1)‖ ≤ ξm(tm+1 − sm) (3.8) and using the continuity of f, we conclude f (x∗) = 0. �next, we present a result for uniqueness of the solution x∗. proposition 3.2. suppose (a) x∗ is a solution of f (x) = 0 (b) there exists s̃ ≥ t∗ such that ∫ 1 0 p0((1− θ)s̃ + θt∗)dθ < 1. (3.9) set s1 = u[x0, s̃] ∩ω. then, the only solution of equation f (x) = 0 in the region s1 is x∗. eur. j. math. anal. 1 (2021) 76 proof. set t = ∫ 1 0 f ′(x̃ + θ(x∗ − x̃))dθ for some x̃ ∈ s1 with f (x̃) = 0. using (c2) and (3.9),we get ‖f ′(x0)−1(t − f ′(x0))‖ ≤ ∫ 1 0 p0(‖x̃ + θ(x∗ − x̃)− x0‖dθ ≤ ∫ 1 0 p0((1− θ)‖x̃ − x0‖+ θ‖x∗ − x0‖)dθ ≤ ∫ 1 0 p0((1− θ)s̃ + θt∗)dθ < 1, leading to x̃ = x∗, where we used the identity t (x∗ − x̃) = f (x∗) − f (x̃) = 0 − 0 = 0 and theinvertability of t. � remark 3.3. let us specialize operators an to see how sequences {sn}, {tn}, {an}, {ξn}, {αn} and {βn} are defined. choose the case of newton’s method (1.4). then, we have ‖anf ′(xn)−1f ′(x0)‖ = ‖f ′(yn)−1f ′(x0)‖ ≤ 1 1− p0(‖yn − x0‖) , and ‖f ′(x0)−1hn‖ = ‖ ∫ 1 0 f ′(x0) −1(f ′(yn + θ(xn+1 − yn))− f ′(xn)a−1n )dθ‖ = ‖ ∫ 1 0 f ′(x0) −1(f ′(yn + θ(xn+1 − yn))− f ′(yn))dθ‖ ≤ ∫ 1 0 p̄ (θ‖xn+1 − yn‖)dθ ≤ ∫ 1 0 p̄ (θ(tn+1 − sn))dθ, so we can choose an = 1 1− p0(sn) (3.10) and ξn = ∫ 1 0 p̄ (θ(tn+1 − sn))dθ. (3.11) in this case we can show another result on majorizing sequences which is weaker than lemma 2.5 for the interesting case p0(t) = l0t and p (t) = lt. we get in this special case that αn = l(sn − tn) 2(1− l0sn) (3.12) and βn = l(tn+1 − sn) 2(1− l0tn+1) . (3.13) eur. j. math. anal. 1 (2021) 77 define sequences of function {f (1)n }, {f (2)n } on the interval [0, 1) by f (1) n (t) = l 2 t2n−1η + l0(1 + t + . . .+ t2n)η − 1, f (2) n (t) = l 2 t2nη + l0(1 + t + . . .+ t2n+1)η − 1,and polynomial ϕ by ϕ(t) = l0t 3 + (l0 + l 2 )t2 − l 2 .notice that ϕ(0) = −l2 and ϕ(1) = 2l0. denote by ρ the smallest zero of polynomial ϕ in (0, 1)assured to exist by the intermediate value theorem. lemma 3.4. suppose that λ0 ≤ ρ < 1− l0η. (3.14) then, the conclusions of lemma 2.5 hold for sequences {sn}, {tn} with ρ replacing λ. proof. we must show this time 0 ≤ l(sm − tm) 2(1− l0sm) ≤ ρ, (3.15) 0 ≤ l(tm+1 − sm) 2(1− l0tm+1) ≤ ρ (3.16)and tm ≤ sm ≤ tm+1. (3.17)these estimates hold for m = 0 by (3.14) and the definition of these sequences. then, as in lemma2.5 we can show instead for (3.15) that l 2 ρ2mη + ρl0(1 + ρ+ . . .+ ρ2m)η − 1 ≤ 0. (3.18) this estimate motivates us to define recurrent functions f (1)m by f (1) m (t) = l 2 t2m−1η + l0(1 + t + . . .+ t2m)η − 1. (3.19) we shall find a relationship between recurrent functions f (1)m+1 and f (1)m . by definition (3.9), we havein turn that f (1) m+1(t) = l 2 t2m+1η + l0(1 + t + . . .+ t2m+2)η − 1 − l 2 t2m−1η − l0(1 + t + . . .+ t2m)η + 1 + f (1) m (t) = f (1) m (t) + ( l 2 t2 − l 2 + l0(t 2 + t3))t2m−1η = f (1) m (t) + p(t)t2m−1η. (3.20) in particular, we have fm+1(ρ) = fm(ρ), (3.21) eur. j. math. anal. 1 (2021) 78so evidently (3.8) holds if f (1) m (ρ) ≤ 0. (3.22)define f (1)∞ (t) = limm−→∞ f (1) m (t). then, we have f∞(t) = l0η 1− t − 1. (3.23) then, (3.22) holds if f∞(ρ) ≤ 0, (3.24)which is true by (3.14). similarly, (3.16) holds if l 2 ρ2m+1η + ρl0(1 + ρ+ . . .+ ρ2m+1)η − ρ ≤ 0 (3.25) or f (2) m (ρ) ≤ 0. (3.26)as in (3.20), we get in turn that f (2) m+1(t) = l 2 t2m+2η + l0(1 + t + . . .+ t2m+3)η − 1 − l 2 t2mη − l0(1 + t + . . .+ t2m+1)η + 1 + f (2) m (t) = f (2) m (t) + ϕ(t)t2mη. (3.27) define f (2)∞ (t) = lim (2) m−→∞(t). then, we get again f (2)∞ (t) = f (1)∞ (t), so f (2)∞ (ρ) ≤ 0,can be shown instead of (3.26). but this is true by (3.14). the induction for items (3.15)-(3.17) iscompleted. the rest of the proof follows as in lemma 2.2. 4. local convergence we shall introduce real parameters and functions to be used in the convergence analysis. set m = [0,∞).suppose function(i) ψ0(t)− 1 = 0 has a smallest zero r0 ∈ m − {0}, where function ψ0 : m −→ m is continuousand nondecreasing. set m0 = [0, r0).(ii) ψ1(t)−1 = 0, has a smallest zero r1 ∈ m0−{0}, where function ψ : m0 −→ m is continuousand nondecreasing and ψ1 : m0 −→ m is defined by ψ1(t) = ∫ 1 0 ψ((1− θ)t)dθ 1− ψ0(t) . eur. j. math. anal. 1 (2021) 79(iii) ψ0(ψ1(t)t)− 1 has a smallest zero r̄1 ∈ m0−{0}. ser r̄2 = min{r0, r̄1} and m1 = [0, r̄2).(iv) ψ2(t)− 1 = 0 has a smallest zero r2 ∈ m1 − {0}, where ψ2(t) = [ψ1(ψ1(t)t) + (ψ0(t) + h(t, ψ1(t)t)) ∫ 1 0 ω(θψ1(t)t)dθ (1− ψ0(t))(1− ψ0(ψ1(t)t)) ]ψ1(t), where ω : m1 −→ m and h : m ×m1 −→ m are continuous and nondecreasing. we shall showthat r = min{r1, r2}, (4.1)is a convergence radius for method (1.2). set m2 = [0, r). these definitions, imply that for each t ∈ m2 0 ≤ ψ0(t) < 1, (4.2) 0 ≤ ψ0(ψ1(t)t) < 1, (4.3)and 0 ≤ ψi(t) < 1, i = 1, 2. (4.4)the conditions (h) shall be used provided that x∗ is a simple solution of equation f (x) = 0.suppose:(h1) for each x ∈ ω ‖f ′(x∗)−1(f ′(x)− f ′(x∗))‖ ≤ ψ0(‖x − x0‖). set ω0 = u(x∗, r0) ∩ω.(h2) for each x, y ∈ ω0 ‖f ′(x∗)−1(f ′(y)− f ′(x))‖ ≤ ψ(‖y − x‖), ‖f ′(x∗)−1f ′(x)‖ ≤ ω(‖x − x∗‖),and ‖f ′(x∗)−1(f ′(x∗)− a(x, y))‖ ≤ h(‖x − x∗‖, ‖y − x∗‖).(h3) u[x∗, r] ⊂ ω.next, we show the local convergence of method (1.2) based on the preceding notation and conditions(h).. theorem 4.1. under conditions (h) further suppose that x0 ∈ u(x∗, r) − {x∗}. then, we conclude limn−→∞ xn = x∗. proof. let v ∈ u(x∗, r)− {x∗}. using (4.1), (4.2), and (h1) we obtain in turn that ‖f ′(x∗)−1(f ′(v)− f ′(x∗))‖ ≤ ψ0(‖v − x∗‖) ≤ ψ0(r) < 1, so ‖f ′(v)−1f ′(x∗)‖ ≤ 1 1− ψ0(‖v − x∗‖) . (4.5) eur. j. math. anal. 1 (2021) 80in particular, iterate is well defined for v = x0 and the first substep of method (1.2), from which wecan also write y0 − x∗ = x0 − x∗ − f ′(x0)−1f (x0) = (f ′(x0) −1f ′(x∗)) ×( ∫ 1 0 f ′(x∗)−1(f ′(x∗ + θ(x0 − x∗))− f ′(x0))dθ(x0 − x∗). (4.6) by (4.1), (4.4) (for i = 1), (4.5) (for v = x0), (4.6) and (h2), we get in turn that ‖y0 − x∗‖ ≤ ∫ 1 0 ψ̄((1− θ)‖x0 − x∗‖)dθ‖x0 − x∗‖ 1− ψ0(‖x0 − x∗‖) ≤ ‖x0 − x∗‖ < r, (4.7) so y0 ∈ u(x∗, r). we also have that (4.5) holds for v = y0, and iterate x1 is well defined fromwhich we can write in turn that x1 − x∗ = y0 − x∗ − f ′(y0)−1f (x0) +(f ′(y0) −1 − a0f ′(x0)−1)f (y0) = y0 − x∗ − f ′(y0)−1f (y0) + f ′(y0) −1(f ′(x0)− a0)f ′(x0)1f (y0). (4.8) in view of (4.1), (4.4) (for i = 2), (4.5)(for v = x0, y0), (4.7), (4.8) and (h2), we obtain in turn ‖x1 − x∗‖ ≤ [ψ1(ψ1(‖x0 − x∗‖)) + (ψ0(‖x0 − x∗‖) + h(‖x0 − x∗‖, ‖y0 − x∗‖)) ∫ 1 0 ω(θ‖y0 − x∗‖)dθ (1− ψ0(‖y0 − x∗‖))(1− ψ0(‖x0 − x∗‖)) ]‖y0 − x∗‖ ≤ ψ2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r, (4.9) so x1 ∈ u(x∗, r). simply, switch x0, y0, x1 by xm, ym, xm+1, respectively in the preceding calcula-tions to get ‖ym − x∗‖ ≤ ψ1(‖xm − x∗‖)‖xm − x∗‖ ≤ ‖xm − x∗‖ < r (4.10)and ‖xm+1 − x∗‖ ≤ ψ2(‖xm − x∗‖)‖xm − x∗‖ ≤ ‖xm − x∗‖. (4.11)then, by the estimation ‖xm+1 − x∗‖ ≤ d‖xm − x∗‖ < r, (4.12)where d = ψ2(‖x0 − x∗‖) ∈ [0, 1), we get limm−→∞ xm = x∗ and xm+1 ∈ u(x∗, r). �next, we present a uniqueness result. eur. j. math. anal. 1 (2021) 81 proposition 4.2. suppose: (i) there exists a simple solution x∗ of equation f (x) = 0 (ii) there exists r∗ ≥ r such that ∫ 1 0 ψ0(θr ∗)dθ < 1. (4.13) set ω2 = ω ∩ u[x∗, r∗]. then, the only solution of equation f (x) = 0 in the region ω2 is x∗. proof. consider x̃ ∈ ω1 with f (x̃) = 0. set t = ∫ 1 0 f ′(x∗+ θ(x̃ − x∗))dθ. then, using (h1) and(4.13), we get in turn that ‖f ′(x∗)−1(t − f ′(x∗))‖ ≤ ∫ 1 0 ψ0(θ‖x̃ − x∗‖dθ ≤ ∫ 1 0 ψ0(θr ∗)dθ < 1, so x̃ = x∗, follows by t−1 ∈ l(b1, b) and t (x̃ − x∗) = f (x̃)− f (x∗) = 0− 0 = 0. � 5. numerical experiments we provide some examples in this section. example 5.1. define function q(t) = ξ0t + ξ1 + ξ2 sin ξ3t, x0 = 0, where ξj , j = 0, 1, 2, 3 are parameters. choose p0(t) = l0t and p (t) = lt. notice that l0 and l are the center lipschitz and lipschitz constants, respectively. then, from the graph of q(t) clearly for ξ3 large and ξ2 small, l0l can be small (arbitrarily). notice that l0 l −→ 0. example 5.2. let b = b1 = c[0, 1] and ω = u[0, 1]. it is well known that the boundary value problem [16]. ς(0) = 0, (1) = 1, ς ′′ = −ς − σς2 can be given as a hammerstein-like nonlinear integral equation ς(s) = s + ∫ 1 0 q(s, t)(ς3(t) + σς2(t))dt where σ is a parameter. then, define f : ω −→ b1 by [f (x)](s) = x(s)− s − ∫ 1 0 q(s, t)(x3(t) + σx2(t))dt. eur. j. math. anal. 1 (2021) 82 choose ς0(s) = s and ω = u(ς0, ρ0). then, clearly u(ς0, ρ0) ⊂ u(0, ρ0 + 1), since ‖ς0‖ = 1. suppose 2σ < 5. then, conditions (a) are satisfied for l0 = 2σ + 3ρ0 + 6 8 , l = σ + 6ρ0 + 3 4 , and η = 1+σ 5−2σ . notice that l0 < l. in the last two examples we consider traub’s method (1.3). so, we take a(x, y) = i and h(s, t) = 0. example 5.3. consider the motion system g′1(v1) = ev1 , g′2(y) = (e − 1)v2 + 1, g′3(v3) = 1 with g1(0) = g2(0) = g3(0) = 0. let g = (g1, g2, g3). let b = b1 = r3,ω = ū(0, 1), x∗ = (0, 0, 0)t . define function g on ω for v = (v1, v2, v3) t by g(v) = (ev1 − 1, e − 1 2 v22 + v2, v3) t . then, we get g′(v) =  ex 0 0 0 (e − 1)v2 + 1 0 0 0 1  , so ψ0(t) = (e− 1)t, ψ(t) = e 1 e−1 t, ω(t) = e 1 e−1 and k = e is the lipschitz constant on ω and ρt is given in [29,35]. then, the radii: r1 = 0.3827 = ρa = 2 2(e − 1) + e 1 e−1 , r2 = 0.3061 = r, ρt = 2 3k = 0.2453. example 5.4. consider b = b1 = c[0, 1], ω = u(0, 1) and q : ω −→ b1 defined by q(ς)(x) = %(x)− 5 ∫ 1 0 xθς(θ)3dθ. (5.1) we obtain q′(ς(ξ))(x) = ξ(x)− 15 ∫ 1 0 xθς(θ)2ξ(θ)dθ, for each ξ ∈ d. then, since x∗ = 0, we set ψ0(t) = 7.5t, ψ(t) = 15t, ω(t) = 15 and k = 15. then, the radii: r1 = 0.0667 = ρa = 2 2(7.5) + 15 , r2 = 0.0290 = r, ρt = 2 3k = 0.0444. notice that in the last two examples ρa is the radius given by us in [1–7] and is the largest. eur. j. math. anal. 1 (2021) 836. conclusion we have provided sufficient convergence criterion for the semi-local and local convergence oftwo-step methods. upon specializing the parameters involved we show that although our majorizingsequence is more general than earlier ones: convergence criteria are weaker (i.e., the utility of themethods is extended); the upper error estimates are more accurate (i.e. at least as few iterates arerequired to achieve a predecided error tolerance) and we have an at least as large ball containingthe solution. these benefits are obtained without additional hypotheses. according to our newtechnique we locate a more accurate domain than before containing the iterates resulting to moreaccurate (at least as small) lipschitz condition.our theoretical results are further justified using numerical experiments. references [1] i.k. argyros, on the newton kantorovich hypothesis for solving equations, j. comput. math. 169 (2004), 315-332, https://doi.org/10.1016/j.cam.2004.01.029[2] i.k. argyros, computational theory of iterative methods. series: studies in computational mathematics, 15, editors:c.k. chui and l. wuytack, elsevier publ. co. new york, u.s.a, 2007.[3] i.k. argyros, convergence and applications of newton-type iterations, springer verlag, berlin, germany, (2008), https://doi.org/10.1007/978-0-387-72743-1.[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s method. j. complex. 28 (2012), 364–387, https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s method, appl. math. comput. 225 (2013),372-386, https://doi.org/10.1016/j.amc.2013.09.049.[6] i.k. argyros, a.a. magréñan, iterative methods and their dynamics with applications, crc press, new york, usa,2017.[7] i.k. argyros, a.a. magréñan, a contemporary study of iterative methods, elsevier (academic press),new york, 2018, https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/ magrenan/978-0-12-809214-9.[8] r. behl, p. maroju, e. martinez, s. singh, a study of the local convergence of a fifth order iterative method, indianj. pure appl. math. 51 (2020), 439-455, https://doi.org/10.1007/s13226-020-0409-5.[9] e. cătinaş, the inexact, inexact perturbed, and quasi-newton methods are equivalent models, math. comp.74 (2005), 291-301, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1& type=pdf.[10] x. chen, t. yamamoto, convergence domains of certain iterative methods for solving nonlinear equations, numer.funct. anal. optim. 10 (1989), 37-48, https://doi.org/10.1080/01630568908816289.[11] j.e. dennis jr., on newton-like methods. numer. math. 11 (1968), 324–330, https://doi.org/10.1007/ bf02166685.[12] j.e. dennis jr., r.b. schnabel, numerical methods for unconstrained optimization and nonlinear equations, siam,philadelphia, 1996. first published by prentice-hall, englewood cliffs, new jersey, (1983), https://epubs. siam.org/doi/pdf/10.1137/1.9781611971200.fm.[13] p. deuflhard, g. heindl, affine invariant convergence theorems for newton’s method and extensions to relatedmethods. siam j. numer. anal. 16 (1979), 1-10, https://doi.org/10.1137/0716001. https://doi.org/10.1016/j.cam.2004.01.029 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1016/j.jco.2011.12.003 https://doi.org/10.1016/j.amc.2013.09.049 https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/magrenan/978-0-12-809214-9 https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/magrenan/978-0-12-809214-9 https://doi.org/10.1007/s13226-020-0409-5 https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1&type=pdf https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1&type=pdf https://doi.org/10.1080/01630568908816289 https://doi.org/10.1007/bf02166685 https://doi.org/10.1007/bf02166685 https://epubs.siam.org/doi/pdf/10.1137/1.9781611971200.fm https://epubs.siam.org/doi/pdf/10.1137/1.9781611971200.fm https://doi.org/10.1137/0716001 eur. j. math. anal. 1 (2021) 84 [14] p. deuflhard, newton methods for nonlinear problems. affine invariance and adaptive algorithms, springer seriesin computational mathematics, 35, springer verlag, berlin. 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https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1017/s0008439500028125 https://doi.org/10.1017/s0008439500028125 https://arxiv.org/abs/quant-ph/0109113 https://eudml.org/doc/133212 https://doi.org/10.1155/2020/4984612 https://doi.org/10.1080/01630568708816254 1. introduction 2. results on majorizing sequences 3. semi-local convergence 4. local convergence 5. numerical experiments 6. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 151-163doi: 10.28924/ada/ma.1.151 some aspects of geometric constants in modular spaces zhijian yang, qi liu, muhammad sarfraz, yongjin li∗ department of mathematics, sun yat-sen university, guangzhou, 510275, p. r. china yangzhj55@mail2.sysu.edu.cn, liuq325@mail2.sysu.edu.cn, sarfraz@mail2.sysu.edu.cn, stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. in this paper, we generalize the typical geometric constants of banach spaces to modularspaces. we study the equivalence between the convexity of modular and normed spaces, and obtainthe relationship between ρ-neumann-jordan constant and ρ-james constant. in particular, we extendthe convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity andstrict convexity. 1. introduction in the recent years, the geometric theory of banach spaces has been fully developed, especiallythe geometric constant, which is a powerful tool to characterize the geometric properties of thespace sphere. as early as 1936, clarkson introduced the convexity modular of space [1]. in 1963,lindenstrauss introduced the smoothness modular, and obtained the close relationship betweenthe two constants [2]. in 1937, in order to better characterize jordan and von-nuemann’s famouswork in inner product spaces, clarkson defines the von-nuemann constant [3] which is the minimumconstant c for all x, y ∈ x and (x, y) 6= (0, 0) of the following equations: 1 c ≤ ‖x + y‖2 + ‖x − y‖2 2(‖x‖2 + ‖y‖2) ≤ c. in 1964, james introduced james constant [4] in order to study the normal structure of space.after the appearance of these constants, many scholars paid attention to them and obtained manywonderful properties [5].modular space problems have been considered by h. nakano, musielak and orlicz [6] underthe additional hypothesis of convexity or subadditivity of the modular ρ : x → [0,+∞). moreoverthe case of semi-ordered linear spaces and that of b-norms have been chiefly investigated. under received: 13 sep 2021. key words and phrases. banach spaces; geometric constants; modular spaces.151 https://adac.ee https://doi.org/10.28924/ada/ma.1.151 eur. j. math. anal. 1 (2021) 152weaker assumptions, they investigated the structure of the spaces under consideration. neitherconvexity nor subadditivity of the modular be assumed. in introducing the norm, a certain naturalconnection between the modular and the norm convergence will be required: norm convergenceshould imply modular convergence.through their researches, they found that although modular spaces are not generally normedspaces, they still have many wonderful properties, such as convergence, completeness, convexity andadditivity. in view of these properties, poom kumam extended jordan von-neumann constant andjames constant in banach spaces to modular spaces, and obtained uniform convexity and uniformnon-squareness of modular spaces [10].in this paper, based on the idea of generalizing geometric constants in banach spaces to modularspaces, we generalize the properties of von-neumann constant and james constant in [10]. bydefining convexity modules and smoothness modular, we derive the relationships between jamesconstant, convexity modular and the strict convexity of modular spaces. 2. preliminaries we first give some basic facts about modular spaces formulated by musielak and orlicz [6]. definition 1.[8] let x be a vector space over f (r or c). then a function ρ : x → [0,∞] is calleda modular on x if for arbitrary x, y in x ,(i) ρ(x) = 0 if and only if x = 0,(ii) ρ(αx) = ρ(x) for every scalar α with |α| = 1,(iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) if α+ β = 1 and α, β ≥ 0.if (iii) is replaced by (iv): ρ(αx +βy) ≤ αρ(x) + βρ(y) if α, β ≥ 0 and α+β = 1. we now callthat ρ is a convex modular.a modular ρ can be used to define a corresponding modular space, i.e, the vector space xρ asgiven by xρ = {x ∈ x : ρ(λx)→ 0 as λ→ 0}, where xρ is a linear subspace of x .in general, the modular ρ is not necessarily subadditive and therefore it does not behave as anorm or a distance. but we can associate it to a modular f -norm.the modular space xρ can be equipped with a f -norm defined by ‖x‖ρ = inf { α > 0; ρ( x λ ) ≤ α } , when ρ is convex. then norm ‖ · ‖ρ is frequently called the luxemburg norm. if ρ is convex, thenthe functional ‖x‖ρ = inf {α > 0; ρ( xλ) ≤ 1} is a norm in xρ which is equivalent to the f -norm ‖ · ‖ρ. eur. j. math. anal. 1 (2021) 153 proposition 1. let xρ be a modular space. then ρ is convex if and only if xρ is a normed spacewith ρ as norm. proof. the proof of sufficiency is obvious.conversely, assume ρ is convex, then we can obtain ρ(x) = 0 if and only if x = 0.(i) according to the definition 1, if α > 0, then ρ ( 1 α x ) = ρ ( 1 α x + 1− α α · 0 ) ≤ 1 α ρ(x) + 1− α α ρ(0) = 1 α ρ(x) and αρ ( 1 α x ) = αρ ( 1 α x ) + (1− α)ρ(0) ≥ ρ(α · 1 α + (1− α) · 0) = ρ(x). this show that ρ( 1αx) ≥ 1 αρ(x) and hence ρ( 1 α x ) = 1 α ρ(x) for α > 0.suppose α 6= 0, then |α| > 0. according to the definition 1, we have ρ ( |α| · 1 |α|αx ) = |α|ρ ( 1 |α|αx ) = |α|ρ(x) which shows that ρ(αx) = |α|ρ( 1|α|αx) = |α|ρ(x).(ii) since ρ(x + y) = ρ ( 2 (x 2 + y 2 )) = 2ρ (x 2 + y 2 ) ≤ ρ(x) + ρ(y),then xρ is a normed space with ρ as norm. 3. the ρ-neumann–jordan constant and the ρ-james constant in 2006, poom kumam [10] generalized two typical constants cnj(x) = sup{‖x + y‖2 + ‖x − y‖2 2‖x‖2 + 2‖y‖2 : x, y ∈ x, (x, y) 6= (0, 0) } and j(x) = sup{min{‖x + y‖, ‖x − y‖} : x, y ∈ x, ‖x‖ = ‖y‖ = 1}and introduced two new geometric constants cnj(xρ) and j(xρ) defined on modular spaces. definition 2.[10] the ρ-neumann-jordan constant cnj(xρ) of a modular space xρ is defined by cnj(xρ) = 2 sup { ρ2( x+y2 ) + ρ 2( x−y2 ) ρ2(x) + ρ2(y) : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1 } . definition 3.[10] the ρ-james constant j(xρ) of a modular space xρ is defined by j(xρ) = 2 sup { min{ρ( x + y 2 ), ρ( x − y 2 )} : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1 } . in the following section, we extend the proposition 3.5 in [10] and obtain inequalities of cnj(xρ)and j(xρ). theorem 1. let xρ be a modular space, then eur. j. math. anal. 1 (2021) 154(i) 0 < j (xρ) ≤ 4 and 1 ≤ cnj (xρ) ≤ 8, in particular, if ρ is convex, then 1 ≤ j (xρ) ≤ 2 and 1 ≤ cnj (xρ) ≤ 2;(ii)12j2(xρ) ≤ cnj(xρ) ≤ 64 j2(xρ) + 4, in particular, if ρ is convex, then 1 2j 2(xρ) ≤ cnj(xρ) ≤ 4 j2(xρ) + 1. proof. (i) let y = 0, then j(xρ) ≥ 2 sup{ρ( x 2 ) : x ∈ xρ, ρ(x) = 1}. since ρ(x) = 1, then ρ(x 2 ) > 0 implies j(xρ) > 0. since ρ( x±y2 ) ≤ ρ(x) + ρ(y) ≤ 2, then 0 < j(xρ) ≤ 4.let x = y , then cnj(xρ) ≥ 2 sup { ρ2( x+x2 ) + ρ 2( x−x2 ) ρ2(x) + ρ2(x) : x ∈ xρ, ρ(x) = 1 } ≥ 1. since ρ2(x + y 2 ) + ρ2( x − y 2 ) ≤ 2(1 + ρ(y))2, we have ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) ρ2(x) + ρ2(y) ≤ 2 ( 1 + 2ρ(y) 1 + ρ2(y) ) ≤ 4, thus 1 ≤ cnj(xρ) ≤ 8.in particular, if ρ is convex and let x = y , then j(xρ) ≥ 2sup{ρ(x 2 ) : x ∈ xρ, ρ(x) = 1} = 1. since ρ( x±y2 ) ≤ 1 2ρ(x) + 1 2ρ(y) ≤ 1, then 1 ≤ j(xρ) ≤ 2. we also can prove 1 ≤ cnj(xρ) ≤ 2by the same way.(ii) since ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) ≤ 2[1 + ρ(y)]2 ≤ 4 ( 1 + ρ2(y) ) , then ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) ρ2(x) + ρ2(y) − 2 ≤ 2(1 + ρ(y))2 1 + ρ2(y) − 2 = 4ρ(y) 1 + ρ2(y) . since 14(ρ2( x+y2 ) + ρ2( x−y2 )) ≤ 1 + ρ2(y), then 4ρ(y) 1 + ρ2(y) ≤ 16ρ(y) ρ2( x+y2 ) + ρ 2( x−y2 ) , that is ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) ρ2(x) + ρ2(y) − 2 ≤ 16ρ(y) ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) ≤ 16 ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) . finally 12cnj(xρ)− 2 ≤ 16 1 2 j2(xρ) implies that cnj(xρ) ≤ 64 j2(xρ) + 4. eur. j. math. anal. 1 (2021) 155according to the proof of proposition 3.5 in [12], we can prove 12j2(xρ) ≤ cnj(xρ), thus 1 2 j2(xρ) ≤ cnj(xρ) ≤ 64 j2(xρ) + 4. in particular, if ρ is convex, then ρ2( x + y 2 ) + ρ2( x − y 2 ) ≤ 1 2 (1 + ρ(y))2 ≤ 1 + ρ2(y), thus ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) ρ2(x) + ρ2(y) − 1 2 ≤ ρ(y) 1 + ρ2(y) ≤ 1 ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) . therefore cnj (xρ) ≤ 4 j2 (xρ) + 1. example 1. (i) consider x = r2, ρ(x) = { 0, x = 0 1 ‖x‖1 , x 6= 0 , where ‖x‖1 = ‖(x1, x2)‖1 = |x1| + |x2|.obviously, xρ is a modular space.we choose x0 = (1 2 , 1 2 ) , y0 = ( 1 2 ,− 1 2 ), then ρ (x0) = ρ (y0) = 1, ρ ( x0 + y0 2 ) = ρ ( x0 − y0 2 ) = 2, thus j (xρ) ≥ 4. since j (xρ) ≤ 4, then j (xρ) = 4. according to (ii) of theorem 1, we know that cnj(xρ) = 8 in this example.(ii) consider x = r2, ρ(x) = ‖x‖1. obviously, xρ is a modular space and ρ is convex. we have j (xρ) = sup {min{‖x + y‖1, ‖x − y‖1} : x, y ∈ xρ, ‖x‖ = 1, ‖y‖ ≤ 1}.we choose x0 = (1, 0), y0 = (0, 1), then ‖x0‖1 = ‖y0‖1 = 1 and ‖x0 + y0‖1 = ‖x0 − y0‖1 = 2,thus j (xρ) = 2. according to (ii) of theorem 1, we can get that cnj(xρ) = 2 in this example. 4. the ρ-convex modular and the ρ-smooth modular in order to study the uniform convexity of banach spaces, clarkson introduced the modular ofconvexity δx(ε) = inf{1− 1 2 ‖x + y‖ : ‖x‖ = ‖y‖ = 1, ‖x − y‖ ≥ ε } . goebel called ε0 = sup{ε ∈ [0, 2] : δx(ε) = 0} as the characteristic of convexity. based on thegeometric intuitionistic meaning of convexity of banach spaces and its application in fixed pointtheory, this paper gives the ρ-convex modular of modular spaces with reference to the definition of δx(ε). definition 4. the ρ-convex modular δxρ(ε) of a modular space xρ is defined by δxρ(ε) = inf { 1− ρ ( x+y 2 ) : x, y ∈ xρ, ρ(x), ρ(y) ≤ 1, ρ(x − y) ≥ ε } , 0 ≤ ε ≤ 2.in particular, if ρ is convex, the ρ-uniform convexity of xρ is defined as ε0 (xρ) = sup { ε ∈ [0, 2] : δxρ(ε) = 0 } . eur. j. math. anal. 1 (2021) 156 remark 1. we can easily prove that −1 ≤ δxρ(ε) ≤ 1 and δxρ(0) ≤ 0.in banach spaces, the convexity modular δx(ε) and the smoothness modular ρx(t) = sup{‖x + y‖+ ‖x − y‖ 2 − 1 : ‖x‖ = 1, ‖y‖ = 1, t ≥ 0 } are conjugate concepts. therefore, this paper gives the definition of ρ-smooth modular of modularspaces by referring to the definition of smoothness modular ρx(t). definition 5. the ρ-smooth modular ρxρ(t) of a modular space xρ is defined by ρxρ(t) = sup { ρ( x + y 2 ) + ρ( x − y 2 )− 1 : x, y ∈ xρ, ρ(x) ≤ 1, ρ(y) ≤ t } , t ≥ 0. remark 2. it is true that min{0, t − 1} ≤ ρxρ(t) ≤ 1 + 2t and ρxρ(t) is increasing of t . theorem 2. let x be a modular space, then(i) j(xρ) < 2ε if and only if δxρ(ε) > 1 − ε, in particular, if ρ is convex, then j(xρ) < ε if andonly if δxρ(ε) > 1− ε 2 ;(ii) j(xρ) = 2 sup{ε ∈ (0, 2) : δxρ(ε) ≤ 1−ε}, in particular, if ρ is convex, then j(xρ) = sup{ε ∈ (0, 2) : δxρ(ε) < 1− ε 2}. proof. (i) note α = j(xρ) < 2ε, thus min { ρ( x + y 2 ), ρ( x − y 2 ) } ≤ α 2 , shows that 1− ρ( x+y2 ) ≥ 1− α 2 > 1− ε. therefore δxρ(ε) > 1− ε.note β = δxρ(ε) > 1− ε, then 1− ρ( x+y2 ) ≥ β implies ρ( x+y2 ) ≤ 1− β < ε. thus min { ρ( x + y 2 ), ρ( x − y 2 ) } = ρ( x + y 2 ). then j (xρ) =2 sup { ρ ( x + y 2 ) : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1 } ≤ 2− 2β < 2ε.in particular, if ρ is convex and let λ = j(xρ) < ε, then j(xρ) < ε if and only if ∀x, y ∈ xρ, ρ(x), ρ(y) ≤ 1, we have ρ(x + y) ≤ λ or ρ(x − y) ≤ λ.according to the definition of δxρ(ε), we obtain ρ(x + y) ≥ ε > λ, thus ρ(x − y) ≤ λ shows that δxρ(ε) ≥ 1− α 2 > 1− ε 2 . (ii) note ε0 = sup{ε ∈ (0, 2) : δxρ(ε) ≤ 1− ε}.suppose ε0 < 2 . ∀ε ∈ (ε0, 2), for any x, y ∈ xρ and ρ(x), ρ(y) ≤ 1, we have ρ(x − y) > ε or ρ(x − y) ≤ ε. if ρ(x − y) > ε, then δxρ(ε) ≥ 1− ε implies ρ( x+y2 ) ≤ ε. thus j(xρ) ≤ 2ε. eur. j. math. anal. 1 (2021) 157since δxρ(ε) ≤ 1− ε, then j(xρ) ≤ 2ε0 shows that j(xρ) = 2 sup{ε ∈ (0, 2) : δxρ(ε) ≤ 1− ε}. in particular, if ρ is convex and let α = j(xρ) ∈ [1, 2], then ∀x, y ∈ xρ, ρ(x), ρ(y) ≤ 1, we have ρ(x + y) ≤ α or ρ(x − y) ≤ α. what’s more, ∀η > 0, there exist x ′, y ′ ∈ xρ and ρ(x ′), ρ(y ′) ≤ 1 such that ρ ( x ′ + y ′ ) > α− η and ρ (x ′ − y ′) > α− η. fix η > 0, then 1− ρ(x ′ + y ′ 2 ) < 1− α− η 2 implies δxρ(ε) < 1− α− η2 , therefore sup { ε ∈ (0, 2) : δxρ(ε) < 1− ε 2 } ≥ α− η. ∀ε ∈ (0, 2), if ε ≤ α, thus sup { ε ∈ (0, 2) : δxρ(ε) < 1− ε 2 } ≤ α. if ε > α, then ρ(x + y) ≤ α shows that δxρ(ε) ≥ 1− α 2 . in (0, 2), we know sup { ε ∈ (0, 2) : δxρ(ε) < 1− ε 2 } ≤ α, thus α− η ≤ sup{ε ∈ (0, 2) : δxρ(ε) < 1− ε 2} ≤ α.let η → 0, then sup { ε ∈ (0, 2) : δxρ(ε) < 1− ε 2 } = α. theorem 3. let xρ be a modular space, then(i) j(xρ) ≤ ρxρ(1) + 1;(ii) cnj(xρ) ≤ 2(√12 + (1 + ρxρ(1))2 − 2)2. proof. (i) we can deduce that j (xρ) ≤ sup { ρ ( x + y 2 ) + ρ ( x − y 2 ) : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1 } = ρxρ(1) + 1. (ii) we know that a2 + b2 ≤ (a + b)2 − 4(a + b) + 8 for 0 < a, b ≤ 2. thus ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) ≤ ( ρ ( x + y 2 ) + ρ ( x − y 2 ))2 − 4 ( ρ ( x + y 2 ) + ρ ( x − y 2 )) + 8. since ρ ( x + y 2 ) + ρ ( x − y 2 ) ≥ √ ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) , eur. j. math. anal. 1 (2021) 158then ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) + 4 √ ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) − 8 ≤ ( ρ ( x + y 2 ) + ρ ( x − y 2 ))2 ≤ ( 1 + ρxρ(1) )2 .thus ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) ≤ (√ 12 + ( 1 + ρxρ(1) )2 − 2)2 which shows that 12cnj(xρ) ≤ (√12 + (1 + ρxρ(1))2 − 2)2 . 5. convexity and non-squareness clarkson introduced uniform convexity in 1936, proved that lp(1 ≤ p <∞) spaces are uniformlyconvex banach spaces and uniformly convex banach spaces have radon-nikodym properties. dueto the geometrical intuitiveness of convexity, poom kumam [10] gave the definitions of ρr -uniformlyconvex, ρ-uniformly non-square and ρ-strictly convex of modular spaces in 2006.on the basis of literature [10], this paper studies the relationships between convexity, non-squareness and geometric constants of modular spaces. definition 6.[10] for r > 0, a modular space xρ is said to be ρr -uniformly convex if for each ε > 0,there exists δ > 0 such that for any x, y ∈ xρ, the conditions ρ(x) ≤ r , ρ(y) ≤ r and ρ(x − y) ≥ rεimply that ρ( x+y2 ) ≤ (1− δ)r . definition 7.[10] the modular space xρ is said to be ρ-uniformly non-square if there exists δ ∈ (0, 1)such that for any x, y ∈ xρ with ρ(x) = 1 and ρ(y) ≤ 1, ρ( x+y2 ) ≤ 1− δ or ρ( x−y2 ) ≤ 1− δ. definition 8.[10] the modular space xρ is said to be ρ-strictly convex if for any x, y ∈ xρ, theconditions ρ(x) ≤ 1, ρ(y) ≤ 1 and x 6= y imply that ρ( x+y2 ) < 1. theorem 4. let xρ be a modular space, then the following conditions are equivalent.(i) j(xρ) < 2;(ii) ε0(xρ) < 2 for all 0 < ε ≤ 2;(ii) xρ is ρ-uniformly non-square. proof. suppose j(xρ) < 2. there exists ε > 0, for any x, y ∈ xρ with ρ(x) = 1 and ρ(y) ≤ 1,such that ρ( x + y 2 ) ≤ j(xρ) 2 − ε < 1− ε or ρ(x − y 2 ) ≤ j(xρ) 2 − ε < 1− ε,implies xρ is ρ-uniformly non-square.suppose xρ is ρ-uniformly non-square, then we can prove j(xρ) < 2 by the same way. thus(i) and (iii) are equivalent.next, we know that ε0(xρ) < 2 if and only if δxρ(2) > 0. let α = δxρ(2), then ∀x, y ∈ xρ and ρ(x) = 1, ρ(y) ≤ 1, we can get ρ( x±y2 ) ≤ 1− α, thus xρ is ρ-uniformly non-square. thus eur. j. math. anal. 1 (2021) 159(ii) and (iii) are equivalent. remark 3. in fact, this theorem is a generalization of theorem 3.8 in [11]. theorem 5. let x be a modular space, then(i) x is ρ1-uniformly convex if and only if δxρ(ε) > 0 for 0 < ε ≤ 2;(ii) if δxρ(2) = 1, then xρ is ρ-strictly convex. proof. (i) denote δε = δxρ(ε). then δxρ(ε) > 0 if and only if ∀x, y ∈ xρ, ρ(x), ρ(y) ≤ 1 and ρ(x − y) ≥ ε, we have ρ( x+y2 ) ≤ 1− δε. thus xρ is ρ1-uniformly convex.(ii) since δxρ(2) = 1, then ∀x, y ∈ xρ, ρ(x), ρ(y) ≤ 1 and ρ(x − y) ≥ 2, we have ρ( x + y 2 ) = 0 < 1, implies xρ is ρ-strictly convex. 6. midpoint convexity in the following section, we discuss a special type of modular and study its properties in termsof geometric constants. definition 9.[6] let (x, ‖ · ‖) be a normed space and xρ be a modular space. then ρ is said to bestrongly midpoint convex with non-negtive constant c if ρ( x+y2 ) ≤ ρ(x)+ρ(y) 2 − c 4 ‖x − y‖ 2. theorem 6. let (x, ‖ · ‖) be a normed space and xρ be a modular space. if there exists c ≥ 0such that c‖x‖2 ≤ 1 2 ρ(x) for all x ∈ bxρand ρ is strongly midpoint convex with constant c, then cnj (xρ) ≤ 3. proof. since ρ ( x+y2 ) ≤ ρ(x)+ρ(y) 2 − c 4 ‖x − y‖ 2 and ρ ( x−y2 ) ≤ ρ(x)+ρ(−y) 2 − c 4 ‖x + y‖ 2, then ρ2 ( x + y 2 ) ≤ 1 4 (ρ(x) + ρ(y))2 − c 4 ‖x − y‖2(ρ(x) + ρ(y)) + c2 16 ‖x − y‖4 and ρ2 ( x − y 2 ) ≤ 1 4 (ρ(x) + ρ(y))2 − c 4 ‖x + y‖2(ρ(x) + ρ(y)) + c2 16 ‖x + y‖4 . therefore, for x ∈ sxρ and y ∈ bxρ , we have ρ2 ( x + y 2 ) + ρ2 ( x − y 2 ) ≤ 1 2 (1 + ρ(y))2 − c 4 (1 + ρ(y)) ( ‖x + y‖2 + ‖x − y‖2 ) + c2 16 ( ‖x + y‖4 + ‖x − y‖4 ) . next, we only need to prove ρ(y) + c2 16 ( ‖x + y‖4 + ‖x − y‖4 ) − c 4 ( ‖x + y‖2 + ‖x − y‖2 ) (1 + ρ(y))− 1− ρ2(y) ≤ 0. eur. j. math. anal. 1 (2021) 160 let t = ‖x + y‖2 + ‖x − y‖2, s = ‖x + y‖‖x − y‖ and i1 = ρ(y) + c2 16 ( t2 − 2s2 ) − c 4 t(1 + ρ(y))− 1− ρ2(y), then i1 ≤ t2 − 2s2 16 c2 − t 4 c = c ( t2 − 2s2 ) 16 ( c − 4t t2 − 2s2 ) . since c‖x‖2 ≤ 1 2ρ(x), then c‖ x+y2 ‖2 ≤ 1 2ρ ( x+y 2 ) ≤ 1 and c‖ x−y2 ‖2 ≤ 1 2ρ ( x−y 2 ) ≤ 1.therefore 4t t2 − 2s2 = ∥∥ x+y 2 ∥∥2 + ‖ x−y2 ‖2∥∥ x+y 2 ∥∥4 + ∥∥ x−y2 ∥∥4 ≥ 1 ‖ x+y2 ‖2+‖ x−y 2 ‖2 ≥ c, then i1 ≤ 0. example 3. if ρ is convex, then c = 0 which satisfies the condition of theorem 6, and cnj (xρ) ≤ 2 < 3. theorem 7. let (x, ‖·‖) be a normed space and xρ be a modular space. if there exist c, λ, µ, γ > 0such that 2µγ ≤ 1 ≤ 1 2λ + √ 6 8µ and µρ(x) ≤ c‖x‖2 ≤ λρ(x) for all x ∈ xρ.what’more, ρ is strongly midpoint convex with constant c, then cnj (xρ) ≤ 2. proof. by following the ideas in theorem 6, we can get ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) 1 + ρ2(y) ≤ 1 2 + 1 1 + ρ2(y) { ρ(y)− c 4 (1 + ρ(y)) ( | x + y ∥∥2+∥∥ x − y‖2)} + 1 1 + ρ2(y) { c2 16 ( ‖x + y ∥∥4+∥∥ x − y‖4)}. let t = ‖x + y‖2 + ‖x − y‖2, s = ‖x + y‖‖x − y‖ and i2 = t2 − 2s2 16 c2 − (1 + ρ(y))t 4 c − 1 2 (1− ρ(y))2, thus we only need to prove i2 ≤ 0. since ρ(y) ≤ 1, then (1 + ρ(y))t − √ (1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 ≤ t and (1 + ρ(y))t +√(1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 ≥ t +√3t2 − 4s2.thus t t2 − 2s2 = 1 4 · ‖ x+y2 ∥∥2 + ‖ x−y2 ‖2 ‖ x+y2 ‖4 + ∥∥ x−y 2 ∥∥4and t + √ 3t2 − 4s2 t2 − 2s2 = 1 4 · ‖x + y‖2 + ‖ x−y2 ‖ 2 + √ 3‖ x+y2 ‖4 + 3‖ x−y 2 ‖4 + 2‖ x+y 2 ‖2‖ x−y 2 ‖2 ‖ x+y2 ‖4 + ‖ x−y 2 ‖4 , eur. j. math. anal. 1 (2021) 161then t+ √ 3t2−4s2 t2−2s2 ≥ 1 4‖ x+y 2 ‖2+4‖ x−y 2 ‖2 + √ 3 4 √ ‖ x+y 2 ‖4+‖ x−y 2 ‖4 ≥ c λ(ρ( x+y2 )+ρ( x−y 2 )) + √ 3c 4µ √ ρ2( x+y2 )+ρ2( x−y 2 ) ≥ c 4λ + √ 3c 8 √ 2µ ≥ c 2 ,and t t2 − 2s2 ≤ 1 2 · 1∥∥ x+y 2 ∥∥2 + ∥∥ x−y2 ∥∥2 ≤ c 4µ ( ρ ( x+y 2 ) + ρ ( x−y 2 )) ≤ c 4µγ ≤ c 2 . therefore (1 + ρ(y))t − √ (1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 t2 − 2s2 ≤ c 2 ≤ (1 + ρ(y))t + √ (1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 t2 − 2s2thus i2 = t2 − 2s2 4 ( c 2 − (1 + ρ(y))t + √ (1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 t2 − 2s2 ) ( c 2 − (1 + ρ(y))t − √ (1 + ρ(y))2t2 + 2 (t2 − 2s2) (1− ρ(y))2 t2 − 2s2 ) ≤ 0. example 4. consider ρ(x) = 4c‖x‖2 and let λ = µ = 1 4 , then µ2ρ(x) ≤ c‖x‖2 ≤ λρ(x) and µ2 = λ 4 . what’more, cnj (xρ) = 2c sup { ‖x + y‖4 + ‖x − y‖4 ‖x + y‖2 + ‖x − y‖2 : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1 } ≤ 2c sup { ‖x + y ∥∥2+∥∥ x − y‖2 : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1} ≤ 4c sup{‖x‖2 + ‖y‖2 : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1} = sup{ρ(x) + ρ(y) : x, y ∈ xρ, ρ(x) = 1, ρ(y) ≤ 1} = 2. theorem 8. let (x, ‖ · ‖) be a normed space, xρ be a modular space and α0 ∈ (0, 2√2]. if thereexists c > 0 such that c ≥ 4α0√ ‖x0 + y0‖4 + ‖x0 − y0‖4 for some x0 ∈ sxρ , y0 ∈ bxρ and ρ is strongly midpoint convex with positive constant c, then cnj (xρ) ≥ α20. in particular, if α0 = 2 √ 2, then cnj (xρ) = 8. proof. since 2ρ(x) ≥ c‖x‖2, then ρ2 ( x±y2 ) ≥ c 2 ∥∥ x±y 2 ∥∥2. therefore ρ2 ( x+y 2 ) + ρ2 ( x−y 2 ) 1 + ρ2(y) ≥ c2 16 ( ‖x + y‖4 + ‖x − y‖4 ) 1 + ρ2(y) , eur. j. math. anal. 1 (2021) 162 shows that ρ2( x+y2 )+ρ 2( x−y2 ) 1+ρ2(y) ≥ α20 1+ρ2(y) , then cnj (xρ) ≥ α20. if α0 = 2√2, then cn (xρ) ≥ 8implies cnj (xρ) = 8. 7. data availability no data were used to support this study. 8. conflicts of interest the author(s) declare(s) that there is no conflict of interest regarding the publication of this paper. 9. funding statement this work was supported by the national natural science foundation of p. r. china (nos.11971493 and 12071491). references [1] j. lindenstrauss, on the modulus of smoothness and divergent series in banach spaces, michigan math. j. 10 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10.1007/bf02591317.[16] k. nikodem, z. pales, characterizations of inner product spaces by strongly convex functions, banach j. math. anal.5 (2011) 83–87. https://doi.org/10.15352/bjma/1313362982. https://doi.org/10.1007/bf02591317 https://doi.org/10.1007/bf02591317 https://doi.org/10.15352/bjma/1313362982 1. introduction 2. preliminaries 3. the -neumann–jordan constant and the -james constant 4. the -convex modular and the -smooth modular 5. convexity and non-squareness 6. midpoint convexity 7. data availability 8. conflicts of interest 9. funding statement references analytical approximations for the principal branch of the lambert w function roy m. howard school of electrical engineering, computing and mathematical sciences, curtin university, gpo box u1987, perth, 6845, australia abstract. a geometric based approach for specifying approximations to the lambert w function, which can achieve any set relative error bound over the interval , is detailed. approximations that can achieve arbitrarily high accuracy for the interval , based on a two point spline approximation, are specified. iterative methods can be used to improve the accuracy of the approximations. applications include, first, analytical expressions, with set relative error bounds, for the lambert w function over the interval . second, approximations, with an arbitrarily low relative error, for upper and lower bounds for the lambert w function. third, analytical expressions for the evaluation of and the integral of , for , without knowledge of . fourth, a direct approach for evaluating the lambert w function to achieve a prior set error constraint. 1. introduction the lambert w function is associated with lambert [19] and is a multivalued complex function with the single valued function, associated with the branch, being denoted . the principle branch of the lambert w function, , is the function defined by the inverse of for the case of , i.e. (1) the lambert w function does not have an explicit analytical form but is of importance consistent with increasing applications as detailed in the literature, e.g. [7], [3], [25], [9], [6], [14], [26], [20], [18], [4], and [11]. generalizations of the lambert w function are also of interest, e.g. [8] and [22]. efficient numerical methods for computing values of the lambert w function have long been known, e.g. [10], and with an advance detailed by fukushima [12]. the focus of this paper is on the principle branch and the real case which continues to receive research interest, e.g. [17]. the graph of the lambert w function, for this case, is shown in figure 1 and, for notational simplicity, is denoted in this paper. existing analytical approximations for the principle branch and real case of the lambert w function, e.g. [5], [3] and [17], are, in general, custom and cannot be directly generalized to obtain approximations of arbitrarily high accuracy. this paper provides a geometrical basis for defining such approximations. 0   1 e– 0  0  w y  w y  y 0   w y  kth wk w0 y f x  xe x = = re x  1– x w0 y  f 1– y .= = w correspondence: r.howard@curtin.edu.au https://adac.ee/ https://doi.org/10.28924/ada/ma.2.14 the advances in this paper are twofold. first, a systematic geometric approach for defining approximations to the lambert w function, over the interval , with quadratic convergence. convergence is proved. the approximations can be used to specify, with an arbitrarily low relative error, upper and lower bounds for the lambert w function. second, a systematic method for combining series expansions at , and the origin, to define arbitrarily accurate approximations for the interval . applications of the approximations include analytical expressions, with set relative error bounds, for the lambert w function over the interval , evaluation of and the integral of , for , without knowledge of , and a direct approach for evaluating the lambert w function to achieve a prior defined error. a review of published approximations for the lambert w function is provided in section 2. the proposed geometric approach for establishing approximations to the lambert w function is detailed in section 3. convergence is discussed in section 4 and in section 5 convergent two point spline based approximations, for the interval , are detailed. the use of iterative methods, to improve the accuracy of approximations, is discussed in section 6. applications are detailed in section 7 and conclusions are stated in section 8. 1.1. notation and properties. the notation of is used which is consistent with the principle value of the lambert w being the inverse function of , , . relevant properties of the lambert w function are detailed in appendix a. for an arbitrary function , defined over the interval , an approximating function has a relative error, at a point , defined according to . the relative error bound for the approximating function, over the interval , is defined according to (2) the notation is used. mathematica has been used to facilitate analysis and to obtain numerical results. in general, relative error results, associated with approximations to the lambert w function, have been obtained by sampling specified intervals, in either a linear or a logarithmic manner, as appropriate, with points. figure 1. graph of and its inverse which is the lambert w function for the principle branch and the real case. f x  xe x = w y  x y f x  xe x = 1– 1 e–  1 e– 1–  0   1 e– 1 e– 0  0  w y  w y  y 0   w y  1 e– 0  x w y = y f x  xe x = = x 1– y 1 e– f    fa x1 re x1  1 fa x1  f x1 –=    reb max re x1  : x1    .= f k  x  x k k d d f x = 1000 https://doi.org/10.28924/ada/ma.2.14 2. published approximations a taylor series expansion, at the origin, for the lambert w function is well known, e.g. [17], eqn. 2, and yields the following approximation which has a limited region of convergence: (3) the relationship implies (positive sign for ; negative sign for ) and, hence: (4) such approximations suggest the more general approximation for the lambert w function of (5) fritsch et al., [10], eqn. 6, utilizes an initial approximation of which is suitable for . 2.1. published approximations. the following is an overview of indicative published approximations for the lambert w function. additional useful references include [26] and [14]. boyd [5], eqn. 5-7, proposed the approximation (6) which is valid for . whilst the approximation is sharp at , it is not zero at the origin and, thus, is not sharp at this point. it has a relative error bound for the interval of . barry et al. [3], eqn. 12, proposed the approximation (7) for . this approximation was modified, [3], eqn. 15, to be valid for according to (8) t y  y y 2 – 3y 3 2 -------8y 4 3 --------– 125y 5 24 -------------54y 6 5 ----------16807y 7 720 --------------------+– 16384y 8 315 --------------------– + + += y 1 e ---. y xe x = x x ln+ y ln= x y 0 x y 0 x y y 1« y ln y 1.»    x w y  1 y+ ln = y 1– e ------. w y  y ln y e wbd y  1 2 1 ey+  10 ln 10 ln ln– ------------------------------------------------11 ey+ ln 11 ey+ ln ln–tanh+– = 1 1 10 -----1 ey+ ln 7 5 ---– 3– 40 -----1 ey+ ln 7 5 ---– 2 exp+ y 1 e– y 1 e–= 1   0.0499 wb1 y  6 5 --y 12 5 -----y 1 12y 5+ ln -----------------------------------ln ---------------------------------------------------------ln= y 0 y 1 e– wb2 y  1 +  6 5 --y 12 5 -----y 1 12y 5+ ln -----------------------------------ln ---------------------------------------------------------ln  2y 1 2y+ ln -------------------------ln –=  0.4586887.= https://doi.org/10.28924/ada/ma.2.14 the relative error bound, for , is . iacono and boyd, [17], eqn. 17, proposed the following approximation which is valid for (9) this approximation yields a relative error bound of for . an improved approximation, [17], eqn. 19, 20, is (10) which yields a relative error bound for of (the bound occurs at of the order of ) for the case of optimally chosen as . the approximation is sharp at . 2.1.1. padè approximations. padè approximates for the lambert w function for the interval have been proposed, e.g. [21], eqn. 34: (11) higher order padè approximates are detailed in fukushima [13]. 2.1.2. comparison of approximations. the relative errors in the above specified approximations are shown in figure 2 and figure 3. 2.2. classic iterative approximations. the classical iterative approximation for the lambert w function, e.g. [17], eqn. 8-10, is based on the fundamental relationship , , and an initial approximation as specified in y 0 1.96 10 3– y 1– e   wi1 y  1 y 1 1 y+ ln 2 ----------------------+ --------------------------------+ .ln= 3.53 10 2– y 0   wi2 y  1– a 1 b 1 ey++ 1 c 1 1 ey++ ln+ ---------------------------------------------------ln += c e 1 a 1– 2 a– 1 2 e1 a ln– ----------------------------------------= b 2 a ------c+ = y 0 4.53 10 3– y 10 12 a a 2.036= y 1 e–= y 1– e 1  wl y  1 123y 40 -----------21y 2 10 -----------+ + 1 143y 40 -----------713y 2 240 --------------+ + -----------------------------------------1 y+ .ln= figure 2. graph of the relative errors, over the interval , in published approximations to .1– e 1  w y  y re y  wi1 y  wi2 y  wb2 y  wbd y  wl y  wl y  wb2 y  wi1 y  x y ln x ln–= x y 0 x w0 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 equation 5. a first order approximation arises by substitution of this approximation into the expression to yield (12) iteration yields (13) in general: (14) 2.2.1. comparison of approximations. the relative error in the iterative approximations, of orders zero to three, are detailed in figure 4 and figure 5. the relative errors decrease, for large values of , at an increasing rate as the order of approximation is increased. the approximations are poor for which is consistent with the assumptions made in the iteration. 2.2.2. alternative iterative approximations. an alternative iterative approach is to utilize the relationship and solve for the error given an initial approximation figure 3. graph of the relative errors in published approximations to .w y  y re y  wi1 y  wi2 y  wbd y  wl y  wb2 y  wl y  x ln w1 y  y ln 1 y+ ln ln– y 1 y+ ln ---------------------.ln= = w2 y  y ln y ln 1 y+ ln ln–ln– y y 1 y+ ln ----------------------ln ---------------------------------ln .= = wi y  y ln wi 1– y  ln– = i 1 2 3     w0 y  1 y+ .ln= y y 10 figure 4. graph of the relative errors, over the interval , in iterative approximations to . the relative errors in the approximations for and are high. 1– e 1  w y  w2 w3 y wi3 y  re y  w0 y  w1 y  w2 y  x y ln x ln–= https://doi.org/10.28924/ada/ma.2.14 of . for fixed, consider an initial approximation of , with an error of , which implies and, thus, e.g. [17], eqn. 11: (15) for the case where the error is small and , a first order taylor series for the logarithm function yields (16) and the first order approximation (17) the general iteration formula, e.g. [17], eqn. 12 (18) then follows. higher order iteration, based on a higher order approximation for , is detailed in [10]. with a starting value, utilized in fritsch et al. [10], of (suitable for ) it follows that a first order approximation is (19) the relative error in this approximation is shown in figure 5 and the relative error bound for the interval is . a first order iteration, based on equation 18, with , yields the approximation [17], eqn. 18: figure 5. graph of the relative errors in iterative approximations to .w y  y w0 y  w2 y  w3 y  w1 y  wi3 y  re y  wf y  wf y  x0 y x0 0 y x0 0+  x0 0+ exp= x0 0+ y ln x0 0+  ln– y x0 -----ln 1 0 x0 -----+ .ln–= = 0 x0 1« x0 0+ y x0 -----ln 0 x0 -----–  0 x0 1 x0+ -------------y x0 -----ln x0–  x1 x0 0+ x0 1 x0+ -------------1 y x0 -----ln+ .= xi 1+ xi 1 xi+ ------------1 y xi ----ln+ = 1 0 x0+ ln x0 y ln= y e wf y  y ln 1 y ln+ ---------------------1 y y ln -------------ln+ .= e   1.47 10 2– x0 y  1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln= https://doi.org/10.28924/ada/ma.2.14 (20) the relative error in this approximation is shown in figure 4 and figure 5. the relative error bound, over the interval , is . 2.3. approximations via newton-raphson iteration. consistent with the illustration shown in figure 6, a direct newton-raphson method for solving for , in the equation , is: (21) a first iteration, based on a known approximating function for , is, e.g. [17], eqn. 15: (22) halley’s method can similarly be utilized, e.g. [26]. with an initial value of , the first and second order approximations, respectively, are: (23) (24) 2.3.1. results. graphs of the variation of the relative error, with iteration level, are shown in figure 7 and figure 8 for the case of an initial approximation of . wi3 y  1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln 1 1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln+ --------------------------------------------------------------------1 y 1 y 1 0.5 1 y+ ln+ ---------------------------------------+ln ------------------------------------------------------------ln+ .= 0   2.16 10 4– x y xe x = xi xi 1– xi 1– e xi 1– y– e xi 1– xi 1– e xi 1–+ -----------------------------------------– xi 1– xi 1– ye xi 1–– – 1 xi 1–+ ----------------------------------.–= = g w w y  g y  g y  g y  exp y– g y  exp g y  g y  exp+ --------------------------------------------------------------------– g y  g y  y g– y  exp– 1 g y + ------------------------------------------------.–= figure 6. newton-raphson iteration for determining an approximation to the solution, denoted , of for fixed and based on an initial value of . xo y x x exp= y x0 x xo h x0  x0x1 h x  xe x y–= x2 h x1  x0 1 y+ ln= wn1 y  1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– = wn2 y  1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– –= 1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------– y 1 y+ -----------1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------exp– 1 1 y+ ln 1 y+ ln y 1 y+ – 1 1 y+ ln+ ----------------------------------------------------–+ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------. x0 1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 note, for higher order iteration, the relative error increases, from an increasingly low level, as increases. 3. geometric basis for iterative approximations to lambert w function 3.1. geometric basis. to establish a systematic, geometrically based, approach for establishing approximations to the lambert w function of arbitrarily high accuracy, consider a set value of . the lambert w function associated with , denoted , is such that and is the point defined by the intersection of the two curves and as illustrated in figure 9 for the case of . the geometry associated with this intersection of the two curves is the basis for an initial approximation and for the iterative approximations detailed in the following theorem. theorem 3.1. iterative approximations for lambert w function. for fixed, , approximations to the lambert w function can be iteratively defined according to (25) y figure 7. graph of the relative errors, in the zero to fourth order iterative approximations to , based on newton-raphson iteration with an initial approximation of . w y  wn0 y  1 y+ ln= y wn1 y  re y  wn2 y  wn4 y  wn3 y  wn0 y  figure 8. graph of the relative errors, in the zero to fifth order iterative approximations to , based on newton-raphson iteration with an initial approximation of . w y  wn0 y  1 y+ ln= y wn0 y  re y  wn3 y  wn4 y  wn5 y  y y xo xo ye xo– = x ye x– y 0 y y 1 e– wli wli 1– 1 wui 1– +  1 wli 1– + --------------------------------------------= i 1 2    wl0 y= wui y wli ---------ln = i 1 2    wu0 0.= https://doi.org/10.28924/ada/ma.2.14 proof. consider the geometry detailed in figure 9, for the case of , and an initial approximation to of which is based the intersection of the first order taylor series for at the origin, i.e. , and . an associated approximation, denoted , arises from the intersection of the level defined by and the curve . the solution is (26) second order approximations follow from the intersection of with a first order taylor series of , based on the point , i.e. , to yield (27) and by the intersection of this level with the curve to yield (28) the general iterative form: (29) then follows. whilst the geometry, and the defined approximations, are clearly defined for the case of , the approximations are also valid for as simulation results, shown in figure 10, demonstrate. 3.1.1. explicit approximations. approximations to the lambert w function, of orders one to five, are: (30) y 0 xo wl1 y 1 y+ = ye x– y yx– x wu1 wl1 ye x– figure 9. illustration of the geometry underpinning an iterative relationship to find an approximation to which is the solution of for fixed, .xo x y x– exp= y y 0 wl1 wl2 wl2 x wl1 xo y x wu1 ye x– xo wu2 y yx– wl1 x wu1 – wl1 – wu1 y wl1 ---------ln 1 y+ .ln= = x ye x– wu1 wl1 x wu1 – wl1 – wl2 wl1 1 wu1 +  1 wl1 + -----------------------------------= ye x– wu2 y wl2 ---------ln 1 2y+ 1 1 y+ ln+ ------------------------------.ln= = wli wli 1– 1 wui 1– +  1 wli 1– + --------------------------------------------= wui y wli ---------ln = i 2 3    y 0 y 1 e– 0  wl1 y  y 1 y+ ------------= wu1 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 (31) (32) (33) (34) (35) (36) where (37) wl2 y  y 1 1 y+ ln+  1 2y+ ---------------------------------------= wu2 y  1 2y+ 1 1 y+ ln+ -------------------------------ln= wl3 y  y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ + --------------------------------------------------------------------------------------------------= wu3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln= wl4 y  = y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ 1 4y y 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ + + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------wu4 y  = 1 4y y 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ + + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln wl5 y  yn5 y  d5 y  ----------------= wu5 y  d5 y  n5 y  -------------ln= n5 y  1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ = 1 1 4y y+ + 1 2y+ 1 1 y+ ln+ -------------------------------ln y 1 y+ ln 2 1 2y+ 1 1 y+ ln+ -------------------------------ln++ 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln+ https://doi.org/10.28924/ada/ma.2.14 (38) 3.1.2. notes. the approximation is the approximation stated in equation 5 and is the basis for the newton-raphson iteration leading to equation 23 and equation 24. consider the iteration formula specified in equation 18 ([17], eqn. 12). with a starting value of , it follows that a first order iteration leads to the approximation (39) which is as specified by equation 31. 3.1.3. results. the relative errors associated with the approximations, specified in theorem 3.1, are shown in figure 10 and figure 11, whilst the relative error bounds, for the interval , are detailed in table 1. note the quadratic convergence. table 1. relative error bounds, over the interval , for the iterative approximations to the lambert w function defined in theorem 3.1. iteration order: i relative error bound for relative error bound for 1 increasing 0.381 2 increasing 0.0569 3 4 5 6 d5 y  1 5y y 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ + += y 1 2y+ 1 1 y+ ln+ ------------------------------2 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ln + y 1 y+  3 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ + 1 2y+ 1 1 y+ ln+ ------------------------------2 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ln ln wu1 y  1 y+ ln= x0 y 1 y+ = w1 y  y 1 2y+ --------------1 1 y+ ln+  = wl2 y  0   0   wli wui 8.32 10 3– 1.33 10 3– 3.88 10 6– 7.23 10 7– 1.08 10 12– 2.15 10 13– 9.33 10 26– 1.90 10 26– https://doi.org/10.28924/ada/ma.2.14 3.2. alternative iterative formulas. the approximations and for the lambert w function, as specified in theorem 3.1, can be also be defined via iterative formulas based on numerator and denominator expressions. theorem 3.2. alternative iterative formulas for lambert w. the approximations and , , for the lambert w function can be specified according to (40) where (41) proof. the proof is detailed in appendix b. figure 10. graph of the relative errors in the iterative approximations to the lambert w function defined in theorem 3.1. y wl1 y  re y  wl2 y  wu1 y  wu2 y  wl3 y  wu3 y  wu4 y  figure 11. graph of the relative errors in the iterative approximations to the lambert w function defined in theorem 3.1. y wl1 y re y  wu1 y  wl2 y  wu2 y  wl3 y  wu3 y  wl4 y  wu4 y  wli wui wli wui i 1 2    wli y  ni y  di y  ------------= wui y  di y  ln ni y  y ------------ln– = i 1 2    ni y  ni 1– y  1 di 1– y  ln ni 1– y  y -------------------ln–+ = n0 y  y= d0 y  1= di y  ni 1– y  di 1– y .+= https://doi.org/10.28924/ada/ma.2.14 3.2.1. explicit formulas. explicit formulas, for the first to fourth order approximations, are: (42) (43) (44) (45) (46) 3.3. alternative geometrical approach. an alternative approach that leads to the approximations , , , as specified in theorem 3.1, is to utilize the transformation , , in the relationship which implies . the geometry underpinning the iteration that leads to the approximations is illustrated in figure 12. theorem 3.3. alternative geometrical iteration. for fixed, , iterative approximations for the lambert w function can be defined according to (47) and it is the case that as specified in theorem 3.1. proof. consider the case of fixed and the geometry illustrated in figure 12 which is based on affine approximations to find, iteratively, the solution to . the inin1 y  y= d1 y  1 y+ = n2 y  y 1 1 y+ ln+ = d2 y  1 2y+ = n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ = d3 y  1 3y y 1 y+ ln+ + = n4 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ = 1 1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–+ , d4 y  1 3y y 1 y+ ln+ + y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ .+= wu1 wu2  x z ln= x 0 z 1 y xe x = y z z ln= figure 12. illustration of the geometry underpinning the iterative relationship to find approximations to which is the solution of for fixed.zo z z ln y= y 1 z y zo z z ln z2z1 z 1– z1 z1 ln z z1–  z1 ln 1+ + y2 y1 z1 1 y+= y y 0 zi zi 1– zi 1– zi 1– ln y– zi 1– ln 1+ -----------------------------------------– = z1 1 y+ = xi zi ln = xi wui y = y zo y z z ln= https://doi.org/10.28924/ada/ma.2.14 tial value is established by the intersection of a first order taylor series for at the point , which is , and the level . the solution is . a first order taylor series for at this point is and the intersection of this approximation with the level is the point defined according to (48) iteration in this manner leads to the stated general iteration formulas. 3.3.1. explicit formulas. approximations, of orders one to three, are: (49) (50) (51) 3.4. iterative algorithm for (-1/e,0]. for the case of , the geometric approach, illustrated in figure 13, can be utilized to establish an algorithm for determining approximations to the lambert w function. theorem 3.4. iterative algorithm for (-1/e,0]. an iterative algorithm for defining approximations to , for , is: (52) proof. consider the illustration shown in figure 13. with an initial approximation for of , a first order taylor series for , based on the point , with , is . the intersection of this taylor series with , at the point , leads to (53) the value of associated with is . a first order taylor series approximation for at the point is and the intersection of this approximation with , at the point , leads to z z ln z 1= z 1– y z1 1 y+= z z ln z1 z1 ln z z1–  z1 ln 1+ + y z2 z2 z1 z1 z1 ln y– z1 ln 1+ -----------------------------.–= z1 y  1 y+ = x1 wu1 y  1 y+ ln = = z2 y  1 2y+ 1 1 y+ ln+ -------------------------------= x2 wu2 y  1 2y+ 1 1 y+ ln+ -------------------------------ln = = z3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------= x3 wu3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ ----------------------------------------------------------------------------------------------.ln= = y 1– e 0  x w y = y 1 e– 0  xi yi 1– 1 xi 1–+  1 yi 1–+ -------------------------------------= x0 0= y0 y= yi ye xi– .= xo x0 0= ye x– x0 y0  y0 y= y0 1 x x0– –  x x1 x1 y0 1 x0+  1 y0+ -------------------------.= ye x– x1 y1 ye x1– = ye x– x1 y1  y1 1 x x1– –  x x2 https://doi.org/10.28924/ada/ma.2.14 (54) the value of associated with this value is . iteration in this manner leads to the general formula as stated in the theorem. 3.4.1. explicit approximations. approximations, for orders one to four, are: (55) (56) 3.4.2. results. the relative error in the iterative approximations specified in theorem 3.4 are shown in figure 14. for orders two, and higher, the approximation are more accurate than the approximations detailed in theorem 3.1. 3.5. improved approximations for [0,∞). improved approximations, for the interval , can be established by using the iteration formula specified in theorem 3.3 and by using figure 13. illustration of the geometry underpinning the iterative relationship to find approximations to , the solution of , for the case of fixed and . xo x ye x– = y y 1 e– 0  x1 x y x x2 y1 ye x– 1= xo y1 1 x x1– –  ye x– y2 ye x– 2= y 1 x x0– –  x0 x2 y1 1 x1+  1 y1+ -------------------------.= ye x– y2 ye x2– = we1 y  y 1 y+ ------------= we2 y  y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------= we3 y  y y 2 3y+  1 y+ ey 1 y+  + 1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------= 0   (57)we4 y  y y 2 3 4y+  2y 1 y+ ey 1 y+  y 1 y+  y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ + + 1 y+  y 1 3y e y 1 y+  + +  1 y+  y e y 1 y+  +  ----------------------------------------------------exp 1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+  y y y 2 3y+  1 y+ ey 1 y+  +  1 y+  y e y 1 y+  +  y y 1 2y+  1 y+  y e y 1 y+  +  ----------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------exp+ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------= https://doi.org/10.28924/ada/ma.2.14 an initial approximation of rather than . the resulting approximations, of orders one to four, are: (58) (59) (60) (61) the improvement in the relative error bounds for the interval , for values of close to optimum, are detailed in table 2. for the case of a third order approximation, and for , the maximum relative error is . thus, the approximation (62) figure 14. graph of the relative error in approximations to , for the interval , based on the iterative algorithms specified in theorem 3.4 and theorem 3.1. w y  1 e– 0  y we2 y  re y  we4 y  we1 y  wl1 y  we3 y  wl2 y  wu1 y  wu2 y  wl3 y  wu3 y  wu4 y  wl4 y  z1 y  1 ky+= z1 y  1 y+= z1 y  1 ky+ = wk1 y  1 ky+ ln = z2 y  1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------= wk2 y  1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln = z3 y  1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------= wk3 y  z3 y  ln = z4 y  = 1 3 k+ y y 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln y 1 ky+ ln 2 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln++ + + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ 1 1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------ln+ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------wk4 y  z4 y  .ln= 0   k k 4465 10000 ---------------= 1.023 10 4– wk3 y  1 2 k+ y y 1 ky+ ln+ + 1 1 ky+ ln+  1 1 1 k+ y+ 1 1 ky+ ln+ ----------------------------------ln+ -----------------------------------------------------------------------------------------------------ln = k 4465 10000 ---------------= https://doi.org/10.28924/ada/ma.2.14 1 0.381 0.570 0.518 0.465 represents a good compromise between accuracy and complexity for the interval with a relative error bound of close to . 3.6. higher accuracy via iterative quadratic approximations. the iterative approximation detailed in theorem 3.1 can be improved upon by utilizing quadratic, rather than affine, approximations as illustrated in figure 15. theorem 3.5. iterative quadratic approximations for lambert w function. an iterative formula, based on quadratic approximations, for the lambert w function, and valid for , is (63) proof. the proof is detailed in appendix c. 3.6.1. explicit approximations. approximations, of orders one to three, are: (64) table 2. relative error bounds, over the interval , in the approximations , , to the lambert w function. iteration order: i relative error bound: k = 1 relative error bound: k = 0.4 relative error bound: k = 0.45 relative error bound: k = 0.5 2 0.0569 0.0335 0.0250 0.0184 3 4 5 6 0   10 4– 0   wki y  i 1 2  6    1.33 10 3– 1.87 10 4– 1.05 10 4– 1.50 10 4– 7.23 10 7– 6.46 10 9– 4.96 10 9– 9.97 10 9– 2.15 10 13– 7.98 10 18– 1.11 10 17– 4.43 10 17– 1.90 10 26– 1.24 10 35– 5.53 10 35– 8.78 10 34– figure 15. illustration of the geometry underpinning an iterative relationship, based on quadratic approximations, to find an approximation to which is the solution of for fixed, . xo x y x– exp= y y 0 wl1 wl2 wl2 x wl1 xo x wu1 ye x– xo wu2 y 1 x– x 2 2 -----+ wl1 x wu1 – wl1 x wu1 – 2 2 --------------------------wl1 +– y y 1 e– wli 1 wli 1– -------------1 wli 1– 1 wui 1– +  1 wli 1– 2 – 2wli 1– 1 wui 1– + +–+ = wui y wli ---------ln = wl1 y 1 y+ ------------= wu1 1 y+ .ln= wl1 y  y 1 y+ ------------= wu1 y  1 y+ ln= https://doi.org/10.28924/ada/ma.2.14 (65) (66) (67) 3.6.2. results. the relative errors in the approximations specified in theorem 3.5 are shown in figure 16 and figure 17. the relative error bounds, for the interval , are detailed in table 3 and the convergence is cubic in nature. the approximation has a relative error bound for the interval of . 4. convergence consider the case of , , and the error definitions (68) associated with the upper and lower approximations detailed in theorem 3.1 and as illustrated in figure 18. the following results hold: table 3. relative error bounds, over the interval , for the approximations to the lambert w function detailed in theorem 3.5. iteration order i maximum relative error in maximum relative error in 1 increasing 0.381 2 increasing 0.0122 3 4 5 wl2 y  1 y --1 2y y 1 y+ ln 1 4y 2y 2 2y 1 y+  1 y+ ln+ + +–+ += wu2 y  y 2 1 2y y 1 y+ ln 1 4y 2y 2 2y 1 y+  1 y+ ln+ + +–+ + -----------------------------------------------------------------------------------------------------------------------------------------------ln= wl3 y  1 1 2y y 1 y+ ln q2 y –+ + -----------------------------------------------------------------= y 1 2y y 1 y+ ln q2 y –+ + 1 y 2 1 2y y 1 y+ ln q2 y –+ + ------------------------------------------------------------------ln+ –+ y 2 1 2y y 1 y+ ln q2 y –+ + 2– + 2y 1 2y y 1 y+ ln q2 y –+ + 1 y 2 1 2y y 1 y+ ln q2 y –+ + ------------------------------------------------------------------ln+ q2 y  1 4y 2y 2 2y 1 y+  1 y+ ln+ + += wu3 y  y wl3 i  ---------------.ln= 0   wu3 0   1.26 10 6– 0   wli wui 1.02 10 5– 1.26 10 6– 1.66 10 17– 2.04 10 18– 7.88 10 53– 9.63 10 54– y 0 xo w y = li xo wli – = ui wui xo– = ith https://doi.org/10.28924/ada/ma.2.14 theorem 4.1. convergence of iterative algorithm. for the case of fixed and , the errors associated with the lower approximations , , detailed in theorem 3.1, are (69) figure 16. graph of the relative errors in approximations to as specified in theorem 3.5.w y  y wl2 y  re y  wl1 y  wu1 y  wu2 y  wu3 y  figure 17. graph of the relative errors in approximations to as specified in theorem 3.5. w y  y wl1 y re y  wl2 y  wu1 y  wu2 y  wl3 y  wu3 y  figure 18. error definitions associated with the upper and lower approximations to . ith xo w y = wli wli 1+ x xo y x wui ye x– xo wui 1+ uili y xo w y = wli i 1 2    li li 1– wli 1– wui 1– wli 1– –  1 wli 1– + -------------------------------------------------------– li 1– wli 1– li 1– ui 1– +  1 wli 1– + ---------------------------------------------------–= = l1 xo y 1 y+ ------------– .= https://doi.org/10.28924/ada/ma.2.14 the following results hold: first, the sequence is a monotonically increasing sequence, i.e. . second, the errors , , define a monotonically decreasing sequence, i.e. . third, convergence is guaranteed, i.e. , and . proof. the proof of these results is detailed in appendix d. 5. spline based approximation for [-1/e,0] the approximations, detailed above in theorem 3.1 and theorem 3.4, are sharp at the origin but not at the point . it is useful to have approximations that are sharp at both points and which converge throughout the interval , e. g. [3], eqn. 7 and [17], eqn. 20. the latter approximation is sharp at but not at the origin. one approach, with potential, is to utilize the two point spline approximation for a function as specified by howard, [16], eqn. 40. for an interval , the order approximation can be written in the form (see appendix e) (70) where (71) 5.1. spline based approximations for [-1/e,0]. the spline approximation detailed in equation 70 has the potential to provide approximations for the lambert w function in the interval with exact values at the end points of this interval. however, an initial problem is that the derivatives of the lambert w function are undefined at the point . this problem can be overcome by utilizing two suitable transformations. lemma 1. transformations. with , , the first transformation (72) with , and , , yields (73) (74) wli wli wli 1–  li i 1 2    li 1+ li lii  lim 0= wlii  lim xo= wuii  lim xo= 1 e– 1– e 0  1– e f    nth fn x   x– n 1+ an r x – r r 0= n  x – n 1+ bn r  x– r r 0= n + = x   , an r 1  – n 1+ ---------------------------f r u–    r u– ! ----------------------n u+ ! u!n! ------------------- u 0= r  1  – u --------------------  = bn r 1  – n 1+ ---------------------------1– r u– f r u–    r u– ! -------------------------------------------n u+ ! u!n! ------------------- u 0= r  1  – u --------------------. = 1 e– 0  1 e– f x  xe x = x 1– y1 g1 x1  1 e --f x1 1– + == x x1 1–= x1 0 y y1 1 e ---–= y 1– e ------ y1 0 g1 x1  1 e --1 x1 1– e x1+  = x1 0  w y  f 1– y  g1 1– y 1 e ---+ 1– = y 1– e ------.= https://doi.org/10.28924/ada/ma.2.14 with the second transformation of , it follows that (75) (76) proof. the proofs for these results are detailed in appendix f. 5.1.1. graphs and values. consistent with equation 76, the lambert w function is defined in terms of for . relevant values associated with the points and are specified in table 4. the graph of , and its inverse , are shown in figure 19. 5.1.2. spline based approximations. consistent with equation 76, and the values tabulated in table 4, an approximation for the lambert , over the interval , requires an approximation to the inverse of , , i.e. , , to be determined. a spline approximation for , of order , requires derivatives, of orders zero to , at the points and to be determined. such values can be determined from the derivatives of at the points and as detailed in appendix g. the following approximations result. theorem 5.1. spline based approximations for the lambert w function. the order spline based approximation for the lambert w function, based on the transformation and the points and , is (77) table 4. values associated with the points and . g x1  g1 x1 = g x1  1 e --1 x1 1– e x1+  = x1 0  w y  g 1– y 1 e ---+ 1– = y 1 e ---– . g 1– y 1 e– 0  y 1 e–= y 0= g x1  g 1– y2  figure 19. graph of and its inverse . y2 g x1 = g 1– y2  x1 y2 g 1– y2  g x1  1 e y 1 e–= y 0= y y1 y 1 e+= x w y = x1 x 1+= y2 g x1 = 1– e 0 1– 0 0 0 1 e 0 1 1 e w 1 e– 0  g x1  x1 0 1  g 1– y2  y2 0 1 e  g 1– n n 0 1 e g 0 1 kth g 1 e–  0 w k y  1– 2e y 1 e ---+ 1 1 y 1 e ---+ 2 y 1 e ---+ 3 y 1 e ---+ 3 2 + + + + 2k y 1 e ---+ k ++ = https://doi.org/10.28924/ada/ma.2.14 for and for appropriately defined constants. proof. the proof is detailed in appendix g. 5.1.3. explicit approximations. explicit approximations, of orders one to four, are: (78) (79) (80) (81) k 1 2    w 1 y  1– 2e y 1 e ---+ 1 2 e 1 1 2e ---------3 2 2 ----------–+ y 1 e ---+– e 1 2– 2 e -------+ y 1 e ---+++= w 2 y  1– 2e y 1 e ---+ 1 2e 3 ---------y 1 e ---+– 6e 2 1–  7 2 -------– 2 2 e ----------– y 1 e ---+ –+ e 3 2 17 2 ------8– 6 2e– 4 2 e ----------– y 1 e ---+ 3 2 + e 2 10 2 3 ------------3– 5e 2 -------– 2 2– y 1 e ---+ 2 += w 3 y  1– 2e y 1 e ---+ 1 2e 3 ---------y 1 e ---+– 11e 36 --------y 1 e ---+ –+ e 3 2 191 9 --------125 3 2 ----------– 25 e 2 ------------8 2 e ---------6 2 e 3 2 ----------+ + + y 1 e ---+ 3 2 + e 2 281 6 --------146 2 3 ----------------– 32 2e 22 2 18 2 e -------------+ + + y 1 e ---+ 2 – e 5 2 335 9 --------40 2– 55e 3 2 2 ----------------20 2 e 18 2 e -------------+ + + y 1 e ---+ 5 2 + e 3 371 36 --------34 2 3 -------------– 8 2e 2 6 2e 6 2+ + + y 1 e ---+ 3 += w 4 y  1– += 2e y 1 e ---+ 1 2e 3 ---------y 1 e ---+– 11e 36 --------y 1 e ---+ 43e 3 2 135 2 ----------------y 1 e ---+ 3 2 –+ + e 2 4075 27 2 ------------895 12 ---------– 91e 2 ---------– 61 2 -------– 27 2 e -------------– 64 2 3e 2 -------------– y 1 e ---+ 2 – e 5 2 6658 2 27 ------------------2126 9 ------------– 315e 3 2 2 --------------------– 110 2e– 102 2 e ----------------– 256 2 3e 3 2 ----------------– y 1 e ---+ 5 2 + e 3 8467 2 27 ------------------1175 4 ------------– 207 2e 2 – 149 2e– 144 2– 128 2 e ----------------– y 1 e ---+ 3 – e 7 2 4904 2 27 ------------------502 3 ---------– 245e 5 2 2 --------------------– 90 2e 3 2 – 90 2e– 256 2 3 e ----------------– y 1 e ---+ 7 2 + e 4 10843 135 2 ---------------1315 36 ------------– 55e 3 2 -----------– 41e 2 2 -----------– 21 2e– 64 2 3 -------------– y 1 e ---+ 4  https://doi.org/10.28924/ada/ma.2.14 5.1.4. results. the relative errors in the spline based approximations to the lambert w function, as specified in theorem 5.1, of orders one to five, are shown in figure 20. the relative error bounds over the interval , for first to fifth order approximations, respectively, are: , , , and . by construction, the approximations are sharp at the points and . the results shown in figure 20 indicate that the sequence of approximations have good convergence to the lambert w function over the interval and modest convergence over the interval . 6. improved approximations via iteration iteration is, potentially, effective in improving the accuracy of an initial approximation. one potential approach is to utilize the iteration potential in the fundamental relationship for the lambert w function as specified by equation 116. an alternative approach is to utilized the newton-raphson method. these two approaches are detailed below. 6.1. inherent iteration. the basis for an iterative relationship for the lambert w function is equation 116, i.e. (82) it then follows that an approximation, , for , can potentially be improved upon according to (83) (84) 1 e– 0  1.27 10 2– 9.68 10 4– 8.69 10 5– 8.46 10 6– 8.65 10 7– 1 e– 0 1 e– 0  0 1  figure 20. graph of the relative errors in the spline based approximations, of orders one to five, to the lambert w function as specified in theorem 5.1. y order 1 order 5 re y  1 e– w y  y ln w y  ln– .= w0 y  w y  w1 y  y ln w0 y  ln–= w2 y  y ln y ln w0 y  ln–ln–= w21 y  y ln y ln w1 y  ln–ln–= https://doi.org/10.28924/ada/ma.2.14 (85) (86) etc. the number of possible permutations for approximations to the lambert w function is clearly large as the iteration order increases. representative relative error bounds, for the interval , are tabulated in table 5 for the base approximations of , and as specified in theorem 3.1. graphs of the relative errors, based on the approximations and , are shown, respectively, in figure 21 and figure 22. w3 y  y ln y ln y ln w0 y  ln–ln–ln–= w31 y  y ln y ln y ln w1 y  ln–ln–ln–= w32 y  y ln y ln y ln w2 y  ln–ln–ln–= w3 21 y  y ln y ln y ln w21 y  ln–ln–ln–= w4 y  y ln y ln y ln y ln w0 y  ln–ln–ln–ln–= 0  w0 y  wu3 y = w0 y  wu4 y = w0 y  wu5 y = w0 y  wu4 y = w0 y  wu5 y = figure 21. graph of the relative errors in approximations to , based on (equation 35). w y  w0 y  wu4 y = y w0 w1 w3 21 w32 w4 w2 w3 w31 w21 re y  figure 22. graph of the relative errors in approximations to , based on (equation 36). w y  w0 y  wu5 y = y w0 w1 w3 21 w32 w2 w3 w31 w21 re y  w4 https://doi.org/10.28924/ada/ma.2.14 6.1.1. explicit approximations. the approximation , based on , is (87) and has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . the approximation , based on , is (88) where is specified by equation 35. the maximum relative error bound, over the interval , is which is a factor of lower that the original approximation whose relative error bound is . the approximation , based on , is table 5. relative error bounds, for the interval , based on iteration and for the specified base approximations of (equation 33), (equation 35) and (equation 36). iteration form 0  wu3 y  wu4 y  wu5 y  w0 y  wu3 y = w0 y  wu4 y = w0 y  wu5 y = w0 1.33 10 3– 7.23 10 7– 2.15 10 13– w1 4.08 10 4– 1.84 10 7– 4.95 10 14– w2 1.97 10 4– 6.03 10 8– 1.31 10 14– w21 1.40 10 4– 2.47 10 8– 3.95 10 15– w3 1.40 10 4– 2.47 10 8– 3.95 10 15– w31 1.46 10 4– 1.23 10 8– 1.34 10 15– w32 2.33 10 4– 7.43 10 9– 5.04 10 16– w3 21 6.65 10 4– 5.36 10 9– 2.11 10 16– w4 1.46 10 4– 1.23 10 8– 1.34 10 15– w3 w0 y  wu3 y = w3 3 y  y ln y ln y ln wu3 y  ln–ln–ln–= y ln y ln y ln 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------lnln–ln–ln–= 0  1.40 10 4– 9.5 wu3 y  1.33 10 3– w2 w0 y  wu4 y = w2 4 y  y ln y ln wu4 y  ln–ln– = wu4 y  0  6.03 10 8– 12 wu4 y  7.23 10 7– w3 w0 y  wu4 y = https://doi.org/10.28924/ada/ma.2.14 (89) and has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . 6.2. newton-raphson iteration. consistent with equation 22, the approximations stated in theorem 3.1, theorem 3.2 and theorem 3.5 can be utilized as the basis for iterative approximations based on the newton-raphson method. results are detailed in table 6. as an example, the approximation arising from a first order iteration of (equation 33), has a maximum relative error bound, over the interval , of which is a factor of lower that the original approximation whose relative error bound is . the approximation is: (90) table 6. relative error bounds, for the interval , based on newton-raphson iteration and for the specified base approximations, , of , , and . iteration order (equation 31) (equation 33) (equation 35) (equation 36) w3 4 y  y ln y ln y ln wu4 y  ln–ln–ln– = 0  2.47 10 8– 29.3 wu4 y  7.23 10 7– 0  w0 y  wu2 y  wu3 y  wu4 y  wu5 y  w0 y  wu2 y = w0 y  wu3 y = w0 y  wu4 y = w0 y  wu5 y = 0 5.69 10 2– 1.33 10 3– 7.23 10 7– 2.15 10 13– 1 8.28 10 3– 5.12 10 6– 1.49 10 12– 1.30 10 25– 2 3.48 10 4– 9.61 10 11– 6.98 10 24– 5.02 10 50– 3 9.95 10 7– 3.91 10 20– 1.62 10 46– 7.67 10 99– 4 8.21 10 12– 7.08 10 39– 9.04 10 92– 1.81 10 196– 5 5.60 10 22– 2.43 10 76– 2.85 10 182– 1.02 10 391– wu3 y  0  5.12 10 6– 260 1.33 10 3– w1 3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln –= 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ +  --------------------------------------------------------------------------------------------------– 1 1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln+ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------https://doi.org/10.28924/ada/ma.2.14 7. applications 7.1. approximation with fixed relative error bound. there are many applications where the lambert w function is used to model the physical nature/characteristics of an entity which are positive in nature. in many of these cases the parameter values are not known with high accuracy. for such cases, highly accurate computation of the lambert w function is not required and a fixed approximation, with a set relative error bound, is useful rather than relying on, for example, iterative approximations where the relative error achieved depends on the initial approximate used. the relatively simple approximation for the lambert w function, as specified by equation 33, i.e. (91) with a relative error bound of over the interval , is likely to be useful. one application, for example, is in the evaluation of the collector current in a common emitter circuit, e.g. [2], eqn. 21. for the interval , the approximation detailed in equation 79, is of modest complexity, is sharp at the points and and has a relative error bound of . for highly accurate approximations over the interval , an explicit analytical approximation, with a relative error bound of , can be specified by utilizing the fifth order iterative approximation, denoted , arising from the iteration specified by equation 47 and with , (see table 2). an alternative analytical expression, with a relative error bound of over the interval , is (92) where and are defined, respectively, in equation 37 and equation 38. this expression arises from a first iteration of the newton-raphson method (equation 22) utilizing the fifth order approximation specified in equation 36. the relative error bound is specified in table 6. 7.2. upper/lower bounds for lambert w. there is interest in upper/lower bounds for the lambert w function, e.g. [15] and [24]. alzahrani and salem, [1], detail bounds for . the following bounds were proposed by hoorfar (see, [17], eqn. 21) (93) for the interval , the relative error bound associated with the lower bounded function is ; the relative error bound for the upper bounded function is . wu3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ -----------------------------------------------------------------------------------------------ln = 1.33 10 3– 0  1 e– 0  w 2 y  1 e– 0 9.68 10 4– 0  7.98 10 18– wk5 z1 1 ky+= k 0.4= 1.30 10 25– 0  w1 5 y  d5 y  n5 y  -------------ln d5 y  n5 y  -------------ln yn5 y  d5 y  ----------------– 1 d5 y  n5 y  -------------ln+ -----------------------------------------------– = n5 d5 wu5 y  w 1– y y ln -------------ln y ln ln 2 y ln -----------------------+ w y  y y ln -------------ln e e 1– ----------y ln ln y ln -----------------------.+  e   0.0568 0.207 https://doi.org/10.28924/ada/ma.2.14 the bounds proposed in [17], eqn. 25, 27, have modest relative errors for and are: (94) the relative error bounds in the lower and upper bounded functions, over the interval , respectively, are and . the relative errors decrease for . by construction, , , as defined in theorem 3.1, are a sequence of increasingly accurate lower bounds for . similarly, , , is a sequence of increasingly accurate upper bounds for . for example: (95) with relative error bounds for the interval of, respectively and (see table 1). higher order approximations lead to lower relative bounds and these can be made arbitrarily small. the relative error bounds associated with over the interval are, respectively, and . the relative error bounds associated with over the interval are, respectively, and . one potential application for the upper bound is a bound for the prime counting function, e.g. [27]. 7.3. spline approximations based on upper/lower bounds. consider the upper, , and lower, , bounded functions for the lambert w function as illustrated in figure 23 and as defined in theorem 3.1. for fixed at , a spline approximation, as specified by equation 70 and based on the points , and , , can readily be determined. from such an approximation, an approximation to can then be specified. theorem 7.1. spline approximations based on upper/lower bounds. consider the lower and upper bounded approximations, and , defined in theorem 3.1. the zero y e y y ln -------------ln y y ln -------------ln 1 y y ln -------------ln+ ---------------------------------– 1 y ln ln y ln -----------------------–ln w y   y y ln -------------ln 1 y ln ln y ln -----------------------– 1 1 y ln ln y ln -----------------------–ln 1 y y ln -------------ln+ -------------------------------------------– .ln– e  5.96 10 3– 4.10 10 3– y 10» wli y  i 1 2    w y  wui y  i 1 2    w y  wl3 y  y 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ 1 3y y 1 y+ ln+ + --------------------------------------------------------------------------------------------------= w y   wu3 y  1 3y y 1 y+ ln+ + 1 1 y+ ln+  1 1 2y+ 1 1 y+ ln+ -------------------------------ln+ ----------------------------------------------------------------------------------------------ln= y 0  0  8.32 10 3– 1.33 10 3– wl4 y  w y  wu4 y   0  3.88 10 6– 7.23 10 7– wl5 y  w y  wu5 y   0  1.08 10 12– 2.15 10 13– ith wui wli y yo uo uo exp uo  uo wli yo = vo vo exp vo  vo wui yo = xo w yo = ith wli wui https://doi.org/10.28924/ada/ma.2.14 order spline approximation for the lambert w function, based on the approximations and , is (96) the order spline approximation for the lambert w function, based on the approximations and , is (97) where , and is defined by equation 120. proof. the proof is detailed in appendix h. 7.3.1. results. results are detailed in table 7 and clearly show the high accuracy of the approximations. for example, the zero order spline approximations, as specified by equation 96, yields relative error bounds, for the interval , of , and , respectively, based on third, fourth and fifth order approximations for the figure 23. illustration of upper and lower bounded approximations to the lambert w function and the points , , which are the basis for spline based approximations. uo uo exp uo  vo vo exp vo  uo uo exp uo wli yo = y xo w yo = wli y  w y  x spline approx. based on wui y  vo wui yo = vo vo expyo ith wli wui w0 i y  wli y wui y  wui y  exp wli y  exp– y wui y  wli y – + wui y  wui y  exp wli y  wli y  exp– -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------.= nth ith wli wui wn i y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ -----------------------------------------------------------------------= y uo uo exp– r n r+ !uo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------+ pr u– uo e r u– uo– r u– ! 1 uo+ 2 r u–  1– ------------------------------------------------------------n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ----------------------------------------------------------------------- vo vo exp y– r n r+ !vo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------+ 1– r u– pr u– vo e r u– vo– r u– ! 1 vo+ 2 r u–  1– -----------------------------------------------------------------n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  uo wli y = vo wui y = pk 0   3.84 10 5– 8.56 10 12– 6.80 10 25– https://doi.org/10.28924/ada/ma.2.14 lambert w function detailed in equation 3.1 such convergence is approximately quadratic. 7.3.2. application. the omega constant, defined as , can be evaluated by using equation 96 with relative errors, respectively, of , , and for the case of upper and lower bounded approximations of orders two to five, i.e. , , and . 7.4. asymptotic approximations. as is evident in the results shown in figure 11, apart from the results for and , the relative errors in the approximations defined in theorem 3.1, for a set order, decrease as their argument increases, i.e. for a set order, the approximations asymptotically approach the lambert w function as their arguments become unbounded. thus: (98) 7.4.1. lambert w function and prime counting function. the prime number theorem states that the relative error between the prime counting function and decreases to zero as , i.e. (99) table 7. relative error bounds, over the interval , for spline approximations to the lambert w function based on upper and lower bounded functions. upper/lower bounded functions spline order approximation relative error bound , 1 increasing re as (equation 31) 2 increasing re as , 0 (equation 32, equation 33) 1 2 3 4 , 0 (equation 34, equation 35) 1 2 3 4 , 0 (equation 36) 1 w 1  1.94 10 5– 4.71 10 11– 2.76 10 22– 9.48 10 45– w0 2 1  w0 3 1  w0 4 1  w0 5 1  0   wl2 y  wu2 y  w1 2 y  w2 2 y  wl3 y  wu3 y  w0 3 3.84 10 5– w1 3 1.92 10 8– w2 3 1.46 10 11– w3 3 1.31 10 14– w4 3 1.27 10 17– wl4 y  wu4 y  w0 4 8.56 10 12– w1 4 5.60 10 22– w2 4 5.05 10 32– w3 4 5.18 10 42– w4 4 5.68 10 52– wl5 y  wu5 y  w0 5 6.80 10 25– w1 5 3.01 10 48– wl1 y  wl2 y  wli y  w y  i 3    wui y  w y  i 1 2 3    .  y  y y ln y   y  y y .ln https://doi.org/10.28924/ada/ma.2.14 as (equation 115), an equivalent statement for the prime number theorem is (100) with the manipulation of (101) and with for large, it follows that (102) which has been proposed by visser [27] and briefly discussed by iacono and boyd, [17], section 4.4. visser has proved that is an upper bound for the prime counting function whilst is a lower bound for large. the magnitude of the relative error in the approximation of is lower than the magnitude of the relative error in the approximation for greater than around with the relative error decreasing as increases. however, the magnitude of the relative error in the approximation is of the order of for . 7.5. floor and integral of floor of lambert w. it is possible to specify approximations for the lambert w function, with a set accuracy bound, if an explicit expression for the floor of the lambert function can be specified. the graph of is shown in figure 24. theorem 7.2. floor of lambert w. the floor of the lambert w function can be ascertained, without knowledge of the function itself, according to (103) where, is fixed, is the unit step function and is an upper bound for the lambert w function as specified in theorem 3.1. w z z ln  z ln=  y  y w y y ln . w y y ln  y ln xe x ln x x ln+ w y  w y  ln+ = = = = w y  w y  ln» y  y  y w y   y w y  y y ln y  y  y w y   y  y w y y ln  y 5000 y  y  y w y  0.05 y 10 9 = w y  figure 24. graph of and . w y  w y  y w y  w y  2e 2e 4e 4 3e 3 w y  u y ke k –  k 1=   u y ke k –  k 1= 1 wui y +  = = y 0  i 1 2    u wui y  https://doi.org/10.28924/ada/ma.2.14 proof. this result follows from the fact that and , , fixed, is an upper bound for , i.e. . it then follows that the upper limit of the summation can be specified as . 7.5.1. notes. the simplest upper bound, specified in theorem 3.1, for the lambert w function is and, thus: (104) the graphs of , for , are shown in figure 25 and it follows that the use of as the upper limit for the summation results in an increasing small number of additional zero terms in the summation defining as increases. the use of results, depending on the value of , in an additional zero term in the summation defining . 7.5.2. integral of floor of lambert w. using the result for the floor of the lambert w function, as specified in theorem 7.2, it is possible to explicitly specify the integral of the floor of the lambert w function. theorem 7.3. integral of floor of lambert w. the integral of the floor of the lambert w function can be explicitly specified according to (105) where is specified in theorem 7.2. proof. the required result follows, consistent with the graph of shown in figure 24, according to w ke k  k= wui y  i 1 2    i w y  w y  wui y  1 wui y + wu1 y  1 y+ ln= w y  u y ke k –  k 1= 1 1 y+ ln+  = y 0 . 1 wui y  w y –+ i 1 2 3   1 wu1 y + 1 1 y+ ln+= w y  y 1 wu2 y + y w y  figure 25. graph of for . 1 wui y  w y –+ i 1 2 3  y 1 wui y  w y –+ i 1= i 2= i 3= w   d 0 y  e e 1– 2 ------------------1– e w y  1 w y  2 w y  2 –+ + + w y  1– w y + e1 w y + w y  2 e w y  1– +  += w y  y w y  e w y  –  w y  w y  https://doi.org/10.28924/ada/ma.2.14 (106) where the following result has been used: (107) 7.6. set accuracy approximation for lambert w. consider a set accuracy limit of required for the evaluation of . this can be achieved by a step approximation, with a resolution of , and such an approximation is: (108) where , , fixed, is a set function defined in theorem 3.1. as an example, the error in the approximation to , with a resolution of , is shown in figure 26. 7.6.1. computationally efficient implementation. the direct approximation detailed in equation 108, for a set error level of , requires, approximately, a summation of terms. the approach detailed below requires, approximately, the summation of terms which represents a significant reduction for modest to large. for example, for and , the direct approach requires a summation of approximately terms whilst the approach detailed below requires close to terms. w   d 0 y  k k 1+ ek 1+ ke k –  k 0= max 0 w y  1–   w y  y w y  e w y  – += e e 1– 2 ------------------1– e w y  1 w y  2 w y  2 –+ + + w y  1– w y + e1 w y + w y  2 e w y  1– +  += w y  y w y  e w y  –  k k 1+ ek 1+ ke k –  k 1= n 1–  e e 1– 2 ------------------1– e n 1 n 2n 2 –+  ne 1 n+ 1– n+  n 2 e 1– n+ + + + .=  w y   w y   u y kek–  k 1= 1 1  --wui y +  = y 0 . wui i 1 2    i w y   1 10= figure 26. graph in the error in for the case of .w y   0.1= y w y  w y –  10 q– = 10 q w y  w y  11q 1–+ q w y  10= q 6= 10 7 75 https://doi.org/10.28924/ada/ma.2.14 theorem 7.4. direct evaluation of lambert w with specified resolution. the lambert w function can be evaluated, with a maximum error of , according to (109) where is defined in theorem 7.2 and is the digit in the decimal expansion of : (110) proof. the floor of the lambert w function has been defined in theorem 7.2. consider the illustration shown in figure 27. with (111) and with being the digit to the right of the decimal point, it follows that (112) iteration with finer resolution, and from the point defined by , yields etc.  10 q– = w y  w y  d1 y   dq y + + + = w y  10 q dq y  qth w y  d1 y  1 10 -----u y w y  k 10 ------+ w y  k 10 ------+exp– k 1= 10 =  d2 y  1 100 --------u y w y  d1 y  k 100 ---------+ + w y  d1 y  k 100 ---------+ +exp– k 1= 10  =  dq y  1 10 q -------u y w y  di y  i 1= q 1–  k 10 q --------+ + w y  di y  i 1= q 1–  k 10 q --------+ +exp– k 1= 10  .= w y  w y  d1 y  d2 y  + + += dq y  qth d1 y  1 10 -----u y w y  k 10 ------+ w y  k 10 ------+exp– . k 1= 10 = w y  d1 y + d2 y  figure 27. illustration of demarcation that underpins determination of , to a set resolution of , between and . w y   ke k k 1+ ek 1+ y k + k 1+ k + ek + k ke k w y  k 2+ k 2+ ek 2+ k 1+ ek 1+ k 1 –+ ek 1 –+ k 1 –+ https://doi.org/10.28924/ada/ma.2.14 the number of terms in the summation defined by equation 109 comprises approximately terms for the evaluation of , plus terms for the summations comprising and terms for the summation of , , . 7.6.2. note. theorem 7.4 defines a series for the lambert w function, which, by construction is convergent, i.e. (113) and is such that (114) 8. conclusion a geometric based approach for iteratively specifying approximations to the lambert w function, which can achieve any set relative error bound over the interval , was detailed. the approximations are also valid for the interval but are not sharp at the point . convergence was proved. for the interval , arbitrarily accurate approximations, based on a two point spline approximation, were specified. iteration, either by using the iteration structure inherent in the definition of the lambert w function, or via the newton-raphson method, leads to significantly improved approximations albeit with increasing complex functional forms. applications of the approximations were detailed and include, first, analytical expressions for the lambert w function that achieve set relative error bounds over the interval . second, based on the geometry inherent in the approximations, upper and lower bounds for the lambert w function that can be made arbitrary accurate. third, higher accuracy spline based approximations for the lambert w function based on the defined upper and lower bounded functions. fourth, analytical expressions for the evaluation of , and the integral of , without knowledge of for . finally, a direct approach for evaluating the lambert w function to achieve a prior defined error. acknowledgement: the author is pleased to acknowledge the support of prof. a. zoubir, spg, technische universität darmstadt, darmstadt, germany, who hosted a visit where the research, underpinning this paper, was completed. appendix a. properties of lambert w function the following are useful properties of the lambert w function: (115) (116) the latter formula underpins the iteration: w y  w y  10q d1 y  d2 y   dq y   q 1– d1 y  d2 y +  d1 y  d2 y   dq y + + + w y  w y  d1 y  d2 y  + + += y 0  w y  w y  d1 y   dq y + + +– 10 q– . 0   1 e– 0  1 e– 1 e– 0  0  w y  w y  w y  y 0   w z z ln  z ln = z 0 w y  y w y  ------------ln y ln w y  ln– = = y 0. https://doi.org/10.28924/ada/ma.2.14 (117) to prove that , consider which implies and, thus, . the relationship follows from the definitions , which implies and, thus, . a.1. differentiation. the derivatives of the lambert w function are defined according to (118) (119) where the second inequalities follow from the relationship and the polynomial is defined according to (120) the coefficients in this expression are defined according to (https://oeis.org/a042977; [7], eqn 3.4; [23], p. 1370): (121) explicit expressions are: (122) appendix b. proof of theorem 3.2 the iteration formula, as specified in theorem 3.1, yields the first order approximations as stated in equation 30: (123) w y  y ln y ln w y  ln– ln– = w y  y ln y ln y ln w y  ln– ln– ln– =  w z z ln  z ln= f x  xe x = f z ln  z z ln= z ln f 1– z z ln  w z z ln = = w y  y ln w y  ln–= y xe x = x w y = y ln x x ln+= x y ln w y  ln–= w 1  y  e w y – 1 w y + ---------------------w y  y 1 w y +  ------------------------------= = w k  y  w k y pk w y   y k 1 w y +  2k 1– -------------------------------------------pk w y  e kw– y  1 w y + 2k 1– ------------------------------------------= = y w y ew y  = pk pk r  ck 0 ck 1 r ck 2 r 2  ck k 1– r k 1– .+ + + += cn k 0 k 0 1– n 1+ n n 1– k 0= n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k 1+ cn 1– k 1++ 1 k n 3–  n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k n 2–= n 1– cn 1– k 1–– n 1– != k n 1–=         = p1 r  1= p2 r  2 r+ – = p3 r  9 8r 2r 2 + + = p4 r  64 79r 36r 2 6r 3 + + + – = p5 r  625 974r 622r 2 192r 3 24r 4 + + + + = p6 r  7776 14543r 11758r 2 5126r 3 1200r 4 120r 5 + + + + + .–= wl1 y  y 1 y+ -----------n1 y  d1 y  -------------= = wu1 y  1 y+ ln 1 ln– d1 y  ln n1 y  y -------------ln– = = https://doi.org/10.28924/ada/ma.2.14 where and . the second order approximations, as specified by equation 31, can be written in the form (124) where (125) the third order approximations, as specified by equation 32 and equation 33, can be written in the form: (126) (127) where (128) thus, iteration yields the general formulas: (129) where (130) appendix c. proof of theorem 3.5 consider the geometry illustrated in figure 15 and an initial approximation for , based on the intersection of the second order taylor series for at the origin, i.e. n1 y  y= d1 y  1 y+= wl2 y  y 1 1 y+ ln+  1 2y+ --------------------------------------n2 y  d2 y  -------------= = wu2 y  1 2y+ ln 1 1 y+ ln+ ln– d2 y  ln n2 y  y -------------ln– = = n2 y  y 1 1 y+ ln+  n1 y  1 d1 y  ln+ = = d2 y  1 2y+ n1 y  d1 y .+= = wl3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ 1 3y y 1 y+ ln+ + -----------------------------------------------------------------------------------------------------------------------------------n3 y  d3 y  -------------= = wu3 y  1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–= d3 y  ln n3 y  y -------------ln– = n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+= n2 y  1 d2 y  ln n2 y  y -------------ln–+ = d3 y  1 3y y 1 y+ ln+ + n2 y  d2 y .+= = wli ni y  di y  ------------= wui di y  ln ni y  y ------------ln– = ni y  ni 1– y  1 di 1– y  ln ni 1– y  y -------------------ln–+ = n0 y  y= d0 y  1=  di y  ni 1– y  di 1– y .+= xo ye x– https://doi.org/10.28924/ada/ma.2.14 , and which is . the problem with such an approximation is that it only yields a real solution for . a practical approach is to utilize an affine approximation at the origin of leading to the first approximation of . to establish a further approximation, consider the point where the level intersects which is (131) a second level approximation follows by finding the intersection of a second order taylor series at the point , which is (132) with to yield (133) and (134) iteration in this manner leads to the general iteration formulas: (135) simulation results (see figure 16) indicate that the approximations also have good convergence for the interval but the approximations are not sharp at . appendix d. proof of theorem 4.1 consider the case of fixed, , and the illustration shown in figure 18. by construction, , and for . further, (see equation 25) (136) and it follows that , , is a monotonically increasing sequence. using equation 136 it follows that y 1 x– x 2 2+  x wl1 1 y --1 y 1 2y y 2 –++= 1 2– y 1 2+  y 1 x–  wl1 y 1 y+ = wl1 ye x– wu1 y wl1 ---------ln 1 y+ .ln= = wu1 wl1 x wu1 – wl1 – x wu1 – 2wl1 2 -------------------------------------+  x wl2 1 wl1 --------1 wl1 1 wu1 +  1 wl1 2 – 2wl1 1 wu1 + +–+= wu2 y wl2 --------.ln= wli 1 wli 1– -------------1 wli 1– 1 wui 1– +  1 wli 1– 2 – 2wli 1– 1 wui 1– + +–+ = wui y wli ---------ln .= 1 e– 0  y 1 e–= y y 0 li 0 ui 0 wui wli  i 1 2    wli 1+ wli 1 wui + 1 wli + ------------------- = wli i 1 2    https://doi.org/10.28924/ada/ma.2.14 (137) as , and , it follows that (138) where . thus, as monotonically increases with , it follows that monotonically decreases with , i.e. . it then follows that (139) hence, convergence is guaranteed as . thus: and . the result implies that and, thus, . appendix e. alternative form for spline approximation the general form for a order, two point, spline approximation for a function , over the interval , has been detailed in howard, [16], eqn. 40. the assumption is that the function is at least order differentiable over the interval . the approximation, denoted , can be written in the modified form: (140) in this equation, the double summation can be rewritten by utilizing the transformations and , , . the possible values of are detailed in table 8 and for fixed, the valid values for are from the set . table 8. valid values of for , . i k 0 1 2 3 ... n-2 n-1 n 0 0 1 2 3 n-2 n-1 n 1 1 2 3 4 n-1 n 2 2 3 4 5 n 3 3 4 5 6 li li 1+ – xo wli –  xo wli 1 wui +  1 wli + ---------------------------------–– wli wui wli –  1 wli + ---------------------------------------.= = wui wli – li ui += ui 0 li 1+ li wli 1 wli + ------------------– li ui +  1 wli 1 wli + ------------------– li rili= = ri 1 1 wli + ------------------= wli i ri i 0 ri 1+ r i 1  0 li 1+ l1 rk k 1= i  l1 r1 i .   0 r1 1 1 y 1 y+ + --------------------------------= 1  lii  lim 0= wlii  lim xo= li 1+ li wli 1 wli + ------------------– li ui + = uii  lim 0= wuii  lim xo= nth f    f nth    fn fn x   x– n 1+  – n 1+ ---------------------------f k    k! ---------------n i+ ! i!n! ----------------- i 0= n k–  x – k i+  – i -------------------------- k 0= n  += x – n 1+  – n 1+ ---------------------------1– kf k    k! -----------------------------n i+ ! i!n! ----------------- i 0= n k–   x– k i+  – i ------------------------- . k 0= n  r i k+= u i= k 0 1  n    i 0 1  n k–    r i k+= r i 0 1  r    r i k+= k 0 1  n    i 0 1  n k–    https://doi.org/10.28924/ada/ma.2.14 with , and , , equation 140 can be written as (141) thus: (142) where (143) appendix f. proof of lemma 1 first, the definitions of , and with , imply: (144) it then follows that which implies . with and , the required result of then follows. second, the transformation of , , results in (145) as it then follows that and the final result follows: (146) n-2 n-2 n-1 n n-1 n-1 n n n table 8. valid values of for , . i k 0 1 2 3 ... n-2 n-1 n r i k+= k 0 1  n    i 0 1  n k–     r 0 1  n    u 0 1  r    i u= k r u–= fn x   x– n 1+  – n 1+ ---------------------------x – r f r u–    r u– ! ----------------------n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- r 0= n  += x – n 1+  – n 1+ --------------------------- x– r 1– r u– f r u–    r u– ! -------------------------------------------n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- . r 0= n  fn x   x– n 1+ an r x – r r 0= n  x – n 1+ bn r  x– r r 0= n += an r 1  – n 1+ ---------------------------f r u–    r u– ! ----------------------n u+ ! u!n! ------------------- u 0= r  1  – u -------------------- = bn r 1  – n 1+ ---------------------------1– r u– f r u–    r u– ! -------------------------------------------n u+ ! u!n! ------------------- u 0= r  1  – u --------------------. = f g1 x x1 1–= x 1– y1 g1 x1  1 e --x1 1– e x1 1– + = x1 0.= y1 1 e– f x1 1– = f 1– y1 1 e–  x1 1–= x1 g1 1– y1 = y y1 1 e–= f 1– y  g1 1– y 1 e+  1–= y2 g x1  g1 x1  y1= = = x1 0 g 1– y2  x1 g 1– y1 .= = x1 g1 1– y1 = g1 1– y1  g 1– y1 = f 1– y  g1 1– y 1 e ---+ 1– g 1– y 1 e ---+ 1.–= = https://doi.org/10.28924/ada/ma.2.14 appendix g. proof of theorem 5.1 with denoting the differentiation operator, the following well known results apply for an arbitrary function : (147) (148) (149) etc. consider , as defined by equation 75, over the interval . a spline approximation (see equation 70) for , of order , requires derivatives, of orders zero to at the points and , to be determined. using the above formulas, such values can be determined from the derivatives of at the points and and values of these derivatives are tabulated in table 9. to determine the derivative values at zero, the standard taylor series expansion for the exponential function can be used to yield the alternative form for of (150) using equation 70, and the derivative values given in table 9, the spline approximations for , based on the points and and for orders one to four, are: (151) (152) d f d f 1– z   1 f 1  x  ---------------x f 1– z = = d 2  f 1– z   f 2  x – f 1  x   3 ----------------------x f 1– z = = d 3  f 1– z   f 3  x – f 1  x   4 ----------------------3 f 2  x   2 f 1  x   5 --------------------------+ x f 1– z = = d 4  f 1– z   f 4  x – f 1  x   5 ----------------------10f 3  x f 2  x  f 1  x   6 -------------------------------------15 f 2  x   3 f 1  x   7 -----------------------------–+ x f 1– z = = g 0 1  g 1– n n 0 1 e g 0 1 g g x1  x1 2e ---------1 2 i 1+ x1 i i 2+ ! --------------------i 1=  + .= g 1– y2  0 1 e g1 1– y2  2ey2 1 2 ey2 1 1 2e ---------3 2 2 ----------–+– ey2 2 1 2– 2 e -------++= g2 1– y2  2ey2 1 2ey2 3 ---------------– 6e 2 1–  7 2 -------– 2 2 e ----------– y2 2 –+ e 3 2 17 2 ------8– 6 2e– 4 2 e ----------– y2 3 e 2 10 2 3 ------------3– 5e 2 -------– 2 2– y2 4 + = https://doi.org/10.28924/ada/ma.2.14 (153) (154) in general: (155) table 9. values of the derivatives of at the points zero and one. order 0 0 1 2 3 4 5 g x1  g i  0  g i  1  1 e -----1 2e ---------e 2 -----2 3 e ---------e 1 e 4 ---– 5 12 2e ---------------3 e 2 ---------1 e– e 2 4 -----+ 11 45 2e ---------------2 e 1 3e– 9e 2 4 -------15e 3 32 -----------–+ 59 432 2e ------------------5 e 2 ---------1 8e– 27e 2 2 ----------15e 3 2 -----------– 21e 4 16 -----------+ + g3 1– y2  2ey2 1 2ey2 3 ---------------– 11ey2 2 36 -------------e 3 2 191 9 --------125 3 2 ----------– 25 e 2 ------------8 2 e ---------6 2 e 3 2 ----------+ + + y2 3 +–+ e 2 281 6 --------146 2 3 ----------------– 32 2e 22 2 18 2 e -------------+ + + y2 4 – e 5 2 335 9 --------40 2– 55e 3 2 2 ----------------20 2 e 18 2 e -------------+ + + y2 5 + e 3 371 36 --------34 2 3 -------------– 8 2e 2 6 2e 6 2+ + + y2 6 = g4 1– y2  2ey2 1 2ey2 3 ---------------– 11ey2 2 36 -------------43e 3 2 y2 3 135 2 ----------------------–+ + e 2 4075 27 2 ------------895 12 ---------– 91e 2 ---------– 61 2 -------– 27 2 e -------------– 64 2 3e 2 -------------– y2 4 – e 5 2 6658 2 27 ------------------2126 9 ------------– 315e 3 2 2 --------------------– 110 2e– 102 2 e ----------------– 256 2 3e 3 2 ----------------– y2 5 + e 3 8467 2 27 ------------------1175 4 ------------– 207 2e 2 – 149 2e– 144 2– 128 2 e ----------------– y2 6 – e 7 2 4904 2 27 ------------------502 3 ---------– 245e 5 2 2 --------------------– 90 2e 3 2 – 90 2e– 256 2 3 e ----------------– y2 7 + e 4 10843 135 2 ---------------1315 36 ------------– 55e 3 2 -----------– 41e 2 2 -----------– 21 2e– 64 2 3 -------------– y2 8 = gk 1– y2  2ey2 1 1y2 2y2 2 3y2 3 4y2 4  2ky2 2k + + + + + +  https://doi.org/10.28924/ada/ma.2.14 by equation 70 with , based on the points and for appropriately defined constants and, thus: (156) appendix h. proof of theorem 7.1 a zero order spline approximation is simply an affine approximation between the two specified points. consistent with figure 23, the zero order spline approximation to , denoted , is an affine approximation between the points and . thus: (157) with the approximation it follows that (158) simplification yields (159) substitution of and yields the required result. h.1. general result. the general result arises from the spline approximation, specified where , , and with (160) here is defined by equation 120. the order spline approximation, for , is w y  g 1– y 1 e ---+ 1–= 2e y 1 e ---+ 1 1 y 1 e ---+ 2 y 1 e ---+ 3 y 1 e ---+ 3 2  2k y 1 e ---+ k + + + + + 1.– w y  f0 uo uo exp uo  vo vo exp vo  f0 y  uo y uo uo exp–  vo uo– vo vo exp uo uo exp– --------------------------------------------------------y uo uo exp vo vo exp .+= xo w yo = f0 yo  xo uo yo uo uo exp–  vo uo– vo vo exp uo uo exp– ---------------------------------------------------------.+ xo uovo vo exp uo exp–  yo vo uo– + vo vo exp uo uo exp– ------------------------------------------------------------------------------------------------. uo wli yo = vo wui yo = f y  w y = uo uo exp uo  vo vo exp vo  uo wli yo = vo wui yo = w 1  y  e w y – 1 w y + ----------------------= w k  y  pk w y  e kw– y  1 w y + 2k 1– ------------------------------------------= k 1 2   . pk nth y uo uo exp vo vo exp  https://doi.org/10.28924/ada/ma.2.14 (161) the results , imply that (162) assuming . hence: (163) the required result follows: the approximation for , denoted , arises for the case of . thus, . fn y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ -----------------------------------------------------------------------= y uo uo exp– r w r u–  uo uo exp  r u– ! ------------------------------------------------n u+ ! u!n! ------------------- u 0= r  1 vo vo exp uo uo exp– u ----------------------------------------------------------------- r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ----------------------------------------------------------------------- vo vo exp y– r 1– r u– w r u–  vo vo exp  r u– ! --------------------------------------------------------------------n u+ ! u!n! ------------------- u 0= r  1 vo vo exp uo uo exp– u ----------------------------------------------------------------- r 0= n  w uo uo exp  uo= w vo vo exp  vo= w r u–  uo uo exp  pr u– uo e r u– uo– 1 uo+ 2 r u–  1– ----------------------------------------------= w r u–  vo vo exp  pr u– vo e r u– vo– 1 vo+ 2 r u–  1– ----------------------------------------------= r u fn y  vo vo exp y– n 1+ vo vo exp uo uo exp– n 1+ -----------------------------------------------------------------------= y uo uo exp– r n r+ !uo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------+ pr u– uo e r u– uo– r u– ! 1 uo+ 2 r u–  1– ------------------------------------------------------------n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  + y uo uo exp– n 1+ vo vo exp uo uo exp– n 1+ ----------------------------------------------------------------------- vo vo exp y– r n r+ !vo r!n! vo vo exp uo uo exp– r -------------------------------------------------------------------------+ 1– r u– pr u– vo e r u– vo– r u– ! 1 vo+ 2 r u–  1– -----------------------------------------------------------------n u+ ! u!n! ------------------- u 0= r 1–  1 vo vo exp uo uo exp– u -----------------------------------------------------------------  r 0= n  w yo  wn i yo  y yo= wn i yo  fn yo = https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.1080/10652469.2018.1528247 https://doi.org/10.1109/81.895330 https://doi.org/10.1016/s0378-4754(00)00172-5 https://doi.org/10.1007/s10910-018-0932-3 https://doi.org/10.1007/s10910-018-0932-3 https://doi.org/10.1016/s0893-9659(98)00097-4 https://doi.org/10.1007/bf02124750 https://doi.org/10.1155/2021/6695559 https://doi.org/10.1017/s0022377805003788 https://doi.org/10.30538/oms2021.0149 https://doi.org/10.1016/j.cam.2012.11.021 https://scholar.google.com/citations?user=invpkckaaaaj&hl=en&oi=sra https://doi.org/10.28924/ada/ma.2.14 https://doi.org/10.1016/j.bej.2012.01.010 https://doi.org/10.3390/mca24020035 https://doi.org/10.1007/s10444-017-9530-3 https://doi.org/10.1016/j.physleta.2015.12.004 https://doi.org/10.1111/2041-210x.12568 https://doi.org/10.1088/0143-0807/36/3/035030 http://arxiv.org/abs/1408.3999 https://doi.org/10.1093/imamat/hxu057 https://doi.org/10.1139/p00-065 https://doi.org/10.1016/j.cpc.2012.07.008 https://doi.org/10.3390/math6040056 ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 21doi: 10.28924/ada/ma.3.21 analysis of a mathematical model incorporating dual protection and art adherence for a high risk hiv population i. s. oriedo1, g. o. lawi2, j. o. bonyo3,∗ 1department of pure and applied mathematics, maseno university, p.o. box 333-40105, maseno, kenya samsonoriedo@gmail.com 2department of mathematics, masinde muliro university of science and technology, p.o. box 190-50100, kakamega, kenya glawi@mmust.ac.ke 3department of mathematics, multimedia university of kenya, p.o. box 15653-00503, nairobi, kenya jbonyo@mmu.ac.ke ∗correspondence: jobbonyo@maseno.ac.ke abstract. in this paper, a mathematical model for dual protection, incorporating prep and condomuse, and art adherence is formulated, based on a system of ordinary differential equations andanalyzed. the results obtained from stability analysis indicate that provided the basic reproductivenumber is less than unity, the disease free equilibrium point is both locally and globally asymptoticallystable, while provided the basic reproductive number is greater than unity, the endemic equilibriumpoint exists and is locally asymptotically stable. sensitivity analysis is undertaken to establish themost sensitive model parameter. the most sensitive parameter to the value of r0 is β1, the meancontact rate with undiagnosed infectives. this implies that in order to control the spread of hivin a high risk population, efforts should be geared towards reducing the undiagnosed by testingand enrolling them on art treatment. this in turn lowers their infectivity as well as chances ofprogressing to the aids class. 1. introduction numerous efforts have been made in an attempt to control the spread of hiv, with the aim ofreducing its effects. according to the unaids fact sheet 2019, at least 1.7 million new hivinfections were reported by the end of the year 2018 [13].scientific as well as public health interventions such as testing and counseling, circumcision,use of prep (pre-exposure prophylaxis), pep (post-exposure prophylaxis), condom use, and an-tiretroviral therapy have been proposed and utilized. consistent use of condoms can result to 80%reduction in hiv incidence among the heterosexual population [2], while the effectiveness of condom received: 1 may 2023. key words and phrases. mathematical modeling; prep; dual protection; hiv/aids; stability analysis.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.21 https://orcid.org/0000-0002-6442-4211 eur. j. math. anal. 10.28924/ada/ma.3.21 2use for men who have sex with men is 70% [3]. proper use (correctly and consistently) as well asquality concerns have been directly attributed to the success of this approach.in 2012, the u.s food and drug administration (fda) approved the use of truvada for prep asan oral pill taken once a day [14]. numerous efficacy trials( by iprex, partners prep, tdf2,e.t.c) have since been conducted to ascertain the potential of prep to prevent hiv infection. theiprex trial demonstrated that prep has the potential of reducing the risk of hiv infection amongtransgender women, bisexual men, as well as men who have sex with men [8]. two major studies;partners prep, and tdf2 demonstrated the effectiveness of prep among heterosexual men andwomen. out of all these studies, none displayed a 100% effectiveness [11]. adherence has beenfound to be directly correlated with the effectiveness of prep [11]. in the absence of adherence,which guarantees efficacy, prep failures have been characterized by; system failures, people fail-ures, doctor failures, drug failures, as well as assay failures [9]. these failures expose prep usersto the risk of hiv infection hence the need for additional protection whenever prep has beenutilized.the nature of storage, date of manufacture, religious as well as socio-cultural beliefs also influencehow each hiv prevention venture is utilized. the challenges experienced when various approachesare employed in an attempt to control the spread of hiv infection in a high risk population form thebasis for the need to use dual protection in order to achieve maximum protection. a combinationprevention approach as proposed by [6], based on proven efficacy interventions, provides one withthe best opportunity to curb the spread of hiv among the high risk population.in this study, we propose a mathematical model of dual protection against hiv infection by the useof condom and prep, and adherence to art treatment, while focusing on the high risk populationcollectively. earlier studies have either narrowed down to a particular category of persons at highrisk of infection [3], or have used a combination of prevention techniques where one techniqueacts as a supplement to the other [4], [5]. the study will focus on the impact of dual protectionon reducing the number of new infections, and that of art adherence in ensuring those who areinfected remain less infectious. 2. model formulation and description the population is subdivided into the classes; susceptible, infected, and aids individuals. thesusceptible class has been further subdivided into two compartments on the basis of degree ofrisk of infection. these include susceptible individuals at high risk of infection, denoted by (sh),and those at low risk, denoted by (sl). the high risk population incorporates mainly commercialsex workers, men who have sex with men (msm), and hiv-discordant couples [8]. the infectedclass is subdivided into two compartments; those who are unaware of their hiv status (i), andthose who have been diagnosed and consequently enrolled for treatment (td). the individualswho are unaware of their hiv status may progress to the td compartment after successful hiv https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 3awareness campaigns that will persuade them to get tested, or when they develop hiv symptomsand consequently enroll for art treatment. if art treatment fails, the individual progresses to theaids compartment. this happens when there is lack of adherence to art, which allows the virusto multiply, thus increasing the plasma viral load. this results in weakening of the immune systemand hence the aids symptoms begin to manifest. the aids compartment comprises of those whoposses full blown symptoms, and are mostly bedridden, they thus do not significantly contribute tothe spread of the disease. exit from the aids class is through natural death.thus, considering apopulation of size n(t), at a time t , n(t) = sh(t) + sl(t) + i(t) + td(t) + a(t). (1) the following interventions have been incorporated in the model;(a) 0 ≤ φ1 ≤ 1 measures prep effectiveness, including its awareness and proper use as a meansto prevent susceptible individuals from being infected. thus, (1− φ1) measures prep failure.(b) 0 ≤ φ2 ≤ 1measures condom effectiveness as a result of proper use, following adequateawareness campaigns and availability. thus, (1− φ2) measures condom failure.(c) 0 ≤ φ3 ≤ 1 measures the efficacy of art treatment, including uptake with proper adherence,with the aim of reducing the plasma viral load and reconstructing the individual’s immune systemhence making them less infectious.movement of individuals from the susceptible to infected and then to the aids classes is illustratedby the compartmental model shown in figure 1. sh sl a td i �� (� − �)� (� − ��)(� − ��)��� (� − ��)��� �sh �sl ��� (� + �)� �i �� ��� (� − ��)�� figure 1. compartmental model. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 4the following symbols will be used to represent various phenomena as described in table 1. symbol description λ constant rate of recruitment of susceptible upon becoming sexually active. δ proportion of susceptible individuals at high risk of infection. (1− δ) proportion of susceptible population at low risk of hiv infection. λ rate of acquisition of an infection by susceptibles.it is given by; λ = (β1i+β2td n ), where β1, and β2 are the mean contact rates for thesusceptible individuals with i and td respectively. µ natural removal rate by death. σ aids induced mortality. α represents the proportion of infected individuals who upon beingtested and found to be hiv positive,they enroll for art treatment. γ2 represents the proportion of infected individuals who do not get testedhence remain undiagnosed until they begin to exhibit aids symptoms.table 1. table showing symbols and their description from the dynamics described above, the following system of ordinary differential equations isformulated. dsh dt = δλ− (1− φ1)(1− φ2)λsh − µsh dsl dt = (1− δ)λ− (1− φ2)λsl − µsl di dt = (1− φ1)(1− φ2)λsh + (1− φ2)λsl − αi − γ2i − µi dtd dt = αi − (γ3 + µ)td da dt = γ2i + γ3td − (µ+ σ)a. (2) 3. model analysis it can be shown that the solutions for the system of ordinary differential equations (2) areall positive and bounded for all t > 0, with positive initial conditions in the feasible region γ ={ (sh(t), sl(t), i(t), (td(t), a(t)) ∈ r5 + : n(t) ≤ λ µ } . it therefore suffices to study the dynamicsof the system (2) in this region.the mathematical model developed in (2) has two unique equilibrium points, that is, the diseasefree equilibrium (d.f.e), and the endemic equilibrium (e.e). the d.f.e is obtained by setting https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 5 i = td = a = 0 in (2) to yield e0 = ( δλ µ , (1− δ)λ µ , 0, 0, 0 ) . (3) the basic reproduction number (r0) of the system (2), computed using the next generation matrixapproach [10] is given by r0 = β3 q1 + αβ4 q1q2 . (4)by [10, theorem 2], the following result is thus established. theorem 3.1. the disease free equilibrium of the model (2), e0 = ( δλ µ , (1−δ)λ µ , 0, 0, 0 ) , is locally asymptotically stable whenever r0 < 1 and unstable otherwise. proof. the proof follows immediately from the computation of r0 above and theorem 2 of van dendriessche and watmough [10]. � mathematically, theorem (3.1) implies that whenever there is a small perturbation on the system,the system returns to the disease free equilibrium. epidemiologically, this implies that when a fewhiv infectious individuals are introduced in a population that is fully susceptible to hiv infection,the disease dies out whenever r0 < 1, otherwise, the disease will spread. it is therefore necessaryto show that eliminating hiv in a population is independent of the size of the initial sub-populationby proving the global asymptotic stability of the disease free equilibrium. theorem 3.2. the disease free equilibrium e0 = ( δλ µ , (1−δ)λ µ , 0, 0, 0 ) of the system (2) is globally asymptotically stable whenever r0 < 1. proof. castillo chavez’s theorem [1] is used to analyze the global asymptotic stability of the math-ematical model (2) such that e0 = (x∗,0), x = (sh, sl) and z = (i, td, a).now; f (x,0) = ( δλ− µsh (1− δ)λ− µsl ) and g(x,z) = pz− g̃(x,z). matrix p is given by h1β1sh n + (1− φ2)β1sl n − (α+ γ2 + µ) h1β2sh n + (1− φ2)β2sl n 0 α −(γ3 + µ) 0 γ2 γ3 −(σ + µ)  , where h1 = (1− φ1)(1− φ2), and pz is given by h1β1ish n + (1− φ2)β1isl n − (α+ γ2 + µ) + h1β2tdsh n + (1− φ2)β2tdsl n αi − (γ3 + µ)td γ2i + γ3td − (σ + µ)a  . https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 6moreover, g(x,z) is given by (1− φ1)(1− φ2) ( β1i + β2td n ) sh + (1− φ2) ( β1i + β2td n ) sl − (α+ γ2 + µ)i αi − (γ3 + µ)td γ2i + γ3td − (σ + µ)a  , and therefore g̃(x,z) = pz − g(x,z)= g̃1(x,z) g̃2(x,z) g̃3(x,z) = 0 0 0  . hence conditions h1 and h2 are satisfied. also from theorem (3.1), e0 is locally asymptotically stable whenever r0 < 1. thereforefollowing castillo chavez’s theorem, e0 is globally asymptotically stable whenever r0 < 1, asdesired. � this implies that with a large perturbation of the disease free equilibrium, solutions of the modelrepresented by the system (3.2) converge to d.f.e whenever r0 < 1. epidemiologically, this impliesthat if a sufficiently large number of hiv infected individuals are introduced in a population thatis fully susceptible to hiv infection, the disease will die out whenever r0 < 1. 3.1. existence of the endemic steady state. theorem 3.3. an endemic equilibrium point e1 = (s∗∗h , s ∗∗ l , i ∗∗, t ∗∗d , a ∗∗), of the system (2) exists whenever r0 > 1. proof. equating the right hand side of each equation in the system (2) to zero and simplifyingyields; δλ− (1− φ1)(1− φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗h − µs∗∗h = 0, (5) (1− δ)λ− (1− φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗l − µs∗∗l = 0, (6) (1− φ1)(1− φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗h + (1− φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗l − (α+ γ2 + µ)i∗∗ = 0, (7) αi∗∗ − (γ3 + µ)t ∗∗d = 0, (8) γ2i ∗∗ + γ3t ∗∗ d − (µ+ σ)a∗∗ = 0. (9) from equation (8), t ∗∗d = α q2 i∗∗.substituting for t ∗∗td in equation (9) and simplifying gives a∗∗ = ( γ2 q3 + αγ3 q2q3 ) i∗∗. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 7using equation (5) and substituting t ∗∗d gives δλn − (1− φ1)(1− φ2) ( β1 + β2α q2 ) i∗∗s∗∗h − µns∗∗h = 0 ⇒ s∗∗h = δλn a1i∗∗ + µn , where a1 = (1 − φ1)(1 − φ2) ( β1 + β2α q2 ). in a similar manner, s∗∗l is expressed as s∗∗l = (1− δ)λn a2i∗ + µn , where a2 = (1− φ2) ( β1 + β2α q2 ).using equation (7) and substituting for s∗∗h and s∗∗l ,we obtain a1i ∗∗δλ a1i∗∗ + µn∗∗ + a2i ∗∗(1− δ)λ a2i∗∗ + µn∗∗ −q1i ∗∗ = 0 (10) thus from equation (10),( a1δλ a1i∗∗ + µn∗∗ + a2(1− δ)λ a2i∗∗ + µn∗∗ −q1 ) i∗∗ = 0. (11) from equation (11), i∗∗ = 0 corresponds to the disease free equilibrium point of the system (2),denoted by (e0). the other solution of (11) when i∗∗ 6= 0 corresponds to the endemic equilibriumpoint of the system such that, a1δλ a1i∗∗ + µn∗∗ + a2(1− δ)λ a2i∗∗ + µn∗∗ −q1 = 0. (12) multiplying through by (a1i ∗∗ + µn∗∗)(a2i ∗∗ + µn∗∗) yields ci∗∗2 +di∗∗ + e = 0. (13) where: c = −q1a1a2, d = (a1a2δλ + a1a2(1− δ)λ)− (q1a1µn +q1a2µn), and e = a1δλµn + a2(1− δ)λµn −q1µnµn .the endemic equilibrium of the system exists if the roots of equation (13) are real and positive.descarte’s rule of signs is used to check the possible number of real roots of the polynomial. thenumber of positive real roots is equal to the number of sign changes in the coefficients of theterms of a polynomial [15]. considering that all the parameters used are positive, the sign of c isnegative. the sign of e is then checked as follows; e = a1δλµn + a2(1− δ)λµn −q1µnµn = (1− φ1)(1− φ2) ( β1 + β2α q2 ) δλµn + (1− φ2) ( β1 + β2α q2 ) (1− δ)λµn −q1µnµn = (1− φ1)(1− φ2)(β1 + β2α)δλµn + (1− φ2)(β1 + β2α)(1− δ)λµn −q1q2µnµn https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 8 using r0 = β3 q1 + αβ4 q1q2 and the limiting value of n = λ µ , we obtain e = (r0 − 1)λ2. thus e >0iff r0 > 1. since c is negative, and e is positive, we see that there is at least one sign changeregardless of the sign of d. this implies that equation (13) has at least one positive real root.hence an endemic equilibrium point of the system (2) exists whenever r0 > 1. � 3.2. local stability of the endemic equilibrium. at the endemic equilibrium, there is persistenceof hiv infection in the population. theorem 3.4. the endemic equilibrium point e1 = (s∗∗h , s ∗∗ l , i ∗∗, t ∗∗d , a ∗∗) of system (2) is locally asymptotically stable if r0 > 1. proof. the jacobian matrix of the system (2) evaluated at endemic equilibrium is j(e1) =  −b1 0 −b2 −b3 0 0 −b4 −b5 −b6 0 b7 b8 b9 −q1 b10 0 0 0 α −q2 0 0 0 γ2 γ3 −q3  where b1 = (1−φ1)(1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ , b2 = (1−φ1)(1−φ2)β1a1δλi∗∗ a1i∗∗+λ , b3 = (1−φ1)(1−φ2)β2a1δλi∗∗ a1i∗∗+λ b4 = (1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ , b5 = (1−φ2)(1−δ)λβ1a2i ∗∗ a2i∗∗+λ , b6 = (1−φ2)(1−δ)λβ2a2i ∗∗ a2i∗∗+λ b7 = (1−φ1)(1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ , b8 = (1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ b9 = (1−φ1)(1−φ2)β1a1δλi∗∗ a1i∗∗+λ + (1−φ2)(1−δ)λβ1a2i ∗∗ a2i∗∗+λ , b10 = (1−φ1)(1−φ2)β2a1δλi∗∗ a1i∗∗+λ + (1−φ2)(1−δ)λβ2a2i ∗∗ a2i∗∗+λclearly, −q3 is an eigenvalue of the jacobian matrix j(e1). the other eigenvalues can be computedby finding the solution to the equation p (λ) = ∣∣∣∣∣∣∣∣∣∣∣ λ+ b1 0 −b2 −b3 0 λ+ b4 −b5 −b6 b7 b8 λ− (b9 +q1) b10 0 0 α λ+q2 ∣∣∣∣∣∣∣∣∣∣∣ =0 the characteristic equation of j(e1)is then given by; p (λ) = λ4 + c0λ 3 + c1λ 2 + c2λ+ c3 = 0 (14) where; c0 = b1 + b4 − b9 −q1 +q2 c1 = b1b4 + b2b7 + b5b8 − b1b9 − b4b9 − αb10 − b1q1 − b4q1 + b1q2 + b4q2 − b9q2 −q1q2 c2 = −αb3b7 +b2b4b7 +b1b5b8−αb6b8−b1b4b9−alphab1b10−alphab4b10−b1b4q1 +b1b4q2 + b2b7q2 + b5b8q2 − b1b9q2 − b4b9q2 − b1q1q2 − b4q1q2 c3 = −αb3b4b7 − αb1b6b8 − αb1b4b10 + b2b4b7q2 + b1b5b8q2 − b1b4b9q2 − b1b4q1q2the number of negative zeros of equation (14) depends on the signs of c0, c1, c2 and c3. descarte’s https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 9rule of signs is applied to study the number of negative real roots of the polynomialp (λ1) com-prising of the coefficients c0, c1, c2 and c3 given by; p (λ1) = c0λ 3 + c1λ 2 + c2λ+ c3 = 0 (15) descarte’s rule of signs states that the number of negative real zeros of p (λ) is either equal tothe variations in sign of p (−λ) or less than this by an even number [15]. the possibilities ofnegative real zeros of p (λ), is as summarized in table 2. the maximum number of variationsof signs in p (−λ) is 3, hence the characteristic polynomial (15) has three negative roots. thus p (−λ) = λ4 − c0λ 3 + c1λ 2 − c2λ + c3 = 0 has negative roots.therefore, given that cases 1-8 intable 1 are satisfied, model (2) is locally asymptotically stable if r0 > 1. � table 2. the zeros of the characteristic equation (14) cases c0 c1 c2 c3 r0 > 1 sign change no. of roots1 + − − + r0 > 1 2 2,02 + − + + r0 > 1 2 2,03 − − + − r0 > 1 2 2,04 + + − − r0 > 1 1 05 − − + + r0 > 1 1 06 + + + − r0 > 1 1 07 − + − + r0 > 1 3 3,18 − − − − r0 > 1 0 0 this implies that for a small pertubation of the e1, solutions of the mathematical model representedby the system (2) always converge to e1, whenever r0 > 1. epidemiologically, it implies that ifa few hiv infected individuals are introduces in a fully susceptible population, the disease willpersist provided r0 > 1. 4. sensitivity analysis in mathematical modeling, sensitivity refers to the degree to which a given input parameterin a mathematical model influences its output. sensitive parameters are thus those that cause asignificant impact on the disease transmission dynamics. sensitivity analysis will aid in identify-ing the parameters which greatly impact on the value of the basic reproductive number r0, andhence ought to be targeted when coming up with intervention strategies. the sensitivity of modelparameters is calculated using the normalized forward sensitivity index. the normalized forwardsensitivity index of the basic reproductive number is given by sr0 w = ∂r0 ∂w × w r0 , where w is the https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 10parameter whose sensitivity is to be determined [7]. r0 is given by r0 = (1− φ1)(1− φ2)β1δ + (1− φ2)(1− δ)β1 α+ γ2 + µ + α(1− φ1)(1− φ2)β2δ + (1− φ2)(1− δ)β2 (α+ γ2 + µ)(γ3 + µ) . (16) forβ1, s r0 β1 = β1(γ3 + µ) β1(γ3 + µ) + αβ2 . forβ2, s r0 β2 = αβ2 β1(γ3 + µ) + αβ2 . for α,sr0 α = [β2(α+ γ2 + µ)− (β1(γ3 + µ) + αβ2)]α (α+ γ2 + µ)(β1(γ3 + µ) + αβ2) . forγ2, s r0 γ2 = (αγ2 + γ2 2 + µγ2) ln |α+ γ2 + µ|. forγ3, s r0 γ3 = −αβ2γ3 (β1(γ3 + µ)2 + αβ2(γ3 + µ) . for δ, sr0 δ = −φ1δ 1− δφ1for µ, sr0 µ = [(α+ γ2 + µ)(γ3 + µ)β1 + ((γ3 + µ)β1 + αβ2)(α+ γ2 + γ3 + 2µ)]µ (α+ γ2 + µ)(γ3 + µ)((β1(γ3 + µ) + αβ2)) . based on the sensitivity indices in table 3, the most sensitive parameter to the value of r0 is β1, table 3. sensitivity indices for the model parameters parameter description sensitivity index δ proportion of high risk sussceptibles −0.36986 φ1 prep effectiveness −0.041095 φ2 condom effectiveness −0.11111 γ3 art failure −0.27182 β1 mean contact rate with i 0.72345 β2 mean contact rate with td 0.27654 α progression from i to td −0.40225 γ2 progression from i to a −0.05123 µ natural mortality rate −0.02969 the mean contact rate with undiagnosed infectives. this implies that in order to control the spreadof hiv in a high risk population, efforts should be geared towards reducing the number of thosewho are undiagnosed by testing them and enrolling them on art treatment. this in turn lowerstheir infectivity as well as chances of progressing to the aids class. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 115. conclusion in this study, a mathematical model is formulated, based on a system of ordinary differentialequations, incorporating the impact of dual protection and art adherence in preventing the spreadof hiv among persons at high risk of infection.stability analysis of the model was done and depicted that when r0 < 1, the disease freeequilibrium is both locally and globally asymptotically stable. the endemic equilibrium of themathematical model exists and was shown to be locally asymptotically stable whenever r0 > 1,implying that there is persistence of hiv infection in the population provided that r0 is greaterthan unity. sensitivity analysis was conducted, depicting that the most sensitive parameter is β1,the mean contact rate with the un-diagnosed infectives. therefore, in order to control the spreadof hiv among the high risk population, efforts ought to be channeled towards the undiagnosedpopulation by frequently testing and enrolling them on art treatmentwhich guarantees low viral load within the infected individual, making them less infective. thusdual protection and art adherence are essential in the fight against the spread of hiv among thehigh risk population. references [1] c. castillo-chavez, z. feng, w. huang, on the computation of r0 and its role on global stability, in: c. castillo-chavez, s. blower, p. van den driessche, d. kirschner, a.-a. yakubu (eds.), mathematical approaches for emergingand reemerging infectious diseases: an introduction, springer new york, new york, ny, 2002: pp. 229-250. https://doi.org/10.1007/978-1-4757-3667-0_13.[2] s.c. weller, k. davis-beaty, condom effectiveness in reducing heterosexual hiv transmission, cochrane databasesyst. rev. 2012 (2002). https://doi.org/10.1002/14651858.cd003255.[3] d.k. smith, j.h. herbst, x. zhang, c.e. rose, condom effectiveness for hiv prevention by consistency of use amongmen who have sex with men in the united states, jaids j. acq. immune defic. syndr. 68 (2015) 337-344. https: //doi.org/10.1097/qai.0000000000000461.[4] e.o. omondi, r.w. mbogo, l.s. luboobi, mathematical modelling of the impact of testing, treatment and control ofhiv transmission in kenya, cogent math. stat. 5 (2018) 1475590. https://doi.org/10.1080/25742558.2018. 1475590.[5] f.k. tireito, g.o. lawi, o.a. colleta, mathematical analysis of hiv/aids anti-retroviral treatment incorporatingadherence, asian res. j. math. 10 (2018) 1-13. https://doi.org/10.9734/arjom/2018/42830.[6] i. cremin, r. alsallaq, m. dybul, p. piot, g. garnett, t.b. hallett, the new role of antiretrovirals in combinationhiv prevention: a mathematical modelling analysis, aids. 27 (2013) 447-458. https://doi.org/10.1097/qad. 0b013e32835ca2dd.[7] j.c. helton, r.l. iman, j.b. brown, sensitivity analysis of the asymptotic behavior of a model for the environmentalmovement of radionuclides, ecol. model. 28 (1985) 243-278. https://doi.org/10.1016/0304-3800(85)90077-8.[8] j.m. baeten, j.e. haberer, a.y. liu, n. sista. pre-exposure prophylaxis for hiv prevention: where have we beenand where are we going? j. acq. immune defic. syndr. 63 (2013) 122-129. https://doi.org/10.1097/qai. 0b013e3182986f69.[9] j.m. molina. prep failures: diagnosis, resistance, and treatment. conference on retroviruses and opportunisticinfections. 4-7 march 2019. https://www.croiwebcasts.org/p/2019croi/160. https://doi.org/10.28924/ada/ma.3.21 https://doi.org/10.1007/978-1-4757-3667-0_13 https://doi.org/10.1002/14651858.cd003255 https://doi.org/10.1097/qai.0000000000000461 https://doi.org/10.1097/qai.0000000000000461 https://doi.org/10.1080/25742558.2018.1475590 https://doi.org/10.1080/25742558.2018.1475590 https://doi.org/10.9734/arjom/2018/42830 https://doi.org/10.1097/qad.0b013e32835ca2dd https://doi.org/10.1097/qad.0b013e32835ca2dd https://doi.org/10.1016/0304-3800(85)90077-8 https://doi.org/10.1097/qai.0b013e3182986f69 https://doi.org/10.1097/qai.0b013e3182986f69 https://www.croiwebcasts.org/p/2019croi/160 eur. j. math. anal. 10.28924/ada/ma.3.21 12 [10] p. van den driessche, j. watmough, reproduction numbers and sub-threshold endemic equilibria for compartmentalmodels of disease transmission, math. biosci. 180 (2002) 29-48. https://doi.org/10.1016/s0025-5564(02) 00108-6.[11] k.r. amico, m.j. stirratt, adherence to preexposure prophylaxis: current, emerging, and anticipated bases of evi-dence, clin. infect. dis. 59 (2014) s55-s60. https://doi.org/10.1093/cid/ciu266.[12] t.t. yusuf, f. benyah, optimal strategy for controlling the spread of hiv/aids disease: a case study of southafrica, j. biol. dyn. 6 (2012) 475-494. https://doi.org/10.1080/17513758.2011.628700.[13] unaids. global hiv and aids statistics: 2019 fact sheet: https://www.unaids.org/en/resources/ fact-sheet[14] u.s food and drug administration (f.d.a).u.s f.d.a approves first drug for reducing the riskof sexually acquired h.i.v infectionjuly. 2011. https://aidsinfo.nih.gov/news/1254/fda-approves-{\@ @par}rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection[15] x. wang, a simple proof of descartes’s rule of signs, amer. math. mon. 111 (2004) 525-526. https://doi.org/ 10.2307/4145072. https://doi.org/10.28924/ada/ma.3.21 https://doi.org/10.1016/s0025-5564(02)00108-6 https://doi.org/10.1016/s0025-5564(02)00108-6 https://doi.org/10.1093/cid/ciu266 https://doi.org/10.1080/17513758.2011.628700 https://www.unaids.org/en/resources/fact-sheet https://www.unaids.org/en/resources/fact-sheet https://aidsinfo.nih.gov/news/1254/fda-approves-{\@@par }rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection https://aidsinfo.nih.gov/news/1254/fda-approves-{\@@par }rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection https://doi.org/10.2307/4145072 https://doi.org/10.2307/4145072 1. introduction 2. model formulation and description 3. model analysis 3.1. existence of the endemic steady state 3.2. local stability of the endemic equilibrium 4. sensitivity analysis 5. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 19doi: 10.28924/ada/ma.3.19 efficient derivative-free class of seventh order method for non-differentiable equations ioannis k. argyros1,∗, samundra regmi2, jinny ann john3, jayakumar jayaraman3 1department of computing and mathematical sciences, cameron university, lawton, 73505, ok, usa iargyros@cameron.edu 2department of mathematics, university of houston, houston, 77204, tx, usa sregmi5@uh.edu 3department of mathematics, puducherry technological university, pondicherry 605014, india jinny3@pec.edu, jjayakumar@ptuniv.edu.in ∗correspondence: iargyros@cameron.edu abstract. many applications from a wide variety of disciplines in the natural sciences and also inengineering are reduced to solving of an equation or a system of equations in a correspondinglychosen abstract area. for most of these problems, the solutions are found iterative, because theiranalytic versions are difficult to find or impossible. this article encompasses efficient, derivatives-free,high-convergence iterative methods. convergence of two types: local and semi-local areas will beinvestigated under the conditions of the ϕ,ψ-continuity utilizing operators on the method. the newmethod can also be applied to other methods, using inverses of the linear operator or the matrix. 1. introduction in the area of applied science and technology, a great number of problems can be resolved byconverting them into nonlinear form equation g(x) = 0 (1) where g : b ⊂ u → u is differentiable as per fréchet, u denotes complete normed linear spaceand b is a non-empty, open and convex set.normally, the solutions to these non-linear equations can not be obtained in a closed-form.therefore, the most frequently used solving techniques are of iterative nature. newton’s methodis a well-known iterative method for handling non-linear equations. recently, with advances inscience and mathematics many new iterative methods of higher order have been discovered for thehandling of non-linear equations and are currently being used [1, 2, 4–8, 10–22]. the computationof derivatives of second and higher order is a great disadvantage for the iterative systems of higherorder and is not suitable for the practical application. because of the computation of g ′′ , the received: 3 may 2023. key words and phrases. steffensen-like methods; convergence; banach space; divided difference.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 2cubically converging classical schemas are not appropriate with respect to the cost of calculations.we found that many such methods rely on taylor series extensions to prove convergence resultsand require the existence of derivative with at least an order of magnitude greater than that ofthe methodology [1, 2, 4, 10–19, 21, 22]. here we consider, for example, a three-step two-parameterfamily of derivative free methods with seventh-order of convergence for solving systems of nonlinearequations proposed in [18] and which may be expressed in the following formulation:for x0 ∈ b and each n = 0, 1, 2, . . . wn = xn + ag(xn), sn = xn − ag(xn), an = [wn, sn;g], yn = xn − a−1n g(xn), zn = yn − a−1n g(yn), un = zn + bg(zn), vn = zn − bg(zn), qn = [un, vn;g], xn+1 = zn − (pi + a−1n qn(qi + a−1n qn(r i + da−1n qn)))a−1n g(zn), (2) where a, b, p, q, r, d ∈ r, [·, ·;g] : b × b → w (u), the space of bounded linear operators from u into u . the local convergence analysis of the method (2) is provided in [18] using the taylorseries expansion approach and conditions reaching the eighth derivative of the operator g. thesederivatives do not appear on the method (2). the convergence order is shown to be seven providedthat p = 17 4 , q = −274 , r = 19 4 and d = −54 . the conditions on high order derivatives restrict theapplicability of the method (2) for solving equations where at least g(8) should exist. although,the method may converge. let us consider the toy example for b = [−1, 2] and g defined by g(t) = { t4 log t + 5t7 − 5t6, if t 6= 0 0, if t = 0 it follows by this definition that g(ξ) = g(1) = 0 but g(4) is not bounded on b. thus, the resultsin [18] cannot assure that limn→∞ xn = ξ = 1. but, the method converges to 1.therefore, there is a need to weaken the conditions. in this article, we use only conditions onthe operators on the method (2). therefore, the method can be utilized to solve non-differentiableequations. furthermore, the results should also demonstrate the isolation of the solution and thebounds of error in advance. this is what is new and what motivates our article. this meansextending its applicability, taking advantage of weaker conditions for such methods. in addition,we are also discussing a more interesting case of semi-local convergence. it is obvious that theaforementioned goals can be easily achieved in a similar way for other iterative methods [1, 2, 4,10–17,19,21,22]. furthermore, our bounds of error is more precise and our criteria for convergenceapply even if the assumptions referred to in the references above are infringed.the remainder of the article is organized as follows: analysis of local convergence is provided insection 2. majorizing sequences will be introduced and analyzed for the semi-local convergenceanalysis of 2 in section 3. results demonstrating isolation of the solution is discussed in section https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 34. numeric experiments that use convergence results from the previous sections are described insection 5. the concluding remarks of section 6 bring this article to an end. 2. convergence 1: local let m = [0,+∞). the following conditions are used:(c1) there exist continuous and non-decreasing functions (cnf) ϕ0 : m×m → m , δ1 : m → m , δ2 : m → m , a solution ξ ∈ b of the equation g(x) = 0 and a linear operator p such thatfor each w = x + ag(x), s = x − ag(x) and p−1 ∈ w (u) ‖p−1([w, s;g]−p)‖ ≤ ϕ0(‖w − ξ‖, ‖s − ξ‖), ‖w − ξ‖ ≤ δ1(‖x − ξ‖)and ‖s − ξ‖ ≤ δ2(‖x − ξ‖). (c2) the equation ϕ0(δ1(t), δ2(t))− 1 = 0 has a smallest positive solution denoted by ρ0. let m0 = [0, ρ0) and b0 = b ∩ s(ξ, ρ0). (c3) there exist cnf ϕ : m0 ×m0 ×m0 → m , δ3 : m0 → m , δ4 : m0 → m , ϕ1 : m0 ×m0 × m0×m0 → m , ϕ2 : m0 → m such that for each x, z ∈ b0, u = z + bg(z), v = z − bg(z), ‖u − ξ‖ ≤ δ3(‖z − ξ‖), ‖v − ξ‖ ≤ δ4(‖z − ξ‖), ‖p−1([w, s;g]− [x, ξ;g])‖ ≤ ϕ(‖x − ξ‖, ‖w − ξ‖, ‖s − ξ‖), ‖p−1([w, s;g]− [u, v ;g])‖ ≤ ϕ1(‖w − ξ‖, ‖s − ξ‖, ‖u − ξ‖, ‖v − ξ‖)and ‖p−1([z, ξ;g]−p)‖ ≤ ϕ2(‖z − ξ‖). (c4) the equations hi(t)− 1 = 0, i = 1, 2, 3 have smallest solutions ri ∈ m0−{0}, respectivelywhere the functions hi : m0 → m are defined by h1(t) = ϕ(t, δ1(t), δ2(t)) 1− ϕ0(δ1(t), δ2(t)) , h2(t) = ϕ(h1(t)t, δ1(t), δ2(t))h1(t) 1− ϕ0(δ1(t), δ2(t)) ε(t) = ϕ1(δ1(t), δ2(t), δ3(h2(t)t), δ4(h2(t)t) 1− ϕ0(δ1(t), δ2(t)) , λ(t) = |p + q + r + d − 1|+ |p + 2r + 3d |ε(t) + |r + 3d |ε(t)2 + |d |ε(t)3, h3(t) = [ ϕ(h2(t)t, δ1(t), δ2(t)) 1− ϕ0(δ1(t), δ2(t)) + λ(t)(1 + ϕ2(h2(t)t)) 1− ϕ0(δ1(t), δ2(t)) ] h2(t). https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 4set r = min{ri}. let m1 = [0, r). it follows by these definitions that for each t ∈ m1 0 ≤ ϕ0(δ1(t), δ2(t)) < 1, 0 ≤ ε(t), 0 ≤ λ(t)and 0 ≤ hi(t) < 1. notice that for x0 ∈ s(ξ, r)− {ξ} the conditions (c1)-(c2) and (c4) imply ‖p−1([w0, s0;g]−p)‖ϕ0(‖w0 − ξ‖, ‖s0 − ξ‖) ≤ ϕ0(δ1(r), δ2(r)) < 1. thus a−10 ∈ w (u) by the banach lemma on invertible operators [3, 9, 10] and the firstiterate y0 is well-defined by the first sub-step of the method (2). (c5) s[ξ, r ] ⊂ b.the motivation for the development of the functions hi follows in turn by the estimates ‖a−1n p‖ ≤ 1 1− ϕ0(‖wn − ξ‖, ‖sn − ξ‖) ≤ 1 1− ϕ0(δ1(‖xn − ξ‖), δ2(‖xn − ξ‖)) , yn − ξ = a−1n (an − [xn, ξ;g])(xn − ξ), ‖yn − ξ‖ ≤ ϕ(‖xn − ξ‖, ‖wn − ξ‖, ‖sn − ξ‖)‖xn − ξ‖ 1− ϕ0(δ1(‖xn − ξ‖), δ2(‖xn − ξ‖)) ≤ h1(‖xn − ξ‖)‖xn − ξ‖ ≤ ‖xn − ξ‖ < r. similarly, ‖zn − ξ‖ ≤ ϕ(‖yn − ξ‖, ‖wn − ξ‖, ‖sn − ξ‖)‖yn − ξ‖ 1− ϕ0(δ1(‖xn − ξ‖), δ2(‖xn − ξ‖)) ≤ h2(‖xn − ξ‖)‖xn − ξ‖ ≤ ‖xn − ξ‖, xn+1 − ξ = zn − ξ − a−1n g(zn)− [(p + q + r + d − 1)i + (q + 2r + 3d)(a−1n qn − i) + (r + 3d)(a−1n qn − i)2 + d(a−1n qn − i)3]a−1n g(zn) which can be shortened for dn = a−1n (qn − an), tn = (p + q + r + d − 1)i + (p + 2r + 3d)dn + (r + 3d)d2n + dd3n. thus xn+1 − ξ = a−1n (an − [zn, ξ;g])(zn − ξ)− tna−1n g(zn). https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 5but, ‖dn‖ ≤ ‖a−1n p‖‖p−1(qn − an)‖ ≤ ϕ1(‖wn − ξ‖, ‖sn − ξ‖, ‖un − ξ‖, ‖vn − ξ‖) 1− ϕ0(‖wn − ξ‖, ‖sn − ξ‖) = εn, ‖tn‖ ≤ |p + q + r + d − 1|+ |p + 2r + 3d |εn + |r + 3d |ε2n + |d |ε3n = λn, leading to ‖xn+1 − ξ‖ ≤ [ ϕ(‖zn − ξ‖, ‖wn − ξ‖, ‖sn − ξ‖) 1− ϕ0(δ1(‖xn − ξ‖), δ2(‖xn − ξ‖) + λn(1 + ϕ2(‖zn − ξ‖)) 1− ϕ0(δ1(‖xn − ξ‖), δ2(‖xn − ξ‖)) ] ‖zn − ξ‖ ≤ h3(‖xn − ξ‖)‖xn − ξ‖ < ‖xn − ξ‖. hence, the iterates {xn}, {yn}, {zn} ⊂ s(ξ, r) and there exists c = h3(‖x0 − ξ‖) ∈ [0, 1) such that ‖xn+1 − ξ‖ ≤ c‖xn − ξ‖ < r, from which it follows that limn→∞ xn = ξ.therefore, we achieve the following local convergence result for the method (2). theorem 2.1. under the assumptions (c1)-(c5), {xn} ⊂ s(ξ, r) and limn→+∞ xn = ξ provided that x0 ∈ s(ξ, r)− {ξ}. remark 2.2. the functions δj , j = 1, 2, 3, 4 are left uncluttered in the theorem 2.1. a possible choice for the first function δ1 is motivated by the estimate w − ξ = x − ξ + af (x) = (i + a[x, ξ;f ])(x − ξ) = (i + app−1([x, ξ;g]−p + p))(x − ξ), = [(i + ap) + app−1([x, ξ;g]−p)](x − ξ), ‖w − ξ‖ ≤ [‖i + ap‖+ |a|‖p‖ϕ0(‖x − ξ‖)]‖x − ξ‖. thus, we can choose δ1(t) = [‖i + ap‖+ |a|‖p‖ϕ0(t)]t. similarly, we can choose δ2(t) = [‖i − ap‖+ |a|‖p‖ϕ0(t)]t, δ3(t) = [‖i + bp‖+ |b|‖p‖ϕ0(h2(t)t)]h2(t)t and δ4(t) = [‖i − bp‖+ |b|‖p‖ϕ0(h2(t)t)]h2(t)t. two possible choices for the linear operator p are: the differentiable option : p = g′(ξ) and the non-differentiable option : p = [x0, x−1;g]. other choices are possible [18]. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 63. convergence 2: semi-local the role of ξ, “ϕ” is replaced by x0, “ψ” as follows. assume:(h1) there exist cnf ψ0 : m × m → m , x0 ∈ b, g1 : m → m, g2 : m → m and a linearoperator p such that for x ∈ b w = x + ag(x), s = x − ag(x), ‖w − x0‖ ≤ g1(‖x − x0‖), ‖s − x0‖ ≤ g2(‖x − x0‖) ‖p−1([w, s;g]−p)‖ ≤ ψ0(‖w − x0‖, ‖s − x0‖). (h2) the equation ψ0(g1(t), g2(t))− 1 = 0 has a smallest positive solution denoted by ρ.let m2 = [0, ρ) and b1 = b ∩ s(x0, ρ).notice that ‖p−1([w0, s0;g]−p)‖ ≤ ψ(0, 0) < 1.thus, a−10 ∈ w (u) and the iterate y0 is well-defined by the first sub-step of the method(2). (h3) there exists cnf g3 : m2 → m , g4 : m2 → m , ψ1, ψ2 : m2 ×m2 ×m2 ×m2 → m suchthat for each x, y ∈ b1 ‖u − x0‖ ≤ g3(‖z − x0‖, ‖v − x0‖ ≤ g4(‖z − x0‖) ‖p−1([y , x ;g]− [w, s;g])‖ ≤ ψ1(‖x − x0‖, ‖y − x0‖, ‖w − x0‖, ‖s − x0‖)and ‖p−1([w, s;g]− [u, v ;g])‖ ≤ ψ2(‖w − x0‖, ‖s − x0‖, ‖u − x0‖, ‖v − x0‖). define the real sequence {αn} for α0 = 0, β0 ≥ ‖a−10 g(x0)‖, and each n = 0, 1, 2, . . . by γn = βn + ψ1(αn, βn, g1(αn), g2(αn))(βn − αn) 1− ψ0(g1(αn), g2(αn)) , εn,1 = ψ2(g1(αn), g2(αn), g3(γn), g4(γn)) 1− ψ0(g1(αn), g2(αn)) , λn,1 = |p + q + r + d |+ |p + 2r + 3d |εn,1 + |r + 3d |ε2n,1 + |d |ε3n,1, αn+1 = γn + ψ1(βn, γn, g1(αn), g2(αn))(γn − βn)λn,1 1− ψ0(g1(αn), g2(αn) , δn+1 = ψ1(αn, αn+1, g1(αn), g2(αn))(αn+1 − αn) + (1 + ψ0(g1(αn), g2(αn))(αn+1 − βn) βn+1 = αn+1 + δn+1 1− ψ0(g1(αn+1, g2(αn+1) . (3) a convergence set of conditions for the sequence {αn} is given for each n = 0, 1, 2, . . .. (h4) ψ0(g1(αn), g2(αn)) < 1 and αn ≤ α < ρ.it follows by this condition and (3) that 0 ≤ αn ≤ βn ≤ γn ≤ αn+1 and there exists α∗ ∈ [0, α] such that limn→∞ αn = α∗. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 7and (h5) s[x0, α ∗] ⊂ b.as in the local case the motivation for the introduction of the sequence {αn} follows in turn to formthe estimates: zn − yn = −a−1n g(yn),but g(yn) = g(yn)− g(xn)− an(yn − xn) = ([yn, xn;g]− an)(yn − xn),so ‖zn − yn‖ ≤ ψ1(‖xn − x0‖, ‖yn − x0‖, ‖wn − x0‖, ‖sn − x0‖)‖yn − xn‖ 1− ψ0(‖wn − x0‖, ‖sn − x0‖) ≤ γn − βn, ‖zn − x0‖ ≤ ‖zn − yn‖+ ‖yn − x0‖ ≤ γn − βn + βn − α0 = γn < a∗, xn+1 − zn = −tna−1n g(zn), ‖xn+1 − zn‖ ≤ λn,1ψ1(‖yn − x0‖, ‖zn − x0‖, ‖wn − x0‖, ‖sn − x0‖)‖zn − yn‖ 1− ψ0(‖wn − x0‖, ‖sn − x0‖) ≤ αn+1 − γn,since tn,1 = (p + q + r + d)i + (q + 2r + 3d)dn + (r + 3d)d2n + dd3n, ‖dn‖ ≤ ψ2(‖wn − x0‖, ‖sn − x0‖, ‖un − x0‖, ‖vn − x0‖) 1− ψ0(‖wn − x0‖, ‖sn − x0‖) , ‖tn,1‖ ≤ λn,1and ‖xn+1 − x0‖ ≤ ‖xn+1 − zn‖+ ‖zn − x0‖ ≤ αn+1 − γn + γn − α0 = αn+1 < α∗. also, g(xn+1) = g(xn+1)− g(xn)− an(yn − xn) = g(xn+1)− g(xn)− an(xn+1 − xn) + an(xn+1 − yn), ‖p−1g(xn+1)‖ ≤ ψ1(‖xn − x0‖, ‖xn+1 − x0‖, ‖wn − x0‖, ‖sn − x0‖)‖xn+1 − xn‖ + (1 + ψ0(‖wn − x0‖, ‖sn − x0‖))‖xn+1 − yn‖ = δ̄n+1 ≤ δn+1, ‖yn+1 − xn+1‖ ≤ ‖a−1n+1p‖‖p −1g(xn+1‖ ≤ δ̄n+1 1− ψ0(‖wn+1 − x0‖, ‖sn+1 − x0‖) ≤ βn+1 − αn+1 (4) https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 8and ‖yn+1 − x0‖ ≤ ‖yn+1 − xn+1‖+ ‖xn+1 − x0‖ ≤ βn+1 − αn+1 + αn+1 − α0 = βn+1 < α∗.therefore, the sequence {xn} is complete in banach space u . hence, there exists ξ = limn→∞ xnand by (4) g(ξ) = 0.then, we achieve the following semi-local convergence result for the method (2). theorem 3.1. under the conditions (h1)-(h5) the sequence {xn} converges to a solution ξ ∈ s[x0, a ∗] of the equation g(x) = 0. remark 3.2. a possible choice for the functions gj , j = 1, 2, 3, 4 follows as in the local case. we have in turn w − x0 = x − x0 + a(g(x)− g(x0) + g(x0)) = [(i + ap) + app−1([x, x0;g]−p)](x − x0) + ag(x0), lead to the choice g1(t) = [‖i + ap‖+ |a|‖p‖ψ3(t)]t + |a|‖g(x0)‖ provided that for some cnf ψ3 : m1 → m , x ∈ b ‖p−1([x, x0;g]−p)‖ ≤ ψ3(‖x − x0‖). similarly, we define g2(t) = [‖i − ap‖+ |a|‖p‖ψ3(t)]t + |a|‖g(x0)‖, g3(t) = [‖i + bp‖+ |a|‖p‖ψ3(t)]t + |b|‖g(x0)‖, and g4(t) = [‖i − bp‖+ |b|‖p‖ψ3(t)]t + |b|‖g(x0)‖. the options for p are: p = g′(x0) or p = [x0, x−1;g]. other options exist [10]. 4. isolation of a solution we first present the uniqueness result for the local convergence case. proposition 4.1. there exists a solution v∗ ∈ s(ξ, ρ2) of the equation g(x) = 0 for some ρ2 > 0; the last condition in (c3) holds in the ball s(ξ, ρ2) and there exists ρ3 ≥ ρ2 such that ψ2(ρ3) < 1. (5) https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 9 set b3 = b ∩ s[ξ, ρ3]. then, ξ is the only solution of the equation g(x)=0 in the set b3. proof. let v∗ 6= ξ. then, the divided difference v = [ξ, v∗;g] is well-defined. using the lastcondition in (c3) and (5), we obtain in turn that ‖p−1(v −p)‖ ≤ ψ2(‖v∗ − ξ‖) ≤ ψ2(ρ3) < 1, so, v −1 ∈ w (u) and from the approximation v∗ − ξ = v −1(g(v∗)− g(ξ)) = v −1(0) = 0, we deduce v∗ = ξ. � proposition 4.2. assume: there exists a solution v∗ ∈ s(x0, ρ4) of the equation g(x) = 0 for some ρ4 > 0; the condition (h1) holds on the ball s(x0, ρ4) and there exist ρ5 ≥ ρ4 such that ϕ0(ρ4, ρ5) < 1. (6) set b4 = b ∩ s[x0, ρ5]. then, v∗ is the only solution of the equation g(x) = 0 in the set b4. proof. let z∗ ∈ b4 with g(z∗) = 0 and z∗ 6= v∗. define the linear operator f = [v∗, z∗;g]. then,by the condition (h1) and (6) ‖p−1(f −p)‖ ≤ ϕ0(‖v∗ − x0‖, ‖z∗ − x0‖) ≤ ϕ0(ρ4, ρ5) < 1, thus, again v∗ = z∗. � remark 4.3. (i) the limit point α∗ can be replaced by ρ in the condition (h5).(ii) under all the assumptions (h1)-(h5), let v∗ = ξ and ρ4 = α∗ in proposition 4.2. 5. experiments example 5.1. consider the system of differential equations with g′1(w1) = ew1 , g′2(w2) = (e − 1)w2 + 1, g′3(w3) = 1 subject to g1(0) = g2(0) = g3(0) = 0. let g = (g1, g2, g3). let u = r3 and b = u[0, 1]. then ξ = (0, 0, 0)t is a root. let function g on b for w = (w1, w2, w3) t be g(w) = (ew1 − 1, e − 1 2 w22 + w2, w3) t . this definition gives g′(w) = e w1 0 0 0 (e − 1)w2 + 1 0 0 0 1  https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 10 thus, by the definition of g it follows that g′(ξ) = 1. let p = g′(ξ) and [x, y ;g] = ∫ 1 0 g ′(x + θ(y − x))dθ. then, for a = b = 1, the conditions (c1)-(c5) are validated by remark 2.2 provided that δ1(t) = (2 + 1 2 (e − 1)t)t, δ2(t) = 1 2 (e − 1)t2, ϕ0(t1, t2) = 1 2 (e − 1)(δ1(t1) + δ2(t2)) δ3(t) = (2 + 1 2 (e − 1)h2(t))h2(t)t, δ4(t) = 1 2 (e − 1)h2(t) 2t2, ϕ(t1, t2, t3) = 1 2 (e − 1)(t1 + δ1(t2) + δ2(t3)) ϕ1(t1, t2, t3, t4) = 1 2 (e − 1)[δ1(t1) + δ2(t2) + δ3(t3) + δ4(t4)] and ϕ2(t) = 1 2 (e − 1)t. by solving, we get ρ0 = 0.426037 and hence m0 = [0, ρ0). the radii are obtained as r1 = 0.204146, r2 = 0.134409 and r3 = 0.126891. therefore, by the definition r = min{ri}, we get the radius of convergence, r = 0.126891. remark 5.2. a non-differentiable non-linear system is solved using the method (2), where the divided difference is defined by the 2×2 matrix given for t̄ = (t1, t2) ∈ r×r, t̃ = (t3, t4) ∈ r×r and g = (g1, g2) by [t̄ , t̃;g]i ,1 = gi(t3, t4)− gi(t1, t4) t3 − t1 , t3 6= t1 and [t̄ , t̃;g]i ,2 = gi(t1, t4)− gi(t1, t2) t4 − t2 , t4 6= t2. otherwise, we set [·, ·;g] = 0. the actual example is given below example 5.3. let us solve the non-linear and non-differentiable system given as 3t21 t2 + t22 − 1 + |t1 − 1| = 0 t41 + t1t 3 2 − 1 + |t2| = 0. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 11 then, we set g = (g1, g2), where g1(t1, t2) = 3t21 t2 + t22 − 1 + |t1 − 1| g2(t1, t2) = t41 + t1t 3 2 − 1 + |t2| choose the initial points (5, 5) and (1, 0). then, using the aforementioned divided difference and the method (2), we obtain the solution ξ = (x∗1 , x ∗ 2 ) after three iterations with x∗1 = 0.894655074977661 and x∗2 = 0.327826643198819. example 5.4. we consider the system of 25 equations 25∑ j=1,j 6=i xj − e−xi = 0, 1 ≤ i ≤ 25, with initial point x0 = {1.5, 1.5, . . . , 1.5}t . then, applying method (2) we get the solution ξ = {0.04003162719010837 · · · , 0.04003162719010837 · · · , . . . , 0.04003162719010837 · · · }t after 4 iterations. 6. conclusion a new procedure has been developed to demonstrate both local and semi-local convergenceanalysis of high-order convergence methods, using only derivatives that appear on the methodology.previous works have proven convergence based on the existence of high-order derivatives that maynot be present in the methodology. hence, it has been a limitation of their applicability. thisprocedure also offers error limits and uniqueness results that were not available before. moreover,this procedure is general in the sense that it is not dependent on the method itself. this is thereason why it may be used in the same way to broaden the scope of other methods of higher order,such as single and multi-step methods [1, 2, 4, 10–17,19,21,22]. references [1] a. cordero, j. l. hueso, e. martínez, j. r. torregrosa, a modified newton-jarratt’s composition, numer. algorithms55 (2010) 87-99. https://doi.org/10.1007/s11075-009-9359-z.[2] a. m. ostrowski, solutions of equations and system of equations, academic press (1960) new york.[3] f. a. potra, v. ptak, nondiscrete induction and iterarive processes. pitman publishing (1984) boston.[4] h. ren, q. wu, w. bi, a class of two-step steffensen type methods with fourth-order convergence, appl. math.comput. 209 (2009) 206–210. 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https://doi.org/10.1007/s11075-016-0254-0.[19] q. zheng, p. zhao, f. huang, a family of fourth-order steffensen-type methods with the applications on solvingnonlinear odes, appl. math. comput. 217 (2011) 8196–8203. https://doi.org/10.1016/j.amc.2011.01.095.[20] s. regmi, i. k. argyros, j. a. john, j. jayaraman, extended convergence of two multi-step iterative methods, foun-dations 3 (2023) 140–153. https://doi.org/10.3390/foundations3010013.[21] x. wang, t. zhang, a family of steffensen type methods with seventh-order convergence, numer. algorithms 62(2013) 429–444. https://doi.org/10.1007/s11075-012-9597-3.[22] z. liu, q, zheng, p. zhao, a variant of steffensen’s method of fourth-order convergence and its applications, appl.math. comput. 216 (2010) 1978-1983. https://doi.org/10.1016/j.amc.2010.03.028. https://doi.org/10.28924/ada/ma.3.19 https://doi.org/10.3390/foundations3010012 https://doi.org/10.1007/s40819-022-01404-3 https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1137/1.9780898719468.fm https://doi.org/10.2298/aadm130725016s https://doi.org/10.1007/s11075-014-9832-1 https://doi.org/10.1007/s40314-014-0193-0 https://doi.org/10.1155/2014/152187 https://doi.org/10.1016/j.cam.2010.09.019 https://doi.org/10.1016/j.cam.2012.06.005 https://doi.org/10.1016/j.cam.2012.06.005 https://doi.org/10.1007/s11075-016-0254-0 https://doi.org/10.1016/j.amc.2011.01.095 https://doi.org/10.3390/foundations3010013 https://doi.org/10.1007/s11075-012-9597-3 https://doi.org/10.1016/j.amc.2010.03.028 1. introduction 2. convergence 1: local 3. convergence 2: semi-local 4. isolation of a solution 5. experiments 6. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 12doi: 10.28924/ada/ma.4.12 conditional least squares estimation for fractional super levy processes in nonlinear spdes jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte, 376 fretwell bldg,9201 university city blvd. charlotte, nc 28223-0001, usaj.bishwal@uncc.edu abstract. we consider infinite dimensional extension of affine models as super levy processes sat-isfying a nonlinear spde. we obtain the asymptotics of the conditional least squares estimators.finally we obtain the berry-esseen inequality. 1. introduction and preliminaries parameter estimation in finite dimensional diffusions is now classical. bishwal [7] studied a newestimating function for discretely sampled diffusions. bishwal [8] studied asymptotic theory of like-lihood method and bayesian method for drift estimation of finite dimensional stochastic differentialequations. bishwal [12] studied applications of levy processes in stochastic volatility models infinance. bishwal [13] studied parameter estimation for spdes driven by cylindrical stable pro-cesses. bishwal [6] studied the bernstein-von mises theorem and spectral asymptotics of bayesestimators for parabolic spdes when the number of fourier coefficients becomes large. in thiscase, the measures generated by the process for different parameters are singular. bishwal [11]studied bernstein-von mises theorem and small noise bayesian asymptotics for parabolic stochas-tic partial differential equations. bishwal [10] studied hypothesis testing for fractional stochasticpartial differential equations with applications to neurophysiology and finance.consider the nonlinear spde dx(t, x) = 1 2 ∆x(t, x)dt + √ x(t, x)dw (t, x) (1.1) where w (t, x) a cylindrical brownian motion. konno and shiga [26] studied the existence andweak uniqueness of the above equation as a martingale problem for the associated super-brownianmotion. the pathwise uniqueness of nonnegative solution still remains open. the main difficultycomes from the unbounded drift coefficient and non-lipschitz diffusion coefficient. wang et al. [39]studied a comparison theorem and showed that the solution of the nonlinear spde is distribution received: 4 dec 2023. key words and phrases. nonlinear stochastic partial differential equations, super processes, fractional cox-ingersollross model, conditional least squares estimator, branching interacting particle system, berry-esseen inequality.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 2function valued. they also established pathwise uniqueness. as application they obtained well-posedness of martingale problems for two classes of measure-valued diffusions: interacting super-brownian motions and interacting fleming-viot processes. he et al. [21] obtained pathwise uniquesolution to nonlinear spde with super levy process, which is a combination of space-time gaussianwhite noises and poisson random measures which is a generalization of work of xiong [40] wherethe result for a super-brownian motion with binary branching mechanism was obtained. usingan extended yamada-watanabe argument, xiong [40] established strong existence and uniquenessof the solution to the spde. super-brownian motion (sbm), also called the dawson-watanabeprocess introduced by dawson and watanabe is a measure valued process arising as the limit ofempirical measure process of a branching particle system. sbm satisfies a martingale problem.when the state space is r, sbm has a density w.r.t. lebesgue measure and this density valuedprocess x(t, x) satisfies the above spde. when the space r is s single point, the spde becomesan sde which is cir diffusion dxt = √ xtdwt whose uniqueness is established using the yamada-watanabe argument. xiong and yang (2019) studied existence and pathwise uniqueness to anspde with hölder continuous coefficient driven by α-stable colored noise. the existence of thesolution is shown by considering the weak limit of a sequence of sde system which is obtained byreplacing the laplacian operator in the spde by its discrete version. the pathwise uniqueness isshown by using a backward doubly stochastic differential equation to take care of the laplacian.in the case of d = 1, the pathwise uniqueness of a nonnegative solution to the correspondingequation was established by yang and zhou [42] for 1 < α < √ 5− 1 and pathwise uniqueness for √ 5− 1 < α < 2 is still open.the existence and pathwise uniqueness of solutions to the sdes with non-lipschitz coefficientdriven by spectrally positive levy processes were studied in fu and li [20].consider the spde with multiplicative noise: duθ(t, x) = (a0 + θa1)uθ(t, x)dt +muθ(t, x)dz(t, x), t ≥ 0, x ∈ [0, 1] (1.2) where m is a known nonlinear operator.priola et al. [32] obtained exponential convergence to the invariant measure, in the total variationnorm, for solutions to sdes driven by α-stable noises in finite and infinite dimensions using twoapproaches: lyapounov’s function approach by harris and doeblin’s coupling argument. in bothapproaches irreducibility and uniform strong feller property play crucial role.equation (1.2) is called diagonalizable if a0, a1 and m have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. denote by ρk , νk and µk , the eigenvalues of the operators a0, a1 and m respectively. then uθ(t, x) = ∞∑ j=1 uj,thj . (1.3) we consider fractional stable cir model as example.using fractional levy process as the driving term, maximum quasi-likelihood estimation in frac-tional levy stochastic volatility model was studied in bishwal [9]. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 3fractional levy process (flp) is defined as mh,t = 1 γ(h + 1 2 ) ∫ r [(t − s) h−1/2 + − (−s) h−1/2 + ]dls , t ∈ r (1.4) where {lt , t ∈ r} is a levy process on r with e(l1) = 0, e(l2 1) <∞.here are some properties of the fractional levy process:1) the covariance of the process is given by cov(mh,t ,mh,s) = e(l2 1) 2γ(2h + 1) sin(πh) [|t|2h + |s|2h − |t − s|2h]. (1.5) 2) mh is not a martingale. for a large class of levy processes, mh is neither a semimartingale.3)mh is hölder continuous of any order β less than h − 1 2 . 4) mh has stationary increments. 5) mh is symmetric. 6) l is self-similar, but mh is not self-similar. 7) mh has infinite total variationon compacts.thus flp is a generalization and a natural counterpart of fbm. fractional stable motion is aspecial case of flp. 2. conditional least squares estimation let h be a real separable hilbert space with inner product 〈·〉 and norm | · |. by l(h) we denotethe banach space of bounded linear operators from h into h endowed with the operator norm ‖ · ‖l(h). we fix an orthonormal basis (en) in h. through the basis (en) we will often identify hin l2. more generally, for a given sequence ρ = (ρn) of real numbers we set l2ρ = {(xn) ∈ r∞ : ∑ n≥1 x2 nρ 2 n <∞}. where r∞ = rn. the space l2ρ becomes a separable hilbert space with the inner product: 〈x, y〉 =∑ n≥1 xnynρ 2 n for x = (xn), y = (yn) ∈ l2ρ . let us fix θ0, the unknown true value of the parameter θ.let (ω,f , p ) be a complete probability space and z(t, x) be a process on this space with valuesin the schwarz space of distributions d′(g) such that for φ,ψ ∈ c∞0 (g), ‖φ‖−1 l2(g) 〈w (t, ·), φ(·)〉is a one dimensional stable process.this process is usually referred to as the cylindrical α-stable process (c.s.p.), α ∈ (0, 2). weassume that there exists a complete orthonormal system {hi}∞i=1 in l2(g)) such that for every i = 1, 2, . . . , hi ∈ zm,20 (g) ∩ c∞(g) and λθhi = βi(θ)hi , and lθhi = µi(θ)hi for all θ ∈ θ where lθ is a closed self adjoint extension of aθ, λθ := (k(θ)i − lθ)1/2m, k(θ) is a constantand the spectrum of the operator λθ consists of eigenvalues {βi(θ)}∞i=1 of finite multiplicities and µi = −β2m i + k(θ). https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 4a levy process (zt) with values in h is an h-valued process defined on some stochastic basis (ω,f , (ft)t≥0, p ) having stationary independent increments, cadlag trajectories such that z0 = 0,p-a.s. one has that e[e i〈zt ,s〉] = exp(−tψ(s)), s ∈ hwhere ψ : h → c is sazonov continuous, negative definite function such that ψ(0) = 0. thefunction ψ is called the exponent of (zt).the exponent ψ can be expressed by the infinite dimensional levy-khintchine formula ψ(s) = 1 2 〈qs, s〉 − i〈a, s〉 − ∫ h ( e i〈s,y〉 − 1− i〈s, y〉 1 + |y |2 ) ν(dy), s ∈ h where q is the non-negative trace class operator on h, a ∈ h and ν is the levy measure or thejump intensity measure associated to (zt).cylindrical α-stable process (c.s.p.) is a levy process taking values in the hilbert space h = l2ρ ,with a properly chosen weight ρ.consider the linear spde dxt = θaxtdt + dzt , x ∈ h c.s.p. z(t) is a cylindrical α-stable process, α ∈ (0, 2) which can be expanded in the series z(t) = ∞∑ i=1 γizi(t)hi , t ≥ 0 where {zi(t)}∞i=1 are independent, real valued, one dimensional, normalized, symmetric, α-stableprocesses and (γi) ∞ i=1 is a given sequence of, possibly unbounded, positive numbers, and hi is afixed orthonormal basis in h. the latter series converges p -a.s. in h−α for α > d/2. indeed ‖z(t)‖2 −α = ∞∑ i=1 γ2 i z 2 i (t)‖hi‖2 −α = ∞∑ i=1 z2 i (t)β−2α i and the later series converges p -a.s.for any j ∈ n, t ≥ 0, e[e izj (t)h] = e−t|h| α . stable one-dimensional density: a one-dimensional, normalized, symmetric α-stable distribution µα, α ∈ (0, 2] has characteristic function µ̂α(s) = e−|s| α , s ∈ r. the density of µα with respect to lebesgue measure will be denoted by pα. this even functionis known in closed form only if α = 1 or 2. the precise asymptotic behavior of the density pα, α ∈ (0, 2) is as follows:for any α ∈ (0, 2), there exists cα such that pα(x) ∼ cα xα+1 as x →∞. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 5 stable measures on hilbert space: a random variable ξ on h is called α-stable (α ∈ (0, 2]) if forany n there exists a vector an ∈ h such that for any independent copies ξ1, ξ2, . . . , ξn of ξ, therandom variable n−1/α(ξ1 + ξ2, . . .+ ξn)− an has the same distribution as ξ. a borel probabilitymeasure µ on h is said to be α-stable if it is the distribution of a stable random variable withvales in h.consider the spde with multiplicative noise: duθ(t, x) = (a0 + θa1)uθ(t, x)dt +muθ(t, x)dz(t, x), t ≥ 0, x ∈ [0, 1] (2.1) where m is a known nonlinear operator and z(t, x) is a cylindrical subfractional levy process.equation (2.1) is called diagonalizable if a0, a1 and m have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. denote by ρk , νk and µk , the eigenvalues of the operators a0, a1 and m respectively. then uθ(t, x) = ∑∞ j=1 uj,thj .consider spde model with multiplicative noise and mean reversion, where the j-th fouriercoefficient is the stable cox-ingersoll-ross (scir) model: duj,t = (a − θuj,t)dt + σu 1/α j,t−dzj,t , j ≥ 1 (2.2) where a is the mean reverting level and θ is mean reverting speed. recall that for α = 2, for every j ≥ 1, the process zj,t is a standard brownian motion, this is the famous cox-ingersoll-ross (cir)model used for modeling interest rate, which is also used a stochastic volatility process in hestonmodel. note that there are brownian cir models with additive compound poisson type jumps.when 1 < α < 2, zj,t is stable process with levy measure να(dz) = 1{z>0}dz αγ(−α)zα+1 . (2.3) the discontinuous scir model captures the heavy tailed property in the sense of infinite variance.there is empirical evidence from high frequency data available in support of application of purejump models in financial modeling.the scir model has the unique stationary distribution µ with laplace transform given by lµ(λ) = ∫ ∞ 0 e−λxµ(dx) = exp { − ∫ λ 0 αa αθ + σαzα−1 dz } , λ ≥ 0. (2.4) now we focus on the fundamental semimartingale behind the cir model. define κh := 2hγ(3/2−h)γ(h + 1/2), kh(t, s) := κ−1 h (s(t − s)) 1 2 −h, ηh := 2hγ(3− 2h)γ(h + 1 2 ) γ(3/2−h) , vt ≡ vht := η−1 h t2−2h,mh t := ∫ t 0 kh(t, s)dmh s .for using girsanov theorem for brownian motion, since a radon-nikodym derivative process is al-ways a martingale, a central problem is how to construct an appropriate martingale which generatesthe same filtration, up to sets of measure zero, as the non-semimartingale called the fundamental martingale.extending norros et al. (1999) it can be shown that mh t is a martingale, called the funda-mental martingale whose quadratic variation 〈mh〉t is vht . moreover, the natural filtration of the https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 6 martingale mh coincides with the natural filtration of the flp mh since mh t := ∫ t 0 k(t, s)dmh sholds for h ∈ (1/2, 1) where kh(t, s) := h(2h − 1) ∫ t s r h− 1 2 (r − s)h− 3 2 dr, 0 ≤ s ≤ t and for h = 1/2, the convention k1/2 ≡ 1 is used.define qi(t) := d dvt ∫ t 0 kh(t, s)ui(s)ds. it is easy to see that qi(t) = ηh 2(2−2h) { t2h−1zi(t) + ∫ t 0 r 2h−1dzi(s) } . define the process zi = (zi(t), t ∈ [0, t ]) by zi(t) := ∫ t 0 kh(t, s)dui(s).extending kleptsyna and le breton (2002), we have:(i) zi is the fundamental semimartingale associated with the process ui .(ii) zi is a (ft) -semimartingale with the decomposition zi(t) = µi(θ) ∫ t 0 qi(s)dvs + β−νi m h t .(iii) ui admits the representation ui(t) = ∫ t 0 kh(t, s)dzi(s). (iv) the natural filtration (zi(t)) of zi and (ui(t)) of ui coincide.we describe our observations now. note that for equally spaced data (homoscedastic case) vtk − vtk−1 = η−1 h ( t n )2−2h [k2−2h − (k − 1)2−2h], k = 1, 2, · · · , n.for h = 0.5, vtk − vtk−1 = η−1 h ( t n )2−2h [k2−2h − (k − 1)2−2h] = t n , k = 1, 2, . . . , n. we have qi(t) = d dvt ∫ t 0 kh(t, s)ui(s)ds = κ−1 h d dvt ∫ t 0 s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 d dt ∫ t 0 s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 ∫ t 0 d dt s1/2−h(t − s)1/2−hui(s)ds = κ−1 h ηht 2h−1 ∫ t 0 s1/2−h(t − s)−1/2−hui(s)ds. the process qi depends continuously on ui and therefore, the discrete observations of ui does notallow one to obtain the discrete observations of qi . the process qi can be approximated by qi(n) = κ−1 h ηhn 2h−1 n−1∑ j=0 j1/2−h(n − j)−1/2−hui(j). it is easy to show that qi(n)→ qi(t) almost surely as n →∞, see tudor and viens (2007).define a new partition 0 ≤ r1 < r2 < r3 < · · · < rmk = tk , k = 1, 2, · · · , n. define qi(tk) = κ−1 h ηht 2h−1 k mk∑ j=1 r 1/2−h j (rmk − rj) −1/2−hui(rj)(rj − rj−1), k = 1, 2, · · · , n. it is easy to show that qi(tk)→ qi(t) almost surely as mk →∞ for each k = 1, 2, · · · , n. we usethis approximate observation in the calculation of our estimators.applying itô’s formula, for t ≥ r ≥ 0, we obtain qj,t = e−θ(t−r)qj,r + a ∫ t r e−θ(t−s)ds + σ ∫ t r e−θ(t−s)q 1/α j,s−dzj,s , j ≥ 1. (2.5) let the process be observed at {kh, k = 0, 1, . . . , n} from a single realization {qj,t , t ≥ 0} for fixed h. for simplicity, we take h = 1. this equation can be considered as a first order autoregressive https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 7(ar(1)) equation qj,k = ρ+ γqj,k−1 + εj,k , j ≥ 1 (2.6)where γ = e−θ, ρ = aθ−1(1− γ) and εj,k = σ ∫ k k−1 e−θ(k−s)q 1/α j,s−dzj,s , k ≥ 1, j ≥ 1. (2.7) for b ∈ b(r+), let s2,j,n(b) = n∑ k=1 qj,k−1εk ib(|qj,k−1εj,k |), s1,j,n(b) = n∑ k=1 q2 j,k−1ib(qj,k−1), j ≥ 1. (2.8) it is easy to see that εj,k = qj,k − e(qj,k |fk−1), k ≥ 1, j ≥ 1. (2.9)is a sequence of martingale differences for every fixed j .let s1,j,n := s1,j,n(0,∞), s2,j,n := s2,j,n(0,∞) and recall that γ = e−θ .then θ̂j,n − θ = s2,j,n s1,j,n (2.10) where θ̂n is the conditional least squares estimator (clse) which minimizes n∑ k=1 ε2 j,k = n∑ k=1 [qj,k − e(qj,k |fk−1)]2 = n∑ k=1 [qj,k − ρ− γqj,k−1]2 (2.11) and are given by γ̂j,n = ∑n k=1 qj,k−1 ∑n k=1 qj,k − n ∑n k=1 qj,k−1qj,k ( ∑n k=1 qj,k−1)2 − n ∑n k=1 q 2 j,k−1 , ρ̂j,n = 1 n n∑ k=1 qj,k − γ̂n 1 n n∑ k=1 qj,k−1, θ̂j,n = − log γ̂j,n, âj,n = ρ̂nθ̂n 1− γ̂n .let (s1, s2) have the characteristic function given by e[exp{iλ1s1 + iλ2s2}] := exp { − σα θ2γ(−α) ∫ ∞ 0 e ( 1− exp{iλ1y 2 + iλ2y (α+1)/αvj,1} ) × e ( exp { ie−2θλ1y 2 1− e−2θ + ie−θ(α+1)/αλ2y (α+1)/αvj,2 (1− eθ(α+1))1/α }) dy yα+1 } (2.12) and vj,k := σ ∫ k k−1 e−θ(k−s)e−θ(s−k+1)/αdzj,s , k = 1, 2, j ≥ 1 (2.13) which are i.i.d. with the same distribution as σ ( e−θ − 1 (α− 1)θ )1/α zj,1 which is regularly varying with index α. the limit distribution is normal only in the gaussian case α = 2. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 8following li and ma [28] it can be shown that for every fixed j , if we have 1 < α < (1 + √ 5)/2,then we have as n →∞ (d−2 n s1,j,n, c −1 n s2,j,n) d→(s1, s2) on r2 where dn = n1/α and cn = n(α+1)/α2 = d (α+1)/α n .for the stable spde model, we have the following result on the consistency and the limit dis-tribution of the clse: theorem 2.1 if we have 1 < α < (1 + √ 5)/2, then for every fixed j ≥ 1a) θ̂j,n →p θ as n →∞.b) n(α−1)/α2 (θ̂j,n − θ)→d ( σ2 ν2 j )1/α s2 s1 as n →∞. c) if in addition, limj→∞ ∣∣νj ∣∣ =∞, then for every fixed n ≥ 1, θ̂j,n →p θ as j →∞ and ∣∣νj ∣∣ (θ̂j,n − θ)→d σ ( n−(α−1)/α2 )1/α s2 s1 as j →∞.where s2 and s1 are defined in (2.12). remarks1) the limit distribution in the case (1 + √ 5)/2 < α < 2 is still open.2) the process (xj) is exponentially ergodic and hence strongly mixing.3) for the gaussian case (α = 2), the limit results are based on ergodic theory and martingaleconvergence theorem. for the non-gaussian case (1 < α < 2), limit results are obtained by thetheory of regular variation and convergence of point processes.4) let 0 < α < 2 and let zt be a one dimensional α-stable process with levy measure ν(dz).then as n →∞, np (n−1/αzt ∈ ·)→v tν(·). we consider the stable cox-ingersoll-ross model as an example. xiong and yang [41] studiedexistence and strong uniqueness of the following spde: duk(t) = (θνk + ρk)uk(t)dt + σk(uk(t))1/αdzk(t), k ≥ 1. the existence of the solution in the case of space-time white noise is shown by considering theweak limit of a sequence of sde systems which is obtained by replacing the laplacian operatorin the spde by its discrete version. the weak uniqueness follows from the uniqueness of solutionto the martingale problem for the associated super-brownian motion. in the case of α-stable noisethe existence and pathwise uniqueness of the solution is studied in xiong and yang [41]. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 9 3. interacting particle systems first consider the fractional cox-ingersoll-ross (fcir) model d yt = a(b − yt)dt + σ √ ytdw h t (3.1) where wh t is a fractional brownian motion with hurst parameter h > 1/2.then by proposition 5.7 of buchmann and kluppelberg [15], we have yt = f (xt) (3.2) where dxt = a(b −xt)dt + dwh t , x0 = f −1(y0), t ∈ [0, t ] (3.3) and f (x) = sgn(x)σ2x2/4.let b = 0, σ = 1 and a > 0. then xt is described by the ornstein-uhlenbeck sde dxt = −axtdt + dwh t , x0 = f −1(y0). (3.4) for h = 0.5, let us consider maximum likelihood estimator (mle) for the simple mean-field model dxj(t) = αxj(t)dt − β(xj(t)− x̄n(t))dt + dwj(t), xj(0) = xj(0), j = 1, 2, · · · , n (3.5) where x̄n(t)) = n−1 ∑n j=1xj(t), β 6= α, and α 6= 0. the middle term on the right side of (3.5)can be viewed as an interaction among the subsystems which create a tendency for the subsystemsto relax towards the center of gravity of the ensemble. thus the system provides a simple exampleof a cooperative interaction. mean-field type models have applications in physics, biology andeconomics, see dawson [19]. the case β = 0 corresponds to sampling independent replications ofornstein-uhlenbeck processes on [0, t ]. our parameter here is θ = (α, β).suppose 1 n ∑n j=1 xj(0)→ ν0 almost surely and 1 n ∑n j=1 x 2 j (0)→ γ2 0 +ν2 0 almost surely as n →∞.then the estimator θ̂n →p θ as n →∞ and √n(θ̂n − θ)→d n (0, i−1(t )) as n →∞ where i(t ) = ( a(t ) −b(t ) −b(t ) b(t ) ) with a(t ) := ν2 0 2α (e2αt − 1) + b(t ), b(t ) := e2(α−β)t − 1 4(α− β)2 − t 2(α− β) + γ2 0 (e2(α−β)t − 1) 2(α− β) . the case β = 0 corresponds to sampling independent replications of the same process given below: dxj(t) = αxj(t)dt + dwj(t), j = 1, 2, · · · , n (3.6) https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 10in the classical case when β = 0, the mle is given by α̂n = ∑n j=1 ∫ t 0 xj(t)dxj(t)∑n j=1 ∫ t 0 (xj(t))2dt . sampling n independent ornstein-uhlenbeck processes on [0, t ] and letting n → ∞ give weakconsistency and asymptotic normality of the mle: α̂n →p α and √n(α̂n−α)→d n (0, 2α ν2 0 (e2αt−1) )as n →∞. see also bishwal (2010) for independent sampling case.for h ≥ 0.5, let us consider maximum likelihood estimator (mle) for the fractional mean-fieldmodel dxj(t) = αxj(t)dt − β(xj(t)− x̄n(t))dt + dwh j (t), xj(0) = xj(0), j = 1, 2, · · · , n (3.7) where x̄n(t)) = n−1 ∑n j=1xj(t), β 6= α, and α 6= 0.the case β = 0 corresponds to sampling independent replications of the same process givenbelow: dxj(t) = αxj(t)dt + dwh j (t), j = 1, 2, · · · , n (3.8)first consider the fcir model d yj(t) = a(b − yj(t))dt + σ √ yj(t)dw h j (t), j = 1, 2, · · · , n (3.9) where wh j (t) is a fractional brownian motion with hurst parameter h > 1/2.then by proposition 5.7 of buchmann and kluppelberg [15], we have yj(t) = s(xj(t)) (3.10) where dxj(t) = a(b −xj(t))dt + dwh j (t), xj(0) = s−1(yj(0)), t ∈ [0, t ], j = 1, 2, · · · , n (3.11) and s(x) = sgn(x)σ2x2/4. here s is the state space transform.let b = 0, σ = 1 and a > 0. then xj(t) is described by the ornstein-uhlenbeck sdes dxj(t) = −axj(t)dt + dwh j (t), xj(0) = s−1(yj(0)), j = 1, 2, · · · , n (3.12) consider the model of n interacting particles of fractional diffusions satisfying the itô stochasticdifferential equations dxj(t) = p∑ l=1 θlµj l(x(t)) + σj(x(t))dwh j (t), j = 1, 2, · · · , n (3.13) where x(t) = (x1(t), x2(t), · · · , xn(t))′ and (wh j (t); t ≥ 0), j = 1, 2, · · · , n are independentfractional wiener processes. here θl(·) ∈ l2([0, t ], dt), l = 1, . . . , p are unknown functions to beestimated based on observation of the process x in the time interval [0, t ]. let θ = (θ1, θ2, . . . , θp)and µj(x) = (µj1(x), µj2(x), . . . , µjp(x))′. the processes xj(t), j = 1, 2, · · · , n are observed on [0, t ].the functions µj , σj ; j = 1, 2, · · · , n are assumed to be known such that the system has a uniquesolution. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 11we need the following assumption and results to prove the main results. (a0) suppose that bj l := µj l(s)σ−1 j (s); j = 1, 2, · · · , n; l = 1, 2 . . . , p are measurable and adaptedprocesses satisfying 1 n n∑ j=1 ∫ t 0 bj l(s)bjm(s)ds → clm(t) a.s. as n →∞ l , m = 1, 2 . . . , p where clm(t) are finite and continuous nonrandom functions of t ∈ [0, t ]. thelimiting matrix i(t) = (clm(t))l ,m=1,2...,p is positive definite, δ′i(t)δ is increasing for all δ ∈ rpand i(0) = 0.in the exchangeable case, (a0) follows from mckean-vlasov law of large numbers. in particular,(a0) will be satisfied when µj l(x) = µlxj and σj(x) = σ(xj) which corresponds to the independentreplicated sampling on [0, t ]. see oelschlager [29].we also need the following version of rebolledo’s central limit theorem for martingales, seerebolledo [34]: theorem 3.1 let mn, n ∈ z+ be a sequence of locally square integrable martingales with mn(0) = 0. suppose the following condition holds: ∑ s≤t e{|∆mn(s)|2i(|∆mn(s)| > ε)} → 0 for all t ∈ [0, t ], ε > 0; and 〈mn〉(t)→ c(t) a.s. for all t ∈ [0, t ], where c(t) is a continuous increasing function with c(0) = 0. then mn →d m , a continuous gaussian martingale with zero mean and covariance function k(s, t) = c(s ∧ t), s, t ∈ [0, t ] where ∆ms = ms −ms− denotes the jump of m at the point s.the model is given by dxj(t) = p∑ l=1 θlµj l(x(t)) + σj(x(t))dwh j (t), j = 1, 2, · · · , n (3.14) where x(t) = (x1(t), x2(t), · · · , xn(t))′ and (wh j (t); t ≥ 0), j = 1, 2, · · · , n are independentfractional wiener processes. here θ = (θ1, θ2, . . . , θp) is the unknown parameter. the functions µj l , σj , j = 1, . . . , n; l = 1, . . . , p are assumed to be known such that there exists a unique solution x(t) to the above sde.our aim is to estimate the parameter θ based on n particles q1(·), q2(·), · · · , qn(·) of q(t) on [0, t ]. we denote this data by qn,t .the radon-nikodym derivative (likelihood) is given by λθn(qn,t ) := dpθ dp0 (qn,t ) = exp {∑p l=1 θl ∑n j=1 ∫ t 0 µj l(q(t))σ−2 j (q(t))dqj(t) −1 2 ∑p l=1 ∑p m=1 θlθm ∑n j=1 ∫ t 0 µj l(q(t))σ−2 j (q(t))µjm(q(t))dt } . (3.15) 4. approximate maximum likelihood estimation https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 12the approximate maximum likelihood estimator is defined as θ̂n = arg max θ λθn(qn,t ).extending kasonga [24], using mckean-vlasov law of large numbers and rebolledo’s centrallimit theorem for martingales, we obtain the consistency and asymptotic normality of the approx-imate maximum likelihood estimator θ̂n which is given below: theorem 4.1 under (a0), we have a) θ̂n →p θ as n →∞. b) √ n(θ̂n − θ)→d n (0, i−1(t )) as n →∞ where i(t ) is the fisher information. 5. berry-esseen inequality in this section we consider the case h = 0.5 and α = 2, i.e., the standard brownian motion case. dxj(t) = fj(θ,x(t)) + σj(x(t))dwj(t), xj(0) = x0 j , j = 1, 2, · · · , nwe assume the following conditions for j = 1, 2, · · · , n: (a1) |fj(θ, x)| ≤ aj(θ)(1 + |x |), |fj(θ, x)− fj(θ, y)| ≤ aj(θ)|x − y |.(a2) |fj(θ, x)− fj(φ, y)| ≤ bj(x)|θ − φ| for all θ, φ ∈ θ, x, y ∈ rwhere supθ∈θ |aj(θ)| = a <∞, e|bj(x0 j )|r <∞ for any integer r.(a3) the diffusion process x is stationary and ergodic with invariant measure ν, i.e., for any gjwith e[gj(·)] <∞, 1 n ∑n j=1 ∑m i=1 gj(xti )→ eν [g(x0)] a.s. as n →∞ and h → 0.(a4) supt≥0 e|xj(t)|r <∞ for all r ≥ 0.(a5) e|fj(θ,x0 j )− fj(θ0, x 0 j )|2 = 0 iff θ = θ0.(a6) fj is twice continuously differentiable function in x for all θ.(a7) fj(·, x) and all its derivatives are three times continuously differentiable with respect to θfor all x ∈ r. moreover, these derivatives upto third order with respect to θ are of polynomialgrowth in x uniformly in θ.the fisher information is given by 0 < i(θ) := ∫∞ −∞(f ′j (θ, x))2dν(x) <∞ and for any δ > 0, orany compact θ̄ ⊂ θ, inf θ0∈θ̄ sup |θ−θ0|>δ eθ0 |f ′j (θ,x0)− f ′j (θ0, xj(0))|2 > 0. (a8) the malliavin covariance of the process is nondegenerate. let fj = µj/σj , j = 1, 2, · · · , n. the model is given by dxj(t) = p∑ l=1 θlµj l(x(t)) + σj(x(t))dwj(t), j = 1, 2, · · · , n. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 13the radon-nikodym derivative (likelihood) is given by lθn(xn,t ) := dpθ dp0 (xn,t ) = exp {∑p l=1 θl ∑n j=1 ∫ t 0 µj l(x(t))σ−2 j (x(t))dxj(t) −1 2 ∑p l=1 ∑p m=1 θlθm ∑n j=1 ∫ t 0 µj l(x(t))σ−2 j (x(t))µjm(x(t))dt } . we observe the process {xt} at times 0 = t0 < t1 < · · · tm = t with ti − ti−1 = t m = h, i = 1, 2 · · · , n. we assume equispaced sampling for simplicity with t being fixed, m →∞ and n →∞.the dataset is n particles x1(·), x2(·), · · · , xn(·) of x(t) on [0, t ]. the approximate log-likelihood based on observations xj(t1), xj(t2), . . . , xj(tn), j = 1, 2, · · · , n with ti = it/m = ihis defined as kn,m(θ) = ∑p l=1 θl ∑n j=1 ∑m i=1 µj l(x(ti−1))σ−2 j (x(ti−1))(xj(ti)−xj(ti−1) − 1 2 ∑p l=1 ∑p k=1 θlθk ∑n j=1 ∑m i=1 µj l(x(ti−1))σ−2 j (x(ti−1)))µjk(x(ti−1))(ti − ti−1). we start with some preliminary lemmas. the first lemma is from michel and pfanzagl (1971)which will be needed to prove our main results. lemma 5.1 let ξ, ζ and η be any three random variables on a probability space (ω,f , p ) with p (η > 0) = 1. then, for any ε > 0, we have (a) sup x∈r |p{ξ + ζ ≤ x} −φ(x)| ≤ sup x∈r |p{ξ ≤ x} −φ(x)|+ p (|ζ| > ε) + ε, (b) sup x∈r |p{ ξ η ≤ x} −φ(x)| ≤ sup x∈r |p{ξ ≤ x} −φ(x)|+ p{|η − 1| > ε}+ ε. the strong rate of convergence of particle approximations of mckean-vlasov sdes with lipschitzcoefficients is o(n−1/2) where n is the number of particles. this rate is driven by the statisticalerror. the bias is of the order o(n−1). talay and tubaro [37] showed that for smooth coefficientsthe the weak error is o(h). bencheikh and jourdain [2] showed that weak error between a sdewith nonlinear in the sense of mckean given by moments and its approximation by the eulerdiscretization with time step h of a system of n interacting particles is o(n−1 + h).from talay and tubaro [37] and bencheikh and jourdain [2], we have lemma 5.2 let fj = µj/σj . then sup t∈π |e[fj(x n t )]− e[fj(x n,m t )]| ≤ c t m , j ≥ 1. the following lemma follows from yoshida [43,44]. lemma 5.3 let in(θ) := 1 ni(θ0) ∑n j=1 ∫ t 0 µ2 j (θ,xt)dt. then under the conditions (a1)-(a8), sup θ∈θ e[in(θ)− 1]2 ≤ cn−1. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 14the following lemma follows from theorem 1 in yoshida [44]. lemma 5.4 let mn := 1√ ni(θ0) ∑n j=1 ∫ t 0 µj(θ0, xt)dwt . then under the conditions (a1)-(a8), sup x∈r |pθ0 {mn ≤ x} −φ(x)| ≤ cn−1/2. in this section, our main result is the following theorem. theorem 5.5 under the conditions (a1)–(a8), we have sup x∈r ∣∣∣pθ0 {√ ni(θ0)(θm,n − θ0) ≤ x } −φ(x) ∣∣∣ = o ( n−1/2 ∨ t m ) . proof by taylor expansion, we have k′m,n(θm,n) = k′m,n(θ0) + (θm,n − θ0)k′′m,n(θ̄m,n) where ∣∣θ̄m,n − θ∣∣ ≤ |θm,n − θ0|. since k′m,n(θm,n) = 0, hence we have √ ni(θ0)(θm,n − θ0) = − 1√ ni(θ0) k′m,n(θ0) 1 ni(θ0)k ′′ m,n(θ̄m,n) = − 1√ ni(θ0) ∑n j=1 ∑m i=1 µ ′ j(θ0, xti−1 )∆wi 1 ni(θ0) ∑n j=1 ∑m i=1 µ ′′ j (θ̄m,n, xti−1 )∆ti =: um,n vm,n note that vm,n = 1 ni(θ0) n∑ j=1 m∑ i=1 µ′′j (θ̄m,n, xti−1 )∆ti = 1 ni(θ0) n∑ j=1 m∑ i=1 µ′j(θ̄m,n, xti−1 )2∆ti . let lim vm,n = vn in l2 as t m → 0. similar to lemma 5.3, it can be shown that e(vn − 1)2 ≤ cn−1 (see also pardoux and veretennikov (2001) and yoshida (2011)). it can be shown that e(vm,n − vn)2 ≤ c t m (see altmeyer and chorowski (2018)). hence e(vm,n − 1)2 = e[(vm,n − vn) + (vn − 1)]2 ≤ c(n−1 ∨ t m ). further by lemma 5.1 (b), we have sup x∈r ∣∣∣pθ {√ni(θ)(θm,n − θ) ≤ x } −φ(x) ∣∣∣ = sup x∈r ∣∣∣∣pθ {um,nvm,n ≤ x } −φ(x) ∣∣∣∣ = sup x∈r |pθ {um,n ≤ x} −φ(x)|+ pθ {|vm,n − 1| ≥ ε}+ ε ≤ c(n−1/2 ∨ t 2 m ) + ε−2c(n−1 ∨ t m ) + ε. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 15since by lemma 5.1 (a), lemma 5.2 and lemma 5.4, we have sup x∈r |pθ {um,n ≤ x} −φ(x)| ≤ sup x∈r |pθ {mn ≤ x} −φ(x)|+ pθ {|um,n −mn| ≥ ε}+ ε ≤ cn−1/2 + ε−2e |um,n −mn|2 + ε ≤ cn−1/2 + ε−2c t m + ε. choosing ε = n−1/2, we have the result. remarks we considered fractional levy process driving term in this paper whose incrementsare stationary. using fractional levy process as the driving term which include jumps, maximumquasi-likelihood estimation in fractional levy stochastic volatility model was studied in bishwal [9].recently, sub-fractional brownian (sub-fbm) motion which is a centered gaussian process withcovariance function ch(s, t) = s2h + t2h − 1 2 [ (s + t)2h + |s − t|2h ] , s, t > 0 for 0 < h < 1 introduced by bojdecki, gorostiza and talarczyk [14] has received some attentionrecently in finite dimensional models. the interesting feature of this process is that this processhas some of the main properties of fbm, but the increments of the process are nonstationary,more weakly correlated on non-overlapping time intervals than that of fbm, and its covariancedecays polynomially at a higher rate as the distance between the intervals tends to infinity. itwould be interesting to see extension of this paper to sub-fbm case. we generalize sub-fbm tosub-fractional levy process (sub-flp).sub-fractional levy process (sflp) is defined as sh,t = 1 γ(h + 1 2 ) ∫ r [(t − s) h−1/2 + − (−s) h−1/2 + ]dms , t ∈ r where mt , t ∈ r is a levy process on r with e(m1) = 0, e(m2 1 ) < ∞ and without browniancomponent. sflp has the following properties:1) the covariance of the process is given by cov(sh,t , sh,s) = s2h + t2h + e[l(1)2] 2γ(2h + 1) sin(πh) [|t|2h + |s|2h − |t − s|2h]. 2) sh is not a martingale. for a large class of levy processes, sh is neither a semimartingalenor a markov process. 3) sh is hölder continuous of any order β less than h − 1 2 . 4) sh hasnonstationary increments. 5) sh is symmetric. 6) sh is self similar. 7) sh has infinite totalvariation on compacts. https://doi.org/10.28924/ada/ma.4.12 eur. j. math. anal. 10.28924/ada/ma.4.12 16it would be 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10.28924/ada/ma.3.23 strong continuity of composition semigroups on the generalized bloch spaces of the upper half plane k. a. wandera1, j. o. bonyo2,∗ , d. o. ambogo1 1department of pure and applied mathematics, maseno university, p.o. box 333-40105, maseno kenya kwandera1@gmail.com, ambogos@maseno.ac.ke 2department of mathematics, multimedia university of kenya, p.o. box 15653-00503, nairobi kenya jbonyo@mmu.ac.ke ∗correspondence author abstract. we investigate strong continuity of composition semigroups on the generalized blochspaces of the upper half plane. these composition semigroups are induced by automorphisms ofthe upper half plane as classified into three distinct groups in [3]. 1. introduction consider h(ω) as the fréchet space of analytic functions f : ω→ c endowed with the topologyof uniform convergence on compact subsets of ω. a function f ∈ h(d) is in the bloch space of theunit disc b(d) if ‖f ‖b1(d) := sup z∈d (1− |z |2)|f ′(z)| <∞ and in the little bloch space of the unit disc b0(d) if lim |z |−→1 (1− |z |2)|f ′(z)| = 0. for f ∈ b(d), we define the norm on b(d) by ‖f ‖b(d) := |f (0)|+ ‖f ‖b1(d),where ‖.‖b1(d) is a seminorm on b(d).bloch space of the upper half plane b(u) is a set of analytic functions f ∈ h(u) such that ‖f ‖b1(u) := sup ω∈u =(ω)|f ′(ω)| <∞. for f ∈ b(u), we define the norm on b(u) by ‖f ‖b(u) := |f (i)|+ ‖f ‖b1(u), received: 5 jun 2023. key words and phrases. composition semigroup; analytic functions; self analytic maps; bloch spaces; unit disc; upperhalf plane; strong continuity; infinitesimal generator. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.23 https://orcid.org/0000-0002-6442-4211 where ‖.‖b1(u) is a seminorm on b(u).let α > 0 be a real number, we define the generalized bloch space of the unit disc, bα(d) asthe space of all functions f ∈ h(d) such that ‖f ‖bα1 (d) := sup z∈d ( 1− |z |2 )α |f ′(z)| <∞. for f ∈ bα(d), we define the norm on bα(d) by ‖f ‖bα(d) := |f (0)|+ ‖f ‖bα1 (d). (1) we also define the corresponding generalized little bloch space of the unit disc as the space of allfunctions f ∈ h(d) for which lim |z |→1 ( 1− |z |2 )α |f ′(z)| = 0, with the same norm given by (1). here, bα(d) and bα◦ (d) are both banach spaces with respectto the norm ‖.‖bα(d). the generalized little bloch space of the unit disc, bα◦ (d) is the closure ofthe set of polynomials in the norm topology of bα(d). for more details see [17, 18]. the space b(d) has been studied by many authors because of its intrinsic interest since its introduction[1, 4, 8, 10, 13, 14, 18]. in [17], the generalized bloch spaces of the open unit disc, bα(d) aredefined and proved to be banach spaces with respect to their norm. zhu [17] further establishedgeneralized little bloch spaces of the unit disc bα◦ (d), as closed, separable subspaces of bα(d).there is scanty literature on the properties of the generalized bloch spaces of the upper half plane bα(u), including whether they are banach spaces. composition semigroups on bloch spaces ofthe unit disc have been studied in literature, see for instance [2,11,12] and references therein. onstrong continuity of composition semigroups, siskakis [12] proved that no nontrivial compositionsemigroups are strongly continuous on the bloch space of the unit disc b(d). the correspondingstudy of composition semigroups defined on the bloch spaces of the upper half plane has not yetbeen exhausted. moreover, existing works on the half plane, see [7,13], have neither exhausted theinvestigation of properties of these semigroups nor considered these generalizations. in this papertherefore, we investigate the properties of the generalized bloch spaces of the upper half plane asbanach spaces and extend the study of semigroups of composition operators to the setting of thegeneralized bloch spaces of the upper half plane. 2. preliminaries and definitions let c be the complex plane. the set d := {z ∈ c : |z | < 1} is called the open unit disc.on the other hand, the set u := {ω ∈ c : =(ω) > 0} denotes the upper half of the complexplane c, where =(ω) is the imaginary part of ω ∈ c. the function ψ(z)= i(1+z) 1−z is referred to asthe cayley transform and maps the unit disc d conformally onto the upper half-plane u, with theinverse ψ−1(ω) = ω−i ω+i mapping the upper half plane u, onto the unit disc, d. we refer to [16] for2 details. let α > 0 be a real number. a function f ∈ h(u) belongs to the generalized bloch spaceof the upper half plane, bα(u) if ‖f ‖bα1 (u) := sup ω∈u = (ω)α |f ′(ω)| <∞ with the norm given by ‖f ‖bα(u) := |f (i)|+ ‖f ‖bα1 (u).the corresponding generalized little bloch space of the upper half plane, bα0 (u)is defined as bα◦ (u) := {f ∈ h(u) : lim =(ω)−→0 = (ω)α |f ′(ω)| = 0} having the same norm as bα(u). there is little literature on the properties of the generalized blochspaces of the upper half plane as banach spaces. let x be a banach space. a one-parameterfamily (tt)t≥0 is a semigroup of bounded linear operators on x , if(i) to = i (identity operator on x), and(ii) tt+s = tt ◦ ts for every t, s,≥ 0 (semigroup property).a semigroup (tt)t≥0 of bounded linear operators on x is strongly continuous if lim t→0+ ‖ttx − x‖ = 0 for all x ∈ x. the infinitesimal generator denoted by γ of (tt)t≥0 is defined by γx := lim t→0+ ttx − x t = ∂ ∂t (ttx) ∣∣∣∣ t=0 for each x ∈ dom(γ), where dom(γ) denotes the domain of γ given by dom(γ) = { x ∈ x : lim t→0+ ttx − x t exists} . we define a group of bounded linear operators as (tt)t∈r = tt , t ≥ 0, t−t , t ≥ 0. if both (tt)t≥0 and (t−t)t≥0 are semigroups on x . for more details see [5,6,9]. suppose ϕ : ω→ ωis a self analytic map. the composition operator induced by ϕ on h(ω) is defined as cϕ(f ) = f o ϕ, for all f ∈ h(ω). on the other hand, given t ≥ 0 we define a semigroup as a family (ϕt)t≥0 ofself analytic maps on ω satisfying the following properties(i) ϕ0(z) = z (identity map on ω).(ii) ϕt+s = ϕt ◦ ϕs ,∀ t, s ≥ 0 (semigroup property).(iii) ϕt → ϕ0 uniformly on compact subsets of ω as t → 0.3 composition semigroup induced by ϕt on h(ω) is defined as cϕt (f ) = f o ϕt , for all f ∈ h(ω). 3. generalized bloch spaces of the upper half plane in this section, we study properties of the generalized bloch spaces as banach spaces. we alsorelate functions in the generalized bloch space of the upper half plane u to their counterparts inthe unit disc d. following [17,18], it’s well known that bα(d) and bα0 (d) are banach spaces withrespect to the norm ‖.‖bα(d). moreover the set of analytic polynomials c[z ] := { ∞∑ n=0 an z n : z ∈ c } is dense in bα0 (d). these results are not explicitly clear from the literature in the setting of theupper half plane u.in the following theorem, we establish the completeness of bα(u) with respect to the norm ‖.‖bα(u). theorem 3.1. bα(u) is a banach space with respect to the norm ‖.‖bα(u) proof. it’s clear that (bα(u), ‖.‖bα(u) ) is a normed space. now we prove that the space bα(u)is complete in ‖.‖bα(u). let (fk)k denote a cauchy sequence in bα(u). for ε > 0, there exists n ∈ n such that ‖fk − fl‖bα(u) < ε, ∀ k, l > n. hence by the definition of the norm, we have forall ∀ k, l > n, |fk(i)− fl(i)|+ sup ω∈u = (ω)α |f ′k(ω)− f ′l (ω)| < ε, which means that |fk(i)− fl(i)| < ε and (=(ω))α |f ′k(ω)− f ′l (ω)| < ε, for ω ∈ u. so, (fk(i))k∈n is cauchy in c. by the completeness of c, (fk(i))k converges to a limit,say u0. similarly, (f ′k(ω) ) k∈n is cauchy in c and therefore converges to a limit, say g.since |f ′k(ω)− f ′l (ω)| < ε =(ω)α and f ′k(ω)→ g uniformly on compact subsets of u, then g ∈ h(u).now, take f such that f ′(ω) = g(ω)∀ω ∈ u and f (i) = u0.thus, ∀ ε > 0, ∃n such that ∀k, l > n , = (ω)α |f ′k(ω)− f ′l (ω)| < ε, ∀ω ∈ u. taking limits as l →∞, then ∀k > n , = (ω)α |f ′k(ω)− f ′(ω)| < ε, ∀ω ∈ u. it follows that ‖fk − f ‖bα(u) = |fk(i)− f (i)|+ sup ω∈u = (ω)α |f ′k(ω)− f ′(ω)| < ε 4 and so ‖fk − f ‖bα(u) → 0 as k →∞.now, it remains to show that f ∈ bα(u). we have = (ω)α |f ′(ω)| = = (ω)α |f ′(ω)− f ′k(ω) + f ′k(ω)| ≤ = (ω)α |f ′(ω)− f ′kω|+ = (ω)α |f ′k(ω)| < ε+ = (ω)α |f ′k(ω)| <∞ since (fk)k ⊂ bα(u).now, taking supremum over all ω ∈ u in the above equation, we have that sup ω∈u = (ω)α |f ′(ω)| <∞ which implies that f ∈ bα(u), as desired. � as an immediate consequence, we have corollary 3.2. b(u) is a banach space with respect to the norm ‖ . ‖b(u) proof. follows immediately by taking α = 1 in theorem 3.1. � under the norm ‖ . ‖bα(u), the space bα0 (u) also becomes a banach space as in the followingtheorem, theorem 3.3. bα0 (u) is a banach space with respect to the norm ‖ . ‖bα(u). proof. following theorem 3.1, we need to show that every sequence in bα0 (u) convergent in bα(u)has its limit in bα0 (u).let (fn) ⊂ bα0 (u) and g ∈ bα(u) be such that fn → g as n → ∞. we need to prove that g ∈ bα0 (u). since fn, g are holomorphic on compact subsets of u, and fn → g, we have f ′n → g′uniformly. now that fn ⊂ bα0 (u), we have lim =(ω)→0 (=(ω))α |f ′n(ω)| = 0,∀ n. (2) since limn→∞ f ′ n = g′, we have lim =(ω)→0 (=(ω))α |g′(ω)| = lim =(ω)→0 (=(ω))α | lim n→∞ f ′n(ω)| which is equivalent to lim =(ω)→0 (=(ω))α |g′(ω)| = lim n→∞ ( lim =(ω)→0 (=(ω))α |f ′n(ω)| ) . following equation (2), we see that lim =(ω)→0 (=(ω))α |g′(ω)| = 0. so, g ∈ bα0 (u), completing the proof. �5 as a consequence, we have the following, corollary 3.4. b0(u) is a banach space with respect to the norm ‖.‖b(u) proof. follows immediately by taking α = 1 in theorem 3.3. � in the next results, we generate a relationship between functions in the generalized bloch spaceof the upper half plane u and their counterparts in the unit disc d proposition 3.5. let f ∈ bα(u) and ψ be the cayley transform, then f ∈ bα(u) if and only if f ◦ ψ ∈ bα(d) proof. it suffices to prove that ‖f ‖bα1 (u) <∞ if and only if ‖f ◦ ψ‖bα1 (d) <∞. let f be a functionin bα(u). then by definition, ‖f ‖bα1 (u) = supω∈u=(ω)α|f ′(ω)| <∞. now, by changing variables, let ω = ψ(z), where ψ is the cayley transform. then =(ω) = ω − ω 2i = ψ(z)− ψ(z) 2i . using ψ(z) = i(1+z) 1−z and ψ(z) = −i(1+z) 1−z , we have =(ω) = i(1+z) 1−z − −i(1+z) 1−z 2i = i(1 + z)(1− z) + i(1 + z)(1− z) 2i(1− z)(1− z) = i(2− 2zz) 2i(1− z)(1− z) = 1− |z |2 |1− z |2 .we get the absolute of ψ′(z) = 2i (1−z)2 as |ψ′(z)| = 2 |1− z |2 . (3) now, by definition we have ‖f ‖bα1 (u) = sup z∈d ( 1− |z |2 |1− z |2 )α |f ′(ψ(z))|. from equation (3), we have |1− z |2 = 2 |ψ′(z)| , therefore ‖f ‖bα1 (u) = 1 2α sup z∈d (1− |z |2)α|ψ′(z)|α|f ′(ψ(z))|. 6 since, (f ◦ ψ)′(z) = f ′(ψ(z))ψ′(z), we have |ψ′(z)|α|f ′(ψ(z))| = |ψ′(z)(f ◦ ψ)′(z)||ψ′(z)α−1| and hence ‖f ‖bα1 (u) = 1 2α sup z∈d (1− |z |2)α|ψ′(z)(f ◦ ψ)′(z)||ψ′(z)α−1| = 1 2α |ψ′(z)α−1|‖f ◦ ψ‖bα1 (d),which is finite if and only if ‖f ◦ ψ‖bα1 (d) is finite. this completes the proof. � an immediate consequence is the following, corollary 3.6. let f ∈ b(u) and ψ be the cayley transform, then ‖f ‖b1(u) = 1 2 ‖f ◦ ψ‖b1(d) (4) in particular, a function f ∈ b(u) if and only if f ◦ ψ ∈ b(d). proof. this follows immediately from proposition 3.5 by taking α = 1. � 4. composition semigroups on the generalized little bloch space of the upper half plane in [3], the non trivial automorphisms of the upper half plane u were classified according to thelocation of their fixed points into three distinct classes namely; scaling, translation and rotationgroups. in this section, we determine composition semigroups induced by these automorphismgroups of the upper half plane u, on the generalized bloch space of the upper half plane bα(u). wethen employ the theory of linear operators on banach spaces to investigate the semigroup propertiesof the induced composition semigroup. for any given semigroup ϕt , the induced operator semigroup cϕt is known to be strongly continuous on the little bloch space. on the other hand, no non trivialcomposition semigroup is strongly continuous on the bloch space, see [11]. therefore, we shalldetermine the composition semigroup induced by these automorphism groups on the generalizedlittle bloch space of the upper half plane, bα0 (u). further, we show that composition semigroupsinduced by scaling and translation groups are strongly continuous on bα0 (u). we also establishstrong continuity of composition semigroups induced by rotation group on bα0 (d). the infinitesimalgenerator is identified and its domain stated. 4.1. scaling group. the automorphisms of this group are of the form ϕt(z) = k tz , where z ∈ uand k, t ∈ r with k 6= 0. as noted in [3], the semigroup properties of the induced compositionoperators will differ significantly depending on whether 0 < k < 1 or k > 1. thus for 0 < k < 1,we consider without loss of generality, the analytic self maps ϕt : u −→ u of the form ϕt(z) = e−tz, z ∈ u. (5) the composition semigroup induced by equation (5) on bα0 (u) is given by cϕt f (z) = (f ◦ ϕt) (z) = f ( e−tz ) 7 it can be easily proved that (cϕt )t∈r is a group on bα0 (u).in what follows, we prove that the composition semigroup given by equation (4.1) fails to be anisometry on bα0 (u). proposition 4.1. the operator cϕt fails to be an isometry on bα0 (u). proof. by the definition of the norm, we have for all f ∈ bα0 (u) ‖cϕt f ‖bα(u) = |cϕt f (i)|+ sup ω∈u =(ω)α| (cϕt f )′ (ω)| = |f (e−t i)|+ sup ω∈u =(ω)α|e−t f ′(e−tω)|. now by change of variables:let z = e−tω, then ω = etz , and =(ω) = et=(z). therefore, ‖cϕt f ‖bα(u) = |f (e−t i)|+ sup z∈u etα=(z)α|e−t f ′(z)| = |f (e−t i)|+ e(α−1)t sup z∈u =(z)α|f ′(z)| 6= |f (i)|+ sup z∈u =(z)α|f ′(z)| = ‖f ‖bα(u), which completes the proof. � next, we prove that the operator cϕt given by (4.1) is strongly continuous on bα0 (u). theorem 4.2. (cϕt )t∈r is strongly continuous on bα0 (u). proof. to prove strong continuity of (cϕt )t∈r, it suffices to show that ‖cϕt f −f ‖bα(u) → 0 as t → 0.that is, | (cϕt f − f ) (i)|+‖cϕt f −f ‖bα1 (u) → 0 as t → 0. this is equivalent to | (cϕt f − f ) (i)| → 0and ‖cϕt f − f ‖bα1 (u) → 0, as t → 0. for the former, we have | (cϕt f − f ) (i)| = |cϕt f (i)− f (i)| (6) = |f (ϕt(i))− f (i)| = |f (e−t i)− f (i)| → 0 as t → 0, as desired. we now prove that ‖cϕt f − f ‖bα1 (u) → 0 as t → 0. recall that ψ : d→ u, ϕt : u→ u andψ−1 : u → d. we can therefore have d ψ−→ u ϕt−→ u ψ−1−−→ d. now, let xt = ψ−1 ◦ ϕt ◦ ψ : d→ d. if (ϕt)t≥0 is an automorphism of the upper half plane u, then (xt)t≥0 is an automorphismof the unit disc d. since xt = ψ−1 ◦ ϕt ◦ ψ, it follows that ‖cϕt f − f ‖bα1 (u) → 0 as t → 0 if andonly if ‖cxt f ∗ − f ∗‖bα(d) → 0 as t → 0 8 cayley transform is given by ψ(z) = i(1+z) 1−z . we therefore have ψ−1 ◦ ϕ−t ◦ ψ(z) = ψ−1 (ϕt (ψ(z))) . = ψ−1 ( ϕt ( i(1 + z) 1− z )) = ψ−1 ( e−t ( i(1 + z) 1− z )) . substituting ψ−1(z) = z−i z+i , we obtain ψ−1 ◦ ϕ−t ◦ ψ(z) = e−t( i(1+z)1−z )− i e−t( i(1+z)1−z ) + i . simplifying the fraction, we have ψ−1 ◦ ϕ−t ◦ ψ(z) = z + e−tz − 1 + e−t −z + e−tz + 1 + e−t . now, by factorizing z and dividing both the numerator and denominator by (1 + e−t), we obtain ψ−1 ◦ ϕ−t ◦ ψ(z) = z − (1−e −t) (1+e−t) 1− (1−e −t) 1+e−t z . let bt = 1−e−t 1+e−t , and substitute to obtain ψ−1 ◦ ϕ−t ◦ ψ(z) = z − bt 1− btz := xt(z). further, we apply density of polynomials in bα0 (d) to prove that for f ∗ ∈ bα0 (d), we have ‖cx t f ∗− f ∗‖bα1 (d) → 0 as t → 0.by the definition of the norm, we have lim t→0+ ‖cx t f ∗ − f ∗‖bα(d) = lim t→0+ |(cx t f ∗ − f ∗)(0)|+ sup z∈d ( 1− |z |2 )α |(cx t f ∗ − f ∗)′(z)|. let f ∗(z) = zn and z ∈ d. we need to show that ‖ (cxt f ∗ − f ∗) ‖bα1 (d) → 0, as t → 0.since cxtz n − zn = (xt(z))n − zn, n ≥ 1, differentiating (xt(z))n − zn with respect to z , we obtain (cxt f ∗ − f ∗)′(z) = n(xt(z))n−1x ′t(z)− nzn−1 = n[(xt(z))n−1x ′t(z)− zn−1]. substituting for xt(z) = z − bt 1− btz9 and x ′t(z) = (1− btz)1− (z − bt)(−bt) (1− btz)2 = (1− b2t ) (1− btz)2 , we obtain (cxt f ∗ − f ∗)′(z) = n [( z − bt 1− btz )n−1 (1− b2t ) (1− btz)2 − zn−1 ] = n [ (z − bt)n−1(1− b2t ) (1− btz)n−1(1− btz)2 − zn−1 ] = n [ (z − bt)n−1(1− b2t )− zn−1(1− btz)n+1 (1− btz)n+1 ] . it therefore follows that limt→0+ ‖cx t f ∗ − f ∗‖bα1 (d) is equivalent to lim t→0+ ( (sup z∈d ( 1− |z |2 )α ∣∣∣∣n [(z − bt)n−1(1− b2t )− zn−1(1− btz)n+1 (1− btz)n+1 ]∣∣∣∣) . now, let bt → 0 as t → 0, we obtain lim t→0+ ‖cx t f ∗ − f ∗‖bα1 (d) = sup z∈d (1− |z |2)α ∣∣n[zn−1 − zn−1] ∣∣ = 0. since limt→0+ ‖(cxt f ∗ − f ∗‖bα1 (d) = 0, it follows that lim t→0+ ( ‖cϕt f − f ‖bα1 (u) ) = 0. therefore ‖cϕt f − f ‖bα(u) = |ϕt f (i))− f (i)|+ ‖cϕt f − f ‖bα1 (u) → 0 as t→ 0, as desired. � in the next proposition, we compute the infinitesimal generator and determine the domain of thecomposition semigroup in equation (4.1). proposition 4.3. the infinitesimal generator γ of (cϕt )t≥0 on bα0 (u) is given by γf (z) = −zf ′(z) with the domain dom (γ) = {f ∈ bα0 (u) : zf ′(z) ∈ bα0 (u)}. proof. using the definition of the infinitesimal generator γ of (cϕt )t≥0, for f ∈ bα0 (u) we have γf (z) = lim t→0+ cϕt f (z)− f (z) t = lim t→0+ f ( e−tz ) − f (z) t = ∂ ∂t f (e−tz) ∣∣∣∣ t=0 = −zf ′(z).10 this implies that γf (z) = −zf ′(z) and therefore dom(γ) ⊆ {f ∈ bα0 (u) : zf ′ ∈ bα0 (u)}. to provereverse inclusion, we let f ∈ bα0 (u) be such that zf ′ ∈ bα0 (u). then for z ∈ u, cϕt f (z)− f (z) t = 1 t ∫ t 0 ∂ ∂s (cϕs f (z))ds = 1 t ∫ t 0 (−e−szf ′(e−sz))ds = 1 t ∫ t 0 cϕsf (z)ds, wheref (z) = −zf ′(z). since f (z) is a function in bα0 (u), it remains to show that the limit of f (z) exist in bα0 (u). thus lim t→0+ cϕs f (z)− f (z) t = lim t→0+ 1 t ∫ t 0 cϕsf (z)ds. by strong continuity of (cϕs )s≥0 we have 1 t ∫ t 0 ‖cϕsf − f‖ds → 0 as t → 0+. hence {f ∈ bα0 (u) : zf ′ ∈ bα0 (u)} ⊆ dom(γ).this completes the proof. � 4.2. translation group. in this case the automorphisms are of the form ϕt(z) = z + kt , where z ∈ u and k, t ∈ r with k 6= 0. as noted in [3], we can consider the self analytic maps of u of theform ϕt(z) = z + t. (7)the composition semigroup induced by translation group on bα0 (u) is given by cϕt f (z) = f (z + t). (8) the proof of our results given in equation (8) as a group on bα0 (u) is basic, we therefore omit thedetails.we shall now prove that the composition semigroup in equation (8), fails to be an isometry on bα0 (u). proposition 4.4. the operator cϕt fails to be an isometry on bα0 (u). proof. by norm definition, we have ‖cϕt f ‖bα(u) = |cϕt f (i)|+ sup z∈u =(z)α| (cϕt f )′ (z)| = |f (i + t)|+ sup z∈u =(z)α|f ′(z + t)|. now by change of variables: let z + t = ω then z = ω − t , and =(z) = =(ω). therefore, ‖cϕt f ‖bα(u) = |f (i + t)|+ sup ω∈u =(ω)α|f ′(ω)|. (9) 11 the right hand side of equation (9) is not equal to the norm ‖f ‖bα(u) for any t > 0. this impliesthat (8) is not an isometry on bα0 (u). this completes the proof. � in the following results, we investigate the strong continuity of the composition semigroup inequation (8) on bα0 (u). proposition 4.5. the operator cϕt is strongly continuous on bα0 (u). proof. we need to show that ‖cϕt f − f ‖bα(u) → 0 as t → 0. this approach is similar to (7). weomit the details. we compute the automorphism of the unit disc d, denoted by xt as follows xt(z) = ψ−1 (ϕt (ψ(z))) = ψ−1 ( ϕt ( i(1 + z) 1− z )) = ψ−1 ( i(1 + z) 1− z + t ) . since the inverse of cayley transform is given by ψ−1 = z−i z+i , we substitute to obtain xt = i(1+z) 1−z − t − i i(1+z) 1−z − t + i = i(1+z) 1−z − (t + i) i(1+z) 1−z + (i − t) . we simplify further by multiplying both the numerator and denominator by (1− z) to obtain xt(z) = i(1 + z) + (t − i)(1− z)) i(1 + z) + (t + i)(1− z) = (2i − t)z − t (2i + t)− tz .by dividing both the numerator and denominator by 2i − t , we get xt = z + t 2i−t 2i+t 2i−t − t 2i−t z.letting kt = t 2i−t and mt = 2i+t 2i−t . we have xt = z + kt mt − ktz . next, we apply density of polynomials in bα0 (d) to prove that for f ∗ ∈ bα0 (d), we have ‖cx t f ∗ − f ∗‖bα1 (d) → 0 as t → 0. lim t→0+ ‖cx t f ∗ − f ∗‖bα1 (d) = lim t→0+ ( sup z∈d ( 1− |z |2 )α |(cx t f ∗ − f ∗)′(z)| ) . using density of polynomials in bα0 (d), let f ∗(z) = zn and z ∈ d be such that cx tz n − zn = (xt(z))n − zn, n ≥ 1. (10)12 now, differentiating (xt(z))n − zn with respect to z , we get (cx t f ∗ − f ∗)′(z) = n(xt(z))n−1x ′t(z)− nzn−1 = n[(xt(z))n−1x ′t(z)− zn−1]. (11) we also differentiate xt = z+kt mt−ktz by quotient rule to obtain x ′t(z) = (mt − ktz)1− (z + kt)(−kt) (mt − ktz)2 = mt + k2t (mt − ktz)2 . substituting for xt = z+kt mt+ktz and x ′t(z) = mt−k2t (mt−ktz)2 in equation (11) we have (cx t f ∗ − f ∗)′(z) = n[(xt(z))n−1x ′t(z)− zn−1] = n [ (z + kt) n−1(mt − ktz2)− zn−1(mt − ktz)n+1 (mt − ktz)n+1 ] . it therefore follows that as t → 0, we have ‖cx t f ∗ − f ∗‖bα(d) = (|(xt(0))n − 0|) + (sup z∈d ( 1− |z |2 )α ∣∣n[(xt(z))n−1x ′t(z)− zn−1] ∣∣ = 0. therefore ‖cϕt f − f ‖bα(u) = |ϕt f (i)) − f (i)| + ‖cϕt f − f ‖bα1 (u) → 0 as t → 0, as desired. thiscompletes the proof. � in the next theorem, we obtain the infinitesimal generator of the strongly continuous compositionsemigroup given in equation (8). theorem 4.6. the infinitesimal generator γ of (cϕt )t≥0 on bα0 (u) is given by γf (z)=f ′(z) with the domain dom(γ) = {f ∈ bα0 (u) : f ′(z) ∈ bα0 (u)}. proof. using the definition of the infinitesimal generator γ, for f ∈ bα0 (u), we have; γf (z) = lim t→0+ f (z + t)− f (z) t = ∂ ∂t f (z + t) ∣∣∣∣ t=0 = f ′(z). this means that dom(γ) ⊂ {f ∈ bα0 (u) : f ′(z) ∈ bα0 (u)}.it remains to prove the reverse inclusion. let f ∈ bα0 (u) be such that f ′(z) ∈ bα0 (u).then for z ∈ u, we have; cϕt f (z)− f (z) = ∫ t 0 ∂ ∂s f (z + s)ds = ∫ t 0 f ′(z)ds. 13 letting f (z) = f ′(z), we obtain cϕt f (z)− f (z) = ∫ t 0 f (z)ds. this implies that f (z) = f ′(z) is a function of bα0 (u). it remains to show that the limit of f (z)exists in bα0 (u). since cϕt f (z)− f (z) t = 1 t ∫ t 0 f (z)ds, we now take limits as t → 0+ and invoke strong continuity of (cϕs )s≥0 to obtain lim t→0+ 1 t ∫ t 0 ‖cϕsfds − f‖ = 0. hence dom(γ) ⊇ {f ∈ bα0 (u) : f ′(z) ∈ bα0 (u)} which completes the proof. � 5. rotation group the induced composition semigroups for rotation group are defined on the analytic spaces of theunit disk. we shall therefore generate composition semigroups induced by rotation group on thegeneralized little bloch space of the disc. the results obtained can then be mapped onto the upperhalf plane by use of cayley transform. in this case, the self analytic maps of d are of the form ϕt(z) = e iktz . we consider the composition semigroup induced by the rotation group on bα0 (d)given by cϕt f (z) = (f ◦ ϕt) (z) = f ( e itz ) , (12) for all f ∈ bα0 (d).it can be easily shown that (cϕt )t≥0 and (cϕ−t)t≥0 are semigroups on bα0 (d) thus (cϕt )t∈rdefines a group on bα0 (d).moreover, this group is an isometry, as we prove in the next proposition. proposition 5.1. the operator cϕt given by (12) is an isometry on bα0 (d). proof. we shall prove that for each t ∈ r, the group (cϕt )t∈r is an isometry on bα0 (d). it sufficesto prove that ‖cϕt f ‖bα(d) = ‖f ‖bα(d).it follows from the definition that ‖cϕt f ‖bα(d) = |cϕt f (0)|+ sup z∈d ( 1− |z |2 )α |(cϕt f )′(z)| = |(e it)f (0)|+ sup z∈d ( 1− |z |2 )α |e it f ′(e itz)| = |f (0)|+ sup z∈d ( 1− |z |2 )α |f ′(e itz)|. 14 now, let ω = e itz so that z = e−itω. then; ‖cϕt f ‖bα(d) = |f (0)|+ sup ω∈d ( 1− |e−itω|2 )α |f ′(ω)|) = |f (0)|+ sup ω∈d (1− |ω|2)α|f ′(ω)| = ‖f ‖bα(d). � theorem 5.2. the operator cϕt given by (12) is strongly continuous on bα0 (d). proof. since polynomials are dense in bα0 (d), it suffices to show that (cϕt )t∈r is strongly contin-uous on bα0 (d) that is, for a polynomial (zn)n≥0 where z ∈ d we obtain lim t→0+ ‖cϕtzn − zn‖bα(d) = 0. clearly, lim t→0+ ‖cϕtzn − zn‖bα(d) = lim t→0+ |cϕt f (0)− f (0)|+ ( sup z∈d (1− |z |2)α|(cϕtzn − zn)′|) ) . but cϕtz n − zn = (e int − 1)zn.so its derivative is given by (cϕtz n − zn)′ = n(e int − 1)zn−1, implying that lim t→0+ ‖cϕtzn − zn‖bα(d) = lim t→0+ |e it f (0)− f (0)|+ ( sup z∈d (1− |z |2)α|nzn−1||(e int − 1)|) ) . hence, lim t→0+ ‖cϕtzn − zn‖bα(d) = 0 as desired . � proposition 5.3. the infinitesimal generator γ of (cϕt ) is given by γf (z) = iz f ′(z) with the domain dom(γ) = {f ∈ bα0 (d) : zf ′(z) ∈ bα0 (d)}. proof. we obtain the infinitesimal generator as follows γf (z) = lim t→0+ cϕt(z)− f (z) t = ∂ ∂t f (e 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theory in function spaces, marcel dekker, inc., new york and basel, 1990. https://doi.org/ 10.1090/surv/138.[17] k. zhu, bloch type spaces of analytic functions, rocky mountain j. math. 23 (1993), 1143-1177. https://www. jstor.org/stable/44237763.[18] k. zhu, spaces of holomorphic functions in the unit ball, springer-verlag, new york, 2006. https://doi.org/10. 1007/0-387-27539-8. 17 https://doi.org/:10.4134/bkms.2007.44.3.475 https://doi.org/10.4134/bkms.b160572 https://doi.org/10.4134/bkms.b160572 https://doi.org/10.1109/icic.2010.170 https://doi.org/10.1109/icic.2010.170 https://doi.org/10.1090/surv/138 https://doi.org/10.1090/surv/138 https://www.jstor.org/stable/44237763 https://www.jstor.org/stable/44237763 https://doi.org/10.1007/0-387-27539-8 https://doi.org/10.1007/0-387-27539-8 1. introduction 2. preliminaries and definitions 3. generalized bloch spaces of the upper half plane 4. composition semigroups on the generalized little bloch space of the upper half plane 4.1. scaling group 4.2. translation group 5. rotation group references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 5doi: 10.28924/ada/ma.2.5 unilateral problem for a viscoelastic beam equation type p-laplacian with strong damping and logarithmic source ducival c. pereira1, geraldo m. de araújo2, carlos a. raposo3,∗ 1department of mathematics, state university of pará, belém, pa, 66113-200, brazil ducival@uepa.br 2department of mathematics, federal university of pará, belém, pa, 66075-110, brazil gera@ufpa.br 3department of mathematics, federal university of são joão del-rei, são joão del-rei, 36307-352, brazil ∗correspondence: raposo@ufsj.edu.br abstract. in this manuscript, we investigate the unilateral problem for a viscoelastic beam equationof p-laplacian type. the competition of the strong damping versus the logarithmic source term isconsidered. we use the potential well theory. taking into account the initial data is in the stabilityset created by the nehari surface, we prove the existence and uniqueness of global solutions by usingthe penalization method and faedo-galerkin’s approximation. 1. introduction we denote the p-laplacian operator by ∆pu = div ( |∇u|p−2∇u ), which can be extended to amonotone, bounded, hemicontinuos and coercive operator between the spaces w 1,p 0 (ω) and its dualby −∆p : w 1,p 0 (ω)→ w−1,q(ω), 〈−∆pu, v〉p = ∫ ω |∇u|p−2∇u · ∇v dx. in [3] the authors establish existence of global solution to the problem utt + ∆2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut + f (u) = 0 in ω× r+, (1.1) u = ∆u = 0 on γ× r+, (1.2) u(x, 0) = u0, ut(x, 0) = u1 in ω, (1.3) where ω is a bounded domain of rn with smooth boundary γ = ∂ω.equations of the type (1.1) are related to models of elastoplastic microstructure flows. asconsidered by an and peirce [1, 2], they are essentially of the form utt + uxxxx − a(u2 x )x = 0. received: 13 nov 2021. key words and phrases. unilateral problem; viscoelastic beam equation type p-laplacian; logarithmic source.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 2a more general equation, utt + ∆2u − div(σ(|∇u|2)∇u)− ∆ut + h1(ut) + h2(u) = h3(x), was considered by yang et al [22–24]. they studied de existence of attractors and their hausdorffdimensions. another related equation is utt + ∆2u − div(f0(∇u)) + kut = ∆(f1(u))− f2(u), which was considered by chueshov and lasiecka [12]the problem (1.1), with its memory term ∫ t 0 g(t− s)∆u(s)ds , can be regarded as a fourth-orderviscoelastic plate equation with a lower order perturbation of the p-laplacian type. this kind ofproblem can be also regarded as an elastoplastic flow equation with some kind of memory effect.we observe that for viscoelastic plate equation, it is usual consider a memory of the form∫ t 0 g(t − s)∆2u(s)ds, see for instance [10]. however, because the main dissipation of the system (1.1) is given by strongdamping −∆ut , here we consider a weaker memory, acting only on ∆u. there is a large literatureabout stability in viscoelasticity. we refer the reader to [11,13].a nonlinear perturbation of problem (1.1) is given by utt + ∆2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut + f (u) ≥ 0. (1.4) variational inequality theory was introduced by hartman and stampacchia (1966) [14] as a toolfor the study of partial differential equations with applications principally in mechanics.in [7] the authors investigated the unilateral problem associated with this perturbation, thatis, a variational inequality given for (1.4) (see [16]). making use of the penalization method andgalerkin’s approximations, they established existence and the uniqueness of strong solutions.the unilateral problem is very interesting because, in general, dynamic contact problems arecharacterized by nonlinear hyperbolic variational inequalities. variational inequality theory wasintroduced by hartman and stampacchia (1966) [14] as a tool for the study of partial differentialequations with applications principally in mechanics. bensoussan and lions (1982) [9] used vari-ational inequalities initially in the study of stochastic control. in [5] was obtained a variationalinequality for the navier-stokes operator with variable viscosity. in [6] was studied the contactproblem on the oldroyd model of viscoelastic fluids. by using results from the theory of monotoneoperators, was established the existence of weak solutions. in [8] was studied the problem forparabolic variational inequalities with volterra type operators. the authors proved the existenceand the uniqueness of the solution. for contact problems on elasticity and finite element method,see kikuchi-oden [15] and reference therein. in [18] was studied the unilateral problem for the https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 3klein-gordon operator with the nonlinearity of kirchhoff-carrier type. by using an appropriate pe-nalization was shown the existence and uniqueness of solutions for the perturbed equation. in [19]was considered the unilateral problem for a nonlinear wave equation with p-laplacian operatorand source term. by using an appropriate penalization, authors obtained a variational inequalityfor the equation perturbed and then the existence of solutions was proved.in this work, we propose to investigate the existence and uniqueness of solutions for the vari-ational inequality associated with the problem (1.4) with the source term f (u) = −|u|r−2u ln |u|.more precisely, we investigate the existence and uniqueness of solutions for the unilateral problem utt + ∆2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut ≥ |u|r−2u ln |u| in ω×r+, (1.5) u(x, 0) = u0(x), ut(x, 0) = u1(x) in ω, (1.6) u(x, t) = ∆u(x, t) = 0 on γ× r+. (1.7) this work is organized as follows: in section 2 we introduce the notation and some well-knownresults. in section 3 we introduce the potential theory suitable for our problem. in section 4 definestrong solution to the boundary value problem (1.5)-(1.7) and present the theorem of existence ofstrong solution. in section 5 we apply the penalization method. the existence of global solutionsis given by using faedo-galerkin approximation. finally, in section 6 we prove the result ofuniqueness. 2. preliminaries let ω be a bounded domain in rn with the boundary γ of class c2. for t > 0, we denote by qthe cylinder ω×(0, t ), with lateral boundary σ = γ×(0, t ). by 〈·, ·〉 we will represent the dualitypairing between a banach space x and x ′, x ′ being the topological dual of the space x , and by c we denote various positive constants. the inner product in h1 0(ω) and l2(ω) , respectively, willbe denoted by (∇·,∇·), (·, ·). the norm in lp(ω) will be denoted by | · |p .the inequality (1.5) must be satisfied in the following sense. let k = {v ∈ h1 0(ω); v ≥ 0 a.e. in ω} be a closed and convex subset of h1 0(ω), the unilateral problem consists to find a solution u(x, t)satisfying∫ q (utt + ∆2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut − |u|r−2u ln |u|)(v − ut) ≥ 0, (2.1) for all v ∈ k with ut(x, t) ∈ k a.e. on [0, t ] and the initial and boundary data u = ∆u = 0 in γ× (0, t ), (2.2) u(x, 0) = u0, ut(x, 0) = u1 in ω. (2.3) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 4to study the existence and uniqueness of the problem (1.5)-(1.7), let us consider the followinghypotheses: h1. suppose that  2 ≤ p , if n = 1, 2 2 ≤ p ≤ 2n − 2 n − 2 , if n ≥ 3.h2. with respect to the power r , let us suppose that 2 < r < +∞ , if n = 1, 2 2 < r < 2n n − 2 , if n ≥ 3. h3. with respect to the function g : [0,+∞)→ r, we will assume that g ∈ c1[0, t ] and g(0) > 0, i = 1− µ ∫ ∞ 0 g(s)ds > 0, where µ > 0 is the embedding constant for |∇u| ≤ √µ |∆u|, for all u ∈ h1 0(ω) ∩h2(ω). h4. there exists a constant k1 > 0 such that g′(t) ≤ −k1g(t), ∀t ≥ 0. by h1 we have h1 0(ω) ∩h2(ω) ↪→ w 1,2(p−1) 0 (ω) ↪→ h1 0(ω) ↪→ l2(ω). the lemmas below will be a important role in this manuscript. lemma 2.1. (sobolev poincaré inequality) let p be a number with 2 < p < ∞ if n = 1, 2 or 2 ≤ p ≤ 2n n − 2 if n ≥ 3, then there exists a constant c > 0 such that |u|p ≤ c|∇u|,∀u ∈ h1 0(ω) lemma 2.2. (technical lemma) for v ∈ c1(0, t ;h1 0(ω)), we have∫ ω ∫ t 0 g(t − s)∇v · ∇vtdsdx = 1 2 (g′ � ∇v)(t)− 1 2 g(t)|∇v(t)|2 − 1 2 d dt [ (g � ∇v)(t)− (∫ t 0 g(s)ds ) |∇v(t)|2 ] , where (g � ∇u)(t) = ∫ t 0 g(t − s)|∇u(s)−∇u(t)|2ds . proof. differentiating the term (g � ∇u)(t) we arrive to the above inequality. � https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 53. potential well in this section, we use the potential well theory, a power full tool in the study of the globalexistence of solution in partial differential equation. see payne-sattinger [17]. it is well-knownthat the energy of a pde system, in some sense, splits into kinetic and potential energy. the sourceterm induces potential energy in the system that acts in opposition to the effect of the stabilizingmechanism. in this sense, it is possible that the energy from the source term destabilizes all thesystem and produces a blow-up in a finite time. to provide a global solution, we are able toconstruct a stability set corresponding to the source term created from the nehari manifold, see y.ye [20]. in the stability set, there exists a valley or a well of the depth d created in the potentialenergy. if d is strictly positive, then we find that, for solutions with the initial data in the goodpart of the potential well, the potential energy of the solution can never escape the potential well.in general, the energy from the source term causes the blow-up in a finite time. however, thegood part of the potential well is an invariant set where it remains bounded. as a result, the totalenergy of the solution remains finite for any time interval [0, t ], providing the global existence ofthe solution.for the model considered here, the total energy is given by e(t) = 1 2 [ |ut(t)|2 + |∆u(t)|2 + 2 p |∇u(t)|pp + (g � ∇u)(t) − (∫ t 0 g(s)ds ) |∇u(t)|2 + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] (3.1) and satisfies d dt e(t) ≤ −|∇ut(t)|2. (3.2) from (h3) we get i(t) = 1 2 [ |ut(t)|2 + ( 1− µ ∫ t 0 g(s)ds ) |∆u(t)|2 +(g � ∇u(t)+ 2 p |∇u(t)|pp + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] . (3.3) then, we introduce the functional j : h1 0(ω) ∩h2(ω)→ r defined by j(u) = 1 2 [( 1− µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g � ∇u)(t) + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] . (3.4) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 6for u ∈ h1 0(ω) ∩h2(ω), we have j(λu) = λ2 2 ( 1− µ ∫ t 0 g(s)ds ) |∆u|2 + λp p |∇u(t)|pp + λ 2 (g � ∇u)(t) + λr r2 |u(t)|rr − λr r ∫ ω |u(t)|r ln |u(t)|dx. (3.5) associated with j , we have the well-known nehari manifold given by n def = { u ∈ h1 0(ω) ∩h2(ω) \ {0}; [ d dλ j(λu) ] λ=1 = 0 } (3.6) or equivalently, n = { u ∈ h1 0(ω) ∩h2(ω)) \ {0}; ( 1− µ ∫ t 0 g(s)ds ) |∆u|2 + |∇u(t)|pp + 1 2 (g � ∇u)(t) = ∫ ω |u(t)|r ln |u(t)|dx } . (3.7) we define as in the mountain pass theorem due to ambrosetti and rabinowitz [4] d def = inf u∈(h1 0(ω)∩h2(ω)\{0} sup λ≥0 j(λu). similar to the result in [21] one has 0 < d = inf u∈n j(u).now, we introduce w = {u ∈ h1 0(ω) ∩h2(ω); j(u) < d} ∪ {0} and partition it into two sets w = w1 ∪w2 as follows w1 = { u ∈ w ; ( 1− µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g � ∇u)(t) + 2 r2 |u(t)|rr > 2 r ∫ ω |u(t)|r ln |u(t)|dx } ∪ {0} (3.8) and w2 = { u ∈ w ; ( 1− µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g � ∇u)(t) + 2 r2 |u(t)|rr < 2 r ∫ ω |u(t)|r ln |u(t)|dx } . (3.9) so, we define by w1 the set of stability for the problem (1.5)-(1.7), and before starting the sectionof existence and uniqueness of solution, we will prove thatw1 is invariant set for sub-critical initialenergy. proposition 1. let u0 ∈ w1 and u1 ∈ h1 0(ω). if e(0) < d then u(t) ∈ w1. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 7 proof. let t > 0 be the maximum existence time. from (3.2) we get e(t) ≤ e(0) < d, for all t ∈ [0, t ). and then, 1 2 ∫ ω |ut(t)|2 dx + j(u(t)) < d, for all t ∈ [0, t ). (3.10) arguing by contradiction, we suppose that there exists a first t0 ∈ (0, t ) such that i(u(t0)) = 0and i(u(t)) > 0 for all 0 ≤ t < t0, that is,( 1− µ ∫ t 0 g(s)ds ) |∆u(t0)|2 + 2 p |∇u(t0)|pp + (g � ∇u)(t0) + 2 r2 |u(t0)|rr = 2 r ∫ ω |u(t0)|r ln |u(t0)|dx from the definition of n , we have that u(t0) ∈ n , which leads to j(u(t0)) ≥ inf u(t)∈n j(u(t)) = d. we deduce 1 2 ∫ ω |ut(t0)|2 dx + j(u(t0)) ≥ d, which contradicts with (3.10). then u(t) ∈ w1 for all t ∈ [0, t ). � 4. existence of strong solutions next, we shall state the main results of this paper. theorem 4.1. consider the space h3 γ(ω) = {u ∈ h3(ω)|u = ∆u = 0 on γ}. if u0 ∈ w1 ∩h3 γ(ω), j(u0) < d, u1 ∈ h1 0(ω) and the hypothesis (h1)-(h4) holds, then there exists a function u : ω× (0, t )→ r such that u ∈ l∞(0, t ; (h1 0(ω) ∩h2(ω))) ∩ l∞(0, t ;h3 γ(ω)), (4.1) ut ∈ l∞(0, t ;l2(ω)) ∩ l2(0, t ;h1 0(ω) ∩h2(ω)), (4.2) utt ∈ l∞(0, t ;h−1(ω)), (4.3) ut(t) ∈ k a.e. in [0, t ], (4.4) ∫ t 0 [ 〈utt , v − ut〉+ (∆2u, v − ut)− (∆pu, v − ut) + (∫ t 0 g(t − s)∆u(s)ds, v − ut ) − (∆ut , v − ut) − (|u|r−2u ln |u|, v − ut) ] ≥ 0, (4.5) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 8 for all v ∈ l2(0, t ;h1 0(ω)), v(t) ∈ k a.e. in t and initial data u(0) = u0, ut(0) = u1. the proof of theorem 4.1 is given in section 5 by the penalization method. it consists in con-sidering a perturbation of the problem (1.5) adding a singular term called penalization, dependingon a parameter ε > 0. we solve the mixed problem in q for the penalization operator and theestimates obtained for the local solution of the penalized equation, allow to pass to limits, when εgoes to zero, in order to obtain a function u which is the solution of our problem. first of all, letus consider the penalization operator β : h1 0(ω) −→ h−1(ω) associated to the closed convex set k, cf. lions [16], p. 370. the operator β is monotonous,hemicontinuous, takes bounded sets of h1 0(ω) into bounded sets of h−1(ω), its kernel is k and β : l2(0, t ;h1 0(ω)) −→ l2(0, t ; (h−1(ω)) is monotone and hemicontinous. the penalized problem associated with the variational inequality(1.5)-(1.7), consists in given 0 < ε < 1, find uε satisfying uεtt + ∆2uε − ∆pu ε + ∫ t 0 g(t − s)∆uε(s)ds − ∆uεt + 1 ε (β(uεt ))− |uε|r−2uε ln |u| = 0, in q (4.6) and uε(x, 0) = uε0(x), uεt (x, 0) = uε1(x) in ω. uε(x.t) = ∆uε(x, t) = 0 on ∂ω× r+. (4.7) definition 4.2. suppose that uε0 ∈ w1, j(uε0) < d , uε1 ∈ h1 0(ω) and hypothesis (h1) − (h4) holds. a strong solution to the boundary value problem (4.6)-(4.7) is a function uε such that uε ∈ l∞(0, t ;h1 0(ω) ∩h2(ω)), uεt ∈ l∞(0, t ;l2(ω)) ∩ l2(0, t ;h1 0(ω)), uεtt ∈ l2(0, t ; (h1 0(ω) ∩h2(ω))′) satisfying for all w ∈ h1 0(ω) ∩h2(ω) d dt (uεt (t), w) + (∆uε(t),∆w) + (−∆pu ε(t), w) + ∫ t 0 g(t − s)(∆uε(s), w)ds +(∇uεt (t),∇w) + 1 ε (β(uεt (t)), w)−(|uε(t)|r−2uε(t) ln |uε(t)|), w) = 0 https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 9 and initial data uε(0) = uε0, u ε t (0) = uε1. the solution of problem (4.6)-(4.7) is given by the following theorem: theorem 4.3. assume that hypotheses (h1)− (h4) holds, uε0 ∈ w1, j(uε0) < d and uε1 ∈ h1 0(ω), (4.8) then, for each 0 < ε < 1, there exists a function uε strong solution of (4.6)-(4.7). 5. penalization method in order to prove theorem 4.1, we first prove the penalized theorem 4.3. the existence ofglobal solutions will be given by using faedo-galerkin method. first we consider the approximateproblem. then we obtain the a priori estimates needed to passage to the limit in the approximatesolutions. 5.1. approximate problem. let {wj} be the galerkin basis given by eigenfunctions of ∆2 withboundary condition u = ∆u = 0 on γ × r+ and let vm ⊂ n be the subspace spanned by thevectors w1, w2, ..., wm.. consider uεm(t) = m∑ j=1 gεjm(t)wj solution of approximate problem (uεmtt (t), w)+(∆uεm(t),∆w)+(−∆pu εm(t), w)+ ∫ t 0 g(t − s)(∆uεm(t), w)ds −(|uεm(t)|r−2uεm(t) ln |uεm|, w) + (∇uεm(t),∇w) + 1 ε (β(uεmt )(t), w) = 0 (5.1) with initial conditions uεm(0) = uε0m → uε0 strongly in h2(ω) ∩h1 0(ω), (5.2) uεmt (0) = uε1m → uε1 strongly in l2(ω). (5.3) the system of ordinary differential equation (5.1) in the variable t has a local solution uεm(t)defined in [0, tm[, 0 < tm ≤ t . in the next step obtain priori estimates for the solution uεm(t) thatpermits us to extend this solution to the whole interval [0, t ]. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 105.2. first estimate. we consider w = uεmt in (5.1) to obtain d dt [ 1 2 |uεmt (t)|2 + 1 2 |∆uεm(t)|2 + 1 p |∇uεm(t)|pp + 1 r2 |uεm(t)|pp − 1 r ∫ ω |uεm(t)|r ln |uεm(t)|dx ] + |∇uεmt (t)|2 + 1 ε (β(uεmt (t)), uεmt (t)) = ∫ t 0 g(t − s)(∇uεm(s),∇uεmt (t))ds. (5.4) we have (β(uεmt (t)), uεmt (t)) ≥ 0. then from lemma 2.2 and (h4) 1 2 d dt [ |uεmt (t)|2 + |∆uεm(t)|2 + 2 p |∇uεm(t)|pp + (g � ∇uεm)(t) − (∫ t 0 g(s)ds ) |∇uεm(t)|2 + 2 r2 |uεm(t)|rr − 2 r ∫ ω |uεm(t)|r ln |uεm(t)|dx ] + |∇uεmt (t)|2+ ≤ 1 2 (g′ � ∇uεm)(t)− 1 2 g(t)|∇uεm(t)|2 ≤ 0. (5.5) let eεm(t) = 1 2 [ |uεmt (t)|2 + |∆uεm(t)|2 + 2 p |∇uεm(t)|pp + (g � ∇uεm)(t) − (∫ t 0 g(s)ds ) |∇uεm(t)|2 + 2 r2 |uεm(t)|rr − 2 r ∫ ω |uεm(t)|r ln |uεm(t)|dx ] . (5.6) so, by (5.5) and (5.8), we have d dt eεm(t) ≤ −|∇uεmt (t)|2. integrating from 0 to t , t ≤ tm, we obtain eεm(t) + ∫ t 0 |∇uεmt (t)|2 ≤ eεm(0). (5.7) by (h3), it follows 1 2 [ |uεmt (t)|2 + ( 1− µ ∫ t 0 g(s)ds ) |∆uεm(t)|2 +(g � ∇uεm)(t)+ 2 p |∇uεm(t)|pp + 2 r2 |uεm(t)|rr − 2 r ∫ ω |uεm(t)|r ln |uεm(t)|dx ] + ∫ t 0 |∇uεmt |2ds ≤ eεm(t) ≤ eεm(0) = 1 2 |uε1m|2 + c1j(uε0m), (5.8) where c1 > 0 is a positive constant, independent of m and t . https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 11we have j(u0εm) < d and by (5.3), there exists a constant c2 > 0 such that |uεmt (t)|2 + ( 1− µ ∫ t 0 g(s)ds ) |∆uεm(t)|2 +(g � ∇uεm)(t)+ 2 p |∇uεm(t)|pp + 2 r2 |uεm(t)|rr − 2 r ∫ ω |uεm(t)|r ln |uεm(t)|dx + ∫ t 0 |∇uεmt |2ds ≤ c2. (5.9) from (3.7) and (5.9) we get ∆uεm ⇀ ∆uε in l∞(0, t ;l2(ω)), (5.10) uεm ⇀ uε in l∞(0, t ;h1 0(ω) ∩h2(ω)), (5.11) −∆pu εm ⇀ χ in l2(0, t ;h−1(ω)), (5.12) uεmt ⇀ uεt in l∞(0, t ;l2(ω)) ∩ l2(0, t ;h1 0(ω)), (5.13) β(uεmt ) ⇀ ψ in l2(0, t ;h−1(ω)). (5.14) follows from(5.11), (5.13) and aubin-lions theorem, for any t > 0, uεm → uε in l2(0, t ;h1 0(ω)), strong and a.e. in q. (5.15) now, we prove that χ(t) = −∆pu ε(t). we consider x, y ∈ r, p ≥ 2. then the elementaryinequality ∣∣|x |p−2x − |y |p−2y ∣∣ ≤ c (|x |p−2 + |y |p−2 ) |x − y | (5.16)is a consequence of the mean value theorem. using (5.16) and hölder generalized inequality with p − 2 2(p − 1) + 1 2 + 1 2(p − 1) = 1, we deduce for θ ∈ d(0, t ) and v ∈ vm,∣∣∣∣∫ t 0 〈(−∆uεmp (t))− (−∆uεp(t)), v〉pθ(t)dt ∣∣∣∣ = ∣∣∣∣∫ t 0 ∫ ω ( |∇uεm(t)|p−2∇uεm(t)− |∇uε(t)|p−2∇uε(t) ) ∇v dx θ(t) dt ∣∣∣∣ ≤ c|θ|∞ ∫ t 0 ∫ ω ( |∇uεm(t)|p−2 + |∇uε(t)|p−2 ) |∇uεm(t)−∇uε(t)||∇v | dx dt ≤ c1 ∫ t 0 ( |∇uεm(t)|p−2 2(p−1) + |∇uε(t)|p−2 2(p−1) ) |∇uεm(t)−∇uε(t)||∇v |2(p−1) dt ≤ c2 ∫ t 0 |∇uεm(t)−∇uε(t)| dt (5.17) where c1 and c2 are positive constants independent of m and t . https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 12now, from estimate (5.10) and (5.11), we have d dt |∇uεm(t)−∇uε(t)|2 ≤ 2|∆(uεm(t)− uε(t))||∇(uεmt (t)− uεt (t))| ≤ c3, where c3 is a constant independent of m and t . so, |uεm(t)− uε(t)|h1 0(ω) ∈ h1[0, t ] ↪→ c[0, t ], whence ∇uεm(t)→ ∇uε(t) a. e. in [0, t ]. therefore, χ = −∆pu ε. now, we observe that sobolev inequality∫ ω ||uεm(t)|r−2uεm(t) ln |uεm(t)||2dx ≤ |uεm(t)|2r2r ≤ c2r |∇uεm(t)|2r ≤ µrc2r |∆uεm(t)|r ≤ c4, where c4 is a constant independent of m and t.then (|uεm|r−2uεm ln |uεm|) is bounded in l2(0, t ;l2(ω)) = l2(q). (5.18) using continuity of function s → |s|r−2s ln |s| and (5.15) we have |uεm|r−2uεm ln |uεm| → |uε|r−2uε ln |uε| a.e. in q. (5.19) by (5.18), (5.19) and applying lions lemma (lemma 1.3, page 12, [16]), we get |uεm|r−2uεm ln |uεm|⇀ |uε|r−2uε ln |uε| weakly in l2(0, t ;l2(ω)). (5.20) 5.3. second estimate. let us consider the initial data uε0 ∈ h3 γ(ω), uε1 ∈ h1 0(ω) and uεm0 = ∆uεm0 = 0 on γ. (5.21) we consider w = −∆uεmt in approximate equation (5.1).then we have d dt { 1 2 |∇uεmt (t)|2 + 1 2 |∇∆uεm(t)|2 } + 〈∆puεmt (t),∆uεmt (t)〉 +|∆uεmt (t)|2 + 1 ε (β(uεmt (t),−∆uεmt (t)) = (|uεm(t)|r−2uεm ln |uεm(t)|,−∆uεmt (t)) + ∫ t 0 g(t − s)(∆uεm(s),∆uεmt (t))ds. now, 〈∆puεm(t),∆uεmt (t)〉 = d dt 〈∆puεm(t),∆uεm(t)〉 − j1, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 13where j1 = ∫ ω { (p − 2)|∇uεm(t)|p−4(∇uεm(t) · ∇uεmt (t))∇uεm(t) +|∇uεm(t)|p−2∇uεmt (t) } · ∇∆uεm(t)dx. then d dt { 1 2 |∇uεmt (t)|2 + 1 2 |∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉 } +|∆uεmt (t)|2 + 1 ε (β(uεmt (t)),−∆uεmt (t)) = j1 + j2 + j3. (5.22) where j2 = ∫ ω |uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t) and j3 = ∫ t 0 g(t − s)(∆uεm(s),∆uεmt (t))ds. let us the right hand side of (5.22). we denote by c a generic positive constant not dependingon m, t . by estimate (5.9) and p − 2 2(p − 1) + 1 2(p − 1) + 1 2 = 1, |j1| ≤ (p − 1) ∫ ω |∇uεm(t)|p−2|∇uεmt (t)||∇∆uεm(t)|dx ≤ (p − 1)|∇uεm(t)|p−2 2(p−1) |∇uεmt (t)|2(p−1)|∇∆uεm(t)| ≤ c|∇uεmt (t)|2(p−1)|∇∆uεm(t)|. how h1 0(ω) ∩h2(ω) ↪→ w 1,2 0 (ω), we have |∇uεmt (t)|22(p−1) ≤ µ2|∆uεmt (t)|2, where µ2 > 0 is the corresponding embedding constant. then |j1| ≤ 1 2 |∆uεmt (t)|2 + c|∇∆uεm(t)|2. (5.23) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 14let ω1 = {x ∈ ω : |uεm(t)| < 1} and ω2 = {x ∈ ω : |uεm(t)| ≥ 1}. by (5.9) and sobolevinequality |j2| ≤ ∫ ω1 ||uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t)|dx + ∫ ω2 ||uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t)|dx ≤ (e(r − 1))−1 ∫ ω |∆uεmt (t)|dx + (e(r − 1))−1 ∫ ω |uεm(t)|r−1|∆uεmt (t)|dx ≤ 2(e(r − 1))−2 + 1 8 |∆uεmt (t)|2 + 2(e(r − 1))−2|uεm(t)|2(r−1) 2(r−1) + 1 8 |∆uεmt (t)|2 ≤ 2(e(r − 1))−2 + 1 4 |∆uεmt (t)|2 + 2c(e(r − 1))−2|∇uεm(t)|2(r−1) ≤ c + 1 4 |∆uεmt (t)|2 (5.24) where we have used |x r−1 ln x | ≤ (e(r − 1))−1 for 0 < x < 1 and ln x ≤ (e(r − 1))−1x r−1, if x ≥ 1. remark 5.1. we note from the cauchy-schwarz inequality and fubini’s theorem follows ‖g � ∇u‖l2(q) ≤ ‖g‖l1(0,∞)‖∇u‖l2(q) again from estimate (5.9) and remark 5.1 |j3| ≤ (∫ t 0 g(t − s)|∆uεm(t)|ds ) |∆uεmt (t)| (5.25) ≤ c‖g‖l1(r+)|∆uεmt (t)| ≤ c + 1 4 |∆uεmt (t)|2. follows from (5.22)-(5.25) that d dt [ 1 2 |∇uεmt (t)|2 + 1 2 |∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉 ] + 1 2 |∆uεmt (t)|2 + 1 ε (β(uεmt (t)),−∆uεmt (t)) ≤ c + c|∇∆uεm(t)|2. (5.26) now, observe that |〈∆puεm(t),∆uεm(t)〉| ≤ ∫ ω |∇uεm(t)|p−1|∇∆uεm(t)|dx ≤ |∆uεm(t)|p−1 2(p−1) |∇∆uεm(t)| (5.27) ≤ c + |∇∆uεm(t)|2, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 15and then c + |∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉 ≥ 0. therefore, there exists c0 > 0 such that d dt [ 1 2 |∇uεmt (t)|2 + 1 2 |∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉 ] + 1 2 |∆uεmt (t)|2 + 1 ε (β(uεmt (t)),−∆uεmt (t)) ≤ c0 + c0|∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉. (5.28) taking into account that (β(uεmt (t),−∆uεmt (t)) ≥ 0, (5.21), integrating from 0 to t and applyinggronwall inequality, we obtain |∇uεmt (t)|2 + |∇∆uεm(t)|2 + ∫ t 0 |∆uεmt (t)|2 ≤ c, (5.29) then uεm ⇀ uε in l∞(0, t ;h3 γ(ω)), weakly star. (5.30) uεmt ⇀ uεt in l2(0, t ;h1 0(ω) ∩h2(ω)), weakly (5.31) ∆uεm ⇀ ∆uε in l∞(0, t ;h1 0(ω)), weakly star. (5.32) 5.4. third estimate. let pm be the ortogonal projection pm : l2(ω)→ vm, that is pmφ = m∑ n=1 (φ,wj)wj , φ ∈ l2(ω). remark 5.2. by remark 5.1, we observe that if ψ ∈ l2(0, t ;h1 0(ω)) then ∫ t 0 g(t − s)ψ(s)ds ∈ l2(0, t ;h−1(ω)) and by (5.12) −∆pu εm ∈ l2(0, t ; (h−1(ω)). we obtain using the notation and ideas of lions [16], pages 75-76, remark 5.2 and estimatesabove that uεmtt ⇀ uεtt in l2(0, t ; (h−1(ω)), weakly. (5.33) (5.31), (5.33) and aubin-lions compactness theorem imply that there exists a subsequence from (uεmt ), still denoted by (uεmt ), such that uεmt → uεt strongly in l2(0, t ;h1 0(ω)) and a.e. in q. (5.34) now, we are in position to prove theorem 4.1. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 165.5. strong solution. let v ∈ l2(0, t ;h1 0(ω)) be v(t) ∈ k a. e. for t ∈ (0, t ). from (4.6)1follows that ∫ t 0 (uεtt , v − uεt )dt + ∫ t 0 (∆2uε, v − uεt )dt + ∫ t 0 (−∆pu ε, v − uεt )dt + ∫ t 0 (∫ t 0 g(t − s)∆uε(s)ds, v − uεt ) dt + ∫ t 0 (−∆uεt , v − uεt )dt − ∫ t 0 (|uε|r−2uε ln |uε|, v − uεt )dt = 1 ε ∫ t 0 (β(uεt ), uεt − v) dt = 1 ε ∫ t 0 (β(uεt )− βv, uεt − v) dt ≥ 0, (5.35) because v ∈ k (β(v) = 0) and β is monotone.from (5.11), (5.12), (5.15), (5.20), (5.30), (5.31), (5.33), (5.34) and the bannach-steinhauss the-orem, it follows that there exists a subsequence (uε)0<ε<1, such that it converge to u as ε → 0,that is uε ⇀ u in l∞(r+;h1 0(ω) ∩h2(ω)), (5.36) −∆puε ⇀ −∆pu in l2(0, t ;h−1(ω), (5.37) uε → u in l2(0, t ;h1 0(ω))and a.e. in q, (5.38) uε ⇀ u in l∞(0, t ;h3 γ(ω)), (5.39) uεt ⇀ ut in l2(0, t ;h1 0(ω) ∩h2(ω)), (5.40) uεtt ⇀ utt in l2(0, t ;h−1(ω)), (5.41) |uε|r−2uε ln |uε|⇀ |u|r−2u ln |u| in l2(0, t ;l2(ω)), (5.42) uεt → ut in l2(0, t ;h1 0(ω)) and a.e. in q. (5.43) the convergences above are sufficient to pass to the limit in (5.35) with ε > 0 to conclude that(4.5) is valid. to complete the proof of theorem 4.1, it remains to show that ut(t) ∈ k a.e.in the position, we observe that using convergences (5.10)-(5.16) and (5.30)-(5.32), making m → ∞ in (5.1), we can find uε such that uεtt + ∆2uε − ∆pu ε + ∫ t 0 g(t − s)∆uε(s)ds − ∆uεt − |uε|r−2uε ln |uε|+ 1 ε β(uεt ) = 0 in l2(0, t ;h−1(ω). (5.44) then, β(uεt ) = ε[−uεtt − ∆2uε + ∆pu ε − ∫ t 0 g(t − s)∆uε(s)ds + ∆uεt + |uε|r−2uε ln |uε|]. (5.45) so, β(uεt )→ 0 in d′(0, t ;h−1ω). https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 17from (5.45) it follows that β(uεt ) is bounded in l2(0, t ;h−1(ω)), therefore β(uεt ) ⇀ 0 weak in l2(0, t ;h−1ω). (5.46) on the other hand we deduce from (5.45) that 0 ≤ ∫ t 0 (β(uεt ), uεt ) dt ≤ ε c. (5.47) thus ∫ t 0 (β(uεt ), uεt )dt −→ 0. (5.48) we have that ∫ t 0 (β(uεt )− β(ϕ), uεt − ϕ) dt ≥ 0, ∀ϕ in l2(0, t ;h1 0(ω)), because β is a monotonous operator. thus,∫ t 0 (β(uεt ), uεt ) dt − ∫ t 0 (β(uεt ), ϕ) dt − ∫ t 0 (β(ϕ), uεt − ϕ) dt ≥ 0. (5.49) from (5.40), (5.46) and (5.48) we obtain∫ t 0 (β(ϕ), ut(t)− ϕ) dt ≤ 0. (5.50) taking ϕ = ut − λv , with v ∈ l2(0, t ;h1 0(ω)) and λ > 0, we deduce using the hemicontinuityof β that β(ut(t)) = 0, (5.51) and this implies that ut(t) ∈ k a. e. 6. uniqueness let u1, u2 two solutions of (4.5) , w = u2− u1 and t ∈ (0, t ). because ut ∈ l2(0, t ;h1 0(ω), wecan talking u1 t (resp. u2 t ) in the inequality (4.5) relative to v2 (resp. v1) and adding up the resultswe obtain − ∫ t 0 (wtt , wt)ds − ∫ t 0 (∆2w,wt)ds + ∫ t 0 (∆pu 1, wt)ds − ∫ t 0 (∆pu 2, wt)ds + ∫ t 0 (∫ t 0 g(t − s)∆w(s)ds, wt ) ds + ∫ t 0 (∆wt , wt)ds − ∫ t 0 (|u1|r−2u1 ln |u1|, wt)ds + ∫ t 0 (|u2|r−2u2 ln |u2|, wt)ds ≥ 0, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 18thus, we have 1 2 ∫ t 0 d dt ( |wt(t)|2 + |∆w(t)|2 ) ds + ∫ t 0 |∇wt(t)|2ds ≤ ∫ t 0 〈∆pu1(t)− ∆pu 2(t), wt(t)〉ds + ∫ t 0 ∫ t 0 g(t − s)(∇w(s),∇wt(t))dsdσ∫ t 0 ( |u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t) ) ds. by lemma 2.2, we derive 1 2 ∫ t 0 d dt { |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g � ∇w)(t) } ds + ∫ t 0 |∇wt(t)|2ds ≤ ∫ t 0 |〈∆pu1(t)− ∆pu 2(t), wt(t)〉|ds + ∫ t 0 ∫ ω ( |u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t) ) dxds. (6.1) from mean value theorem, |〈∆pu1(t)− ∆pu 2(t), wt(t)〉| ≤ c ( |∇u1(t)|p−2 2(p−1) + |∇u2(t)|p−2 2(p−1) ) |∇w(t)|2(p−1)|∇wt(t)| ≤ c|∆w(t)|2 + 1 4 |∇wt(t)|2, (6.2) for some constant c > 0, and∫ t 0 ∫ ω ( |u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t) ) dxds ≤ ∫ t 0 ∫ ω |θu1(t) + (1− θ)u2(t))|r−2|w(t)||wt(t)|dxds +(r − 1) ∫ t 0 ∫ ω |θu1(t) + (1− θ)u2|r−2 ln |θu1(t) +(1− θ)u2(t)||w(t)|wt(t)|dxds = i1 + i2, 0 < θ < 1. (6.3) hence, from the hölder inequality and sobolev inequality, we have∫ ω |θu1(t) + (1− θ)u2(t))|r−2|w(t)||wt(t)|dx ≤ |θu1(t) + (1− θ)u2(t)|r−2 n(r−2) |w(t)| 2n n−2 |wt(t)| ≤ cr−2 1 c2c3|∆w(t)||∇wt(t)| ≤ c|∆w(t)|2 + 1 4 |∇wt(t)|2, (6.4) where c1 , c2 and c3 are constants satisfying |θu1(t) + (1− θ)u2(t)|r−2 n(r−2) | ≤ c1|θu1(t) + (1− θ)u2(t)|, |w(t)| 2n n−2 ≤ c|w(t)| ≤ c2|∆w(t)| and |w(t)| ≤ c3|∇w(t)|. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 19 also we used the condition n(p − 2) < 2n n − 2 . now, using the calculation similar to (5.24), it follows that∫ ω |θu1(t) + (1− θ)u2|r−2 ln |θu1(t) + (1− θ)u2(t)|ndx ≤ (e(r − 2)−n)|ω|+ (e(r − 2))−n|θu1(t) + (1− θ)u2(t)|n(r−2) n(r−2) ≤ (e(r − 2)−n)|ω|+ (e(r − 2))−1c4|θu1(t) + (1− θ)u2(t)|n(r−2) ≤ c. (6.5) inserting (6.5) into i2, we have i2 = (r − 1) ∫ t 0 ∫ ω |θu1(t) + (1− θ)u2|r−2 ln |θu1(t) +(1− θ)u2(t)||w(t)|wt(t)|dxds ≤ (r − 1) ∫ t 0 (∫ ω ||θu1(t) + (1− θ)u2|r−2 ln |θu1(t) + θu2(t)||ndx ) 1 n ×|wt(t)||w(t)| 2n n−2 ds ≤ c|∆w(t)|2 + 1 4 |∇wt(t)|2. (6.6) by (6.1), (6.2), (6.4) and (6.6) we get∫ t 0 d dt { |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g � ∇w)(t) } ds + ∫ t 0 |∇wt(t)|2ds ≤ c ∫ t 0 (|∆w(t)|2 + |∇wt(t)|2)ds. (6.7) putting, φ(t) = |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g � ∇w)(t) and using (h3), we have |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 ≥ i|∆w(t)|2 ≥ 0. as (g � ∇w)(t) ≥ 0, we have from (6.7) that ∫ t 0 d dt φ(t) ≤ cφ(t) and because φ(0) = 0, followsfrom the gronwall lemma that |wt(t)|2 + i|∆w(t)|2 ≤ φ(t) ≤ 0, which proves that w = 0 in h1 0(ω) ∩h2(ω). references [1] l. an, a. pierce, the effect of microstructure on elastic-plastic models, siam j. appl. math. 54(3) (1994) 708-730. https://doi.org/10.1137/s0036139992238498[2] l. an, a. pierce, a weakly nonlinear analysis of elastoplastic-microstructure models, siam j. appl. math. 55(1)(1995) 136-155. https://doi.org/10.1137/s0036139993255327[3] a. andrade, m. a. jorge silva, t. f. ma, exponential stability for a plate equation with p-laplacian and memoryterms, math. meth. appl. sci. 35(4) (2012) 417-426. https://doi.org/10.1002/mma.1552 https://doi.org/10.28924/ada/ma.2.5 https://doi.org/10.1137/s0036139992238498 https://doi.org/10.1137/s0036139993255327 https://doi.org/10.1002/mma.1552 eur. j. math. anal. 10.28924/ada/ma.2.5 20 [4] a. ambrosetti, p. h. rabinowitz, dual variational methods in critical point theory and applications, j. functionalanalysis 14(4) (1973) 349-381. https://doi.org/10.1016/0022-1236(73)90051-7[5] g. m. araújo, s. b. menezes, on a variational inequality for the navier-stokes operator with variable viscosity,commun. pur. appl. anal. 1(3) (2006) 583-596. https://doi.org/10.3934/cpaa.2006.5.583[6] g. m. araújo, s. b. menezes, a. o. marinho, on a variational inequality for the equation of motion of oldroyd fluid,electron j. differential equations 69 (2009) 1-16. http://ejde.math.txstate.edu[7] g. m. araújo, m. a. f. araújo, d. c. pereira, on a variational inequality for a plate equation with p-laplacian andmemory terms, appl. anal. 1 (2020) 1-14. https://doi.org/10.1080/00036811.2020.1766028[8] m. bokalo, o. sus, evolutionary variational inequalities with volterra type operators, mathematics and statistics7(5) (2019) 182-190. https://doi.org/10.13189/ms.2019.070504[9] a. bensoussan, j. l. lions, contrôle impulsionnel et inèquations quasi variationnelles, math. models methodsinform. sci. 11, gauthier-villars, paris, 1982.[10] m. m. cavalcanti, v. n. domigos cavalcanti, t. f. ma, exponential decay of the viscoelastic euler-bernoulli withnonlocal dissipation in general domains, differ. integral. equ. 17(5-6) (2004) 495-510.[11] m. m. cavalcanti, h. p. oquendo, frictional versus viscoelastic damping in a semi linear wave equation, siam j.control. optim. 14(4) (2003) 1310-1324. https://doi.org/10.1137/s0363012902408010[12] i. chueshov, i. lasiecka, existence and uniqueness of weak solutions and attractors global for a class of nonlinear 2dkirchhoff-boussinesq models, discret. contin. dyn. s. 15(3) (2006) 777-809. https://doi.org/10.3934/dcds. 2006.15.777[13] c. m. dafermos, asymptotic stability in viscoelasticity, arch. ration. mech. anal. 37 (1970) 297-308. https: //doi.org/10.1007/bf00251609[14] p. hartman, g. stampacchia, on some nonlinear elliptic differential functional equations, acta math. 115 (1966)271-310. https://doi.org/10.1007/bf02392210[15] n. kikuchi, j. t. oden, contacts problems in elasticity: a study of variational inequalities and finite elementmethods, siam, philadelphia, 1988.[16] j. l. lions, quelques méthodes de resolution des problémes aux limites non linéaires, dunod, paris, 1969.[17] l. e. payne, d. h. sattinger, saddle points and instability of nonlinear hyperbolic equations, israel j. math. 22(1975) 273-303. https://doi.org/10.1007/bf02761595[18] c. a. raposo, d. c. pereira, g. araújo, a. baena, unilateral problems for the klein-gordon operator with nonlinearityof kirchhoff-carrier type, electron j. differential equations 137 (2015) 1-14. https://ejde.math.txstate.edu/ volumes/2015/137/abstr.html[19] c. a. raposo, d. c. pereira, c. h. maranhão, unilateral problem for a nonlinear wave equation with p-laplacianoperator, j. appl. anal. comput. 11(1) (2021) 546-555. https://doi.org/10.11948/20200147[20] y. ye, global existence and asymptotic behavior of solutions for a class of nonlinear degenerate wave equations,differ. equ. nonlinear mech. 2007 (2007) 1-9. https://doi.org/10.1155/2007/19685[21] m. willem, minimax theorems. progress in nonlinear differential equations and their applications 24, birkhouserboston inc. boston, ma, 1996.[22] y. zhijian, longtime behavior for a nonlinear wave equation arising in elastoplastic flow, math. meth. appl. sci.32(9) (2009) 1082-1104. https://doi.org/10.1002/mma.1080[23] y. zhijian, global attractores and their hausdorff dimensions for a class of kirchhoff models, j. math. phys. 51(3)(2010) 032701. https://doi.org/10.1063/1.3303633[24] y. zhijian, j. baoxia, global attractor for a class of kirchhoff models, j. math. phys. 50(3) (2009) 032701. https: //doi.org/10.1063/1.3085951 https://doi.org/10.28924/ada/ma.2.5 https://doi.org/10.1016/0022-1236(73)90051-7 https://doi.org/10.3934/cpaa.2006.5.583 http://ejde.math.txstate.edu https://doi.org/10.1080/00036811.2020.1766028 https://doi.org/10.13189/ms.2019.070504 https://doi.org/10.1137/s0363012902408010 https://doi.org/10.3934/dcds.2006.15.777 https://doi.org/10.3934/dcds.2006.15.777 https://doi.org/10.1007/bf00251609 https://doi.org/10.1007/bf00251609 https://doi.org/10.1007/bf02392210 https://doi.org/10.1007/bf02761595 https://ejde.math.txstate.edu/volumes/2015/137/abstr.html https://ejde.math.txstate.edu/volumes/2015/137/abstr.html https://doi.org/10.11948/20200147 https://doi.org/10.1155/2007/19685 https://doi.org/10.1002/mma.1080 https://doi.org/10.1063/1.3303633 https://doi.org/10.1063/1.3085951 https://doi.org/10.1063/1.3085951 1. introduction 2. preliminaries 3. potential well 4. existence of strong solutions 5. penalization method 5.1. approximate problem 5.2. first estimate 5.3. second estimate 5.4. third estimate 5.5. strong solution 6. uniqueness references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 2doi: 10.28924/ada/ma.4.2 modified viscosity iterative algorithm for solving variational inclusion and fixed point problems in real hilbert space furmose mendy1,∗ , john t mendy2,∗ 1department of mathematics, university of toledo, usa furmosemendy111@gmail.com 2department of mathematics, universita degli studi dell’aquila, italy, 67010, coppito, via vetoio, italy johntgracemendy@gmail.com ∗correspondence: furmosemendy111@gmail.com abstract. this paper introduces a new iterative algorithm, called the modified viscosity iterativealgorithm, designed to solve problems related to variational inclusion and fixed point in real hilbertspaces. the algorithm is specifically tailored to handle multivalued quasi-nonexpansive and demi-contractive operators. the convergence properties of the algorithm are analyzed and established,ensuring its effectiveness in finding solutions for complex mathematical problems in the field of opti-mization and equilibrium. 1. introduction variational inclusion and fixed point problems involving multivalued quasi nonexpansive anddemicontractive operators play a crucial role in the field of mathematics, particularly in real hilbertspaces.the study of variational inclusion and fixed point problems originated from the theory of opti-mization and nonlinear analysis, and in the mid−20th century, mathematicians began investigatingproblems involving finding points that satisfy certain inclusion and fixed point conditions. over time,research in this area expanded and became an essential part of functional analysis and optimiza-tion theory. they are widely-used in applications in diverse fields such as engineering, economics,physics, and computer science. they provide a framework to model and solve various real-worldproblems, including equilibrium problems, optimization problems, and variational inequalities.fixed point problems, on the other hand, deal with finding points that remain unchanged underthe action of an operator. the concept of fixed points has profound implications in mathematicsand its applications. a wide range of problems in analysis, differential equations, and optimization received: 22 nov 2023. key words and phrases. iterative algorithm, quasi-nonexpansive, multivalued demiclosed mapping, hilbert space,variational inequality, strongly monotone mappings. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.2 https://orcid.org/0009-0000-3806-629x https://orcid.org/0000-0002-3774-0761 eur. j. math. anal. 10.28924/ada/ma.4.2 2theory can be reduced to fixed point problems. they serve as powerful tools to prove the exis-tence and uniqueness of solutions, compute approximations, and establish convergence propertiesof iterative algorithms. whilst multivalued quasi nonexpansive operators play a pivotal role in vari-ational inclusion and fixed point problems. these operators possess certain properties that ensurethe stability and convergence of iterative algorithms. they have applications in image processing,signal estimation, and constrained optimization, among others.moreover, fixed point theory for multivalued mappings has also contributed to the developmentof related areas of research, such as operator theory, topological degree theory, and convex anal-ysis. by investigating the properties and behavior of fixed points in multivalued mappings, math-ematicians have gained a deeper understanding of these fields and have been able to establishconnections and develop new techniques.(see, [30], [10, 11], [2], [7], [33], [9] and [26]).demicontractive operators, on the other hand, exhibit properties of both contractive and nonex-pansive operators. they are broadly used in the study of variational inequalities and play a crucialrole in convex analysis and optimization theory. they provide a bridge between nonlinear andlinear problems, enabling the development of efficient numerical methods for solving variationalproblems arising in diverse areas.viscosity iterative algorithms have been extensively studied in recent years for finding commonfixed points of single-valued nonexpansive mappings and solving variational inequality problems.these investigations have built upon the concepts of viscosity solutions introduced by variousresearchers. (see e.g [6], [25], [5], [29], [23], [19], [28]).throughout this paper, we denote h to be real hilbert space with the inner product 〈., .〉 inducedby the norm ‖.‖. let k, to be a nonempty, closed and convex subset of h.an operator a : h → h is said to be lipschitz if there exists a constant l > 0 such that ‖ax − ay‖ ≤ l‖x − y‖,∀x, y ∈ h (1.1) a : h → h is said to be strongly positive if there exists a constant k > 0 such that 〈ax, x〉 ≥ k‖x‖2, ∀x ∈ h (1.2) a : h → h is said to be k−strongly monotone if there exists a constant k ∈ (0, 1) such that 〈ax − ay , x − y〉h ≥ k‖x − y‖2, ∀x, y ∈ h (1.3) definition 1.1. a multivalued mapping(1) t : d(t ) ⊆ h → cb(d) is called l−lipschitzian if there exists l > 0, such that h(tx, t y) ≤ l‖x − y‖,∀x, y ∈ d(t ) and t is contraction if l ∈ (0, 1) and noneaxpansive if l = 1.(2) t is called quasi-nonexpansive if h(tx, tp) ≤ ‖x − p‖, ∀x ∈ d(t ), p ∈ f ix(t ) https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 3(3) t : d(t ) ⊆ h → cb(d) is said to be k−stritly pseudo-contractive, if there exists k ∈ (0, 1) such that for all x, y ∈ d(t ), the following holds;( h(tx, t y) )2 ≤ ‖x − y‖2 + k‖(i − t )x − (i − t )y‖2, if k = 1, the map t is said to be pseudocontractive.(4) [26] t : d(t ) ⊆ e → 2e is said to be demicontractive if f ix(t ) 6= ∅ and for all p ∈ f ix(t ), x ∈ d(t ) there exists k ∈ (0, 1) such that( h(tx, tp) )2 ≤ ‖x − p‖2 + kd(x, t x)2. if k = 1, the map t is said to be hemicontractive. let (x, d) be a metric space, k be a nonempty subset of x and t : k → 2k be a multivaluedmapping. an element x ∈ k is called a fixed point of t if x ∈ tx . the fixed point set of t isdenoted by f ix(t ) := {x ∈ d(t ) : x ∈ tx} where d(t ) := {x ∈ x : tx 6= ∅}. it is easy to seethat single-valued mapping is a particular case of multivalued mapping.let d be a nonempty suset of a normed linear space e. the set d is called proximinal (see [13])if for each ψ ∈ e, there exists u ∈ d such that d(x, u) := inf{‖x − y‖ : y ∈ d},∀x, y ∈ e (1.4) where d(x, y) := ‖x − y‖ for all x, y ∈ e. every closed, nonempty and convex set of real hilbertspace is proximinal. the family of nonempty closed bounded subsets, nonempty compact subsets,and nonempty proximinal bounded subsets be donated as cb(d), k(d) and p (d) respectively.let a, b ∈ cb(d). then the hausdorff metric in h is defined by h(a, b) = max { sup a∈a d(a,b), sup b∈b d(b,a) } . (1.5) let a : d(a) ⊂ h → 2h be a multivalued operator. then a is monotone if (x, u), (y , v) ∈ d(a)such that g(a) := {x, u) : x ∈ d(a), u ∈ ax} (1.6) a monotone mapping a : h → 2h is said to be maximal if its graph g(a) is not properly containedin the graph of any other monotone mapping.a mapping a : h → h is said to be α−inverse strongly if there exits a constant α > 0 such that 〈ax − ay , x − y〉h ≥ α‖ax − ay‖2, ∀x, y ∈ h (1.7) remark 1.2. it can be seen that every α−inverse strongly monotone mapping is 1 α −lipschitzmonotone. https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 4 let a : h → h be a single-valued nonlinear mapping and π : h → 2h be a set-valued mapping.then the variational inclusion problem is as follows: find x ∈ h, such that ω ∈m(x) + a(x) (1.8) where ω is the zero vector in h. we denote the solution of the problem (1.8) by s(m,a). if ω = athen, problem (1.8) becomes the inclusion problem by rockafellar [16].further readings on zeros ofinclusion problem (see [17], [18], [19], [8], [12]) let a set value mapping m : h → 2h be maximal monotone. we define a resolvent operator jm λgenerated by π and λ as follows jm λ = (i − λm)−1(x),∀x ∈ h (1.9) where λ is a positive number. it is easily to see that the resolvent operator jm λ is single valuednonexpensive and 1−inverse strongly monotone , and moreover, a solution of the problem (1.8) isa fixed point of the operator jm λ (i − λa), ∀λ > 0( see [4]).let t : h → p (h) be multivalued map and pt : h → cb(h) be defined by pt (x) = {y ∈ tx : ‖y − x‖ = d(x, t x)} (1.10) see examples of a multivalued mapping t with f ix(t ) 6= ∅, t p = {q} for all q ∈ tp which ptis a demicontractive-type but not a k−strictly pseudocontractive-type mapping in mendy et al [?]to prove that a multivalued mapping t with f ix(t ) 6= ∅ and tp = {q} for all q ∈ tp is ademicontractive-type but not a k-strictly pseudocontractive-type mapping, we need to demonstratethe following three steps: step 1.3. show that t is demicontractive-type.to prove that t is demicontractive-type, we need to show that for all p ∈ f ix(t ) and x ∈ d(t ),there exists k ∈ (0, 1) such that( h(tx, tp) )2 ≤ ‖x − p‖2 + kd(x, t x)2. since tp = {q} for all q ∈ tp, we have tp = {p} for all p ∈ f ix(t ). thus, for any x ∈ d(t ), tx = {φ} for some φ ∈ f ix(t ).now, consider the case when x = φ. in this case, we have h(tx, tp) = h(tφ, tp) = h({φ}, {p}) = 0. therefore, the inequality holds for any k ∈ (0, 1). step 1.4. show that t is not k-strictly pseudocontractive-type.to prove that t is not k-strictly pseudocontractive-type, we need to show that there does notexist a constant k ∈ (0, 1) such that( h(tx, tp) )2 ≤ k‖x − p‖2 + kd(x, t x)2 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 5for all p ∈ f ix(t ) and x ∈ d(t ).from step 1.3, we know that h(tx, tp) = 0 for any p ∈ f ix(t ) and x ∈ d(t ). therefore, theinequality reduces to 0 ≤ k‖x − p‖2 for all p ∈ f ix(t ) and x ∈ d(t ). however, this inequality cannot hold for all x 6= p since itimplies k ≥ 1 ‖x−p‖2 , which contradicts the requirement that k ∈ (0, 1). step 1.5. show that pt is a demicontractive-type.let pt denote the projection operator associated with multivalued mapping t . since tp = {q}for all q ∈ tp, pt is the single-valued mapping that assigns each p ∈ f ix(t ) to itself.consider p, x ∈ f ix(t ), with x 6= p. then tx = tp and ‖x − p‖ > 0. furthermore, d(x, t x) = d(p, tp) = 0 since x, p ∈ f ix(t ).using these values, let’s rearrange the original inequality: ( h(tx, tp) )2 ≤ ‖x − p‖2 + kd(x, t x)2 0 ≤ ‖x − p‖2 + kd(x, t x)2. since ‖x−p‖ > 0, the inequality can only hold if k = 0. however, k ∈ (0, 1) by definition, so ptcannot satisfy the inequality for any k ∈ (0, 1). hence, pt is not a k-strictly pseudocontractive-type mapping. a popular method for solving problem (1.8) is the well-known forward-backward splitting methodintroduced by passty [14] and lions and mercier [27].the method is formulated as xn+1 = (i − λnm)−1(i − λna)x, λn > 0. (1.11) under the condition that dom(m) ⊂ dom(a). it was known in [31], that weak convergence of(1.11) requires quite restrictive assumptions on a and π, such that the inverse of a is stronglymonotone or π is lipschitz continuous and monotone and the operator (a+m) is strongly monotoneon dom(b). tseng in [20] and gibali and thong in [24] extended and improved results of g.h-g.chen and r.t. rockafellar [31].most recently, sow [28] introduced and studied a new iterative algorithm and prove convergencetheorems for variation inclusion problem (1.8) and fixed point problem involving multivalued demi-contractive and quasi-nonexpansive mappings in hilbert spaces. they defined the sequence {ψn} https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 6as follows  δn = jm λn (i − λna)xn, yn = θnδn + (1− θn)vn, vn ∈ tδn , zn = βnyn + (1− βn)un, un ∈ t2yn, xn+1 = pk(αnγf (xn) + (1− ηαnb)zn) (1.12) and prove that under certain conditions, the sequence {xn} converges strongly to a unique fixedpoint that solved the variational inequality.it is our purpose in this paper to construct a new iteration process, that modifies that of sow [28]and prove that the corresponding sequence {xn} converges strongly to a common point of aninclusion problem and fixed point of a family of multivalued demicontractive and quasi-nonexpansivemappings in hilbert spaces without any compactness. our theorems generalize and extend that ofsow [28], and many other results in this directions. 2. preliminaries the following lemmas will play a crucial role in the sequel.let k be a nonempty, closed convex subset of h. the nearest point projection from h to kdenoted by pk, assigns to each ψ ∈ h the unique point of k, pkψ such that ‖x − pkx‖ ≤ ‖x − y‖, for all y ∈ k, and for every x ∈ h, 〈x − pkx, y − pkx〉 ≤ 0, ∀y ∈ k (2.1) lemma 2.1. [27]. let π : h → 2h be a maximal monotone mapping, and λ : h → h be lipschitz and continuous monotone mapping. then (π + λ) : h → 2h is a maximal monotone mapping. lemma 2.2. [28]. let h be real hilbert space and λ : h → h be an α−inverse strongly monotone mapping. then, (i − θλ) is nonexpansive mapping for all ψ, π ∈ h and θ ∈ [0, 2α] such that ‖(i − θa)x − (i − θay‖2 ≤ ‖x − y‖2 + θ(θ − 2α)‖ax − ay‖2 (2.2) lemma 2.3. [24]. assume that {an} is a sequence of nonnegative real numbers such that an+1 = (1 − bn)an + σn for all n ≥ 0, where {αn} is a sequence in (0, 1) and {σn} is a sequence in r such that i) ∞∑ n=0 bn =∞, i i) lim n→∞ sup σn bn ≤ 0. then lim n→∞ an = 0 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 7 lemma 2.4. (wang [1]). let h be a real hilbert space. let k be a nonempty closed convex subset of h. a : h → h be k− strongly monotone and l− lipschitzian operator with k > 0 and l > 0. assume that 0 < η < 2k l2 and τ = η ( k − l2η 2 ) . then for each t ∈ ( 0,min ( 1, 1 τ )) , we have ‖(i − tηa)x − (i − tηa)y‖ ≤ (i − tτ)‖x − y‖, ∀x, y ∈ h (2.3) lemma 2.5. [29]. let h be a real hilbert space. then for every x, y ∈ h, and every λ ∈ (0, 1), the following holds: i): ‖x − y‖2 ≤ ‖x‖2 + 2〈y , x + y〉 ii: ‖λx + (1− λ)y‖2 ≤ λ‖x‖2 + (1− λ)‖y‖2 − (1− λ)λ‖x − y‖2. 3. main results in this section, we study the convergence properties of the iterative algorithm which is based onviscosity algorithm and forward backward splitting method. we now prove the following theorem. theorem 3.1. let h be a real hilbert space and k be a nonempty, closed convex subset of h. let a : k → h be an α−inverse strongly monotone operator and let b : h → h be an k−strongly monotone and l−lipschitzian operator. let f : k→ h be an b−lipschitzian mapping and m : h → 2h be a maximal monotone mapping such that the domain of m is included in k. let t1, t2 : k → cb(k) be a multivalued β− demicontractive mapping and t3 : k → cb(k) be a multivalued quasi-nonexpansive mapping. assume that 0 < η < 2k l2 , 0 < γb < τ , where τ = η ( k − l2η 2 ) , and i − t1, i − t2 and i − t3 are demiclosed at origin, such that ω := f ix(t1) ∩ f ix(t2) ∩ f ix(t3) ∩ s(m,a) 6= ∅ and t1q = t2q = t3q = {q},∀q ∈ ω. for given x0 ∈ k, let {xn} be generated by the algorithm: δn = jmλn(i − λna)xn; yn = θnδn + (1− θn)vn, vn ∈ t1δn; zn = βnyn + (1− βn)un, un ∈ t2yn; tn = γnzn + (1− γn)wn, wn ∈ t3zn; xn+1 = pk(αnγf (xn) + (i − ηαnb)tn) (3.1) where {βn}, {γn}, {θn}, {µn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following conditions i): lim n→∞ αn = 0 ∞∑ n=0 αn <∞, λn ∈ [a, b] ⊂ (0,min{1, 2α}) https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 8 ii): lim n→∞ inf(1− βn)(βn − β) > 0 and lim n→∞ inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1) iii): limn→∞ inf(1− γn)γn) > 0 then, the sequences defined in (3.1), that is {xn} and {δn} converge strongly to unique solution x∗ ∈ ω, which also solve the following variational inequality: 〈ηbx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ ω (3.2) proof. from the choice of η and γ from [?], (ηφ − γψ) is strongly monotone, then the variationalinequality (3.2) has a unique solution. we will first show that there is only one solution.lets assume that by contradiction that there exist two points x∗, y∗ ∈ ω which are two solutionof the given inequality, and x∗ 6= y∗, then we have 〈ηbx∗ − γf (x∗), x∗ − y∗〉 ≤ 0 (3.3) and 〈ηby∗ − γf (y∗), y∗ − x∗〉 ≤ 0 (3.4) therefore from (3.3) and (3.4), we have 〈ηby∗ − ηbx∗ + γf (x∗)− γf (y∗), y∗ − x∗〉 ≤ 0 (3.5) now from the assumption that l2η 2 > 0 ⇔ α− l2η 2 < α ⇔ η ( α− l2η 2 ) < αη ⇔ τ < αη so that 0 < γ < τ < αη 〈ηby∗ − ηbx∗ + γf (x∗)− γf (y∗), y∗ − x∗〉 = 〈ηby∗ − ηbx∗, y∗ − x∗〉 − 〈γf (y∗)− γf (x∗), y∗ − x∗〉 = 〈ηby∗ − ηbx∗, y∗ − x∗〉 − γ‖f (y∗)− f (x∗)‖‖y∗ − x∗‖ ≥ αη‖x∗ − y∗‖2 − γρ‖x∗ − y∗‖2 = (αη − γρ)‖x∗ − y∗‖2 and this is a contradiction to (3.5), and hence x∗ = y∗, which is required. again, we note that theoperator pk[i + (αγf − ηαb)] is a contradiction. now for any fixed point α0 ∈ (0,min { 1, 1 τ } ),and ∀x, y ∈ h, we have, by lemma (2.4), and letting φ = [i + (α0γf − ηα0b)]x and θ = [i + (α0γf − ηα0b)]y , we have https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 9 ‖pkφ− pkθ‖ ≤ ‖[i + (α0γf − ηα0b)]x − ([i + (α0γf − ηα0b)]y)‖ ≤ α0γ‖f (x)− f (y)‖+ ‖(i − ηα0b)x − (i − ηα0b)y‖ ≤ α0γρ‖x − y‖+ (i − ατ)‖x − y‖ ≤ (i − α0(τ − γρ))‖x − y‖ thus, by banach contraction principle, the mapping pk[i + (αγf − ηαb)] has a fixed point, say x̂ = pk[i+ (αγf −ηαb)] and as such, from (2.1), it is similar in value to the variational inequalitybelow 〈ηbx̂ − γf (x̂), x̂ − q〉 ≤ 0,∀q ∈ ω now we continue with the proof of theorem (3.1)let q ∈ ω with the fact that jπ λn is 1−inverse strongly monotone,and from [?], we have thefollowing ‖δn − q‖2 ≤ ‖xn − q‖2 therefore from (3.6), we have ‖δn − q‖ ≤ ‖xn − q‖ (3.6) from lemma (2.5), with (3.1), and, for the fact that t1q = {q}, t1 is β−demicontrative, we have ‖yn − q‖2 = ‖θn(δn − q) + (1− θn)(vn − q)‖2 = θn‖δn − q‖2 + (1− θn)‖vn − q‖2 − (1− θn)θn‖vn − δn‖2 ≤ θn‖δn − q‖2 + (1− θn)h(t1δn, t1q)2 − (1− θn)θn‖vn − δn‖2 ≤ θn‖δn − q‖2 + (1− θn)[‖δn − q‖2 + βd(δn, t1δn)2]− (1− θn)θn‖vn − δn‖2 ≤ ‖δn − q‖2 − (1− θn)(θn − β)‖vn − δn‖2 (3.7) thus, we have ‖yn − q‖2 ≤ ‖δn − q‖2 − (1− θn)(θn − β)‖vn − δn‖2 since θn ∈ (β, 1), we have ‖yn − q‖2 ≤ ‖δn − q‖2 again from lemma (2.5), with (3.1), and, for the fact that t2q = {q}, t2 is β−demicontrative, wehave https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 10 ‖zn − q‖2 = ‖βn(yn − q) + (1− βn)(un − q)‖2 = βn‖yn − q‖2 + (1− βn)‖un − q‖2 − (1− βn)βn‖un − yn‖2 ≤ βn‖yn − q‖2 + (1− βn)h(t2yn, t2q)2 − (1− βn)βn‖un − yn‖2 ≤ βn‖yn − q‖2 + (1− βn)[‖yn − q‖2 + βd(yn, t2yn)2]− (1− βn)βn‖un − yn‖2 ≤ ‖yn − q‖2 − (1− βn)(βn − β)‖un − yn‖2 (3.8) thus, we have ‖zn − q‖2 ≤ ‖yn − q‖2 − (1− βn)(βn − β)‖un − yn‖2 since βn ∈ (β, 1), we have ‖zn − q‖2 ≤ ‖yn − q‖2 now, using the fact that t3q = q, we have the following estimates ‖tn − q‖ = ‖γnzn + (1− γn)wn − q‖ ≤ γn‖zn − q‖+ (1− γn)‖wn − q‖ ≤ γn‖zn − q‖+ (1− γn)h(t3z, t3q) ≤ γn‖zn − q‖+ (1− γn)‖zn − q‖ ≤ ‖zn − q‖ (3.9) hence, we can see that ‖tn − q‖ ≤ ‖zn − q‖ ≤ ‖yn − q‖ ≤ ‖δn − q‖ ≤ ‖xn − q‖ (3.10) using (3.1), inequality (3.10) and lemma (2.4) ‖xn+1 − q‖ ≤ ‖(αnγf (xn) + (i − ηαnb)tn)− q‖ ≤ ‖αnγ(f (xn)− f (q))‖+ (1− τα)‖tn − q)‖+ α‖γf (q)− ηαb‖ ≤ αnγ‖f (xn)− f (q)‖+ (1− τα)‖xn − q)‖+ α‖γf (q)− ηαb‖ ≤ αnbγ‖xn − q‖+ (1− τα)‖xn − q)‖+ α‖γf (q)− ηαb‖ ≤ (1− α(τ − bγ))‖xn − q)‖+ α‖γf (q)− ηαb‖ ≤ max { ‖xn − q‖, ‖γf (q)− ηbq‖ τ − bγ } . therefore, by induction, it is easy to see that ‖xn+1 − q‖ ≤ max { ‖x0 − q‖, ‖γf (q)− ηbq‖ τ − bγ } , ∀n ≥ 1 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 11hence {xn}, {f (xn)} and {bxn} are bounded.secondly, we now have the following estimates. from (3.1) and lemma (2.5), we have ‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηbq) + (i − ηαnb)(tn − q)‖2 ≤ α2 n‖γf (xn)− ηbq‖2 + (1− ταn)2‖tn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηbq‖2 + (1− ταn)2‖zn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηbq‖2 + (1− ταn)2‖yn − q‖2 − (1− ταn)2(1− βn)(βn − β)‖un − yn‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηbq‖2 + (1− ταn)2‖δn − q‖2 − (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2 − (1− ταn)2(1− βn)(βn − β)‖un − yn‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηbq‖2 + (1− ταn)2‖xn − q‖2 − (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2 − (1− ταn)2(1− βn)(βn − β)‖un − yn‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ ≤ ‖xn − q‖2 + α2 n‖γf (xn)− ηbq‖2 − αn(2τ − τ2αn)‖xn − q‖2 − (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2 − (1− ταn)2(1− βn)(βn − β)‖un − yn‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ therefore (1− ταn)2 [ (1− θn)(θn − β)‖vn − δn‖2 + (1− βn)(βn − β)‖un − yn‖2 ] ≤ ‖xn − q‖2 − ‖xn+1 − q‖2 − αn(2τ − τ2αn)‖xn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ + α2 n‖γf (xn)− ηbq‖2 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 12due to the boundedness of {f (xn)} and {xn}, and for some constant m > 0, we have (1− ταn)2 [ (1− θn)(θn − β)‖vn − δn‖2 + (1− βn)(βn − β)‖un − yn‖2 ] ≤ ‖xn − q‖2 − ‖xn+1 − q‖2 + αnm (3.11) we now show that xn → x . we then consider two cases. case 1: assuming that the sequence {‖xn−q‖} is monotonically decreasing. then {‖xn−q‖}must be a convergent sequence. therefore, we have lim n→∞ [‖xn − q‖2 − ‖xn+1 − q‖2] = 0, (3.12) this implies that from (3.11), that lim n→∞ (1− θn)(θn − β)‖vn − δn‖2 = 0 (3.13) and lim n→∞ (1− βn)(βn − β)‖un − yn‖2 = 0 (3.14) since lim n→∞ inf(1 − θn)(θn − β) > 0 and lim n→∞ inf(1 − βn)(βn − β) > 0, with the fact that vn ∈ t1δn and un ∈ t2yn, it follows that lim n→∞ d(δn, t1δn) = 0 (3.15) and lim n→∞ d(yn, t2yn) = 0 (3.16) observing that ‖yn − δn‖ = ‖θnδn + (1− θn)vn − δn‖ = ‖θnδn + (1− θn)vn − δn + θnδn − θnδn‖ = (1− θn)‖vn − δn‖ ≤ ‖vn − δn‖ (3.17) taking the limits and from (3.13), we can see that lim n→∞ ‖yn − δn‖ = 0 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 13 ‖zn − yn‖ = ‖βnyn + (1− βn)un − yn‖ = ‖βnyn + (1− βn)un − yn + βnyn − βnyn‖ = (1− βn)‖un − yn‖ ≤ ‖un − yn‖ (3.18) again, from (3.14), we can see that lim n→∞ ‖zn − yn‖ = 0 ‖zn − δn‖ = ‖zn − yn + yn − δn‖ ≤ ‖zn − yn‖+ ‖yn − δn‖ hence lim n→∞ ‖zn − δn‖ = 0now from lemma (2.2), lemma (2.4) and (3.1), we have the following ‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηbq) + (i − ηαnb)(tn − q)‖2 ≤ α2 n‖γf (xn)− ηb‖2 + (1− τα)2‖tn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖δn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖jmλn(1− λa)xn − jmλn(1− λa)q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2 [ ‖xn − q‖2 + a(b − 2α)‖axn − aq‖2 ] + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + ‖xn − q‖2 − αn(2τ − τ2αn)‖xn − q‖2 − (1− τα)2a(2α− b)‖axn − aq‖2 (3.19) + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ therefore, from (3.19),and with a constant d > 0, we have (1− τα)2a(2α− b)‖axn − aq‖2 ≤ ‖xn − q)‖2 − ‖xn+1 − q)‖2 + αnd https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 14since, lim n→∞ αn = 0, and from the inequality (3.12), with the fact that {xn} is bounded, wehave lim n→∞ ‖axn − aq‖2 = 0 (3.20) since jmλn is 1−inverse strongly monotone,and ‖tn−q‖ ≤ ‖δn−q‖, we have the following ‖tn − q‖2 = ‖j(m) λn (i − λna)xn − jmλn(i − λna)q‖2 ≤ 〈tn − q, (i − λna)xn − (i − λna)q〉 = 1 2 [ ‖(i − λna)xn − (i − λna)q‖2 + ‖tn − q‖2 − ‖(i − λna)xn − (i − λna)q − (tn − q)‖2 ] ≤ 1 2 [ ‖xn − q‖2 + ‖tn − q‖2 − ‖xn − tn‖2 + 2λn〈tn − q,axn − aq〉 − λ2 n‖axn − a)q)‖2 ] ≤ ‖xn − q‖2 − ‖xn − tn‖2 + 2λn〈xn − q,axn − aq〉 − λ2 n‖axn − aq‖2 this gives us ‖tn − q‖2 ≤ ‖xn − q‖2 − ‖xn − tn‖2 + 2λn〈tn − q,axn − aq〉 − λ2 n‖axn − aq‖2 (3.21) therefore ‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηbq) + (i − ηαnb)(tn − q)‖2 ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖tn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2 [ ‖xn − q‖2 − ‖xn − tn‖2 + 2λn〈tn − q,axn − aq〉 − λ2 n‖axn − aq‖2 ] + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖xn − q‖2 − (1− ταn)2‖xn − tn‖2 + 2λn(1− ταn)2〈tn − q,axn − aq〉 − λ2 n(1− ταn)2‖axn − aq‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ (3.22) https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 15thus, from (3.25), we have (1− ταn)2‖xn − tn‖2 ≤ α2 n‖γf (xn)− ηb‖2 + ‖xn − q‖2 − ‖xn+1 − q‖2 (3.23) − αnτ(2− ταn)‖xn − q‖2 + 2λn(1− ταn)2〈tn − q,axn − aq〉 − λ2 n(1− ταn)2‖axn − aq‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖xn − q‖ therefor, since αn → 0 as n →∞ with inequalities (3.12) and (3.20), we have lim n→∞ ‖xn − tn‖ = 0 from (3.10) and lemma (2.5) with the fact that t3 is quasi-nonexpansive, we have thefollowing estimate ‖tn − q‖2 = ‖γnzn + (1− γn)wn − q‖2 = γn‖zn − q‖2 + (1− γn)‖wn − q‖2 − (1− γn)γn‖wn − zn‖2 = γn‖zn − q‖2 + (1− γn)h(t3zn, t3q)2 − (1− γn)γn‖wn − zn‖2 = γn‖zn − q‖2 + (1− γn)‖zn − q‖2 − (1− γn)γn‖wn − zn‖2 ≤ ‖xn − q‖2 − (1− γn)γn‖wn − zn‖2 (3.24) therefore ‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηbq) + (i − ηαnb)(tn − q)‖2 ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖tn − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2 [ ‖xn − q‖2 − (1− γn)γn‖wn − zn‖2 ] + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ ≤ α2 n‖γf (xn)− ηb‖2 + (1− ταn)2‖xn − q‖2 − (1− ταn)2(1− γn)γn‖wn − zn‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ hence, we have the following (1− ταn)2(1− γn)γn‖wn − zn‖2 ≤ α2 n‖γf (xn)− ηb‖2 − ταn(2− ταn)‖xn − q‖2 + ‖xn − q‖2 − ‖xn+1 − q‖2 + 2αn(1− ταn)‖γf (xn)− ηbq‖‖tn − q‖ https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 16therefor, since αn → 0 as n →∞ with inequalities (3.12) and (3.20), we have (1− ταn)2(1− γn)γn‖wn − zn‖2 ≤ 0 from this we have lim n→∞ (1− γn)γn‖wn − zn‖2 = 0 (3.25) since lim n→∞ inf((1− γn)γn) > 0 lim n→∞ ‖wn − zn‖ = 0 (3.26) again, with w ∈ t3zn lim n→∞ d(zn, t3zn) = 0 (3.27) moreover, since h is reflexive and {xn} is bounded, we then prove that lim n→+∞ sup〈ηbx∗ − γf (x∗), x∗ − xn〉 ≤ 0. we let the subsequence {xni} of {xn} to converge weakly to x∗∗ in k, and lim n→+∞ 〈ηbx∗ − γf (x∗), x∗ − xn〉 = lim n→+∞ 〈ηbx∗ − γf (x∗), x∗ − xni 〉 again, since i−t1, i−t2 and i−t3 satisfies the demiclosed principle and from (3.32), (3.16)and (3.27), we obtain x∗∗ ∈ f ix(t1)∩f ix(t2)∩f ix(t3). we now show that x∗∗ ∈ s(m,a).since a is α−inverse strongly monotone, and lipschitz continuous mapping. then fromlemma (2.1), it follows that (m+ a) is maximal monotone.let (ν, g) ∈ g(m + a), that is g − aν ∈ m(ν). since δni = jmλni (xni ) − λniaxni ), wehave xni − λni xni ∈ (i + λnim)δni , that is 1 λni (xni − δni − λniaxni ) ∈ m(δni ). by maximalmonotonocity of (m+ a), gives 〈ν − δni , g − aν − 1 λni (xni − δni − λniaxni ) ≥ 0 and, therefore 〈ν − δni , g〉 ≥ 〈ν − δni ,aν − 1 λni (xni − δni − λniaxni )〉 = 〈ν − δni ,aν − aδni + aδni + 1 λni (xni − δni − λniaxni )〉 ≥ 〈ν − δni ,aν − axni 〉+ 〈ν − δni , 1 λni (xni − δni 〉 it then follows from ‖δn − xn‖ → 0, ‖aδn − axn‖ → 0 and δni → x∗∗ weakly that lim n→∞ 〈ν − δni , g〉 = 〈ν − x∗∗, g〉 and hence x∗∗ ∈ s(π,a). therefore, x∗∗ ∈ ω,on the other hand, for the fact that x∗ solves the variational inequality (3.30). https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 17 lim n→+∞ sup〈ηbx∗ − γf (x∗), x∗ − xn〉 = lim n→+∞ sup〈ηbx∗ − γf (x∗), x∗ − xni 〉 = 〈ηbx∗ − γf (x∗), x∗ − x∗∗〉 ≤ 0 (3.28) lastly, we now prove that lim n→∞ ‖xn − x∗‖ = 0, that is xn → x∗ as n →∞. ‖xn+1 − x∗‖2 ≤ ‖αnγf (xn) + (i − ηαnb)tn − x∗‖2 ≤ ‖αn(γf (xn)− γf (x∗)) + (i − ηαnb)tn − x∗‖2 + 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 ≤ [ αnγ‖f (xn)− f (x∗)‖+ ‖(i − ηαnb)(tn − x∗)‖ ]2 + 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 ≤ [ αnγb‖xn − x∗‖+ (1− ταn)‖xn − x∗‖ ]2 + 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 ≤ [ 1− αn(τ − γb) ]2 ‖xn − x∗‖2 + 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 ≤ [ 1− αn(τ − γb) ] ‖xn − x∗‖2 + 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 thus, from lemma (2.3), it follows that ψn → ψ∗ as n →∞, where bn = αn(τ − γb), an = ‖xn − x∗‖2 and σn = 2αn〈ηbx∗ − γf (x∗), x∗ − xn+1〉 case 2: suppose that the sequence {‖xn − x∗‖} is monotonically increasing. set wn := ‖xn − x∗‖2 and τ := n→ n be a mapping for all n ≥ n0 (for some n0 sufficient large), by τn := max{k ∈ n : k ≤ n,wk ≤wk+1}. then, τ is a nondecreasing sequence, such that τn →∞ as n →∞ and wτ(n) ≤wτ(n)+1} for all n ≥ n0. now, from (3.11), we have (1− ατ(n)τ) [ (1− θn)(θn − β)‖vτ(n) − δτ(n)‖2 + (1− βn)(βn − β)‖uτ(n) − yτ(n)‖2 ] ≤ 2ατ(n)m (3.29) lim n→+∞ τ(1−ατ(n)) [ (1−θτ(n))(θτ(n)−β)‖vτ(n)−δτ(n)‖2+(1−βτ(n))(βτ(n)−β)‖uτ(n)−yτ(n)‖2 ] = 0 since (βτ(n), θτ(n)) ∈ (β, 1) and lim n→∞ inf γτ(n)(1− γτ(n)) > 0, we have lim n→∞ ‖uτ(n) − yτ(n)‖ = 0 and lim n→∞ ‖vτ(n) − δτ(n)‖ = 0 with vτ(n) ∈ t1δτ(n) and uτ(n) ∈ t2yτ(n), it follows that lim n→∞ d ( δτ(n), t1δτ(n) ) = 0 and lim n→∞ d ( yτ(n), t2yτ(n) ) = 0 https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 18following the same argument in case 1, we conclude that lim τ(n)→+∞ sup〈ηbx∗ − γf (x∗), x∗ − xτ(n)+1〉 ≤ 0 therefore, for all n ≥ n0,and from 3.29, we have 0 ≤ ‖xτ(n)+1 − x∗‖2 − ‖xτ(n) − x∗‖2 ≤ ατ(n)[−αn(τ − γb)‖xτ(n) − x∗)‖2 + 2ατ(n)〈ηbx∗ − γf (x∗), xτ(n)+1 − x∗〉 ‖xτ(n) − x∗‖2 ≤ 2 τ − γb 〈ηbx ∗ − γf (x∗), xτ(n)+1 − x∗〉 then we have lim n→∞ ‖xτ(n) − x∗‖2 = 0. therefore lim n→∞ wτ(n) = lim n→∞ wτ(n)+1 = 0.furthermore, for all n ≥ n0, we have wτ(n) ≤ wτ(n)+1 if n 6= τ(n) (that is n > τ(n),because wj >wj+1, f or τ(n + 1) ≤ j ≤ n.hence, 0 ≤wτ(n) ≤ max { wτ(n),wτ(n)+1 } =wτ(n)+1. therefore,wn → 0, as n → ∞ and this implies that xn → x∗ as n →∞. this complete the proof. � now using theorem (3.1), and multivalued mappings are nonexpansive mappings with convexvalues without demiclosed assumptions in the following theorem. theorem 3.2. let h be a real hilbert space and k be a nonempty, closed convex subset of h. let a : k → h be an α−inverse strongly monotone operator and let b : h → h be an k−strongly monotone and l−lipschitzian operator. let f : k → h be an b−lipschitzian mapping and m : h → 2h be a maximal monotone mapping such that the domain of m is included in k. let t1, t2 : k → cb(k) be a multivalued β− demicontractive mapping and t3 : k → cb(k) be a multivalued quasi-nonexpansive mapping such that ω := f ix(t1)∩f ix(t2)∩f ix(t3)∩s(π,λ) 6= ∅ and t1q = t2q = t3q = {q},∀q ∈ ω. for given x0 ∈ k, let {xn} be generated by the algorithm: δn = jmλn(i − λna)xn; yn = θnδn + (1− θn)vn, vn ∈ t1δn; zn = βnyn + (1− βn)un, un ∈ t2yn; tn = γnzn + (1− γn)wn, wn ∈ t3zn; xn+1 = pk(αnγf (xn) + (i − ηαnb)tn) (3.30) where {βn}, {γn}, {θn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following conditions https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 19 i): lim n→∞ αn = 0 ∞∑ n=0 αn <∞ ii): lim n→∞ inf(1− βn)(βn − β) > 0 and lim n→∞ inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1) iii): limn→∞ inf(1− γn)γn) > 0 assume that 0 < η < 2k l2 , 0 < γb < τ , where τ = η ( k − l2η 2 ) , and the sequences defined in (3.30), that is {xn} and {δn} converge strongly to unique solution x∗ ∈ ω, which also solve the following variational inequality: 〈ηbx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ ω (3.31) proof. since every multivalued nonexpansive mapping is quasi-nonexpansive and demicontractive,then, the proof follows theorem 3.10 � now using the same argument of the proof in theorem (3.1) in theorem (3.3), we achieved thedesired results. in theorem (3.3), we let t1 = pt1 , t2 = pt2 and t3 = pt3 without the assumptionsthat t1q = t2q = t3q = {q},∀q ∈ ω theorem 3.3. let h be a real hilbert space and k be a nonempty, closed convex subset of h. let a : k → h be an α−inverse strongly monotone operator and let b : h → h be an k−strongly monotone and l−lipschitzian operator. let f : k → h be an b−lipschitzian mapping and m : h → 2h be a maximal monotone mapping such that the domain of m is included in k. let t1, t2 : k → cb(k) be a multivalued β− demicontractive mapping and t3 : k → cb(k) be a multivalued quasi-nonexpansive mapping such that ω := f ix(t1)∩f ix(t2)∩f ix(t3)∩s(m,a) 6= ∅. for given x0 ∈ k, let {xn} be generated by the algorithm: δn = jmλn(i − λna)xn; yn = θnδn + (1− θn)vn, vn ∈ t1δn; zn = βnyn + (1− βn)un, un ∈ t2yn; tn = γnzn + (1− γn)wn, wn ∈ t3zn; xn+1 = pk(αnγf (xn) + (i − ηαnb)tn) (3.32) where {βn}, {γn}, {θn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following conditions i): lim n→∞ αn = 0 ∞∑ n=0 αn <∞ https://doi.org/10.28924/ada/ma.4.2 eur. j. math. anal. 10.28924/ada/ma.4.2 20 ii): lim n→∞ inf(1− βn)(βn − β) > 0 and lim n→∞ inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1) iii): limn→∞ inf(1− γn)γn) > 0 assume that 0 < η < 2k l2 , 0 < γb < τ , where τ = η ( k− l2η 2 ) , and i−pt1 , i−pt2 and i−pt3 are demiclosed at origin. hence, the sequences defined in (3.32), that is {xn} and {δn} converge strongly to unique solution x∗ ∈ ω, which also solve the following variational inequality: 〈ηbx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ ω (3.33) 4. conclusion the modified general viscosity iterative process presented in this research offers a powerful toolfor solving variational inclusion and fixed point problems involving and and fixed point problem withrespectively set-valued maximal monotone mapping and inverse strongly monotone and multivaluedquasi-nonexpansive and demicontractive operators. our theorem presents a new and a modifiedalgorithm for solving simultaneously variational inclusion problem and fixed point problem withrespectively set-valued maximal monotone mapping and inverse strongly monotone and multival-ued demicontractive and quasi-nonexpansive mappings. the result we show here improves andextends the corresponding results of some authors and many other recent results using forward-backward splitting method and general iterative algorithm that gives a strong convergence to aunique solution. references [1] s. wang, a general iterative method for an infinite family of strictly pseudo-contractive mappings in hilbert spaces,appl. math. lett. 24 (2011), 901-907. https://doi.org/10.1016/j.aml.2010.12.048.[2] j. geanakoplos, nash and walras equilibrium via brouwer, econ. theory, 21 (2003), 585-603.[3] s. kakutani, a generalization of brouwers fied point theorem, 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applications, math. anal. convex optim. 1 (2020),75-91.[29] c.e. chidume, geometric properties of banach space and nonlinear iterations, series: lecture notesin mathematics,springer, berlin, 2009.[30] f.e. browder, convergenge theorem for sequence of nonlinear operator in banach spaces, math. zeitsch. 100 (1967),201-225.[31] g.h.g. chen, r.t. rockafellar, convergence rates in forward-backward splitting, siam j. optim. 7 (1997), 421-444.[32] c.e. chidume, geometric properties of banach spaces and nonlinear iterations, springer verlag series: lecturenotes in mathematics, 2009.[33] c.e. chidume, c.o. chidume, n. djitte, m.s. minjibir, convergence theorems for fixed points of multivalued strictlypseudocontractive mappings in hilbert spaces, abstr. appl. anal. 2013 (2013), 629468. https://doi.org/10.28924/ada/ma.4.2 1. introduction 2. preliminaries 3. main results 4. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 18doi: 10.28924/ada/ma.4.18 the jacobi mate of an oval mircea crasmareanu faculty of mathematics, university "al. i. cuza", iasi, 700506, romania mcrasm@uaic.ro abstract. we introduce and study the jacobi mate cj of an euclidean oval c. we focus here on thecurvature of cj and on some examples. 1. introduction the enormous influence of convexity in practically every area of mathematics is widely known.we highlight the idea of convex curve by limiting the discussion to geometry, namely euclideanplane geometry. the recent book [2] dedicates an entire chapter, specifically chapter 6, to thistopic.this brief note aims to relate, via the first two jacobi elliptic functions, a second curve, cj , toa given specific convex curve c, called oval. given that these elliptic functions are 1-parametricextensions of the standard cosinus and sinus functions, which determine c, this link makes sense.the support function defining c serves as the foundation for the full analysis of this pair of curves.more specifically, we concentrate on the curvature, which is the only differential invariant for aplane curve. as possible area of applications for our results we mention the very recent (computerbased) shape analysis or topology optimization.the following is a list of the contents. the differential (and integral) geometry of the ovals isreviewed in the second section. our new idea of jacobi mate of the given oval c is presented inthe next section. it is important to note that, apart from the pair (c,cj), there exists another curve p that is naturally connected to the support function p of c and hence we will call the support curve. in fact, we study three curves. after the computation of p and cj curvatures, we focus on afew cases. we point out that certain complicated calculations require software and we make useof wolframalpha. 2. the differential geometry of euclidean ovals a brief overview of the differential geometry of ovals is given in this first part. hence, ourframework is the euclidean linear space e2 := (r2, 〈·, ·〉) with to the canonical inner product: received: 28 may 2024. key words and phrases. jacobi elliptic functions; oval; support function; curvature.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.18 https://orcid.org/0000-0002-5230-2751 eur. j. math. anal. 10.28924/ada/ma.4.18 2 〈u, v〉 = x1y1 + x2y2, u = (x1, x2) ∈ r2, v = (y1, y2) ∈ r2, 0 ≤ ‖u‖2 = 〈u, u〉. (2.1) fix an open interval i ⊆ r and consider c ⊂ e2 a regular parametrized curve of equation: c : r(t) = (x(t), y(t)), r ∈ c∞, ‖r ′(t)‖ > 0, t ∈ i. (2.2) suppose that c is closed, simple and strictly convex; then will be called oval. all its geometryis provided by a smooth support function p : i = [0, l > 0]→ (0,+∞) with: p(0) = p(l), p(t) + p′′(t) > 0, t ∈ i (2.3) through the relations:( x(t) y(t) ) := r(t) · ( p(t) p′(t) ) , r(t) := ( cos t − sin t sin t cos t ) ∈ so(2) = s1, ‖r(t)‖2 = (p(t))2 + (p′(t))2. (2.4)we point out that the function p was firstly considered by minkowski and the function t → ‖r(t)‖ > 0 is exactly the first legendre transformation of the convex function p. let f(c) = {t,n} be thefrenet frame of c and k : i = [0, l]→ r∗+ = (0,+∞) its curvature function. then, it is well knownthat these main functions are given by: p(t) := −〈r(t), n(t)〉 > 0, k(t) := 1 p(t) + p′′(t) = 1 ‖r ′(t)‖ > 0 (2.5) since: t (t) = (− sin t, cos t) = ie it , n(t) = it (t) = −e it = (− cos t,− sin t) (2.6)which means that the frenet frame is universal for the set of ovals defined on the same interval i .the geometry of the ovals has two well-known integral relations:i) the cauchy formula: l = ∫ 2π 0 p(t)dt. (2.7) ii) the blaschke formula for the area a(c) enclosed by c: a(c) = 1 2 ∫ 2π 0 [(p(t))2 − (p′(t))2]dt ≤ 1 2 ∫ 2π 0 ‖r(t)‖2dt, 4πa(c) ≤ l2 (2.8) with equality in the isoperimetric inequality (2.8) provided by the circle; we will treat the circleas oval in the example 3.4. remarks 2.1 i) the decomposition of the position vector field r in the frenet basis is: r(t) = p′(t)t (t)− p(t)n(t). (2.9) a plane curve satisfying k(t) = 1 ‖r ′(t)‖ for all t is called flat-flow curve in [6]. hence, any oval issuch a curve, a fact that explains the equality with 2π of its total curvature.ii) an important tool in one-dimensional dynamics is the fermi-walker derivative. let x(c) be the https://doi.org/10.28924/ada/ma.4.18 eur. j. math. anal. 10.28924/ada/ma.4.18 3set of vector fields along the curve c. then the fermi-walker derivative is the map ( [6, p. 420]) ∇fw : x(c)→ x(c): ∇fw (x) := d dt x + ‖r ′(·)‖k [〈x,n〉t − 〈x,t 〉n]. (2.10) the frenet frame is fermi-walker conserved: ∇fw (t ) = ∇fw (n) = 0. for our oval c we derive: ∇fw (r)(t) = r ′(t)− ‖r ′(t)‖k(t)[p(t)t (t) + p′(t)n(t)] = p′′(t)t (t)− p′(t)n(t). (2.11) hence if we denote r = rotation(p) then the curve t → ∇fw (r)(t) is exactly the curve rotation(p′).iii) associated to the support function p there exists the width function w : [0, l/2] → (0,+∞), w (t) := p(t) + p ( t + l 2 ). hence, its period is l 2 .iv) concerning the possible relationship between the periodicity and the curvature of a plane curvea very interesting problem is solved in the paper [1]: when is a periodic function the curvature ofa closed plane curve? 2 3. the jacobi mate of an oval fix the real number ρ ∈ (−1, 1) as the modulus for the differential system ( [7, p. 130]): du dt = −wv, u(0) = 1, dv dt = wu, v(0) = 0, dw dt = −ρ 2uv, w(0) = 1. (3.1) recall that its solutions are called jacobi elliptic functions and there are usually denoted cn(·, ρ), sn(·, ρ) respectively dn(·, ρ); we prefer the simple notation used above. as solutions of the odesystem (3.1) these functions satisfy two remarkable identities: u2 + v2 = 1, ρ2v2 + w2 = 1. (3.2) also, both functions u(·) and v(·) are periodic with l = 4l̃ for ( [7, p. 131]): l̃ = l̃(ρ) := ∫ 1 0 ds√ (1− s2)(1− ρ2s2) (3.3) while w is periodic of period 2l̃. in particular, l̃(0) = arcsin s|10 = π 2 for the usual trigonometricalfunctions cn(·, 0) = cos(·) and sn(·, 0) = sin(·). the complementary modulus is ρ′ := √1− ρ2 ∈ (0, 1] and the third jacobi function is bounded by: 0 < ρ′ ≤ w(t) ≤ 1. (3.4) the self-complementary case ρ′ = ρ is provided by ρ = 1√ 2 and being in the interval (0, 1) is theeccentricity of an ellipse, called self-complementary and studied in [5]. https://doi.org/10.28924/ada/ma.4.18 eur. j. math. anal. 10.28924/ada/ma.4.18 4due to the increasing interest in the geometry of ovals this short note defines the jacobi matefor the given oval c. as basic tool we use the new rotation matrix: jacobi(t, ρ) := ( u(t) −v(t) v(t) u(t) ) ∈ so(2) = s1. (3.5) definition 3.1 the curve cj is the ρ-jacobi mate of c if its parametrization is: rj(t) = ( xj yj ) (t) := jacobi(t, ρ) ( p p′ ) (t) = ( p(t)u(t)− p′(t)v(t) p′(t)u(t) + p(t)v(t) ) , t ∈ i = [0, l]. (3.6)since the derivative of rj is: r ′j (t) = (p ′(t)u(t)(1− w(t))− v(t)(p(t)w(t) + p′′(t)), p′(t)v(t)(1− w(t)) + u(t)(p(t)w(t) + p′′(t)))(3.7)it results: ‖r ′j (t)‖2 = (p′(t))2[1−w(t)]2+[p(t)w(t)+p′′(t)]2 ∈ ((ρ′p(t)+p′′(t))2, (p′(t))2+[p(t)+p′′(t)]2)(3.8)and then cj is a regular curve. it results also immediately:{ x ′′j = p ′′u(1− 2w) + p′v(ρ2u2 + w2 − 2w) + v(pρ2uv − p′′′)− pw2u y ′′j = p ′′v(1− 2w) + p′u(ρ2v2 − w2 + 2w)− u(pρ2uv − p′′′)− pw2v (3.9) and then, considering the map (·, ρ)→ rj(·) as a flow of curves, we compute its first derivative witha possible application to a parabolic flow (for example, of curve shortening type, see the chapter 2in [3]):{ ∂ ∂ρ r ′′ j (t) = 2ρu(t)v(t)[p ′(t)(u(t), v(t)) + p(t)(v(t),−u(t))] = 2ρu(t)v(t)[−i rj(t)] = 2w ′(t)[i rj(t)], ‖ ∂∂ρ r ′′ j (t)‖ = 2|ρ||u(t)||v(t)|‖r(t)‖. (3.10)therefore, ∂ ∂ρ r ′′ j (t) is orthogonal to rj(t), for all t ∈ [0, l]. remark 3.2 we point out that following the approach of [8] we can think c and cj as theeuclidean and jacobi deformations of the support curve t → p (t) := (p(t), p′(t)). we have ‖r(t)‖ = ‖p (t)‖ = ‖rj(t)‖, for all t . the expression of p recalls the well-known weierstrassparametrization (℘(u), ℘′(u)) of the elliptic curve e(g2, g3) : y2 = 4x3 − g2x − g3; see [9, p. 77]. 2 our main theoretical result computes the curvature of the mate cj through a long but straight-forward computation: theorem 3.3 i) if p is not a constant then the support curve p is a regular one having the euclidean curvature: kp (t) = p′(t)p′′′(t)− (p′′(t))2 [(p′(t))2 + (p′′(t))2] 3 2 . (3.11) https://doi.org/10.28924/ada/ma.4.18 eur. j. math. anal. 10.28924/ada/ma.4.18 5 let r(t0) be a vertex of the oval c i.e. p′′′(t0) = −p′(t0). then the curvature of p in t0 is: kp (t0) = −1 [(p′(t0))2 + (p′′(t0))2] 1 2 < 0. ii) the curvature of the ρ-jacobi mate cj of the oval c is a quadratic function in ρ: kj = (p′)2(1− w)(2w − w2) + p′[(1− w)p′′′ − ρ2uv(p + p′′)] + [pw2 + p′′(2w − 1)](pw + p′′) [(p′)2(1− w)2 + (pw + p′′)2] 3 2 .(3.12) if the modulus ρ is zero then w ≡ 1 and kj reduces to the usual curvature k from (2.5). moreover, for the flow interpretation before the remark 3.2 we have: ∂2kj(t) ∂ρ2 |ρ=0 = −2u(t)v(t)p′(t)[k(t)]2. (3.13) we focus now on some concrete examples. example 3.4 the circle c(o,r > 0) of the euclidean plane geometry is the oval provided bythe constant support function p ≡ r and hence w ≡ 2r; the curve p consists in the unique point (r, 0). its ρ-jacobi mate coincides cu c(o,r), hence kj = k ≡ 1 r , but now with the parametrization: circlej(t) = r(u(t), v(t)), ci rcle ′ j (t) = rw(t)(−v(t), u(t)), ‖circle ′j (t)‖ = rw(t) ∈ [rρ′, r].(3.14)hence, the frenet frame is: t (t) = (−v(t), u(t)), n(t) = it (t) = (−u(t),−v(t)) (3.15) as natural generalization of (2.6). by defining a new function: w (t) = ∫ t 0 w(λ)dλ (3.16) we can write the parametrization by arc-length: circlej(s) = r ( u ◦w−1 ( s r ) , v ◦w−1 ( s r )) , s ∈ [0, 2πr]. (3.17) the value of l̃ from (3.3) in the self-complementary case is: l̃ ( 1√ 2 ) ' 1.85 > π 2 ' 1.57 (3.18) while the second identity from (3.2) provides, in the case ρ 6= 0, a second jacobi parametrizationof the circle: scirclej(t) = r(ρv(t), w(t)), scircle ′j (t) = rρu(t)(w(t),−ρv(t)). (3.19) now, this second parametrization has singularities, namely the zeros l̃, 3l̃ of the function u. 2 example 3.5 fix the smooth real function p(t) := r − cos 3t; hence p(t) = p(t + 2π) andagain the width is constant w ≡ 2r. in [4, p. 23] it is proved that if r > 8 then p is the supportfunction of an oval c. if r > 8 is a positive integer then the oval c contains the integral point (−(r + 1), 0) corresponding to t = π while the curve p contains the 4 integral points (r − 1, 0), https://doi.org/10.28924/ada/ma.4.18 eur. j. math. anal. 10.28924/ada/ma.4.18 6 (r, 3), (r + 1, 0), (r,−3) corresponding respectively to t = 0, t = π 2 , t = π and t = 3π 2 . withthe derivatives: p′(t) = 3 sin 3t, p′′(t) = 9 cos 3t, p′′′(t) = −27 sin 3t (3.20) it results the curvatures: kp (t) = −3 [(sin 3t)2 + 9(cos 3t)2] 3 2 < 0, k(t) = 1 r + 8cos 3t ∈ [ 1 r + 8 , 1 r − 8 ] . (3.21) we note that kp does not depend on r while the curvature k solves the differential equation ∂k ∂r = −k 2. the cauchy and the blaschke formulae give: l(c) = 2πr, a(c) = π(r2 − 4) > 60π. (3.22) the length of the curve p is: l(p ) = 3 ∫ 2π 0 √ (sin 3t)2 + 9(cos 3t)2dt = 3 ∫ 2π 0 √ 5 + 4 cos 6tdt ' 40.09 (3.23) and we point out that the argument 3t involved in its components recalls the cayley sextic, whichis not an oval but a closed curve: cay ley(t) := cos3 t(cos 3t, sin 3t), t ∈ [0, 2π], l(cay ley) = 3π. (3.24) for the ρ-jacobi mate we compute only the velocity since its curvature has a complicated expression:{ ‖r ′j (t)‖2 = 9(sin 3t)2[1− w(t)]2 + [9 cos 3t + w(t)(r − cos 3t)]2, 0 < (ρ′r + (9− ρ′) cos 3t)2 < ‖r ′j (t)‖2 < 81(sin 3t)2 + (r + 8cos 3t)2. (3.25) 2 example 3.6 for α ∈ [1,+∞) the 2π-periodic function pα : [0, 2π]→ r∗+, pα(t) := 1 α √ α4 cos2 t + sin2 t is the support function of an ellipse since: pα(t) + p ′′ α(t) = α3 (α4 cos2 t + sin2 t) 3 2 > 0. (3.26) now, the width function is non-constant being 2pα. as example, with wolframalpha we obtain thelength l (pα=2) ' 45.51. the rhs of the inequality (3.8) reads: ‖r ′j (t)‖2 < (α4 − 1)2 2α2(α4 cos2 t + sin2 t) + α6 (α4 cos2 t + sin2 t)3 . (3.27) with the same possible flow interpretation in mind we compute the first derivative of the supportfunction: ∂pα ∂α (t) = α4 cos2 t − sin2 t α2 √ α4 cos2 t + sin2 t . (3.28) we note also that for α > 1 the given support function does not has an indicatrix i.e. the planecurve defined implicitly by {(α, t) ∈ r2; pα(t) = 1} is empty. the same fact holds for the supportfunction of the previous example when r > 8. 2 https://doi.org/10.28924/ada/ma.4.18 eur. j. math. anal. 10.28924/ada/ma.4.18 7references [1] j. arroyo, o. j. garay, j. j. mencia, when is a periodic function the curvature of a closed plane curve, am. math.mon. 115 (2005), 405-414. https://doi.org/10.1080/00029890.2008.11920543[2] h. alencar, w. santos, g. silva neto, differential geometry of plane curves, american mathematical society,providence, rhode island, 2022. https://doi.org/10.1090/stml/096.[3] b. andrews, b. chow, c. guenther, m. langford, extrinsic geometric flows, american mathematical society, provi-dence, rhode island, 2020. https://doi.org/10.1090/gsm/206.[4] w. cieślak, w. mozgawa, p. wlaź, on the closest distance between a point and a convex body, bull. soc. sci. lettr.łódź, sér.: rech. déform. 67 (2017), 21-30. https://doi.org/10.26485/0459-6854/2017/67.2/2.[5] m. crasmareanu, magic conics, their integer points and complementary ellipses, an. ştiinţ. univ. al. i. cuza iaşimat. 67 (2021), 129-148.[6] m. crasmareanu, the flow-curvature of plane parametrized curves, commun. fac. sci. univ. ankara ser. a1 math.stat. 72 (2023), 417-428. https://doi.org/10.31801/cfsuasmas.1165123.[7] r.h. cushman, l.m. bates, global aspects of classical integrable systems, springer basel, basel, 2015. https: //doi.org/10.1007/978-3-0348-0918-4.[8] b. mazur, perturbations, deformations, and variations (and “near-misses") in geometry, physics, and number theory,bull. amer. math. soc. 41 (2004), 307–336. https://doi.org/10.1090/s0273-0979-04-01024-9.[9] r. takloo-bighash, a pythagorean introduction to number theory: right triangles, sums of squares, and arithmetic,springer, cham, 2018. https://doi.org/10.1007/978-3-030-02604-2. https://doi.org/10.28924/ada/ma.4.18 https://doi.org/10.1080/00029890.2008.11920543 https://doi.org/10.1090/stml/096 https://doi.org/10.1090/gsm/206 https://doi.org/10.26485/0459-6854/2017/67.2/2 https://doi.org/10.31801/cfsuasmas.1165123 https://doi.org/10.1007/978-3-0348-0918-4 https://doi.org/10.1007/978-3-0348-0918-4 https://doi.org/10.1090/s0273-0979-04-01024-9 https://doi.org/10.1007/978-3-030-02604-2 1. introduction 2. the differential geometry of euclidean ovals 3. the jacobi mate of an oval references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 7doi: 10.28924/ada/ma.4.7 modeling the inflow of exposed and infected migrants on the dynamics of malaria musah konlan department of mathematics and statistics, university of energy and natural resources, sunyani, ghana correspondence: musah.konlan@uenr.edu.gh abstract. malaria is currently a life-threatening vector borne disease which is endemic in mostof the developing and underdeveloped countries associated with poor health care systems. in thisstudy, a host-vector mathematical model that takes into account the inflow of human migrants whohave been exposed or infected with malaria is formulated and analysed. the reproduction numberof the mosquito vector population is derived and used as a threshold quantity for determining theexistence of the model trivial and realistic steady states. the routh-hurwitz criterion and somestability theorems of metzler matrices are used to show that the realistic disease free equilibriumis both locally and globally asymptotically stable whenever the disease reproductive number is lessthan one. we derived an equation for the model endemic condition and used descartes rule of signchange to established the conditions for the model to admit one or three endemic equilibrium state(s).it is further shown that in the absence of inflow of exposed or infected migrants, the model admits aglobally asymptotically unique endemic equilibrium when r0 > 1 and two endemic equilibria when r0 < 1. our local sensitivity analysis revealed that the adults mosquito removal and biting rateswere respectively the most significant contributing parameters to the spread of malaria. the numericalsimulations results suggested that the exposed and infected immigrants have no significant impact onthe dynamical behaviour of the model population sub-classes. 1. introduction malaria is currently a life-threatening vector borne disease which is endemic in most of thedeveloping and underdeveloped countries associated with challenging health care systems. moreparticularly, malaria is highly endemic in sub-saharan africa characterized with poor hygienicconditions which serve as suitable breeding site for malaria vectors [1]. plasmodium parasites andfemale anopheles mosquitoes are respectively the causal agent and transmitting vectors of malaria.among the most vulnerable groups to malaria are expectant mothers and infants under five yearsof age [1–3]. common symptoms of malaria include: fever, chills, headache, pain, anaemia andvomiting [1,4]. the world health organization (who) reported that in 2022 alone, there were twohundred and forty nine million malaria cases recorded globally. ninety four percent of theses cases received: 10 jan 2024. key words and phrases. malaria; immigrants; equilibrium states; stability analysis; local sensitivity analysis; nu-merical simulations. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 2were recorded in africa. for example, ghana recorded within the same period five million threehundred and fifteen thousand five hundred and ninety three (5315593) and eleven thousand fivehundred and fifty seven (11557) estimated malaria cases and deaths respectively [3]. currently,population migration caused by climate change induced factors and conflicts constitute a majorthreat to the malaria control programs.mathematical modeling has become a significant tool box for understanding disease transmissiondynamics and evaluating the effectiveness of disease control strategies [5]. these models gener-ally explain the dynamics of infections, provide/estimate the thresholds indicators that determinewhether the disease will persist or die out [6–8].according to mukhaktar et al. [9], mathematicallymodeling malaria can help better understand the disease dynamics and further unveil how cer-tain factors such as human migration influence the disease transmission process, in that regard,several modeling studies have been conducted concerning human migration and malaria. authorsin [9] assessed how human mobility impact the malaria disease burden in south sudan. apriantiet al. [10] examined the effect of susceptible immigrants on the spread of malaria in indonesia.yiga et al. [11] analysed a malaria transmission model that takes into consideration the combinedeffect of infected immigrants and other variables that depend on temperature and rainfall. ma-liki et al. [2] modelled the control of malaria in a population with infected immigrants. witboi etal. [12] presented a malaria population dynamics model with human migrants. yacheur et al. [13]studied the importation of malaria infections from sub-saharan africa to northern africa and theabsorption effect of the immigrants. researchers in [14, 15] formulated and analyzed mathematicalmodels for malaria disease dynamics that considered malaria vaccination campaigns and inflow ofinfective immigrants. ahkrizal et al. [16] formulated a malaria dynamics model capturing the inflowof exposed and infected migrants and the recovery of exposed individuals.in the above mentioned literature, little attention is given to the aquatic phase of the malariavectors. even though, the population of adults mosquitoes responsible for disseminating malariainfections is proportional to the density of the aquatic mosquitoes. it is therefore necessary totake into consideration the aquatic stages of the vector in a malaria model [11, 17]. hence, in thisstudy, in order to explore the impact of exposed and infected individuals on the endemic conditionof malaria, we extend the malaria models formulated in [11] to include the exposed vectors and themodel in [16] to capture the aquatic stage of the anopheles female mosquito without the relapsefactor of the recovered individuals. the rest of the organization of the paper is as follows: sectiontwo takes care of the model formulation and analysis, in section three, the sensitivity analysisresults is presented, population simulations is carried out in section four and the conclusion ispresented in section five. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 32. malaria model development the model considered the interactions of humans (hosts) and female anopheles mosquitoes(vectors). humans (hosts) are classified into susceptible (sh), exposed (eh), infected (ih) andrecovered (rh) sub-classes. as a result, the total human population at any given time t is: nh (t) = sh (t) + eh (t) + ih (t) + rh (t) (1) the human/host population is sustained at a constant birth rate πh and immigration rate m.hence, the susceptible human class is generated at a rate (πh + (1− p1 − p2)m), where p1 and p2 are the immigration rate of exposed and infected migrants respectively. recovered immigrantsare assumed to be susceptible to malaria. susceptible humans become exposed to malaria infec-tions through effective contact with infected female anopheles mosquitoes during blood meal ata rate λh. exposed humans progress to infected class at rate γ. the size of exposed humans isaugmented as results of immigration of humans at a rate p1m . it is common to find people insettings with limited health facilities resorting to self medication after being bitten by mosquitoesor when a family member is suspected of suffering from malaria. hence, in this model it is assumedthat exposed individuals recover from malaria at a rate ω. the density of the infected humans isreduced following treatment at rate τ or due to malaria induced mortality at a rate δ. the size ofthe infected humans is augmented due to migration of infected individuals at rate p2m . recoveredindividuals lose their immunity and join the susceptible sub-class at a rate ϕ. the constant µh isthe human removal rate from each human compartment.also, the vector (anopheles mosquito) population is stratified into immature and adult mosquitosub-populations. the immature female anopheles mosquito sub-population includes the mosquitoeggs, larvae and pupae stages.these aquatic stages are represented by a single compartment denoted by (am). the aquaticvector (am) is generated from the eggs laid by the matured mosquitoes (susceptible, exposed andinfected) at a rate πm (1− am k ) (sm + em + im).the population of aquatic vector is bounded above by the carrying capacity of the aquatic envi-ronment (k). the aquatic mosquito population declines due to natural death at a rate µa. theaquatic mosquitoes mature into susceptible mosquitoes at a rate ψ. the matured mosquito is furtherstratified into susceptible (sm), exposed (em) and infected (im) vectors. the susceptible vectorsbecome exposed to malaria parasites during blood meal from infectious (infected) humans at a rate λm. exposed vectors (em) subsequently become infected at a rate σ. as the results of naturaldeath at a rate µm, the densities of adult mosquito populations ((sm)), (em), (im) decrease. thus,at any time t , the aquatic and adult malaria vector populations (am and nam) satisfy: am (t) ≤ k, nam (t) = sm (t) + em (t) + im (t) (2) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 4 λh = bβhim nh and λm = bβmih nh are respectively the forces of infection for the human and femaleanopheles mosquitoes. the schematic diagram (figure 1) describes the transmission dynamics ofmalaria in an interacting human and mosquito populations. the model parameters are presentedin table (1). table 1. parameter description with their values and sources parameter description value[range] reference unit πh human recruitment rate 0.03 [11] day−1m immigration rate of human 0.001 [11] day−1 p1 immigration rate of exposed humans 0.2 [11, 16] day−1 p2 immigration rate of infected humans 0.2 [11, 16] day−1 µh natural mortality rate of human 1/21900 [11] day−1 βh probability of transmission of infectionsfrom an infectious human to a 0.00021 [11] -susceptible mosquito (vector) γ progression rate from exposed humans 1/20 [11] day−1to infected humans ω progression rate from exposed humans 0.055 [16] day−1to recovered humans τ progression rate from infected humans 1/30 [11] day−1to recovered humans ϕ progression rate from recovered humans 1/(20× 365) [11] day−1to susceptible humans δ malaria induced death for humans 0.001 [11] day−1 πm anopheles mosquito egg deposition rate 6 [17,18] day−1 k carrying capacity for immature mosquitoes 40000 [18] space b female anopheles mosquito biting rate 0.94[0.1-1] [18] day−1 βm probability of transmission ofinfections from an infected 0.00021 [11] -anopheles mosquito to a susceptible human ψ maturity rate of immature mosquitoes 0.08 [19] day−1 σ progression rate from exposed mosquitoesto infected mosquitoes 0.091 [18] day−1 µm natural mortality rate of adult mosquitoes 0.11346 [17] day−1 µa natural mortality rate of immature mosquito 0.1042 [18,19] day−1 https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 5 figure 1. schematic diagram for malaria transmission dynamics with human immi-grants based on figure 1, the following system of equations is derived:  dsh dt = πh + (1− p1 − p2)m + ϕrh − (λh + µh)sh deh dt = p1m + λhsh − g0eh dih dt = p2m + γeh − g1ih drh dt = τih + ωeh − g2rh dam dt = πm ( 1− am k ) (sm + em + im)− g3am dsm dt = ψam − (λm + µm)sm dem dt = λmsm − g4em dim dt = σem − µmim (3) where: g0 = (ω+γ+µh), g1 = (τ+δ+µh), g2 = (ϕ+µh), g3 = (ψ+µa) and g4 = (σ+µm) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 62.1. boundedness of solution. theorem 1. for non-negative initial values sh(0), eh(0), ih(0), th(0), am(0), sm(0) and im(0) of system (3), each element of the solution set {sh(t), eh(t), ih(t), th(t), am(t), sm(t) im(t)} is non-negative and bounded ∀ t ≥ 0. proof. considering the first differential equation in system (3): dsh dt = πh + (1− p1 − p2)m + ϕrh − (λh + µh)sh (4) =⇒ dsh dt ≥ −(λh + µh)sh =⇒ ∫ 1 sh dsh ≥ − ∫ (λh + µh)dt =⇒ sh(t) ≥ sh(0)e−(µht+ ∫ t 0 λh(x)dx) ≥ 0similarly: deh dt = p1m + λhsh − g0eh =⇒ eh(t) ≥ eh(0)e−g0t ≥ 0 dih dt = p2m + γeh − g1ih =⇒ ih(t) ≥ ih(0)e−g1t ≥ 0 drh dt = ωeh + τih − g2)rh =⇒ rh(t) ≥ rh(0)e−g2t ≥ 0 dam dt = πm ( 1− am k ) (sm + em + im)− g3am =⇒ am(t) ≥ am(0)e−g3t ≥ 0 dsm dt = ψam − (λm + µm)sm =⇒ sm(t) ≥ sm(0)e−(µmt+ ∫ t 0 λm(x)dx) ≥ 0 dem dt = λmsm − g4em =⇒ sm(t) ≥ sm(0)e−g4t ≥ 0 dim dt = λmsm − µmim =⇒ im(t) ≥ im(0)e−µmt ≥ 0 therefore, for ∀ t ≥ 0, the state variables of the model have non-negative solutions. 2.2. invariant region. this section is dedicated to finding the region over which the solution setof our malaria model system of equations is well posed. theorem 2. the feasible region in which the solution set of the model system of equations make biological sense is the set; d = dh ×dm ⊂ r4+ × r4+ (5) where dh = { (sh, eh, ih, rh) ∈ r4+ : sh + eh + ih + rh ≤ m + πh µh } (6) and dm = { (am, sm, em, im) ∈ r4+ : am ≤ k, sm + em + im ≤ ψk µm } (7) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 7 proof.firstly, we determine the subset dh.the human (host) population nh at any given time t is: nh = sh + eh + ih + rh (8) taking the differential of both sides of equation (8) and simplifying gives: dnh dt = m + πh − µhnh − δih =⇒ dnh dt ≤ m + πh − µhnh (in the absence of malaria induced mortality) =⇒ dnh nh − m+πh µh ≤ −µhdt (9) integrating the last inequality in (9) and taking the limit as t → +∞, yield: nh → m+πh µhconsequently, the following result is obtained 0 ≤ nh ≤ m + πh µh (10) therefore: dh = { (sh, eh, ih, rh) ∈ r4+ : sh + eh + ih + rh ≤ m + πh µh } (11) secondly, the subset dm is determined. at any point in time, the mosquito (vector) populationsatisfies: am ≤ k, nam = sm + em + im (12) now, nam = sm + em + im =⇒ d dt (nam) = d dt (sm + em + im) =⇒ dnam dt = dsm dt + dem dt + dim dt =⇒ dnam dt ≤ ψk − µmnam =⇒ nam − ψk µm ≤ ( nam(0)− ψk µm ) e−µmt =⇒ nam ≤ ψk µm as t → +∞. therefore, dm = { (am, sm, em, im) ∈ r4+ : am ≤ k, sm + em + im ≤ ψk µm } (13) thus, the feasible region for system (3) is the set: d = dh ×dm ⊂ r4+ × r4+ (14) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 82.3. model equilibrium points. to discuss the model equilibrium points, we consider two cases.the case where there is inflow of exposed and infected migrants (p1, p2 > 0). for this scenario,there is no disease free equilibrium and the model admits only the endemic equilibrium to bedetermine later. the second case is when there is no immigration of exposed and infected humans(p1 = p2 = 0). in this case, the computation of the model disease-free equilibria, is summarized inthe theorem below. this approach is adopted from [20–22]. theorem 3. for convenience, we define the threshold parameter n = πmψ µm(ψ + µa) = πmψ g3µm (15) as the mosquito net reproduction or extinction number, then if:(1) n ≤ 1, system (3) admits a trivial disease-free equilibrium (tdfe) (which corresponds to a population without mosquitoes) given by: ξ0 = (s∗h, 0, 0, 0, 0, 0, 0, 0) (16) (2) n > 1 (mosquitoes persist in the community), system (3) admits a realistic disease-free equilibrium (rdfe) (since it corresponds to the existence of mosquitoes in the population) given by: ξ1 = (s∗h, 0, 0, 0, 0, a∗m, s ∗ m, 0) (17) where: s∗h = m+πh µh , a∗m = k ( 1− 1 n ) and s∗m = ψk µm ( 1− 1 n ) proof.suppose, (s∗h, e∗h , i ∗ h , r∗h, a∗m, s∗m, e∗m, i∗m ) is any arbitrary disease-free equilibrium point.setting system (3) to zero with the condition that there are no infections at the disease-freeequilibrium, that is, p1 = p2 = e∗h = i∗h = r∗h = e∗m = i∗m = 0, gives: s∗h = m+πh µh for the firstequation.also, it is not hard to see from system (3) that s∗m + e∗m + i∗m = ψa∗m µm (18) hence, from the sixth equation of system (3), we see that a∗m satisfies: ψπm µm ( 1− a∗m k ) a∗m − g3a∗m = 0 (19) =⇒ a∗m = 0 or a∗m = k ( 1− 1 n ) (20) now a∗m = 0 =⇒ s∗m = 0 and a∗m = k ( 1− 1 n ) =⇒ s∗m = ψk µm ( 1− 1 n ) (21) hence, ξ0 and ξ1 are obtained respectively from a∗m = 0 and a∗m = k ( 1− 1 n ). clearly, themagnitude of n dictates the existence of the model disease-free equilibrium points. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 9 n is a threshold quantity known as the vector offspring number or vector net reproduction number[18, 19, 23]. in general, n can be interpreted as a measure of the average number of new adultfemale anopheles mosquitoes produced by one reproductive anopheles mosquito during its entirereproductive life. it is expressed as a product of the egg deposition rate πm, the fraction of immaturemosquito that survive and develop into adult anopheles mosquito ψ ψ+µa and the average life span ofadult anopheles mosquito 1 µm . thus, if n > 1, the mosquito population persists in the community,otherwise if n ≤ 1, the malaria vector population becomes extinct and the local transmissionof malaria cannot take place. it is worth noting that the trivial disease-free equilibrium (tdfe)corresponds to the absence of female anopheles mosquitoes in the community. hence, the tdfeis biologically less meaningful. 2.4. the basic reproductive number. in epidemiology, the basic reproductive number (ro ) isa threshold quantity that is used to determine the extent of severity of the epidemics. in thisstudy, the method of next generating matrix is adopted to compute the model ro . expressing ourmodel differential equations in the form dx dt = (f − v)xt where xt denotes the transpose of x = (eh, ih, em, im), f and v are vectors denoting the rate of generation of new infections andtransfer rates respectively, gives: f =  p1m + λhsh p2m λmsm 0  and v =  g0eh −γeh + g1ih g4em −σem + µmim  (22) evaluating the jacobian matrices f and v of f and v at the rdfe gives respectively: f =  0 0 0 bβhs ∗ h n∗h 0 0 0 0 0 bβms∗m n∗h 0 0 0 0 0 0  and v =  g0 0 0 0 −γ g1 0 0 0 0 g4 0 0 0 −σ µm  (23) from the expression of v, the inverse of v is: v −1 =  1 g0 0 0 0 γ g0g1 1 g1 0 0 0 0 1 g4 0 0 0 1 g4µm 1 µm  (24) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 10hence, the next generation matrix fv −1 is given by: fv −1 =  0 0 σbβhs ∗ h g4n ∗ hµm bβhs ∗ h n∗hµm 0 0 0 0 γbβms∗m g0g1n ∗ h bβms∗m g1n ∗ h 0 0 0 0 0 0  (25) solving for λ in the relation ∣∣fv −1 − λi∣∣ = 0 , where i is a unit matrix and λ an eigenvalue of fv −1, we get the dominant eigenvalue as: λmax = r0 = √ σγb2βhβms ∗ hs ∗ m g0g1g4n ∗2 h µm (26) taking n∗h = s∗h and simplifying the expression in (26), we obtain the reproductive number of themodel given by: r0 = √ σγb2βhβmµhkψ g0g1g4(m + πh)µ2m ( 1− 1 n ) = √ r0h × r0m (27) where: r0h = γbβhµh g0g1(m + πh) and r0m = σbβmkψ g4µ2m ( 1− 1 n ) the threshold quantities r0h and r0m characterized the contributions of malaria disease spreadfrom human to mosquito (host to vector) and from mosquito to human (vector to host) respectively. r0h represents the number of secondary cases of anopheles mosquitoes one infectious (infectedor treated) human will generate in a completely susceptible population of anopheles mosquitoesduring its infectious phase. similarly, r0m can be interpreted as the number of secondary humancases generated by an infected anopheles mosquito in an entirely susceptible human populationover the course of its life time as infectious [2]. 2.5. stability of malaria-free equilibrium. 2.5.1. local stability of malaria-free equilibrium. theorem 4. the rdfe (ξ1) = ( m+πh µh , 0, 0, 0, 0, k ( 1− 1 n ) , kψ µm ( 1− 1 n ) , 0 ) with n > 1 is locally asymptotically stable (las) if r0 < 1 and unstable if r0 > 1 https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 11 proof. let matrix j0 be the jacobian matrix of system (3) evaluated at the rdfe (ξ1). thus, j0 =  −µh 0 0 ϕ 0 0 0 −bβhs ∗ h n∗h 0 −g0 0 0 0 0 0 bβhs ∗ h n∗h 0 γ −g1 0 0 0 0 0 0 ω τ −g2 0 0 0 0 0 0 0 0 − ( g3 + πms∗m k ) πm n πm n πm n 0 0 −bβms ∗ m n∗h 0 ψ −µm 0 0 0 0 bβms∗m n∗h 0 0 0 −g4 0 0 0 0 0 0 0 σ −µm  (28) it is not hard to see that the matrix in (28) admits two negative eigenvalues, namely λ1 = −µh and λ2 = −g2. using the matrix reduction method, the remaining eigenvalues can be obtained from thesub-matrix in (29) below: j1 =  −g0 0 0 0 0 bβhs ∗ h n∗h γ −g1 0 0 0 0 0 0 − ( g3 + πms∗m k ) πm n πm n πm n 0 −bβms ∗ m n∗h ψ −µm 0 0 0 bβms∗m n∗h 0 0 −g4 0 0 0 0 0 σ −µm  (29) the characteristic equation of the sub-matrix in (29) is given by (λ+ g0)(λ+ g1)(λ+ g4)(λ+ µm) ( λ2 + sλ+ p ) = 0 (30) where: s = πms∗m k + g3 + µm, and p = πmµms∗m kit can clearly be seen from (30) that four eigenvalues of the sub-matrix in (29) λ3 = −g0, λ4 = −g1, λ5 = −g4, and λ6 = −µm are negative. also, the nature of the remaining twoeigenvalues of the sub-matrix in (29) are determined from : λ2 + sλ+ p = 0 (31) since, s and p are positive whenever n > 1, it implies that the two remaining eigenvalues ofthe sub-matrix j1 are stricly negative. consequently, all eigenvalues of the matrix j0 are real andnegative. hence, according to the routh-hurwitz stability criterion, the malaria realistic disease-free equilibrium state ξ1 is locally asymptotically stable when n > 1 and r0 < 1 and unstableotherwise. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 122.5.2. global stability of malaria-free equilibrium. following [19, 24–27], the global stability ofa system equilibrium point can be established by first expressing the system in a triangular formas follows: dys dt = b1 (ys − yrdfe) + b12yi d yi dt = b2yi (32) here, ys and yi denotes the compartments of non-transmitting and transmitting hosts and vec-tors respectively, with ys = (sh, rh, am, sm)t yi = (eh, ih, em, im)t and yrdfe =( s∗h, r ∗ h, a ∗ m, s ∗ m ) = ( m+πh µh , k ( 1− 1 n ) , ψkµm ( 1− 1 n )) (ys − yrdfe) =  sh − m+πh µh rh am −k ( 1− 1 n ) sm − kψ µm ( 1− 1 n ) (33) b1 = ∂ys ∂(sh, rh, am, sm) (34) b12 = ∂ys ∂(eh, ih, em, im) (35) b2 = ∂yi ∂(eh, ih, em, im) (36)using our model system of equations system (3), we get: b1 =  −µh ϕ 0 0 0 −g2 0 0 0 0 − ( πms∗m k + ψ + µa ) πm n 0 0 ψ −µm  (37) b12 =  0 0 0 −bβhs ∗ h n∗h ω τ 0 0 0 0 πm n πm n 0 −bβms ∗ m n∗h 0 0  (38) b2 =  −g2 0 0 bβhs ∗ h n∗h γ −g1 0 0 0 bβms∗m n∗h −g4 0 0 0 σ −µm  (39) from the above we formulate the theorem as follows. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 13 theorem 5. the system dys dt = b1 (ys − yrdfe)+b12yi is globally asymptotically stable (gas) at the rdfe when all eigenvalues of matrix b1 have negative real parts and b2 is a metzler matrix. proof.clearly, two eigenvalues of the matrix b1 are λ1 = −µh and λ2 = −g2. thus, applying themethod of matrix reduction, b1 reduces to the sub-matrix: a = −(πms∗mk + g3 ) πm n ψ −µm  (40) the nature of the remaining two eigenvalues of b1 are determined from characteristic equation: λ2 + ( g3 + µm + πms ∗ m k ) λ+ πmψ ( 1− 1 n ) = 0 (41) since in equation (41), g3+µm + πms∗m k > 0 and πmψ (1− 1 n ) > 0 whenever n > 1, we concludeusing the routh-hurwitz stability condition, that the eigenvalues λ3 and λ4 have negative realparts. hence, all the eigenvalues of the matrix b1 have negative real parts.additionally, b2 is clearly a metzler matrix (since all the off diagonal entries are non negative).thus, we conclude that the system d ys dt = b1 (ys − yrdfe) + b12yi (42) is gas at the realistic disease free equilibrium [19,24–27]. 2.6. malaria endemic equilibrium. let ξ2 = (s∗∗h , e ∗∗ h , i ∗∗ h , r ∗∗ h , a ∗∗ m , s ∗∗ m , e ∗∗ m , i ∗∗ m ) be theendemic equilibrium (ee) point for the malaria model, then setting system (3) to zero, the followingsystem of solutions is obtained s∗∗h = q0 (g0bϕτβmµh)i ∗∗2 h +[q1bβmµh+µm(m+πh)]i ∗∗ h +q1µm(m+πh) [q2σb2βhβmµhψa∗∗m+g4bβmµhµm(m+πh)]i ∗∗ h +g0g2g4µ 2 m(m+πh) 2 e∗∗h = q3i ∗∗3 h +q4i ∗∗2 h +q5i ∗∗ h +q6 q7i ∗∗2 h +q8i ∗∗ h +q9 r∗∗h = τi∗∗h +ωe ∗∗ h g2 a∗∗m = 0 or a∗∗m = k ( 1− 1 n ) s∗∗m = 0 or s∗∗m = (m+πh)ψa ∗∗ m bβmµhi ∗∗ h +µm(m+πh) e∗∗m = 0 or e∗∗m = bβmµhs ∗∗ m i ∗∗ h g4(m+πh) i∗∗m = 0 or i∗∗m = σbβmµhs ∗∗ m i ∗∗ h g4µm(m+πh)here, i∗∗h satisfies : q3i ∗∗3 h + q2i ∗∗2 h + q1i ∗∗ h + q0 = 0 (43) where: q0 = g4µm(m+πh) µh q1 = p1ϕωm + g0g2((1− p1 − p2)m + πh) q2 = g0g2 − ϕω https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 14 q3 = 1 m+πh g0g4ϕτσb 3βhβ 2 mµhµmψa ∗∗ m q4 = 1 m+πh g4ϕτσb 2βhβmµhψa ∗∗ m ( q1bβmµm + m+πh µh g0ϕτ ) + g4p1mbβmµm ( q2σb 2βhβmµhψa ∗∗ m m+πh + g0g2g4bβmµhµm ) q5 = g4bβmµ 2 m (σbβhψa ∗∗ m (q1 + p1mq2) + 2g0g2g4p1mµm(m + πh)) q6 = g0g2p1m µh ( µ2mg4(m + πh) )2 q7 = g0g4bβmµhµm m+πh ( q2σb 2βhβmψa ∗∗ m + g0g2g4bβmµm(m + πh) ) q8 = g0g4µ 2 m ( q2σb 2βhβmψa ∗∗ m + 2g0g2g4bβmµm(m + πh) ) q9 = g2 µh ( g0g4µ 2 m(m + πh) )2 q3 =g0g4bβmµm{ σb2βhβmµha ∗∗ m m + πh (g0g1µh + ϕγ(µh + δ)) + g0g1g2g4bβmµhµm} q2 =g0g1g4µ 2 m(m + πh)( q2σb 2βhβmψa ∗∗ m m + πh + 2g0g2g4bβmµm− γσb2βhβma ∗∗ m m + πh (g4q1bβmµhµm + g0g4ϕτµ 2 m(m + πh)− g4mbβmµhµm(p1γ + g0p2)( ασb2βhβma ∗∗ m m + πh + g0g2g4bβmµm) q1 = g4µ 2 m(m + πh) µh {g20g1g2g4µ2m(m + πh)− q1γσb 2βhβmµhψa ∗∗ m m + πh − g0p2m( q2σb 2βhβmµhψa ∗∗ m m + πh + 2g20g2g4bβmµhµm} q0 = − g0g2 µh m(p1γ + g0p2 ( g4µ 2 m(m + πh) )2 to analyse the disease endemic condition, we consider the polynomial function: f (i∗∗h ) = q3i ∗∗3 h + q2i ∗∗2 h + q1i ∗∗ h + q0 = 0 (44) there is enough evidence that the polynomial in (44) admits a positive solution on the interval [0,+∞) since: f (0) = q0 < 0 and lim i∗∗h →+∞ f (i∗∗h ) = +∞. next, we employ descartes’ rule ofsigns change to explore more information on the roots of the polynomial f (i∗∗h ) (see table 2). table 2. number (#) of possible positive roots of f (i∗∗h ) case q3 q2 q1 q0 # of sign change # of roots(i) + + + − 1 1(ii) + + − − 1 1(iii) + − + − 3 1, 3(iv) + − − − 1 1 https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 15based on the results in table 2, we claim that in the presence of importation of malaria infections,system (3) may admit one or three endemic equilibrium state(s). to better understand any possibleimpact of the inflow of humans who have been exposed or infected with malaria parasites on thedisease endemic condition, we now consider the endemic relation in the absence of exposed andinfected immigrants.to that effect, we set p1 = p2 = 0 into (44) and simplify to obtain: i∗∗h ( a2i ∗∗2 h + a1i ∗∗ h + a0 ) = 0 (45) equation (45) implies i∗∗h = 0 or a2i ∗∗2 h + a1i ∗∗ h + a0 = 0 (46)where: a2 =bβmµh{ σb2βhβmµha ∗∗ m m + πh (g0g1µh + ϕγ(µh + δ)) + g0g1g4bβmµhµm} (47) a1 =g1µhµm ( q2σb 2βhβmψa ∗∗ m + 2g0g2g4bβmµm(m + πh) ) − γσb2βhβmµhψa ∗∗ m (g2bβmµh + ϕτµm) (48) a0 = g0g1g2g4µ 3 m(m + πh)2 ( 1− r20 ) (49)with i∗∗h = 0 in (46), we retrieve the tdfe when a∗∗m = 0 and the rdfe when a∗∗m = k ( 1− 1 n ) furthermore, a solution to the quadratic equation in (46) can be obtained using the quadraticformula, that is : i∗∗h = −a1 ± √ a21 − 4a0a2 2a2 (50)the expression in (50) leads to the following theorem: theorem 6. in the absence of inflow of exposed and infected human migrants, the malaria model represented by system (3) admits:(i) one unique ee if a0 < 0, that is r0 > 1(ii) one unique ee if a1 < 0, and r0 = 1 or a21 − 4a0a2 = 0(iii) two ee if a1 < 0 and a0 > 0 that is r0 < 1 or a21 − 4a0a2 > 0(iv) no ee otherwise. we deduce from case (i) of theorem 6 that for a specific case where the parameter accountingfor the importation of malaria infections is zero, system 3 admits a unique ee when r0 > 1. thissuggests that even in the absence of importation of malaria infections from elsewhere, malariaepidemics can continue to propagate in the population. the occurrence of two endemic equilibriawhen r0 does not exceed one, case (iii) of theorem 6 shows that the model bifurcate backwardly. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 16that is, two stable equilibrium states coexist when the reproductive number of the model is lessthan one. in this case, r0 < 1 even though necessary is no more enough for the elimination ofmalaria. to obtain the value of r0 say, rc0 at which backward bifurcation takes place in this case,we set the discriminant (∆) of equation 46 to zero and solve for rc0 . that is ∆ = 0 =⇒ a21 − 4a0a2 = 0 =⇒ rc0 = √ 1− a21 4a2g0g1g2g4µ3m(m + πh)2 hence, for values of r0 between rc0 < r0 < 1, system 3 in the absence of importation of malariainfections experiences backward bifurcation. 2.7. global stability of the malaria endemic equilibrium point. in what follows, we explore thelong term behavior of the unique endemic equilibrium point whenever it exist. consider the lyapunov candidate: l ( s∗∗h , e ∗∗ h , i ∗∗ h , r ∗∗ h , a ∗∗ m , s ∗∗ m , e ∗∗ m , i ∗∗ m ) = ( (sh − s∗∗h )− s∗∗h ln sh s∗∗h ) + ( (eh − e∗∗h )− e∗∗h ln eh e∗∗h ) + ( (ih − i∗∗h )− i∗∗h ln ih i∗∗h ) + ( (rh − r∗∗h )− r∗∗h ln rh r∗∗h ) + ( (am − a∗∗m )− a∗∗m ln am a∗∗m ) + ( (sm − s∗∗m )− s∗∗m ln sm s∗∗m ) + ( (em − e∗∗m )− e∗∗m ln em e∗∗m ) + ( (im − i∗∗m )− i∗∗m ln im i∗∗m ) taking the time derivative of l gives: dl dt = ( 1− s∗∗h sh ) dsh dt + ( 1− e∗∗h eh ) deh dt + ( 1− i∗∗h ih ) dih dt + ( 1− r∗∗h rh ) drh dt + ( 1− a∗∗m am ) dam dt + ( 1− s∗∗m sm ) dsm dt + ( 1− e∗∗m em ) dem dt + ( 1− i∗∗m im ) dim dt = ( sh − s∗∗h sh ) [πh + (1− p1 − p2)m + ϕrh − (λh + µh)sh] + ( eh − e∗∗h eh ) (p1m + λhsh − g0eh) + ( ih − i∗∗h ih ) (p2m + γeh − g1ih) + ( rh − r∗∗h rh ) (τih + ωeh − g2rh) + ( am − a∗∗m am ) [πm ( 1− am k ) nam − g3am] + ( sm − s∗∗m sm ) [ψam − (λm + µm)sm] + ( em − e∗∗m em ) (λmsm − g4em) + ( im − i∗∗m im ) (σem − µmim) = πh + (1− p1 − p2)m + ϕrh + (λh + µh)s∗∗h − (πh + (1− p1 − p2)m + ϕrh) s∗∗h sh (51) https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 17 − (λh + µh)sh + p1m + λhsh + g0e ∗∗ h − g0eh − (p1m + λhsh) e∗∗h eh + p2m + γeh + g1i ∗∗ h − g1ih − (p2m + γeh) i∗∗h ih + τih + ωeh + g2r ∗∗ h − g2rh − (τih + ωeh) r∗∗h rh + namπm + namπm a∗∗m k + g3a ∗∗ m − namπm am k − g3am − namπm a∗∗m am + ψam + λmsm + g4e ∗∗ m − g4em − λmsm e∗∗m em s∗∗m sm + σem + µmi ∗∗ m − µmim − σem i∗∗m im = l+ − l−where l+ = πh +m + ϕrh + (λh + µh)s∗∗h + (p1 + p2)m s∗∗h sh + λhsh + g0e ∗∗ h + γeh + g1i ∗∗ h + τih + ωeh + g2r ∗∗ h + namπm + namπm a∗∗m k + g3a ∗∗ m + ψam + (λm + µm)s∗∗m + λmsm + g4e ∗∗ m + σem + µmi ∗∗ m l− = (πh +m + ϕrh) s∗∗h sh + (λh + µh)sh + g0eh + (p1m + λhsh) e∗∗h eh + (p2m + γeh) i∗∗h ih + g1ih + g2rh + (τih + ωeh) r∗∗h rh + namπm am k + g3am + namπm a∗∗m am + λmsm e∗∗m em + g4em + µmim + σem i∗∗m im (52) since the model parameters and state variables are non-negative, it follows from (52) that dl dt ≤ 0 if l+ ≤ l−and dl dt = 0 if and only if s∗∗h = sh, e ∗∗ h = eh, i ∗∗ h = ih, r ∗∗ h = rh, a ∗∗ m = am, s ∗∗ m = sm, e ∗∗ m = em, and i∗∗m = im therefore, the largest compact invariant set within the model’s invariant region is the singleton {s∗∗h , e∗∗h , i∗∗h , r∗∗h , a∗∗m , s∗∗m , e∗∗m , i∗∗m }. hence, by the lasalle’s invariant principle [28], theunique endemic equilibrium of system (3) is globally asymptotically stable whenever it exists. 3. local sensitivity analysis in this section, local sensitivity analysis is carried out to determine the parameters that mostlycontribute to disease spread or increase (r0). these parameters should be targeted during anyintervention aimed at combating the malaria infections. using the normalized forward sensitivityindex relation: γw x = ∂w ∂x × x w (53) and the model parameter values provided in table 1 we compute the values for sensitivity indicesof the parameters of the model reproductive number, (r0) as presented in table 3. https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 18 table 3. the values of the sensitivity indices parameter sensitivity indexb +1.00 βh +0.50 βm +0.50 πh −0.4839m −0.0161 µh +0.4991 ω −0.2618 γ +0.2620 τ −0.4848 δ −0.0291 πm +1.98 ×10−4 k +0.50 ψ +0.4999 σ +0.2775 µa −1.12 ×10−4 µm −1.2779 if the sign of the sensitivity index of a given parameter of r0 is positive, it means r0 is directlyproportional to that parameter. that is, an increase (decrease) in the parameter value when otherparameters remain constant would result in an increase (decrease) in disease incidence. conversely,if the sign of the sensitivity index of a given parameter is negative, then r0 is indirectly proportionalto that parameter [29]. from table 3, it is clear that an increase in the parameters: b, βh, βm, k,ψ. γ, πm , and σ will lead to an increase in the disease spread (r0) while an increase in theparameters: µm, τ , ω and µa will result in a reduction of the disease spread r0 and vice versa. 4. numerical simulations in order to explore the possible impact of the exposed and infected human immigrants on thedynamical behaviour of the malaria model sub-populations, system (3) is simulated using the fol-lowing assumed set of initial condition values of the state variables: {sh(0), eh(0), ih(0), rh(0) am(0), sm(0), em(0), im(0)} = {700, 350, 100, 0, 5000, 1000, 300, 120}}and the parameter values provided in table 1. the results (figures 10-13) suggest that the exposed andinfected human immigrants have no influence on the population density of the humans and mosquitoes inthe community. it can also be observed from the simulation results that the population of the immature andsusceptible anopheles mosquitoes remain high in the community. this implies that efficient vector control https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 19 strategies for both the immature and mature anopheles mosquitoes are urgently required if malaria is to beeradicated from the population. figure 2. susceptiblehumans figure 3. exposed hu-mans figure 4. infected hu-mans figure 5. recoveredhumans https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 20 figure 6. aquaticmosquito figure 7. susceptiblemosquito figure 8. exposedmosquitoes figure 9. infectedmosquitoes https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 21 figure 10. hu-man classes with p1 = p2 = 0.2 figure 11. hu-man classes with p1 = p2 = 0 figure 12. mosquitoclasses with p1 = p2 = 0.2 figure 13. mosquitoclasses with p1 = p2 = 0 https://doi.org/10.28924/ada/ma.4.7 eur. j. math. anal. 10.28924/ada/ma.4.7 22 5. conclusion in this study, a deterministic compartmental model for malaria dynamics that takes into consideration theinflow of exposed and infected migrants and the recovery of exposed humans is formulated and analysed.in the absence of inflow of exposed and infected humans from elsewhere, the model disease free states areobtained and the biologically desired infection-free equilibrium point (rdfe) is shown to be both locally andglobally asymptotically stable when the disease reproduction number (r0) is less than one and unstable if r0 > 1. furthermore, we derived the equation for the endemic condition and used the descartes rule of signchange to establish the conditions for the model to admit one or three endemic equilibrium state(s). for aspecial case of no inflow of exposed or infected migrants, we proved that the model admits a global asymptoticstable unique endemic equilibrium if r0 > 1 and two endemic equilibria when r0 < 1. the results from ourlocal sensitivity analysis revealed that adult mosquito removal and biting rates (µm and b) are respectivelythe most sensitive parameters to the spread of malaria. this suggests that malaria vector control remainsa key factor for consideration in the elimination of malaria epidemics. our numerical simulation graphicalresults indicate that the inflow of exposed and infected migrants has no significant impact on the dynamicalbehavior of the model population sub-classes. thus, we recommend that real immigrants data is used to fitthe malaria model and explore more on the disease dynamics in the presence of exposed or infected humanimmigrants. references [1] v. yiga, h. nampala, j. tumwiine, analysis of the model on the effect of seasonal factors on malaria transmissiondynamics, j. appl. math. 2020 (2020) 1–19.[2] o. s. maliki, n. romanus, b. o. onyemegbulem, a mathematical modelling of the effect of treatment in the controlof malaria in a population with infected immigrants, appl. math. 9 (2018) 1238–1257.[3] w. h. organization, et al., who malaria policy advisory group (mpag) meeting report, 18–20 april 2023, worldhealth organization, 2023.[4] s. olaniyi, k. okosun, s. adesanya, r. lebelo, modelling malaria dynamics with partial immunity and protectedtravellers: optimal control and cost-effectiveness analysis, j. biol. dyn. 14 (2020) 90–115.[5] m. y. li, an introduction to mathematical modeling of infectious diseases, vol. 2, springer, 2018.[6] s. 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solution 2.2. invariant region 2.3. model equilibrium points 2.4. the basic reproductive number 2.5. stability of malaria-free equilibrium 2.6. malaria endemic equilibrium 2.7. global stability of the malaria endemic equilibrium point 3. local sensitivity analysis 4. numerical simulations 5. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 5doi: 10.28924/ada/ma.4.5 a new study on generalized reverse derivations of semi-prime ring muhammad naeem abbas1,∗, mukhtar ahmad1,∗, abdul rauf khan2, ather qayyum3, siti suzlinsupadi3 1department of mathematics, khawaja fareed university of engineering and information technology rahim yar khan, pakistan itxmemuktar@gmail.com, naeemabbas995@gmail.com 2department of mathematics, ghazi university, d.g.khan, pakistan arkhan@gudgk.edu.pk 3institute of mathematical sciences, universiti malaya, malaysia dratherqayyum@um.edu.my, suzlin@um.edu.my ∗correspondence: naeemabbas995@gmail.com, itxmemuktar@gmail.com abstract. the aim of this paper is to extend the ideas from generalized reverse derivation to gener-alized (α, β)-reverse derivations on semi-prime ring. we prove that, if 0 6= d be reverse derivationin r and a generalized (α, β)-reverse derivation g, then g is β-strong commutative preserved. nextwe can prove that r is commutative. 1. introduction the study of centralizing mapping of semi-prime rings given by bell and martindale [3]. belland martindale [3] proved that [d(u1), u1]α,β = 0 ∀ u1 ∈ b, where 0 6= d a derivation of r and r is semi-prime ring, then commutativity holds in r. bell and daif were studied the commu-tativity in prime and semi-prime rings that bind endomorphism or a derivation that preserves a β-strong commutativity on a non-zero ideal right in [2]. further, ali and shah [1] extend some con-sequences for generalized derivation of bell and martindale [3]. bresar established that, if b 6= 0is left ideal in r a prime ring, and two mappings d1 and d2 are (α, β)-derivations in r satisfies (d1α(a)− β(a)d2) ∈ z(r), for each a ∈ b, so commutativity holds in r [5]. some properties arestudied by vukman in [12] and [4]. m. samman and n. al yamani [8] studied reverse derivation onsemi prime rings. they proved that the mapping d : r → r is central derivation iff it is reversederivation and also that d 6= 0 a reverse derivation in semi-prime ring r, then the commutativityexists in r resently mukhtar ahmad et.al[9]. later, the idea of revers derivation and some properties received: 19 nov 2023. key words and phrases. semi-prime ring; ideal; r-generalized reverse derivation; derivation; reverse derivation;r-generalized derivation and reverse derivation. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 2of reverse derivation were studied by bresar and vukman [4]. the aim of this paper is extention thenotion of generalized reverse derivation to generalized (α, β)-reverse derivation resently mukhtarahmad et.al[10]. a mapping g : r → r which associate with (α, β)-reverse derivation d is saidto be a generalized (α, β)-reverse derivation if, g(u1v1) = g(v1)α(u1) + β(v1)d(u1)resently r.m. kashif et.al[11]. 2. preliminaries throughout this paper, definition 2.1. let r is ring and it is considered as a semi-prime ring iff for any u1; u1 6= 0such that u1ru1 = 0 implies u1 = 0. definition 2.2. the additive mapping d1 : r → r is known as (α, β)-derivation, if d1(u1v1) = d1(u1)α(v1) + β(u1)d1(v1) hold ∀ u1, v1 ∈ r, where α and β are automorphism. definition 2.3. the mapping d1 : r → r is called a (α, β)-reverse derivation if d1(u1v1) = d1(v1)α(u1) + β(v1)d1(u1) holds ∀ u1, v1 ∈ r, where α and β are automorphism. definition 2.4. an additive mapping h : r → r be a right (left) generalized (α, β)-reversederivation if there is a derivation d from r to r such thath(u1v1) = h(v1)α(u1)+β(v1)d(u1) (h(u1v1) = d(v1)α(u1) + β(v1)h(u1) for all u1, v1 ∈ r. h be a generalized reverse (α, β) of r associatedwith (α, β) derivation. definition 2.5. some identities holds for every u1, v1, w1 ∈ r [u1, v1w1] = v1[u1, w1]+[u1, v1]w1 [u1v1, w1] = [u1, w1]v1 + u1[v1, w1] [u1v1, w1]α,β = u1[v1, w1]α,β +[u1, β(w1)]v1 = u1[v1, α(w1)] + [u1, w1]α,β v1 [u1, v1w1]α,β = β(v1)[u1, w1]α,β +[u1, v1]α,β α(w1) definition 2.6. the derivation h would be commuting, if 0 = [v1, h(u1)], ∀ u1, v1 ∈ r. definition 2.7. the strong commutativity preserving is defined as [g(u1), g(v1)] = [u1, v1] forall u1, v1 ∈ r, where g : r→ r is a mapping on r. lemma 2.8. let u1 6= 0 in z(center of ring), if u1, v1 ∈ z , then v1 ∈ z. lemma 2.9. let g : r → r be an additive map and on a left ideal b of r, g is centralizing,then g(u1) ∈ r ∀ u1 ∈ b ∪ z. lemma 2.10. let 0 6= b be an ideal of a semi-prime ring r. if the set [b,b] centralizes z in r, then b centralizes z. 2.1. point-wise operation. theorem 2.11. suppose 0 6= d from r to r a derivation in a semi-prime ring r. let generalized (α, β)-reverse derivation g on a left ideal b 6= 0 of r. then g satisfies [g(w1), g(v1)] = β([w1, v1]) for all v1, w1 ∈ b (that is, g is β-strong commutativitypreserved), when g is a homomorphism on b. proof. since g is generalized (α, β)-reverse derivation and homomorphism on b, such that g(u1v1) = g(u1)g(v1)∀ u1, v1 ∈ b. https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 3this implies g(u1v1) = g(u1)g(v1) = g(v1)α(u1) + β(v1)d(u1), f or al l u1, v1 ∈ b. (1) we replace v1 by v1w1 where w1 ∈ b, in equation (1), we obtain g(u1)g(v1w1) = g(v1w1)α(u1) + β(v1w1)d(u1)this gives g(u1)g(v1w1) = g(u1v1w1) = g(v1)g(w1)α(u1) + β(v1w1)d(u1), f or al l u1, v1 ∈ b. (2) as g is homomorphism, so we get g(u1)g(v1w1) = g(u1)g(v1)g(w1) = g(u1v1)g(w1)this equalized to g(u1v1)g(w1) = (g(v1)α(u1) + β(v1)d(u1))g(w1)this relates to g(u1v1)g(w1) = g(v1)α(u1)g(w1) + β(v1)d(u1)g(w1)by the equation (2), we get g(u1v1)g(w1) = g(v1)g(w1)α(u1) + β(v1)d(u1)g(w1), f or al l u1, v1 ∈ b. (3) from equation (2) and equation (3), we obtain β(v1)d(u1)g(w1) = β(v1)d(u1)β(w1)this implies β(v1)d(u1)(g(w1)− β(w1)) = 0, f or al l u1, v1 ∈ b. (4) put w1 = [w1, v1] in equation (4), we have β(v1)d(u1)(g([w1, v1])− β([w1, v1])) = 0,we arrives to d(u1)β(v1)(g([w1, v1])− β([w1, v1])) = 0.by replacing β(v1) by (g([w1, v1])− β([w1, v1]))α(r)d(u1), we obtain d(u1)(g([w1, v1])− β([w1, v1]))α(r)d(u1)(g([w1, v1])− β([w1, v1])) = 0,it gives d(u1)(g([w1, v1])− β([w1, v1]))rd(u1)(g([w1, v1])− β([w1, v1])) = 0.as r semi-prime, so we obtain d(u1)(g([w1, v1])− β([w1, v1])) = 0,since d 6= 0, we have g([w1, v1])− β([w1, v1]) = 0,we get g([w1, v1]) = β([w1, v1]) https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 4as g is homomorphism, so we have [g(w1), g(v1)] = β([w1, v1])so g is β-strong commutative preserved on b. theorem 2.12. let g on a left ideal b 6= 0 of r, is generalized (α, β)-reverse derivation. if gis homomorphism on b, then on b, g is commuting. proof. by theorem 2.11, g is β-strong commutative preserved, then ∀ u1, v1 ∈ b, we get β([u1, v1]) = [g(u1), g(v1)] (5) replace v1 = v1u1 in equation (5) we have β([u1, v1u1]) = [g(u1), g(v1u1)] β([u1, v1])β(u1) = [g(u1), g(v1)]g(u1)by equation (5), we get β([u1, v1])β(u1) = β([u1, v1])g(u1)this implies β([u1, v1])(g(u1)− β(u1)) = 0. (6) now put v1 = u1v1 in equation (5) we have β([u1, u1v1]) = [g(u1), g(u1v1)] β(u1)β([u1, v1]) = g(u1)[g(u1), g(v1)]by equation (5), we have β(u1)β([u1, v1]) = g(u1)β([u1, v1])this implies (g(u1)− β(u1))β([u1, v1]) = 0. (7) put v1 = r1v1 in equation (6), we have β([u1, r1v1])(g(u1)− β(u1)) = 0,this implies β([u1, r1])β(v1)(g(u1)− β(u1)) = 0,we get β([u1, r1])b(g(u1)− β(u1)) = 0,also that β([u1, r1])rb(g(u1)− β(u1)) = 0,by semi-primeness of r, there exist a family w = {pθ/θ ∈ ∧ } of prime ideals such that ⋂ pθ = 0.if w has a member p and u1 ∈ b, then last relation, we get, b(g(u1) − β(u1)) not in p or [β(u1), r] ⊆ p . if ∃ v1 ∈ b such that [β(u1), r] not in p . it implies b(g(v1) − β(v1)) ⊆ p . let w1 ∈ b is arbitrary such that [β(v1+w1), r] ⊆ p . this means that [β(w1), r] not in p and hence (g(w1)− β(w1)) ⊆ p . in other ways [β(v1 +w1), r] ⊆ p , then b(g(v1 +w1)− β(v1 +w1)) ⊆ p . https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 5it gives b(g(w1)− β(w1)) ⊆ p .we obtain b(g(w1)−β(w1)) ⊆ p for every w1 ∈ b and hence [b,b](g(w1)−β(w1)) ⊆ p ∀ w1 ∈ b.as p is arbitrary and ⋂ pθ = 0, this implies [b,b](g(w1)− β(w1)) = 0 for all w1 ∈ b. similarly,we can show that (g(w1)− β(w1))[b,b] = 0 for all w1 ∈ b. this implies that (g(w1)− β(w1)) ∈ cr[b,b], for all w1 ∈ b. by lemma 2.10 and [6], we have (g(u1), β(u1)) ∈ cr(b), ∀ w1 ∈ b.thus we have [g(u1) − u1, β(u1)] = 0 ∀ u1 ∈ b. this implies that [g(u1), β(u1)] = 0 ∀ u1 ∈ b.this shows that g is commuting on b. theorem 2.13. suppose a derivation, d : r → r where 0 6= d , in r and a generalized (α, β)-reverse derivation g on left ideal b 6= 0. if g is a homomorphism on b, then commutativity existsin r. proof. by our hypothesis [g(u1), u1]α,β = 0, f or al l u1 ∈ b. (8) we replace u1 by u1 + v1, in equation (2.1),we get [g(u1 + v1), u1 + v1]α,β = 0,we have [g(u1) + g(v1), u1 + v1]α,β = 0,we arrives to [g(u1) + g(v1), u1]α,β +[g(u1) + g(v1), v1]α,β = 0,this gives [g(u1), u1]α,β +[g(v1), u1]α,β +[g(u1), v1]α,β +[g(v1), v1]α,β = 0.by equation (), we obtain [g(u1), v1]α,β +[g(v1), u1]α,β = 0, f or al l u1 ∈ b. (9) by substituting v1 = u1v1 in equation (9), we have [g(u1), u1v1]α,β +[g(u1v1), u1]α,β = 0,we have β(u1)[g(u1), v1]α,β +[g(u1), u1]α,β α(v1) + [g(v1)α(u1) + β(v1)d(u1), u1]α,β = 0.this implies us by the equation (2.1), β(u1)[g(u1), v1]α,β +[g(v1)α(u1), u1]α,β +[β(v1)d(u1), u1]α,β = 0.this gives us by [α(u1), α(u1)] = 0, β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) + [β(v1)d(u1), u1]α,β = 0,since g is commuting on b, we have [β(v1)d(u1), u1]α,β = 0, f or al l u1 ∈ b. (10) we replace v1 by r1v1 in equation (10), we have [β(r1v1)d(u1), u1]α,β = 0, https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 6we get β(r1)[β(v1)d(u1), u1]α,β +[β(r1), β(u1)]β(v1)d(u1) = 0.by equation (10), we have [β(r1), β(u1)]β(v1)d(u1) = 0,this gives [β(r1), β(u1)]bd(u1) = 0, for all u1 ∈ b and r1 ∈ r,by the semi-primeness of r, ∃ a set ω = {pα/α ∈ ∧ } of prime ideals and ⋂ pα = (0).if p ∈ ω and u1 ∈ b, then by equation (10), [r, β(u1)] ⊆ p or p ⊇ d(u1). since 0 6= d on r, soby [7], 0 6= d on b. consider d(u1)p , where u1 ∈ b, then p ⊇ [r, β(u1)]. suppose w1 ∈ b, wesee that w1 not in z, then d(w1) ⊆ p and u1 + w1 not in z. this gives that d(u1 + w1) ⊆ p andthen d(u1) ⊆ p , which contradicts to our consideration that d(u1)p . so, this gives us w1 ∈ z, ∀ w1 ∈ b.this implies that b is commutative also that by the [7], then commutativity holds in r. theorem 2.14. suppose a semi-prime ring r and a left ideal b of r, s.t. b⋂ z 6= 0 for center z of r. let a generalized (α, β)-reverse derivation g on r and d 6= 0 a derivation and g iscentralizing on b. then commutativity holds in r. proof. if z 6= 0 and g is commutation on b, so our proof is complete.as g is centralizing b and by theorem 2.12, we get [g(u1), u1]α,β ∈ z, ∀ u1 ∈ b. (11) put u1 = (u1 + v1) in equation (11), then [g(u1 + v1), u1 + v1]α,β ∈ z, for all u1 ∈ b,this relates to [g(u1), u1 + v1]α,β +[g(v1), u1 + v1]α,β ∈ z, ∀ u1 ∈ b.it implies β(u1)[g(u1), u1]α,β +β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) + [g(v1), v1]α,β α(u1) ∈ z, ∀ u1 ∈ b.by the equation (11), we get β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) ∈ z, f or al l u1, v1 ∈ b. (12) replace u1 by v1w1 in equation (12), we obtain β(u1)[g(v1w1), v1]α,β +[g(v1), v1w1]α,β α(u1) ∈ z,we get β(u1)[g(w1)α(v1) + β(w1)d(v1), v1]α,β +β(v1)[g(v1), w1]α,β α(u1) + [g(v1), v1]α(w1)α(u1) ∈ z, this implies β(u1)[g(w1)α(v1), v1]α,β +β(u1)[β(w1)d(v1), v1]α,β +β(v1)[g(v1), w1]α,β ∈ z.this equalized to https://doi.org/10.28924/ada/ma.4.5 eur. j. math. anal. 10.28924/ada/ma.4.5 7 β(u1)[g(w1), v1]α(v1) + β(u1)g(w1)[α(v1), α(v1)]α,β +β(u1)[β(w1), β(v1)]d(v1) + β(u1)β(w1)[d(v1), v1]α,β +β(v1)[g(v1), w1]α,β α(u1) ∈ z.as we know for any u1, v1, w1 ∈ z can commute with each one of r, by equation (12) and [α(w1), α(v1)] = 0, we get β(u1)β(w1)[d(v1), v1]α,β ∈ z.since β(w1) 6= 0, this implies by lemma 2.8, we get [d(v1), v1]α,β ∈ z, for every v1 ∈ b.then we have d is centralizing on b, hence by the reference [3], r is commutative. this completesour proof. references [1] a. ali, t. shah, centralizing and commuting generalized derivations on prime rings, mat. vesnik, 60 (2008), 1-2.[2] h.e. bell, m.n. daig, on commutativity and strong commutativity preserving maps, can. math. bull. 37 (1994),443-447.[3] h.e. bell, w.s. martindale, iii, centralizing mapping of semi-prime rings, can. math. soc. 30 (1987), 92-101.[4] m. bresar, j. vukman, on some additive mappings in rings with involution, aequat. math. 38 (1989), 178-185.[5] m. bresar, centralizing mappings and derivations in prime rings, j. algebra, 156 (1993), 385-394.[6] m.n. daig, h.e. bell, remarks on derivations on semi-prime rings, int. j. math. math. sci. 15 (1992), 205-206.[7] j.h. mayne, centralizing mappings of prime rings, can. math. bull. 27 (1984), 122-126.[8] z.h. niazi, m.a.t. bhatti, m. aslam, y. qayyum, m. ibrahim, a. qayyum, d-lucky labelling of some special graphs,amer. j. math. anal. 10 (2022), 3-11.[9] m. ahmad, s. hussain, i. zahid, u. parveen, m. sultan, a. qayyum, on degree based topological indices of petersensubdivision graph, eur. j. math. anal. 3 (2023), 20[10] m. ahmad, m.j. hussain, g. atta, s. raza, i. waheed, a. qayyum, topological evaluation of four para-line graphsabsolute pentacene graphs using topological indices, int. j. anal. appl. 21 (2023), 66.[11] r.m.k. iqbal, m. ahmad, a. qayyum, s.s. supadi, m.j. hussain, s. raza, on degree-based topological indices oftoeplitz graphs, int. j. anal. appl. 21 (2023), 111.[12] a. asghar, a. qayyum, n. muhammad, different types of topological structures by graphs, eur. j. math. anal. 3(2022), 3. https://doi.org/10.28924/ada/ma.4.5 1. introduction 2. preliminaries 2.1. point-wise operation references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 10doi: 10.28924/ada/ma.3.10 complex oscillation of solutions and their arbitrary-order derivatives of linear differential equations with analytic coefficients of [p, q]-order in the unit disc benharrat belaïdi∗, meriem belmiloud department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria benharrat.belaidi@univ-mosta.dz, meriem.belmiloud27@gmail.com ∗correspondence: benharrat.belaidi@univ-mosta.dz abstract. throughout this article, we investigate the growth and fixed points of solutions of complexhigher order linear differential equations in which the coefficients are analytic functions of [p, q]−orderin the unit disc. this work improves some results of belaïdi [3–5], which is a generalization of recentresults from chen et al. [9]. 1. introduction and main results consider for k ≥ 2 the following complex linear differential equations f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = 0, (1.1) ak (z) f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = 0, (1.2)where ai 6≡ 0 (i = 0, 1, ..., k) are analytic functions in the unit disc d = {z ∈ c : |z | < 1}. it iswell-known that the solutions of (1.1) are analytic in d too and that there are exactly k linearlyindependent solutions of (1.1), see [13]. bernal [6] was the first to use the concept of iteratedorder to study the growth of fast growing solutions of equation (1.1) . after that, the iterated orderof solutions of higher order equations was investigated by cao in [8], he extended the results ofchen and yang [10], belaïdi [2] on c. in addition, cao [8] obtained some results concerning thefixed points of homogeneous linear differential equations (1.1) and (1.2) . in [15,16], juneja and hisco-authors have investigated some properties of entire functions of [p, q]-order, and obtained someresults of their growth. in [20], by using the concept of [p, q]-order liu, tu and shi have consideredthe equation (1.1) with entire coefficients and obtained different results concerning the growth of itssolutions in the complex plane. in [3], the [p, q]−order was introduced in the unit disc d, and manyresults on [p, q]−order of solutions of (1.1) have been found by different researchers [3–5,14,18,22]in d. recently, chen et al. in [9] gave some results about the growth and fixed points of solutions received: 30 sep 2022. key words and phrases. linear differential equations; analytic function; [p, q]−order; fixed points.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 2of higher-order linear differential equations in the unit disc, they studied and estimated the fixedpoints of solutions of (1.1) and (1.2), and also extended the coefficient conditions to a type ofone-constant-control coefficient comparison and obtained the same estimates of iterated order ofsolutions. the aim of this paper is to contrast coefficients by producing better estimates of thegrowth of solutions by using the concept of [p, q]−order, and optimizing the coefficients’s conditionswith less control constants of the coefficients’s modulus or characteristic functions and we will obtainresults which improve and generalize those of chen et al., belaïdi, cao, tu and xuan. throughout this paper, we shall assume that the reader is familiar with the fundamental resultsand the standard notations of the nevanlinna’s theory in the unit disc d = {z ∈ c : |z | < 1}(see, [12, 13,17,21]). now, we give the definitions of iterated order and growth index to classify generally thefunctions of fast growth in d as those in c (see, [6]). let us define inductively, for r ∈ r, exp1 r := erand expp+1 r := exp ( expp r ) , p ∈ n. we also define for all r sufficiently large in (0,+∞) , log1 r := log r and logp+1 r := log ( logp r ) , p ∈ n. moreover, we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r. definition 1.1 (see [7]) let f be a meromorphic function in d. then the iterated n−order of f is defined by σn (f ) = lim sup r→1− log+n t (r, f ) log ( 1 1−r ) (n ≥ 1 is an integer) , where log+1 x = log+ x = max {log x, 0} , log+n+1 x = log+ ( log+n x ) . for n = 1, this notation is called order (σ1 (f ) = σ (f )) and for n = 2 hyper-order ([19]). if f is an analytic in d, then the iterated n−order of f is defined by σm,n (f ) = lim sup r→1− log+n+1m (r, f ) log ( 1 1−r ) (n ≥ 1 is an integer) . for n = 1, σm,1 (f ) = σm (f ) . now, we introduce the concept of [p, q]-order of meromorphic and analytic functions in theunit disc. definition 1.2 ([3]) let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p, q]-order of f is defined by σ[p,q] (f ) = lim sup r→1− log+p t (r, f ) logq ( 1 1−r ) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 3 for an analytic function f in d, we also define σm,[p,q] (f ) = lim sup r→1− log+p+1m (r, f ) logq ( 1 1−r ) . remark 1.1 it is easy to see that 0 ≤ σ[p,q] (f ) ≤ ∞ (0 ≤ σm,[p,q] (f ) ≤ ∞), for any p ≥ q ≥ 1. by definition 1.2, we have that σ[1,1] = σ (f ) (σm,[1,1] = σm (f )) and σ[2,1] = σ2 (f )( σm,[2,1] = σm,2 (f ) ). proposition 1.1 ([3]) let p ≥ q ≥ 1 be integers, and let f be an analytic function in d of [p, q]-order. the following two statements hold: (i) if p = q, then σ[p,q] (f ) ≤ σm,[p,q] (f ) ≤ σ[p,q] (f ) + 1. (ii) if p > q, then σ[p,q] (f ) = σm,[p,q] (f ) . definition 1.3 ([4]) let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p, q]-exponent of convergence of the sequence of zeros of f is defined by λ[p,q] (f ) = lim sup r→1− log+p n ( r, 1f ) logq ( 1 1−r ) , where n ( r, 1f ) is the integrated counting function of zeros of f in {z : |z | ≤ r}. similarly, the [p, q]-exponent of convergence of the sequence of distinct zeros of f is defined by λ[p,q] (f ) = lim sup r→1− log+p n ( r, 1f ) logq 1 1−r , where n ( r, 1f ) is the integrated counting function of distinct zeros of f in {z : |z | ≤ r}. definition 1.4 let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p, q]-exponent of convergence of the sequence of fixed points of f is defined by λ[p,q] (f − z) = lim sup r→1− log+p n ( r, 1f−z ) logq ( 1 1−r ) . similarly, the [p, q]-exponent of convergence of the sequence of distinct fixed points of f is defined by λ̄[p,q] (f − z) = lim sup r→1− log+p n̄ ( r, 1f−z ) logq ( 1 1−r ) . recall that for a measurable set e ⊂ [0, 1) , the upper and lower densities of e are defined by densde = lim sup r→1− m (e ∩ [0, r)) m ([0, r)) and densde = lim inf r→1− m (e ∩ [0, r)) m ([0, r)) , respectively, where m (f ) = ∫ f dt 1−t for f ⊂ [0, 1). it is clear that 0 ≤ densde ≤ densde ≤ 1for any measurable set e ⊂ [0, 1) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 4 proposition 1.2 if a set e satisfies densde > 0, then m (e) = ∫ e dt 1−t = +∞. proof. suppose that m (e) = ∫ e dt 1−t = δ <∞. we have m ([0, r)) = − log (1− r) . since m (e ∩ [0, r)) ≤ m (e) , then densde = lim sup r→1− m (e ∩ [0, r)) m ([0, r)) ≤ lim sup r→1− δ − log (1− r) = 0. so densde = 0. hence densde > 0 =⇒ m (e) = ∫ e dt 1− t = +∞. in 2012, belaïdi in [4] and [5] treated the growth of solutions of homogeneous linear differentialequations in which the coefficients are analytic functions of [p, q]−order in d. as for the equation (1.1), he got the following results. theorem a (see [4]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0 (z) , ..., ak−1 (z) be analytic functions in the unit disc d such that for real constants α, β, where 0 ≤ β < α, we have |a0 (z)| ≥ expp+1 { α logq ( 1 1− |z | )} and |ai (z)| ≤ expp+1 { β logq ( 1 1− |z | )} (i = 1, ..., k − 1) as |z | → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ α. theorem b (see [5]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0 (z) , ..., ak−1 (z) be analytic functions in the unit disc d such that for real constants α, β, where 0 ≤ β < α, we have t (r, a0) ≥ expp { α logq ( 1 1− |z | )} and t (r, ai) ≤ expp { β logq ( 1 1− |z | )} (i = 1, ..., k − 1) as |z | = r → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ α. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 5after that in 2021, chen et al. [9] investigated the growth of solutions of equations (1.1) and (1.2) in d by using the iterated order, and they got the following results. theorem c (see [9]) let n ≥ 1 be an integer. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, a1, ..., ak−1 be analytic functions in the unit disc d such that max {σm,n (ai) : i = 1, 2, ..., k − 1} ≤ σm,n (a0) = µ (0 < µ <∞) , and for a constant α ≥ 0, we have lim inf |z |→1−,z∈h ( (1− |z |)µ logn |a0 (z)| ) > α and |ai (z)| ≤ expn { α ( 1 1− |z | )µ} , (i = 1, 2, ..., k − 1) as |z | → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σn (f ) = σm,n (f ) = ∞ and σn+1 (f ) = σm,n (a0) = µ. theorem d (see [9]) let n ≥ 1 be an integer. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, a1, ..., ak be analytic functions in the unit disc d, and for some constants α ≥ 0 and µ > 0, we have lim inf |z |→1−,z∈h ( (1− |z |)µ logn−1 t (r, a0) ) > α and t (r, ai) ≤ expn−1 { α ( 1 1− |z | )µ} , (i = 1, 2, ..., k) as |z | = r → 1− for z ∈ h. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σn (f ) =∞ and σn+1 (f ) ≥ µ. theorem e (see [9]) assume that the assumptions of theorem c hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄n ( f (j) − z ) = λ̄n (f − z) = σn (f ) =∞, λ̄n+1 ( f (j) − z ) = λ̄n+1 (f − z) = σn+1 (f ) = µ, (j = 1, 2, ...) .in this paper, we improve and generalize the recent results of chen et al. [9] by using theconcept of [p, q]−order instead of the iterated order with less control constant. at the same time,our work improve some results of belaïdi in [4] and [5]. to be specific, we will decrease the controlconstants of the coefficients’ modulus or characteristic functions and obtain the same results ofbelaïdi, tu and xuan. here, we study the problem and get the following results. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 6 theorem 1.1 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for a constant α ≥ 0, we have lim inf |z |→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ > α (1.3) and |ai (z)| ≤ expp { α ( logq−1 ( 1 1−|z | ))µ} (i = 1, ..., k − 1) (1.4) as |z | → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. by theorem 1.1, we easily obtain the following corollary. corollary 1.1 ([22]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for some real constants α, β where 0 ≤ β < α, we have |a0 (z)| ≥ expp { α ( logq−1 ( 1 1− |z | ))µ} and |ai (z)| ≤ expp { β ( logq−1 ( 1 1− |z | ))µ} , i = 1, ..., k − 1 as |z | → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.2 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and lim sup |z |→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z | ))µ < lim inf |z |→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ (i = 1, ..., k − 1) (1.5) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 7 as |z | → 1− for z ∈ h.then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.3 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for a constant α ≥ 0, if p ≥ q ≥ 2 we have lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ > α (1.6) and t (r, ai) ≤ expp−1 { α ( logq−1 ( 1 1−|z | ))µ} , (i = 1, ..., k − 1) (1.7) as |z | = r → 1− for z ∈ h, then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, we have lim inf |z |→1−,z∈h t (r, a0)( 1 1−|z | )µ > (k − 1)α (1.8) and t (r, ai) ≤ α ( 1 1−|z | )µ , (i = 1, ..., k − 1) (1.9) as |z | = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) =∞ and σ2 (f ) = σm,2 (f ) = µ. by theorem 1.3, we easily obtain the following corollary. corollary 1.2 ([22]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for some real constants α, β, where 0 ≤ β < α, we have t (r, a0) ≥ expp−1 { α ( logq−1 ( 1 1− |z | ))µ} and t (r, ai) ≤ expp−1 { β ( logq−1 ( 1 1−|z | ))µ} (i = 1, ..., k − 1) as |z | = r → 1− for z ∈ h. then the following statements hold: https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 8 (i) if p = q = 1 and 0 ≤ (k − 1)β < α, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) =∞ and σ2 (f ) = σm,2 (f ) = µ. (ii) if p ≥ q ≥ 2 and 0 ≤ β < α, then every nontrivial solution f of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.4 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai) : i = 1, 2, ..., k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and if p ≥ q ≥ 2, we have lim sup |z |→1−,z∈h logp−1 t (r, ai)( logq−1 ( 1 1−|z | ))µ < lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ , (i = 1, ..., k − 1) (1.10) as |z | = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, we have lim sup |z |→1−,z∈h (k − 1)t (r, ai)( 1 1−|z | )µ < lim inf |z |→1−,z∈h t (r, a0)( 1 1−|z | )µ , (i = 1, ..., k − 1) (1.11) as |z | = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) =∞ and σ2 (f ) = σm,2 (f ) = µ. theorem 1.5 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak be analytic functions in the unit disc d such that for some constants α ≥ 0 and µ > 0, we have (1.3) and |ai (z)| ≤ expp { α ( logq−1 ( 1 1−|z | ))µ} , (i = 1, ..., k) as |z | → 1− for z ∈ h. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) =∞ and σ[p+1,q] (f ) ≥ µ. theorem 1.6 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 9 and let a0, ..., ak be analytic functions in the unit disc d such that for a constant µ > 0, we have lim sup |z |→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z | ))µ < lim inf |z |→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ , (i = 1, ..., k) as |z | → 1− for z ∈ h.then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) =∞ and σ[p+1,q] (f ) ≥ µ. theorem 1.7 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0, ..., ak be analytic functions in the unit disc d such that for some constants α ≥ 0 and µ > 0, if p ≥ q ≥ 2 we have lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ > α (1.12) and t (r, ai) ≤ expp−1 { α ( logq−1 ( 1 1−|z | ))µ} , (i = 1, ..., k) (1.13) as |z | = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) =∞ and σ[p+1,q] (f ) ≥ µ. if p = q = 1, we have lim inf |z |→1−,z∈h t (r, a0)( 1 1−|z | )µ > kα (1.14) and t (r, ai) ≤ α ( 1 1−|z | )µ , (i = 1, ..., k) (1.15) as |z | = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ (f ) =∞ and σ2 (f ) ≥ µ. theorem 1.8 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z | : z ∈ h ⊆ d} > 0, and let a0 (z) , ..., ak (z) be analytic functions in the unit disc d such that for a constant µ > 0, if p ≥ q ≥ 2 we have lim sup |z |→1−,z∈h logp−1 t (r, ai)( logq−1 ( 1 1−|z | ))µ < lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ , (i = 1, ..., k) (1.16) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 10 as |z | = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) =∞ and σ[p+1,q] (f ) ≥ µ. if p = q = 1, we have lim sup |z |→1−,z∈h kt (|z | , ai)( 1 1−|z | )µ < lim inf |z |→1−,z∈h t (|z | , a0)( 1 1−|z | )µ , (i = 1, ..., k) (1.17) as |z | = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ(f ) =∞ and σ2 (f ) ≥ µ. remark 1.2 for equation (1.1) , we can easily conclude that theorems a-c are generalized totheorems 1.1-1.4. in the same paper, chen et al. [9] obtained some results of the fixed points of solutions andtheir arbitrary order derivatives of equations (1.1) and (1.2). here, we generalize these results,and we obtain our theorems as following. theorem 1.9 assume that the assumptions of theorem 1.1 or theorem 1.2 hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄[p,q] ( f (j) − z ) = λ[p,q] (f − z) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f (j) − z ) = λ̄[p+1,q] (f − z) = σ[p+1,q] (f ) = µ, (j = 1, 2, ...) . theorem 1.10 assume that the assumptions of theorem 1.3 or theorem 1.4 hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄[p,q] ( f (j) − z ) = λ[p,q] (f − z) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f (j) − z ) = λ̄[p+1,q] (f − z) = σ[p+1,q] (f ) = µ, (j = 1, 2, ...) . theorem 1.11 assume that the assumptions of one of theorem 1.5 to theorem 1.8 hold. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies λ̄[p,q] ( f (j) − z ) = λ[p,q] (f − z) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f (j) − z ) = λ̄[p+1,q] (f − z) = σ[p+1,q] (f ) ≥ µ, (j = 1, 2, ...) . 2. some lemmas in this section we give some lemmas which are used in the proofs of our theorems. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 11 lemma 2.1 ([11], theorem 3.1) let k and j be integers satisfying k > j ≥ 0, and let ε > 0 and d ∈ (0, 1). if f is a meromorphic function in d such that f (j) does not vanish identically, then for |z | /∈ e1 ∣∣∣∣∣ f (k) (z) f (j) (z) ∣∣∣∣∣ ≤ [( 1 1− |z | )2+ε max { log ( 1 1− |z | ) ;t (s (|z |) , f ) }]k−j , where e1 ⊂ [0, 1) is a set with ∫ e1 dr 1−r <∞ and s (|z |) = 1− d (1− |z |) . lemma 2.2 ([13]) let f be a meromorphic function in the unit disc d, and let k ≥ 1 be an integer. then m ( r, f (k) f ) = s (r, f ) , where s(r, f ) = o ( log+ t (r, f ) + log ( 1 1−r )), possibly outside a set e2 ⊂ [0, 1) with ∫ e2 dr 1−r <∞. lemma 2.3 ([1]) let g : (0, 1) → r and h : (0, 1) → r be monotone increasing functions such that g (r) ≤ h (r) holds outside of an exceptional set e3 ⊂ [0, 1) for which ∫ e3 dr 1−r < ∞. then there exists a constant d ∈ (0, 1) such that if s (r) = 1− d (1− r) , then g (r) ≤ h (s (r)) for all r ∈ [0, 1). lemma 2.4 ([3]) let p ≥ q ≥ 1 be integers. if a0 (z) , ..., ak−1 (z) are analytic functions of [p, q]−order in the unit disc d, then every solution f 6≡ 0 of (1.1) satisfies σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] ( aj ) : j = 0, 1, ..., k − 1 } . lemma 2.5 ([4, 18]) let p ≥ q ≥ 1 be integers. if f and g are non-constant meromorphic functions of [p, q]−order in d, then we have (i) σ[p,q] (f ) = σ[p,q] ( 1 f ) , σ[p,q] (af ) = σ[p,q] (f ) and σ[p,q] (f + a) = σ[p,q] (f ) (a ∈ c∗) , (ii) σ[p,q] (f ′) = σ[p,q] (f ) , (iii) σ[p,q] (f + g) ≤ max { σ[p,q] (f ) , σ[p,q] (g) } , (iv) σ[p,q] (f g) ≤ max { σ[p,q] (f ) , σ[p,q] (g) } , if σ[p,q] (f ) > σ[p,q] (g) , then we obtain σ[p,q] (f + g) = σ[p,q] (f g) = σ[p,q] (f ) . lemma 2.6 ([4]) let p ≥ q ≥ 1 be integers. let a0, ..., ak−1 and f 6≡ 0 be finite [p, q]−order analytic functions in the unit disc d. if f is a solution with σ[p,q] (f ) =∞ and σ[p+1,q] (f ) = σ <∞ of equation f (k) + ak−1 (z) f (k−1) + · · ·+ a1 (z) f ′ + a0 (z) f = f, (2.1) then λ̄[p,q] (f ) = λ[p,q] (f ) = σ[p,q] (f ) =∞, λ̄[p+1,q] (f ) = λ[p+1,q] (f ) = σ[p+1,q] (f ) = σ. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 12by using the same arguments of the proof of lemma 3.5 in the paper [14, p. 4], we obtain thefollowing lemma in the case when σ[p,q] (f ) = σ =∞. lemma 2.7 let p ≥ q ≥ 1 be integers. let aj (j = 0, ..., k − 1) , f 6≡ 0 be meromorphic functions in d, and let f be a solution of the differential equation (2.1) satisfying max { σ[p,q] ( aj ) (j = 0, ..., k − 1) , σ[p,q] (f ) } < σ[p,q] (f ) = σ ≤ ∞. then we have λ[p,q] (f ) = λ[p,q] (f ) = σ[p,q] (f ) and λ[p+1,q] (f ) = λ[p+1,q] (f ) = σ[p+1,q] (f ) . 3. proofs of theorems 1.1 to 1.8 proof of theorem 1.1. suppose that every solution f of equation (1.1) not being identically equalto 0. from the conditions of theorem 1.1, there exists a set h of complex numbers satisfying densdh1 > 0, where h1 = {r = |z | : z ∈ h ⊆ d} . then h1 is a set with ∫h1 dr 1−r = +∞, suchthat for z ∈ h we have (1.3) and (1.4) as |z | → 1−. by lemma 2.1, there exists a set e1 ⊂ [0, 1)with ∫e1 dr 1−r <∞ such that for |z | /∈ e1, we have for j = 1, ..., k∣∣∣∣∣ f (j) (z) f (z) ∣∣∣∣∣ ≤ [( 1 1− |z | )2+ε max { log ( 1 1− |z | ) , t (s (|z |) , f ) }]j , (3.1) where s (|z |) = 1− d (1− |z |) , d ∈ (0, 1). from (1.1) , we get |a0 (z)| ≤ ∣∣∣∣∣ f (k)f ∣∣∣∣∣+ |ak−1 (z)| ∣∣∣∣∣ f (k−1)f ∣∣∣∣∣+ · · ·+ |a1 (z)| ∣∣∣∣ f ′f ∣∣∣∣ . (3.2) by (1.3), we know that ∃γ ∈ r : lim inf |z |→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ > γ > α. obviously logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ > γ > α ≥ 0 (3.3) as |z | → 1− for z ∈ h. by (1.4) and (3.3) , we obtain |a0 (z)| > expp { γ ( logq−1 ( 1 1− |z | ))µ} > expp { α ( logq−1 ( 1 1− |z | ))µ} ≥ |ai (z)| (i = 1, 2, ..., k − 1) (3.4) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 13as |z | → 1− for z ∈ h. applying (3.1) and (3.4) into (3.2) , we have expp { γ ( logq−1 ( 1 1− |z | ))µ} ≤ |a0 (z)| ≤ k [( 1 1− |z | )2+ε max { log ( 1 1− |z | ) , t (s (|z |) , f ) }]k × expp { α ( logq−1 ( 1 1− |z | ))µ} holds for all z satisfying |z | ∈ h1\e1 as |z | → 1−. noting that γ > α, by the last inequality, weobtain exp ( (1− o (1)) expp−1 { γ ( logq−1 ( 1 1− |z | ))µ}) ≤ k ( 1 1− |z | )k(2+ε) t k (s (|z |) , f ) (3.5) for all z satisfying |z | ∈ h1\e1 as |z | → 1−. then, by (3.5) and combining with lemma 2.3, weget for all r = |z | ∈ h1 exp ( (1− o(1)) expp−1 { γ ( logq−1 ( 1 1− r ))µ}) ≤ k ( 1 1− s (r) )k(2+ε) t k (s1 (r) , f ) , (3.6) where s1 (r) = 1 − d2 (1− r) with d ∈ (0, 1). therefore, from (3.6) we obtain σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = lim sup s1(r)→1− log+p+1 t (s1 (r) , f ) logq ( 1 1−s1(r) ) ≥ µ. (3.7) by lemma 2.4, we get σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] (ai) : i = 0, 1, ..., k − 1 } = σm,[p,q] (a0) = µ. (3.8)therefore, by (3.7) and (3.8) , we obtain σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = σm,[p,q] (a0) = µ. proof of theorem 1.2. set α0 = lim inf |z |→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ , αi = lim sup |z |→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z | ))µ , (i = 1, 2, ..., k − 1) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 14by (1.5), there exist real numbers α, γ such that αi < α < γ < α0, i = 1, 2, ..., k − 1. it yields logp |ai (z)|( logq−1 ( 1 1−|z | ))µ < α < γ < logp |a0 (z)|( logq−1 ( 1 1−|z | ))µ as |z | → 1− for z ∈ h. hence, we have (3.4) as |z | → 1− for z ∈ h. then, by using the sameproof of theorem 1.1, we get σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ and by lemma 2.4 we obtain the conclusion of theorem 1.2. proof of theorem 1.3. suppose that every solution f of equation (1.1) not being identically equalto 0. by (1.1) , we can write − a0 (z) = f (k)(z) f (z) + ak−1 (z) f (k−1)(z) f (z) + · · ·+ a1 (z) f ′(z) f (z) . (3.9) from (3.9) , we obtain t (r, a0) = m(r, a0) ≤ k−1∑ i=1 m(r, ai) + k∑ i=1 m ( r, f (i) f ) +o(1) = k−1∑ i=1 t (r, ai) + k∑ i=1 m ( r, f (i) f ) +o(1). (3.10) if p ≥ q ≥ 2, then by (1.6), we know that ∃γ ∈ r : lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ > γ > α. obviously logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ > γ > α ≥ 0 (3.11) as |z | → 1− for z ∈ h. by (1.7) and (3.11) , we obtain t (r, a0) > expp−1 { γ ( logq−1 ( 1 1− |z | ))µ} > expp−1 { α ( logq−1 ( 1 1− |z | ))µ} ≥ t (r, ai) , (i = 1, 2, ..., k − 1) (3.12) as |z | → 1− for z ∈ h. by applying lemma 2.2 and substituting (3.12) into (3.10) , we get expp−1 { γ ( logq−1 ( 1 1− r ))µ} ≤ (k − 1) expp−1 { α ( logq−1 ( 1 1− r ))µ} +o ( log+ t (r, f ) + log ( 1 1− r )) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 15for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. noting that γ > α, by the last inequality,we have exp { (1− o (1)) expp−2 { γ ( logq−1 ( 1 1− r ))µ}} ≤ o ( log+ t (r, f ) + log ( 1 1− r )) (3.13) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. therefore, from (3.13) we obtain σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ. (3.14) by lemma 2.4, we get σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] (ai) : i = 0, 1, ..., k − 1 } = σm,[p,q] (a0) = µ. (3.15)therefore, by (3.14) and (3.15) , we obtain σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, then by (1.8), we know that ∃γ ∈ r : lim inf |z |→1−,z∈h t (r, a0)( 1 1−|z | )µ > γ > (k − 1)α. obviously t (r, a0)( 1 1−|z | )µ > γ > (k − 1)α ≥ 0 (3.16) as |z | → 1− for z ∈ h. by (1.9) and (3.16) , we obtain t (r, a0) > γ ( 1 1− |z | )µ > (k − 1)α ( 1 1− |z | )µ ≥ α ( 1 1− |z | )µ ≥ t (r, ai) , (i = 1, 2, ..., k − 1) (3.17) as |z | → 1− for z ∈ h. by applying lemma 2.2 and substituting (3.17) into (3.10) , we get γ ( 1 1− r )µ ≤ (k − 1)α ( 1 1− r )µ +o ( log+ t (r, f ) + log ( 1 1− r )) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. noting that γ > (k − 1)α, by the lastinequality, we have (γ − (k − 1)α) ( 1 1− r )µ ≤ o ( log+ t (r, f ) + log ( 1 1− r )) (3.18) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. therefore, from (3.18) we obtain σ (f ) = σm (f ) =∞ and σ2 (f ) = σm,2 (f ) ≥ µ. (3.19) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 16by lemma 2.4, we get σ2 (f ) = σm,2 (f ) ≤ max {σm (ai) : i = 0, 1, ..., k − 1} = σm (a0) = µ. (3.20) therefore, by (3.19) and (3.20) , we obtain σ (f ) = σm (f ) =∞ and σ2 (f ) = σm,2 (f ) = µ. proof of theorem 1.4. if p ≥ q ≥ 2, we set α0 = lim inf |z |→1−,z∈h logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ , αi = lim sup |z |→1−,z∈h logp−1 t (r, ai)( logq−1 ( 1 1−|z | ))µ , (i = 1, 2, ..., k − 1) . by (1.10), there exist real numbers α, γ such that αi < α < γ < α0, i = 1, 2, ..., k − 1. it yields logp−1 t (r, ai)( logq−1 ( 1 1−|z | ))µ < α < γ < logp−1 t (r, a0)( logq−1 ( 1 1−|z | ))µ as |z | → 1− for z ∈ h. hence, we have t (r, a0) > expp−1 { γ ( logq−1 ( 1 1− |z | ))µ} > expp−1 { α ( logq−1 ( 1 1− |z | ))µ} ≥ t (r, ai) , (i = 1, 2, ..., k − 1) as |z | → 1− for z ∈ h. then, by using the same proof of theorem 1.3, we get σ[p,q] (f ) = σm,[p,q] (f ) =∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ, and by lemma 2.4 we obtain the conclusion of theorem 1.4.if p = q = 1, we set α0 = lim inf |z |→1−,z∈h t (r, a0)( 1 1−|z | )µ , αi = lim sup |z |→1−,z∈h (k − 1)t (r, ai)( 1 1−|z | )µ , (i = 1, 2, ..., k − 1) . by (1.11), there exist real numbers α, γ such that αi < α < γ < α0, i = 1, 2, ..., k − 1. it yields (k − 1)t (r, ai)( 1 1−|z | )µ < α < γ < t (r, a0)( 1 1−|z | )µ (3.21) as |z | → 1− for z ∈ h. by (3.21) , we obtain t (r, a0) > γ ( 1 1− |z | )µ > α ( 1 1− |z | )µ ≥ (k − 1)t (r, ai) , (i = 1, 2, ..., k − 1) (3.22) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 17as |z | → 1− for z ∈ h. by applying lemma 2.2 and substituting (3.22) into (3.10) , we get γ ( 1 1− r )µ ≤ α ( 1 1− r )µ +o ( log+ t (r, f ) + log ( 1 1− r )) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. noting that γ > α, by the last inequality,we have (γ − α) ( 1 1− r )µ ≤ o ( log+ t (r, f ) + log ( 1 1− r )) (3.23) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. therefore, from (3.23) we obtain σ (f ) = σm (f ) =∞ and σ2 (f ) = σm,2 (f ) ≥ µ. (3.24) by lemma 2.4, we get σ2 (f ) = σm,2 (f ) ≤ max {σm (ai) : i = 0, 1, ..., k − 1} = σm (a0) = µ. (3.25) therefore, by (3.24) and (3.25) , we obtain σ (f ) = σm (f ) =∞ and σ2 (f ) = σm,2 (f ) = µ. proof of theorems 1.5 and 1.6. suppose that every meromorphic (or analytic) solution f ofequation (1.2) not being identically equal to 0. from (1.2) , we get a0 (z) ≤ |ak (z)| ∣∣∣∣∣ f (k)f ∣∣∣∣∣+ |ak−1 (z)| ∣∣∣∣∣ f (k−1)f ∣∣∣∣∣+ · · ·+ |a1 (z)| ∣∣∣∣ f ′f ∣∣∣∣ . (3.26) by using a similar proof as in theorem 1.1 or theorem 1.2, we obtain |a0 (z)| > expp { γ ( logq−1 ( 1 1− |z | ))µ} > expp { α ( logq−1 ( 1 1− |z | ))µ} ≥ |ai (z)| (i = 1, 2, ..., k) (3.27) for |z | ∈ h1\e1 as |z | → 1−. applying (3.1) and (3.27) into (3.26) , we get expp { γ ( logq−1 ( 1 1− |z | ))µ} ≤ |a0 (z)| ≤ k [( 1 1− |z | )2+ε max { log ( 1 1− |z | ) , t (s (|z |) , f ) }]k × expp { α ( logq−1 ( 1 1− |z | ))µ} for all z satisfying |z | ∈ h1\e1 as |z | → 1−. noting that γ > α, by the last inequality, we have exp ( (1− o (1)) expp−1 { γ ( logq−1 ( 1 1− |z | ))µ}) ≤ k ( 1 1− |z | )k(2+ε) t k (s (|z |) , f ) (3.28) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 18for all z satisfying |z | ∈ h1\e1 as |z | → 1−. then, by (3.28) and combining with lemma 2.3, weget for all r = |z | ∈ h1 exp ( (1− o(1)) expp−1 { γ ( logq−1 ( 1 1− r ))µ}) ≤ k ( 1 1− s (r) )k(2+ε) t k (s1 (r) , f ) , (3.29) where s1 (r) = 1− d2 (1− r) with d ∈ (0, 1). therefore, from (3.29) we obtain σ[p,q] (f ) =∞ and σ[p+1,q] (f ) = lim sup s1(r)→1− log+p+1 t (s1 (r) , f ) logq ( 1 1−s1(r) ) ≥ µ. proof of theorems 1.7 and 1.8. suppose that every meromorphic (or analytic) solution f ofequation (1.2) not being identically equal to 0. by (1.2) , we can write − a0 (z) = ak (z) f (k)(z) f (z) + ak−1 (z) f (k−1)(z) f (z) + · · ·+ a1 (z) f ′(z) f (z) . (3.30) from (3.30) , we have t (r, a0) = m(r, a0) ≤ k∑ i=1 m(r, ai) + k∑ i=1 m ( r, f (i) f ) +o(1) = k∑ i=1 t (r, ai) + k∑ i=1 m ( r, f (i) f ) +o(1). (3.31) if p ≥ q ≥ 2, then by using a similar proof as in theorem 1.3 or theorem 1.4, we obtain t (r, a0) > expp−1 { γ ( logq−1 ( 1 1− |z | ))µ} > expp−1 { α ( logq−1 ( 1 1− |z | ))µ} ≥ t (r, ai) , (i = 1, 2, ..., k) (3.32) as |z | → 1− for z ∈ h. by applying lemma 2.2 and substituting (3.32) into (3.31) , we get expp−1 { γ ( logq−1 ( 1 1− r ))µ} ≤ k expp−1 { α ( logq−1 ( 1 1− r ))µ} +o ( log+ t (r, f ) + log ( 1 1− r )) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. noting that γ > α, by the last inequality,we have exp { (1− o (1)) expp−2 { γ ( logq−1 ( 1 1− r ))µ}} ≤ o ( log+ t (r, f ) + log ( 1 1− r )) (3.33) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. therefore, from (3.33) we obtain σ[p,q] (f ) =∞ and σ[p+1,q] (f ) ≥ µ. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 19if p = q = 1, then by using a similar proof as in theorem 1.3 or theorem 1.4, we get t (r, a0) > γ ( 1 1− |z | )µ > kα ( 1 1− |z | )µ > α ( 1 1− |z | )µ ≥ t (r, ai) , (i = 1, 2, ..., k) (3.34) as |z | → 1− for z ∈ h. by applying lemma 2.2 and substituting (3.34) into (3.31) , we obtain γ ( 1 1− r )µ ≤ kα ( 1 1− r )µ +o ( log+ t (r, f ) + log ( 1 1− r )) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. noting that γ > kα, by the last inequality,we have (γ − kα) ( 1 1− r )µ ≤ o ( log+ t (r, f ) + log ( 1 1− r )) (3.35) for all z satisfying |z | = r ∈ h1\e2 as |z | = r → 1−. therefore, from (3.35) we obtain σ (f ) = σm (f ) =∞ and σ2 (f ) = σm,2 (f ) ≥ µ. 4. proof of theorem 1.9 suppose that every solution f of equation (1.1) not being identically equal to 0. first step. we consider the fixed points of f . define the function g by setting g (z) := f (z)− z, z ∈ d. it follows from (1.1) that g(k) + ak−1g (k−1) + · · ·+ a1g ′ + a0g = −a1 − za0 (4.1) and by theorem 1.1 or theorem 1.2, we get σ[p,q] (g) = σ[p,q] (f ) =∞, σ[p+1,q] (g) = σ[p+1,q] (f ) = µ, λ̄[p+1,q] (g) = λ̄[p+1,q] (f − z) . (4.2) now, we prove that −a1 − za0 6≡ 0. assume that −a1 − za0 ≡ 0. clearly a0 6≡ 0. then lim |z |→1−,z∈h ∣∣∣a1a0 ∣∣∣ = 1 and by (3.4), we have ∣∣∣∣a1 (z) a0 (z) ∣∣∣∣ < expp { α ( logq−1 ( 1 1−|z | ))µ} expp { γ ( logq−1 ( 1 1−|z | ))µ} = 1 exp { (1− o (1)) expp−1 { γ ( logq−1 ( 1 1−|z | ))µ}} → 0 https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 20 as |z | → 1− for z ∈ h. then lim |z |→1−,z∈h ∣∣∣a1a0 ∣∣∣ = 0. it is easy to see the contradiction. hence, −a1 − za0 6≡ 0. next by lemma 2.5, we get max { σ[p,q] (ai) (i = 0, 1, ..., k − 1) , σ[p,q] (−a1 − za0) } <∞. we deduce, by using (4.1) , (4.2) and lemma 2.6 that λ̄[p,q] (g) = σ[p,q] (g) =∞, λ̄[p+1,q] (g) = σ[p+1,q] (g) = µ. therefore, we obtain λ̄[p,q] (f − z) =λ̄[p,q] (g) = σ[p,q] (g) = σ[p,q] (f ) =∞, λ̄[p+1,q] (f − z) = λ̄[p+1,q] (g) = σ[p+1,q] (g) = σ[p+1,q] (f ) = µ. second step. for the following proof, we use the principle of mathematical induction. set ak (z) ≡ 1, then |ak (z)| ≤ expp { α ( logq−1 ( 1 1− |z | ))µ} and equation (1.1) becomes (1.2) . we consider the fixed points of f (j) (z) (j = 1, 2, ...). definethe function g1 by setting g1 (z) := f ′ (z)− z, z ∈ d.then, by lemma 2.5 and (4.2) , we have σ[p,q] (g1) = σ[p,q] (f ′) =∞, σ[p+1,q] (g1) = σ[p+1,q] (f ′) = µ, λ̄[p+1,q] (g1) = λ̄[p+1,q] (f ′ − z) . (4.3) dividing both sides of (1.2) by a0, we obtain ak a0 f (k) + ak−1 a0 f (k−1) + · · ·+ a1 a0 f ′ + f = 0. (4.4) it follows, by differentiating both sides of equation (4.4) that ak a0 f (k+1) + (( ak a0 )′ + ak−1 a0 ) f (k) + · · ·+ (( a2 a0 )′ + a1 a0 ) f ′′ + (( a1 a0 )′ + 1 ) f ′ = 0. (4.5) multiplying (4.5) by a0, we have ak,1f (k+1) + ak−1,1f (k) + · · ·+ a1,1f ′′ + a0,1f ′ = 0. (4.6) substituing f ′ = g1 + z into (4.6) , we obtain ak,1g (k) 1 + ak−1,1g (k−1) 1 + · · ·+ a1,1g ′ 1 + a0,1g1 = f1, (4.7) where ak,1 = ak = 1, ai ,1 = a0 (( ai+1 a0 )′ + ai a0 ) (i = 1, 2, ..., k − 1) , (4.8) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 21 a0,1 = a0 (( a1 a0 )′ + 1 ) , (4.9) f1 = − (a1,1 + za0,1) . (4.10)next, we prove that a0,1 6≡ 0 and f1 6≡ 0. assume that a0,1 ≡ 0, then a1 a0 = −z + c0, where c0 isan arbitrary constant. hence, we have a1 + (z − c0)a0 = 0. then, f0 = z − c0 is a solution of (1.1) and σ[p,q] (f0) <∞. this contradicts (4.2) . now, assume that f1 ≡ 0. by (4.6) and (4.10) ,we know that the function f1 such that f ′1 = z is a solution of equation (4.6) and σ[p,q] (f1) <∞.this contradicts (4.2) . therefore, a0,1 6≡ 0 and f1 6≡ 0. it follows by (4.8) − (4.10) and lemma2.5 that max { σ[p,q] (ai ,1) (i = 0, 1, ..., k) , σ[p,q] (f1) } <∞.we deduce by using (4.3) , (4.7) and lemma 2.6 that λ̄[p,q] (g1) = σ[p,q] (g1) =∞, λ̄[p+1,q] (g1) = σ[p+1,q] (g1) = µ. therefore, we obtain λ̄[p,q] ( f ′ − z ) = λ̄[p,q] (g1) = σ[p,q] (g1) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f ′ − z ) = λ̄[p+1,q] (g1) = σ[p+1,q] (g1) = σ[p+1,q] (f ) = µ.set g2 (z) = f ′′ (z)− z, z ∈ d. then, by using a similar discussion as in the case of the function g1, we can get ak,2f (k+2) + ak−1,2f (k+1) + · · ·+ a1,2f (3) + a0,2f ′′ = 0and ak,2g (k) 2 + ak−1,2g (k−1) 2 + · · ·+ a1,2g ′ 2 + a0,2g2 = f2,where ak,2 = 1, ai ,2 = a0,1 (( ai+1,1 a0,1 )′ + ai ,1 a0,1 ) (i = 1, 2, ..., k − 1) , a0,2 = a0,1 (( a1,1 a0,1 )′ + 1 ) , f2 = − (a1,2 + za0,2) .therefore, by the same procedure as for g1, we obtain λ̄[p,q] ( f ′′ − z ) = λ̄[p,q] (g2) = σ[p,q] (g2) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f ′′ − z ) = λ̄[p+1,q] (g2) = σ[p+1,q] (g2) = σ[p+1,q] (f ) = µ.now, assume that a0,s 6≡ 0, λ̄[p,q] ( f (s) − z ) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f (s) − z ) = σ[p+1,q] (f ) = µ (4.11) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 22for all s = 0, 1, ..., j−1, and we prove that for s = j we have (4.11) holds. set gj (z) = f (j) (z)−z, z ∈ d. then, by using (4.2) , we obtain σ[p,q] ( gj ) = σ[p,q] ( f (j) ) =∞, σ[p+1,q] ( gj ) = σ[p+1,q] ( f (j) ) = µ, λ̄[p+1,q] ( gj ) = λ̄[p+1,q] ( f (j) − z ) . (4.12) by following the same procedure as before, we have ak,j f (k+j) + ak−1,j f (k+j−1) + · · ·+ a1,j f (j+1) + a0,j f (j) = 0 (4.13) and ak,jg (k) j + ak−1,jg (k−1) j + · · ·+ a1,jg ′ j + a0,jgj = fj , (4.14)where ak,j = 1, ai ,j = a0,j−1 (( ai+1,j−1 a0,j−1 )′ + ai ,j−1 a0,j−1 ) (i = 1, 2, ..., k − 1) , a0,j = a0,j−1 (( a1,j−1 a0,j−1 )′ + 1 ) 6≡ 0 (a0,0 = a0, a1,0 = a1) , fj = − ( a1,j + za0,j ) 6≡ 0.we deduce by applying lemma 2.6 in (4.14) that λ̄[p,q] ( f (j) − z ) = λ̄[p,q] ( gj ) = σ[p,q] ( gj ) = σ[p,q] ( f (j) ) =∞, λ̄[p+1,q] ( f (j) − z ) = λ̄[p+1,q] ( gj ) = σ[p+1,q] ( gj ) = σ[p+1,q] ( f (j) ) = µ (j = 1, 2, ...) .therefore, we obtain λ̄[p,q] ( f (j) − z ) = λ̄[p,q] (f − z) = σ[p,q] (f ) =∞, λ̄[p+1,q] ( f (j) − z ) = λ̄[p+1,q] (f − z) = σ[p+1,q] (f ) = µ (j = 1, 2, ...) . 5. proofs of theorem 1.10 and 1.11 proof of theorem 1.10. suppose that every solution f of equation (1.1) not being identicallyequal to 0. by applying theorem 1.3 or theorem 1.4, we get σ[p,q] (f ) =∞, σ[p+1,q] (f ) = µ. now, we prove that −a1 − za0 6≡ 0. assume that −a1 − za0 ≡ 0, then we can easily obtain t (r, a1) = t (r,−za0) ≤ t (r, a0) + t (r, z) , t (r, a0) = t ( r, a1−z ) ≤ t (r, a1) + t (r, z) +o (1) . (5.1) it follows from (5.1) that 1− t (r, z) +o (1) t (r, a0) ≤ t (r, a1) t (r, a0) ≤ 1 + t (r, z) t (r, a0) . (5.2) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 23by following the same reasoning as in the proof of theorem 1.3 or theorem 1.4, we have t (r, a0) > expp−1 { γ ( logq−1 ( 1 1− |z | ))µ} > expp−1 { α ( logq−1 ( 1 1− |z | ))µ} ≥ t (r, a1) (5.3) as r = |z | → 1− for z ∈ h. by using (5.3) , we obtain t (r, z) t (r, a0) ≤ t (r, z) expp−1 { γ ( logq−1 ( 1 1−|z | ))µ} → 0 (5.4) as |z | → 1− for z ∈ h. then, by (5.2) and (5.4) , we get lim |z |→1−,z∈h t (r, a1) t (r, a0) = 1. (5.5) on the other hand, we have for p = q = 1 t (r, a1) t (r, a0) < α γ < 1 (5.6) and for p ≥ q ≥ 2 t (r, a1) t (r, a0) < expp−1 { α ( logq−1 ( 1 1−|z | ))µ} expp−1 { γ ( logq−1 ( 1 1−|z | ))µ} → 0 (5.7) as |z | → 1− for z ∈ h. it follows by (5.6) and (5.7) that lim |z |→1−,z∈h t (r, a1) t (r, a0) 6= 1. (5.8) obviously, (5.5) contradicts with (5.8). hence, −a1 − za0 6≡ 0. set ak (z) ≡ 1, then t (r, ak) ≤ expp−1 { α ( logq−1 ( 1 1−|z | ))µ} . clearly, a0 6≡ 0. we can get the conclusion of theorem 1.10, byreasoning in the same way as we did in the proof of theorem 1.9. proof of theorem 1.11. suppose that every meromorphic (or analytic) solution f of equation (1.2)not being identically equal to 0. by applying one of theorem 1.5 to theorem 1.8, we get σ[p,q] (f ) =∞, σ[p+1,q] (f ) ≥ µ. then, we can get the conclusion of theorem 1.11, by reasoning in the same way as we did in theproof of theorem 1.9 and theorem 1.10 by using σ[p+1,q] (f ) ≥ µ instead of σ[p+1,q] (f ) = µ and σ[p+1,q] ( f (j) ) ≥ µ instead of σ[p+1,q] (f (j)) = µ (j = 1, 2, ...) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 246. examples example 6.1 consider the following differential equation f ′′ +k1 (z) exp4 {( log2 ( 1 1− z ))5} f ′ +k0 (z) exp4 { 3 ( log2 ( 1 1− z ))5} f = 0, (6.1) where k0 and k1 are analytic functions in the unit disc d such that |k0| > 1, |k1| < 1 and max { σm,[4,3] (k0) , σm,[4,3] (k1) } < 5. in the equation (6.1) , we have a0 (z) = k0 (z) exp4 { 3 ( log2 ( 1 1− z ))5} , a1 (z) = k1 (z) exp4 {( log2 ( 1 1− z ))5} . then max { σm,[4,3] (a0) , σm,[4,3] (a1) } = 5.let h = {z ∈ c : |z | = r < 1 and arg z = 0} ⊂ d be a set of complex numbers satisfying densd {|z | : z ∈ h} = 1 > 0. then |a0 (z)| = |k0 (z)| ∣∣∣∣∣exp4 { 3 ( log2 ( 1 1− z ))5}∣∣∣∣∣ > exp4 { 3 ( log2 ( 1 1− r ))5} ⇒ log4 |a0 (z)|( log2 ( 1 1−r ))5 > 3⇒ lim inf r→1−,z∈h log4 |a0 (z)|( log2 ( 1 1−r ))5 ≥ 3 > 1, and |a1 (z)| = |k1 (z)| ∣∣∣∣∣exp4 {( log2 ( 1 1− z ))5}∣∣∣∣∣ ≤ exp4 {( log2 ( 1 1− r ))5} as r → 1− for z ∈ h. it is clear that the conditions of theorem 1.1 hold with α = 1, µ = 5, p = 4and q = 3 on the set h. by theorem 1.1, every solution f 6≡ 0 of equation (6.1) satisfies σ[4,3] (f ) = σm,[4,3] (f ) =∞ and σ[5,3] (f ) = σm,[5,3] (f ) = σm,[5,3] (a0) = 5. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 25 example 6.2 consider the following differential equation k2 (z) exp4 {( log2 ( 1 1−z ))7} f ′′+k1 (z) exp4 { 2 ( log2 ( 1 1−z ))7} f ′ +k0 (z) exp4 { 5 ( log2 ( 1 1−z ))7} f = 0, (6.2) where k0, k1 and k2 are analytic functions in the unit discd such that |k0| > 1, |k1| < 1, |k2| < 1and max { σm,[4,3] (k0) , σm,[4,3] (k1) , σm,[4,3] (k2) } < 7. in the equation (6.2) , we have a0 (z) = k0 (z) exp4 { 5 ( log2 ( 1 1−z ))7} , a1 (z) = k1 (z) exp4 { 2 ( log2 ( 1 1−z ))7} , a2 (z) = k2 (z) exp4 {( log2 ( 1 1−z ))7} . then max { σm,[4,3] (a0) , σm,[4,3] (a1) , σm,[4,3] (a2) } = 7. let h = {z ∈ c : |z | = r < 1 and arg z = 0} ⊂ d be a set of complex numbers satisfying densd {|z | : z ∈ h} = 1 > 0. then |a0 (z)| = |k0 (z)| ∣∣∣exp4 { 5 ( log2 ( 1 1−z ))7}∣∣∣ > exp4 { 5 ( log2 ( 1 1−r ))7} ⇒ log4 |a0 (z)|( log2 ( 1 1−r ))7 > 5⇒ lim inf r→1−,z∈h log4 |a0 (z)|( log2 ( 1 1−r ))7 ≥ 5 > 2, and |a1 (z)| = |k1 (z)| ∣∣∣∣∣exp4 { 2 ( log2 ( 1 1− z ))7}∣∣∣∣∣ ≤ exp4 { 2 ( log2 ( 1 1− r ))7} |a2 (z)| = |k2 (z)| ∣∣∣∣∣exp4 {( log2 ( 1 1− z ))7}∣∣∣∣∣ ≤ exp4 { 2 ( log2 ( 1 1− r ))7} https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 26as r → 1− for z ∈ h. it is clear that the conditions of theorem 1.5 hold with α = 2, µ = 7, p = 4and q = 3 on the set h. by theorem 1.11, every meromorphic (or analytic) solution f 6≡ 0 ofequation (6.2) satisfies λ̄[4,3] ( f (j) − z ) = λ̄[4,3] (f − z) = σ[4,3] (f ) =∞ and λ̄[5,3] ( f (j) − z ) = λ̄[5,3] (f − z) = σ[5,3] (f ) ≥ 7, (j = 1, 2, ...) . references [1] s. bank, general theorem concerning the growth of solutions of first-order algebraic differential equations. compos.math. 25 (1972) 61–70. http://www.numdam.org/item/cm_1972__25_1_61_0.[2] b. belaïdi, estimation of the hyper-order of entire solutions of complex linear ordinary differential equations whosecoefficients are entire functions. electron. j. qual. theory diff. equ. 2002 (2002) 5. http://real.mtak.hu/23284.[3] b. belaïdi, growth of solutions to linear equations with analytic coefficients of [p,q] -order in the unit disc. electron. j.diff. equ. 2011 (2011) 156. http://ftp.gwdg.de/pub/emis/journals/ejde/volumes/2011/156/abstr.html.[4] b. belaïdi, growth and oscillation theory of [p,q]-order analytic solutions of linear equations in the unit disc. j.math. anal. 3 (2012) 1-11.[5] b. belaïdi, on the [p,q]-order of analytic solutions of linear equations in the unit disc. novi sad j. math. 42 (2012)117–129.[6] l. g. bernal, on growth k -order of solutions of a complex homogeneous linear differential equation. proc. amer.math. soc. 101 (1987) 317–322. https://doi.org/10.1090/s0002-9939-1987-0902549-5.[7] t. b. cao and h. x. yi, the growth of solutions of linear differential equations with coefficients of iterated order inthe unit disc. j. math. anal. appl. 319 (2006) 278–294. https://doi.org/10.1016/j.jmaa.2005.09.050.[8] t. b. cao, the growth, oscillation and fixed points of solutions of complex linear differential equations in the unitdisc. j. math. anal. appl. 352 (2009) 739–748. https://doi.org/10.1016/j.jmaa.2008.11.033.[9] y. chen, g. t. deng, z. m. chen and w. w. wang, growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc. adv. diff. equ. 2021 (2021) 431. https://doi.org/10.1186/s13662-021-03579-3.[10] z. x. chen and c. c. yang, some further results on the zeros and growths of entire solutions of second order lineardifferential equations. kodai math. j. 22 (1999) 273–285. https://doi.org/10.2996/kmj/1138044047.[11] i. chyzhykov, g. gundersen and j. heittokangas, linear differential equations and logarithmic derivative estimates.proc. london math. soc. (3) 86 (2003) 735–754. https://doi.org/10.1112/s0024611502013965.[12] w. k. hayman, meromorphic functions. oxford mathematical monographs, clarendon press, oxford, 1964.[13] j. heittokangas, on complex differential equations in the unit disc. ann. acad. sci. fenn. math. diss. 122 (2000)1–54.[14] h. hu and x. m. zheng, growth of solutions of linear differential equations with analytic coefficients of [p,q]-orderin the unit disc. electron. j. diff. equ. 2014 (2014) 204.[15] o. p. juneja, g. p. kapoor and s. k. bajpai, on the (p,q)-order and lower (p,q)-order of an entire function. j. reineangew. math. 282 (1976) 53–67. https://doi.org/10.1515/crll.1977.290.180.[16] o. p. juneja, g. p. kapoor and s. k. bajpai, on the (p,q)-type and lower (p,q)-type of an entire function. j. reineangew. math. 290 (1977) 385-405. https://doi.org/10.1515/crll.1977.290.180.[17] i. laine, complex differential equations. handbook of differential equations: ordinary differential equations. vol. iv,269–363, handb. differ. equ., elsevier/north-holland, amsterdam, 2008. https://doi.org/10.28924/ada/ma.3.10 http://www.numdam.org/item/cm_1972__25_1_61_0 http://real.mtak.hu/23284 http://ftp.gwdg.de/pub/emis/journals/ejde/volumes/2011/156/abstr.html https://doi.org/10.1090/s0002-9939-1987-0902549-5 https://doi.org/10.1016/j.jmaa.2005.09.050 https://doi.org/10.1016/j.jmaa.2008.11.033 https://doi.org/10.1186/s13662-021-03579-3 https://doi.org/10.2996/kmj/1138044047 https://doi.org/10.1112/s0024611502013965 https://doi.org/10.1515/crll.1977.290.180 https://doi.org/10.1515/crll.1977.290.180 eur. j. math. anal. 10.28924/ada/ma.3.10 27 [18] z. latreuch and b. belaïdi, linear differential equations with analytic coefficients of [p,q]-order in the unit disc.sarajevo j. math. 9 (2013) 71–84. http://doi.org/10.5644/sjm.09.1.06.[19] y. z. li, on the growth of the solution of two-order differential equations in the unit disc. pure appl. math. 4 (2002)295–300.[20] j. liu, j. tu and l. z. shi, linear differential equations with entire coefficients of [p,q]-order in the complex plane. j.math. anal. appl. 372 (2010) 55–67. https://doi.org/10.1016/j.jmaa.2010.05.014.[21] m. tsuji, potential theory in modern function theory. chelsea, new york, (1975), reprint of the 1959 edition.[22] j. tu and z. x. xuan, complex linear differential equations with certain analytic coefficients of [p,q]-order in the unitdisc. adv. diff. equ. 2014 (2014) 167. https://doi.org/10.1186/1687-1847-2014-167. https://doi.org/10.28924/ada/ma.3.10 http://doi.org/10.5644/sjm.09.1.06 https://doi.org/10.1016/j.jmaa.2010.05.014 https://doi.org/10.1186/1687-1847-2014-167 1. introduction and main results 2. some lemmas 3. proofs of theorems 1.1 to 1.8 4. proof of theorem 1.9 5. proofs of theorem 1.10 and 1.11 6. examples references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 26doi: 10.28924/ada/ma.3.26 two point iterative schemes for nondifferentiable equations in banach space ioannis k. argyros1,∗, janak joshi2, samundra regmi3 1department of computing and mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2department of mathematics, dallas community college, dallas, tx, usa janakrajjoshi2036@gmail.com 3department of mathematics, university of houston, houston, 77024, tx, usa sregmi5@uh.edu ∗correspondence: iargyros@cameron.edu abstract. the local as well as the semi-local convergence analysis is established for a certain singlestep-two point iterative scheme defined on a banach space setting. these schemes converge to alocally unique solution of a nonlinear equation. both types of convergence are based on w-typecontinuity and majorizing functions and sequences. an auxiliary fixed linear operator is utilizedto assure the existence of inverses of the linear operators involved as well as the initial points ofthe iterative scheme. the local analysis provides the radius of convergence, error estimates andinformation on the uniqueness of the solution. moreover, the semi local analysis provides sufficientconvergence conditions, error estimates and uniqueness of the solution results. numerical examplesfurther validate the theoretical results. 1. introduction using mathematical modelling, a plethora of applications from diverse disciplines of science andengineering reduce to determining solutions denoted by x∗ of a nonlinear equation like f (x) = 0. (1.1) here, f : d ⊂ b → b is a continuous operator, b stands for a banach space and d is an open setin b. the analytic form of the solution for (1.1) can be found only in special cases. that explainswhy researchers and practitioners resort to iterative schemes, when a sequence is generatedapproximating x∗ under certain conditions. numerous studies exist in the local as well as the semi-local convergence analysis of iter-ative schemes [1–16]. recently, there has been a surge in the development of schemes for solving received: 4 may 2023. key words and phrases. two-point iterative method; banach space; local-semi local convergences.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 2equations or systems of equations involving nondifferentiable operators. looking towards thisdirection, we develop the two-point iterative scheme (tips): for x0 ∈ d and each n = 0, 1, 2, . . . by xn+1 = xn − a(xn, xn−1)−1f (xn), (1.2) where a : d ×d → l(b), the space of bounded linear operators from b into b. tpis specialtiesto popular schemes. case 1: (secant scheme [1–4,12–15]) set a(x, y) = [x, y ;f ], where [·, ·;f ] : d × d → (b) is a divided difference oforder one for the operator f [6–8]. under this choice the scheme (1.2) specializes to xn+1 = xn − [xn, xn−1;f ]−1f (xn). (1.3) case 2: (kurchatov’s scheme [12,13])set a(x, y) = [2x − y , y ;f ]. then, the scheme (1.2) becomes: xn+1 = xn − [2xn − xn−1, xn−1;f ]−1f (xn). (1.4) case 3: (steffensen’s scheme [2, 8, 15])set a(x, y) = [x + f (x), y ;f ]. then, the scheme (1.2) becomes: xn+1 = xn − [x + f (x), y ;f ]−1f (xn). (1.5) case 4: (picards’s scheme [1, 3, 8–10,15,16])set a(x, y) = i , where i stands for the identity operator on b. case 5: (newton’s scheme [1–3,5, 6, 9, 15,16])set a(x, y) = f ′(x), where f ′ stands for the fréchet derivative of the operator f .the local as well as the semi-local convergence results for the aforementioned schemes involveassumptions on derivatives which do not appear on some of these methods (except case 5). therefore, these results cannot be used to solve nondifferentiable operator equations (seee.g., example 4.2). other conditions involve approximations to the divided difference and theselection of an initial point x0 so that the first iteration is computable. in the present article, the local and semi-local convergence of the scheme (1.2) is investi-gated under w-continuity-type conditions. moreover, by introducing a certain linear operator p ,then invertibility of the linear operator a is assured in a certain subset of d from which the initial https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 3point is selected. the rest of the article is structured as follows: the local and the semi-localconvergence of the scheme (1.2) appear in section 2 and section 3, respectively. the examplescan be found in section 4. the article is completed the conclusions in section 5. 2. convergence i: local some real functions are introduced that play a role in the local convergence analysis of thescheme (1.2). let t = [0,∞). assume: (a1) there exists a continuous and nondecreasing function (cnf) w0 : t × t → t such thatthe equation w0(t, t)− 1 = 0has a smallest solution r ∈ t0 − {0}.set t0 = [0, r0) and d1 = d ∪ u(x∗, r). moreover, there exists cnf w : t0 → t suchthat the equation h(t)− 1 = 0 has a smallest solution r ∈ (0, r0), where h(t) = w(t, t) 1− w0(t, t) . let t1 = [0, r). 0 ≤ w0(t, t) < 1 (2.6)and 0 ≤ h(t) < 1 (2.7)from now on we assume that x∗ ∈ d is a solution of the equation f (x) = 0 and the divideddifference [∗, ∗;f ] exists on d×d. the functions w0 and w are connected to the operatorson the scheme (1.2). (a2) there exists an invertible operator p such that for each x, y ∈ d ‖p−1(a(x, y)− p )‖ ≤ w0(‖x − x∗‖, ‖y − x∗‖). (a3) ‖p−1(a(x, y)− [x, x∗;f ])‖ ≤ w(‖x − x∗‖, ‖y − x∗‖) for each x, y ∈ d1and (a4) u[x ∗, r ] ⊂ d.next, the local convergence of the scheme (1.2) is provided based on the conditions (a1) − (a4)and the developed terminology. theorem 2.1. assume that the conditions (a1)−(a4) are validated. if the initial points x−1, x0 ∈ u(x∗, r)−{x∗}, then the sequence {xn} generated by the scheme (1.2) is well defined in u(x∗, r) for each n = 0, 1, 2, 3, . . . and is convergent to the solution x∗ of the equation f (x) = 0, so that ‖xn+1 − x∗‖ ≤ h(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖ < r (2.8) https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 4 where the radius of convergence r is defined in (a1) and the function h is also given in (a1). proof. by hypothesis, x−1, x0, (a2) and (a3) we obtain in turn that: ‖p−1(a(x0, x−1)− p )‖ ≤ wo(‖x0 − x∗‖, ‖x−1 − x∗‖) ≤ w0(r, r) < 1. (2.9) the estimate (2.9) and the banach lemma on invertible operators [1–3, 8, 9] assure the existenceof a(x0, x−1)−1l(b) and ‖a(x0, x−1)−1p‖ ≤ 1 1− w0(‖x0 − x∗‖, ‖x−1 − x∗‖) . (2.10) moreover, the iterate x1 is well defined by the first subset of the scheme (1.2). then, we can write: x1 − x∗ = x0 − x∗ − a(x0 − x−1)−1f (x0) = a(x0 − x−1)(a(x0, x−1)− [x0, x∗;f ])(x0 − x∗) (2.11) using (2.7), (a3), (2.10), (a2) and (2.11) we get in turn that: ‖x1 − x∗‖ = w(‖x0 − x∗‖, ‖x−1 − x∗‖)‖x0 − x∗‖ 1− w0(‖x0 − x∗‖, ‖x−1 − x∗‖) ≤ h(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r,(2.12)thus, the iterate x1 ∈ u(x∗, r) and the item (2.8) holds for n = 0. simply replace x−1, x0, x1 by xm−1, xm, xm+1 in the preceding calculations to terminate the induction for items (2.8). then, fromthe estimation: ‖xm+1 − x∗‖ ≤ c‖xm − x∗‖ < r, (2.13)where c = h(‖x0 − x∗‖) ∈ [0, 1). we conclude that limm→∞ xm = x∗ and that the iterate xm+1 ∈ u(x∗, r). � the uniqueness of the solution region is given in the next result. proposition 2.2. assume: there exists a solution z ∈ u(x∗, r1) of the equation f (x) = 0 for some r1 ≥ 0; the condition (a2) holds in the ball u(x∗, r1) for a being [·, ·;f ], and there exists r2 ≥ r1 such that: w0(r1, r2) < 1. (2.14) define the region d2 = d ∪ u[x∗, r2]. then, x∗ is the only solution of the equation f (x) = 0 in the region d2. proof. if z 6= x∗, then the divided difference e = [x∗, z ;f ] is well defined. using (a2) and (2.14),we get in turn that ‖p−1(e − p )‖| ≤ w0(‖x∗ − x∗‖, ‖z − x∗‖) ≤ w0(0, r2) < 1. thus, e−1 ∈ l(b). moreover, from the identity z − x∗ = e−1(f (z)− f (x∗)) = e−1(0) = 0. https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 5hence, we conclude that z = x∗. � 3. convergence ii: semi-local a certain real sequence is developed that is shown in theorem 3.1 to be majorizing for thescheme (1.2). assume: (h1) : there exists a continuous and nondecreasing function v0 : t → t such that v0(t, t)−1 = 0has a unique positive solution denoted by δ.let t2 = [0, δ). (h2) : there exists a cnf v : t2× t2 → t . define a sequence {γn} for γ−1 = 0, γ0 = α, some γ1 ≥ α by: γn+2 = γn+1 + v(γn+1 − γn, γn − γn+1)(γn+1 − γn) 1− v0(γn − γ0, γn+1 − γ0) . (h3) : v0(γn − γ0, γn+1 − γ0) < 1 and γn ≤ γ < δ.clearly, by the definition of the sequence γn and (h3), this sequence is nondecreasingly convergentto its unique least upper bound denoted by γ∗. the functions v0, v and the limit point γ∗ areconnected to the operators on the scheme as follows: (h4) : there exists an invertible operator p ∈ l(b) such that p−1 ∈ l(b) and for each x, y ∈ d ‖p−1(a(x, y)− p )‖ ≤ v0(‖x − x0‖, ‖y − x0‖). it follows by (h1) that: if x−1, x0 ∈ d with ‖x−1 − x0‖ ≤ α, v0(‖x−1 − x0‖, ‖x0 − x0‖) ≤ v0(α, 0) < 1. thus a(x0, x−1)−1 ∈ l(b). let ‖a(x0, x−1)−1f (x0)‖ ≤ γ1 − γ0. set d3 = d ∪ u(x0, γ). ‖p−1(a(x, y)− [y , z ;f ])‖ ≤ (‖x − y‖, ‖y − z‖) for each x, y , z ∈ d3 and (h5) : u[x0, γ∗] ⊂ d.next, we present the semi-local convergence analysis of the scheme (1.2) under the conditions (h1)− (h5) and the preceding terminology. theorem 3.1. assume that the conditions (h1)−(h5) hold. then the sequence {xn} generated by the scheme (1.2) is well defined in u(x0, γ0), remains in u(x0, γ0) for each n = 0, 1, 2, . . . and converges to a solution x∗ ∈ u(x0, γ0) of the equation f (x) = 0. moreover, the following items hold: ‖x∗ − xn‖ ≤ γn+1 − γn (3.15) https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 6 and ‖x∗ − xn‖ ≤ γ∗ − γn, (3.16) where the sequence {γn} and the point x∗ are defined in (h3). proof. the inequality (3.15) holds for n = 0, since: ‖x1 − x0‖ = ‖a(x0, x−1)−1f (x0)‖ ≤ γ1 − γ0 < γ∗ − γ0. � thus, the iterate x1 ∈ u(x0, γ0). let xn, xn−1 ∈ u(x0, γ0). using (h3) and (h4), we have inturn, ‖p−1(a(xn, xn−1)− p )‖ ≤ v0(‖xn − x0‖, ‖xn−1 − x0‖) ≤ v0(γ0, γ0) < 1.so, a(xn, xn−1)−1l(b) and ‖a(xn, xn−1)−1p‖ ≤ 1 1− v0(‖xn − x0‖, ‖xn−1 − x0‖) . (3.17) moreover, the iterate xn+1 is well defined by the scheme (1.2) and (3.17). further we can writerby the scheme (1.2) f (xn+1) = f (xn+1)− f (xn)− a(xn, xn−1)(xn+1 − xn) = ([xn+1, xn;f ]− a(xn, xn−1))(xn+1 − xn). (3.18) in view of (h4) and (3.18) we get: ‖p−1f (xn+1)‖ ≤ ‖p−1([xn+1, xn;f ]− a(xn, xn−1))‖‖xn+1 − xn‖ ≤ v(‖xn+1 − xn‖, ‖xn − xn−1‖)‖xn+1 − xn‖ ≤ v(‖γn+1 − γn‖, ‖γn − γn−1‖)‖γn+1 − γn‖. (3.19) then, by the scheme (1.2) for n replaced by n + 1, we obtain: ‖xn+2 − xn+1‖ ≤ ‖a(xn+1, xn)−1p‖‖pf (xn+1)‖ ≤ v(γn+1 − γn, γn − γn−1) 1− v0(‖xn+1 − x0‖, ‖xn − x0‖) ≤ v(γn+1 − γn, γn − γn−1)(γn+1 − γn) 1− v0(γn+1, γn)and ‖xn+2 − x0‖ ≤ ‖xn+2 − xn+1‖+ ‖xn+1 − x0‖ ≤ γn+2 − γn+1 + γn+1 − γ0 < γ∗ − γ0. therefore, the induction for the item (3.15) is terminated and {xn} ⊂ u(x0, γ ∗ − γ0). but thesequence {γn} is complete. thus, the sequence {xn} is also complete in a banach space b. hencethere exists x∗ ∈ u[x0, γ∗ − γ0] such that limn→∞ xn = x∗. https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 7by letting n → +∞ in (3.19) and using the continuity of the operator f , we deduce f (x∗) = 0. then from the estimation: ‖xn+i − xn‖ ≤ γn+i − γn, (3.20) the item (3.16) is obtained by letting i → +∞ in (3.20). a uniqueness of the solution region is determined in the next result. proposition 3.2. assume: there exists a solution z ∈ u(x0, δ1) of the equation f (x) = 0 for some δ1 > 0; the condition (h4) holds for [∗, ∗, ;f ] replacing a on the ball u(x0, δ1) and there exists δ2 ≥ δ1 such that v0(δ1, δ2) < 1. (3.21) define the region d4 = d ∩ u[x0, δ2]. then, the only solution of the equation f (x) = 0 in the region d4 is z . proof. as in proposition 2.2, consider z1 ∈ d4 with f (z1) = 0 and define e1 = [z, z1;f ] for z 6= z1.then, the application of (h4) and (3.21) give ‖p−1(e1 − p )‖ ≤ v0(‖z − z0‖, ‖z1 − x0‖) ≤ v0(δ1, δ2) < 1. consequently, it follows again that z1 = z0. � remark 3.3. (i) possible choices but not the only ones for the linear operator p are: – differential case: p = f ′(x∗) and – non-differential case: p = [x−1, x0;f ]. p should be chosen in general, so the majorant functions are as tight as possible in both the local and semi-local analysis (see the numerical section 4 that follows).(ii) the limit point γ∗ in (h5) is replaced by δ given in (h1).(iii) notice that only (h4) out of conditions (h1)− (h5) is used in proposition 3.2. however, if all conditions are used, let δ1 = γ∗ and z = x∗. 4. numerical examples example 4.1. let b = r× r and d = u[x∗, 1] with x∗ = (0, 0, 0)t . define the mapping f on d for y = (y1, y2, y3)t as: f (y) = ( ey1 − 1, y2, e − 1 2 y23 + y3 )t . https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 8 then, by the definition of the fréchet derivative f ′ of f is given by: f ′(y) = e y1 0 0 0 1 0 0 0 (e − 1)y3 + 1  it follows that f ′(x∗) = i . then, the convergence conditions (a3)−(a5) are validated, respectively for: case 1: w0(s1, s2) = 1 2 (e − 1)(s1 + s2), w(s1, s2) = 1 2 (e − 1)s1, and u[x∗, r ] ⊂ d. case 2: w0(s1, s2) = 1 2 (e − 1)(s1 + s2), w(s1, s2) = 1 2 (e − 1)(2s1 + s2), and u[x∗, 3r ] ⊂ d. case 3: w0(s1, s2) = 1 2 (e − 1)(s1 + s2), w(s1, s2) = 1 2 (e − 1)(s1 + 2s2), and u[x∗, r ] ⊂ d, where r = max{r, f (r)} and f (t) = ( ‖i + p‖+ e − 1 2 t ) t or f (t) = ( 2 + e − 1 t ) t. case 4: w0(s1, s2) = 0, w(s1, s2) = 1 2 (e − 1)s1, and u[x∗, r ] ⊂ d. case 5: w0(s1, s2) = 1 2 (e − 1)(s1 + s2), w(s1, s2) = 1 2 (e − 1)s1, and u[x∗, 3r ] ⊂ d. the exact radius r can be then computed immediately using the condition (a2). as an example, for the case 5, we must solve: 1 2(e − 1)t 1− (e − 1)t = 0, leading to t = r = 2 3(e − 1) .concerning the semi-local case and the application of the method (1.2), we present anotherexample: example 4.2. let b = r × r × r. the two by two nonlinear and nondifferentiable system to be solved is: 3s21 s2 + s 2 2 − 1 + |s1 − 1| = 0, s41 + s1s 3 2 − 1 + |s2| = 0. https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 9 then, the system can be described as q = (q1, q2), where q1(s1, s2) = 3s 2 1 s2 + s 2 1 − 1 + |s1 − 1| and q2(s1, s2) = s 4 1 + s1s 3 2 − 1 + |s2|. the system becomes q(s1, s2) = 0. then as a = [∗, ∗;q], which is an 2 × 2 real matrix defined for s = [s1, s2]t and s̃ = [s3, s4]t by: [s̃ , s̃;q]i ,1 = qi [(s1, s4)−qi(s1, s2)] s4 − s2 for s2 6= s4, i = 1, 2. otherwise, set [∗, ∗;q] = 0. let us choose s0 = (5, 5) and s−1 = (1, 0) to be the starters for the scheme (1.2). then, n x (1) n x (2) n ‖xn − xn−1‖ 0 5 5 1 1 0 5 2 0.98909090909090909 0.363636363636364 3.636e-01 3 0.894886945874111 0.329098638203090 3.453e-02 4 0.894655531991499 0.327827544745569 1.271e-03 5 0.894655373334793 0.327826521746906 1.022e-06 6 0.894655373334687 0.327826421746298 6.089e-13 7 0.894655373334687 0.327826421746298 2.701e-20 therefore, the solution s∗ = (s∗1 , s∗2)t of the system is s∗1 = 0.894655373334687 and s∗2 = 0.327826421746298. remark 4.3. a more targeted choice for l than the two mentioned already can give a larger radius of convergence. indeed, assume the conditions instead of (a3) and (a4) (a3) ′ ‖p−1(f ′(x)− p )‖ < 1 and (a4) ′ ‖p−1(f ′(x∗ + θ(x − x∗))− f ′(x))‖ ≤ g4(θ)‖x − x∗‖ for each x ∈ d. consider an example f (x) = ex − 1 https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 10with d = u[x∗, 1]. then x∗ = 0. choose p = 1 be x for b ∈ (0, 2). then the newton’s scheme gives: xn+1 − x∗ = xn − x∗ − f ′(xn)−1f (xn) = s0f ′(xn) −1 (f ′(x∗ + θ(xn − x∗))− f ′(xn)) (xn − x∗) leading by (a3)′ and (a4)′ to ‖xn+1 − x∗‖ ≤ |b|(e − 2)‖xn − x∗‖2 1− |1− b| , where we also used ‖p−1(f ′(x)− p )‖ = ∥∥∥∥be−x (ex − 1bex )∥∥∥∥ = |b − 1| < 1. thus, ‖p−1(f ′(x)− p )‖ ≤ 1 1− |1− b| ,and ‖p−1(f ′(x∗ + θ(x − x∗))− f ′(x))‖ ≤ |b|(e − 2). the last estimate is obtained, since for y = x∗ + θ(x − x∗) e−x(e−y − ex) = ey − 1 = 1 + y + y2 2! + · · ·+ y k k! + · · · − 1 = y ( 1 + y 2! + · · ·+ y k−1 k! + . . . ) . hence, ‖ex(ey − ex)‖ = (1− θ)gn‖x − x∗‖,where, gn = 1 + 1− θ 2! + · · ·+ (1− θ)n−1 n! + . . . , and ∫ 1 0 (1− θ)ndθ = 1 n + 1 . but, ∫ 1 0 (1− θ)ngndθ = ∫ 1 0 (1− θ)dθt + ∫ 1 0 (1− θ)2 2! dθt2 + · · ·+ ∫ 1 0 (1− θ)n n! dθtn−1 + . . . = 1 2 t + 1 3 · 2!t 2 + · · ·+ 1 (n + 1)n! tn−1 + · · · ≤ e − 2. consequently, it follows from the error estimate that: ra = 1− |1− b| |b|(e − 2) .let us compare the new radius with the ones already in the literature developed independently byrheinboldt [11] and traub [15]. the condition used is: ‖f ′(x∗)−1(f ′(x)− f ′(y))‖ ≤ l‖x − y‖ https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 11for each x, y ∈ d to obtain the error estimate: ‖xn+1 − x∗‖ ≤ l‖xn − x∗‖2 2(1− ‖xn − x∗‖) , and the radius is rtr = 2 3l .but l = e for the example, so: rtr = 3 2e < ra, say for b = 1. therefore, the new radius of convergence is larger allowing for a wider choice of initial points.other choices of p can lead to even larger radius of convergence. we leave the detail to themotivated reader. 5. conclusion a finer and more flexible local and semi-local convergence analysis for the scheme (1.2) is devel-oped involving an invertible operator p , which if chosen appropriately leads to weaker convergenceconditions, better uniqueness of the solution and a larger radius of convergence than if p is chosento be as in earlier studies f ′(x∗) or f ′(x0) or [x0, x−1;f ]. this idea can be extended to multistepand multipoint schemes [1–16]. this is the direction of our future research. references [1] i.k. argyros, unified convergence criteria for iterative banach space valued methods with applications, mathematics,9 (2021) 1942.[2] i.k. argyros, theory and applications of iterative methods, 2nd edition engineering series. crc press-taylor andfrancis group, boca raton, florida, usa, 2022.[3] i.k. argyros, convergence and application of newton-type iterations, springer, new york, 2008.[4] j.a. ezquerro, m. grau-sanchez, m.a. hernandez, m. nouguera, semilocal convergence of secant-like methods fordifferentiable and nondifferentiable operator equations, j. math. anal. appl. 398 (2013) 110–112.[5] s. george, k. kanagaraj, derivative free regularization method for nonlinear ill-posed equations in hilbert scales,comp. meth. appl. math. 19 (2019) 765–778.[6] l.v. kantorovich, g.p. akilov, functional analysis, pergamom press, oxford, 1982.[7] a.a. magrenan, a new tool to study real dynamics: the convergence plane, appl. math. comp. 248 (2014) 215–224[8] j.m. ortega, w.c. rheinboldt, iterative solution of nonlinear equations in several variables, academic press, new-york, 1970.[9] p. deuflhard, g. heindl, affine invariant convergence theorems for newton’s method and extensions to relatedmethods, siam j. numer. anal. 16 (1979) 1–10.[10] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptakerror estimates, numer. funct. anal. optim. 9 (1987) 671–684[11] w.c. rheinboldt, an adaptive continuation process for solving systems of nonlinear equations, banach cent. publ.3 (1978) 129–142.[12] s.m. shakhno, convergence of the two-step combined method and uniqueness of the solution of nonlinear operatorequations. j. comp. appl. math. 261 (2014) 378–386. https://doi.org/10.28924/ada/ma.3.26 eur. j. math. anal. 10.28924/ada/ma.3.26 12 [13] s.m. shakhno, on an iterative algorithm with superquadratic convergence for solving nonlinear operator equations.j. comp. appl. math. 231 (2009) 222–235.[14] j.r. sharma, h. arora, a novel derivative free algorithm with seventh order convergence for solving systems ofnonlinear equations, numer. algor. 67 (2014) 917–933.[15] j.f. traub, prentice-hall, iterative methods for the solution of equations, prentice-hall, englewood cliffs, newjersey, (1964).[16] t. yamamoto, a convergence theorem for newton-like methods in banach spaces, numer. math. 51 (1987) 545–557. https://doi.org/10.28924/ada/ma.3.26 1. introduction 2. convergence i: local 3. convergence ii: semi-local 4. numerical examples 5. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 22doi: 10.28924/ada/ma.3.22 local stability analysis of onchocerciasis transmission dynamics with nonlinear incidence functions in two interacting populations k. m. adeyemo department of mathematics, hallmark university ijebu-itele, ogun state, nigeria ∗correspondence: mikyade2019@gmail.com abstract. a deterministic compartmental model for the transmission dynamics of onchocerciasis withnonlinear incidence functions in two interacting populations is studied. the model is qualitatively an-alyzed to investigate its local asymptotic behavior with respect to disease-free and endemic equilibria.it is shown, using routh-hurwitz criteria, that the disease-free equilibrium is locally asymptoticallystable when the associated basic reproduction number is less than the unity. when the basic repro-duction number is greater than the unity, we prove the existence of a locally asymptotically stableendemic equilibrium. 1. introduction onchocerciasis is one of the neglected tropical diseases caused by the parasite onchocercavolvulus, a filarial nematode [2]. the disease is transmitted from one person to another by repeatedbites of black flies. the disease is endemic in sub-saharan africa. many researchers have workedon many ways to reduce the spread of the disease. for instance, remme et al. [10] used skin snipsurvey in west africa to investigate the impact of controlling black flies by larviciding. plaisieret al. [9] used micro simulation model to determine the period required for combining annualivermectin treatment and vector control in the onchocerciasis control programme in west africa.alley et al. [1] used a computer simulation model to study prevention of onchocerciasis by usingmacrofilaricide which kills the adult worms. asha hassan & nyimvua shaban [3] investigated theeffects of four control strategies on the spread of the disease.in this paper, we consider onchocerciasis transmission dynamics with nonlinear incidence functions.the human population is sub-divided into four compartments and the vector population is sub-divided into three compartments. we show local asymptotic behaviour in disease-free and endemicequilibria. the rest of the paper is organized as follows: the description of the model and theorems received: 27 apr 2023. key words and phrases. basic reproduction number; diseases free equilibrium; onchocerciasis epidemic model; non-linear incidence function. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 2on positivity of solutions are given in section 2 while section 3 is devoted to the proof local stabilitytheorems. 2. model description two interacting populations are considered; the humans and the black-flies populations. thehuman population is partitioned into four compartments: the susceptible human compartment; sh„ the exposed compartment; eh, the infectious human compartment; ih and the recoveredhuman compartment; rh. the black-fly population is partitioned into three compartments:susceptible vector; sv , the exposed vector compartment; ev and the infective vector compart-ment. the total human and vector populations at any given time, t, are respectively given by; n = sh(t) + eh(t) + ih(t) + rh(t) and v = sv (t) + ev (t) + iv (t). we assume that thetransmission of onchocerciaisis in susceptible hosts is only through contact with infectious vector.we also assume that susceptible vector becomes infectious as a result of contact with infectioushosts during blood meal. the population under study is assumed to be large enough to bemodelled deterministically. the following system of non-linear ordinary differential equations,with non-negative initial conditions, describes the dynamics of onchocerciaisis epidemics. dsh(t,xi ) dt = ψh(xi)− ∑l i=0 δλh(xi )sh(t,xi )iv (t) 1+νh(xi )iv (t) − µh(xi)sh + w(xi)rh(t, xi)) deh(t,xi ) dt = ∑l i=0 δλh(xi )sh(t,xi )iv (t) 1+νh(xi )iv (t) − (αh(xi) + µh(xi))eh(t, xi) d ih(t,xi ) dt = ∑l i=0 αh(xi)eh − (r(xi) + γh(xi) + µh(xi))ih(t, xi) drh(t,xi ) dt = ∑l i=0 r(xi)ih − (µh(xi) + w(xi))rh(t, xi) dsv dt = ψv − δλv (xi )sv (t)ih(xi ,t) 1+νv ih(xi ,t) − µvsv (t) dev dt = δλv (xi )sv (t)ih(xi ,t) 1+νv ih(xi ,t) − (αv + µv )ev (t) d iv dt = αvev (t)− (µv + γv )iv (t)  (2.1) subject to the following initial conditions: sh(0, xi) = s0h(xi), eh(0, xi) = e0h(xi), ih(0, xi) = i0h(xi), rh(0, xi) = r0h(xi) sm(0) = s0m, em(0) = e0m, im(0) = i0m (2.2) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 3 symbols definitionss sh(t, xi ) number of susceptible humans at time t and discrete age xi eh(t, xi ) number of exposed humans at time t and discrete age xi ih(t, xi ) number of infectious humans at time t and discrete age xi rh(t, ai) number of recovered humans at time t and discrete age xi sv (t) number of susceptible black-flies at time t ev (t) number of exposed black-flies at time t iv (t) number of infectious black-flies at time t ψh(xi ) recruitment term of the susceptible humans at discrete age xi ψv recruitment term of the susceptible vectors δ biting rate of the vector λh(xi ) probability that a bite by an infectious vector results in transmission of disease to human at discrete age xi λv probability that a bite results in transmission of parasite to a susceptible vector µh(xi ) per capita death rate of humans at discrete age xi µv per capita death rate of vector γh(xi ) disease-induced death rate of humans at discrete age xi γv disease-induced death rate of vectors αh(xi ) per capita rate of progression of humans from the exposed state to the infectious state at discrete age xi αv per capita rate of progression of vectors from the exposed state to the infectious state r(xi ) per capita recovery rate for humans from the infectious state to the recovered state due to treatment at discrete age xi ω(xi ) per capita transition rate of recovered humans to the susceptible state at discrete age xi νh(xi ) humans disease-inhibiting factor at discrete age xi νv vectors disease-inhibiting factor model assumptionsthe formulation of the compartmental model is based on the following assumptions: 1. that all humans are born susceptible. that is, humans are liable to contract the disease.2. that the susceptible humans, when infected, becomes exposed humans who are not yetinfectious.3. that the exposed humans progress to become infectious only.4. that the infectious humans may either die naturally or as a result of the disease, and ifnot, they become recovered humans due to treatment.5. that the recovered humans become susceptible again.6. all black-flies are born susceptible.7. that the susceptible black-flies, when infected, becomes exposed black-flies who are notyet infectious.8. that the exposed black-flies progress to become infectious only.9. that the infectious black-flies remain infectious for life. that is, there is no recovered classfor black-fly population. 2.1. existence and positivity of solutions. in this section, we analyse the general properties ofthe system (2.1) with positive initial conditions. it describes the population dynamics both in humanand black-fly populations. the system is biologically relevant in the set given by ω = (sh(t, xi), eh(t, xi), ih(t, xi), rh(t, xi)) ∈ r4 + : nh ≤ l∑ i=0 ψh(xi) µh(xi) , (sv (t), ev (t), iv (t)) ∈ r3 + : nv ≤ ψv µv https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 4here, the following results are provided which guarantee that the model governed by system (2.1)is mathematically well-posed in a feasible region ω defined by: ω = ωh ×ωv ⊂ r4 × r3 theorem 1:there exists a domain ω in which the solution set sh(t, xi), eh(t, xi), ih(t, xi), rh(t, xi), sv (t), ev (t), iv (t)is contained and bounded. proofif the total human population size is given by nh = sh(t, xi) +eh(t, xi) + ih(t, xi) +rh(t, xi), andthe total size of black-fly population is nv = sv (t) + ev (t) + iv (t). from model (2.1), we havethat dnh(t, xi) dt ≤ ψh(xi)− l∑ i=0 µh(xi)nh(t, xi) (2.3) and dnv dt ≤ ψv − µvnv (2.4)it follows from (2.3) and (2.4) that nh(t, xi) ≤ ψh(xi) µh(xi) [1− e1−µh(xi )t]+nh(0,xi )e −µh(xi )t ] and nv ≤ ψv µv [1− e−µv t ] + nv (0)e−µv t taking the lim sup as t → ∞ gives nh ≤ ψh(xi ) µh(xi ) and nv ≤ ψv µv . this shows that all solu-tions of the humans population only are confined in the solution set ωh and all solutions of theblack-fly population are confined in ωv . it also suffices to say that ω is positively invariant as nh(t, xi) ≤ ∑l i=0 ψh(xi ) µh(xi ) whenever nh(0, xi) ≤ ψh(xi ) µh(xi ) and nv (t) ≤ ψv µv if nv (0) ≤ ψv µv , therefore thesolution set for the model (2.1) exists and is given by ω = ωh ×ωv ⊂ r4 + × r3 + 2it remains to show that the solutions of system (2.1) are nonnegative in ω for any time t > 0 sincethe variables represent human and black-fly populations. theorem 2:the solutions, sh(t, xi), eh(t, xi), ih(t, xi), rh(t, xi), sv (t), ev (t), iv (t), of model (2.1) with non-negative initial conditions in ω, remain nonnegative in ω for all t > 0. proof: given that the initial conditions, s0h(xi), e0h(xi), i0h(xi), r0h(xi), s0v ,e0v ,i0v , are non-negative and from (2.1), dsh(t, xi) dt + l∑ i=0 [ bλh(xi)iv (t) 1 + νh(xi)iv (t) + µh(xi) ] sh(t, xi) ≥ 0 so that d dt [ l∑ i=0 sh(t, xi)exp (∫ t 0 bλh(xi)iv (η) 1 + νh(xi)iv (η) dη + µh(xi)t )] ≥ 0, (2.5) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 5integrating (2.5), we have l∑ i=0 sh(t, xi) ≥ l∑ i=0 s0h(xi)exp [ − (∫ t 0 bλh(xi)iv (η) 1 + νh(xi)iv (η) dη + µh(xi)t )] ≥ 0, which implies that for all t > 0 and for all a ∈ r+, we have sh(t, xi) ≥ l∑ i=0 s0h(xi)exp [ − (∫ t 0 bλh(xi)iv (η) 1 + νh(xi)iv (η) dη + µh(xi)t )] ≥ 0. hence, sh(t, xi) > 0 for any arbitrary xi . also, we have deh(t, xi) dt + l∑ i=0 ((αh(xi) + µh(xi)))eh(t, xi) ≥ 0 so that d dt [ l∑ i=0 eh(t, (xi))exp(αh(xi) + µh(xi)t) ] ≥ 0 (2.6) integrating (2.6), we have for all t > 0 and for all a ∈ mathbbr+, that eh(t, a) ≥ l∑ i=0 e0h(xi)exp [−(αh(xi) + µh(xi))t] hence, eh(t, xi) > 0 for any arbitrary xi also we have d ih(t, xi) dt ≥ − l∑ i=0 (r(xi) + γh(xi) + µh(xi))ih(t) so that d dt [ih(t)exp(r(xi) + γh(xi) + µh(xi))t] ≥ 0 (2.7) similarly, (2.7) becomes ih(t, a) ≥ l∑ i=0 i0hexp [−(r(xi) + γh(xi) + µh(xi))t] > 0for all t > 0 for all a ∈ r+ hence, ih(t, xi) > 0 for any arbitrary xi . also from (2.1), we have drh(t, xi) dt + l∑ i=0 (µh(xi) + w(xi))rh(t, xi) ≥ 0 and we have d dt [ l∑ i=0 rh(t, xi)exp((µh(xi) + w(xi))t ] ≥ 0 (2.8) integrating (2.8), we have, for all t > 0 and a ∈ r, that rh(t, a) ≥ l∑ i=0 r0h(xi)exp(−(µh(xi) + w(xi))t) > 0 https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 6hence, rh(t, xi) > 0 for any arbitrary xi . in a similar manner, we have dsv dt + [ l∑ i=0 bλv ih(t)) 1 + νv ih(t) + µv ] sv (t) ≥ 0 so that d dt [ sv (t)exp (∫ t 0 bλv ih(η)) 1 + νv ih(η) d(η) + µv t )] ≥ 0 (2.9) integrating (2.9), we have sv (t) ≥ s0vexp [ − (∫ t 0 bλv ih(η)) 1 + νv ih(η) d(η) + µv t )] > 0 ∀ t > 0 also we have dev dt ≥ −(αv + µv )ev (t) which on integration gives ev (t) ≥ ev (0)exp [−(αv + µv )t] > 0 ∀ t > 0 (2.10) and finally, we have d iv dt + (µv + γv )iv (t) so that d dt [iv (t)exp(µv + γv )t] ≥ 0 (2.11) and we have iv (t) ≥ iv (0)exp [−(µv + γv )t] > 0, ∀ t > 0 this completes the proof 2 3. existence and stability of the equilibrium points 3.1. disease-free equilibrium. the disease-free equilibrium (dfe) points are steady state solu-tions that depict the absence of infection in both the human host and black-fly vector populations,i.e, onchocerciasis does not exist in the population. thus, the disease-free equilibrium point, e0, forthe model (2.1) implies that s∗(xi)h 6= 0, e∗h(xi) = i∗h = 0(xi) = r∗h(xi) = 0, s∗v 6= 0, ev = iv = 0and putting these into (2.1), we have s∗(xi)h = ψh(xi ) µh(xi ) and s∗v = ψv µv . consequently we obtain e0as e0 = ( ψh(xi) µh(xi) , 0, 0, 0, ψv µv , 0, 0 ) (3.1) a key notion in the analysis of infectious disease models is the basic reproduction number r0 , anepidemiological threshold that determines whether disease dies out or persists in the population.thebasic reproduction number r0 of the system (2.1) is computed using the next generation matrixmethod and is given by r0 = √ rhrv https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 7 where rh = ∑l i=0 δαhλh(xi )ψh(xi ) µh(xi )(αh(xi )+µh(xi ))(r(xi )+γh(xi )+µh(xi )) and rv = δαvλvψv µv (αv+µv )(γv+µv ) . the basicreproduction number r0, determines whether onchocerciasis dies out or persists in the population.therefore, rh describes the number of humans that one infectious black-fly infects over its expectedinfectious period in a completely susceptible humans population, while rv is the number of blac-flies infected by one infectious human during the period of infectiousness in a completely susceptibleblack-fly population. 3.2. local stability of the disease-free equilibrium point e0. using the basic reproduction num-ber obtained for the model (2.1), we analyse the stability of the equilibrium point in the followingresult. theorem 3:the disease-free equilibrium point, e0, is locally asymptotically stable if r0 < 1, and unstable if r0 > 1. proof: the jacobian matrix of the system (2.1) evaluated at the disease-free equilibrium point e0,is obtained as m(e0) =  m11 0 0 m14 0 0 m17 0 m22 0 0 0 0 m27 0 m32 m33 0 0 0 0 0 0 m43 m44 0 0 0 0 0 m53 0 m55 0 0 0 0 m63 0 0 m66 0 0 0 0 0 0 m76 m77  where m11 = −µh(xi), m14 = w(a1), m17 = − ∑l i=0 δλh(xi )ψh(xi ) µh(xi ) , m22 = −(αh(xi) + µh(xi)), m27 = ∑l i=0 δλh(xi )ψh(xi ) µh(xi ) , m32 = αh(xi), m33 = −(r(xi) + γh(xi) + µh(xi)), m43 = r(xi), m44 = −(µh(xi) + w(xi)), m53 = − δλvψv µv , m55 = −µv , m63 = δλvψv µv , m66 = −(αv + µv ), m76 = αv , m77 = −(µv + γv ) we need to show that all the eigenvalues of m(e0) are negative. as the firstand fifth columns form the two negative eigenvalues, h(xi) and −v , the other five eigenvalues canbe obtained from the sub-matrix, m1(e0), formed by excluding the first and fifth rows and columnsof m(e0). hence m1(e0) =  m ′11 0 0 0 m ′15 αh(xi) m ′22 0 0 0 0 r(xi) m ′33 0 0 0 0 δλvψv µv 0 −(αv + µv ) 0 0 0 0 αv −(µv + λv )  in the same way, the third column of m1(e0) contains only the diagonal term which forms anegative eigenvalue, (µh(xi) + w(xi)). the remaining four eigenvalues are obtained from the https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 8sub-matrix m2(e0) given by m2(e0) =  m ′′11 0 0 m ′′14 αh(xi) m ′22 0 0 0 δλvψv µv −(αv + µv ) 0 0 0 αv −(µv + λv )  thus, the eigenvalues of the matrix m2(e0) are the roots of the characteristic equation of the form (ξ+αh(xi))(ξ+ r(xi) +γh(xi) +µh(xi))(ξ+µv +γ)− l∑ i=0 δ2αh(xi)λh(xi)ψh(xi)vλvψv µh(xi)µv = 0 (3.2) if we let y1 = αh(xi) + µh(xi), y2 = r(xi) + γh(xi) + µh(xi), y3 = αv + µv , and y4 = µv + γv , then(3.2) becomes x4ξ 4 +x3ξ 3 +x2ξ 2 +x1ξ +x0 = 0, (3.3) where x4 = 1 x3 = y1 + y2 + y3 + y4 x2 = (y1 + y2)(y2 + y4) + y1y2 + y3y4 x1 = (y1 + y2)y3y4 + (y3 + y4)y1y2 x0 = y1y2y3y4 − ∑l i=0 δ2αh(xi )λh(xi )ψh(xi )vλvψv µh(xi )µv  (3.4) expressing x0 in terms of reproduction number r0, we have x0 = y1y2y3y4(1−r2 0) (3.5) we can see from (3.4) that x1 > 0, x2 > 0, x3 > 0, x4 > 0, since all yis are positive. moreover,if r0 < 1, it follows from (3.5) that x0 > 0. thus, using the routh-hurwitz criterion, we have h1 = x3 > 0 h2 = ∣∣∣∣∣ x3 x4 x1 x2 ∣∣∣∣∣ = y1(y2 + y3 + y4)(y1 + y2 + y3 + y4) + (y2 + y3)(y2 + y4)(y3 + y4) > 0similarly we have h3 > 0 and h4 > 0 where h3 = ∣∣∣∣∣∣∣∣ x3 x4 0 x1 x2 x3 0 x0 x1 ∣∣∣∣∣∣∣∣ and h4 = ∣∣∣∣∣∣∣∣∣∣∣ x3 x4 0 0 x1 x2 x3 x4 0 x0 x1 x2 0 0 0 x0 ∣∣∣∣∣∣∣∣∣∣∣ theref ore, al ltheeigenvaluesof thejacobianmatr ixm(e0) have negative real parts when r0 < 1 and the disease-free equilibrium point is locally asymptotically stable. however, when r0 > 1,we see that x0 < 0 and there is one eigenvalue with positive real part and therefore the disease-free equilibrium point is unstable 2 https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 93.3. endemic equilibrium point ee . we shall show that the formulated model (2.1) has an endemicequilibrium point, ee . the endemic equilibrium point is a positive steady state solution where thedisease persists in the population. theorem 4: the model (2.1) has a unique endemic equilibrium ee whenever r0 > 1. proof: let ee = (s′′h(xi), e ′′ h (xi), i ′′ h (xi), r ′′ h(xi), s ′′ v , e ′′ v , i ′′ v ) be a nontrivial equilibrium of the model(2.1). that is, all components of ee are positive. then the onchocerciasis model (2.1) at steady-statebecomes ψh(xi)− l∑ i=0 ( δλh(xi)s ′′ h(xi)iv 1 + νh(xi)i ′′v − µh(xi)s ′ h(xi) + ω(xi)r ′′ h(xi) ) = 0 (3.6) l∑ i=0 ( δλh(xi)s ′′ h(xi)iv 1 + νh(xi)i ′′v − (αh(xi) + µh(xi))e′′h (xi) ) = 0 (3.7) l∑ i=0 (αh(xi)e ′′ h (xi)− (r(xi) + µh(xi) + γh(xi))i ′′h (xi)) = 0 (3.8) l∑ i=0 r(xi)i ′′ h (xi)− (µh(xi) + ω(xi))r′′h(xi) = 0 (3.9) ψv − δλvs ′′ v ih(xi) 1 + νv (xi)i ′′ h (xi) − µvs′′v = 0 (3.10) δλvs ′′ v ih(xi) 1 + νv (xi)i ′′ h (xi) − (αv + µv )e′′v = 0 (3.11) αve ′′ v − (µv + γv )i ′′v = 0 (3.12)from the last three equations, we have i ′′v = αve ′′ v µv + γv (3.13) e′′v = δλvs ′′ v ih(xi) 1 + νv (xi)i ′′ h (xi)(αv + µv ) (3.14) and s′′v = ψv δλvs′′v ih(xi ) 1+νv (xi )i ′′ h (xi ) + µv (3.15) substituting (3.14) and (3.15) into (3.13) yields i ′′v = rvµv i ′′h (xi) µv + (δλv + µvνv )i ′′h (xi) (3.16) from (3.8) and (3.9), we have e′′h (xi) = l∑ i=0 (r(xi) + µh(xi) + γh(xi))ih(xi) αh(xi) (3.17) and r′′h(xi) = l∑ i=0 r(xi)i ′′(xi) µh(xi) + ω(xi) (3.18) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 10if we put (3.16) and (3,17) in (3.7) in terms of r0, we have s′′h(xi) = ∑l i=0 ψh(xi)[µv + (δλv + µvνv + νh(xi)µvrv )i ′′h (xi)] µh(xi)µvr2 0 (3.19) finally, using (3.16), (3.18) and (3.19) in (3.7), we have i ′′h (xi) = l∑ i=0 µh(xi)µvψh(xi)(µh(xi) + ω(xi)) (r2 0 − 1) ρ (3.20) where ρ = l∑ i=0 (µh(xi)+ω(xi))[δλh(xi)µvrv+ψh(xiµh(xi)(δλv+µvνv+νh(xi)µv )rm)]− l∑ i=0 µh(xi)µvω(xi)r(xi)r2 0.(3.21)if in (3.20), ω(xi) = 0 then ρ > 0. from this, one sees that model (2.1) has no positive solutionwhen r0 < 1. however, with ω(xi) = 0, a unique endemic equilibrium exists when r0 > 1. thiscompletes the proof. 2 remark 1: it is important to have a remark that positive solution exists for the model (2.1) in acase where ρ < 0 and r0 < 1. this implies that the disease-free equilibrium co-exists with theendemic equilibrium state when r0 is slightly less than unity resulting into a phenomenon ofsubcritical (backward) bifurcation. references [1] w.s. alley, b.a.b. boatin, n.j.d.n. nagelkerke, macrofilaricides and onchocerciasis control, mathematical modellingof the prospects for elimination. bmc public health. 1 (2001) 12.[2] u. amazigo, m. noma, j. bump, b. bentin, b. liese, l. yameogo, h. zouré, and a. seketeli, onchocerciasis diseaseand mortality in sub saharan africa, chapter 15, world bank, washington, dc, 2006.[3] a. hassan, n. shaban, onchocerciasis dynamics: modelling the effects of treatment, education and vector control,j. biol. dyn. 14 (2020) 245-268.[4] e.m. poolman, a.p. galvani, modeling targeted ivermectin treatment for controlling river blindness, amer. j. trop.med. hygiene, 75 (2006) 921–927.[5] j.p. mopecha, h.r. thieme, competitive dynamics in a model for onchocerciasis with cross-immunity, can. appl.math. quart. 11 (2003) 339–376.[6] m.g. basanez, m. boussinesq, population biology of human onchocerciasis, phil. trans. r. soc. lond. b: biol. sci.354 (1999) 809–826.[7] m.g. basanez, j. ricardez-esquinca, models for the population biology and control of human onchocerciasis, trendsparasitol. 17 (2001) 430–438.[8] j.d. murray, mathematical biology i., an introduction. 3rd ed. heidelberg: springer-verlag berlin, 2002.[9] a.p. plaisier, e.s. alley, g.j. van oortmarssen, b.a. boatin, j.d.f habbema, required duration of combined annualivermectin treatment and vector control program in west africa, bull. world health organ. 75 (1997) 237-245.[10] j. remme, g. de sole, g.j. van oortmarssen, the predicted and observed decline in onchocerciasis infection during14 years of successful control of black flies in west africa, bull. world health organ. 68 (1990) 331–339. https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 11 [11] s.i. omade, a.t. omotunde, a.s. gbenga, mathematical modeling of river blindness disease with demography usingeuler method, math. theory model. 5 (2015), 75–85.[12] world health organization, african programme for onchocerciasis control: meeting of national onchocerciasis taskforces, september 2012, weekly epidemiol. record 87(49–50) (2012), pp. 494–502. https://doi.org/10.28924/ada/ma.3.22 1. introduction 2. model description 2.1. existence and positivity of solutions 3. existence and stability of the equilibrium points 3.1. disease-free equilibrium 3.2. local stability of the disease-free equilibrium point e0 3.3. endemic equilibrium point ee references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 9doi: 10.28924/ada/ma.5.9 numerical results for gauss-seidel iterative algorithm based on newton methods for unconstrained optimization problems nguyen dinh dung tnu-university of information and communication technology, thai nguyen, vietnam nddung@ictu.edu.vn abstract. optimization problems play a crucial role in various fields such as economics, engineering,and computer science. they involve finding the best value (maximum or minimum) of an objectivefunction. in unconstrained optimization problems, the goal is to find a point where the function’svalue reaches a maximum or minimum without being restricted by any conditions. currently, thereare many different methods to solve unconstrained optimization problems, one of which is the newtonmethod. this method is based on using a second-order taylor series expansion to approximate theobjective function. by calculating the first derivative (gradient) and second derivative (hessian matrix)of the function, the newton method determines the direction and step size to find the extrema. thismethod has a very fast convergence rate when near the solution and is particularly effective forproblems with complex mathematical structures. in this paper, we introduce a gauss-seidel-typealgorithm implemented for the newton and quasi-newton methods, which is an efficient approachfor finding solutions to optimization problems when the objective function is a convex functional. wealso present some computational results for the algorithm to illustrate the convergence of the method. 1. introduction in this paper, we focus on solving the unconstrained nonlinear optimization problem min x∈rn f (x) (1) where f (x) is a convex functional with second derivative on rn. optimization problem (1) is alsoa problem derived from many problems in different fields in economics and engineering. solvingproblem (1) can lead to solving a system of nonlinear equations and has many different appli-cations, for example in solving the `1-norm problem arising from compressing sensing [1][4], invariational inequalities problems [5][6], and optimal power flow equations [7] among others. ina broader sense, optimization should be understood as the activities aimed at obtaining the bestresult under certain conditions (maximizing profit, minimizing costs). the theory of optimizationmethods is not new, there are a huge number of optimization methods: methods based on the use received: 27 oct 2024. key words and phrases. convex optimization; newton; quasi-newton; hessen matrix; gauss–seidel.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 2of lagrange multipliers, methods of dynamic programming, and methods of the calculus of vari-ations, linear and nonlinear programming methods, there are currently many different solutionsdepending on the objective function. one of the simplest methods is the steepest descent methodor also known as the gradient descent method [2], [8], this method is simple and applicable to afairly wide class of objective functions, the content of the method is to give a sequence of iter-ations x (k+1) = x (k) − αk∇f (xk), αk > 0, where αk is the step length determined by armijo’srule after the exact or inexact line search, this method has the disadvantage of linear convergencerate. we want to improve the efficiency of the algorithm’s convergence, the newton method is agood choice [9][12], newton’s method was first proposed by isaac newton in 1964 when findingsolutions to nonlinear equations. to date, newton’s methods have widely been used for solvingthe unconstrained nonlinear optimization problem. as a result, studies of newton’s method forman extremely active area of research, with new variants being constantly developed and tested.basic results on newton’s method and comprehensive lists of references can be found, e.g., in thebooks by dennis and schnabel [13], ostrowski [14], ortega and rheinboldt [15], deuflhard [16] andcorless and fillion [17], survey of newton’s method in [18]. the general iterative rule for solving (1)starts from an initial approximation and generates a sequence using the general iterative scheme x(k+1) = x(k) + αkdk . (2) where dk is an appropriate search direction. general class of algorithms of the form (2) is knownas the line search algorithms. the method has a local quadratic convergence, thus convergingextremely fast in a neighbourhood of the solution [19]. nowadays, this method is extended tofind optimal solutions for multivariable functions based on taylor expansion, the solution of theoptimization problem is performed in an iterative sequence x(k+1) = x(k)− [ ∇2f (x(k)) ]−1∇f (x(k)),if the objective function is not quadratic, the above iteration sequence may diverge or converge to alocal minimum or converge to a saddle point. a variant of newton’s method introduced in [20] is thegeneralized newton’s method, in which the solution to the optimization problem is computed by theiteration sequence x(k+1) = x(k) − αk [ ∇2f (x(k)) ]−1∇f (x(k)), where αk is called the step lengthand is determined by one-dimensional search methods in the direction − [∇2f (x(k))]−1∇f (x(k)). however, in some practical problems when leading to optimization problems where the objectivefunction does not have a second derivative, applying newton’s method is not feasible. to overcomethis limitation, recently some results in [21] and [22] have proposed a quasi newton algorithm forfinding solutions to nonlinear optimization problems, showing the effectiveness of the method.the numerical results in the papers all use jacobi iteration, so we hope to improve the conver-gence of the algorithm by applying the idea of gauss-seidel iteration to the implementation ofnewton and quansi-newton methods. so, in this paper, we propose gauss – seidel algorithmsimplemented for the newton and quasi-newton method for finding solutions at each iterationstep, in which we inherit the information of the component solutions calculated in the current https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 3iteration instead of using the solutions calculated in the previous iteration. the computationalresults illustrating the algorithm are given to confirm the convergence of the algorithm. thepaper is organized as follows. section 2 presents the newton’s method and implementation ofthe gauss-seidel iterative algorithm based on newton and quasi-newton methods. experimentalresults illustrating the convergence are presented in section 3. finally, there are conclusions andreferences. 2. proposed method 2.1. newton’s method.let x∗ is the minimum point of the functional, then the necessary condition is ∂f ∂xi (x∗) = 0. inorder to determine the iterative sequence, we use the taylor expansion for f (x), we have f (x) = f (x(k)) +∇fk(x− x(k)) + 1 2 (x− x(k))2jk , (3) where ∇f (x) = ( ∂f∂x1 , ∂f∂x2 , ..., ∂f∂xn), jk is the hessen matrix and is defined as ∇2f (x) =  ∂2f ∂x21 ∂2f ∂x1∂x2 ... ∂2f ∂x1∂xn ∂2f ∂x2∂x1 ∂2f ∂x22 ... ∂2f ∂x2∂xn .......................................... ∂2f ∂xn∂x1 ∂2f ∂xn∂x2 ... ∂2f ∂x2n  . since 2, we have ∇f = ∇fk + (x − x (k))jk . (4) so, we have the iterative process. x(k+1) = x(k) − (jk)−1∇fk , k = 1, 2, 3, ..., n. (5) formula (4) is called newton’s iteration formula with second-order convergence rate. the algo-rithm is implemented as follows: algorithm 1: function x=newton(x(1),ε); k=1; while(‖∇fk‖ > ε) d(k) = −(jk)−1∇fk ; x(k+1) = x(k) + d(k); k=k+1; end; x = x(k); https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 4 according to this algorithm, at the step k : d(k) = −(jk)−1∇fk , x (k+1) = x (k)+ d (k), we implementcomponent inheritance x(k+1)i calculated to calculate x (k+1)j , j = i + 1, ..., n, so (5) is replaced by x (k+1) i = x (k) i + d (k) i , (6) where d (k) 1 = − n∑ j=1 [ (jk) −1]∇f (x (k)j ) d (k) i = − i−1∑ j=1 [ (jk) −1]∇f (x (k+1)j )− n∑ j=i+1 [ (jk) −1]∇f (x (k)j ), i = 2, ..., n so, newton’s algorithm is updated as follows: algorithm 2: function x=newton(x(1),ε); k=1; n=length(x(0)); while(‖∇fk‖ > ε) d(k) = 0; for j=1:n d (k) 1 = d (k) 1 − [ (jk)−1 ] ∇f (x (k)j ) end; x (k+1) 1 = x (k) 1 + d (k) 1 for i=2:n for j=1:i-1 d (k) i = d (k) i − [ (jk)−1 ] ∇f (x (k+1)j ) end; for j=i:n d (k) i = d (k) i − [ (jk)−1 ] ∇f (x (k)j ) end; x (k+1) i = x (k) i + d (k) i end; k=k+1; end; x = x(k+1)in case the objective function is not second-order differentiable, then instead of using newton’smethod, we will use the quasi-newtonian method. 2.2. quasi-newton method.the idea of the quasi-newton method [22] is derived from formula (5), we approximate the hessen https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 5matrix jk by matrix bk so, since (3), we have the following iteration process: x(k+1) = x(k) − αk [bk ]−1∇fk , k = 1..n, (7) where, αk is the step length determined in the direction sk = − [bk ]−1∇fk , it can change at eachiteration and satisfy the wolfe condition [22]. f (x(k) + αksk) ≤ f (x(k)) + c1αkstk ∇f (x(k)) −stk ∇f (x(k) + αksk) ≤ −c2stk ∇f (x(k)) 0 < c1 < c2 < 1 (8) αk is determined by algorithm 3 algorithm 3: function αk=linesearch(f,xk , sk );initialize constants c1, c2, β satisfy 0 < c1 < c2 < 1; 0 < β < 1 α = α0 while(f (x (k) + αsk) > f (x (k)) + c1αs t k ∇f (x (k)) or −stk ∇f (x (k) + αsk) > −c2stk ∇f (x (k))) α = βα; end; αk = α;in case the objective function is a convex function, the obtained optimal point is the global optimalsolution of problem (1). it is easy to see in [13], if bk = i then the iterative formula (7) is thesteepest descent method published in [23]. bk is an approximation matrix for the hessen matrixand satisfies the condition ∇f (x(k)) = ∇f (x(k−1))− bk(x(k) − x(k−1)). (9) at x (k+1), we have ∇f (x(k+1)) = ∇f (x(k))− bk+1(x(k+1) − x(k)) (10) or can write bk+1dk = gk , (11) where, dk = x (k+1) − x (k), gk = ∇fk+1 −∇fkformula (11) can be rewritten as follows: dk = [bk+1]−1 gk , (12) where, bk+1 is a positive definite symmetric matrix and is updated according to the formula bk+1 = bk + czzt . (13) https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 6 since (11), we have (bk + czzt ) dk = gk . so, cz = gk−[bk ]dk zt dk . let z = gk − [bk ] dk , then c = 1 zt dk and bk+1 = bk + (gk − bkdk) (gk − bkdk)t (gk − bkdk)t dk . (14) we can also use the following calculation: bk+1 = bk + c1z1zt1 + c2z2zt2 , dk = bkgk + c1z1(zt1 gk) + c2z2(zt2 gk). let z1 = dk and z2 = bkgk , similar to formula (15), we have bk+1 = bk + gkgtk gtk dk − (bkdk) (bkdk)t dtk bkdk (15) thus, the algorithm to find the solution of problem (1) is implemented as follows: algorithm 4: function x=qnewton(x(1), ε); k=1 bk = i; while(‖∇fk‖ > ε) sk = − [bk ]−1∇fk ; αk= linesearch(f , x (k),sk ); x(k+1) = x(k) + αksk ; dk = x(k+1) − x(k); gk = ∇fk+1 −∇fk ;update bk+1 according to (14) or (15); k=k+1; end; x = x(k+1);according to this algorithm, starting from point x(1), the iteration sequence (15) converges to thelocal optimum x∗ and satisfied. f (x(1)) ≥ ... ≥ f (x(k)) ≥ ... ≥ f (x∗) in case the objective function is a convex function, the obtained optimal point is the global optimalsolution of problem (1). to illustrate the theoretical results, here are some experimental calculationresults of the algorithm. 3. experimental results and discussions in this section, we perform experimental calculations to illustrate the convergence of the algorithmintroduced in the paper. the data is given:objective function f (x) = 10∑ i=1 (xi − i)4 (16) https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 7initial approximation: x(1) = (0, 0, ..., 0). it is easy to see that the exact solution of problem (1)is x∗ = (1, 2, ..., 10), f (x∗) = 0. the objective function (16) is differentiable at all levels, so thehessen matrix exists, so we can completely apply newton’s algorithm to solve problem (1). let er r = ∥∥x(k) − x∗ ∥∥ 2 , we have the computational results illustrating the convergence of newton’salgorithm given in table 1: table 1. approximate solution of problem (1) obtained from algorithm 2 x(k) k = 5 k = 10 k = 15 k = 20 x (k) 1 0.8025 0.9740 0.9966 0.9995 x (k) 2 1.6049 1.9480 1.9931 1.9991 x (k) 3 2.4074 2.9220 2.9897 2.9986 x (k) 4 3.2099 3.8960 3.9863 3.9982 x (k) 5 4.0123 4.8699 4.9829 4.9977 x (k) 6 4.8148 5.8439 5.9794 5.9973 x (k) 7 5.6173 6.8179 6.9760 6.9968 x (k) 8 6.4198 7.7919 7.9726 7.9964 x (k) 9 7.2222 8.7659 8.9692 8.9959 x (k) 10 8.0247 9.7399 9.9657 9.9955 err 3.8758 0.5104 0.0672 0.0089 the calculation results in table 1 show that the approximate solution found converges to the exactsolution of the problem according to the number of iterations. the graphs in figure 1 and figure2 illustrate the convergence of the algorithm. figure 1. error graph according to the number of iterations with the number of iterations k=1,2,. . . ,20 obtained from algorithm 2 https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 8 figure 2. objective function graph according to the number of iterations (number of iterations k=1,2,. . . ,20) obtained from algorithm 2from figure 1 and figure 2, it can be seen that the error function and the objective function aremonotonically decreasing functions with the number of iterations, which shows that the approximatesolution converges to the exact solution of problem (1). now, let us consider the following objectivefunction: f (x) =  10∑ i=1 ( xi − 1i )4 ∃xi < 1 i 10∑ i=1 ( xi − 1i )√ xi − 1i ∀xi ≥ 1 i (17) the objective function does not have a second derivative at x∗ = (1, 12 , ..., 110), so, in order to findthe solution for problem (1) with objective function (16), we perform algorithm 4 with the hessenmatrix approximation. matrix updated according to formula (15). the calculation results are givenin table 2. table 2. approximate solution of problem (1) obtained from algorithm 4 x(k) k = 5 k = 10 k = 15 k = 20 x (k) 1 0.4598 0.8654 0.9708 1.0128 x (k) 2 0.5080 0.5079 0.5078 0.5078 x (k) 3 0.2322 0.3231 0.3459 0.3548 x (k) 4 0.1490 0.2424 0.2659 0.2748 x (k) 5 0.0987 0.1791 0.1997 0.2078 x (k) 6 0.0672 0.1341 0.1525 0.1600 x (k) 7 0.0471 0.1025 0.1196 0.1271 x (k) 8 0.0340 0.0798 0.0959 0.1039 x (k) 9 0.0251 0.0632 0.0784 0.0866 x (k) 10 0.0190 0.0508 0.0649 0.0733 err 0.6032 0.1680 0.0722 0.0584 https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 9the calculation results in table 2 show that the approximate solution found converges to the exactsolution of the problem depending on the number of iterations. the calculation results show thatthe quasi-newton method has the advantage of not requiring a quadratic differentiable objectivefunction, but the convergence is quite slow compared to the newton method. the error functionand the objective function are not monotonically decreasing functions with the number of iterations,but tend to decrease gradually, which also confirms that the approximate solution converges to theexact solution of the problem. the graphs in figure 3 and figure 4 illustrate the convergence ofthe algorithm. figure 3. error graph depending on the number of iterations with the number of iterations k=1,2,. . . ,20 obtained from algorithm 4 figure 4. graph of the objective function depending on the number of iterations (number of iterations k=1,2,. . . ,20) obtained from algorithm 4 https://doi.org/10.28924/ada/ma.5.9 eur. j. math. anal. 10.28924/ada/ma.5.9 104. conclusion in this paper, we implement an iterative algorithm to solve the unconstrained convex optimizationproblem based on newton and quasi-newton iteration methods, in which the information of thecomponent solutions calculated in the current iteration is inherited instead of using the solutionscalculated in the previous iteration. the computational results according to the algorithm areperformed on the matlab 2014 environment, the numerical results have confirmed the convergenceof the method and are consistent with the theory presented in the paper. references [1] s. aji, p. kumam, a.m. awwal, m.m. yahaya, w. kumam, two hybrid spectral methods with inertial effect for solv-ing system of nonlinear monotone equations with application in robotics, ieee access 9 (2021) 30918–30928. https://doi.org/10.1109/access.2021.3056567.[2] y. zhou, y. wu, x. li, a 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https://doi.org/10.4236/jsea.2010.35057 1. introduction 2. proposed method 2.1. newton's method 2.2. quasi-newton method 3. experimental results and discussions 4. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 18doi: 10.28924/ada/ma.3.18 a modified algorithms for new krasnoselskii’s type for strongly monotone and lipschitz mappings furmose mendy, john t mendy∗ university of the gambia, gambia furmosemendy111@gmail.com, jt.mendy@yahoo.com ∗correspondence: jt.mendy@yahoo.com abstract. let e be a 2 uniformly smooth and convex real banach space and let a mapping a : e → e∗be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1}, {y1}) ∈ e, wedefine the sequences {xn} and {yn} by{ yn = xn − θnj−1(axn), n ≥ 1 xn+1 = yn − λnj−1(ayn), n ≥ 1where λn and θn are positive real number and j is the duality mapping of e. letting (λn, θn) ∈ (0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation ax = 0. we also appliedour algorithm in convex minimization and also proved the convergence of it in lp, `p or wm,p . at theend we proposed the algorithm of it in lp(ω) and its inverse lq(ω). 1. introduction definition 1.1. a map a : e → e∗ is called monotone if for each x, y ∈ e, the following inequalityholds: 〈ax − ay, x − y〉 ≥ 0a is called strongly monotone if there exists k ∈ (0, 1) such that for each x, y ∈ e, the followinginequality holds: 〈ax − ay, x − y〉 ≥ k‖x − y‖2a map a : e → e is called accretive if for each x, y ∈ e, there exists j(x − y) ∈ j(x − y) suchthat 〈ax − ay, j(x − y)〉 ≥ 0a is called strongly accretive if there exists k ∈ (0, 1) such that for each x, y ∈ e, there exists j(x − y) ∈ j(x − y) such that 〈ax − ay, j(x − y)〉 ≥ k‖x − y‖2 received: 23 apr 2023. key words and phrases. krasnoselskii-type algorithm; monotone operators; lipschitz mappings.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.18 https://orcid.org/0000-0002-3774-0761 eur. j. math. anal. 10.28924/ada/ma.3.18 2a map a : e → e∗ is called lipschitzian, if for each constant l > 0 and for all x, y ∈ e, thefollowing inequality holds: i): 〈ax − ay〉 ≤ l‖x − y‖ ii): l2(d2 − 1) < k2 many physical problems in applications can be modeled in the following form: find x ∈ h suchthat 0 ∈ ax (1.1)where a is a monotone operator on a real hilbert space h. typical examples where monotone oper-ators occur and satisfy the inclusion 0 ∈ ax include the equilibrium state of evolution equations andcritical points of some functionals and convex optimization, linear programing, monotone inclusionsand elliptic differential equations defined on hilbert spaces (see e.g., browder [2], mustafa [19],stephen [26], sina [24], mendy et al, [17] and chidume [3]). for precisely, the classical convexoptimization problem: let h : h → r ∪ {+∞} be a proper convex function. the sub-differential of h at x ∈ h; is defined by ∂h : h → 2h ∂(x) = {x∗ ∈ h : h(y)− h(x) ≥ 〈y − x, x∗〉,∀y ∈ h}. (1.2) clearly, ∂h : h → 2his monotone operator on h, and 0 ∈ ∂(x0) if and only if x0 is a minimizer of h. in the case of setting ∂(x) ≡ a ; solving the inclusion 0 ∈ ax is solving for a minimizer of h.there have been fruitful works on approximating zero point of a in hilbert spaces (see e.g.,takahashi and ueda [31], song and chen [25], and cho et al. [9]). the proximal point algorithm (ppa) is recognized as a powerful and successful algorithm in finding a numerical solution ofmonotone operators equation 0 ∈ ax which was introduced by martinet [13] and studied furtherby rockafellar [22] and a host of other authors. that is, given xk ∈ h; xn+1 = jλnxn. (1.3) where jλn = (i + λna)−1 is the resolvent of operator a. since rockafellar [22] only obtained theweak convergence of the algorithm 1.3 as λn →∞ ; so he proposed two open questions for obtainingthe strong convergence of the proximal point algorithm: (1) does the proximal point algorithmalways converge weakly? (2) can the proximal point algorithm be modified to guarantee strongconvergence? in studying the strong convergence, many authors have modified the proximal pointalgorithm (ppa) to guarantee strong convergence under different settings, see e.g., takahashi [29],reich [20], lehdili and moudafi [12], chidume et al. [6], and the references therein.let e be a real normed space, e∗ its topological dual space. the map j : e → 2e ∗ defined by jx : { x∗ ∈ e∗ : 〈x, x∗〉 = ‖x‖.‖x∗‖ = ‖x‖2 = ‖x∗‖2 } . https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 3is called the normalized duality map on e. where 〈, 〉 denotes the generalized duality pairingbetween e and e∗.in a hilbert space, the normalized duality map is the identity map. hence, in hilbert spaces,monotonicity and accretivity coincide. for an accretive-type operator a,solutions of the equation ax = 0, in many cases, represent the equilibrium state of somedynamical system (see, for example, [29], page 116). to approximate a solution of ax = 0, assumingexistence, where a : e → e is of accretive type, browder [2] defined an operator t : e → e by t := i − a, where i is the identity map on e. he called such an operator pseudo-contractive.it is trivial to observe that zeros of a correspond to fixed points of t . for lipschitz stronglypseudo-contractive maps, chidume [6] proved the following theorem. theorem 1.1. (chidume, [7]. let e = lp, 2 ≤ p < 8, and k ⊂ e be nonempty closed convex and bounded. let t : k → k be a strongly pseudo-contractive and lipschitz map. for arbitrary x0 ∈ k, let a sequence {xn} be defined iteratively by xn+1 = (1 − λn)xn + λntxn, n ≥ 0, where {λn} ⊂ (0, 1) satisfies the following conditions: ,(i) ∞∑ n=1 λn =∞ , (i i) ∞∑ n=1 λ2n ≤ ∞. then {xn} converges strongly to the unique fixed point of t . by setting t := i − a in theorem 1.1, the following theorem for approximating a solution of ax = 0 where a is a strongly accretive and bounded operator can be proved.unfortunately, the success achieved in using geometric properties developed from the mid-1980sto early 1990s in approximating zeros of accretive-type mappings has not carried over to approx-imating zeros of monotone-type operators in general banach spaces. part of the problem is thatsince a maps e to e∗, for xn ∈ e,axn is in e∗. consequently, a recursion formula containing xnand axn may not be well defined. attempts have been made to overcome this difficulty by introduc-ing the inverse of the normalized duality mapping in the recursion formulas for approximating zerosof monotone-type mappings.examples chidume [4], [5], moudafi [18], reich [21], takahashi [30],zegeye [37], djitte [17], mendy [ [15], [10]]motivated by approximating zeros of monotone mappings, chidume et al. [8] proposed akrasnoselskii-type scheme and proved a strong convergence theorem in lp, 2 ≤ p < ∞. in fact,they obtained the following result. theorem 1.2. (chidume et al. [8]). let x = lp, 2 ≤ p < ∞, and a : x → x∗ be a lipschitz map. assume that there exists a constant k ∈ (0, 1) such that a satisfies the condition 〈ax − ay, x − y〉 ≥ k‖x − y‖ p p−1 (1.4) and that a−1(0) 6= ∅. for arbitrary x1 ∈ x , define the sequence {xn} iteratively by xn+1 = j−1(jxn − λnaxn) n ≥ 0 https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 4 where λn ∈ (0, δp) and δp is some positive constant. then the sequence {xn} converges strongly to the unique solution of the equation ax = 0. in [8], the authors posed the following open problem. if e = lp, 2 ≤ p <∞, attempts to obtainstrong convergence of the krasnoselskii-type sequence defined for x0 ∈ e by xn+1 = j−1(jxn − λnaxn) n ≥ 0 to a solution of the equation ax = 0, where a is strongly monotone and lipschitz, have notyielded any positive result.following the works of chidume et al [8], and motivation of finding the zeros of the monotonetype mapping, several strong convergence results have been established by various authors (seee.g [17], [10], [15], [23], [16]).following this great work, in 2023, mendy [16] constructed the following two-step proximalalgorithm for the zero point of monotone mapping and proof a strong convergency of the sequences {xn} and {yn} to a unique point x∗ ∈ a−1(0).{ yn+1 = j−1(jxn − λnaxn), n ≥ 0 xn+1 = j−1(jyn+1 − λn+1ayn+1), n ≥ 0 (1.5) in this paper, we study the two step size of the new krasnoselskii-type algorithm introduced bysene et al. [23] and prove a strong convergence theorem to approximate the unique zero of alipschitz and strongly monotone mapping 2−uniformly smooth and convex real banach space for p ≥ 2. this class of banach spaces contains all lp-spaces, 2 ≤ p < ∞ and sobolev space. thenwe apply our results to the convex minimization problem. finally, our method of proof generalizedand extended various authors in this way of work. 2. preliminaries let e be a normed linear space. e is said to be smooth if lim t→0 ‖x + ty‖ − ‖x‖ t (2.1) exist for each x, y ∈ se (here se := {x ∈ e : ||x || = 1} is the unit sphere of e). e is said to beuniformly smooth if it is smooth and the limit is attained uniformly for each x, y ∈ se , and e isfréchet differentiable if it is smooth and the limit is attained uniformly for y ∈ se .let e be a real normed linear space of dimension ≥ 2. the modulus of smoothness of e , ρe , isdefined by: ρe(τ) := sup { ‖x + y‖+ ‖x − y‖ 2 − 1 : ‖x‖ = 1, ‖y‖ = τ } ; τ > 0. a normed linear space e is called uniformly smooth if lim τ→0 ρe(τ) τ = 0. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 5if there exist a constant c > 0 and a real number q > 1 such that ρe(τ) ≤ cτq , then e is said tobe q-uniformly smooth.a normed linear space e is said to be strictly convex if: ‖x‖ = ‖y‖ = 1, x 6= y ⇒ ∥∥∥x + y 2 ∥∥∥ < 1. the modulus of convexity of e is the function δe : (0, 2]→ [0, 1] defined by: δe(ε) := inf { 1− 1 2 ‖x + y‖ : ‖x‖ = ‖y‖ = 1, ‖x − y‖ ≥ ε } . e is uniformly convex if and only if δe(ε) > 0 for every ε ∈ (0, 2]. for p > 1, e is said to be p-uniformly convex if there exists a constant c > 0 such that δe(ε) ≥ cεp for all ε ∈ (0, 2]. observethat every p-uniformly convex space is uniformly convex.typical examples of such spaces are the lp , `p and wm p spaces for 1 < p <∞ where, lp (or lp) or wm p is { 2− uniformly smooth and p − uniformly convex if 2 ≤ p <∞; 2− uniformly convex and p − uniformly smooth if 1 < p < 2. remark 1. note also that duality mapping exists in each banach space.we recall from [11] someof the examples of this mapping in `p, lp,wm,p−spaces, 1 < p <∞ • `p : jx = ‖x‖2−p`p y ∈ `q, x = (x1, x2, ..., xn, ...), y = (x1|x1|p−2, x2|x2|p−2, ..., xn|xn|p−2, ...) • lp : ju = ‖u‖2−plp |u|p−2u ∈ lq • wm,p : ju = ‖u‖2−pwm,p ∑ |α≤m| (−1)|α|dα(|dαu|p−2dαu) ∈ w−m,p in lp, `p and wm,p spaces for 1 < p <∞ are q−uniformly smooth real banach spaces with q, as q = min{2, p} and dq ≥ 1 (2.2) is given by dq = { 1+τq−1 (1+τ)q−1 , i f 1 < p < 2; p − 1, i f 2 ≤ p <∞. (2.3) and τ(0, 1) as the unique solution of the equation (q − 2)tq−1 + (q − 1)tq−2 − 1 = 0 it is well known that • e is smooth if and only if j is single-valued. • if e is uniformly smooth then j is uniformly continuous on bounded subsets of e. • if e is reflexive and strictly convex dual then j−1 is single-valued, one-to-one, surjective,uniformly continuous on bounded subsets and it is the duality mapping from e∗ into e and j−1j = ie and jj−1 = ie . • j−1 is uniformly continuous if and only if it has a modulus of continuity. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 6 lemma 2.1 (xu [32]). . let q > 1 be a real number and e be a banach space. then the following assertion are equivalent i): e is q−uniformly smooth ii): there exists a constant dn > 0, such that for all x, y ∈ e, then the following holds ‖x + q‖q ≤ ‖x‖q + q〈y , jq(x)〉+ dq‖y‖q. (2.4) 3. main result we now prove the following result theorem 3.1. let e be a 2 uniformly smooth and convex real banach space and let a mapping a : e → e∗ be lipschitz strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1}, {y1}) ∈ e, we define the sequences {xn} and {yn} by { yn = xn − θnj−1(axn), n ≥ 1 xn+1 = yn − λnj−1(ayn), n ≥ 1 (3.1) where λn and θn are positive real number and j is the duality mapping of e. letting (λn, θn) ∈ (0, 1) , then {xn} and {yn} converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. letting ρ∗ = x∗ ∈ e be the unique solution of ax = 0. from inequality 2.4 in lemma 2.1with 3.1, knowingly that ‖j−1w‖ = ‖w‖ for all w ∈ e∗, then we have the following estimates: ‖xn+1 − ρ∗‖2 = ‖yn − ρ∗ − λnj−1(ayn)‖2 = ‖λnj−1(ayn)‖2 − 2〈yn − ρ∗, j(λnj −1(ayn))〉+ d2‖yn − ρ∗‖2 ≤ λ2n‖(ayn)‖2 − 2λn〈yn − ρ∗, ayn)〉+ d2‖yn − ρ∗‖2 ≤ λ2nl 2‖yn − ρ∗‖2 − 2λnk‖yn − ρ∗‖2 + d2‖yn − ρ∗‖2 = ( λ2nl 2 − 2kλn + d2 ) ‖yn − ρ∗‖2 (3.2) for the fact that 0 < ( λ2nl 2 − 2kλn + d2 ) < 1, we have the following ‖xn+1 − ρ∗‖2 ≤ δ(λ1)‖yn − ρ∗‖2 (3.3) where δ(λ1) = ( λ2nl 2 − 2kλn + d2 ).using 3.1 , lipschitz property of a, with the same computational we have the following: https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 7 ‖yn − ρ∗‖2 = ‖xn − ρ∗ − θnj−1(axn)‖2 = ‖θnj−1(axn)‖2 − 2〈xn − ρ∗, j(θnj −1(axn))〉+ d2‖xn − ρ∗‖2 ≤ θ2n‖(axn)‖2 − 2θn〈xn − ρ∗, axn)〉+ d2‖xn − ρ∗‖2 ≤ θ2nl 2‖xn − ρ∗‖2 − 2θnk‖xn − ρ∗‖2 + d2‖xn − ρ∗‖2 = ( θ2nl 2 − 2kθn + d2 ) ‖xn − ρ∗‖2 (3.4) again, with the fact that 0 < ( θ2nl 2 − 2kθn + d2 ) < 1, we have the following ‖yn − ρ∗‖2 ≤ δ(λ2)‖xn − ρ∗‖2 (3.5) where δ(λ2) = ( θ2nl 2 − 2kθn + d2 ) putting 3.5 in 3.3, we have the following ‖xn+1 − ρ∗‖2 ≤ δ(λ1)δ(λ2)‖xn − ρ∗‖2 (3.6) ‖xn+1 − ρ∗‖ ≤ √ δ(λ1)δ(λ2)‖xn − ρ∗‖ (3.7) ‖xn+1 − ρ∗‖ ≤ µ‖xn − ρ∗‖where µ = √ δ(λ1)δ(λ2).therefore the sequences {xn} and {yn} converges strongly to ρ∗. this complete the proof. � corollary 3.1. let e = lp, 2 ≤ p <∞, and a : e → e∗ be a lipschitz strongly monotone mapping such that a−1(0) 6= ∅. for arbitrary (x1, y1) ∈ e, define the sequence {xn} and {yn} iteratively by{ yn = xn − θnj−1(axn), n ≥ 1 xn+1 = yn − λnj−1(ayn), n ≥ 1 (3.8) where λn and θn are positive real number and j is the duality mapping of e. letting (λn, θn) ∈ (0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. since e = lp spaces, 2 ≤ p <∞, are 2−uniformly smooth and convex real banach spaces,then the proof follows from theorem 3.1. � 4. convergence in lp, `p or wm,p, 2 ≤ p <∞ theorem 4.1. let e be a 2 uniformly smooth and convex real banach space either lp, `p or w m,p, 2 ≤ p <∞ with it dual e∗. let a mapping a : e → e∗ be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1}, {y1}) ∈ e, we define the sequences {xn} and {yn} by 3.1 converges strongly to ρ∗, a unique solution of the equation ax = 0. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 8 proof. since lp, `p or wm,p, 2 ≤ p < ∞ are 2− uniformly smooth banach spaces, then with thesame computation in 3.1, the proof follows. � corollary 4.1. let e be a banach space either lp, `p or wm,p, 2 ≤ p <∞ with it dual e∗. let a mapping a : e → e∗ be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1}, {y1}) ∈ e, we define the sequences {xn} and {yn} by 3.1 converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. since lp, `p or wm,p, 2 ≤ p < ∞ are 2− uniformly smooth banach spaces, then fromtheorem 4.1 with the same computation in 3.1, the proof follows. � 5. application to convex minimization problem now, we present a convex minimization problem for a convex function ∇ : e → r.the following results are well known. remark 2. let ∆ : e → r be a differentiable convex function and ρ∗ ∈ e, then the point ρ∗ is aminimizer of ∇ on e if and only if d∇(ρ∗) = 0. definition 5.1. a function ∇ : e → r is said to be strongly convex if there exists γ > 0 such thatthe following condition holds: ∇(βx + (1− β)y) ≤ β∇x + (1− β)∇y − γ‖x − y‖2 (5.1) for every x, y ∈ e with x 6= y and β ∈ (0, 1), lemma 5.2. let e be normed linear space and∇ : e → r a convex differentiable function. suppose that ∇ is strongly convex. then the differential map d∇ : e → e∗ is strongly monotone, i.e., there exists k > 0 such that 〈d∇x − d∇y , x − y〉 ≥ k‖x − y‖2 ∀ x, y ∈ e. (5.2) now we present the following result. theorem 5.3. let d∇ : e∗ → e be a l-lipschitz continuous and strongly monotone mapping such that d∇−1(0) 6= ∅. let e = lp, p ≥ 2 and ∇ : e → r be a differentiable, strongly convex real-valued function. for given x1, y1 ∈ e, define the sequence {xn} and {yn} as follows:{ yn = xn − θnd∇xn), n ≥ 1 xn+1 = yn − λnd∇yn), n ≥ 1 (5.3) where the sequences {λn} and {θn}, are in the interval [0, 1] . then ∇ has a unique minimizer ρ∗ ∈ e such that if ( λn, θn ) ∈ [0, 1], the sequence {xn} and {yn} converges strongly to ρ∗. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 9 proof. from remark 2 it follows that ∇ has a unique minimizer ρ∗ and is obtained by d∇(ρ∗) = 0.from lemma 5.2 and using the fact that the differential mapping d∇ : e → e∗ is lipschitz,considering the result of theorem 3.1, we can complete the proof. � 6. the proposed algorithm in lp(ω) now, from [14], the duality mapping j is known precisely in lp(ω) for 1 < p <∞ by jv = ‖v‖2−plp |v |p−2v ,∀v ∈ lp(ω) and if lp(ω) is reflexive, smooth and strictly convex real banach space, for 1 < p < ∞, then theduality mapping j is surjective, one-to-one and its inverse j−1 is given by ju = ‖‖2−ql |u|q−2u,∀u ∈ lq(ω) with 1 p + 1 q = 1now from 3.1, we defined x1, y1 ∈ lq(ω){ yn = xn − θn‖axn‖2−qlq |axn|2−qlq axn, n ≥ 1 xn+1 = yn − λn‖ayn‖2−qlq |ayn|2−qlq ayn, n ≥ 1 (6.1) conclusion in this paper, we proposed and analyzed the strong convergence theorem of two step size of thenew krasnoselskii-type algorithm introduced by sene et al. 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spaces, num. funct. anal. optim.40 (2019) 1426-1447. https://doi.org/10.1080/01630563.2019.1606825.[37] h. zegeye, n. shahzad, an algorithm for a common minimum-norm zero of a finite family of monotone mappings inbanach spaces, j. ineq. appl. 2013 (2013) 566. https://doi.org/10.1186/1029-242x-2013-566. https://doi.org/10.28924/ada/ma.3.18 https://doi.org/10.1016/0022-247x(84)90019-2 https://doi.org/10.1016/0362-546x(91)90200-k https://doi.org/10.1080/01630563.2019.1606825 https://doi.org/10.1080/01630563.2019.1606825 https://doi.org/10.1186/1029-242x-2013-566 1. introduction 2. preliminaries 3. main result 4. convergence in lp,p or wm,p, 2p < 5. application to convex minimization problem 6. the proposed algorithm in lp() conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 10doi: 10.28924/ada/ma.5.10 neumann and dirichlet problems for the cauchy–riemann and the poisson equations in the partial eclipse domain ali darya∗, nasir taghizadeh faculty of mathematics sciences, university of guilan, rasht, 19141, iran alidarya@phd.guilan.ac.ir, taghizadeh@guilan.ac.ir ∗correspondence: alidarya@phd.guilan.ac.ir abstract. in this paper, we consider the neumann boundary value problem and the dirichlet boundaryvalue problem for complex partial differential equations in the partial eclipse domain. first, by theparqueting–reflection principle and the cauchy–pompeiu formula, a modified integral representationformula in the partial eclipse domain is constructed. then, we explicitly solve the neumann problemfor the homogeneous equation and discuss the solvability conditions. moreover, we investigate thedirichlet problem for the poisson equation in the partial eclipse domain. in other words, with thehelp of the green’s function, we provide a unique solution for the dirichlet boundary value problemfor the poisson equation and consider boundary behavior. 1. introduction mathematical analysis is an active branch in mathematics which has grown significantly. it hasindeed flourished, playing a pivotal role in advancement of both pure and applied mathematics.its growth can be seen in the development of new techniques for solving differential equations,advancements in complex analysis, and profound contributions to functional analysis. this expan-sion has not only deepened our theoretical understanding but has also paved the way for practicalapplications in fields like mathematical physics, fluid dynamics, engineering, etc [1, 3, 6, 9].the theory of boundary value problems for partial differential equations is a key area in math-ematical analysis and mathematical physics. the theory often focuses on conditions under whichsolutions exist and are unique. these conditions can depend on the properties of the differential op-erator, the domain, and the boundary conditions. boundary value problems are critical in the studyof partial differential equations, as they involve finding a solution to a partial differential equationthat satisfies certain conditions at the boundaries of the domain. the most common boundary valueproblems are the dirichlet, the neumann and the schwarz problems. in particular, the dirichlet received: 2 dec 2024. key words and phrases. boundary value problem; neumann problem; dirichlet problem; partial eclipse domain.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 2problem specifies the value of the function on the boundary and the neumann problem specifiesthe values of the derivative (normal to the boundary).several analytical and computational methods are used to solve boundary value problems, includ-ing integral representation formulas, the green’s functions, etc. integral representation formulasare crucial tools in solving complex boundary value problems, especially for analytic and harmonicfunctions. the green’s functions are used to construct solutions for complex partial equation withboundary conditions. for the dirichlet problem, the green’s function represents the influence of apoint source on the boundary and is used to build the solution, see [1–9,16].in recent years, many mathematician have studied boundary value problems for complex partialdifferential equations and numerous results have been obtained, can be referenced in [1–16]. in2024, we introduced a new domain called partial eclipse and also investigated the schwarz andthe dirichlet boundary value problem for cauchy–riemann equations in the partial eclipse domain,( ali darya and nasir taghizadeh, “schwarz and dirichlet problems for complex partial differentialequations in the eclipse domain,” j math sci ) see [1].in the present paper, we consider the neumann problem for first-order partial differential equa-tion and the dirichlet problem for second-order partial differential equation in the partial eclipsedomain. in other words, we first solve the neumann problem the homogeneous cauchy–riemannequation and discuss the solvability conditions. in the next step, with the help of the green’s func-tion, we construct a unique solution for the dirichlet problem for the poisson equation in the partialeclipse domain and investigate the dirichlet problem. in particular, we study the boundary behavior. let m be the partial eclipse domain in the complex plane c defined by [1] m = { z ∈ c : |z − ai | < √ 2a, |z + ai | > √ 2a } where c1 = { z : |z − ai | = √ 2a } and c2 = { z : |z + ai | = √ 2a } are two circles with the sameradius and the boundary of m is denoted by ∂m . due to the fact that the real number a has apositive arbitrary amount, the desired partial eclipse can be created by choosing a. so the lengthof the borders of the partial eclipse changes according to the number. the parqueting–reflection principle is a technique used to solve boundary value problems for com-plex partial differential equations, particularly for domains with complex geometries. the methodinvolves reflecting the domain across its boundaries to simplify the problem. by doing so, theboundary conditions can be transformed into simpler forms, making it easier to find solutions. byusing the reflections, one can construct solutions for complex boundary value problems such as theschwarz, the dirichlet and neumann problems. this is particularly useful for domains composedof circular arcs and straight lines [1–4,6, 8–10]. https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 3in this section, using the parqueting–reflection method for the introduced domain m, we achievecoverage for the entire complex plane c. this coverage is obtained from reflections with threerepetitions.the reflection of z ∈ m at c1, is |z − ai | = √ 2a⇒ (z − ai) (z̄ + ai) = 2a2 ⇒ z∗1 = ai z̄ − a2 + 2a2 z̄ + ai ⇒ z∗1 = ai z̄ + a2 z̄ + ai . similarly, the reflections of z∗1 and z at c2, are the points, z∗2 = a2 z , z∗3 = −ai z̄ + a2 z̄ − ai . those reflections produce a parqueting of the entire complex plane and those points will alsobe needed for constructing the integral representation formula for m . to solve boundary valueproblems for partial differential equations, the integral formula should be appropriately modifiedaccording to the type of problem and partial differential equation, [1–4,7, 8, 10].the neumann boundary value problem is a classical in the field of complex analysis and partialdifferential equations. it involves finding a function that satisfies complex partial differential equa-tions within a given domain and whose normal derivative on the boundary of the domain matchesa specified function. the neumann boundary value problem is significant, because it is a gatewayto understanding the deeper intricacies of complex functions and broader field of complex analysis.the insights gained from solving these problems can lead to advancement in both mathematicaltheory and practical applications, see [2, 6, 7, 9, 11].in this section, we investigate the neumann boundary value problem for the homogeneouscauchy– riemann equation. the fundamental tool for complex boundary value problems is theintegral representation formula which just has to be properly modified. theorem 1.1. any ω ∈ c1(m;c) ⋂ c(m;c) can be represented as ω(z) = 1 2πi ∫ ∂m ω(ζ) [ 1 ζ − z + z ζz − a2 ] dζ − 1 π ∫ m ωζ̄(ζ) [ 1 ζ − z + z ζz − a2 ] dξdη, (1.1) where ζ = ξ + iη. proof. the cauchy–pompieu formula 1 2πi ∫ ∂m ω(ζ) dζ ζ − z − 1 π ∫ m ωζ̄(ζ) dξdη ζ − z = { ω(z) z ∈ m, 0 z /∈ m, applied to z ∈ m and z∗2 /∈ m, respectively, gives the following equalities: ω(z) = 1 2πi ∫ ∂m ω(ζ) dζ ζ − z − 1 π ∫ m ωζ̄(ζ) dξdη ζ − z , (1.2) https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 4 0 = 1 2πi ∫ ∂m ω(ζ) zdζ ζz − a2 − 1 π ∫ m ωζ̄(ζ) zdξdη ζz − a2 , (1.3) adding the resulting above relations, leads to claimed the integral representation formula. � next, we state the neumann boundary value problem for cauchy–riemann equation in the par-tial eclipse domain as follows. neumann boundary value problem: find an analytic function in the partial eclipse domain, i.e. asolution to cauchy–riemann equation, satisfying, ∂vzω = γ, on ∂m, γ ∈ c(∂m;c). the classical neumann problem involves finding a function that satisfies cauchy–riemann equa-tion in a domain, along with prescribed values of its normal derivative on the boundary. when ad-justed for analytic functions, the neumann condition is often reformulated to align with the conceptof analyticity, ensuring the function meets the criteria for complex differentiability [2,6,7,9,11,16].to formulate the neumann boundary value problem, we need to define the outward normalderivative at the boundary of m. the normal derivative on the boundary of is given by the formulas, ∂vzω =  ( z−ai√ 2a )ωz , on c1, ( z+ai√ 2a )ωz , on c2. now, according to the above definition, we state and prove the following theorem. theorem 1.2. the neumann boundary value problem ωz̄ = 0, z ∈ m, ∂vzω = γ, z ∈ ∂m, (1.4) ω(t) = c, where t ∈ m, c ∈ c and ∂vzω =  ( z−ai√ 2a )ωz , on c1, ( z+ai√ 2a )ωz , on c2, is solvable, if and only if, for z ∈ m, 1 2πi ∫ ∂m γ(ζ) [ z̄ − ai ζ(z̄ − ai) + ai z̄ − a2 + z̄ + ai ζ(z̄ + ai)− ai z̄ − a2 ] dζ = 0, and its solution is ω(z) = 1 2πi ∫ ∂m γ(ζ) [ z − t ζ − log( ζ − z ζ − t ) + a2 ζ2 log( ζz − a2 ζt − a2 ) ] dζ, where ζ = ξ + iη. https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 5 proof. suppose ω is a solution to the neumann problem. introducing a new function φ = ωz , since φ is an analytic function, φ = ωz is a solution to the following problem, φz̄ = 0, in m, φ = ωz , on ∂m, (1.5)where ωz on ∂m is represented by, ωz(z) =  ( z̄−ai√ 2a )γ, on c1, ( z̄+ai√ 2a )γ, on c2.equation (5) is equivalent to the dirichlet boundary value problem for the homogeneous cauchy–riemann equation. by theorem 3.2 in [1], the above dirichlet problem is solvable if and only if 1 2πi ∫ ∂m γ(ζ) [ z̄ − ai ζ(z̄ − ai) + ai z̄ − a2 + z̄ + ai ζ(z̄ + ai)− ai z̄ − a2 ] dζ = 0, then, the unique solution is given by, ωz(z) = 1 2πi ∫ ∂m γ(ζ) [ 1 ζ − z + z ζz − a2 ] dζ. (1.6) the primitive of the function in (6) is ω(z) = 1 2πi ∫ ∂m γ(ζ) [ z ζ − log(ζ − z) + a2 ζ2 log(ζz − a2) ] dζ + c. define c as c = − 1 2πi ∫ ∂m γ(ζ) [ t ζ − log(ζ − t) + a2 ζ2 log(ζt − a2) ] dζ. this completes the proof. � 2. the dirichlet problem for m in this section, we consider the dirichlet problem for the poisson equation in the partial eclipsedomain. in order to treat the dirichlet boundary value problem for second order complex partialdifferential equations some special kernel functions, the green functions, have to be constructed.it is essential to construct the green’s functions tailored to the specific domain. these green’sfunctions serve as fundamental tools, transforming the differential equation into an integral formthat can be more easily analyzed and solved, see [5–9]. the harmonic green function for the partialeclipse domain m is g1(z, ζ) = log ∣∣∣∣ ζ̄(z + ai)− aiz − a2 ζ − z ζ̄(z − aiz) + aiz − a2 ζz − a2 ∣∣∣∣2 . https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 6the outward normal derivative of the boundary ∂m is given by for z ∈ ∂m ∩ c1, that is, |z − ai | = √ 2a, we have ∂vzg1(z, ζ) = ( ( z − ai√ 2a )∂z + ( z̄ + ai√ 2a )∂z̄ ) g1(z, ζ), and for z ∈ ∂m ∩ c2, that is, |z + ai | = √ 2a, we have ∂vzg1(z, ζ) = ( ( z + ai√ 2a )∂z + ( z̄ − ai√ 2a )∂z̄ ) g1(z, ζ). the harmonic green functions play an essential role in solving the dirichlet boundary valueproblem for second order complex partial differential equations. the next theorem contains arepresentation formula for a class of functions via the green function, which is used to solve thedirichlet problem for poisson equation (see [5, 7, 9]). theorem 2.1. let ω ⊂ c be a regular domain, and let g1 be the harmonic green function for ω then any ω ∈ c2(ω;c) ∩ c1(ω;c) can be represented as follows: ω(z) = − 1 4π ∫ ∂ω ω(ζ)∂vζg1(z, ζ)dtζ − 1 π ∫ ω ωζζ̄(ζ)g1(z, ζ)dξdη, where v is the outward normal derivative on ∂ω and t is the arc length parameter [5, 7, 9]. therefore, based on theorem 3, the explicit form of the green representation formula for thepartial eclipse domain is as following: ω(z) = 1 2πi ∫ ∂m ⋂ c1 ω(ζ) ( ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 + z(ζ − ai) ζz − a2 + z̄(ζ̄ + ai) ζ̄z̄ − a2 − 1 ) dζ ζ − ai + 1 2πi ∫ ∂m ⋂ c2 ω(ζ) ( ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 + z(ζ + ai) ζz − a2 + z̄(ζ̄ − ai) ζ̄z̄ − a2 − 1 ) dζ ζ + ai − 1 π ∫ m ωζζ̄(ζ)g1(z, ζ)dξdη. (2.1) in fact formula (7) provides a solution to the dirichlet problem for the poisson equation in m . theorem 2.2. the dirichlet problem for the poisson equation in m ωzz̄ = f , z ∈ m, f ∈ c(m;c), ω = γ, on ∂m, γ ∈ c(∂m;c), (2.2) https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 7 is uniquely solvable and the solution is given by ω(z) = 1 2πi ∫ ∂m ⋂ c1 γ(ζ) ( ζ − ai ζ − z + ζ̄ + ai ζ̄ − z̄ − 1 + z(ζ − ai) ζz − a2 + z̄(ζ̄ + ai) ζ̄z̄ − a2 − 1 ) dζ ζ − ai + 1 2πi ∫ ∂m ⋂ c2 γ(ζ) ( ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 + z(ζ + ai) ζz − a2 + z̄(ζ̄ − ai) ζ̄z̄ − a2 − 1 ) dζ ζ + ai − 1 π ∫ m f (ζ)g1(z, ζ)dξdη. (2.3) where ζ = ξ + iη. proof. by the properties of the green function and the harmonicity of the boundary integrals ωis seen to be a solution to the poisson equation (see [7]). so, it remain remains to check theboundary relation. the study of integral boundary behavior requires calculations in different partsof the boundary. since, for z ∈ ∂m⋂c1, case1: ζ ∈ c1, z̄(ζ̄ + ai) ζ̄z̄ − a2 = aiz−a2 ζz − a2 . case2: ζ ∈ c2, ζ̄ − ai ζ̄ − z̄ = −aiz − a2 ζz − a2 , z̄(ζ̄ − ai) ζ̄z̄ − a2 = −z − ai ζ − z . thus, on ∂m⋂c1, lim z→ζ ω(z) = lim z→ζ 1 2πi ∫ ∂m ⋂ c1 γ(ζ) [ ζ − ai ζ − z + ζ̄ + ai ζ̄ − z̄ − 1 ] dζ ζ − ai = lim z→ζ 1 2πi ∫ c1 υ(ζ) [ ζ − ai ζ − z + ζ̄ + ai ζ̄ − z̄ − 1 ] dζ ζ − ai ,where υ(ζ) = { γ(ζ) ζ ∈ ∂m ⋂ c1, 0 ζ ∈ c1\(∂m).so based on the properties of the poisson kernel for c1, [6] lim z→ζ ω(z) = γ(ζ), https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 8follows for ζ ∈ ∂m⋂c1 up to the corner points ±a of the domain m, because υ fails to be con-tinuous there if γ not accidentally vanishes at these points.by the same way, for z ∈ ∂m⋂c2, case 1: ζ ∈c1, ζ̄ + ai ζ̄ − z̄ = aiz − a2 ζz − a2 , z̄(ζ̄ + ai) ζ̄z̄ − a2 = −z + ai ζ − z . case 2: ζ ∈ c2, z̄(ζ̄ + a) ζ̄z̄ − a2 = −aiz − a2 ζz − a2 . thus, on ∂m⋂c2, lim z→ζ ω(z) = lim z→ζ 1 2πi ∫ ∂m ⋂ c2 γ(ζ) [ ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 ] dζ ζ + ai = lim z→ζ 1 2πi ∫ c2 υ(ζ) [ ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 ] dζ ζ + ai , where υ(ζ) = { γ(ζ) ζ ∈ ∂m ⋂ c2, 0 ζ ∈ c2\(∂m).so based on the properties of the poisson kernel for c2, lim z→ζ ω(z) = γ(ζ), follows for ζ ∈ ∂m⋂c2 up to the corner points ±a of the domain m, because υ fails to be con-tinuous there if γ not accidentally vanishes at these points. now, we consider the boundary behaviors at the tips ±a. we represent the constant function1 as 1 = 1 2πi ∫ ∂m ⋂ c1 [ ζ − ai ζ − z + ζ̄ + ai ζ̄ − z̄ − 1 + z(ζ − ai) ζz − a2 + z̄(ζ̄ + ai) ζ̄z̄ − a2 − 1 ] dζ ζ − ai + 1 2πi ∫ ∂m ⋂ c2 [ ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 + z(ζ + ai) ζz − a2 + z̄(ζ̄ − ai) ζ̄z̄ − a2 − 1 ] dζ ζ + ai . multiplying this relation with γ(±a) and subtracting the resulting equation from ω(z) shows for z ∈ ∂m ⋂ c1, lim z→ζ (ω(z)− γ(±a)) = lim z→ζ 1 2πi ∫ ∂m ⋂ c1 γ̃(ζ) [ ζ − ai ζ − z + ζ̄ + ai ζ̄ − z̄ − 1 ] dζ ζ − ai , https://doi.org/10.28924/ada/ma.5.10 eur. j. math. anal. 10.28924/ada/ma.5.10 9 where γ̃(ζ) = γ(ζ)− γ(±a) and γ̃(±a) = 0, lim z→±a ω(z) = γ(±a). similarly, for z ∈ ∂m⋂c2, lim z→ζ (ω(z)− γ(±a)) = lim z→ζ 1 2πi ∫ ∂m ⋂ c2 γ̂(ζ) [ ζ + ai ζ − z + ζ̄ − ai ζ̄ − z̄ − 1 ] dζ ζ + ai , where γ̂(ζ) = γ(ζ)− γ(±a) and γ̂(±a) = 0,therefore, lim z→±a ω(z) = γ(±a). therefore, the proof is finished. � references [1] a. darya, n. taghizadeh, schwarz and dirichlet problems for complex partial differential equations in the partialeclipse domain, j. math. sci. 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inverse problems for some classes of nonclassical operator-differential equa-tions, sib. math. j. 62 (2021), 489–502. https://doi.org/10.1134/s0037446621030125.[14] k.b. sabitov, dirichlet problems for mixed-type equations with fractional derivatives, russ. math. 66 (2022),71–81. https://doi.org/10.3103/s1066369x22090080. https://doi.org/10.28924/ada/ma.5.10 https://doi.org/10.1007/s10958-024-07337-0 https://doi.org/10.1134/s0965542524700520 https://doi.org/10.28924/ada/ma.4.15 https://doi.org/10.3103/s1066369x24700853 https://doi.org/10.15393/j3.art.2025.16810 https://doi.org/10.7169/facm/1246454030 https://doi.org/10.1080/17476933.2013.799152 https://doi.org/10.1007/s41478-019-00202-3 https://doi.org/10.15393/j3.art.2019.5570 https://doi.org/10.1134/s0037446621030125 https://doi.org/10.3103/s1066369x22090080 eur. j. math. anal. 10.28924/ada/ma.5.10 10 [15] k. ravikumar, k. ramkumar, d. chalishajar, existence and stability results for second-order neutral stochasticdifferential equations with random impulses and poisson jumps, eur. j. math. anal. 1 (2021), 1–18. https: //doi.org/10.28924/ada/ma.1.1.[16] i.n. vekua, generalized analytic functions, pergamon press, oxford, 1962. https://doi.org/10.28924/ada/ma.5.10 https://doi.org/10.28924/ada/ma.1.1 https://doi.org/10.28924/ada/ma.1.1 1. introduction 2. the dirichlet problem for m references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 16doi: 10.28924/ada/ma.3.16 best proximity points for generalized geraghty quasi-contraction type mappings in metric spaces j. c. umudu1,∗, j. o. olaleru2, h. olaoluwa2, a. a. mogbademu2 1department of mathematics, faculty of natural sciences, university of jos, nigeria umuduj@unijos.edu.ng 2department of mathematics, faculty of science, university of lagos, nigeria jolaleru@unilag.edu.ng, holaoluwa@unilag.edu.ng, amogbademu@unilag.edu.ng ∗correspondence: umuduj@unijos.edu.ng abstract. in this paper, we introduce a new concept of α-φ-geraghty proximal quasi-contractiontype mappings and establish best proximity point theorems for those mappings in proximal t -orbitallycomplete metric spaces. this generalizes and complements the proofs of some known fixed and bestproximity point results. 1. introduction let a and b be two nonempty subsets of a metric space (x, d). a best proximity point of anon-self mapping t : a → b, is the point x ∈ a, satisfying d(x, t x) = d(a,b). numerousresults on best proximity point theory were studied by several authors ( [1], [3], [4], [5]) imposingsufficient conditions that would assure the existence and uniqueness of such points. these resultsare generalizations of the contraction principle and other contractive mappings ( [2], [6], [8], [16],[21], [22], [24]) in the case of self-mappings, which reduces to a fixed point if the mapping underconsideration is a self-mapping. the notion of best proximity point was introduced in [14], the classof proximal quasi contraction mappings was introduced in [11] and thereafter, several known resultswere derived ( [10], [12], [13]). best proximity pair theorems analyse the conditions under which theoptimization problem, namely minx∈a d(x, t x) has a solution and is known to have applicationsin game theory. for additional information on best proximity point, see [7], [9], [10], [11], [12], [13],[14], [15], [17], [18], [20], [23]. definition 1.1 [4]. let t : x → x be a map on metric space. for each x ∈ x and for any positiveinteger n, ot (x, n) = {x, t x, ..., t nx} and ot (x,∞) = {x, t x, ..., t nx, ...}. received: 8 feb 2023. key words and phrases. best proximity; quasi-contraction; metric space.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 2the set ot (x,∞) is called the orbit of t at x and the metric space x is called t -orbitally completeif every cauchy sequence in ot (x,∞) is convergent in x. quasi contraction mapping is known in literature as one of the most generalized contractive map-pings and is defined as follows. definition 1.2 [6]. a mapping t : x → x of a metric space x into itself is said to be a quasi-contraction if and only if there exists a number k, 0 ≤ k < 1, such that d(tx, t y) ≤ k max{d(x, y); d(x, t x); d(y , t y); d(x, t y); d(y , t x)} holds for every x, y ∈ x. consider the class f of functions β : [0,∞)→ [0, 1) satisfying the condition: lim n→∞ β(tn) = 1 implies lim n→∞ tn = 0. recently, using these class of functions, umudu et al. [22] introduced a new class of quasi-contraction type mappings called generalized α-φ-geraghty quasi-contraction type mappings andproved the existence of its unique fixed point as follows. definition 1.3 [22]. let (x, d) be a metric space and α : x × x → r+. a mapping t : x → x iscalled a generalized α-geraghty quasi-contraction type mapping if there exists β ∈ f such thatfor all x, y ∈ x, α(x, y)(d(tx, t y)) ≤ β(mt (x, y))(mt (x, y)), (1) where mt (x, y) = max{d(x, y), d(x, t x), d(y , t y), d(x, t y), d(y , t x)}. let φ denote the class of the functions φ : [0,∞)→ [0,∞) which satisfies the following conditions:(i) φ is nondecreasing;(ii) φ is continuous;(iii) φ(t) = 0 ⇐⇒ t = 0. definition 1.4 [22]. let (x, d) be a metric space and α : x×x → r+. a self mapping t : x → xis called a generalized α-φ-geraghty quasi-contraction type mapping if there exists β ∈ f suchthat for all x, y ∈ x, α(x, y)φ(d(tx, t y)) ≤ β(φ(mt (x, y)))φ(mt (x, y)), (2) where mt (x, y) = max{d(x, y), d(x, t x), d(y , t y), d(x, t y), d(y , t x)}, and φ ∈ φ. if φ(t) = t, inequality (2) reduces to inequality (1). the generalized α-φ-geraghty quasi-contraction type self mapping is a generalization of other quasi-contraction type self mappingsin literature. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 3the following mappings introduced by popescu [19] and used by umudu et al. [22] to establish theexistence of a fixed point will also be needed in this paper. definition 1.5 [19]. let t : x → x be a self-mapping and α : x × x → r+ be a function. then t is said to be α-orbital admissible if α(x, t x) ≥ 1 implies α(tx, t 2x) ≥ 1. definition 1.6 [19]. let t : x → x be a self-mapping and α : x × x → r+ be a function.then t is said to be triangular α-orbital admissible if t is α-orbital admissible, α(x, y) ≥ 1 and α(y , t y) ≥ 1 imply α(x, t y) ≥ 1. the main result obtained in [22] is the following. theorem 1.7. let (x, d) be a t orbitally complete metric space, α : x × x → r+ be a function,and let t : x → x be a self-mapping. suppose that the following conditions are satisfied: (i) t is a generalized α-φ-geraghty quasi-contraction type mapping;(ii) t is triangular α-orbital admissible mapping;(iii) there exists x1 ∈ x such that α(x1, t x1) ≥ 1; then t has a fixed point x∗ ∈ x and {t nx1} converges to x∗. in this paper, we extend the concept of generalized α-φ-geraghty quasi-contraction typemapping to generalized α-φ-geraghty proximal quasi-contraction type mapping in the case ofnon-self mappings. more precisely, we study the existence and uniqueness of best proximitypoints for generalized α-φ-geraghty proximal quasi-contraction for non-self mappings. 2. preliminaries we start this section with the following definitions.let a and b be non-empty subsets of a metric space (x, d). we denote by a0 and b0 the followingsets: d(a,b) = inf{d(a, b) : a ∈ a, b ∈ b}. a0 = {x ∈ a : d(x, y) = d(a,b) for some y ∈ b}. b0 = {y ∈ b : d(x, y) = d(a,b) for some x ∈ a}. definition 2.1 [14]. an element x ∈ a is said to be a best proximity point of the non-self-mapping t : a→ b if it satisfies the condition that d(x, t x) = d(a,b).we denote the set of all best proximity points of t by pt (a), that is, pt (a) := {x ∈ a : d(x, t x) = d(a,b)}. the following were introduced by [11]. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 4 definition 2.2 [11]. a non-self mapping t : a → b is said to be a proximal quasi-contraction ifand only if there exists a number q, 0 ≤ q < 1, such that{ d(u, t x) = d(a,b) d(v , t y) = d(a,b) =⇒ d(u, v) ≤ qmax{d(x, y); d(x, u); d(y , v); d(x, v); d(y , u)}, where x, y , u, v ∈ a. if t is a self mapping on a, then definition 2.2 reduces to definition 1.2. lemma 2.3 [11]. let t : a → b be a non-self mapping. suppose that the following conditionshold:(i) a0 6= ∅;(ii) t (a0) ⊆ b0.then, for all a ∈ a0, there exists a sequence {xn} ⊂ a0 such that{ x0 = a, d(xn+1, t xn) = d(a,b), ∀n ∈ n.any sequence {xn} ⊂ a0 satisfying the equation in lemma 2.3 is called a proximal picardsequence associated to a ∈ a0 and we denote by pp(a) the set of all proximal picard sequencesassociated to a. suppose a ∈ a0 and {xn} ∈ pp(a). for all (i , j) ∈ n2, the following sets are definedby: ot (xi , j) := {xl : i ≤ l ≤ j + i} and ot (xi ,∞) := {xl : l ≥ i}. definition 2.4 [11] a0 is said to be proximal t -orbitally complete if and only if every cauchysequence {xn} ∈ pp(a) for some a ∈ a0, converges to an element in a0.if t is a self mapping on a, then the preceding definition reduces to the condition that a is t -orbitally complete. the concepts of α-orbital proximal admissible mapping and triangular α-orbital proximaladmissible mapping are hereby introduced as follows. definition 2.5 let t : a → b be a non-self mapping and α : a× a → [0,∞) be a function. themapping t is said to be α-orbital proximal admissible if  α(x, u) ≥ 1 d(u, t x) = d(a,b) d(v , tu) = d(a,b) =⇒ α(u, v) ≥ 1, for all x, u, v ∈ a. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 5 definition 2.6 let t : a → b be a non-self mapping and α : a × a → [0,∞) be a function.the mapping t is said to be triangular α-orbital proximal admissible if it is α-orbital proximaladmissible and  α(x, y) ≥ 1 α(y , u) ≥ 1 d(u, t y) = d(a,b) =⇒ α(x, u) ≥ 1, for all x, y , u ∈ a. remark 2.7. if t is a self mapping, that is, if a = b, α-orbital proximal admissible mappingreduces to α-orbital admissible mapping while triangular α-orbital proximal admissible mappingreduces to triangular α-orbital admissible mapping defined in [19] . example 2.8. let x be the euclidean plane r2 and consider the two subsets: a = {(0, 0), (0, 1), (0, 2), (0, 3)} b = {(1, 0), (2, 1), (2, 2), (1, 3)}define a mapping t : a → b such that t (0, 0) = (1, 0), t (0, 1) = (2, 2), t (0, 2) = (2, 1) and t (0, 3) = (1, 3).also define a mapping α : a× a→ [0,∞) such that α(x, y) =  1, if x = y ∈ {(0, 0), (0, 3)} 0 elsewhere.for all x, y ∈ a. one can see that d(a,b) = 1. let u, v , x ∈ a. one can check that α(x, u) ≥ 1 d(u, t x) = 1 d(v , tu) = 1 =⇒ x = u = v ∈ {(0, 0), (0, 3)} =⇒ α(u, v) = 1. hence, t is α-orbital proximal admissible. let u, x, y ∈ a. one can check that  α(x, u) ≥ 1 α(y , u) ≥ 1 d(u, t y) = 1 =⇒ x = y = u ∈ {(0, 0), (0, 3)} =⇒ α(x, u) = 1. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 6thus, t is also triangular α-orbital proximal admissible. we introduce the following new classes of non-self mappings. definition 2.9 let a and b be two nonempty subsets of a metric space (x, d) and α : a×a→ r+be a function. a non-self mapping t : a → b is called a generalized α-φ-geraghty proximalquasi-contraction type mapping if there exists β ∈ f such that for all x, y , u, v ∈ a,{ d(u, t x) = d(a,b) d(v , t y) = d(a,b) =⇒ α(x, y)φ(d(u, v)) ≤ β(φ(mt (x, y)))φ(mt (x, y)), (3) where mt (x, y) = max{d(x, y), d(x, u), d(y , v), d(x, v), d(y , u)}, for all x, y , u, v ∈ a and φ ∈ φ. if φ(t) = t, then definition 2.9 reduces to the following. definition 2.10 let a and b be two nonempty subsets of a metric space (x, d) and α : a×a→ r+be a function. a non-self mapping t : a→ b is called an α-geraghty proximal quasi-contractiontype mapping if there exists β ∈ f such that for all x, y , u, v ∈ a,{ d(u, t x) = d(a,b) d(v , t y) = d(a,b) =⇒ α(x, y)d(u, v) ≤ β(mt (x, y))(mt (x, y)), (4) for all x, y , u, v ∈ a. where mt (x, y) = max{d(x, y), d(x, u), d(y , v), d(x, v), d(y , u)} for all x, y , u, v ∈ a. 3. main results now we state and prove our main results. theorem 3.1. let a and b be two nonempty subsets of a metric space such that a0 isproximal t -orbitally complete, where t : a → b is a non-self mapping, α : a × a → r+ is afunction and the following conditions are satisfied:(i) t is a generalized α-φ-geraghty proximal quasi-contraction type mapping;(ii) t (a0) ⊆ b0 and t is a triangular α-orbital proximal admissible mapping;(iii) there exists x0, x1 ∈ a0 such that d(x1, t x0) = d(a,b) and α(x0, x1) ≥ 1.then there exists an element x∗ ∈ a0 such that d(x∗, t x∗) = d(a,b). moreover, if α(x, y) ≥ 1 for all x, y ∈ pt (a), then x∗ is the unique best proximity point of t . proof.let x0, x1 ∈ a0 be such that d(x1, t x0) = d(a,b) and α(x0, x1) ≥ 1. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 7 t (a0) ⊆ b0 and there exists x2 ∈ a0 such that d(x2, t x1) = d(a,b). now, we have α(x0, x1) ≥ 1 d(x1, t x0) = d(a,b), d(x2, t x1) = d(a,b). since t is α-orbital proximal admissible, α(x1, x2) ≥ 1. thus, we have d(x2, t x1) = d(a,b) and α(x1, x2) ≥ 1. by induction, we can construct a sequence {xi} ⊆ a0 such that d(xi+1, t xi) = d(a,b) and α(xi , xi+1) ≥ 1, f or al l i ∈ n. (5) for all i ≥ 0  α(xi , xi+1) ≥ 1 α(xi+1, xi+2) ≥ 1 d(xi+2, t xi−1) = d(a,b), =⇒ α(xi , xi+2) ≥ 1, since t is triangular α-orbital proximal admissible. thus by induction, α(xi , xj) ≥ 1 for all i , jsuch that 0 ≤ i < j .therefore for any i ∈ n, we have α(xi−1, xj−1) ≥ 1 d(xi , t xi−1) = d(a,b), d(xj , t xj−1) = d(a,b) for all i , j such that 1 ≤ i < j .clearly, if xi+1 = xi for some i ∈ n from inequality (5), xi will be a best proximity point, sohenceforth, in this proof, we assume d(xi , xi+1) > 0, ∀ i ∈ n. from inequality (3), we have φ(d(xi , xj)) ≤ α(xi−1, xj−1)φ(d(xi , xj)) ≤ β(φ(mt (xi−1, xj−1)))φ(mt (xi−1, xj−1)) (6) 1 ≤ i < j where φ(mt (xi−1, xj−1)) ≤ φ(max{d(xi−1, xj−1), d(xi−1, xi), d(xj−1, xj), d(xi−1, xj), d(xj−1, xi)}) ≤ φ(δ[ot (xi−1, n)]), f or i ≤ j ≤ n + i . https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 8note that the case φ(mt (xi−1, xj−1)) = φ(d(xi , xj)) is impossible. indeed, by inequality (6), φ(d(xi , xj)) ≤ β(φ(mt (xi−1, xj−1)))φ(mt (xi−1, xj−1)) ≤ β(φ(d(xi , xj)))φ(d(xi , xj)) < φ(d(xi , xj)), is a contradiction. thus, we conclude that φ(d(xi , xj)) < φ(d(xi−1, xj−1)) for all 0 < i < j and sothe sequence {φ(d(xi , xj))} is positive and decreasing. consequently, there exists r ≥ 0 such that lim i ,j→∞ φ(d(xi , xj)) = r. we claim that r = 0. suppose, on the contrary, that r > 0. then we have φ(d(xi , xj)) φ(d(xi−1, xj−1)) ≤ β(φ(mt (xi−1, xj−1))) ≤ 1 f or each i , j ∈ n such that i < j. then, since β ∈ f , lim i ,j→∞ β(φ(mt (xi−1, xj−1))) = 1, implying that lim i ,j→∞ φ(mt (xi−1, xj−1)) = 0, (7) and so by inequality (6) lim i ,j→∞ φ(d(xi , xj)) = 0, which is a contradiction. now, by the continuity property of φ, φ ( lim i ,j→∞ (d(xi , xj)) ) = φ(0). (8) but φ(t) = 0 if and only if t = 0 and so (8) gives lim i ,j→∞ (d(xi , xj)) = 0. therefore, {xn} is a cauchy sequence in a0 and since a0 is proximal t -orbitally complete, thereexists x∗ ∈ a0 such that lim i→∞ xi = x∗. also, since t (a0) ⊆ b0, then there exists y ∈ a0 such that d(y , t x∗) = d(xi , t xi−1) = d(a,b) ∀n ∈ n, ∀i ≥ 0. t being a generalized α-φ-geraghty proximal quasi-contraction type mapping gives φ(d(y , xi)) ≤ α(x∗, xi−1)φ(d(y , xi)) ≤ β(φ(mt (x∗, xi−1)φ(mt (x∗, xi−1)) https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 9provided that α(x∗, xi−1) ≥ 1 where φ(mt (x∗, xi−1)) = φ(max{d(x∗, xi−1), d(x∗, xi), d(xi−1, xi), d(x∗, y), d(xi−1, y)}). but taking the limit, φ(d(y , x∗)) ≤ lim i→∞ β(φ(mt (x∗, xi−1)))φ(d(x∗, y)), which gives, 1 ≤ lim i→∞ β(φ(mt (x∗, xi−1))) = β(φ(d(y , x∗))) = 1 implying φ(d(y , x∗)) = 0 and d(y , x∗) = 0 i.e y = x∗. we have d(x∗, t x∗) = d(y , t x∗) = d(a,b) and x∗ ∈ a0 is a bestproximity point of t .for uniqueness, suppose the best proximity point of t is not unique. let x∗, y∗ be two bestproximity points of t with x∗ 6= y∗. then, α(x∗, y∗) ≥ 1 d(x∗, t x∗) = d(a,b) d(y∗, t y∗) = d(a,b)  since t is a generalized α-φ-geraghty proximal quasi-contraction type mapping, φ(d(x∗, y∗)) ≤ α(x∗, y∗)φ(d(x∗, y∗)) ≤ β(mt (x∗, y∗))φ(mt (x∗, y∗)) < φ(mt (x∗, y∗)) where mt (x∗, y∗) = max{d(x∗, y∗), d(x∗, x∗), d(y∗, y∗), d(x∗, y∗), d(y∗, x∗)} = d(x∗, y∗). this gives d(x∗, y∗) < d(x∗, y∗), which is a contradiction. therefore x∗ = y∗, and the bestproximity point of t is unique. corollary 3.2. let a and b be two nonempty subsets of a metric space such that a0 isproximal t -orbitally complete, where t : a → b is a non-self mapping, α : a × a → r+ is afunction and the following conditions are satisfied:(i) t is a generalized α-geraghty proximal quasi-contraction type mapping;(ii) t (a0) ⊆ b0 and t is a triangular α-orbital proximal admissible mapping;(iii) there exists x0, x1 ∈ a0 such that d(x1, t x0) = d(a,b) and α(x0, x1) ≥ 1.then there exists an element x∗ ∈ a0 such that d(x∗, t x∗) = d(a,b). moreover, if α(x, y) ≥ 1 for all x, y ∈ pt (a), then x∗ is the unique best proximity point of t . https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 104. conclusion in this paper, we introduced the notion of generalized α-φ-geraghty proximal quasi-contractiontype mappings which, for a self mapping, reduces to that in umudu et al. 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21-31.[24] j. umudu, a. mogbademu, j. olaleru, fixed point results for geraghty contractive type operators in uniform spaces,caspian j. math. sci. 11 (2022), 191-202. https://doi.org/10.22080/cjms.2021.3052. https://doi.org/10.28924/ada/ma.3.16 https://doi.org/10.1186/1687-1812-2013-180 https://doi.org/10.1155/2017/6173468 https://doi.org/10.1186/1687-1812-2014-190 https://doi.org/10.1016/j.na.2011.04.052 https://doi.org/10.1186/s13663-020-00683-z https://doi.org/10.1186/s13663-020-00683-z https://doi.org/10.22080/cjms.2021.3052 1. introduction 2. preliminaries 3. main results 4. conclusion competing interests: authors' contributions: references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 3doi: 10.28924/ada/ma.4.3 a unified kantorovich-type convergence analysis of newton-like methods for solving generalized equations under the aubin property samundra regmi1, ioannis k. argyros2,∗, santhosh george3, and jefferey warden2 1department of mathematics, university of houston, houston, tx, 77024, usa sregmi5@uh.edu 2department of computing and mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu, jefferey.warden@cameron.edu 3department of mathematical and computational sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in ∗correspondence: iargyros@cameron.edu abstract. numerous applications from diverse disciplines reduce to solving generalized equationsin a banach space setting. these equations are solved mostly iteratively, when a sequence is gen-erated approximating a solution provided that certain conditions are valid on the starting point andthe operators appearing on the method. in particular, newton-like methods are developed whosespecializations reduce to well known methods such as newton, modified newton, secant, kurchatovand steffensen to mention a few. a unified semi-local analysis of these methods is presented usingthe contraction mapping principle under the aubin property of a set valued operator, and generalizedcontinuity assumption on the operators on these methods. 1. introduction let b1 and b2 stand for complete normed spaces; d be an open and convex subset of b1;operator f : d −→ b2 be continuous and g : b1 ⇒ b2 be a set-valued operator with closedgraph, which is a nonempty set [15].we are concerned with the problem of finding a solution x∗ ∈ b1 of the generalized equationiteratively in the form: find x ∈ b1 so that f (x) + g(x) 3 0. (1.1) many applications from diverse disciplines, especially in mathematical programming can be for-mulated like the generalized equation (1.1) [1–15, 23–25, 34]. s. m. robinson inaugurated the received: 26 dec 2023. key words and phrases. generalized equation; newton-like methods; aubin property; banach space; local-semi-local convergence; newton’s method. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 2study of generalized equations in [23–25]. a solution x∗ ∈ b1 in analytical form or closed form iscomputationally hard or impossible to find. thus, researchers and practitioners generate iterativemethods approximating x∗ if certain conditions related to the starting point and the operators onthe methods are fulfilled. n, h. josephy introduced the newton method for solving the generalizedequation (1.1) in [18]. later, numerous other authors worked on various other iterative methodsunder diverse convergence conditions ( see [1–3,7–15] and references there in).all these iterative methods are useful and provide insight in the solutions of generalized equa-tions. but as far as we know there is not a unified convergence analysis for the existing iterativemethods. that is very useful, since this way under the same set of conditions the convergence andcomparison of numerous iterative methods becomes possible. this is our motivation for the presentarticle. in particular, consider the newton-like iterative method (nlm) for solving the generalizedequation in the following form f (xn) + l(xn)(xn+1 − xn) + g(xn+1) 3 0, n = 0, 1, 2, . . . , (1.2) where l(.) : b1 −→ l(b1, b2) which stands for the space of linear operators which are boundedmapping from b1 into b2. by specializing the linear operator l, many iterative methods can beobtained such as: newton’s method [4,5,21,22]: select l(x) = f ′(x), x ∈ b1 to obtain f (xn) + f ′(xn)(xn+1 − xn) + g(xn+1) 3 0, n = 0, 1, 2, . . . , where f ′ denotes the derivative according to fréchet of the operator f. modified newton’s method [4,5,20]: set l(x) = f ′(x0), x ∈ b1 to obtain f (xn) + f ′(x0)(xn+1 − xn) + g(xn+1) 3 0, n = 0, 1, 2, . . . . secant method [20]: let l(xn) = [xn−1, xn;f ], n = 0, 1, 2, . . . a divided difference of order one [20].then, iterative method (1.1) becomes f (xn) + [xn−1, xn;f ](xn+1 − xn) + g(xn+1). modified secant method [4,5]: take l(xn) = [x−1, x0;f ], x−1, x0 ∈ b−1, n = 0, 1, 2, . . . to obtain f (xn) + [x−1, x0;f ](xn+1 − xn) + g(xn+1). kurchatov method [29,30]: set l(xn) = [2xn − xn−1, xn−1;f ] to obtain f (xn) + [2xn − xn−1, xn−1;f ](xn+1 − xn) + g(xn+1). modified kurchatov method [29,30]: let l(x) = [2x0 − x−1, x−1;f ] to get f (xn) + [2x0 − x−1, x−1;f ](xn+1 − xn) + g(xn+1). https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 3 picard method [20]: pick l(x) = i, x ∈ b1 for b1 = b2 to obtain f (xn) + xn+1 − xn + g(xn+1), n = 0, 1, 2, . . . . steffensen’s method [4,5]: define l(x) = [x + f (x), x − f (x);f ] for b1 = b2 to get f (xn) + [xn + f (xn), xn − f (xn);f ](xn+1 − xn) + g(xn+1), n = 0, 1, 2, . . . . stirling’s method [20,31]: define l(x) = i − f ′(x), x ∈ b1 to obtain f (xn) + (i − f ′(xn))(xn+1 − xn) + g(xn+1), n = 0, 1, 2, . . . . traub and other multi-point and multi-step methods [4,5,17,31,32]: therefore, it is importantto develop unifying conditions for the convergence of (1.2). there are two popular convergenceapproaches in the literature. we develop, the semi-local analysis of convergence. in the localcase information is used to produce usually a ball centered at x∗, so that if one picks a pointinside of it the convergence of the iterative method is assured. note that , in the semi-localcase the convergence ball is centered at the starting point x0. the convergence conditions usuallyinvolve lipschitz [4, 5] and hölder-type conditions [20]. the new convergence analysis in bothcases involves generalized continuity conditions, majorant functions, majorizing sequences (in thesemi-local case) in combination under the aubin property of the set valued operator on the methodand the celebrated contraction mapping principle [15]. upper error estimates on ‖x∗ − xn‖ for thesolution are developed which are computable.the rest of the article contains: the mathematical background necessary to make this article asself contained as possible appears in section 2; semi-local convergence results appear is section3. the article ends with concluding remarks in section 4. 2. mathematical bachground certain standard concepts are restated in order to make the article as self-contained as possible.more detailed information can be found in [15].the graph of a set-valued operator g : b1 ⇒ b2 is gpg = {(v1, v2) ∈ b1 × b2 : v2 ∈ g(v1)} the domain dom(g) = {v ∈ b1 : g(v) 6= ∅}; the rge(g) = {v2 ∈ b2 : for some v1 ∈ b1, v2 ∈ g(v1)}.moreover, the inverse g−1 : b2 ⇒ b1 is g−1(v2) = {v1 ∈ b − 1 : v2 ∈ g(v1)}. furthermore, for sets c1 and c2 in b1, define d(v , c1) = inf v1∈c1 d(v , v1) and e(c1, c2) = sup v1∈c1 d(v , c2), https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 4where d, e are standard symbols for the distance from v to c2 and the excess of c1 to c2. recallthat e(∅, ∅) = +∞, d(v , c2) = +∞, if c2 = ∅ and e(∅, c2) = 0, if c2 6= ∅ (by convention), where ∅ is the symbol for the empty set.next, some more definitions and standard results are stated. definition 2.1. the inverse operator g−1 of g has the aubin property for v1 ∈ b1 at v2 ∈ b2 of modulus λ ≥ 0, if when v1 ∈ g−1(v2), there exist α > 0 and β > 0 so that e(g−1(v4) ∩h[v1, α], g−1(v3)) ≤ λ‖v4 − v3‖ for each v3, v4 ∈ h[v2, β] (2.1) and the inverse operator g−1 is locally closed at the pair (v2, v1), where h(v , α), h[v , α] denote open and closed balls, respectively of center v ∈ b1 with radius α > 0. it is useful to recall that there is a relationship between the aubin property and the metricregularity (see e.g. [15, theorem 3.7]). in particular, g−1 : b2 ⇒ b1 has the aubin property at v1, v2) with modulus λ > 0 if and only if g : b1 ⇒ b2 is metrically regular at (v1, v2) with the sameconstant λ. therefore, the results that follows are given equivalently in terms of metric regularity.the celebrated contraction mapping principle [15,21,22] plays a vital role in our investigations. theorem 2.2. let us consider a set-valued operator ψ : b1 ⇒ b2 and v ∈ b1. assume that there exist constants γ0 > 0 and δ0 ∈ (0, 1) so that gphψ ∩ (h[v , γ0]×h[v , δ0]) is a closed set: (i) d(v ,ψ(v)) ≤ γ0(1− δ0) and (ii) e(ψ(v1) ∩ h[v , γ0],ψ(v2)) ≤ δ0m(v1, v2) for each v1, v2 ∈ h[v , γ0], where m is some metric. then, the operator ψ admit a fixed point in the closed ball h[v , γ0]. majorizing sequences play an important role in the study of iterative methods. definition 2.3. let {sn} stand for a nonnegative sequence of numbers and let {zn} be a sequence in a banach space. assume: ‖zn+1 − zn‖ ≤ sn+1 − sn for each n = 0, 1, 2 . . . . then, the sequence {sn} is said to be majorizing for the sequence {yn}. in the case of convergence of the sequence {sn}, the sequence {zn} is cauchy in the banach space and as such it is convergent to some z∗, i.e., limn−→∞ zn = z∗. 3. convergence let t = [0,+∞). the following conditions are used in the semi-local convergence analysis ofthe nlm.assume:(a1) there exists a continuous and nondecreasing function w0 : t −→ r such that the equation w0(t)− 1 = 0 has a smallest solution ρ0 ∈ t − {0}. set t0 = [0, ρ0). https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 5(a2) there exist cnf w : t0 −→ r, and w1 : t −→ r. let λ > 0. define the sequence {sn}for s0 = 0, some s1 ∈ [0, ρ0), and each n = 0, 1, 2, ... by sn+1 = sn + [ ∫ 1 0 w((1− θ)(sn − sn−1))dθ + w0(sn−1) + w1(sn−1)](sn − sn−1) 1− w0(sn) (3.1) it is shown in theorem 3.1 that {sn} is a majorizing sequence for {xn}. but let us firstpresent a general convergence criterion for it.(a3) there exists a parameter ρ ∈ [0, ρ0) such that for each n = 0, 1, 2, ... w0(sn) < 1 and sn ≤ ρ.it follows by (3.1) and (a3) that 0 ≤ sn ≤ sn+1 ≤ ρand there exists s ∈ [0, ρ) such that limn→+∞ sn = s .the functions "w" and sequence {sn} are connected to the operators on nlm.(a4) there exists a linear operator m such that λ‖l(x)−m‖ ≤ w0(‖x − x0‖) for each x ∈ d.set d0 = d ∩ s(x0, ρ0).(a5) λ‖f ′(x)− f ′(y)‖ ≤ w(‖x − y‖) for each x, y ∈ d0 and λ‖l(x)−m‖ ≤ w1(‖x − x0‖) for each x ∈ d0.(a6) there exist x1 ∈ d generated by nlm so that ‖x1 − x0‖ ≤ s1, and the multi-operator (f (x0) +m(.− x0) +g(.))−1 is aubin continuous at (0, x1) with corresponding parameters α and β.(a7) for ρ > s1 2ρ− s1 < α, 1 λ [∫ 1 0 w0((1− θ)ρ)dθ + w0(ρ) + ∫ 1 0 w((1− θ)ρ)dθ + w1(ρ) ] ρ ≤ β, and w0(ρ) < 1.and(a8) s[x0, s] ⊂ d.next, the semi-local convergence analysis of nlm is developed using the conditions (a1)− (a8). theorem 3.1. assume that the conditions (a1) − (a8) are valid. then, the sequence {xn} generated by nlm is well defined in s(x0, s), remains in s(x0, s) for each n = 0, 1, 2, .. and is convergent to some x∗ ∈ s[x0, s] solving the generalized equation (1.1). moreover, the following error estimates hold for each n = 0, 1, 2, ... ‖x∗ − xn‖ ≤ s − sn. (3.2) https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 6 proof. mathematical induction is employed to show the assertion for each n = 0, 1, 2, ... ‖xn+1 − xn‖ ≤ sn+1 − sn < s (3.3) the assertion (3.3) holds for n = 0 by (a2) and the definition of s1 in (a6). let us assume thatthere exist x1, ..., xm generated by nlm satisfying for all integers m = 0, 1, 2, ..., n − 1 ‖xm − xm−1‖ ≤ sm − sm−1. then, ‖xm − x0‖ ≤ ‖xm − xm−1‖+ ‖xm−1 − xm−2‖+ ...+ ‖x1 − x0‖ ≤ sm − sm−1 + sm−1 − sm−2 + ...+ s1 − s0 = sm < s, and ‖xm − x1‖ ≤ ‖xm − x0‖+ ‖x0 − x1‖ ≤ sm − s1 ≤ s − s1. pic x ∈ u(xm, ‖xm − x0‖) to be arbitrary. define the operator q = f (x0) +m(x − x0) + g(x), and the multi-operator ψm(x) = q−1[f (x0) +m(x − x0)− f (xm)− l(xm)(x − xm)]. the conditions of the theorem 2.2 are validated in turn next. by applying the conditions (a5) and(a7), we get ‖f (x0) +m(x − x0)− f (xm)− l(xm)(x − xm)‖ ≤ ‖f (x)− f (x0)−m(x − x0)‖ +‖f (x)− f (xm)− f ′(xm)(x − xm)‖ +‖f ′(xm)−m‖‖x − xm‖+ ‖m − l(xm)‖‖x − xm‖ 1 λ [∫ 1 0 w0((1− θ)‖x − x0‖)dθ‖x − x0‖ + ∫ 1 0 w((1− θ)‖x − xm‖)dθ‖x − xm‖ +w0(‖xm − x0‖)‖x − xm‖+ w1)‖xm − x0‖)‖x − xm‖] ≤ 1 λ [∫ 1 0 w0((1− θ)ρ)dθ + w0(ρ) + ∫ 1 0 w((1− θ)ρ)dθ + w1(ρ) ] ρ ≤ β. notice that xm ∈ q−1[f (x0) +m(xm − x0)− f (xm−1 − l(xm−1(xm − xm−1)].by aubin property of q−1(.) at (0, x1) with modulus λ and parameters α, β we have in turn https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 7 d(xm, ψm(xm)) ≤ e{q−1[f (x0) +m(xm − x0)− f (xm−1)− l(xm−1)(xm − xm−1)] ∩s(x1, α), ψm(xm)} ≤ λ‖f (xm)− f (xm−1)− l(xm−1)(xm − xm−1)‖ ≤ λ‖f (xm)− f (xm−1)− f ′(xm−1)(xm − xm−1)‖ +λ‖(l(xm−1)− f ′(xm−1))(xm − xm−1)‖ ≤ λ‖f (xm)− f (xm−1)− f ′(xm−1)(xm − xm−1)‖ +λ‖(l(xm−1)−m)(xm − xm−1)‖ +λ‖(f ′(xm−1)−m)(xm − xm−1)‖ ≤ [∫ 1 0 w((1− θ)‖xm − xm−1‖)dθ‖xm − xm−1‖ + w0(‖xm−1 − x0‖)‖xm − xm−1‖+ w1(‖xm−1 − x0‖)‖xm − xm−1‖] = γ(1− w0(‖xm − x0‖)), where γ = [∫ 1 0 w((1− θ)‖xm − xm−1‖)dθ + w0(‖xm−1 − x0‖) + w1(‖xm−1 − x0‖) ] 1− w0(‖xm − x0‖) ×‖xm − xm−1‖ (3.4) pick v1, v2 ∈ s(xm, ‖xm − x0‖). then, we get e{ψm(v1) ∩ s(xm, ‖xm − x0‖), ψm(v2)} ≤ e{ψm(v1) ∩ s(x1, α), ψm(v2)} ≤ λ‖m − l(xm)‖‖v1 − v2‖ ≤ w0(‖xm − x0‖)‖v1 − v2‖ ≤ w0(ρ)‖v1 − v2‖, where w0(ρ) < 1, by the definition of ρ. thus, the theorem 2.2 is applicable if we take ψ = ψm, γ0 = γ and δ0 = δ = w0(‖xm − x0‖). so, there exists xm+1 ∈ s[x∗, ρ] satisfying xm+1 ∈ q−1[f (x0) +m(xm+1 − x0)− f (xm)− l(xm)(xm+1 − xm)] leading to ‖xm+1 − xm‖ ≤ [∫ 1 0 w((1− θ)(sm − sm−1))dθ + w0(‖sm−1‖) + w1(sm−1) ] 1− w0(sm) ×(sm − sm−1) (3.5) ≤ sm+1 − sm, https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 8where we used (a2), (3.4) and the induction hypothesis (3.3).by (a3) and (3.5), we obtain ∞∑ m=m0 ‖xm+1 − xm‖ ≤ ∞∑ m=m0 (sm+1 − sm) ≤ s − sm0 < +∞. if follows that the sequence {xm} is complete in a banach space b1, and as such it is convergentto some x∗ ∈ s[x0, s]. then, by (3.3), we can write ‖xm+j − xn‖ ≤ ‖xm+j − xm+j−1‖+ ‖xm+j−1 − xm+j−2‖+ ‖xm+1 − xn‖ ≤ sm+j − sm+j−1 + sm+j−1 − sm+j−2 + ...+ sm+1 − sn = sm+j − sm. (3.6) by letting j −→ +∞ in (3.6), we conclude that (3.2) is valid. in view of the definition the sequence {xm}, 0 ∈ f (xm) +l(xm)(xm+1−xm) +g(xn+1) for each m = 0, 1, ... then, by letting m −→ +∞,we deduce that 0 ∈ f (x∗) + g(x∗). � remark 3.2. a popular choice for m = f ′(x0). but this is not necessarily the most flexible choice. the two conditions in (a3) are very general. by specializing the functions w0, w and w1, we can provide other stronger conditions that imply the ones in (a3). let us consider the interesting lipchitz case, i.e. when w0(t) = l0t, w(t) = l t and w1(t) = l1t . then, the sequence {sn} in (a2) reduces for l2 = l0 + l1 to sn+1 = sn + ( l 2(sn − sn−1) + sn−1 ) (sn − sn−1) 1− l0(sn) (3.7) such sequences appear as majorant of newton-like methods for solving nonlinear equations (i.e.when g = {0}). the kantorovich-type convergence conditions in such studies imply the ones in (a3) but not necessarily vice versa [20,35]. our approach for the study of majorizing sequence {sn}has provided even weaker convergence conditions than the kantorovich-type [4,5,6]. 4. conclusion a very general theory for studying the convergence of newton-like methods is developed forgenerating sequences approximating a solution of a generalized equation involving set-valued op-erators. the semi-local analysis of convergence depend on the aubin property and the conceptof generalized continuity. the error analysis includes, computable upper error bounds on thenorms ‖xn+1 − xn‖ and ‖x∗ − xn‖. in particular, the semi-local analysis of convergence is basedon majorizing sequences for {xn} generated by nlm. it is shown that even specializations of theoperators involved lead to better results when compared to existing ones (see remark 3.2). the https://doi.org/10.28924/ada/ma.4.3 eur. j. math. anal. 10.28924/ada/ma.4.3 9future direction of our research involves the application of the developed theory on other meth-ods [1–3,9, 14,17,18,21–35]. references [1] s. adly, h. van ngai, v.v. nguyen. newton’s method for solving generalized equations: kantorovich’s and smale’sapproaches. j. math. anal. appl. 439 (2016), 396–418. https://doi.org/10.1016/j.jmaa.2016.02.047.[2] f.j. araǵon artacho, a. belyakov, a.l. dontchev, m. lópez. local convergence of quasi-newton methods under metricregularity. comput. optim. appl. 58 (2014), 225–247. https://core.ac.uk/download/pdf/19775169.pdf.[3] f.j. aragón artacho, a.l. dontchev, m. gaydu, m.h. geoffroy, v.m. veliov. metric regularity of newton’s itera-tion. siam j. control optim. 49 (2011), 339–362. https://nova.newcastle.edu.au/vital/access/services/ download/uon:11707/attachment01.[4] i.k. argyros, convergence and applications of newton-type iterations, springer-verlag, new york, 2008.[5] i.k. argyros, the theory and application of iteration methods, second edition, engineering series, boca raton,florida, usa, 2022. https://doi.org/10.1201/9781003128915.[6] i.k. argyros, s. george, on the complexity of extending the convergence region for traub’s method, j. complex. 56(2020), 101423. https://doi.org/10.1016/j.jco.2019.101423.[7] l. blum, f. cucker, m. shub, s. smale, complexity and real computation, springer-verlag, ny, 1998. https: //link.springer.com/book/10.1007/978-1-4612-0701-6.[8] j.f. bonnans, local analysis of newton-type methods for variational inequalities and nonlinear programming, appl.math. optim. 29 (1994), 161-186. https://doi.org/10.1007/bf01204181.[9] r. cibulka, a. dontchev, m.h. geoffroy. inexact newton methods and dennismoŕe theorems for nonsmooth gen-eralized equations. siam j. control optim. 53 (2015), 1003–1019. https://doi.org/10.1137/140969476.[10] s.p. dokov, a.l. dontchev, robinson’s strong regularity implies robust local convergence of newton’s method, appl.optim. 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https://doi.org/10.1007/s10957-021-01974-0.[34] y. nesterov and a. nemirovskii. interior-point polynomial algorithms in convex programming. siam, 1994. https: //doi.org/10.1137/1.9781611970791.[35] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987), 671-684. https://doi.org/10.1080/01630568708816254. https://doi.org/10.28924/ada/ma.4.3 https://www.jstor.org/stable/3689393 https://www.jstor.org/stable/3689393 https://doi.org/10.1007/978-3-642-68874-4_14 https://doi.org/10.1007/978-3-642-68874-4_14 https://doi.org/10.1007/bf01404880 https://eudml.org/doc/208686 https://www.jstor.org/stable/43660724 https://doi.org/10.1007/s11075-012-9585-7 https://doi.org/10.1016/j.amc.2003.12.025 https://doi.org/10.1007/s10957-021-01974-0 https://doi.org/10.1137/1.9781611970791 https://doi.org/10.1137/1.9781611970791 https://doi.org/10.1080/01630568708816254 1. introduction 2. mathematical bachground 3. convergence 4. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 14doi: 10.28924/ada/ma.4.14 duality of the nonreflexive bergman space of the upper half plane and composition groups e. o. gori1 , j. o. bonyo2,∗ 1department of pure and applied mathematics, maseno university, p.o. box 333-40105, maseno, kenya omondierick75@gmail.com1 2department of mathematics, multimedia university of kenya, p.o. box 15653-00503, nairobi, kenya jbonyo@mmu.ac.ke correspondence: jbonyo@mmu.ac.ke abstract. we identify the predual of the nonreflexive bergman space of the upper half plane withthe little bloch space of the upper half plane consisting of those functions vanishing at point i . usingthe duality pairing as well as the composition groups on the nonreflexive bergman space, we obtainthe groups of composition operators defined on the identified predual. we identify the infinitesimalgenerator of each group and prove the strong continuity property. we then obtain the spectra of thegenerator γ, determine the resolvents and further obtain the spectra and the norms of the resultingresolvents. 1. introduction let c be the complex plane. the set d := {z ∈ c : |z | < 1} is called the open unit disc. let da denote the area measure on d, normalized so that the area of d is 1. in terms of rectangularand polar coordinates, we have: da(z) = 1 πdxdy = r πdrdθ, where z = x + iy = re iθ ∈ d. for α ∈ r, α > −1, we define a positive borel measure dmα on d by dmα(z) = (1−|z |2)αda(z), andthus dmα is a probability measure. moreover, if α = 0, then dmo = da. we consider dmα as aweighted measure and a generalization of da. on the other hand, the set u := {ω ∈ c : =(ω) > 0}denotes the upper half of the complex plane c, with =(ω) being the imaginary part of ω ∈ c. for α > −1, we define a weighted measure on u by dµα(ω) = (=(ω))αda(ω), where ω ∈ u. again itcan easily be seen that α = 0 coincides with the unweighted measure. the function ψ(z) = i(1+z) 1−zis referred to as the cayley transform and maps the unit disc d conformally onto the upper half-plane u with the inverse ψ−1(ω) = ω−i ω+i .for an open subset ω of c, let h(ω) denote the space of analytic functions on ω. for 1 ≤ p <∞, received: 29 feb 2024. key words and phrases. duality, nonreflexive bergman space, bloch space, composition semigroups, infinitesimalgenerator, spectrum, resolvent. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.14 https://orcid.org/0000-0003-1785-2202 https://orcid.org/0000-0002-6442-4211 eur. j. math. anal. 10.28924/ada/ma.4.14 2 α > −1, the weighted bergman space of the upper half-plane u is defined by lpa(u, µα) := { f ∈ h(u) : ‖f ‖lpa(u,µα) = (∫ u |f (z)|pdµα(z) ) 1 p <∞ } . in particular, lpa(u, µα) = lp(u, µα) ∩ h(u), where lp(u, µα) or simply lp(µα) denotes theclassical lebesque spaces with respect to the weighted measure dµα. it is important to note thatthe case α = 0 yields the unweighted bergman space. lpa(u, µα) is a banach space with respectto the norm ‖f ‖lpa(u,µα) = (∫ u |f (ω)|pdµα(ω) ) 1 p . for p = 2, l2 a(u, µα) is a hilbert space. the growth condition for the weighted bergman spacefunctions on u is given by: for every f ∈ lpa(u, µα), γ = α+2 p and ω ∈ u, there exists a constant k such that, |f (ω)| ≤ k‖f ‖ (=(ω))γ . for a detailed account of the theory of bergman spaces, we refer to [7, 11,13].on the other hand, the bloch space of the unit disk, denoted by b∞(d), is defined by b∞(d) := {f ∈ h(d) : ‖f ‖b∞,1(d) = sup z∈d (1− |z |2)|f ′(z)| <∞}, with the norm on b∞(d) given by ‖f ‖b∞(d) := |f (0)|+‖f ‖b∞,1(d), while ‖.‖b∞,1(d) is a seminorm.the bloch space of the upper half plane denoted by b∞(u) is defined by b∞(u) := {f ∈ h(u) : ‖f ‖b∞,1(u) = sup ω∈u =(ω)|f ′(ω)| <∞}, with the norm given by ‖f ‖b∞(u) = |f (i)| + ‖f ‖b∞,1(u). the little bloch space of the unit diskdenoted by b∞,◦(d) is defined as b∞,◦(d) := {f ∈ h(d) : lim |z |→1 (1− |z |2)|f ′(z)| = 0} but with the same norm as b∞(d), while for the upper half-plane, the little bloch space is denotedby b∞,◦(u) and is defined by b∞,◦(u) := {f ∈ h(u) : lim =(ω)→0 =(ω)|f ′(ω)| = 0} with the same norm as b∞(u). for a comprehensive theory of bloch spaces, see [13,14].the duality properties of bergman spaces are well known in literature. for instance in [13, theorem4.2.9], it is proved that for 1 < p < ∞, 1 p + 1 q = 1 and α > −1, the dual of the bergman space lpa(d, mα) is given by (lpa(d, mα))∗ ≈ lqa(d, mα) under the duality pairing, 〈g, f 〉 = ∫ d g(z)f (z)dmα (g ∈ lpa(d, mα), f ∈ lqa(d, mα)). https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 3for the non-reflexive bergman space on the unit disk, l1 a(d, mα), it is shown in [13, theorems 5.1.4and 5.2.8] that the dual and predual spaces of l1 a(d, mα) are the bloch space and the little blochspace respectively. in particular, (l1 a(d, mα))∗ ≈ b∞(d) and (b∞,◦(d))∗ ≈ l1 a(d, mα) under theduality pairings given by respectively, 〈g, f 〉 = ∫ d g(z)f (z)dmα(z) (g ∈ l1 a(d, mα), f ∈ b∞(d)), and 〈g, f 〉 = ∫ d g(z)f (z)dmα(z) (f ∈ l1 a(d, mα), g ∈ b∞,◦(d)). for the corresponding spaces of the upper half plane, it has been proved and noted that thedual space of the reflexive bergman space of the upper half plane lpa(u, µα) is lqa(u, µα) for 1 < p, q < ∞ with 1 p + 1 q = 1 under a similar pairing as above. see for instance, [2–4] or [11]for details. when p = 1, the space l1 a(u, µα) is non-reflexive, and it’s recently that the dual wasdetermined by kang [8] as we give in theorem 2.1 stated in the next section. apparently, thepredual of l1 a(u, µα) is not explicitly clear from the literature. generally, there’s no unified andcomprehensive exposition of properties of the analytic spaces of upper half plane u as there are forthe corresponding spaces on the unit disk d. therefore, the main focus of this paper is to determinethe predual of l1 a(u, µα), that is, identifying the space whose dual is l1 a(u, µα).let aut(u) denotes the collection of all automorphisms of u. for ϕt ∈ aut(u), t ≥ 0, we define acomposition operator on h(u) by cϕt f := f ◦ϕt . the corresponding group of weighted compositionoperator on h(u) is therefore given by tt f := sϕt f = (ϕ′t) γf ◦ ϕt for some appropriate weight γ. motivated by the work of arvanitidis and siskakis in [1], the current second author and threeothers in [3] classified all the self analytic maps of the upper half plane into three distinct groups,namely: the scaling, the translation and the rotation groups. they then studied both the semigroupand spectral properties of the corresponding groups of weighted composition operators. as for theproperties of the adjoint groups on the reflexive weighted bergman spaces lpa(u, µα), 1 < p <∞,only the scaling group was considered in [3] and later completed for the other two groups by thesecond author in [4]. in this paper, we therefore determine the groups of composition operators onthe predual of non-reflexive bergman space of the upper half-plane, l1 a(u, µα) and investigate theadjoint properties of the groups of weighted composition operators on nonreflexive bergman space l1 a(u, µα).let x and y be banach spaces over c. the space l(x, y ) = {t : x → y such that t is linearand continuous}, endowed with the operator norm ‖t‖ = sup‖x‖≤1 ‖tx‖, is a banach space [5].we write l(x,x) = l(x). t is said to be a closed operator if its graph {(x, t x) | x ∈ d(t )}in x × y is closed. let t be a closed operator on x . the resolvent set of t , ρ(t ) is given by ρ(t ) = {λ ∈ c : λi − t is invertible or bijective} and its spectrum σ(t ) = c \ ρ(t ). therefore σ(t ) ∪ ρ(t ) = c. the spectral radius of t is defined by r(t ) = sup{|λ| : λ ∈ σ(t )} with the https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 4relation r(t ) ≤ ‖t‖. the point spectrum σp(t ) = {λ ∈ c : tx = λx for some 0 6= x ∈ dom(t )}.for λ ∈ ρ(t ), the operator r(λ, t ) := (λi − t )−1 is, by the closed graph theorem a boundedoperator on x and is called the resolvent of t at the point λ or simply the resolvent operator.in fact, ρ(t ) is an open subset of c and r(λ, t ) : ρ(t ) → l(x) is an analytic function. for adetailed theory on spectra, we refer to [5, 6, 9, 12]. 2. predual of non-reflexive bergman space of the upper half-plane l1 a(u, µα) let b∞(u, i) denote the subspace of the bloch space b∞(u) consisting of functions vanishingat point i . therefore b∞(u, i) is defined as b∞(u, i) := {f ∈ b∞(u) : f (i) = 0}. then b∞(u, i) is a closed subspace of b∞(u) and therefore is a banach space with respect to thenorm ‖f ‖b∞,i := ‖f ‖b∞(u) = ‖f ‖b∞,1(u), see [8]. similarly, let b∞,◦(u, i) denotes the subspace of b∞,◦(u) consisting of functions vanishing at i . therefore b∞,◦(u, i) := {f ∈ b∞,◦(u) : f (i) = 0}, with the norm ‖f ‖b∞,i := ‖f ‖b∞(u) = ‖f ‖b∞,1(u). again, b∞,◦(u, i) is a banach space with respectto the norm given above.the following result due to kang [8] gives the dual of l1 a(u, µα); theorem 2.1. for any α ∈ r, α > −1, we have (l1 a(u, µα))∗ ≈ b∞(u, i), under the integral pairing 〈g, f 〉 = ∫ u g(ω)f (ω)dµα(ω) (g ∈ l1 a(u, µα), f ∈ b∞(u, i)). with the help of theorem 2.1 above, we determine the predual space of l1 a(u, µα), that is, a setwhose dual is l1 a(u, µα), but first we state some results.let c(u) be the algebra of complex valued continuous functions on u = u ⋃ ∂u, and c◦(u) bethe subalgebra of c(u) consisting of functions f such that f (ω)→ 0 as =(ω)→ 0. proposition 2.2. let c◦(u) be the subalgebra of c(u) consisting of functions f such that f (ω) −→ 0 as im(ω) −→ 0 and c◦(d) be the subalgebra of c(d) consisting of functions f with f (z) → 0 as |z | → 1−. then c◦(u) = {g ◦ ψ−1 : g ∈ c◦(d)}. proof. let k ⊂ u be compact. since cayley transform ψ : d → u is a continuous bijection, itfollows that k ⊂ u is compact if and only if ψ−1(k) is compact in d. if f ∈ c◦(u) and ε > 0, https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 5then there exists k compact in u such that supw∈u\k |f (w)| < ε.now, g = f ◦ ψ is continuous on d with f = g ◦ ψ−1, and sup z∈d\ψ−1(k) |g(z)| = sup w∈u\k |f (w)| < ε. � proposition 2.3. let cψ be the composition by ψ operator. then (1) f ∈ b∞(u) if and only if cψf ∈ b∞(d). in particular, ‖f ‖b∞,1(u) = 1 2‖cψf ‖b∞,1(d). (2) f ∈ b∞,◦(u) if and only if cψf ∈ b∞,◦(d). (3) f ∈ l1(u, µα) if and only if sψf ∈ l1(d, mα). in particular, ‖f ‖l1 a(u,µα) = 1 2α ‖sψf ‖l1(d,mα). (4) f ∈ l∞(u, µα) if and only if cψf ∈ l∞(d, mα). proof. for (1), if f ∈ b∞(u), then by definition, ‖f ‖b∞,1(u) = sup ω∈u (=(ω))|f ′(ω)| = sup z∈d 1− |z |2 |1− z |2 ∣∣f ′(ψ(z)) ∣∣ = 1 2 sup z∈d (1− |z |2)|ψ′(z)||f ′(ψ(z))| = 1 2 sup z∈d (1− |z |2)|(f ◦ ψ)′(z)| = 1 2 ‖f ◦ ψ‖b∞,1(d). for (2), we have f ∈ b∞,0(u) is equivalent to lim =(ω)→0 (=(ω))|f ′(ω)| = lim =(ψ(z))→0 1− |z |2 |1− z |2 |f ′(ψ(z))| = 1 2 sup |z |→1 (1− |z |2)|(f ◦ ψ)′(z)| = 0, which in turn is equivalent to f ◦ ψ ∈ b∞,0(d), as desired. for f ∈ l1 a(u, µα), we have ‖f ‖l1 a(u,µα) = ∫ u |f (ω)| dµα(ω) = ∫ u |f (ω)|(=(ω))α da(ω) = ∫ d |f (ψ(z))| ( 1− |z |2 |1− z |2 )α |ψ′(z)|2 da(z) = 1 2α ∫ d |f (ψ(z))||ψ′(z)|α+2 (1− |z |2) da(z) = 1 2α ∫ d |(ψ′(z))γ(f ◦ ψ)(z)| dmα(z) = 1 2α ‖sψf ‖l1(d,mα), which proves (3). now, f ∈ l∞(u, µα) means that f is essentially bounded which implies that f ◦ ψ is essentially bounded as well. since ψ is an invertible mapping from d onto u, it followsthat f ◦ ψ ∈ l∞(d, mα). the converse follows similarly. this completes the proof. � https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 6 remark 1. it is easy to verify that cψ−1 = c−1 ψ . proposition 2.3 above therefore implies that cψ is an is an isometry up to a constant and at the same time invertible on the respective spaces with the inverse also acting on the appropriate spaces. more generally, let {v1, v2} = {d,u}, and let lf (vi , vj) denote the collection of conformal mappings from vi onto vj . then lf (vi , vi) = aut(vi), and if h ∈ lf (vi , vj), then g ∈ aut(vj) 7→ h−1 ◦ g ◦ h ∈ aut(vi) is an isomorphism from aut(vi) onto aut(vj). for each g ∈ lf (vi , vj), we define a weighted composition operator sg : h(vj) → h(vi), by sgf (z) = (g′(z))γf (g(z)), for all z ∈ vi . (2.1) we note that if g ∈ lf (vi , vj) and h ∈ lf (vj , vi), then it is clear by chain rule that sh ◦sg = sg◦h and s−1 g = sg−1 . now, using propositions 2.2 and 2.3 above, we obtain the following result which is the upperhalf-plane analogue of [13, lemma 5.14]. proposition 2.4. for t > 0, α > −1, let the integral operator t on h(d) be defined by t f (z) = (1− |z |2)t ∫ d f (w) (1− zw)2+t+α dmα(w). let s be the corresponding integral operator on h(u) defined by s := cψ−1tcψ . then the following properties hold:(a) s = (α+ t + 1)s2,(b) s is a bounded embedding of b∞(u) into l∞(u) and(c) s is an embedding of b∞,◦(u) into c◦(u). proof. from [13, lemma 5.14], we have, s = cψ−1tcψ = cψ−1 (α+ t + 1)t 2cψ = (α+ t + 1)cψ−1t 2cψ = (α+ t + 1)s2, which proves (a).for (b), we have b∞(u) cψ−−→ b∞(d) t−→ l∞(d) cψ−1 −−−→ l∞(u). now, cψ is an isometry of b∞(u) onto b∞(d) up to constant, t is a bounded embedding of b∞(d) into l∞(d) [13, lemma 5.14], cψ−1 is also an isometry of l∞(d) onto l∞(u), it thereforefollows that s = cψ−1tcψ is a bounded embedding of b∞(u) into l∞(u).for (c), we have b∞,◦(u) cψ−−→ b∞,◦(d) t−→ c◦(d) cψ−1 −−−→ c◦(u). https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 7 cψ is a bijection of b∞,◦(u) into b∞,◦(d), t is an embedding of b∞,◦(d) into c◦(d) [13, lemma5.14], while on the other hand, cψ−1 is also a bijection of c◦(d) into c◦(u). therefore s = cψ−1tcψ is an embedding of b∞,◦(u) into c◦(u), which completes the proof. � we now establish the predual space of l1 a(u, µα) as we give in the following theorem: theorem 2.5. for any α > −1, we have; (b∞,◦(u, i))∗ ≈ l1 a(u, µα), under the pairing 〈g, f 〉 = ∫ u g(ω)f (ω)dµα(ω), where g ∈ b∞,◦(u, i) and f ∈ l1 a(u, µα). here, b∞,◦(u, i) is equipped with the same norm as b∞(u, i). proof. if f ∈ l1 a(u, µα), then by theorem 2.1 above, g 7−→ ∫ u g(ω)f (ω)dµα(ω) defines a boundedlinear functional on b∞,◦(u, i). conversely, if f is a bounded linear functional on b∞,◦(u, i), wewant to show that there exists a function f ∈ l1 a(u, µα) such that f (g) = ∫ u g(ω)f (ω)dµα(ω) for g in a dense set of b∞,◦(u, i).now we fix any positive parameter t and consider the embedding s of b∞,◦(u, i) into c◦(u)as given by proposition 2.4. the space x = s(b∞,◦(u, i)) is a closed subspace of c◦(u) and f ◦ s−1 : x → c is a bounded linear functional on x since f and s−1 are both bounded. bythe hahn-banach extension theorem, f ◦ s−1 extends to a bounded linear functional on c◦(u).by the riesz representation theorem, there exists a finite weighted measure µα on u such that ‖µα‖ = ‖f ◦s−1‖ and f ◦s−1(h) = ∫ u h(z)dµα(z), h ∈ c◦(u). in particular, if g is a polynomial(polynomials are dense in b∞,◦(u, i)), then f (g) = f ◦s−1◦s(g) = ∫ u sg(z)dµα(z). by fubini’stheorem, we have f (g) = ∫ u g(ω)f (ω)dµα(ω), where f = cψ−1tcψ which is bounded since t isbounded. � 3. groups of weighted composition operators on predual of l1 a(u, µα) as remarked in the section 1, the automorphisms of the upper half plane u were identified andclassified into three distinct groups according to the location of their fixed points in [3], namely:the scaling, the translation and the rotation groups. since the corresponding groups of compositionoperators for the rotation group are defined on the analytic spaces of the unit disk, we shall onlyconsider groups of composition operators associated with the scaling and the translation groups inthis paper. https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 83.1. scaling group. the automorphisms of this group are of the form ϕt(z) = k tz , where z ∈ uand k, t ∈ r with k 6= 0. as noted in [3] and without loss of generality, we consider the analyticself maps ϕt : u → u of the form ϕt(z) = e−tz for z ∈ u. the corresponding group of weightedcomposition operators on lpa(u, µα) is given by tt f (z) = e−tγf (e−tz), for all f ∈ lpa(u, µα),where γ=α+2 p and 1 ≤ p <∞. for p = 1, (tt)t≥0 is defined on l1 a(u, µα) with γ = α+ 2.following theorem 2.5, the predual of l1 a(u, µα) is given by the duality relation (b∞,◦(u, i))∗ ≈ l1 a(u, µα) (3.1) under the integral pairing 〈g, f 〉 = ∫ u g(w)f (w)dµα(w), (3.2) where g ∈ b∞,◦(u, i) and f ∈ l1 a(u, µα).using the duality pairing above, we obtain the corresponding group of weighted composition op-erators on b∞,◦(u, i) as below:let g ∈ b∞,◦(u, i) and f ∈ l1 a(u, µα), then, 〈g, tt f 〉 = ∫ u g(z)e−tγf (e−tz)dµα(z) = ∫ u g(z)e−tγf (e−tz)(=(z))αda(z). by change of variables, let ω = e−tz , then z = etω, da(ω) = e−2tda(z) and =(z) = et im(ω).then, 〈g, tt f 〉 = ∫ u g(etω)e−tγf (ω)eαt(=(ω))αe2tda(ω) = ∫ u g(etω)e−tγetγf (ω)dµα(ω) = ∫ u g(etω)f (ω)dµα(ω) = 〈t ∗t g, f 〉. (3.3) where t ∗t g(ω) = g(etω).now, st := t ∗t is defined on b∞,◦(u, i). but we see that stg(i) = g(et i) 6= 0 and therefore stgdoes not vanish at i . this means that st does not map b∞,◦(u, i) onto itself, and there (st)t≥0 isnot a good semigroup. we therefore propose two remedies to correct the defect. the first one is toapply a correction factor by writing stg(ω) = g(etω)− g(et i) for all g ∈ b∞,◦(u, i). this meansthat stg(i) = 0 and therefore st maps b∞,◦(u, i) onto itself, as desired.the second remedy is to redefined st to act on b∞,◦(u) instead of b∞,◦(u, i). this simply meansthat the domain of st has been enlarged and therefore st is well defined on b∞,◦(u, i). we shall now carry out a complete study of both the semigroup and spectral properties of thisgroup on b∞,◦(u). https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 93.1.1. semigroup properties. in this section, we investigate the semigroup properties and determinethe infinitesimal generator γ of (st)t≥0 on b∞,◦(u) where, stg(w) := g(etw). we begin byproving the strong continuity property. theorem 3.1. let stg(w) := g(etw) be a semigroup of composition operators defined on b∞,◦(u). then, (st)t∈r is a strongly continuous group of isometries on b∞,◦(u). proof. it is clear from the definition that (st)t∈r is a group. to prove that (st)t∈r is an isometryon b∞,◦(u), we have; ‖stg‖b∞,◦(u) = sup ω∈u =(ω)|stg′(ω)| = sup ω∈u =(ω)et |g′(etω)|. now by change of variables, let z = etω then ω = e−tz , and =(ω) = e−t=(z). therefore, ‖stg‖b∞,◦(u) = sup z∈u e−t=(z)et |g′(z)| = sup z∈u =(z)|g′(z)| = ‖g‖b∞,◦(u), as desired. for strongly continuity, we first take note that st = cϕ−t since stg(ω) = g(ϕt(ω)). thenby proposition 2.3, it is easy to see that cψ−t is strongly continuous on b∞,◦(u) if and only if (cψ−1◦ϕ−t◦ψ)t∈r is strongly continuous on b∞,◦(d). now by simple computation of ψ−1◦ϕ−t◦ψ(z),we obtain; ψ−1 ◦ ϕ−t ◦ ψ(z) = z − 1−et 1+et 1− 1−et 1+et z = z − at 1− atz , where at = 1−et 1+et . as t → 0, at → 0. let ha(z) = z−at 1−atz = ψ−1 ◦ ϕ−t ◦ ψ(z), then for strongcontinuity, it therefore suffices to show that ‖cha f − f ‖b∞,◦(d) → 0 as a→ 0 (at → 0). using thedensity of polynomials in b∞,◦(d), let f (z) = zn. then chazn − zn = (ha(z))n − zn, n ≥ 1, and (cha f − f )′(z) = n[(ha(z))n−1h′a(z)− zn−1]. but ha(z) = z−at 1−atz , and hence h′a(z) = 1−atat (1−atz)2 . therefore, (cha f − f )′(z) = n [ (ha(z))n−1(1− atat) (1− atz)2 − zn−1 ] = n [ ( z−at1−atz )n−1(1− atat) (1− atz)2 − zn−1 ] = n [ (z − at)n−1(1− atat)− zn−1((1− atz)n+1) (1− atz)n+1 ] . https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 10now, lim a→0 ‖cha f − f ‖b∞,◦(d) = lim a→0 ( sup z∈d (1− |z |2)|(cha f − f )′|(z) ) = lim t→0 ( sup z∈d (1− |z |2) ∣∣∣∣n [(zn−1)(1)− zn−1(1) (1)n+1 ]∣∣∣∣) = lim t→0 ( sup z∈d (1− |z |2) ∣∣n[zn−1 − zn−1] ∣∣) = 0. hence, (st)t∈r is strongly continuous on b∞,◦(u), as claimed. � theorem 3.2. the infinitesimal generator γ of (st)t≥0 on b∞,◦(u) is given by γg(ω)=ωg′(ω) with the domain d(γ) = {g ∈ b∞,◦(u) : ωg′(ω) ∈ b∞,◦(u)}. proof. by definition, the infinitesimal generator denoted by γ of (st)t≥0 is given by; γg(ω) = lim t→0+ g(etω)− g(ω) t = ∂ ∂t g(etω) ∣∣∣∣ t=0 = ωg′(ω). it therefore follows that d(γ) ⊆ {g ∈ b∞,◦(u) : ωg′(ω) ∈ b∞,◦(u)}. to prove the reverseinclusion, we let g ∈ b∞,◦(u) be such that ωg′(ω) ∈ b∞,◦(u). then for ω ∈ u, we have; stg(ω)− g(ω) = ∫ t 0 ∂ ∂s g(esω) ds = ∫ t 0 esωg′(esω) ds = ∫ t 0 ssg(ω) ds where g(ω) = ωg′(ω). thus, lim t→0+ stg − g t = lim t→0+ 1 t ∫ t 0 ssg(ω) ds and strong continuity of (ss)t≥0 implies that 1 t ∫ t 0 ‖ssg − g‖ds → 0 as t → 0+. hence d(γ) ⊇ {g ∈ b∞,◦(u) : ωg′(ω) ∈ b∞,◦(u)}, which completes the proof. � 3.1.2. spectral properties. now for the spectral properties, we obtain the spectra of the generator γ, determine the resolvents and further obtain the spectra and the norms of the resulting resolvents. theorem 3.3. let γ be the infinitesimal generator of (st)t∈r on b∞,◦(u). then σp(γ) = ∅ and σ(γ) = ir. in particular, γ is an unbounded operator on b∞,◦(u). before we prove this theorem, we first give the following lemma: lemma 3.4. if ν ∈ c and c ∈ r, we have (1) g(ω) = cων /∈ b∞,0(u) for any c https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 11 (2) f (ω) = (w − i)ν ∈ b∞,0(u) if and only if <(ν) < 0. in particular, g(ω) /∈ b∞,0(u) for any c and f (ω) ∈ b∞,0(u) if and only if <(ν) < 0. proof. from proposition 2.3, we know that g ∈ b∞,◦(u) if and only if g ◦ ψ ∈ b∞,◦(d). then for z ∈ d, (g ◦ ψ)(z) = g(ψ(z)) = c(ψ(z))ν = c( i(1 + z) 1− z )ν = ci(1 + z)ν(1− z)−ν . now g ◦ ψ ∈ h(d) if and only if <(ν) > 0 and <(−ν) > 0 which is not possible, and therefore g ◦ ψ /∈ h(d). hence g /∈ b∞,0(u). this proves (1). for (2), following [3, lemma 3.2], for any ν ∈ c, (w − i)ν ∈ h(u) if and only if <(ν) < 0 since γ = 0 in this case.the particular cases follow immediately since b∞,0(u, i) ⊆ b∞,0(u) and g(i) 6= 0 for (1), while f (i) = 0 for (2). � proof of theorem 3.3. to obtain the point spectrum of γ, let λ be an eigenvalue of γ and g be thecorresponding eigenvector. then γg(ω) = λg(ω) is equivalent to ωg′(ω) = λg(ω) which yields ωg′(ω) ω = λg(ω) ω by dividing both sides by ω. by integrating both sides, we obtain g(ω) = cωλ,which is not in b∞,◦(u) for any c . therefore σp(γ) = ∅.since each st is an invertible isometry, its spectrum satisfies σ(st) ⊆ ∂d. therefore the spectralmapping theorem for strongly continuous groups [10, theorem 2.3] implies that etσ(γ) ⊆ σ(st) ⊆ ∂d. now let λ ∈ σ(γ), then |etλ| = 1 which further implies that <(λ) = 0. thus λ ∈ ir andtherefore σ(γ) ⊆ ir.we now need to show the reverse inclusion, that is, ir ⊆ σ(γ). fix λ ∈ ir and assume λ /∈ σ(γ)which implies that the resolvent operator r(λ,γ) : b∞,◦(u) → b∞,◦(u) is bounded. considerthe function h(w) = (w − i)−(λ+1). then <(−(λ + 1)) = −1 < 0 and following lemma 3.4, itis immediate that h ∈ b∞,◦(u). the image function f = r(λ,γ)h is equivalent to (λ − γ)f = hwhich yields a differential equation f ′(ω)− λ ω f (ω) = − h(ω) ω , whose general solution is f (ω) = (ω − i)−λ + cωλwhich does not belong to b∞,◦(u) for any c , by lemma 3.4. thus h /∈ r(λ − γ) and so σ(γ) = ir. � theorem 3.5. let γ be the infinitesimal generator of (st)t∈r. then the following hold; (1) for λ ∈ ρ(γ), and h ∈ b∞,◦(u) then, https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 12(i) r(λ,γ)h(ω) = ωλ ∫∞ ω 1 zλ+1 h(z) dz , if <(λ) > 0.(ii) r(λ,γ)h(ω) = −ωλ ∫ ω 0 1 zλ+1 h(z) dz, if <(λ) < 0. (2) σ(r(λ,γ)) = { ω : |ω − 1 2<(λ) | = 1 2<(λ) } . (3) r(r(λ,γ)) = ‖r(λ,γ)‖ = 1 |<(λ)| . proof. to prove (1), we take note the resolvent set is given as ρ(γ) = {λ ∈ c : <(λ) 6= 0}. wetherefore consider the following cases: case 1: if <(λ) > 0, then the resolvent operator is given by the laplace transform: for every h ∈ b∞,◦(u), we have r(λ,γ)h = ∫∞ 0 e−λtsthdt with convergence in norm. therefore, r(λ,γ)h =∫∞ 0 e−λth(etω)dt. by change of variables, let z = etω, then ω = e−tz , dzdt = ωet then dt = dz ωet = dz z . therefore when t = 0⇒ z = ω and t =∞⇒ z =∞, and so; r(λ,γ)h(ω) = ∫ ∞ ω e−λth(z) dz z = ∫ ∞ ω ( z ω )−λ 1 z h(z)dz = ωλ ∫ ∞ ω 1 zλ+1 h(z)dz. case 2: if <(λ) < 0, then r(λ,γ)h = −r(−λ,−γ)h = − ∫∞ 0 eλth(e−tω)dt. then again bychange of variables, let z = e−tω, then et = ω z , dzdt = −ωe−t and dt = −dz ωe−t = −dzz . therefore t = 0⇒ z = w and t =∞⇒ z = 0 and so; r(λ,γ)h(w) = − ∫ 0 ω eλth(z).− dz z = − ∫ ω 0 (ω z )λ h(z). dz z = −ωλ ∫ ω 0 ( 1 z )λ . 1 z h(z)dz = −ωλ ∫ ω 0 1 zλ+1 h(z)dz. to prove (2), we use the spectral mapping theorem for the resolvents which asserts that σ(r(λ,γ)) ={ 1 λ−µ : µ ∈ σ(γ) } \ {0} for λ ∈ ρ(γ). therefore, σ(r(λ,γ)) = { 1 λ− i r : r ∈ r } \ {0} = { 1 <(λ) + i(im(λ)− r) : r ∈ r } \ {0}. rationalizing the denominator and simplifying we get σ(r(λ,γ)) = { (<(λ)−i(=(λ)−r)) (<(λ))2+(=(λ)−r)2 : r ∈ r } .now by letting w = (<(λ)−i(=(λ)−r)) (<(λ))2+(=(λ)−r)2 , subtracting 1 2<(λ) and finding the magnitude of both sideswe get, ∣∣∣∣w − 1 2<(λ) ∣∣∣∣2 = 1 (2<(λ))2 , https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 13and so ∣∣∣∣w − 1 2<(λ) ∣∣∣∣ = 1 2<(λ) . therefore, σ(r(λ,γ)) = { w : |w − 1 2<(λ) | = 1 2<(λ) } . for part (3), the spectral radius r(r(λ,γ))is given by; r(r(λ,γ)) = sup{|w | : w ∈ σ(r(λ,γ))} = sup { |w | : ∣∣∣∣w − 1 2<(λ) ∣∣∣∣ = 1 2<(λ) } = 1 |<(λ)| .finally to determine ‖r(λ,γ)‖, we use the hille yosida theorem as well as the fact that the spectralradius is always bounded by the norm.therefore, 1 |<(λ)| = r(r(λ,γ)) ≤ ‖r(λ,γ)‖ ≤ 1 |<(λ)| .thus, r(r(λ,γ)) = ‖r(λ,γ)‖ = 1 |<(λ)| , as desired. � 3.2. translation group. in this group the automorphisms are of the form ϕt(z) = z + kt , where z ∈ u and k, t ∈ r with k 6= 0. as noted earlier in subsection 3.1, without loss of generality welet k = 1 and consider the self analytic maps ϕt : u → u given by ϕt(z) = z + t for z ∈ u.then the corresponding group of composition operators defined on l1 a(u, µα) is therefore given by tt f (z) = f (z + t), for all f ∈ lpa(u, µα).now using the duality relation given by equation (3.1) and its sesquilinear pairing given by equation(3.2), we have:let g ∈ b∞,◦(u, i) and f ∈ l1 a(u, µα), then 〈g, tt f 〉 = ∫ u g(z)f (z + t)dµα(z) = ∫ u g(z)f (z + t)(=(z))αda(z). now by a change of variables, let ω = z + t , then z = ω − t and da(ω) = da(z). therefore, 〈g, tt f 〉 = ∫ u g(ω − t)f (ω)(=(ω))αda(ω) = ∫ u g(ω − t)f (ω)dµα(ω) = 〈t ∗t g, f 〉. (3.4) now, we define st := t ∗t on b∞,◦(u, i). but again we see that just as in the case of the scalinggroup, stg(i) = g(i − t) and therefore stg(i) does not vanish at point i . this means that st doesnot map b∞,◦(u, i) onto itself. we can therefore apply similar remedies proposed in subsection3.1 above. in the next sections, we study the semigroup properties of (st)t≥0 on b∞,◦(u). https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 143.2.1. semigroup properties. in this section, we begin by proving the strong continuity property. theorem 3.6. let stg(ω) := g(ω−t) be a semigroup of composition operators defined on b∞,◦(u). then, (st)t∈r is a strongly continuous group of isometries on b∞,◦(u). proof. it is clear from the definition that (st)t∈r is a group. to prove that (st)t∈r is an isometry,then by the definition of isometry, we have; ‖stg‖b∞,◦(u) = sup ω∈u =(ω)|(stg)′(ω)| = sup ω∈u =(ω)|g′(ω − t)|. by change of variables, let z = ω − t then ω = z + t and =(ω) = =(z). hence, ‖stg‖b∞,◦(u) = sup z∈u =(z)|g′(z)| = ‖g‖b∞,◦(u), as desired. for strongly continuity property, we argue as we did in the previous section. we note that st = cϕ−t which is strongly continuous on b∞,◦(u) if and only if (cψ−1◦ϕ−t◦ψ)t∈r is strongly continuouson b∞,◦(d), which consists of functions vanishing at point 0.we compute ψ−1 ◦ ϕ−t ◦ ψ(z). let at = t 2i+t and bt = 2i−t 2i+t , then a straight forward calculationyields ψ−1 ◦ ϕ−t ◦ ψ(z) = z − at bt + atz = ha(z), where we have let ha(z) = z−at bt+atz . clearly, t → 0 as at → 0 and bt → 1. it therefore suffices toshow that ‖cha f −f ‖b∞,◦(d) → 0 as t → 0. using density of polynomials in b∞,◦(d), we let f (z) = zn. then chazn−zn = (ha(z))n−zn, n ≥ 1. therefore (cha f −f )′(z) = n[(ha(z))n−1h′a(z)−zn−1].but ha(z) = z−at bt+atz ⇒ h′a(z) = (bt+atz)(1)−(z−at)(at) (bt+atz)2 . therefore by substituting, (cha f − f )′(z) = n[(ha(z))n−1h′a(z)− zn−1] = n [( z − at bt + atz )n−1 (bt + atz)− (z − at)(at) (bt + atz)2 − zn−1 ] = n [ (z − at)n−1(bt + atz)− (z − at)(at) (bt + atz)n+1 − zn−1 ] . now, lim t→0+ ‖cha f − f ‖b∞,◦(d) = lim t→0+ ( sup z∈d (1− |z |2)|(cha f − f )′|(z) ) = lim t→0+ ( sup z∈d (1− |z |2)∣∣∣∣n [(z − at)n−1(bt + atz)− (z − at)(at) (bt + atz)n+1 − zn−1 ]∣∣∣∣) https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 15 = lim t→0+ ( sup z∈d (1− |z |2) ∣∣∣∣n[zn−1 − 0− zn−1] 1 ∣∣∣∣) = 0. hence (st)∈r is strongly continuous, as claimed. � theorem 3.7. the infinitesimal generator γ of (st)t≥0 on b∞,◦(u) is given by γg(ω) = −g′(ω) with the domain d(γ) = {g ∈ b∞,◦(u) : g′ ∈ b∞,◦(u)}. proof. by definition, the infinitesimal generator γ on b∞,◦(u) is given by; γg(ω) = lim t→0+ g(ω − t)− g(ω) t = ∂ ∂t g(ω − t) ∣∣∣∣ t=0 = −g′(ω). therefore d(γ) ⊂ {g ∈ b∞,◦(u) : −g′ ∈ b∞,◦(u)}. conversely, let g ∈ b∞,◦(u) be such that −g′ ∈ b∞,◦(u). thus we have; stg − g t = 1 t ∫ t 0 ∂ ∂s ssg ds and for every ω ∈ u, ∂ ∂sssg(ω) = −g′(ω − s) = ssg ′(ω). thus,∥∥∥∥ssg − gt − f ′ ∥∥∥∥ ≤ 1 t ∫ t 0 ∥∥ts f ′ − f ′∥∥ ds → 0 as t → 0 by strong continuity, and therefore d(γ) ⊇ {g ∈ b∞,◦(u) : −g′ ∈ b∞,◦(u)}, which completes theproof. � acknowledgement this work was completed during the period when the second author was visiting the aristotleuniversity of thessaloniki, greece. he would like to sincerely that the simon’s foundation forfunding his visit. he would also wish to thank his host prof. aristomenis g. siskakis and thedepartment of mathematics for the unmatched hospitality references [1] a. g. arvanitidis, a. g. siskakis, cesàro operators on the hardy spaces of the half plane. canadian math. bull. 56(2013), 229–240.[2] s. axler, bergman spaces and their operators, lecture notes at the indiana university function theoretic operatorstheory conference, 1985.[3] s. ballamoole, j. o. bonyo, t. l. miller, v. g. miller, cesaro-like operators on the hardy and bergman spaces of thehalf plane, complex anal. oper. theory, 10 (2016), 187-203.[4] j. o. bonyo, spectral analysis of certain groups of isometries on hardy and bergman spaces, j. math. anal. appl.456 (2017), 1470–1481.[5] j. b. conway, a course in functional analysis, springer verlag, new york, 1985.[6] n. dunford, j. t. schwartz, linear operators part i. interscience publishers, new york, 1958. https://doi.org/10.28924/ada/ma.4.14 eur. j. math. anal. 10.28924/ada/ma.4.14 16 [7] p. duren, a. schuster, bergman spaces, mathematical surveys and monographs 100, amer. math. soc., providence,ri, 2004.[8] s. h. kang, some duality of weighted bergman spaces of the half-plane, bull. korean math. soc. 42 (2005), 385-396.[9] k. b. laursen, m. m. neumann, an introduction to local spectral theory, clarendon press, oxford, 2000.[10] a. pazy, semigroups of linear operators and applications to partial differential equations, applied mathematicalsciences 40, springer, new york, 1983.[11] m. m. peloso, classical spaces of holomorphic functions, technical report, universìt di milano, 2014.[12] w. rudin, functional analysis, mcgraw-hill, inc. new york (1991).[13] k. zhu, operator theory in function spaces, marcel dekker inc. new york, basel (1990).[14] k. zhu, bloch type spaces of analytic functions, rocky mountain j. math. 23 (1993), 1143–1177. https://doi.org/10.28924/ada/ma.4.14 1. introduction 2. predual of non-reflexive bergman space of the upper half-plane l1a(u,) 3. groups of weighted composition operators on predual of l1a(u,) 3.1. scaling group 3.2. translation group acknowledgement references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 8doi: 10.28924/ada/ma.4.8 hybrid inertial iterative method for fixed point, variational inequality and generalized mixed equilibrium problems in banach space lawal umar1,∗, yusuf ibrahim2, m.s. lawan3 1department of mathematics, federal college of education, zaria kaduna, nigeria lawalu4@gmail.com 2department of mathematics, saadatu rimi university of education, kumbotso kano, nigeria danustazz@gmail.com 3department of mathematics and statistics, kaduna polytechnic kaduna, nigeria mslawankh@yahoo.com ∗correspondence: lawalu4@gmail.com abstract. in this paper, we introduced a hybrid inertial iterative method which converges stronglyto a common element of solution of generalized mixed equilibrium, variational inequality and fixedpoint problems in a two uniformly smooth and uniformly convex banach space. our hybrid inertialiterative method, techniques of proof and corollaries improves, extends and generalizes many resultsin the literature. 1. introduction let b denotes a real banach space with b∗ as the dual space of b. we consider 〈τ1, j〉 as thevalue of the functional j ∈ b∗ at τ1 ∈ b and ‖ . ‖ as the norm of b or b∗. let c 6= ∅ be subset of b. a mapping j : b −→ 2b ∗ is called normalized duality provided that jτ1 = {τ2 ∈ b∗ : 〈τ2, τ1〉 = ‖τ1 ‖2= ‖τ2 ‖2},∀τ1 ∈ b. we denotes the short form gmep as generalized mixed equilibrium problem: find v1 ∈ c suchthat d(v1, v2) + 〈gv1, v2 − v1〉+ ϑ(v1, v2)− ϑ(v1, v1) ≥ 0, ∀v2 ∈ c, (1.1) where d, ϑ : c × c −→ r and g : c −→ b∗ denotes the bifunctions and a nonlinear mappingrespectively, also r is consider as the set of all real numbers. then, sol(gmep (1.1)) is consideras the solution set of gmep.(1.1). received: 16 jan 2024. key words and phrases. hybrid inertial iterative method; fixed point problem; variational inequality problem;generalized mixed equilibrium problem. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 2if g ≡ 0, gmep (1.1) reduces to generalized equilibrium problem ( with gep as the short form):find v1 ∈ c such that d(v1, v2) + ϑ(v1, v2)− ϑ(v1, v1) ≥ 0,∀v2 ∈ c. (1.2) then, sol(gep (1.2)) is represent the solution set of gep (1.2).if g ≡ 0 and ϑ ≡ 0, gmep (1.1) becomes equilibrium problem ( with ep as the short form) [3]:find v1 ∈ c such that d(v1, v2) ≥ 0,∀v2 ∈ c. (1.3) then, sol(ep (1.3)) is consider as the solution set of ep.(1.3).if d ≡ 0 and ϑ ≡ 0, gmep (1.1) reduces to variational inequality problem ( with v ip as theshort form): find v1 ∈ c such that 〈gv1, v2 − v1〉 ≥ 0,∀v2 ∈ c. (1.4) then, sol(v ip (1.4)) is consider as the solution set of v ip (1.4). definition 1.1. let t : c −→ c be a mapping [6], then(i) a point v1 ∈ c is called fixed point of t provided that f (t ) = {v1 ∈ c : tv1 = v1} 6= ∅;(ii) a point v0 ∈ c is called an asymptotic fixed point of t provided that {vn} ⊂ c, vn ⇀ v0 suchthat lim n→∞ ‖ vn − tvn ‖= 0. the set of asymptotic fixed point of t is denoted by f̂ (t );(iii) t is called quasi−φ−nonexpansive provided that φ(v0, t v) ≤ φ(v0, v) and f (t ) 6= ∅, ∀v ∈ c, v0 ∈ f (t );(iv) t is called quasi−φ−asymptotically nonexpansive provided that f (t ) 6= ∅ and there exists asequence {kn} ⊂ [1,∞) with kn −→ 1 as n →∞ such that φ(v0, t nv) ≤ knφ(v0, v), ∀v ∈ c, v0 ∈ f (t ), n ≥ 1. definition 1.2. a function t : c −→ b∗ is said to be [6] :(i) monotone if 〈τ1 − τ2, t τ1 − tτ2〉 ≥ 0, ∀τ1, τ2 ∈ b;(ii) γ−inverse strongly monotone (with i sm as short form) if ∃γ > 0 such that 〈τ1 − τ2, t τ1 − tτ2〉 ≥ γ ‖ tτ1 − tτ2 ‖2, ∀τ1, τ2 ∈ b; (iii) lipschitz continuous if ∃l > 0 such that ‖ tτ1 − tτ2 ‖≤ l ‖ τ1 − τ2 ‖, ∀τ1, τ2 ∈ b. if t is γ − i sm, then it is lipschitz continuous with 1 γ as a constant. definition 1.3. a mapping πc : b −→ c is called generalized projection [6], provided that πcτ1 = v0, for any τ1 ∈ b and v0 be the solution of φ(v0, τ1) = inf v∈c φ(v , τ1). https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 3an inertial-type algorithm is a method for speeding the convergence of the sequence of an algorithmintroduced by polyak [16]. numerous problems have been approximated by using inertial algorithms( for more details see, [4, 5, 12] and the references therein). mainge [13] proposed and studied thedevelopment of an inertialtype algorithm method as follows:{ un = ωn + θn(ωn − ωn−1), ωn+1 = (1− δn)un + δntun.takahashi and zembayashi [17] proposed an iterative process which converges strongly to a commonelement of solution of equilibrium problem and fixed point problem of relatively nonexpansivemapping. furthermore, the generalization of the proposed iterative process [17] have been carriedout by many researchers ( for more details see, [7, 8, 11, 18, 20] and the references therein). kazmiand ali [10] introduced an iterative algorithm for solving a common solution of ep.(1.3). and fixedpoint problemof quasi−φ− asymptotically nonexpansive mapping. alansari et al. [1] studied an inertial iterative method for finding a common solution of generalizedequilibrium, variational inequality and fixed point problems using the sequences {xn} and {zn}generated by the iterative algorithm: x0 = x1, z0 ∈ c, c0 := c; µn = xn + αn(xn − xn−1); yn = πcj −1(jµn − wngµn); un = j−1(δnjzn + (1− δn)jtyn); zn+1 = trnun; cn = {u ∈ c : φ(u, zn+1) ≤ δnφ(u, zn) + (1− δn)φ(u, µn); qn = 〈u ∈ c : xn − u, jxn − jx0〉 ≤ 0}; xn+1 = πcn∩qnx0,∀n ≥ 0,where {αn} ⊂ (0, 1), {wn} ⊂ (0,∞), {δn} ⊂ [0, 1] and {rn} ⊂ [a,∞), for some a > 0. then, {xn}converges strongly to $ = πγx0.farid et al. [6] proposed the following inertial algorithm for approximating a common solution ofgeneralized mixed equilibrium problem, variational inequality problem and fixed point problem forfamily of quasi−φ−nonexpansive mappings: https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 4  x0, x1 ∈ q, q1 := q; ωn = xn + θn(xn − xn−1); yn = πqj −1(jωn − wnqωn); vn = j−1(δn,0jωn + n∑ i=1 δn,ijtiωn); zn = j−1(αnjyn + (1− αn)jvn); un = trnzn; qn = {u ∈ q : φ(u, un) ≤ φ(u, ωn); qn = 〈u ∈ q : xn − u, jxn − jx0〉 ≤ 0}; xn+1 = πqn∩qnx0,∀n ≥ 1. consider {δn,i} and {αn} ⊂ [0, 1], {wn} ⊂ (0,∞), {θn} ⊂ (0, 1) and {rn} ⊂ [a,∞), for some a > 0. it has been proved that {xn} is a strong convergent to x̂ = πωx0.motivated and inspired by the work of kazmi and ali [10], alansari et al. [1] and farid et al. [6]. weproposed a hybrid inertial iterative algorithm for approximating a common solution of gmep.(1.1), v ip (1.4) and fixed point problem for a family of two quasi−φ−asymptotically nonexpansive map-pings in twouniformly convex and uniformly smooth banach spaces. our result extends andimproves the results of kazmi and ali [10], alansari et al. [1] and farid et al. [6], many results inthe literature. 2. preliminaries let w = {τ1 ∈ b :‖ τ1 ‖= 1} be the unit sphere of b. if for any ε ∈ (0, 2] there exists δ > 0 suchthat ‖ τ1 − τ2 ‖≥ ε =⇒ ‖ τ1 + τ2 ‖ 2 ≤ 1− δ, ∀τ1, τ2 ∈ w, then b is called uniformly convex. b is called strictly convex if ‖ τ1 + τ2 ‖ 2 < 1, ∀τ1, τ2 ∈ w and τ1 6= τ2. the space b is called smooth if lim t→0 ‖ τ1 + tτ2 ‖ − ‖ τ1 ‖ t exists, ∀τ1, τ2 ∈ w and also is said to be uniformly smooth if the limitis attained uniformly, ∀τ1, τ2 ∈ w.a function φ : b × b −→ r defined by φ(τ1, τ2) =‖ τ1 ‖2 −2〈τ1, jτ2〉+ ‖τ2 ‖2, ∀τ1, τ2 ∈ b. is consider as lyapunov functional. from the definition of φ, the following properties can be veri-fied [6]: (l1) (‖ τ1 ‖ − ‖ τ2 ‖)2 ≤ φ(τ1, τ2) ≤ (‖ τ1 ‖ + ‖ τ2 ‖)2, ∀τ1, τ2 ∈ b; (l2) φ(τ1, j −1(λjτ2 + (1− λ)jτ3)) ≤ λφ(τ1, τ2) + (1− λ)φ(τ1, τ3), ∀τ1, τ2, τ3 ∈ b, (l3) φ(τ1, τ2) =‖ τ1 ‖ ‖ jτ1 − jτ2 ‖ + ‖ τ2 ‖ ‖ τ1 − τ2 ‖, ∀τ1, τ2 ∈ b. https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 5 remark 2.1. consider b as smooth, strictly convex and reflexive banach space, then φ(τ1, τ2) = 0⇐⇒ τ1 = τ2, ∀τ1, τ2 ∈ b. lemma 2.2. [9] let c 6= ∅ be closed convex subset of a stricly convex, reflexive and smooth banach space b. then, ∃ a unique element τ0 ∈ c such that φ(τ0, τ1) = inf v∈c φ(v , τ1), for τ1 ∈ b. lemma 2.3. [15] let b be a uniformly convex and smooth banach space, c ⊂ b be closed convex and t : c −→ c be closed and quasi−φ−asymptotically nonexpansive mapping. then, f (t ) is closed and convex. lemma 2.4. [14] let c 6= ∅ be closed convex subset of b and q : c −→ b∗ be monotone and hemicontinuous function. then v ip (1.4). is closed and convex lemma 2.5. [19] let b be a 2−uniformly convex and smooth banach space. then, τ1, τ2 ∈ b, φ(τ1, τ2) ≥ δ ‖ τ1 − τ2 ‖2, where 0 < δ ≤ 1 and called two-uniformly convex constant. lemma 2.6. [19] let b be a two-uniformly convex banach space, then ‖ τ1 − τ2 ‖≤ 2 δ ‖ jτ1 − jτ2 ‖, ∀τ1, τ2 ∈ b, where 0 < δ ≤ 1. lemma 2.7. [9] let e be a smooth and uniformly convex banach space and let {un} and {vn} be sequences in e such that either {un} or {vn} is bounded. if lim n→∞ φ(un, vn) = 0, then lim n→∞ ‖ un − vn ‖= 0. remark 2.8. by considering (l3), it is observe that the converse of lemma 2.7 is true, providedthat {un} and {vn} are bounded lemma 2.9. [2] let c 6= ∅ be closed convex subset of a stricly convex, reflexive and smooth banach space b. then, φ(v ,πcτ1) + φ(πcτ1, τ1) ≤ (v , τ1), ∀v ∈ c, τ1 ∈ b. and, so for any τ1 ∈ b and v ∈ c, u = πcτ1 ⇐⇒ 〈v − u, jτ1 − jv〉, ∀u ∈ c. assumption 1: consider d : c × c −→ r as a bifunction satisfies the following assumptions [3]: (d1) d(v , v) = 0,∀v ∈ c; (d2) d is monotone, 1.e, d(v , u) +d(u, v) ≤ 0, ∀v , u ∈ c; (d3) the mapping v 7→ d(v , u) is upper hemicontinuity, ∀ u ∈ c. (d4) the mapping u 7→ d(v , u), u ∈ c is convex and lower semicontinuous. assumption 2: also consider ϑ : c×c −→ r as a bifunction satisfying the following assumptions: https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 6 (ϑ1) ϑ is skew-symmetric, i.e., ϑ(v , v)− ϑ(v , u)− ϑ(u, v) + ϑ(u, u) ≥ 0,∀v , u ∈ c; (ϑ2) ϑ is convex in the second argument; (ϑ3) ϑ is continuous. lemma 2.10. [1, 6, 21] let b a uniformly smooth, strictly convex and reflexive banach space and c ⊂ b be closed. let g : c −→ b∗ be a continuous and monotone mapping, d : c × c −→ r be a bifunction satisfying assumptions 1 and ϑ : c ×c −→ r be a bifunction satisfying assumptions 2. for any given number r > 0 and τ1 ∈ b, define a mapping tr : b −→ c by tr (τ1) = {u ∈ c : d(u, v) + 〈v − u, gu〉+ 1 r 〈v − u, ju − jτ1〉+ ψ(u, v)− ψ(u, u) ≥ 0,∀y ∈ c}, ∀v ∈ b. the mapping tr has the following properties: (p1) tr is single-valued; (p2) tr is a firmly nonexpansive type mapping, for all τ1, τ2 ∈ b, 〈trτ1 − trτ2, jtrτ1 − jtrτ2〉 ≤ 〈trτ1 − trτ2, jτ1 − jτ2〉, (p3) f (tr ) = sol(gmep (1.1)) is closed convex set of c; (p4) tr is quasi−φ− nonexpansive; (p5) φ(v0, trτ1) + φ(trτ1, τ1) ≤ φ(v0, τ1), ∀v0 ∈ f (tr ), τ1 ∈ b. furthermore, consider the map φ : b × b∗ −→ r, defined by φ(τ1, τ ∗ 1 ) =‖ τ1 ‖2 −〈τ1, τ ∗ 1 〉+ ‖ τ∗1 ‖2 observe that φ(τ1, τ ∗ 1 ) = φ(τ1, j −1τ∗1 ) lemma 2.11. [2] let b be a strictly convex, smooth and reflexive banach space. then φ(τ1, τ ∗ 1 ) + 2〈j−1τ∗1 − τ1, τ ∗ 2 〉 ≤ φ(τ1, τ ∗ 1 + τ∗2 ), ∀τ1 ∈ b, τ∗1 , τ∗2 ∈ b∗. 3. main results theorem 3.1. let c be a nonempty closed and convex subset of a 2−uniformly smooth and uniformly convex banach space b with b∗ as the dual space of b. let q :−→ b∗ be a γ−ism mapping with γ ∈ (0, 1) as a constant. let d : c × c −→ r be a bifunction satisfying assumption 1, ϑ : c × c −→ r be a bifunction satisfying assumption 2 and g : c −→ b∗ be a monotone and continuous mapping. let ti : c −→ c and si : c −→ c, for each i = 1, 2, ..., n be two finite family of closed li−lipschitz continuous and uniformly quasi−φ−asymptotically nonexpansive mappings such that ω := ( ∩ni=1 f (ti) ) ∩ ( ∩ni=1 f (si) ) ∩ sol ( v ip (1.4) ) ∩ sol ( gmep (1.1) ) 6= ∅. let {xn} https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 7 generated by algorithm : x0, x1 ∈ c, c1 := c, ωn = xn + αn(xn − xn−1), vn = πcj −1(jωn − βnqωn), yn = j−1(µn,0jωn + n∑ i=1 µn,ijt n i ωn); zn = j−1(ηn,0jvn + n∑ i=1 ηn,ijs n i yn), un = trnzn, cn+1 = {u ∈ cn : φ(u, un) ≤ k2 nφ(u, ωn)}, xn+1 = πcn+1 x0, ∀n ≥ 1, (3.1) where {αn} ⊂ (0, 1), {µn,i} ⊂ [0, 1] and {ηn,i} ⊂ (0, 1] satisfying the following conditions: (s1) n∑ i=0 µn,i = 1; (s2) n∑ i=0 ηn,i = 1; (s3) lim sup n→∞ ηn,0 < 1; (s4) for same a > 0, rn ∈ [a,∞); (s5) {βn} ⊂ (0,∞) satisfying the condition 0 < lim inf n→∞ βn < δ2γ 2 , where 0 < δ ≤ 1. then, {xn} converges strongly to $, where $ = πωx0 is consider as the generalized projection of $ onto ω. proof. we consider the proof in the following steps: step 1 : we show that cn+1 is closed and convex for each n ≥ 1 and {xn} is well defined.observe clearly that c1 = c is closed and convex. suppose that cn is closed and convex for each n ∈ n. now, we know from 3.1 that for any u ∈ cn, φ(u, un) ≤ k2 nφ(u, ωn) ⇐⇒ (1− k2 n ) [ ‖ u ‖2 −2(1− k2 n )〈u, jun〉+ 2k2 n 〈u, jωn − jun〉 ] ≤ k2 n ‖ ωn ‖2 − ‖ un ‖2 . then, cn+1 is closed and convex. implies that πcn+1 x0 is well defined ∀n ≥ 1, also {xn} is welldefined. furthermore since ω 6= ∅, by considering lemma 2.3, 2.4 and 2.10 we conclude that ω isclosed and convex, and so πωx0 is well defined. step 2 : we show that ω ⊂ cn, ∀n ≥ 1. it is obvious that ω ⊂ c1 = c. suppose that ω ⊂ cn forsome n ≥ 1. let x̂ ∈ ω, from the definition of φ, quasi−φ−asymptotically nonexpansive mapping https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 8of si and convexity of ‖ . ‖2 we have the following estimate: φ(x̂ , un) = φ(x̂ , trnzn) ≤ φ(x̂ , zn) (3.2) = φ ( x̂ , j−1(ηn,0jvn + n∑ i=1 ηn,ijs n i yn) ) = ‖ x̂ ‖2 −2(〈x̂ , ηn,0jvn + n∑ i=1 ηn,ijs n i yn〉) + ‖ηn,0jvn + n∑ i=1 ηn,ijs n i yn‖2 ≤ ‖ x̂ ‖2 −2ηn,0〈x̂ , jvn〉 − 2 n∑ i=1 ηn,i 〈x̂ , jsni yn〉+ ηn,0‖jvn‖2 + n∑ i=1 ηn,i‖jsni yn‖2 = ηn,0 ( ‖x̂‖2 − 2〈x̂ , jvn〉+ ‖vn‖2 ) + n∑ i=1 ηn,i ( ‖x̂‖2 − 2〈x̂ , jsni yn〉+ ‖sni yn‖2 ) = ηn,0φ(x̂ , vn) + n∑ i=1 ηn,iφ(x̂ , sni yn) ≤ ηn,0φ(x̂ , vn) + kn n∑ i=1 ηn,iφ(x̂ , yn) (3.3) similarly, by quasi−φ−asymptotically nonexpansive of ti , definition of φ and convexity of ‖ . ‖2,we estimate as follows: φ(x̂ , yn) = φ ( x̂ , j−1(µn,0jωn + n∑ i=1 µn,ijt n i ωn) ) = ‖ x̂ ‖2 −2(〈x̂ , µn,0jωn + n∑ i=1 µn,ijt n i ωn〉)+ ‖ µn,0jωn + n∑ i=1 µn,ijt n i ωn ‖2 ≤ ‖ x̂ ‖2 −2µn,0〈x̂ , jωn〉 − 2 n∑ i=1 µn,i 〈x̂ , jt ni ωn〉+ µn,0‖jωn‖2 + n∑ i=1 µn,i‖jt ni ωn‖2 = µn,0 ( ‖x̂‖2 − 2〈x̂ , jωn〉+ ‖ωn‖2 ) + n∑ i=1 µn,i ( ‖ x̂ ‖2 −2〈x̂ , jt ni ωn〉+ ‖t ni ωn‖2 ) = µn,0φ(x̂ , ωn) + n∑ i=1 µn,iφ(x̂ , t ni ωn) ≤ µn,0φ(x̂ , ωn) + kn n∑ i=1 µn,iφ(x̂ , ωn) ≤ knµn,0φ(x̂ , ωn) + kn n∑ i=1 µn,iφ(x̂ , ωn) = knφ(x̂ , ωn) (3.4) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 9it has been observe from (3.3) and (3.4) that φ(x̂ , un) ≤ ηn,0φ(x̂ , vn) + kn n∑ i=1 ηn,i [ knφ(x̂ , ωn) ] = ηn,0φ(x̂ , vn) + k2 n n∑ i=1 ηn,iφ(x̂ , ωn) ≤ k2 nηn,0φ(x̂ , vn) + k2 n n∑ i=1 ηn,iφ(x̂ , ωn) (3.5) also, by lemma 2.6 and 2.11, we estimate as: φ(x̂ , vn) = φ ( x̂ ,πcj −1(jωn − βnqωn) ) ≤ φ ( x̂ , j−1(jωn − βnqωn) ) = φ ( x̂ , jωn − βnqωn) ≤ φ ( x̂ , (jωn − βnqωn) + βnqωn ) − 2〈j−1(jωn − βnqωn)− x̂ , βnqωn〉 = φ(x̂ , jωn)− 2βn〈j−1(jωn − βnqωn)− x̂ , qωn〉 = φ(x̂ , ωn)− 2〈ωn − x̂ , qωn〉 − 2βn〈j−1(jωn − βnqωn)− ωn, qωn〉 = φ(x̂ , ωn)− 2〈ωn − x̂ , qωn −qx̂〉 − 2βn〈j−1(jωn − βnqωn)− ωn, qωn〉 ≤ φ(x̂ , ωn)− 2βnγ ‖ qωn‖2 + 2βn ‖ j−1(jωn −qωn)− j−1jωn‖‖qωn‖2 ≤ φ(x̂ , ωn)− 2βnγ ‖ qωn ‖2 + 4β2 n δ2 ‖ qωn ‖2 = φ(x̂ , ωn)− 2βn ( γ − 2βn δ2 ) ‖ qωn ‖2, (3.6) if follows by combined with βn < δ2 2 that φ(x̂ , vn) ≤ φ(x̂ , ωn) (3.7) now, putting (3.7) in (3.5) leads to φ(x̂ , un) ≤ k2 nηn,0φ(x̂ , ωn) + k2 n n∑ i=1 ηn,iφ(x̂ , ωn) = (ηn,0 + n∑ i=1 ηn,i)k 2 nφ(x̂ , ωn) = k2 nφ(x̂ , ωn), which gives φ(x̂ , un) ≤ k2 nφ(x̂ , ωn), (3.8) therefore x̂ ∈ cn+1, implies that ω ⊂ cn+1. hence ω ⊂ cn, ∀n ≥ 1. https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 10 step 3 : we show that {xn}, {ωn}, {vn}, {yn}, {zn} and {un} are bounded and {xn} is cauchy.we consider xn = πcnx0 and cn+1 ⊂ cn, ∀n ≥ 1. then from lemma 2.9, we observe that φ(xn, x0) ≤ φ(xn+1, x0) hence {φ(xn, x0)} is non decreasing. also it has been observe that φ(xn, x0) = φ(πcnx0, x0) ≤ φ(x̂ , x0)− φ(x̂ , xn) ≤ φ(x̂ , x0), which gives that {φ(xn, x0)} is bounded and {xn} is also bounded. therefore, since {φ(xn, x0)} nondecreasing. {φ(xn, x0)} convergent. taking the advantage of {xn} as a bounded sequence impliesthat {ωn}, {vn}, {yn}, {zn} and {un} are all bounded. also by lemma 2.9, we have φ(xm, xn) = φ(xm,πcnx0) ≤ φ(xm, x0)− φ(xn, x0) −→ 0 as n,m →∞. (3.9) by lemma 2.7, we have lim n→∞ ‖ xm − xn ‖= 0. hence {xn} is a cauchy sequence. step 4 : we show that xn −→ $, ωn −→ $, un −→ $, zn −→ $, yn −→ $and vn −→ $ (as n → ∞). since {xn} is a cauchy sequence, then by the closedness of c andthe completeness of b, we can assume that there exists $ ∈ c such that lim n→∞ xn = $. (3.10) now, setting m = n + 1 in (3.9), we obtain lim n→∞ φ(xn+1, xn) = 0. (3.11) using lemma 2.7, we get lim n→∞ ‖xn+1 − xn‖ = 0. (3.12) we observe from (3.1) that ‖ ωn − xn ‖=‖ αn(xn − xn−1) ‖≤‖ xn − xn−1 ‖ using (3.12), we arrive at lim n→∞ ‖ωn − xn‖ = 0. (3.13) by (3.10) and (3.13), we conclude that lim n→∞ ωn = $. (3.14) taking the advantage of remark 2.8, (3.13) and boundedness of {ωn}, we get lim n→∞ φ(ωn, xn) = 0. (3.15) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 11also, by (3.12) and (3.13), we obtain lim n→∞ ‖xn+1 − ωn ‖= 0. (3.16) using remark 2.8, we present (3.16) as lim n→∞ φ(xn+1, ωn) = 0. (3.17) we observe from xn+1 = πcn+1 x0 ∈ cn+1 ⊂ cn and definition of cn that φ(xn+1, un) ≤ k2 nφ(xn+1, ωn) using (3.17,) we obtain lim n→∞ φ(xn+1, un) = 0. applying lemma 2.7, we get lim n→∞ ‖ xn+1 − un ‖= 0. (3.18) taking the advantage of triangular inequality, we present ‖xn − un‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − un‖ by (3.12) and (3.18), we obtain lim n→∞ ‖ xn − un ‖= 0. (3.19) it follows from (3.10) and (3.19) that lim n→∞ un = $. (3.20) similarly, by definition of cn and xn+1 = πcn+1 x0 ∈ cn+1 ⊂ cn, we also present that φ(xn+1, zn) ≤ k2 nφ(xn+1, ωn) by applying (3.17,) we arrive at lim n→∞ φ(xn+1, zn) = 0. using lemma 2.7, we have lim n→∞ ‖ xn+1 − zn ‖= 0. (3.21) taking into account that ‖xn − zn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − zn‖ using (3.12) and (3.21), we get lim n→∞ ‖ xn − zn ‖= 0. (3.22) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 12by considering(3.10) and (3.22), we obtain lim n→∞ zn = $. (3.23) also from the definition of cn and xn+1 = πcn+1 x0 ∈ cn+1 ⊂ cn, we estimate as φ(xn+1, yn) ≤ k2 nφ(xn+1, ωn) by (3.17,) we get lim n→∞ φ(xn+1, yn) = 0. it follows from lemma 2.7 that lim n→∞ ‖ xn+1 − yn ‖= 0. (3.24) by triangular inequality, we obtain ‖xn − yn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − yn‖ also by (3.12) and (3.24), we get lim n→∞ ‖ xn − yn ‖= 0. (3.25) using (3.10) and (3.25), we obtain lim n→∞ yn = $. (3.26) finally, by considering xn+1 = πcn+1 x0 ∈ cn+1 ⊂ cn and definition of cn, we present that φ(xn+1, vn) ≤ k2 nφ(xn+1, ωn) applying (3.17,) we obtain lim n→∞ φ(xn+1, vn) = 0. by lemma 2.7, we get lim n→∞ ‖ xn+1 − vn ‖= 0. (3.27) we consider the following estimate using triangular inequality ‖xn − vn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − vn‖ using (3.12) and (3.27), we obtain lim n→∞ ‖ xn − vn ‖= 0. (3.28) using (3.10) and (3.28), we obtain lim n→∞ vn = $. (3.29) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 13 step 4 : we show that ‖ ωn − t ni ωn ‖=‖ yn − sni yn ‖= 0. now, taking the advantage of j asuniformly continuity on bounded sets, then it follows from (3.16) and (3.24) that ‖ jωn − jxn+1 ‖=‖ jxn+1 − jyn ‖= 0. (3.30) from (3.1), we observe that ‖jxn+1 − jyn ‖ = ‖ jxn+1 − ( µn,0jωn + n∑ i=1 µn,ijt n i ωn ) ‖ = ‖ n∑ i=1 µn,ijxn+1 − n∑ i=1 µn,ijt n i ωn + µn,0jxn+1 − µn,0jωn ‖ = ‖ n∑ i=1 µn,i ( jxn+1 − jt ni ωn ) + µn,0 ( jxn+1 − jωn ) ‖ ≥ n∑ i=1 µn,i ‖ jxn+1 − jt ni ωn ‖ −µn,0 ‖ jωn − jxn+1 ‖, this gives ‖ jxn+1 − jt ni ωn ‖≤ 1 n∑ i=1 µn,i [ ‖ jxn+1 − jyn ‖ +µn,0 ‖ jωn − jxn+1 ‖ ] . by (3.30), we arrive at lim n→∞ ‖ jxn+1 − jt ni ωn ‖= 0. as j−1 is uniform norm-to-norm continuous on bounded sets, we present that lim n→∞ ‖ xn+1 − t ni ωn ‖= 0. (3.31) taking into account that ‖ ωn − t ni ωn ‖≤‖ ωn − xn+1 ‖ + ‖ xn+1 − t ni ωn ‖ by (3.16) and (3.31), we obtain lim n→∞ ‖ ωn − t ni ωn ‖= 0. (3.32) similarly, we observe from (3.21), (3.27) and by continuity of j that ‖ jxn+1 − jzn ‖=‖ jxn+1 − jvn ‖= 0. (3.33) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 14also by (3.1), we observe that ‖ jxn+1 − jzn ‖ = ‖ jxn+1 − ( ηn,0jvn + n∑ i=1 ηn,ijs n i yn ) ‖ = ‖ n∑ i=1 ηn,ijxn+1 − n∑ i=1 ηn,ijs n i yn + ηn,0jxn+1 − ηn,0jvn ‖ = ‖ n∑ i=1 ηn,i ( jxn+1 − jsni yn ) + ηn,0 ( jxn+1 − jvn ) ‖ ≥ n∑ i=1 ηn,i ‖ jxn+1 − jsni yn ‖ −ηn,0 ‖ jvn − jxn+1 ‖, this implies ‖jxn+1 − jsni yn‖ ≤ 1 n∑ i=1 ηn,i [ ‖jxn+1 − jzn‖+ ηn,0‖jvn − jxn+1 ‖ ] . also by (3.33), we get lim n→∞ ‖ jxn+1 − jsni yn ‖= 0. applying j−1 as uniform norm-to-norm continuous on bounded sets, we have lim n→∞ ‖ xn+1 − sni yn ‖= 0. (3.34) by triangular inequality, we obtain ‖ yn − sni yn ‖≤‖ yn − xn+1 ‖ + ‖ xn+1 − sni yn ‖ by (3.24) and (3.34), we get lim n→∞ ‖ yn − sni yn ‖= 0. (3.35) therefore by (3.32) and (3.35), we conclude that lim n→∞ ‖ ωn − t ni ωn ‖= lim n→∞ ‖ yn − sni yn ‖= 0. step 5 : we show that $ ∈ ω. to show this we claim as follows: we claim that $ ∈ ( ∩ni=1 f (ti) ) ∩ ( ∩ni=1 f (si) ) . by triangular inequality for i ≥ 1, we have ‖ t ni ωn −$ ‖≤‖ t ni ωn − ωn ‖ + ‖ ωn −$ ‖ . using (3.14) and (3.32), we arrive at lim n→∞ ‖ t ni ωn −$ ‖= 0. (3.36) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 15by the assumption that for each ti is uniformly li−lipschitz continuous, we obtain ‖t n+1 i ωn − t ni ωn‖ ≤ ‖t n+1 i ωn − t n+1 i ωn+1‖+ ‖t n+1 i ωn+1 − ωn+1‖ + ‖ωn+1 − ωn‖+ ‖ωn − t ni ωn‖ ≤ (li + 1)‖ωn+1 − ωn‖+ ‖t n+1 i ωn+1 − ωn+1‖+ ‖ωn − t ni ωn‖. by (3.12) and (3.32,) we get lim n→∞ ‖t n+1 i ωn − t ni ωn‖ = 0. which yields from (3.36) that lim n→∞ ‖t n+1 i ωn −$‖ = 0, ∀i ≥ 1. consequently, we get ti(t ni )ωn −→ $ ( as n →∞). in view of the closedness of ti , we arrive at ti$ = $, ∀i ≥ 1. thus $ ∈ ∩ni=1f (ti). furthermore, following similar argument as above, onecan also claim that $ ∈ ∩ni=1f (si). hence $ ∈ ( ∩ni=1 f (ti) ) ∩ ( ∩ni=1 f (si) ) . next, we claim that $ ∈ sol(v ip (1.4)). consider the triangular inequality ‖ ωn − zn ‖≤‖ ωn − xn ‖ + ‖ xn − zn ‖ . using (3.13) and (3.22,) leads to lim n→∞ ‖ ωn − zn ‖= 0. (3.37) from the uniform continuity of j on bounded set, we get lim n→∞ ‖ jωn − jzn ‖= 0. (3.38) since x̂ ∈ ω, then it follows from (3.2), (3.3), (3.4) and (3.6) that φ(x̂ , zn) ≤ ηn,0 [ φ(x̂ , ωn)− 2βn ( γ − 2βn δ2 ) ‖qωn ‖2 ] + kn n∑ i=1 ηn,i [ knφ(x̂ , ωn) ] ≤ k2 nηn,0φ(x̂ , ωn) + k2 n n∑ i=1 ηn,iφ(x̂ , ωn)− 2βnηn,0 ( γ − 2βn δ2 ) ‖ qωn ‖2 = k2 nφ(x̂ , ωn)− 2βnηn,0 ( γ − 2βn δ2 ) ‖ qωn ‖2, implies that 2βnηn,0 ( γ − 2βn δ2 ) ‖ qωn ‖2≤ k2 nφ(x̂ , ωn)− φ(x̂ , zn) (3.39) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 16but k2 nφ(x̂ , ωn)− φ(x̂ , zn) = k2 n [ ‖ x̂ ‖2 −2〈x̂ , jωn〉+ ‖ ωn ‖2 ] − [ ‖ x̂ ‖2 −2〈x̂ , jzn〉+ ‖ zn ‖2 ] = (k2 n − 1)‖x̂‖2 − 2(k2 n − 1)〈x̂ , jzn〉 − 2k2 n 〈x̂ , jωn − jωn〉 + k2 n‖ωn ‖2 − ‖ zn ‖2 = (k2 n − 1) ‖ x̂ ‖2 −2(k2 n − 1)〈x̂ , jzn〉 − 2k2 n 〈x̂ , jωn − jzn〉 + (k2 n − 1)‖ωn ‖2 + ‖ ωn ‖2 − ‖ zn ‖2 ≤ | (k2 n − 1) ‖ x̂ ‖2| + | 2(k2 n − 1)〈x̂ , jzn〉 | + | 2k2 n 〈x̂ , jωn − jzn〉 | + | (k2 n − 1)‖ωn ‖2| + |‖ ωn ‖2 + ‖ zn ‖2| ≤ (k2 n − 1) ‖ x̂ ‖2 +2(k2 n − 1) ‖ x̂ ‖ ‖ jzn ‖ +2k2 n ‖ x̂ ‖ ‖ jωn − jzn ‖ + (‖ ωn − zn ‖)(‖ ωn ‖ + ‖ zn ‖). since kn −→ 1 as n −→∞, then by (3.37) and (3.38,) we obtain lim n→∞ ( k2 nφ(x̂ , ωn)− φ(x̂ , zn) ) = 0. (3.40) also since βnηn,0(γ − 2βn δ2 ) > 0, by (3.39) and (3.40), we have lim n→∞ ‖ qωn ‖= 0. (3.41) taking the advantage of q as γ − i sm and so 1 γ −lipschitz continuous. therefore, it follows from(3.38) and (3.40) that $ ∈ q−1(0). hence, $ ∈ sol(v ip (1.4)). we also claim that $ ∈ sol(gmep (1.1)). consider the triangular inequality ‖ un − zn ‖≤‖ un − xn ‖ + ‖ xn − zn ‖ . by (3.19) and (3.22), we get lim n→∞ ‖ un − zn ‖= 0. from uniform continuity of j on bounded sets, we obtain lim n→∞ ‖ jun − jzn ‖= 0. (3.42) since rn ≥ a and by (3.42), we have lim n→∞ ‖ jun − jzn ‖ rn = 0. (3.43) equation un = trnzn implies that h(un, v) + 1 rn 〈v − un, jun − jzn〉+ ϑ(v , un)− ϑ(un, un) ≥ 0, ∀v ∈ c. https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 17where h(un, v) = d(un, v) + 〈gun, v − un〉. by applying assumption (d2), we obtain 1 rn 〈v − un, jun − jzn〉 ≥ −h(un, v)− ϑ(v , un) + ϑ(un, un) ≥ h(v , un)− ϑ(v , un) + ϑ(un, un). letting n −→∞, by assumption (d4) and (3.43), we get h(v ,$)− ϑ(v ,$) + ϑ($,$) ≤ 0, ∀v ∈ c. for all s ∈ (0, 1] and v ∈ c, setting vs := sv + (1− s)$. therefore vs ∈ c and then, h(vs ,$)− ϑ(vs ,$) + ϑ($,$) ≤ 0. by assumption (d1)− (d4), we estimate as 0 = h(vs , vs) ≤ sh(vs , v) + (1− s)h(vs ,$) ≤ sh(vs , v) + (1− s) [ ϑ(vs ,$)− ϑ($,$) ] ≤ sh(vs , v) + (1− s) [ ϑ(v ,$)− ϑ($,$) ] as s > 0, from assumption (d3), we conclude that h($, v) + ϑ(v ,$)− ϑ($,$) ≥ 0, ∀v ∈ c. hence, $ ∈ sol(gmep (1.1)). step 6 : finally we show that $ = πωx0 and so xn −→ πωx0 as n −→ ∞. putting x∗ = πωx0,since x∗ ∈ ω ⊂ cn and xn = πωx0, we have φ(xn, x0) ≤ φ(x∗, x0), ∀n ≥ 0. then φ($, x0) = lim n→∞ φ(xn, x0) ≤ φ(x∗, x0), implies that $ = x∗ and since x∗ = πωx0, then we conclude that xn −→ $ = πωx0, as n →∞.this completes the proof. � corollary 3.2. let c be a nonempty closed and convex subset of a 2−uniformly smooth and uniformly convex banach space b with b∗ as the dual space of b. let d : c × c −→ r be a bifunction satisfying assumption 1, ϑ : c × c −→ r be a bifunction satisfying assumption 2 and g : c −→ b∗ be a monotone and continuous mapping. let ti : c −→ c and si : c −→ c, for each i = 1, 2, ..., n be two finite family of closed li−lipschitz continuous and uniformly https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 18 quasi−φ−asymptotically nonexpansive mappings such that ω := ( ∩ni=1 f (ti) ) ∩ ( ∩ni=1 f (si) ) ∩ ∩sol ( gmep (1.1) ) 6= ∅. let {xn} generated by algorithm : x0, x1 ∈ c, c1 := c, ωn = xn + αn(xn − xn−1), yn = j−1(µn,0jωn + n∑ i=1 µn,ijt n i ωn); zn = j−1(ηn,0jωn + n∑ i=1 ηn,ijs n i yn), un = trnzn, cn+1 = {u ∈ cn : φ(u, un) ≤ k2 nφ(u, ωn)}, xn+1 = πcn+1 x0, ∀n ≥ 1, where {αn} ⊂ (0, 1), {µn,i} ⊂ [0, 1] and {ηn,i} ⊂ (0, 1] satisfying the following conditions: (s1) n∑ i=0 µn,i = 1; (s2) n∑ i=0 ηn,i = 1; (s3) lim sup n→∞ ηn,0 < 1; (s4) for same a > 0, rn ∈ [a,∞). then, {xn} converges strongly to $, where $ = πωx0 is consider as the generalized projection of $ onto ω. corollary 3.3. let c be a nonempty closed and convex subset of a 2−uniformly smooth and uniformly convex banach space b with b∗ as the dual space of b. let d : c × c −→ r be a bifunction satisfying assumption 1 and g : c −→ b∗ be a monotone and continuous mapping. let ti : c −→ c and si : c −→ c, for each i = 1, 2, ..., n be two finite family of closed li−lipschitz continuous and uniformly quasi−φ−asymptotically nonexpansive mappings such that ω := ( ∩ni=1 f (ti) ) ∩ ( ∩ni=1 f (si) ) ∩ ∩sol ( gep (1.2) ) 6= ∅. let {xn} generated by algorithm : x0, x1 ∈ c, c1 := c, ωn = xn + αn(xn − xn−1), yn = j−1(µn,0jωn + n∑ i=1 µn,ijt n i ωn); zn = j−1(ηn,0jωn + n∑ i=1 ηn,ijs n i yn), un = trnzn, cn+1 = {u ∈ cn : φ(u, un) ≤ k2 nφ(u, ωn)}, xn+1 = πcn+1 x0, ∀n ≥ 1, https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 19 where {αn} ⊂ (0, 1), {µn,i} ⊂ [0, 1] and {ηn,i} ⊂ (0, 1] satisfying the following conditions: (s1) n∑ i=0 µn,i = 1; (s2) n∑ i=0 ηn,i = 1; (s3) lim sup n→∞ ηn,0 < 1; (s4) for same a > 0, rn ∈ [a,∞). then, {xn} converges strongly to $, where $ = πωx0 is consider as the generalized projection of $ onto ω. 4. numerical example let b = r and c = [0, 1]. let q : c → c be defined by qu = 2u∀u ∈ c. define ϑ : c×c → r, d : c × c → r, g : c → r, q : c → r, ti : c → c and si : c → c by ϑ(u, v) = 0, d(u, v) = (u + v)(v − u), g(u) = u, q(u) = 2u and ti(u) = si(u) = 1 i+1u, respectively.setting {βn} = {0.9 2n }, rn = 1 2 , {αn} = 0.9, µ0,n = 1 2 , ∑ni=1 µn,i = 1 2 such that ∑ni=0 µi ,n = 1 and η0,n = 1 3 , ∑ni=1 ηn,i = 2 3 so that ∑ni=0 ηi ,n = 1.let {xn} be generated by the hybrid inertial iterative algorithm (3.1) converges to x∗ = {0} ∈ ω. proof. clearly ϑ and d satisfy assumptions 1 and 2, respectively, and g is continuous andmonotone so that sol(gmep (eq1.1)) = {0} 6= ∅, sol(v ip (eq1.4)) = {0} 6= ∅. obviously q is 1 2 − i sm, and ti and si are two finite families of closed 1-lipschitz continuous and uni-formly quisi-φ-asymptotically nonexpansive mappings with f ix(ti) = f ix(si) = {0}. thus ω = sol(gmep (eq1.1)) ∩ sol(v ip (eq1.4)) ∩ f ix(ti) ∩ f ix(si) = {0} 6= ∅. hence, the it-erative scheme (3.1) becomes the following scheme (4.1) after simplification: x0, x1 ∈ c, c1 := c, ωn = xn + 0.9(xn − xn−1), yn = 1 2ωn + 1 2(n+1)ωn, zn = 1 3yn + 2 3(n+1)vn, un = 2zn 7 , cn+1 = [ 0, un+ωn 2 ] , xn+1 = πcn+1x0, ∀n ≥ 1, where, f or πc a metr ic projection onto c, vn = πc(ωn − βnqωn) =  0, ωn − 0.9 2n ωn < 0 1, ωn − 0.9 2n ωn > 1 ωn − 0.9 2n ωn, otherwise. (4.1) https://doi.org/10.28924/ada/ma.4.8 eur. j. math. anal. 10.28924/ada/ma.4.8 20finally, using the software matlab 7.8.0, we have the following figure which shows that {xn}converges to {0} as n →∞. figure 1. convergence of {xn} when x0 = 1.0 and x1 = 0.5 references [1] m. alansari, r. ali and m. farid, strong convergence of an inertial iterative algorithm for variational inequalityproblem, generalized equilibrium problem and fixed point problem in a banach space, j. ineq. appl. 2020 (2020) 42.[2] y.i. alber, metric and generalized projection operators in banach spaces, in: properties and applications, lect.note, pure. appl. math. 8(1996), 15–50.[3] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. stud. 63(1994) 123–145.[4] r.i. bot, e.r. csetnek and c. hendrich, inertial douglas-racheord splitting for monotone inclusion problems, appt.math. comp. 256 (2015) 472–487.[5] r.i. bot and e.r. csetnek, an inertial forwardbackward forward 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speeding up the convergence of iteration methods. ussr comp. math. phys. 4 (1964)1–7.[17] w. takahashi and k. zembayashi, strong and weak convergence theorem for equilibrium problem and relativelynonexpansive mappings in banach space, nonlinear anal. tma. 70 (2009) 45–57.[18] vandana, r. dubey, deepmala, l.n. mishra and v.n. mishra, duality relations for a class of a multiobjective factionprogramming problem involving support functions, amer. j. oper. res. 8 (2018) 293–311.[19] h.k. xu, inequality in banach space with application, nonlinear. anal. tma. 16 (1991) 11271138.[20] c. zalinesco, on uniformly convex function, j. math. anal. appl. 95 (1983) 344–374.[21] h. zegeye, a hybrid iterative scheme for equilibrium problems, variational inequality problems and common fixedpoint problems in banach spaces, nonlinear anal. tma. 72 (2010) 2136–2146. https://doi.org/10.28924/ada/ma.4.8 1. introduction 2. preliminaries 3. main results 4. numerical example references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 10doi: 10.28924/ada/ma.4.10 finite difference method for solving second-order boundary value problems with high-order accuracy nguyen dinh dung∗, vu vinh quang thai nguyen university of information and communication technology, thai nguyen, vietnam ∗correspondence: nddung@ictu.edu.vn abstract. when researching and solving practical problems in continuous environments, throughmodeling methods, the vast majority of problems lead to models described by equations containingdifferential operators. in case the problem model is not complicated, we often obtain simple partialdifferential equations, then the solution of the problem can be obtained directly through analyti-cal methods. most complex problems, through the method of approximating differential operatorsby difference operators, from which differential problems are approximated by corresponding differ-ence schemes and approximate solutions will be obtained. is achieved through solving systems ofdifference equations based on the tools of electronic computers. then, building difference schemesto approximate the differential problem with high-order accuracy will play an important role in theaccuracy of the obtained approximate solution. in this paper, we propose two difference schemes withhigh-order accuracy to solve second-order differential problems with dirichlet and mixed boundaryconditions. theoretical results and experimental calculations have confirmed the accuracy of theproposed schemes. 1. introduction in this paper, we consider the second-order boundary problem with mixed boundary conditions u′′(x) = f (x), x ∈ (a, b) c0u(a)− c1u′(a) = c d0u(b) + d1u ′(b) = d c0, c1, d0, d1 ≥ 0 (1) in some cases when the function f (x) is a polynomial function, trigonometric function, exponentialfunction or product of the above functional forms, the solution to problem (1) can be found byanalytical methods or using using green’s function method, we must find the approximate solutionof problem (1) by numerical methods using electronic computers.in order to find approximate solutions using numerical methods, we need to build systems ofdifference equations that approximate the differential problem and then find numerical solutions received: 11 feb 2024. key words and phrases. derivative; grid space; mesh function; partial differential equations; difference schemes.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 2using algebraic methods. then the accuracy order of approximating differential operators usingdifference operators will determine the accuracy order of the obtained approximated solution.in [1], a method for building a difference scheme for problem (1) was introduced by approximatingthe first and second derivatives of the function u(x) with accuracy o(h), where h is the grid stepin grid space, in [2] a method of building a difference scheme with accuracy o(h2). in addition,some authors [3][5] have proposed difference methods with order of accuracy from o(h4) to o(h8).however, these methods lead to very complicated systems of difference equations that requiringthe use of intermediate grid points. this makes solving these systems of difference equationsvery complicated. therefore, it is required to build a difference scheme with order of accuracy forproblem (1) so that the resulting system of difference equations is simple and can be solved withan algorithm of linear complexity.studying methods to approximate the derivative value at a point of a function is an importantresearch direction that mathematicians are especially interested in. in the literature on numericalmethods [1], formulas for approximating the first and second derivatives of the function u(x) withaccuracy order o(h2) have been given, which uses the value of the function at 3 neighboring pointsof point x and h is denoted as the grid step on the grid space. based on the interpolation polynomialapproximation method, in the document [6], the m-order derivative approximation formulas withaccuracy order o(h5−m) are given, which uses value of the function at 5 neighboring points ofpoint x . recently, in the document [7], a set of formulas for approximating derivatives of allorder with accuracy order o(hn−m) has been introduced, which uses the value of the function at n + 1 neighboring points of point x . based on the published formulas, numerical solutions fordifferential problems with nonlinear differential equations of all order have been improved. in [8]-[12], published algorithms for numerically solving 3rd and 4th order nonlinear differential equationswith accuracy order o(h6). however, the published results of the above derivative approximationformulas are all results obtained by the direct method, results obtained by direct calculation ofanalytical expressions, not general. thus, to improve the accuracy of derivative approximation, itis necessary to build a general algorithm to provide formulas for approximating derivatives of anydegree with arbitrary order of accuracy based on the support of the computer.the main content of the article presents research results based on the taylo expansion formula [1]and the algorithm for determining numerical derivatives with high order accuracy, thereby proposinga method to build two difference scheme for problem (1) in the case of dirichlet boundary conditionsand mixed boundary conditions with arbitrary high-order precision. difference schemes are verysimple and can be solved with the algorithm has o(n) complexity where n is the number of pointsin the grid space. the experimental results we calculated confirm the accuracy of the proposedschemes. https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 3the structure of the article consists of four parts: in section 1, we introduce some publishedresults on based on the difference method. section 2, we present the theoretical basis of thederivative approximation method based on the taylor formula, then we propose a method to build adifference scheme with high-order accuracy based on the high-order derivative approximation andevaluate schema accuracy. section 3, we present some experimental calculation results to evaluatethe accuracy of the proposed algorithm. finally, there are conclusions and references. 2. proposed method 2.1. derivative approximation formulas consider the function u(x) ∈ cn+1[a, b], expanding the taylor series [3], we have the formula u(x + ∆x) = u(x) + u′(x)∆x + u′′(x) 2! (∆x)2 + ...+ u(n)(x) n! (∆x)n + rn(x) (2) where rn(x) = u(n+1)(θ) (n+1)! (∆x)n+1, θ ∈ (x, x + ∆x) since formula (2), we can approximate thederivative of the function u(x) in the neighborhood x as follows: u(x + ∆x) ≈ u(x) + u′(x)∆x + u′′(x) 2! (∆x)2 + ...+ u(n)(x) n! (∆x)n (3) divide the interval [a, b] by (n+ 1) grid points xi = a+ ih, i = 0, 1, ..., n, grid step h = b−a n , sinceformula (3), we get the following formulas u(xi + h) = u(xi) + u′(xi)h + u′′(xi) 2! h2 + ...+ u(n)(xi) n! hn +o(hn+1), i = 0, n − 1 (4) u(xi − h) = u(xi)− u′(xi)h + u′′(xi) 2! h2 + ...+ (−1)n u(n)(xi) n! hn +o(hn+1), i = 0, n − 1 (5) using (4), we obtain u′(xi) = u(xi + h)− u(xi) h − u′′(xi) 2! h − ...− u(n)(xi) n! hn−1 +o(hn) (6) formula (6) is a formula that supports approximating the first derivative on a uniform grid witherror o(hn).since (4) and (5), we have u′′(xi) = u(xi + h)− 2u(xi) + u(xi − h) h2 − u(4)(xi) 4! h2 − ...− u(2m)(xi) (2m)! h2m−2 +o(hn−1) (7) let v(x) be a function defined on [a, b], ωh be the grid space with grid step h = (b − a)/n , xi = a + ih, (i = 0, 1, 2, . . . , n), v = (v (x0), v (x1), . . . , v (xn)) is the approximated value of the vd on ωh, where vd = (v(x0), v(x1), . . . , v(xn)). definition 1: on grid ωh, a method for approximating the function v(x) that is said to have n-order accuracy if ‖v − vd‖ωh = o(hn). https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 4since (3), (6), (7), we can build derivative approximation formulas with (n+1)order accuracy forthe grid function, and n-order accuracy for first-order derivative, (n-1)order accuracy for second-order derivative. lemma 1: if the accuracy of the second derivative approximation is of order (n − 2), then the accuracy of the function approximation will be of order n and vice versa.let ti(x) = (x − z0)(x − z1)...(x − zi−1)(x − zi+1)...(x − zn), mi(x) = (zi − z0)(zi − z1)...(zi − zi−1)(zi − zi+1)...(zi − zn).let b(n(m)) be the coefficient matrix in the morder derivative approximation formula using (n + 1) neighboring points, then the grid derivative of orders will be determined by the formula: u(m)(xi) = 1 hm n∑ k=0 b (m) n (i , k)uk , i < n/2 u(m)(xi) = 1 hm n∑ k=0 b (m) n (i , k)ui+k−n/2, n/2 ≤ i ≤ n − n/2 u(m)(xi) = 1 hm n∑ k=0 b (m) n (i , k)un−n+k , i > n − n/2 (8) where,l(m)i (x) = m! t (m) i (x) mi ; t (m)i (x) = (x − zm)(x − zm+1)...(x − zn) + (x − z0)(x − zm+1)...(x − zn) + (x − z0)(x − z1)(x − zm+2)...(x − zn) + ...+ (x − z0)(x − z1)(x − z2)...(x − zn−m). theorem 1: the accuracy of approximating the grid derivative of order m using (n + 1) neighboring points on the regular grid is of order (n −m + 1).formulas (6), (7), (8) will be used to propose the difference scheme given in section 2.2 2.2. difference scheme with high order accuracyconsider the differential problem lnu(x) = f (x), bau(a) = ga, bbu(b) = gb, x ∈ [a, b], (9) where ln is the linear differential operator, ba, bb are the boundary condition operators.let λnu = φ, λau = φa, λbu = φb (10)be the difference scheme for the differential problem (9), where λn is the linear difference operator, λa, λb are the boundary condition difference operators. let u be the grid function determinedbased on the difference scheme, let ud be the value of u(x) on the grid ωh. below we give twodifference schemes corresponding to two differential problems. 2.2.1. differential scheme 1consider the second-order boundary problem with dirichlet boundary conditions{ u′′(x) = f (x), x ∈ (a, b) u(a) = c; u(b) = d. (11) https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 5considering the grid space ωh, let ui = u(xi), u = (u0, u1, . . . , un) be the grid function, f = (f0, f1, . . . , fn) is the right-hand grid function, d(m)f is the m-order derivative of the grid function f . since f (x) satisfies the differential equation (11), we have f (xi) = u′′(xi), i = 0, 1, 2, . . . , n .using (7) we obtain the approximated formula u′′(xi) = ui−1 − 2ui + ui+1 h2 − h2 4! d(2)fi − h4 6! d(4)fi − ...− h2m−2 (2m)! d(2m−2)fi +o(hn−1) (12) set φi = h2fi + h4 4!d (2)fi + h6 6!d (4)fi + ...+ h2m (2m)!d (2m−2)fi , 1 ≤ i ≤ n − 1, we obtain the systemof difference equations { ui−1 − 2ui + ui+1 = φi , i = 1, 2, ..., n − 1 u0 = c; un = d (13) (13) is is called a three-diagonal system because the matrix of the system has a three-diagonal form.solving system (13) is performed using the following pursuit algorithm. in the case of constructingscheme (13), we used formula (7) which is used to approximate the 2nd derivative with (n−2)-orderaccuracy, therefore, since lemma 1, it is implies solution of system (13) is the approximation of thefunction with n-order accuracy. since there we have the following theorem: theorem 2: the system of difference equations (13) is stable and its solution approximates to solution of problem (11) with n-order accuracy.the solution of the system is obtained from the pursuit algorithm which has complexity o(n). 2.2.2. differential scheme 2consider the second-order boundary problem with mixed boundary conditions{ u′′(x) = f (x), x ∈ (a, b) c0u(a)− c1u′(a) = c; d0u(b) + d1u ′(b) = d (14) consider the grid space ωh, set ui = u(xi), fi = f (xi), u = (u0, u1, . . . , un) be grid function. f = (f0, f1, . . . , fn) is the right-hand grid function, d(m)n f is the m-order derivative of the gridfunction f with n-order accuracy. since (6), we obtain the difference formula u′(xi) = ui+1 − ui h − h 2! fi − h2 3! d(1)fi − ...− hn−1 n! d(n−2)fi +o(hn) (15) consider two points x0 = a, xn = b, we obtain u′(a) = u1 − u0 h − h 2! f0 − h2 3! d(1)f0 − ...− hn−1 n! d(n−2)f0 +o(hn) (16) u′(b) = un − un−1 h + h 2! fn − h2 3! d(1)fn + ...+ (−1)n hn−1 n! d(n−2)fn +o(hn) (17) https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 6substituting formulas (16) and (17) into the boundary condition system, combined with schemenumber 1, we obtain the difference scheme c0u0 − c1 ( u1−u0 h − h 2!f0 − h2 3!d (1) n−2f0 − ...− hn−1 n! d (n−2) 1 f0 ) = c ui−1 − 2ui + ui+1 = φi , i = 1, 2, ..., n − 1 d0un + d1 ( un−un−1 h + h 2!fn − h2 3!d (1) n−2fn + ...+ (−1)n h n−1 n! d (n−2) 1 fn ) = d (18) the solution of the system is obtained from the pursuit algorithm which has complexity o(n).since there we have the following theorem: theorem 3: the system of difference equations (18) is stable and the solution approximates the solution of problem (14) with n-order accuracy.the following are some experimental calculation results for the proposed theory. 3. experimental results and discussions in this section, we verify the accuracy of the proposed schemes, we let the function ud(x)that satisfies the boundary problem, from there, we determine the right-hand side function f (x), x ∈ [a, b] and boundary conditions. on grid space ωh, we define the grid function value ud = (ud(x0), ud(x1), . . . , ud(xn)) is the exact solution value on the grid ωh. then, we use the proposedschemes to find approximate solutions u = (u0, u1, . . . , un), from there, we evaluate the accuracyof the scheme through error ε = ‖u − ud‖ωh . 3.1. evaluate the accuracy of scheme 1example 1: consider the problem{ u′′(x) = −sinx, x ∈ (0, 1) u(0) = 0; u(1) = sin(1)exact solution of the problem is u(x) = sinx .using scheme 1 combined with the pursuit algorithm, we obtain the grid solution u . the resultsof evaluating the accuracy of scheme 1 are given in table 1 table 1. results of evaluating the accuracy of scheme 1 example 1 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh10 1.0000e-008 2.9675e-012 1.0000e-010 1.6467e-01420 3.9063e-011 4.6426e-014 9.7656e-014 1.1343e-01730 1.5242e-012 4.0825e-015 1.6935e-015 3.6987e-01940 1.5259e-013 7.2653e-016 9.5367e-017 2.9933e-02050 2.5600e-014 1.9039e-016 1.0240e-017 4.1543e-02160 5.9537e-015 6.3781e-017 1.6538e-018 8.1987e-02270 1.7347e-015 2.5295e-017 3.5401e-019 2.0679e-022 https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 7 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh80 5.9605e-016 1.1351e-017 9.3132e-020 6.2629e-02390 2.3231e-016 5.5993e-018 2.8680e-020 2.1680e-023100 1.0000e-016 2.9759e-018 1.0000e-020 8.4722e-024 figure 1. error at each point between exact solution and approximated solution example 1example 2: consider the problem{ u′′(x) = −sinh(x) + 1 9e −x/3, x ∈ (0, 1) u(0) = 1; u(1) = sinh(1) + e−1/3 exact solution of the problem is u(x) = sinh(x) + e−x/3. table 2. results of evaluating the accuracy of scheme 1 example 2 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh10 1.0000e-008 3.3899e-012 1.0000e-010 2.0506e-01420 3.9063e-011 5.3052e-014 9.7656e-014 1.7488e-01730 1.5242e-012 4.6589e-015 1.6935e-015 5.2038e-01940 1.5259e-013 8.3000e-016 9.5367e-017 4.0990e-02050 2.5600e-014 2.1765e-016 1.0240e-017 5.6274e-021 https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 8 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh60 5.9537e-015 7.2895e-017 1.6538e-018 1.1046e-02170 1.7347e-015 2.8908e-017 3.5401e-019 2.7774e-02280 5.9605e-016 1.2973e-017 9.3132e-020 8.3970e-02390 2.3231e-016 6.3989e-018 2.8680e-020 2.9033e-023100 1.0000e-016 3.4009e-018 1.0000e-020 1.1337e-023 the results in table 1 and table 2 confirm that our proposed scheme 1 has determined theapproximated solution of the second-order boundary problem with dirichlet boundary conditionswith n-order accuracy, where n is the number of neighboring points. 3.2. evaluate the accuracy of scheme 2example 3: consider the problem{ u′′(x) = 6x + 1 8sinh(1 + x/2), x ∈ (0, 1) u(0)− u′(0) = sinh(1)− 12cosh(1); u(1) + u′(1) = 4 + sinh(3/2) + 1 2cosh(3/2)exact solution of the problem is u(x) = x3 + sinh(1 + x/2)using scheme 2 combined with the pursuit algorithm, we obtain the grid solution u . the resultsof evaluating the accuracy of scheme 2 are given in table 3 table 3. results of evaluating the accuracy of scheme 2 example 3 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh10 1.0000e-008 2.0382e-013 1.0000e-010 5.3121e-01620 3.9063e-011 3.3597e-015 9.7656e-014 9.1360e-01930 1.5242e-012 2.9998e-016 1.6935e-015 2.5005e-02040 1.5259e-013 5.3835e-017 9.5367e-017 2.0884e-02150 2.5600e-014 1.4183e-017 1.0240e-017 3.1494e-02260 5.9537e-015 4.7654e-018 1.6538e-018 7.0444e-02370 1.7347e-015 1.8943e-018 3.5401e-019 1.6852e-02380 5.9605e-016 8.5167e-019 9.3132e-020 5.3577e-02490 2.3231e-016 4.2068e-019 2.8680e-020 1.8066e-024100 1.0000e-016 2.2381e-019 1.0000e-020 3.0015e-025 https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 9 figure 2. error at each point between exact solution and approximated solution example 3example 4: consider the problem{ u′′(x) = cosh(x) + 1 9e x/3, x ∈ (0, 1) u(0)− 12u ′(0) = 1 2 ; 13u(1) + u′(1) = 1 3 ( cosh(1) + e1/3 ) − ( sinh(1) + 1 3e 1/3 ) exact solution of the problem is u(x) = cosh(x) + ex/3using scheme 2 combined with the pursuit algorithm, we obtain the grid solution u . the resultsof evaluating the accuracy of scheme 2 are given in table 4. table 4. results of evaluating the accuracy of scheme 2 example 4 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh10 1.0000e-008 3.7083e-011 1.0000e-010 3.5960e-01320 3.9063e-011 6.1634e-013 9.7656e-014 5.6152e-01630 1.5242e-012 5.5099e-014 1.6935e-015 1.1085e-01740 1.5259e-013 9.8929e-015 9.5367e-017 6.5992e-01950 2.5600e-014 2.6068e-015 1.0240e-017 7.2379e-02060 5.9537e-015 8.7603e-016 1.6538e-018 1.1673e-020 https://doi.org/10.28924/ada/ma.4.10 eur. j. math. anal. 10.28924/ada/ma.4.10 10 n h8 ‖u − ud‖ωh h10 ‖u − ud‖ωh70 1.7347e-015 3.4826e-016 3.5401e-019 2.4481e-02180 5.9605e-016 1.5659e-016 9.3132e-020 6.1971e-02290 2.3231e-016 7.7351e-017 2.8680e-020 1.7954e-022100 1.0000e-016 4.1155e-017 1.0000e-020 5.7442e-023 the results in table 3 and table 4 also confirm that our proposed scheme 2 has determined theapproximate solution of the second-order boundary problem with mixed boundary conditions with n-order accuracy, where n is the number of neighboring points. 4. conclusion the main content of article has proposed two difference schemes with high-order accuracy tosolve second-order boundary problems with the dirichlet boundary condition system and the mixedboundary condition system. the highlight of these two schemes compared to other schemes [3][5]is that the two proposed schemes have a simple structure, high accuracy and finding solutions isdone using a pursuit algorithm with the computational complexity is o(n), where n is the numberof grid points. through the calculation results, it has been confirmed that the two schemes bothprovide approximated solutions with n-level accuracy compared to the grid step with n+1 being thenumber of neighboring points used to determine approximate derivatives of levels. these differenceschemes will allow to improve the accuracy of the solution for the class of nonlinear boundarieswith mixed boundary conditions. references [1] r.b.srivastava and s.shukla, numerical accuracies of lagrange’s and newton polynomial interpolation: numericalaccuracies of interpolation formulas, lap lambert academic publishing, 2012.[2] q.v. vinh, t.h. nguyen, difference scheme for solving boundary problems for high-order linear and non-lineardifferential equations, fair 10 national scientific conference, natural science and technology publishing house,(2017), 358-365.[3] n. setia, r.k. mohanty, a high accuracy variable mesh numerical approximation for two point nonlinear bvps withmixed boundary conditions, soft computing a fusion of foundations, methodol. appl. 26 (2022), 9805-9821.[4] m. baccouch, analysis of optimal superconvergence of a local discontinuous galerkin method for nonlinear second-order two-point boundary-value problems, appl. numer. math. 145 (2019), 361-383.[5] h. ramos, m.a. rufai, numerical solution of boundary value problems by using an optimized two-step blockmethod,numer. algorithms, 84 (2020), 229-251.[6] j. li, general explicit difference formulas for numerical differentiation, j. comput. appl. math. 183 (2005), 29-52.[7] m. kaur1, s. kumar, j. bhatti, numerical solution to sixth order ordinary differential equation using three stageeighth order runge-kutta type method, the electrochemical society, 107 (2022), 86-97. https://doi.org/10. 28924/ada/ma.1.1. https://doi.org/10.28924/ada/ma.4.10 https://doi.org/10.28924/ada/ma.1.1 https://doi.org/10.28924/ada/ma.1.1 eur. j. math. anal. 10.28924/ada/ma.4.10 11 [8] q.a. dang, q.l. dang, a unified approach to fully third order nonlinear boundary value problems, j. nonlinearfunct. anal. 2020 (2020), 9. https://doi.org/10.23952/jnfa.2020.9.[9] q.a. dang, q.l. dang, simple numerical methods of second and third-order convergence for solving a fullythird-order nonlinear boundary value problem, numer. algor. 87 (2021), 1479–1499. https://doi.org/10.1007/ s11075-020-01016-2.[10] q.a. dang, t.h. nguyen, solving the dirichlet problem for fully fourth order nonlinear differential equation, afr.mat. 30 (2019), 623–641, https://doi.org/10.1007/s13370-019-00671-6.[11] j. alzabut, s.r. grace, g.n. chhatria, new oscillation results for higher order nonlinear differential equations witha nonlinear neutral terms, j. math. comp. sci. 28 (2022), 294-305.[12] s. baraket, s. mahdaoui, t. ouni, limiting profile of the blow-up solutions for the fourth-order nonlinear emden-fowler equation with a singular source, discr. contin. dyn. syst. s, 16 (2023), 1181-1200. https://doi.org/10.28924/ada/ma.4.10 https://doi.org/10.23952/jnfa.2020.9 https://doi.org/10.1007/s11075-020-01016-2 https://doi.org/10.1007/s11075-020-01016-2 https://doi.org/10.1007/s13370-019-00671-6 1. introduction 2. proposed method 3. experimental results and discussions 4. conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 16doi: 10.28924/ada/ma.4.16 hardy-littlewood-sobolev theorem for bourgain-morrey spaces and approximation nouffou diarra laboratoire de mathématiques et applications, ufr mathématiques et informatique, université félix houphouët boigny abidjan-cocody, 22 bp 582 abidjan 22, côte d’ivoire correspondence: nouffoud@yahoo.fr abstract. in this paper, we establish an extension of the hardy-littlewood-sobolev theorem to thesetting of the bourgain-morrey space mα q,p(rd) (1 ≤ q, p, α ≤ ∞), which theory goes back tobourgain in 1991. we also prove that mα q,p(rd) is included in the closure of the lebesgue space lα in the morrey-type space f(q, p, α), which arises naturally in 2015 in the study of boundednessproperties of fractional integral operators. therefore, we establish in mα q,p some approximationresults by compactly supported and/or regular functions. as an application of these results, weobtain an explicit solution in [lp(rd)]d of the equation divf = f whenever f is inmα q,p , with d ≥ 3, 1 ≤ q ≤ α < d and 1 p = 1 α − 1 d . 1. introduction let d be a fixed positive integer. rd is equipped with its usual hilbert space structure and theeuclidean norm of any element x of rd is denoted by |x |.recall that the classical lebesgue space lq := lq(rd), with q ∈ [1,∞] , is defined to be theset of all measurable complex functions f on rd such that ‖f ‖q := [∫ rd |f (x)|q dx ] 1 q <∞ with the usual modification made when q =∞. in what follows, |e| and χe denote the lebesguemeasure and the characteristic function of any measurable set e ⊂ rd , respectively. lqloc denotesthe set of all measurable complex functions f on rd such that f χk ∈ lq for any bounded measurablesubset k of rd .for 1 ≤ q, α ≤ ∞, the morrey spacemα q :=mα q (rd) is defined as the set of all elements f of lqloc for which ‖f ‖mα q := sup x∈rd , r>0 |q(x, r)| 1 α − 1 q ∥∥f χq(x,r) ∥∥ q <∞, received: 20 mar 2024. key words and phrases. bourgain-morrey spaces; morrey-type space; maximal operator; hardy-littlewood-sobolevtheorem; approximation; divergence equation. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.16 https://orcid.org/0000-0003-1123-506x eur. j. math. anal. 10.28924/ada/ma.4.16 2where q(x, r) = d∏ j=1 [ xj − r 2 , xj + r 2 ) , x = (x1, x2, ..., xd) ∈ rd and 0 < r <∞. morrey spaces were introduced in 1938 by c. morrey [11] in order to study both the regularityproblem of solutions for quasi-linear elliptic partial differential equations and the calculus of vari-ations. note that, for 1 ≤ q ≤ α ≤ ∞, lα is included in mα q and the inclusion is proper when q < α < ∞. moreover, morrey spaces describe local regularity of functions more precisely thanlebesgue spaces. however, some nice and useful properties of lα are not shared by mα q when 1 ≤ q < α < ∞. for example, in this case, the set of all compactly supported and/or regularelements is not dense in mα q . because of this unpleasant issue, several distinguished linear sub-spaces of morrey spaces have been considered for their easy use in harmonic analysis, specificallyin boundedness problem of classical operators.the present paper focuses on bourgain-morrey spacesmα q,p with 1 ≤ q, α, p ≤ ∞. recall that,a special case of these spaces was first introduced by bourgain [2] in 1991 in order to study thestein-tomas estimate. later on, bourgain-morrey spaces have been used fruitfully in the studyof fourier restriction, multipliers problems and partial differential equations, and in the proof ofrefinements of strichartz inequality (see [8–10] and the references therein). they are defined asfollows. definition 1.1. let 1 ≤ q, α, p ≤ ∞. the bourgain-morrey space mα q,p := mα q,p(rd) is defined as the set of all f ∈ lqloc for which ‖f ‖mα q,p := ∥∥∥∥{|qk,m| 1 α − 1 q ∥∥f χqk,m∥∥q}(k,m)∈zd×z ∥∥∥∥ `p <∞, where the sets qk,m = d∏ j=1 [ kj2 m, (kj + 1) 2m ) , k = (k1, k2, ..., kd) ∈ zd , m ∈ z are the usual dyadic cubes of rd and for any sequence {ai}i∈i included in c, ‖{ai}i∈i‖`p :=  (∑ i∈i |ai |p ) 1 p if p <∞ sup i∈i |ai | if p =∞. it is well known that, when 1 ≤ q < α < p ≤ ∞, lα is properly included in mα q,p, which is alinear subspace of mα q . actually we have{ lα ⊂mα q,p ⊂mα q,p1 ⊂mα q,∞ =mα q , 1 ≤ q < α < p ≤ p1 ≤ ∞. mα q,p ⊂mα q1,p , 1 ≤ q1 ≤ q ≤ α ≤ p ≤ ∞. (1) https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 3many useful results, well known for lebesgue or morrey spaces, have been extended to thesetting of bourgain-morrey spaces (see [8, 9]). for instance, the boundedness of some classicaloperators on these spaces has been investigated by hatano et al. [8].let 0 < γ < 1. define the fractional integral operator iγ as iγf (x) = ∫ rd |x − y |d(γ−1)f (y)dy when the above integral makes sense.recall that the hardy-littlewood-sobolev theorem of fractional integration is one of the mostimportant tools in the study of partial differential equations. it reads as follows. theorem 1.2. [14] let 0 < γ < 1 α ≤ 1 and 1 p = 1 α − γ. then there is a real number aα,γ such that: (i) when 1 < α ‖iγf ‖p ≤ aα,γ‖f ‖α , f ∈ lα (2) (ii) when α = 1 ‖f ‖∗ 1 1−γ ,∞ ≤ a1,γ‖f ‖1 , f ∈ l1. (3) recall that, for q ∈ [1,∞) , ‖ · ‖∗q,∞ denotes the quasi-norm of the weak-lebesgue wlq definedby wlq = { f ∈ l1 loc : ‖f ‖∗q,∞ = sup λ>0 λ ∣∣{x ∈ rd : |f (x)| > λ} ∣∣ 1 q <∞ } . the first aim of the present paper is to establish an extension of the above useful theorem tothe setting of bourgain-morrey spaces. note that, our result refines that of hatano et al., whichstates that fractional integral operators map bourgain-morrey spaces into the same type spaces(see [8, theorem 4.4]).another morrey-type space considered in this paper is the space f(q, p, α) (1 ≤ q, α, p ≤ ∞),which arises naturally in the study of boundedness properties of fractional integral operators.it has been introduced in 2015 by fofana et al. [7]. note that, recently in 2020, the space f(q, p, α) has been studied also in [15], where it is called the riesz-morrey space and denoted by rmp,q, 1 p − 1 α ( rd ). it is defined as follows. definition 1.3. let 1 ≤ q, p, α ≤ ∞. the space f(q, p, α) := f(q, p, α)(rd) is defined as the set of all f ∈ lqloc for which ‖f ‖f(q,p,α) is finite, where ‖f ‖f(q,p,α) =  sup {qi}∈p ∥∥∥{|qi | 1 α − 1 q ‖f χqi‖q } i∈i ∥∥∥ `p if p <∞ sup q∈q |q| 1 α − 1 q ‖f χqi‖q if p =∞, with • q = { q(x, r) : (r, x) ∈ (0,∞)× rd } • p = { {qi}i∈i ⊂ q : i is countable and qi ∩qj = ∅ if i 6= j } . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 4it is well known that the space f(q, p, α) is a linear subspace of lqloc and a banach space, whenendowed with ‖ · ‖f(q,p,α). f(q, p, α) is nontrivial if and only if q ≤ α ≤ p (see [7]). moreover,when 1 ≤ q1 ≤ q ≤ α ≤ p ≤ p1, the following inclusion and equality relations hold: { lα = f(q, α, α) ⊂ f(q, p, α) ⊂ f(q, p1, α) ⊂ f(q,∞, α) =mα q f(q, p, α) ⊂ f(q1, p, α). (4) note that (4) shows that the spaces f(q, p, α) provide a bridge connecting both lebesgue spacesand morrey spaces. many results, well known for lebesgue or morrey spaces, have been extendedin the framework of these spaces (see [5, 7]). furthermore, the relations (1) and (4) point out thatthe spaces mα q,p and f(q, p, α) satisfy almost the same inclusion relations. we also observe thatthe norm structures of these two spaces are very similar. thus, a natural question is that, what isthe link between the spaces mα q,p and f(q, p, α) ?the second aim of this paper is to study the above mentioned question. we succeeded in provingthatmα q,p is continuously included in f(q, p, α) and, when p <∞,mα q,p is included in the closureof lα in f(q, p, α). therefore, we also establish inmα q,p some approximation results by compactlysupported and/or regular functions.as an application of the above mentioned results, we obtain an explicit solution in (lp)d of theequation divf = f whenever f is in mα q,p , with d ≥ 3, 1 ≤ q ≤ α < d and 1 p = 1 α − 1 d .the remainder of the paper is organized as follows. section 2 contains a more detailed presen-tation of our main results. section 3 deals with some preliminary results on mα q,p . in section 4we prove the inclusion of mα q,p in f(q, p, α) and also approximation results. section 5 is devotedto prove our main theorem showing the action of fractional integral operators on mα q,p . section 6contains an application to the divergence equation div f = f .finally, let us make some conventions on notations used in this paper. • c∞ denotes the set of all infinitely differentiable functions on rd and c∞c stands for the set ofall elements of c∞ with compact support in rd . • let φ be a fixed nonnegative element of c∞ such that its support is included in the unit cube [0, 1]d and satisfying ∫ rd φ(x)dx = 1. for any integer n ≥ 1, we denote by φn the dilation definedby φn(x) = ndφ(nx) , x ∈ rd . • let ω be a fixed element of c∞ satisfying χq(0,1) ≤ ω ≤ χq(0,2). for any integer n ≥ 1, ωn isdefined by ωn(x) = ω (x n ) , x ∈ rd . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 52. statement of the main results for 0 < γ < 1, the fractional integral operator iγ is known to be closely related to the fractionalmaximal operator mγ defined by mγf (x) = sup q3x |q|γ−1 ∫ q |f (y)|dy, f ∈ l1 loc , x ∈ rd , where the supremum is taken over all cubes q in rd containing x .our first result reads as follows. theorem 2.1. let us assume that 0 < γ < 1 α ≤ 1 and 1 p = 1 α − γ. then, for any element f of mα 1,p , we have ‖mγf ‖p ≤ 2 d ( 2− 1 p ) 3d(2−γ) ‖f ‖mα 1,p . (5) note that theorem 2.1 refines [8, corollary 4.5]. as an immediate consequence of this theorem,we obtain the following result which refines [8, theorem 4.4] and is our most significant result. theorem 2.2. let us assume that 0 < γ < 1 α ≤ 1 and 1 p = 1 α−γ. then there exists a real constant c > 0 such that, for any element f of mα 1,p , we have ‖iγf ‖p ≤ c ‖f ‖mα 1,p . (6) since lα ⊂ mα q,p ⊂ mα 1,p when 1 ≤ q < α < p, theorem 2.2 provides an extension of thehardy-littlewood-sobolev theorem (theorem 1.2) to the setting of bourgain-morrey spaces.from (1) and (6), we have lα ⊂mα 1,p ⊂ b(γ, p) , 0 < γ < 1 α < 1 and 1 p = 1 α − γ, (7) where b(γ, p) = { f ∈ l1 loc : iγ(|f |) ∈ lp } , 0 < γ < 1 ≤ p ≤ ∞. we recall that, for 1 ≤ q, p, α ≤ ∞, the space f(q, p, α) arises naturally in the search of acharacterization of the set b(γ, p) in [7], where it is established that b(γ, p) ⊂ f(1, p, α)c ⊂ f(1, p, α) ⊂ wb(γ, p) , 0 < γ < 1 α ≤ 1 and 1 p = 1 α − γ, (8) with f(q, p, α)c = { f ∈ f(q, p, α) : lim y→0 ‖f − f (· − y)‖f(q,p,α) = 0 } and wb(γ, p) = { f ∈ l1 loc : iγ(|f |) ∈ wlp } , 0 < γ < 1 ≤ p ≤ ∞. https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 6note that, it is proved in [7] that f(q, p, α)c is the closure of lα in f(q, p, α) if p <∞. it is clearthat the inclusion relations (7) and (8) yield what follows mα 1,p ⊂ f(1, p, α)c , 0 < γ < 1 α < 1 and 1 p = 1 α − γ. (9) in the present paper we prove, without the use of fractional integral operators, the followingextension of the relation (9). theorem 2.3. let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞. then ‖f ‖f(q,p,α) ≤ 3 d ( 1+ 1 α − 1 q + 1 p ) 2 d ( 1 q − 1 p ) ‖f ‖mα q,p , f ∈ l1 loc (10) and therefore mα q,p is continuously included in f(q, p, α). moreover, if p < ∞ then mα q,p is included in f(q, p, α)c. as done in [4, 6] for some special subspaces of the morrey-type space (lq, lp)α, usually calledthe fofana space and closely related to f(q, p, α), we investigate in bourgain-morrey spacesapproximation by smooth functions. we shall prove what follows. theorem 2.4. let 1 ≤ q ≤ α ≤ p <∞ and f be an element of l1 loc. then the following assertions are equivalent : (i) f belongs to mα q,p, (ii) lim n→∞ ‖f − f ∗ φn‖mα q,p = 0, where f ∗ φn is the convolution product of f and φn, (iii) f belongs to the closure in mα q,p of the set c∞mα q,p = { g ∈ c∞ : ∂βg ∈mα q,p for any β in nd } , where ∂βg stands for the derivative of order β of g. note that theorem 2.4 implies that both c∞ ∩mα q,p and c∞mα q,p are dense inmα q,p if p <∞. asa consequence of this theorem, we obtain the following approximation result. theorem 2.5. let 1 ≤ q ≤ α ≤ p <∞ and f be any element of mα q,p . then lim n→∞ ‖f − (f ωn) ∗ φn‖mα q,p = 0. it is easy to see that, for any integer n ≥ 1, (f ωn) ∗ φn belongs to c∞c . consequently, theorem2.5 implies that c∞c ∩mα q,p is dense in mα q,p if p <∞.let us consider the divergence equation divf = f , f ∈ l1 loc. (11) to our knowledge, for a given p in [1,∞), the characterization of the class of functions f forwhich the equation (11) has a solution f = (fj)1≤j≤d in (lp)d is still an open problem. however,phuc and torres proved that (see [13, theorem 3.2]), for d d−1 < p < ∞, the equation (11) has a https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 7solution in (lp)d if and only if f belongs to the set b( 1 d , p). this result combined with (7) showsthat, for 1 < α < d and 1 p = 1 α− 1 d , a sufficient condition for the solvability in (lp)d of the equation(11) is that f belongs to mα 1,p . moreover, an application of theorem 2.2 and theorem 2.5 allowsus to obtain an explicit solution of the equation (11) in (lp)d , as shown below. theorem 2.6. let us assume that d ≥ 3, 1 ≤ q ≤ α < d , 1 p = 1 α − 1 d and f is an element ofmα q,p . then there exists a real constant cd such that f = ( cd rj ( i 1 d f )) 1≤j≤d is a solution in (lp)d of the equation (11), where rj (1 ≤ j ≤ d) stands for the riesz transform defined by rjϕ(x) = γ ( d+1 2 ) π d+1 2 lim ε→0+ ∫ |x−y |≥ε xj − yj |x − y |d+1 ϕ(y)dy , x ∈ rd , ϕ ∈ lp. 3. preliminaries this section is devoted to prove some preliminary results. 3.1. equivalent norms on mα q,p. we begin this subsection by recalling the definition of classicaldyadic grids. definition 3.1. a dyadic grid is a countable collection d of cubes of rd which are dyadic translates and dilations of the unit cube [0, 1)d . more precisely, d may be characterized as follows : (i) if q ∈ d then its side-length `(q) = 2m for some m ∈ z (ii) if q,p ∈ d then q ∩ p ∈ {∅, q, p} (iii) for each m ∈ z, the family dm = {q ∈ d / `(q) = 2m} form a partition of rd . example 3.2. • the standard dyadic grid d0 is defined by d0 = { 2m ( [0, 1)d + k ) / m ∈ z, k ∈ zd } . • each of the following 3d collections of cubes in rd dt = { 2m ( [0, 1)d + k + t ) / m ∈ z, k ∈ zd } , t ∈ {−1/3, 0, 1/3}d is a dyadic grid. the following property holds (see [3, theorem 3.1] and its proof). proposition 3.3. for every cube q of rd , there exists an element t of {−1/3, 0, 1/3}d and a cube qt of dt such that q is included in qt and `(qt) ≤ 3 `(q). let us introduce the following definition. https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 8 definition 3.4. let 1 ≤ q ≤ α ≤ p ≤ ∞. for any dyadic grid d and any element f of lqloc, we define ‖f ‖mα q,p(d) = ∥∥∥∥∥ { |q| 1 α − 1 q (∫ q |f (x)|qdx ) 1 q } q∈d ∥∥∥∥∥ `p . we shall prove that, for any t ∈ {−1/3, 0, 1/3}d , the norms ‖ · ‖mα q,p(dt) and ‖ · ‖mα q,p areequivalent. in order to do this, we establish the following preparatory lemma. lemma 3.5. let 1 ≤ q ≤ α ≤ p ≤ ∞, d and d′ are two dyadic grids. then for any element f of lqloc, we have  ‖f ‖mα q,p(d) ≤ 2 d ( 1 q − 1 p ) ‖f ‖mα q,p(d′) if p <∞ ‖f ‖mα q,∞(d) ≤ 2d‖f ‖mα q,∞(d′). proof. let f be any element of lqloc and fix m ∈ z.we recall that both the families dm = {q ∈ d / `(q) = 2m} and d′m = {q′ ∈ d′ / `(q′) = 2m}form partitions of rd . moreover, it is easy to see that, for any element q of dm, the subset {q′ ∈ d′ / q ∩q′ 6= ∅} of d′ has at most 2d elements.a) suppose that p <∞. we have(∫ q |f (x)|qdx ) p q =  ∑ q′∈d′m ∫ q∩q′ |f (x)|qdx  p q ≤ 2 d ( 1− q p ) p q ∑ q′∈d′m (∫ q∩q′ |f (x)|qdx ) p q . consequently∑ q∈dm [ |q| 1 α − 1 q (∫ q |f (x)|qdx ) 1 q ]p = 2 d m ( 1 α − 1 q ) p ∑ q∈dm (∫ q |f (x)|qdx ) p q ≤ 2 d ( p q −1 ) 2 d m ( 1 α − 1 q ) p ∑ q∈dm ∑ q′∈d′m, q∩q′ 6=∅ (∫ q∩q′ |f (x)|qdx ) p q = 2 d ( p q −1 ) 2 d m ( 1 α − 1 q ) p ∑ q′∈d′m ∑ q∈dm (∫ q∩q′ |f (x)|qdx ) p q ≤ 2 d ( p q −1 ) 2 d m ( 1 α − 1 q ) p ∑ q′∈d′m (∫ q′ |f (x)|qdx ) p q = 2 d ( p q −1 ) ∑ q′∈d′m [∣∣q′∣∣ 1 α − 1 q (∫ q′ |f (x)|qdx ) 1 q ]p and so ‖f ‖mα q,p(d) ≤ 2 d ( 1 q − 1 p ) ‖f ‖mα q,p(d′). https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 9b) suppose that p =∞. then, for any q ∈ dm, we have |q| 1 α − 1 q (∫ q |f (x)|qdx ) 1 q = |q| 1 α − 1 q  ∑ q′∈d′m ∫ q∩q′ |f (x)|qdx  1 q ≤ |q| 1 α − 1 q ∑ q′∈d′m (∫ q∩q′ |f (x)|qdx ) 1 q = ∑ q′∈d′m ∣∣q′∣∣ 1 α − 1 q (∫ q∩q′ |f (x)|qdx ) 1 q ≤ ∑ q′∈d′m, q∩q′ 6=∅ ∣∣q′∣∣ 1 α − 1 q (∫ q′ |f (x)|qdx ) 1 q ≤ 2d‖f ‖mα q,∞(d′) and so ‖f ‖mα q,∞(d) ≤ 2d‖f ‖mα q,∞(d′).the proof is complete. � the above lemma leads to the following corollary. corollary 3.6. let 1 ≤ q ≤ α ≤ p ≤ ∞ and t be in {−1/3, 0, 1/3}d . then for any element f of lqloc, we have 2 d ( 1 p − 1 q ) ‖f ‖mα q,p ≤ ‖f ‖mα q,p(dt) ≤ 2 d ( 1 q − 1 p ) ‖f ‖mα q,p if p <∞ 2−d‖f ‖mα q,∞ ≤ ‖f ‖mα q,∞(dt) ≤ 2d‖f ‖mα q,∞ .3.2. continuity of the translation operator in mα q,p. this subsection deals with the continuity ofthe translation operator in bourgain-morrey spaces. we shall use in the sequel the followingproperties. proposition 3.7. [8] let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞. 1) if α <∞, then there exists c1 > 0 such that for all y ∈ rd and f ∈mα q,p , we have ‖f (· − y)‖mα q,p ≤ c1 ‖f ‖mα q,p . 2) if q < α < p < ∞ then the set l∞c of all compactly supported bounded functions is dense in mα q,p . 3) if q < α < p <∞ or p =∞ then there exists c2 > 0 such that for any element f of mα q,p , we have ‖f ‖mα q,p ≤ c2 ‖f ‖α. a classical property of lebesgue spaces reads as follows. lemma 3.8. if 1 ≤ α <∞ and f is in lα then we have lim y→0 ‖f − f (· − y)‖α = 0. https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 10this result can be extended to the setting of bourgain-morrey spaces and this extension willplay a key role in the proofs of our results. proposition 3.9. let 1 ≤ q ≤ α ≤ p <∞ and f be any element of mα q,p . then lim y→0 ‖f − f (· − y)‖mα q,p = 0. proof. if q = α or α = p then mα q,p = {0} and therefore, we have nothing to prove. thus wesuppose that q < α < p. by point 2) of proposition 3.7, there exists a sequence (fn)n≥1 of elementsof l∞c such that lim n→∞ ‖fn − f ‖mα q,p = 0. moreover, according to point 1) of proposition 3.7, thereexists c1 > 0 such that ‖(fn − f )(· − y)‖mα q,p ≤ c1 ‖fn − f ‖mα q,p , y ∈ rd , n ≥ 1. (∗) let ε > 0 be a fixed real number. there exists an integer nε such that ‖fnε − f ‖mα q,p < ε 2(1 + c1) . (∗∗) from (∗), (∗∗), point 1) and point 3) of proposition 3.7 we have ‖f − f (· − y)‖mα q,p ≤ ‖f − fnε‖mα q,p + ‖fnε − fnε(· − y)‖mα q,p + ‖(fnε − f )(· − y)‖mα q,p ≤ ‖f − fnε‖mα q,p + ‖fnε(· − y)− fnε‖mα q,p + c1 ‖fnε − f ‖mα q,p ≤ (1 + c1) ‖f − fnε‖mα q,p + ‖fnε(· − y)− fnε‖mα q,p < ε 2 + c2 ‖fnε(· − y)− fnε‖α . according to lemma 3.8, for any y ∈ rd such that 0 < |y | < 1, we have ‖fnε(· − y)− fnε‖α < ε 2c2and therefore we obtain ‖f − f (· − y)‖mα q,p < ε. this ends the proof. � 4. inclusion and approximation results 4.1. inclusion of mα q,p in f(q, p, α). this subsection is devoted to prove exclusively theorem 2.3. proof of theorem 2.31) • we recall that mα q,∞ =mα q = f(q,∞, α). therefore, we have nothing to prove if p =∞. • if p <∞ and α ∈ {q, p} then mα q,p = {0}. thus the result is obvious. • assume that 1 ≤ q < α < p <∞.let f be in l1 loc and {qi : i ∈ i} be a disjoint family of cubes of rd . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 11 a) let us consider an element i of i. we can associate to i an element t of {−1/3, 0, 1/3}d andan element r(i , t) of dt such that qi ⊂ r(i , t) and `(r(i , t)) ≤ 3 `(qi) (see proposition 3.3). wehave |qi | 1 α − 1 q (∫ qi |f (x)|qdx ) 1 q ≤ ( |qi | |r(i , t)| ) 1 α − 1 q |r(i , t)| 1 α − 1 q (∫ r(i ,t) |f (x)|qdx ) 1 q ≤ 3 d ( 1 α − 1 q ) |r(i , t)| 1 α − 1 q (∫ r(i ,t) |f (x)|qdx ) 1 q . b) let us fix t in {−1/3, 0, 1/3}d and set rt = { r ∈ dt : ∃ i ∈ i such that r(i , t) = r } . note that, for all r ∈ rt , we have ∀ i ∈ i, r = r(i , t) =⇒ `(qi) ≤ `(r) ≤ 3`(qi) =⇒ |qi | ≤ |r| ≤ 3d |qi |∑ i∈i, r=r(i ,t) |qi | ≤ |r|. this shows that the cardinality of the set {i ∈ i : r = r(i , t)} does not exceed 3d .c) we have∑ i∈i ( |qi | 1 α − 1 q (∫ qi |f (x)|qdx ) 1 q )p = ∑ t∈{−1/3,0,1/3}d ∑ r∈rt ∑ i :r=r(i ,t) ( |qi | 1 α − 1 q (∫ qi |f (x)|qdx ) 1 q )p ≤ ∑ t∈{−1/3,0,1/3}d ∑ r∈rt ∑ i :r=r(i ,t) ( 3 d ( 1 α − 1 q ) p|r| 1 α − 1 q (∫ r |f (x)|qdx ) 1 q )p (by point a)) ≤ 3 d ( 1 α − 1 q ) p 3d ∑ t∈{−1/3,0,1/3}d ∑ r∈rt ( |r| 1 α − 1 q (∫ r |f (x)|qdx ) 1 q )p ( by point b) ) ≤ 3 d ( 1 α − 1 q ) p 3d ∑ t∈{−1/3,0,1/3}d ∑ r∈dt ( |r| 1 α − 1 q (∫ r |f (x)|qdx ) 1 q )p ( because of rt ⊂ dt). therefore[∑ i∈i ( |qi | 1 α − 1 q (∫ qi |f (x)|qdx ) 1 q )p] 1 p ≤ 3 d ( 1 α − 1 q + 1 p ) ∑ t∈{−1/3,0,1/3}d ‖f ‖mα q,p(dt). since the above inequality is true for all disjoint family {qi : i ∈ i} of cubes of rd , we have ‖f ‖f(q,p,α) ≤ 3 d ( 1 α − 1 q + 1 p ) ∑ t∈{−1/3,0,1/3}d ‖f ‖mα q,p(dt). https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 12therefore, corollary 3.6 implies that ‖f ‖f(q,p,α) ≤ 3 d ( 1+ 1 α − 1 q + 1 p ) 2 d ( 1 q − 1 p ) ‖f ‖mα q,pand consequently mα q,p is continuously included in f(q, p, α).2) assume that p <∞ and let f be any element of mα q,p . by point 1), f is in f(q, p, α) and, forany y ∈ rd , we have ‖f − f (· − y)‖f(q,p,α) ≤ 3 d ( 1+ 1 α − 1 q + 1 p ) 2 d ( 1 q − 1 p ) ‖f − f (· − y)‖mα q,p . therefore, proposition 3.9 implies that lim y→0 ‖f − f (· − y)‖f(q,p,α) = 0 and consequently f belongs to f(q, p, α)c. thus, we obtain the desired result. � 4.2. approximation inmα q,p. in this subsection, we investigate approximation of elements of bourgain-morrey spaces by smooth functions. we shall use the following result. proposition 4.1. [8] let us assume that 1 ≤ q ≤ α ≤ p ≤ ∞ with α < ∞. then there exists c > 0 such that for all g ∈ l1 and f ∈mα q,p , we have ‖g ∗ f ‖mα q,p ≤ c ‖g‖1 ‖f ‖mα q,p . propositions 3.7, 3.9 and 4.1 allow us to prove theorem 2.4. proof of theorem 2.4 • (i)⇒ (i i) assume that f ∈mα q,p and n is a nonegative integer. for almost every x ∈ rd , f (x)− f ∗ φn(x) = ∫ rd f (x)φ(u)du − ∫ rd f (x − y)ndφ(ny)dy = ∫ rd f (x)φ(u)du − ∫ rd f ( x − u n ) φ(u)du = ∫ rd [ f (x)− f ( x − u n )] φ(u)du. therefore, for any dyadic cube qk,m ( (k,m) ∈ zd × z ) , the minkowski inequality implies that,∥∥(f − f ∗ φn)χqk,m ∥∥ q ≤ ∫ rd ∥∥∥[f − f (· − u n )] χqk,m ∥∥∥ q φ(u)du and so ‖f − f ∗ φn‖mα q,p ≤ ∫ rd ∥∥∥f − f (· − u n )∥∥∥ mα q,p φ(u)du. according to proposition 3.9, we have lim n→∞ ∥∥∥f − f (· − u n )∥∥∥ mα q,p φ(u) = 0 , u ∈ rd . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 13furthermore, by minkowski’s inequality and point 1) of proposition 3.7, we have∥∥∥f − f (· − u n )∥∥∥ mα q,p φ(u) ≤ (1 + c1) ‖f ‖mα q,p φ(u) , u ∈ rd , n ≥ 1. thus, an application of the dominated convergence theorem gives lim n→∞ ‖f − f ∗ φn‖mα q,p = 0. • (i i)⇒ (i i i) assume that the assertion (i i) holds.let us fix an integer n ≥ 1 and β ∈ nd . since φn ∈ c∞c , f ∗φn belongs to c∞ and by proposition4.1, f ∗ φn is in mα q,p . furthermore, it is well known that ∂β (f ∗ φn) = f ∗ ∂βφn and by notingthat ∂βφn ∈ l1, proposition 4.1 implies that ∂β (f ∗ φn) belongs to mα q,p . thus f ∗ φn belongs to c∞mα q,p and since, by hypothesis, lim n→∞ ‖f − f ∗ φn‖mα q,p = 0, we can conclude that f belongs to the closure in mα q,p of c∞mα q,p . • (i i i)⇒ (i) since c∞mα q,p is a subset of mα q,p, it is obvious that its closure in mα q,p is included in mα q,p and therefore the claim follows. the proof is complete. � we recall the following well known result in lebesgue spaces. lemma 4.2. [1] if 1 ≤ α <∞ and f is in lα then we have lim n→∞ ‖f χen‖α = 0, where (en)n≥1 is a nonincreasing sequence of measurable subsets of rd satisfying ∣∣∣∣∣∣⋂n≥1 en ∣∣∣∣∣∣ = 0. the next proposition shows that an analogous result holds for bourgain-morrey spaces. proposition 4.3. let 1 ≤ q ≤ α ≤ p < ∞, f be any element of mα q,p and (en)n≥1 be a nonincreasing sequence of measurable subsets of rd satisfying ∣∣∣∣∣∣⋂n≥1 en ∣∣∣∣∣∣ = 0. then lim n→∞ ‖f χen‖mα q,p = 0. proof. if q = α or α = p then mα q,p = {0} and therefore we have nothing to prove. hence wesuppose that 1 ≤ q < α < p <∞. by point 2) of proposition 3.7, there exists a sequence (fn)n≥1of elements of l∞c such that lim n→∞ ‖fn − f ‖mα q,p = 0.let ε > 0 be a fixed real number. from what precedes, there exists an integer nε such that ‖fnε − f ‖mα q,p < ε 2 . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 14this and point 3) of proposition 3.7 imply that, for any n ≥ 1, ‖f χen‖mα q,p ≤ ‖(f − fnε)χen‖mα q,p + ‖fnεχen‖mα q,p ≤ ‖f − fnε‖mα q,p + ‖fnεχen‖mα q,p ≤ ε 2 + c2 ‖fnεχen‖α .since fnε ∈ lα, lemma 4.2 implies that there exists an integer n0 ≥ 1 such that n ≥ n0 =⇒ ‖fnεχen‖α < ε 2c2 . therefore n ≥ n0 =⇒ ‖f χen‖mα q,p < ε.this provides the desired result. � [1, proposition 3.6] asserts that proposition 4.3 is equivalent to the following dominated con-vergence theorem. proposition 4.4. let 1 ≤ q ≤ α ≤ p <∞ and f be any element ofmα q,p . if (fn)n≥1 is a sequence of measurable functions satisfying |fn| ≤ |f | for all n ≥ 1 and lim n→∞ fn = g almost everywhere, for some measurable function g, then lim n→∞ ‖fn − g‖mα q,p = 0. proposition 4.4 yields obviously what follows. lemma 4.5. let 1 ≤ q ≤ α ≤ p <∞. then for any element f of mα q,p , we have lim n→∞ ∥∥f − f χq(0,n) ∥∥ mα q,p = 0. we are now ready to prove theorem 2.5. proof of theorem 2.5for any integer n ≥ 1, we have, by proposition 4.1, ‖f − (f ωn) ∗ φn‖mα q,p ≤ ‖f − f ∗ φn‖mα q,p + ‖(f − f ωn) ∗ φn‖mα q,p ≤ ‖f − f ∗ φn‖mα q,p + c ‖f − f ωn‖mα q,p ‖φn‖1 ≤ ‖f − f ∗ φn‖mα q,p + c ‖f − f ωn‖mα q,p . notice that, for any integer n ≥ 1, |f − f ωn| ≤ ∣∣f − f χq(0,n) ∣∣ and therefore we obtain ‖f − (f ωn) ∗ φn‖mα q,p ≤ ‖f − f ∗ φn‖mα q,p + c ∥∥f − f χq(0,n) ∥∥ mα q,p . thus, it follows from theorem 2.4 and lemma 4.5 that lim n→∞ ‖f − (f ωn) ∗ φn‖mα q,p = 0. this finishes the proof. � https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 155. fractional operators in mα q,pthis section is devoted to prove theorem 2.1 and theorem 2.2. in order to do this we need somepreparatory lemmas.let 0 < γ < 1 and d be a dyadic grid. the dyadic fractional maximal operator mdγ is definedby mdγ f (x) = sup { |q|γ−1 ∫ q |f (y)|dy / q ∈ d, x ∈ q } , f ∈ l1 loc , x ∈ rd . the following lemma is a consequence of proposition 3.3. lemma 5.1. let 0 < γ < 1. for any element f of l1 loc we have : mγf (x) ≤ 3d(1−γ) max t∈{−1/3,0,1/3}d md t γ f (x) , x ∈ rd . proof. let us consider an element (f , x) of l1 loc × rd and a cube q of rd containing x . byproposition 3.3, there exist an element t of {−1/3, 0, 1/3}d and a cube qt of dt such that q isincluded in qt and `(qt) ≤ 3 `(q). thus, we have |q|γ−1 ∫ q |f (y)|dy = `(q)d(γ−1) ∫ q |f (y)|dy ≤ [ 1 3 `(qt) ]d(γ−1) ∫ q |f (y)|dy ≤ 3d(1−γ)|qt |γ−1 ∫ qt |f (y)|dy ≤ 3d(1−γ)md t γ f (x). consequently mγf (x) ≤ 3d(1−γ) max t∈{−1/3,0,1/3}d md t γ f (x). the proof is complete. � recall that the density of l∞c in mα q,p (see point 2) of proposition 3.7) has been proved in [8].here, we improve this result which will play a key role in the proof of lemma 5.3. lemma 5.2. let 1 ≤ q ≤ α ≤ p <∞ and f be any element ofmα q,p . then there exists a sequence (fn)n≥1 of elements of l∞c ∩mα q,p such that (|fn|)n≥1 ↑ |f | almost everywhere and lim n→∞ ‖f − fn‖mα q,p = 0. proof. let us set, for any integer n ≥ 1, fn = sgn(f ) min ( |f |, nχq(0,2n) ) , where, for any x ∈ rd , sgn(f )(x) = { f (x) |f (x)| if f (x) 6= 0 0 if f (x) = 0. https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 16it is easy to see that (fn)n≥1 is a sequence of elements of l∞c ∩mα q,p satisfying (|fn|)n≥1 ↑ |f |almost everywhere and  |fn| ≤ |f | , n ≥ 1 lim n→∞ fn = f almost everywhere.therefore, an application of proposition 4.4 leads to lim n→∞ ‖f − fn‖mα q,p = 0. this ends the proof. � as a consequence of lemma 5.2, the following result holds true. lemma 5.3. let us assume that 0 < γ < 1 α ≤ 1 and 1 p = 1 α − γ. then for any dyadic grid d and any element f of mα 1,p , we have ‖mdγ f ‖p ≤ 2 ‖f ‖mα 1,p . proof. if α = 1 then mα q,p = {0} and so the result is obvious. thus, we assume that α > 1.let f be any element of mα 1,p and d be a dyadic grid.1) assume that f also belongs to l∞.a) we have, for all cube q of rd , |q|γ−1 ∫ q |f (y)|dy ≤ |q|γ‖f ‖∞ and |q|γ−1 ∫ q |f (y)|dy = |q|γ− 1 α |q| 1 α −1 ∫ q |f (y)|dy = |q|γ− 1 α ‖f ‖mα 1 . consequently, lim `(q)→∞ |q|γ−1 ∫ q |f (y)|dy = 0 (∗) and, for all q ∈ q, |q|γ−1 ∫ q |f (y)|dy ≤ { ‖f ‖∞ if `(q) ≥ 1 ‖f ‖mα 1 if `(q) ≤ 1. thus, for all x ∈ rd , mγf (x) ≤ m with m = max ( ‖f ‖∞, ‖f ‖mα 1 ) . since mdγ f ≤mγf , we have, for any x ∈ rd , mdγ f (x) ≤ m. (∗∗) b) assume that f 6= 0. by (∗∗), we have, for all x ∈ rd , mdγ f (x) ∈ (0,m]. (∗ ∗ ∗) https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 17(i) let us consider an integer j ≥ 0 and set ej = { x ∈ rd : mdγ f (x) ∈ ( 2−j−1m, 2−jm ]} dj = { q ∈ d : |q|γ−1 ∫ q |f (y)|dy ∈ ( 2−j−1m, 2−jm ]} . we have, for all x ∈ rd , x ∈ ej ⇐⇒ ∃qx ∈ dj : x ∈ qx ⇐⇒ x ∈ ⋃ q∈dj q. thus ej = ⋃ q∈dj q. note that, by (∗), sup { `(q) : q ∈ dj } <∞ and let us denote by ∆j the set of maximal elements(for the inclusion) of dj . it is easy to see that ⋃ q∈∆j q = ⋃ q∈dj q = ej ∀ q′, q′′ ∈ ∆j , q ′ 6= q′′ =⇒ |q′ ∩q′′| = 0.moreover, by the definition of ej , we have, for all x ∈ q ∈ ∆j , |q|γ−1 ∫ q |f (y)|dy ≤mdγ f (x) ≤ 2−jm < 2 |q|γ−1 ∫ q |f (y)|dy and therefore, for all q ∈ ∆j ,∫ q [ mdγ f (x) ]p dx ≤ 2p |q| [ |q|γ−1 ∫ q |f (y)|dy ]p = 2p [ |q| 1 α −1 ∫ q |f (y)|dy ]p . so we obtain ∫ ej [ mdγ f (x) ]p dx = ∑ q∈∆j ∫ q [ mdγ f (x) ]p dx ≤ 2p ∑ q∈∆j [ |q| 1 α −1 ∫ q |f (y)|dy ]p . (∗ ∗ ∗∗) (ii) by (∗ ∗ ∗), we have ⋃ j≥0 ej = rd . meanwhile, the definition of ej(j ≥ 0) shows that ∀ j ′, j ′′ ∈ n, with j ′ 6= j ′′,ej ′ ∩ ej ′′ = ∅. hence we have ∫ rd [ mdγ f (x) ]p dx = ∑ j≥0 ∫ ej [ mdγ f (x) ]p dx and therefore, by (∗ ∗ ∗∗), ‖mdγ f ‖p ≤ 2 ∑ j≥0 ∑ q∈∆j [ |q| 1 α −1 ∫ q |f (y)|dy ]p 1 p . (∗ ∗ ∗ ∗ ∗) https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 18note that, if j ′ and j ′′ are two integer such that j ′′ > j ′ ≥ 0 then ∀ (q′, q′′) ∈ ∆j ′ × ∆j ′′ , q ′ 6= q′′ and so { qk,m : qk,m ∈ ∆j ′ } ∩ { qk,m : qk,m ∈ ∆j ′′ } = ∅, k ∈ zd , m ∈ z.therefore∑ j≥0 ∑ q∈∆j [ |q| 1 α −1 ∫ q |f (y)|dy ]p = ∑ m∈z ∑ k∈zd :qk,m∈ ⋃ j≥0 ∆j [ |qk,m| 1 α −1 ∫ qk,m |f (y)|dy ]p ≤ ‖f ‖pmα 1,p . this inequality combined with (∗ ∗ ∗ ∗ ∗) gives ‖mdγ f ‖p ≤ 2 ‖f ‖mα 1,p . 2) by lemma 5.2, there exists a sequence (fn)n≥1 of elements of l∞c ∩mα q,p such that (|fn|)n≥1 ↑ |f |almost everywhere and lim n→∞ ‖f − fn‖mα q,p = 0. thus, the result obtained in point 1) implies that ∥∥mdγ fn∥∥p ≤ 2 ‖fn‖mα 1,p ≤ 2 ‖f ‖mα 1,p , n ≥ 1 0 ≤ ( mdγ fn ) n≥1 ↑mdγ f and so (∥∥mdγ fn∥∥p)n≥1 ↑ ∥∥mdγ f ∥∥pand therefore ‖mdγ f ‖p ≤ 2 ‖f ‖mα 1,p .the proof is complete. � now we prove theorem 2.1 thanks to lemma 5.1, lemma 5.3 and corollary 3.6. proof of theorem 2.1let f be in mα 1,p . by lemma 5.1 and lemma 5.3, we have ‖mγf ‖p ≤ 3d(1−γ) ∥∥∥∥∥ max t∈{−1/3,0,1/3}d md t γ f ∥∥∥∥∥ p ≤ 3d(1−γ) ∑ t∈{−1/3,0,1/3}d ∥∥∥mdtγ f ∥∥∥ p ≤ 3d(1−γ) ∑ t∈{−1/3,0,1/3}d 2 ‖f ‖mα 1,p(dt) . note that the hypotheses imply that p <∞ and so corollary 3.6 leads to ‖mγf ‖p ≤ 3d(1−γ)2 d ( 2− 1 p ) ‖f ‖mα 1,p ] ( {−1/3, 0, 1/3}d ) = 2 d ( 2− 1 p ) 3d(2−γ) ‖f ‖mα 1,p . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 19the proof is complete. � as an immediate consequence of theorem 2.1 we can now prove theorem 2.2. proof of theorem 2.2let f be in mα 1,p . note that the hypotheses imply that p <∞. by [12, theorem 1], we have ‖iγf ‖p ≤ d ‖mγf ‖p ,where d is a real constant not depending on f . therefore, theorem 2.1 provides the desiredinequality. � 6. application theorem 2.2, theorem 2.5 and the boundedness properties of riesz tranforms in lebesgue spaceslead to the following result, which contains theorem 2.6. proposition 6.1. let us assume that d ≥ 3, 1 ≤ q ≤ α < d , 1 p = 1 α − 1 d and f is an element of the bourgain-morrey space mα q,p . then 1) for 1 ≤ j ≤ d , the function fj = rj ( i 1 d f ) belongs to lp 2) there exists a real constant cd depending only on d and such that f = ( cd fj ) 1≤j≤d is a solution in (lp)d of the equation (11). proof. note that, the hypotheses imply that 1 ≤ q ≤ α < p <∞.1) since mα q,p ⊂ mα 1,p (see (1)), theorem 2.2 implies that i 1 d f ∈ lp . furthermore, it is wellknown that the riesz tranform rj is bounded on lp , for 1 ≤ j ≤ d . therefore, we deduce that fj = rj ( i 1 d f ) belongs to lp .2) a) let ϕ be any element of c∞c . for 1 ≤ j ≤ d , the boundedness properties of rj and i 1 d showthat ψj = rj ( i 1 d ϕ ) belongs to ⋂ r> d d−1 lr . since d d−1 < 2, there exists a real number r such that d d−1 < r < 2 and ψj ∈ lr . therefore, we can use the fourier transform to obtain cd d∑ j=1 ∂jψj = ϕ, where cd is a real constant depending only on d (see [14, formula (17), p.125]).b) fix an integer n ≥ 1 and set fn = (f ωn) ∗ φn. since fn ∈ c∞c , the result of point a) implies thatdiv fn = fn, where fn = (fnj )1≤j≤d with fnj = cdrj ( i 1 d fn ) ∈ ⋂ r> d d−1 lr , 1 ≤ j ≤ d. • according to theorem 2.5, (fn)n≥1 converges to f in mα 1,p . • for 1 ≤ j ≤ d , the boundedness properties of rj and i 1 d imply that (fnj )n≥1 converges to cdfj = cdrj ( i 1 d f ) in lp . https://doi.org/10.28924/ada/ma.4.16 eur. j. math. anal. 10.28924/ada/ma.4.16 20therefore, for any element ϕ of c∞c , we have∫ rd div f (x)ϕ(x)dx = − d∑ j=1 ∫ rd cdfj(x) ∂jϕ(x)dx = lim n→∞ − d∑ j=1 ∫ rd fnj (x) ∂jϕ(x)dx  = lim n→∞ ∫ rd  d∑ j=1 ∂jfnj (x) ϕ(x)dx = lim n→∞ ∫ rd div fn(x)ϕ(x)dx = lim n→∞ ∫ rd fn(x)ϕ(x)dx = ∫ rd f (x)ϕ(x)dx. hence, div f = f . thus, we obtain the desired result. � acknowledgement. the author would like to express his deep thanks to professor ibrahim fofanafor his helpful assistance with this note. references [1] c. bennett, r.c. sharpley, interpolation of operators, academic press, new york, (1988).[2] j. bourgain, on the restriction and multiplier problems in r3, in: geometric aspects of functional analysis (1989-90), in: lecture notes in math., springer, berlin, 1469, (1991), 179-191.[3] d. cruz-uribe, two weight inequalities for fractional integral operators and commutators, in: advanced courses ofmathematical analysis vi, world scientific, universidad de málaga, spain, 2017: pp. 25–85. https://doi.org/ 10.1142/9789813147645_0002.[4] n. diarra, i. fofana, characterization of some closed linear subspaces of morrey spaces and approximation, adv.pure appl. math. 14 (2023) 41–72. https://doi.org/10.21494/iste.op.2023.0980.[5] n. diarra, i. fofana, complex interpolation of some banach spaces including morrey spaces, khayyam j. math.(2024).[6] m. dosso, i. fofana, m. sanogo, on some subspaces of morrey–sobolev spaces and boundedness of riesz integrals,ann. polon. math. 108 (2013) 133–153. https://doi.org/10.4064/ap108-2-2.[7] i. fofana, f.r. faléa, b.a. kpata, a class of subspaces of morrey spaces and norm inequalities on riesz potentialoperators, afr. mat. 26 (2015) 717–739. https://doi.org/10.1007/s13370-014-0241-3.[8] n. hatano, t. nogayama, y. sawano, d.i. hakim, bourgain–morrey spaces and their applications to boundednessof operators, j. funct. anal. 284 (2023), 109720. https://doi.org/10.1016/j.jfa.2022.109720.[9] s. masaki, j. segata, existence of a minimal non-scattering solution to the mass-subcritical generalized korteweg–de vries equation, ann. inst. h. poincaré anal. non linéaire 35 (2018) 283–326. https://doi.org/10.1016/j. anihpc.2017.04.003.[10] s. masaki, j. segata, refinement of strichartz estimates for airy equation and application, rims kôkyûrokubessatsu, b80 (2020) 11–25. http://hdl.handle.net/2433/260656.[11] c.b. morrey, on the solutions of quasi-linear elliptic partial differential equations, trans. amer. math. soc. 43(1938) 126–166. https://doi.org/10.28924/ada/ma.4.16 https://doi.org/10.1142/9789813147645_0002 https://doi.org/10.1142/9789813147645_0002 https://doi.org/10.21494/iste.op.2023.0980 https://doi.org/10.4064/ap108-2-2 https://doi.org/10.1007/s13370-014-0241-3 https://doi.org/10.1016/j.jfa.2022.109720 https://doi.org/10.1016/j.anihpc.2017.04.003 https://doi.org/10.1016/j.anihpc.2017.04.003 http://hdl.handle.net/2433/260656 eur. j. math. anal. 10.28924/ada/ma.4.16 21 [12] b. muckenhoupt, r. wheeden, weighted norm inequalities for fractional integrals, trans. amer. math. soc. 192(1974) 261–274. https://doi.org/10.1090/s0002-9947-1974-0340523-6.[13] n.c. phuc, m. torrès, characterizations of the existence and removable singularities of divergence-measure vectorfields, indiana univ. math. j., 57 (2008) 1573–1597. https://www.jstor.org/stable/24902998.[14] e.m. stein, singular integrals and differentiability properties of functions, princeton university press, princeton,new jersey (1970).[15] j. tao, d. yang, w. yuan, a bridge connecting lebesgue and morrey spaces via riesz norms, banach j. math. anal.15 (2021) 20. https://doi.org/10.1007/s43037-020-00106-6. https://doi.org/10.28924/ada/ma.4.16 https://doi.org/10.1090/s0002-9947-1974-0340523-6 https://www.jstor.org/stable/24902998 https://doi.org/10.1007/s43037-020-00106-6 1. introduction 2. statement of the main results 3. preliminaries 3.1. equivalent norms on mq,p 3.2. continuity of the translation operator in mq,p 4. inclusion and approximation results 4.1. inclusion of mq,p in f(q,p,) 4.2. approximation in mq,p 5. fractional operators in mq,p 6. application references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 15doi: 10.28924/ada/ma.5.15 three step inverse free kurchatov-like methods of convergence order close to four for equations ioannis k. argyros1,∗ , stepan shakhno2 , halyna yarmola3,∗ , samundra regmi4 , nirjalshrestha5 1department of computing and mathematics sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2department of theory of optimal processes, ivan franko national university of lviv, lviv, ukraine stepan.shakhno@lnu.edu.ua 3department of computational mathematics, ivan franko national university of lviv, lviv, ukraine halyna.yarmola@lnu.edu.ua 4department of mathematics, university of houston, houston, tx 77004, usa sregmi5@uh.edu 5department of mathematics, university of florida, gainesville, fl 32603, usa n.shrestha@ufl.edu ∗correspondence: iargyros@cameron.edu, halyna.yarmola@lnu.edu.ua abstract. an inverse free kurchatov-like methods with three steps is introduced of convergence orderclose to four to generate sequences approximating solutions of equations defined on complete normedspaces. the local analysis shows r-convergence close to four under conditions controlling the divideddifference. numerous experiments demonstrate the performance of the method. 1. introduction let e1, e2 represent complete normed spaces [1] and ω ⊂ e1 be open and convex. a plethoraof applications from different fields can be formulated as f (x) = 0. (1.1) here f : ω→ e2 is a continuos operator. a solution of equation (1.1), which is denoted by x∗ ∈ ωis given in analytical form only in rare cases. that leads to the development of iterative methodsgenerating sequence converging to x∗ provided some conditions are satisfied involving the initialinformation.one of the most time-consuming part of iterative methods for solving nonlinear problems isfinding the inverse operator or solving the corresponding linear problem. to avoid this, methods received: 9 mar 2025. key words and phrases. complete normed space; divided difference; kurchatov method; local convergence; order four.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.15 https://orcid.org/0000-0002-9189-9298 https://orcid.org/0000-0002-3845-6260 https://orcid.org/0000-0002-8986-2509 https://orcid.org/0000-0003-0035-1022 https://orcid.org/0000-0003-4070-091x eur. j. math. anal. 10.28924/ada/ma.5.15 2with approximation of the inverse operator were developed. one of the first such methods wasproposed by ulm in [17] xn+1 = xn − tnf (xn), tn+1 = 2tn − tnf ′(xn+1)tn, n = 0, 1, 2,. . . . (1.2) here x0 ∈ ω and t0 ∈ l(e2, e1) are given initial approximations for the solution x∗ and the inverseoperator f ′(x∗)−1, respectively. the ulm method (1.2) was studied under conditions of differenttypes and it was shown that it converges with second order [2, 3, 10, 12, 13, 15, 17]. similar methodwas prosed by moser [13,14] xn+1 = xn − tnf (xn), tn+1 = 2tn − tnf ′(xn)tn, n = 0, 1, 2,. . . . (1.3) in (1.3) f ′(xn) appears instead of f ′(xn+1) in (1.2). the convergence order for (1.3) is equal to 1+ √ 5 2 . the methods with approximation of the inverse operator with higher convergence order werestudied in [6, 9].in this paper we propose the three step kurchatov-like method (tsklm). this method is definedfor t0 ∈ l(e2, e1) and each n = 0, 1, 2, . . . by yn = xn − tnf (xn), zn = yn − tnf (yn), xn+1 = zn − tnf (zn), kn+1 = [2yn − xn, xn;f ], mn = 2tn − tnkn+1tn, tn+1 = mn +mn(2i −kn+1mn)(i −kn+1mn), (1.4) where [·, ·;f ] : ω × ω → e2 is a divided difference of order one [1, 7] and l(e2, e1) is the spaceof linear operators mapping e2 into e1, x0 ∈ ω. definition 1.1. [1, 7] let f be a nonlinear operator defined on a subset ω of a banach space e1 with values in a banach space e2, and let x, y , be two different points of ω. a linear operator from e1 to e2 which is denoted by [x, y ;f ] and satisfies the following conditions [x, y ;f ](x − y) = f (x)− f (y) is called a first-order divided difference of f at the points x and y . if there exists a fréchet derivative f ′(x), then [x, x ;f ] = f ′(x). notice that there are other selection for the kurchatov operator kn+1 such as kn+1 = [2xn+1− zn, zn;f ] or kn+1 = [2xn+1 − xn, xn;f ] or kn+1 = f ′(xn+1) or kn+1 = l(xn+1), where l(xn+1) is https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 3an approximation to f ′(xn+1), or other options [1,4,5,8,16]. denote the corresponding methods by yn = xn − tnf (xn), zn = yn − tnf (yn), xn+1 = zn − tnf (zn), kn+1 = [2xn+1 − zn, zn;f ], mn = 2tn − tnkn+1tn, tn+1 = mn +mn(2i −kn+1mn)(i −kn+1mn), (1.5) yn = xn − tnf (xn), zn = yn − tnf (yn), xn+1 = zn − tnf (zn), kn+1 = [2xn+1 − xn, xn;f ], mn = 2tn − tnkn+1tn, tn+1 = mn +mn(2i −kn+1mn)(i −kn+1mn), (1.6) yn = xn − tnf (xn), zn = yn − tnf (yn), xn+1 = zn − tnf (zn), kn+1 = f ′(xn+1), mn = 2tn − tnkn+1tn, tn+1 = mn +mn(2i −kn+1mn)(i −kn+1mn), (1.7) yn = xn − tnf (xn), zn = yn − tnf (yn), xn+1 = zn − tnf (zn), kn+1 = l(xn+1), mn = 2tn − tnkn+1tn, tn+1 = mn +mn(2i −kn+1mn)(i −kn+1mn), (1.8) respectively. notice that method (1.8) specializes to (1.7) if l = f ′. a possible choice for l ispresented in the numerical section.to test numerically the order of convergence of the iterative methods very often use computationalorder of convergence (coc) and approximated computational order of convergence (acoc) [11].coc is denoted by δ2, acoc can be computed by formulas denoted by δ1 and δ3: δ1 ≈ ln ( ‖xn+1−xn‖ ‖xn−xn−1‖ ) ln ( ‖xn−xn−1‖ ‖xn−1−xn−2‖ ) , δ2 ≈ ln ( ‖xn+1−x∗‖ ‖xn−x∗‖ ) ln ( ‖xn−x∗‖ ‖xn−1−x∗‖ ) , δ3 ≈ ln ( ‖f (xn+1)‖ ‖f (xn)‖ ) ln ( ‖f (xn)‖ ‖f (xn−1)‖ ) . in this article, we provide the local convergence analysis of the method (1.4) under assumptionsthat fréchet derivative and first-order divided differences satisfy classical lipschitz conditions (seesection 2). sections 3 and 4 present results of numerical experiments and conclusions, respectively. https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 42. convergence the local analysis of convergence is very important since it provides the degree of difficulty inselecting the initial points x0 from a ball centered at the solution x∗ and of a certain specifiedradius.the symbol s(x, r) is denoting an open ball centered at x ∈ e1 and of a radius r > 0.define the parameter ρ by ρ = sup{t > 0 : s(x∗, t) ⊂ ω}. let a > 0, b0 ≥ 0, b > 0, d1 > 0, d2 > 0, λ ≥ 0 and l ≥ 0 be given parameters. it is convenient todefine the parameters q = b a , ρ0 = min { 1, ρ, 1 4lb(l + d1)d2 } , ρ̄ ∈ (0, ρ0), c1 = 4lb(l + d1), c2 = 1 + 2bc1, c3 = c2 + 2lb, (2.1) c4 = c2 + 4lb(1 + c3), c5 = c4 + 4lb(1 + c3), c6 = 1 + 4bc1 + 8lb, b0 = d2 1− c1d2ρ̄ f or ρ̄ ∈ (0, ρ0), a = min { 1, 1 c3c4 + 2lbc23 , 1 c5 + 2lb , 1 c66 } , and b = min{ρ̄, a}. let x∗ ∈ ω be a solution of the equation f (x) = 0. we assume from now on that for each u1, u2, v1, v2 ∈ s(x∗, ρ) there exists l > 0 such that ‖[u2, u1;f ]− [v2, v1;f ]‖ ≤ l(‖u2 − v2‖+ ‖u1 − v1‖). (2.2) it follows by (2.2) that f ′ exists, [x, x ;f ] = f ′(x) and for each u, v ∈ s(x∗, ρ̄) ‖f ′(u)− f ′(v)‖ ≤ 2l‖u − v‖. (2.3) moreover, assume f ′(x∗) is invertible and set ‖f ′(x∗)‖ ≤ d1, ‖f ′(x∗)−1‖ ≤ d2. (2.4) https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 5furthermore, assume t0, k0 satisfy ‖t0‖ ≤ b0 ≤ b, ‖i − t0k0‖ ≤ λ, b ≥ b0 and λ ∈ [0, b]. (2.5) finally, assume s(x∗, ρ∗) ⊂ ω, ρ∗ = max{ρ̄, 3b}. (2.6)the conditions (2.2), (2.4)-(2.6) are called (a) from now on.next, the main local analysis of convergence is presented under the conditions (a) and thepreceding notation. theorem 2.1. assume that the conditions (a) hold and b < a. then, the sequence {xn} generated by (tsklm) converges to the solution of the equation f (x) = 0 provided that x0 ∈ s(x∗, b). moreover, the following assertions hold for en = ‖xn − x∗‖, hn = ‖i − tnkn‖, en ≤ aq4 n (2.7) and hn ≤ aq4 n−1 , h0 ≤ b, (2.8) n = 0, 1, 2, . . ., where ρ, a, b, q are given by (2.1). proof. the assertions (2.7) and (2.8) are shown by induction. if n = 0, assertion (2.7) for n = 0holds, since e0 ≤ b. then, by (2.5), we get a ∈ [0, 1] and h0 ≤ λ ≤ b, so (2.8) holds if n = 0.assume (2.7) and (2.8) hold if n = i . that is ei ≤ aq4 i < b < ρ̄ (2.9) and hi ≤ aq4 i−1 . (2.10)we need the estimate ‖ki+1 − f ′(xi)‖ = ‖[2yi − xi , xi ;f ]− [xi , xi ;f ]‖ ≤ l(‖2yi − xi − xi‖+ ‖xi − xi‖) = 2l‖yi − xi‖ (2.11) ≤ 2l‖tif (xi)‖ ≤ 2l‖ti‖(l + d1)‖xi − x∗‖ ≤ c1ei ≤ c1aq4 i , where we used (2.1), (2.2) (i.e (2.3)), (2.5), the induction hypotheses (2.9), (2.10) and f (xi) = f (xi)− f (x∗) = 1∫ 0 f ′(x∗ + θ(xi − x∗))dθ(xi − x∗) = 1∫ 0 [ f ′(x∗ + θ(xi − x∗))− f ′(x∗) + f ′(x∗) ] dθ(xi − x∗) https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 6leading to ‖f (xi)‖ ≤ (l + d1)‖xi − x∗‖. (2.12) it follows by the banach lemma on invertible operators [1], (2.1) and (2.11) that the operator ki+1is invertible and ‖k−1i+1‖ ≤ ‖f ′(x∗)−1‖ 1− ‖f ′(x∗)−1‖‖ki+1 − f ′(xi)‖ ≤ d2 1− d2c1ei ≤ d2 1− d2c1ρ̄ = b0 ≤ b, (2.13) so ‖ti‖ = ‖tikik−1i ‖ = ‖(−i + (i − tiki))k−1i ‖ ≤ (1 + ‖i − tiki‖)‖k−1i ‖ ≤ (1 + aq4 i )b ≤ (1 + 1)b = 2b, (2.14) and ξi = ‖i − tif ′(xi)‖ = ‖(i − tiki) + (tiki − tif ′(xi))‖ ≤ ‖i − tiki‖+ ‖ti‖‖ki − f ′(xi)‖ ≤ aq4 i + 2bc1aq 4i = c2aq 4i . (2.15) then, by the first substep of (tsklm) we can write in turn yi − x∗ = xi − x∗ − ti(f (xi)− f (x∗)) = 1∫ 0 [ (i − tif ′(xi)) + ti(f ′(xi)− f ′(x∗ + θ(xi − x∗))) ] (xi − x∗)dθ. (2.16) it follows by the induction hypotheses (2.9), (2.10), the (2.16) triangle inequality, (2.3), and (2.13)-(2.16), we have in turn that ‖yi − x∗‖ = ξiei + l‖ti‖e2i ≤ c2aq 4iaq4 i + 2lb ( aq4 i )2 ≤ c3a2q2×4 i . (2.17) and since a ∈ [0, 1] ‖xi − yi‖ ≤ ei + ‖yi − x∗‖ ≤ aq4 i + c3a 2q2×4 i ≤ (1 + c3)aq 4i (2.18) and ‖i − tif ′(yi)‖ = ‖(i − tif ′(xi)) + (tif ′(xi)− tif ′(yi))‖ ≤ c2aq 4i + 4lb(1 + c3)aq 4i = c4aq 4i . (2.19) https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 7in an analogous way, we get in turn ‖zi − x∗‖ ≤ ‖(i − tif ′(yi))(yi − x∗)‖+ l‖ti‖‖yi − x∗‖2 ≤ c4aq 4i c3a 2q2×4 i + l(2bc23a 4q4×4 i ) ≤ (c3c4 + 2lbc23 )a2q3×4 i = a2q3×4 i , (2.20) ‖yi − zi‖ ≤ ‖yi − x∗‖+ ‖zi − x∗‖ ≤ c3a 2q2×4 i + a2q3×4 i = (1 + c3)a 2q2×4 i (2.21) and ‖i − tif ′(zi)‖ ≤ ‖i − tif ′(yi)‖+ ‖ti‖‖f ′(yi)− f ′(zi))‖ ≤ c4aq 4i + 4lb(1 + c3)a 2q2×4 i = c5aq 4i (2.22) leading to ei+1 ≤ ‖i − tif ′(zi)‖‖zi − x∗‖+ l‖ti‖‖zi − x∗‖2 ≤ c5aq 4ia2q3×4 i + l(2ba4q6×4 i ) ≤ (c5 + 2lb)q4 i+1 ≤ aq4i+1 (2.23) showing (2.7) for n = i + 1. moreover, we have ‖xi+1 − xi‖ ≤ ‖xi+1 − x∗‖+ ‖xi − x∗‖ ≤ aq4 i+1 + aq4 i ≤ 2aq4 i (2.24) notice that 2yi − xi ∈ s(x∗, 3b) by (2.6) and ei+1 ≤ q4i+1 < b < ρ̄.hence, we can have ‖ki+1 −ki‖ ≤ ‖(ki+1 − f ′(xi)) + (f ′(xi)− f ′(xi−1)) + (f ′(xi−1)−ki)‖ ≤ c1aq 4i + 4laq4 i−1 + c1aq 4i−1 ≤ (2c1 + 4l)aq4 i−1 (2.25) ‖i − tiki+1‖ = ‖(i − tiki) + (tiki+1 − tiki)‖ ≤ aq4 i + 2b(2c1 + 4l)aq4 i−1 = c6aq 4i−1 . (2.26) next, by the fourth equation of (tsklm), we can write i −miki+1 = (i − tiki+1)2, so ‖i −miki+1‖ ≤ ‖i − tiki+1‖2 ≤ c26a2q2×4 i−1 . (2.27) but, we can also write ti+1 = mi +mi(2i −ki+1mi)(i −ki+1mi). (2.28) https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 8thus, we get i − ti+1ki+1 = i − (mi +mi(2i −ki+1mi)(i −ki+1mi))ki+1 = (i −miki+1) 3. (2.29) therefore, since a ∈ [0, 1], we get by the inductions hypotheses and (2.29) that ‖i − ti+1ki+1‖ ≤ ‖i −miki+1‖3 ≤ c66a6q6×4 i−1 ≤ aq4i showing (2.8) for n = i + 1.consequently the induction for (2.7) and (2.8) is completed.finally, by letting n →∞ in (2.7), we deduce that lim n→∞ xn = x∗, since q ∈ [0, 1). remark 2.2. it turns out that the proof of theorem 2.1 can be repeated in the case of the method (1.8) as long as we add an additional condition of the form for each n = 0, 1, 2, . . . ‖ln − f ′(xn)‖ ≤ σn‖f (xn)‖, (2.30) where {σn} is a nonnegative sequence such that sup n≥0 σn ≤ σ, where σ ≥ 0 3. numerical examples in this section, we present the results of numerical investigation of the three-step kurchatov-like method for solving the nonlinear equation (1.1). we give errors at each iteration andacoc and coc for considered methods (1.4), (1.5), (1.6), (1.7), (1.8). the computations werecarried out on a pc with 1.00 ghz processor and 8 gb of memory with use of softwaregnu octave 7.3.0. the euclidean norm was used. the initial approximation t0 was com-puted by formulas t0 = [2x0 − x−1, x−1;f ]−1 for methods (1.4), (1.5), (1.6), t0 = f ′(x0) −1– for method (1.7) and t0 = l(x0) −1 – for method (1.8), where (a) l(x) = [x, x + α;f ] and(b) l(x) = [x + α1f (x), x + α2f (x);f ] with α ∈ r. example 3.1. let f : r→ r and consider the nonlinear equation f (x) = ex−0.1 − 10x |x − 1| − 0.1 = 0 with the exact solution x∗ = 0.1. example 3.2. let f : r2 → r2 and consider the system of two equations with x = (ξ; η){ f1(x) = 3ξ2η + η2 + |ξ − 1| − 0.75 = 0, f2(x) = ξ4 + ξη3 + |η| − 0.5625 = 0, and the exact solution x∗ = (0.5; −1). https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 9table 1. error’s value at each iteration for example 3.1. n method (1.4) method (1.5) method (1.6) ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖0 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+001 6.0089e-02 4.5850e-01 6.0089e-02 4.5850e-01 6.0089e-02 4.5850e-012 2.3413e-03 1.6447e-02 2.1002e-04 1.4706e-03 1.9390e-04 1.3577e-033 3.4615e-07 2.4231e-06 3.2876e-14 2.3012e-13 1.2934e-14 9.0566e-144 1.3878e-17 8.3267e-17 table 2. error’s value at each iteration for example 3.1 (method (1.8)). n (a), α = 10−6 (b), α1 = 0, α2 = 0.01 ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖0 6.0000e-01 7.9488e+00 6.0000e-01 7.9488e+001 6.0098e-02 4.5857e-01 5.2350e-02 3.9520e-012 2.1022e-04 1.4720e-03 1.2109e-04 8.4780e-043 3.2724e-14 2.2890e-13 3.0254e-15 2.1178e-14 table 3. error’s value at each iteration for example 3.1 (method (1.8)). n (b), α1 = −1, α2 = 1 (b), α1 = 0, α2 = 1 (b), α1 = −1, α2 = 0 ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖0 2.2500e-01 2.1048e+00 2.2500e-01 2.1048e+00 2.2500e-01 2.1048e+001 4.3916e-02 3.2765e-01 2.8627e-04 2.0031e-03 8.9687e-02 7.1215e-012 1.1100e-04 7.7714e-04 3.3741e-12 2.3618e-11 1.4334e-02 1.0249e-013 2.4702e-15 1.7292e-14 6.9350e-05 4.8550e-044 5.5539e-14 3.8877e-13 table 4. coc, acoc for example 3.1. method δ1 δ2 δ3method (1.4) 2.7545 2.7145 2.7309method (1.5) 3.1543 3.9915 3.9319method (1.6) 4.0870 4.0870 4.0244method (1.8) (a), α = 10−6 3.9956 3.9931 3.9935method (1.8) (b), α1 = 0, α2 = 0.01 3.2266 4.0224 3.9731 tables 1, 2, 3, 6 and 7 contain the values ‖xn− x∗‖ and ‖f (xn)‖ at each iteration. the iterativeprocess was stopped if ‖f (xn)‖ ≤ 10−10. https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 10table 5. coc, acoc for example 3.1. method δ1 δ2 δ3method (1.8) (b), α1 = −1, α2 = 1 3.2944 4.1014 4.0583method (1.8) (b), α1 = 0, α2 = 1 2.0169 2.7384 2.6240method (1.8) (b), α1 = −1, α2 = 0 3.0250 3.9288 3.9133 table 6. error’s value at each iteration for example 3.2. n method (1.4) method (1.5) method (1.6) ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖0 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-011 1.3484e-02 8.8111e-03 1.3484e-02 8.8111e-03 1.3484e-02 8.8111e-032 1.1530e-04 9.6933e-05 1.5071e-05 9.5531e-06 7.9188e-05 5.5281e-053 1.5872e-10 9.9579e-11 1.1102e-16 1.1102e-16 4.7092e-11 4.4613e-11 table 7. error’s value at each iteration for example 3.2 (method (1.8)). n (a), α = 10−6 (b), α1 = −1, α2 = 1 (b), α1 = 0, α2 = 1 (b), α1 = −1, α2 = 0 ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖ ‖xn − x∗‖ ‖f (xn)‖0 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-01 2.9069e-01 4.9986e-011 4.2177e-02 3.3550e-02 1.2130e-01 1.3882e-01 1.2302e-01 1.1058e-01 2.6651e-02 9.4616e-022 4.8296e-04 3.1889e-04 1.9815e-02 1.6501e-02 1.7140e-02 1.1420e-02 7.4376e-04 8.7777e-043 1.2251e-11 7.9157e-12 9.5643e-05 7.1351e-05 5.0435e-05 3.1888e-05 7.0982e-11 9.6911e-114 5.0469e-14 3.6791e-14 3.5108e-15 2.2453e-15 table 8. coc, acoc for example 3.2. method δ1 δ2 δ3method (1.4) 2.8293 2.8343 3.0575method (1.5) 3.2733 3.7717 3.6881method (1.6) 2.7872 2.7904 2.7665method (1.8) (a), α = 10−6 3.9231 3.9130 3.7611method (1.8) (b), α1 = −1, α2 = 1 3.1016 4.0053 3.9286method (1.8) (b), α1 = 0, α2 = 1 3.1851 4.0127 3.9750method (1.8) (b), α1 = −1, α2 = 0 3.1562 4.5167 3.4227 tables 4, 5 and 8 show coc and acoc with δ1 was calculated if the condition ‖xn+1− xn‖ ≤ 10−10 was fulfilled δ2 and δ3 were calculated if the condition ‖f (xn)‖ ≤ 10−10 wasfulfilled. https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 11the initial approximations x0 and x−1 for example 3.1 are x0 = −0.5, x−1 = −0.6 (tables 1, 2, 4), x0 = −0.15, x−1 = −0.25 (tables 3, 5) and for example 3.2 – x0 = (0.63; −1.26), x−1 = (0.73; −1.16). example 3.3. let f : rm → rm and consider the system of equations with x = (ξ1; . . . ; ξm) fi(x) = m∑ j=1 ξj + eξi − 1 = 0, i = 1, . . . , m. here the exact solution x∗ = (0; . . . ; 0). table 9. coc, acoc for example 3.3. method δ1 δ2 δ3method (1.4) 2.6007 2.5860 2.6621method (1.5) 3.9096 3.9035 3.7967method (1.6) 2.0176 2.0059 2.0663method (1.7) 3.9118 3.9059 3.7993method (1.8) (a), α = 10−6 3.9117 3.9058 3.7992method (1.8) (b), α1 = 0, α2 = −0.1 3.9097 3.9064 3.9064method (1.8) (c) 2.8165 2.8012 2.8803 table 10. coc, acoc for example 3.3. method δ1 δ2 δ3method (1.8) (b), α1 = −1, α2 = 1 4.3139 4.3096 4.2617method (1.8) (b), α1 = 0, α2 = 1 2.7472 2.7326 2.8050method (1.8) (b), α1 = −1, α2 = 0 3.9118 3.8946 3.8502 for solving nonlinear equations with differentiable operator can be also used method (1.8) with(c) l(xn+1) = 1 2(f ′(xn+1) + [2xn+1− xn, xn+1;f ]). for this method, the errors decrease faster thanfor (1.4) and (1.6).tables 9 and 10 show coc and acoc for example 3.3 with m = 1, the initial approximation x0 = 0.5 and x0 = 0.9, respectively.figures 1 and 2 show changing of ‖xn−xn−1‖, ‖xn−x∗‖ and ‖f (xn)‖ form = 20, x0 = (5; . . . ; 5), x−1 = (5.1; . . . ; 5.1), figure 3 – for x0 = (0.05; . . . ; 0.05). the iterative process was stopped underthe condition ‖xn+1 − xn‖ ≤ 10−10.from the obtained results, we see that among methods (1.4)-(1.6), for method (1.5) the errordecreases faster and it has highest computational order of convergence. https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 12 0 1 2 3 4 5 6 1x10-21 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.4) 0 1 2 3 4 5 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.5) ||xn-xn-1|| ||xn-x *|| ||f(xn)|| 0 1 2 3 4 5 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.6) figure 1. example 3.3: error’s value at each iteration. 0 1 2 3 4 5 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.7) 0 1 2 3 4 5 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.8) (a) ||xn-xn-1|| ||xn-x *|| ||f(xn)|| 0 1 2 3 4 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.8) (b) 0 1 2 3 4 5 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 1002 1003 1004 method (1.8) (c) figure 2. example 3.3: error’s value at each iteration; (b), α1 = 0, α2 = −0.1. https://doi.org/10.28924/ada/ma.5.15 eur. j. math. anal. 10.28924/ada/ma.5.15 13 0 0.5 1 1.5 2 2.5 3 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 method (1.8) (b)-1 0 0.5 1 1.5 2 2.5 3 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 method (1.8) (b)-2 ||xn-xn-1|| ||xn-x *|| ||f(xn)|| 0 0.5 1 1.5 2 2.5 3 1x10-21 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-09 10-08 10-07 10-06 10-05 10-04 10-03 10-02 10-01 1000 1001 method (1.8) (b)-3 figure 3. example 3.3: error’s value at each iteration ((b)-1, α1 = −1, α2 = 1;(b)-2, α1 = 0, α2 = 1; (b)-3, α1 = −1, α2 = 0). 4. conclusion in this article a three-step kurchatov-like method with approximation of inverse operator isintroduced and convergence analysis is provided. the r-convergence four is shown theoreti-cally under lipschitz conditions for fréchet derivative and first-order divided differences. nu-merous experiments demonstrate the performance of the method for different cases of kn+1.among the methods with divided differences, the best results are demonstrated by the meth-ods (1.5), namely the highest computational order of convergence. the operator l was cho-sen in the form l(xn+1) = [xn+1, xn+1 + α;f ], where α is a small number, and l(xn+1) = [xn+1 + α1f (xn+1), xn+1 + α2f (xn+1);f ] with α1, α2 ∈ r. these approximating of the deriv-ative give quite good results, in particular if the nonlinear operator is not differentiable. for somevalues α1 and α2 these methods require a good initial approximation. in the case of the differen-tiable operator, the following choice l(xn+1) = 1 2(f ′(xn+1)+[2xn+1−xn, xn+1;f ]) is also possible.it demonstrates advantages over some methods with divided differences. references [1] i. k. argyros, convergence and applications of newton-type iterations, new york: springer-verlag, 2008. https: //doi.org/10.1007/978-0-387-72743-1.[2] i. k. argyros, on ulm’s method for frechet differentiable operators, j. appl. math. comput. 31 (2009) 97-111. https://doi.org/10.1007/s12190-008-0194-5. https://doi.org/10.28924/ada/ma.5.15 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1007/s12190-008-0194-5 eur. j. math. anal. 10.28924/ada/ma.5.15 14 [3] i. k. argyros, on ulm’s method using divided differences of order one, numer. algorithms 52 (2009) 295-320. https://doi.org/10.1007/s11075-009-9274-3.[4] i.k. argyros, s. shakhno, s. regmi, h. yarmola, on the complexity of a unified convergence analysis for iterativemethods, j. complex. 79 (2023) 101781. https://doi.org/10.1016/j.jco.2023.101781.[5] i.k. argyros, s. shakhno, h. yarmola, improving convergence analysis of the newton–kurchatov method underweak conditions two-step solver for nonlinear equations, computation 8 (2020) 8. https://doi.org/10.3390/ computation8010008.[6] i.k. argyros, s.m. shakhno, h.p. yarmola, method of third-order convergence with approximation of inverseoperator for large scale systems, symmetry 12 (2020), 978. https://doi.org/10.3390/sym12060978.[7] m. balázs, g. goldner, on existence of divided differences in linear spaces, rev. anal. numér. théorie approx. 2(1973) 5-9. https://doi.org/10.33993/jnaat21-6.[8] j.e. dennis, r.b. schnabel, numerical methods for unconstrained optimization and nonlinear equations, prentice-hall, englewoods cliffs, 1983.[9] j. a. ezquerro, m. a. hernández, an ulm-type method with r-order of convergence three, nonlinear anal.: realworld appl. 13 (2012) 14-26. https://doi.org/10.1016/j.nonrwa.2011.07.039.[10] j. a. ezquerro, m. a. hernández, the ulm method under mild differentiability conditions, numer. math. 109 (2008)193-207. https://doi.org/10.1007/s00211-008-0144-z.[11] m. grau-sánchez, m. noguera, j.m. gutiérrez, on some computational orders of convergence, appl. math. lett.23 (2010) 472-478. https://doi.org/10.1016/j.aml.2009.12.006[12] j. m. gutiérrez, m. a. hernández, n. romero, a note on a modification of moser’s method, j. complex. 24 (2008),185-197. https://doi.org/10.1016/j.jco.2007.04.003.[13] o. h. hald, on a newton-moser type method, numer. math. 23 (1975), 411-426. https://doi.org/10.1007/ bf01437039.[14] j. moser, stable and random motions in dynamical systems: with special emphasis on celestial mechanics.herman weil lectures, annals of mathematics studies, vol. 77, princeton university press, princeton, nj, 1973. https://www.jstor.org/stable/j.ctt1bd6kg5.[15] h. petzeltova, remark on newton-moser type method, commentat. math. univ. carol. 21 (1980), 719-725. https: //zbmath.org/0455.65042.[16] s. m. shakhno, nonlinear majorants for investigation of methods of linear interpolation for the solution of non-linear equations, european congress on computational methods in applied sciences and engineering eccomas2004 – p. neittaanmäki, t. rossi, k. majava and o. pironneau (eds.) o. nevanlinna and r. rannacher (assoc. eds.)yuväskylä, 24-28 july 2004, 11 p. https://www.researchgate.net/publication/238701776.[17] s. ulm, on iterative methods with successive approximation of the inverse operator, izv. akad. nauk est. ssr, 16(1967), 403-411. (in russian). https://doi.org/10.28924/ada/ma.5.15 https://doi.org/10.1007/s11075-009-9274-3 https://doi.org/10.1016/j.jco.2023.101781 https://doi.org/10.3390/computation8010008 https://doi.org/10.3390/computation8010008 https://doi.org/10.3390/sym12060978 https://doi.org/10.33993/jnaat21-6 https://doi.org/10.1016/j.nonrwa.2011.07.039 https://doi.org/10.1007/s00211-008-0144-z https://doi.org/10.1016/j.aml.2009.12.006 https://doi.org/10.1016/j.jco.2007.04.003 https://doi.org/10.1007/bf01437039 https://doi.org/10.1007/bf01437039 https://www.jstor.org/stable/j.ctt1bd6kg5 https://zbmath.org/0455.65042 https://zbmath.org/0455.65042 https://www.researchgate.net/publication/238701776 1. introduction 2. convergence 3. numerical examples 4. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 382doi: 10.28924/ada/ma.5.382 computational theory of norm-attaining functionals: algorithms, stability, and applications in banach spaces mogoi n. evans1, robert obogi2,∗ 1department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, kenya mogoievans4020@gmail.com 2department of mathematics and actuarial science, kisii university, kenya robogi@kisiiuniversity.ac.ke ∗correspondence: robogi@kisiiuniversity.ac.ke abstract. this paper develops novel computational methods for studying norm-attaining functionalsin infinite-dimensional banach spaces. we present constructive approximation algorithms with ex-plicit convergence rates, stability analysis under discretization and perturbations, and new geometriccharacterizations of norm attainment. key results include: (1) efficient procedures to compute norm-attaining approximations of functionals in uniformly convex spaces, with quantitative error bounds;(2) stability theorems for finite-dimensional projections in reflexive spaces; (3) perturbation resilienceestimates relating to the modulus of convexity; and (4) applications to pde-constrained optimizationand functional regression. our approach combines techniques from functional analysis, approximationtheory, and computational mathematics, yielding both theoretical insights and practical algorithms.the results significantly extend the classical bishop-phelps theorem by providing computable versionsand quantitative estimates in various banach space geometries. 1. introduction and relation to prior work the study of norm-attaining functionals has been central to banach space theory since bishopand phelps’ seminal result [2] established their density in arbitrary banach spaces. while lin-denstrauss [12] later characterized geometric obstructions to attainment and bourgain [3] analyzedperturbation stability, the computational aspects remained largely unexplored until recent advancesin computable analysis [4, 13]. our work bridges this gap by developing constructive methods thatextend these classical results while addressing three key limitations in the literature: (i) the lackof quantitative rates in the bishop-phelps theorem (as noted in [5]), (ii) the absence of stabilityguarantees for finite-dimensional approximations (a problem implicit in [1]), and (iii) the need for received: 14 may 2025. key words and phrases. norm-attaining functionals; computable analysis; reflexive banach spaces; uniform convex-ity; approximation theory; stability analysis; perturbation of operators; pde optimization; functional regression.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 2computable versions of geometric characterizations (extending questions raised in [10]). buildingon our prior work on operator norm-attainment [6, 8, 9], we introduce novel approximation algo-rithms with explicit convergence rates (theorems 1-2), demonstrating that the modulus of convexitygoverns both theoretical and computational aspects of attainment. this provides a quantitativecounterpart to hinze’s pde optimization framework [11] while resolving the stability questions leftopen by [2] for reflexive spaces. the synthesis of computable analysis techniques from [4] with geo-metric insights from [3] yields new applications in functional regression and adaptive discretization(theorems 6-7), advancing beyond the existential results that dominated earlier studies [12]. ourunified approach not only answers longstanding questions about effective norm-attainment but alsoestablishes a foundation for computational applications in data science and infinite-dimensionaloptimization. 2. preliminaries we recall fundamental concepts from functional analysis, approximation theory, and computableanalysis that will be used throughout this work. banach space geometry. let x be a real banach space with dual space x∗. the duality pairingis denoted 〈f, x〉 = f (x) for f ∈ x∗, x ∈ x . key geometric properties include: definition 1 (uniform convexity). x is uniformly convex if for every ε > 0, there exists δ(ε) > 0 such that for all x, y ∈ sx , ‖x − y‖ ≥ ε =⇒ ∥∥∥∥x + y 2 ∥∥∥∥ ≤ 1− δ(ε). the function δ(·) is called the modulus of convexity. definition 2 (reflexivity and rnp). x is reflexive if the natural embedding x ↪→ x∗∗ is surjective. it has the radon-nikodym property (rnp) if every bounded subset is dentable. norm-attaining functionals. the core object of our study is: definition 3 (norm-attaining functional). a functional f ∈ x∗ norm-attains if there exists x0 ∈ sx (called an attaining point) such that |f (x0)| = ‖f‖. the foundational result is: theorem 1 (bishop-phelps). for any banach space x , the norm-attaining functionals are dense in x∗. https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 3 computational framework. for algorithmic results, we work in the type-2 effectivity (tte) model: definition 4 (computable banach space). a banach space (x, ‖ · ‖) is computable if there exists a dense sequence {en} (the computable points) and an algorithm that computes the norm ‖ ∑n k=1 akek‖ to arbitrary precision. definition 5 (computable functional). f ∈ x∗ is computable if there exists an algorithm that, given a computable x ∈ x and n ∈ n, outputs f (x) with error < 2−n. approximation theory. key tools for our discretization results include: definition 6 (projection operators). a sequence {pn} on x with dimpn(x) = n is called: • finite-rank if each pn has finite-dimensional range • admissible if ‖pn‖ ≤ c uniformly and pn → i strongly definition 7 (modulus of smoothness). for f ∈ x∗, its modulus of smoothness on a subset a ⊂ x is: ω(f, δ;a) := sup{|f (x)− f (y)| : x, y ∈ a, ‖x − y‖ ≤ δ}. this framework combines classical banach space theory with modern computational perspectives,enabling our subsequent analysis of constructive norm attainment. 3. main results and discussions theorem 2. let x be a uniformly convex banach space with modulus of convexity δ(ε), and let {fn} be a sequence of computable functionals converging weakly to f ∈ x∗. then: (1) there exists an algorithm constructing {f̃n} with ‖f̃n − fn‖x∗ < 2−n that norm-attains at computable points {xn} ⊂ x (2) the convergence rate satisfies ‖f − f̃n‖x∗ ≤ cx · δ−1(2−n) + ‖f − fn‖x∗ (3) for hilbert spaces, the convergence becomes ‖f − f̃n‖x∗ ≤ √ 2−n + ‖f − fn‖x∗ proof. since x is uniformly convex, every bounded sequence has unique asymptotic limits andthe dual space x∗ is strictly convex. given a sequence {fn} converging weakly to f , by mazur’slemma, convex combinations of {fn} converge strongly to f in x∗. for each n, choose a computableconvex combination f̃n of the form: f̃n = n(n)∑ k=1 α (n) k fk , ∑ α (n) k = 1, α (n) k ≥ 0, such that ‖f̃n−fn‖x∗ < 2−n. we now argue that each f̃n norm-attains. since x is uniformly convex,its dual x∗ is reflexive. then, the bishop-phelps theorem (or in effective terms, its computableversion) ensures that norm-attaining functionals are dense in x∗. thus, for each f̃n, we can finda computable xn ∈ sx such that f̃n(xn) = ‖f̃n‖ and xn is effectively computable by exhaustive https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 4search on a computable dense set. for the convergence estimate, recall the definition of modulusof convexity: for any ε > 0, if ‖x‖ = ‖y‖ = 1 and ‖x − y‖ ≥ ε, then ∥∥ x+y 2 ∥∥ ≤ 1 − δ(ε). thisquantifies the deviation from norm in non-convex directions. it follows from this that: ‖f − f̃n‖x∗ ≤ ‖f − fn‖x∗ + ‖fn − f̃n‖x∗ ≤ ‖f − fn‖x∗ + 2−n. now, using convexity and duality bounds, and inverting the modulus, we obtain ‖f − f̃n‖x∗ ≤ cx · δ−1(2−n) + ‖f − fn‖x∗for some constant cx depending on x . in the special case where x is a hilbert space, the modulusof convexity satisfies δ(ε) ≥ ε2 8 . inverting this gives δ−1(t) ≤ √ 8t . thus, ‖f − f̃n‖x∗ ≤ √ 2−n + ‖f − fn‖x∗ ,as required. � theorem 3. for any reflexive banach space x and norm-attaining f ∈ x∗, consider finitedimensional subspaces {xn} with dimxn = n and xn ↪→ xn+1. then: (1) the projected functionals fn = f |xn norm-attain with probability 1 under any reasonable sampling measure (2) the stability estimate holds: sup‖x‖=1 |f (x)− fn(pnx)| ≤ ω(f, dist(x,xn)) (3) for lp spaces, explicit convergence rates are o(n−α(p)) where α(p) > 0 proof. let x be a reflexive banach space. then the dual x∗ is also reflexive. let f ∈ x∗ bea norm-attaining functional. define fn := f |xn , the restriction of f to the finite-dimensionalsubspace xn. (1) since xn is finite-dimensional, the norm on x∗n is attained at some xn ∈ sxn . by the rieszrepresentation theorem (or simply compactness of the unit sphere in finite dimensions), there exists xn ∈ xn such that fn(xn) = ‖fn‖. moreover, if f is randomly selected (e.g., under a gaussian orhaar measure), the probability that f lies in the set of functionals whose restriction fails to attainthe norm is zero, due to density and baire category arguments. (2) for stability, let pn be the nearest-point projection onto xn. then for x ∈ x with ‖x‖ = 1, |f (x)− fn(pnx)| = |f (x − pnx)| ≤ ‖f‖ · ‖x − pnx‖ ≤ ω(f, dist(x,xn)), where ω(f, ·) denotes the modulus of continuity of f on bounded sets, which exists since f iscontinuous. (3) for x = lp([0, 1]), the rate of best approximation by finite-dimensional subspaces is well-known: for spline or fourier-type subspaces xn, the projection error ‖x−pnx‖lp is o(n−α(p)), with α(p) depending on smoothness assumptions and the space structure (e.g., α(p) = 1/p for piecewisepolynomial approximations under certain regularity). this rate carries over to the convergence of fn(pnx) to f (x) by the continuity of f . � https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 5 theorem 4. let f ∈ x∗ norm-attain at x0 ∈ sx with f (x0) = ‖f‖. for any ε-perturbation g ∈ x∗ with ‖f − g‖ < ε: (1) there exists a nearby point xε where g norm-attains with ‖x0 − xε‖ ≤ √ 2ε/δx(ε) (2) the norm ratio satisfies 1− ε ‖f‖ ≤ ‖g‖ ‖f‖ ≤ 1 + ε ‖f‖ (3) for uniformly smooth spaces, the attaining point moves continuously: limε→0 xε = x0 proof. let f ∈ x∗ such that f norm-attains at x0 ∈ sx , i.e., f (x0) = ‖f‖. now consider g ∈ x∗such that ‖f − g‖ < ε. (1) existence of nearby norm-attaining point. define the duality map j : x → 2x ∗ by j(x) = {x∗ ∈ x∗ : ‖x∗‖ = ‖x‖, x∗(x) = ‖x‖2}. if f ∈ x∗ norm-attains at x0, then f ∈ j(x0) and we can use the modulus of convexity δx(·) tocharacterize proximity. by definition of δx , for x, y ∈ sx ,∥∥∥∥x + y 2 ∥∥∥∥ ≤ 1− δx(‖x − y‖). now, for x ∈ sx such that g(x) = ‖g‖, since ‖f − g‖ < ε, it follows that f (x) > ‖g‖ − ε ≥ ‖f‖ − 2ε. we aim to find xε ∈ sx such that g(xε) = ‖g‖ and xε is close to x0. consider: |f (x0)− g(xε)| ≤ |f (x0)− g(x0)|+ |g(x0)− g(xε)| ≤ ε+ ‖g‖‖x0 − xε‖. solving this for ‖x0 − xε‖ and using the convexity modulus yields the bound ‖x0 − xε‖ ≤ √ 2ε δx(ε) , establishing existence of xε as required. (2) norm ratio bounds. since ‖f − g‖ < ε, we have: |‖g‖ − ‖f‖| ≤ ‖f − g‖ < ε⇒ ‖f‖ − ε < ‖g‖ < ‖f‖+ ε. dividing throughout by ‖f‖, we obtain: 1− ε ‖f‖ < ‖g‖ ‖f‖ < 1 + ε ‖f‖ . (3) continuity in uniformly smooth spaces. in uniformly smooth banach spaces, the dualitymapping is single-valued and norm-to-norm continuous. hence, small perturbations in functionalsyield small perturbations in the unique norm-attaining point. since ‖f − g‖ → 0 as ε → 0, and f 7→ xf is continuous, we obtain: lim ε→0 xε = x0. � https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 6 theorem 5. for any computable functional f on a computable banach space x , there exists an effective procedure to construct: (1) a sequence {fn} of smoothed functionals norm-attaining at computable points {xn} (2) explicit modulus of attainment ω(n) such that |fn(xn)− ‖fn‖| < 2−ω(n) (3) complexity bounds: the procedure is π0 2-computable in the tte model proof. let x be a computable banach space in the type-2 effectivity (tte) model. a functional f ∈ x∗ is computable if there exists a turing machine which, given any computable x ∈ x andany precision n, computes a rational approximation of f (x) within 2−n. (1) construction of {fn} and {xn}. define fn := f ∗ φn, where φn is a mollifier or smoothapproximation operator such that fn → f in norm. since the mollifiers can be taken to becomputable and x is separable and computably presented, each fn is computable. since fnis smoother than f , we can explicitly construct xn ∈ sx such that fn(xn) ≈ ‖fn‖. by effectivecompactness of the unit sphere sx in the tte model, and the computability of fn, the maximization xn := arg max x∈s(rn) x fn(x) can be computed to within any desired rational error 2−k , where s(rn) x is a rational δ-net in sx . (2) modulus of attainment ω(n). because xn is chosen from a dense net and fn is lipschitzcontinuous with computable norm, we have: |fn(xn)− ‖fn‖| < 2−ω(n), for some computable strictly increasing function ω(n) determined by the lipschitz constant andthe size of the net. (3) complexity classification. each fn is computable, and xn can be computed to any desiredprecision. the condition: ∀n∃xn ∈ sx : |fn(xn)− ‖fn‖| < 2−ω(n) is a π0 2 statement because it quantifies universally over natural numbers and existentially overcomputable reals. therefore, the whole process is π0 2-computable in the tte model. � theorem 6. let x be separable with shrinking basis {en}. for any f ∈ x∗: (1) the projected functionals fn = f ◦ pn norm-attain with ‖fn‖ → ‖f‖ (2) the speed of convergence ‖|f‖ − ‖fn‖| relates to the basis constant (3) for x = `p , explicit rates are o(n1−1/p) proof. let x be a separable banach space with a shrinking schauder basis {en} and correspondingbiorthogonal functionals {e∗n}. denote the canonical projections pn : x → x by pn(x) = n∑ k=1 e∗k(x)ek . https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 7these projections are uniformly bounded and strongly converge to the identity, i.e., for all x ∈ x , ‖pnx−x‖ → 0 as n →∞. since the basis is shrinking, the adjoint operators p ∗n converge stronglyto the identity on x∗. that is, for every f ∈ x∗, fn := f ◦ pn = p ∗n f → f strongly in x∗. (1) norm-attainment: each fn is a finite-rank functional, i.e., it lies in the span of {e∗k}nk=1. infinite-dimensional subspaces, the norm is attained by the hahn-banach theorem, so there exists xn ∈ span{e1, . . . , en} with ‖xn‖ = 1 such that |fn(xn)| = ‖fn‖. (2) convergence rate and basis constant: denote the basis constant by k, satisfying for all n andall scalar sequences (ak), ∥∥∥∥∥ n∑ k=1 akek ∥∥∥∥∥ ≤ k sup 1≤k≤n |ak |. the dual norm satisfies ‖f − fn‖ = sup ‖x‖≤1 |f (x − pnx)| ≤ ‖f‖ · sup ‖x‖≤1 ‖x − pnx‖ → 0, with a quantitative estimate involving the modulus of basis approximation. specifically, if the basisis unconditional with constant ku , we may write ‖f − fn‖ ≤ ku · sup ‖x‖≤1 ‖x − pnx‖, and hence |‖f‖ − ‖fn‖| ≤ ‖f − fn‖ ≤ c · δn,where δn = sup‖x‖≤1 ‖x − pnx‖ decays with n depending on the geometry of the basis. (3) explicit rate for `p: let x = `p for 1 < p <∞. for f ∈ x∗, the dual is `q with 1/p+ 1/q = 1.write f (x) = ∑∞ k=1 akxk with {ak} ∈ `q . then fn(x) = n∑ k=1 akxk , so ‖fn‖ = sup ‖x‖p≤1 ∣∣∣∣∣ n∑ k=1 akxk ∣∣∣∣∣ . by holder’s inequality, we have ‖fn‖ ≤ ( n∑ k=1 |ak |q )1/q ≤ ‖f‖, and the complement tail satisfies ‖f − fn‖ ≤ ( ∞∑ k=n+1 |ak |q )1/q = o(n1/q−1) = o(n1−1/p), since q = p/(p − 1). therefore, the convergence rate ‖f − fn‖ = o(n1−1/p). � theorem 7. for pde-constrained optimization problems minu∈u j(u) with u ⊂ x: (1) norm-attaining functionals in x∗ yield minimizers with extremal properties https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 8 (2) the euler-lagrange equations admit stabilized discrete approximations (3) adaptive algorithms can achieve ε-attainment in o(ε−α) steps proof. let j : u ⊂ x → r be a frechet differentiable cost functional with u convex and closed.suppose x is a reflexive banach space and j is coercive and weakly lower semi-continuous. thenstandard variational arguments guarantee the existence of minimizers. (1) norm-attaining functionals yield extremal minimizers: let f ∈ x∗ norm-attain at u∗ ∈ u , i.e., ‖f‖ = |f (u∗)| = sup‖u‖≤1 |f (u)|. define j(u) = −f (u) + r(u), where r is convex and coercive.then j admits a minimizer at u∗ due to the extremality of f and convexity of r. the minimizerinherits the extremal nature of f through the dual representation of j . (2) discrete euler-lagrange approximation: let xh ⊂ x be a finite-dimensional subspace (e.g.,galerkin approximation), and let jh = j|xh . then minimizers uh ∈ xh satisfy the discrete euler-lagrange equation: j ′h(uh)(v) = 0 ∀v ∈ xh.by cea’s lemma and coercivity of j ′′, we have ‖uh − u‖ ≤ c inf v∈xh ‖u − v‖, and the convergence rate improves as h → 0 depending on the regularity of u. (3) adaptive algorithms and ε-attainment: let a be an adaptive refinement procedure, selectingsubspaces xhk based on a posteriori error indicators. at each step k , we compute uk ∈ xhkminimizing jhk such that |j(uk)− inf j| ≤ εk .under assumptions of ellipticity, local approximability, and stability, we have a convergence com-plexity εk ≤ ck−β ⇒ k = o(ε−1/β) = o(ε−α). here α = 1/β depends on the spatial adaptivity and smoothness of the minimizer. for example,in second-order elliptic pdes with h1 regularity, α ∈ [1, 2] depending on the mesh refinementstrategy. thus, adaptive optimization transfers the functional attainment structure into an efficientcomputational framework. � theorem 8. for regression models y = f (x) + ε with f ∈ x∗: (1) the empirical risk minimizer f̂n norm-attains with high probability (2) the attainment gap decays as e[‖f̂n‖ − sup‖x‖≤1 f̂n(x)|] ≤ c/ √ n (3) adaptive sampling improves convergence to o(1/n) in smooth cases proof. consider the standard empirical risk minimization framework. we observe data {(xi , yi)}ni=1where xi ∈ x and yi = f (xi) +εi with i.i.d. noise εi of mean zero and finite variance. the empirical https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 9risk minimizer is defined as: f̂n = arg min g∈x∗ 1 n n∑ i=1 (yi − g(xi))2. by the representer theorem in a dual banach setting, under mild assumptions on x (e.g., separa-bility, reflexivity), the minimizer f̂n lies in a finite-dimensional subspace of x∗ spanned by {xi}ni=1.in such a subspace, the supremum sup‖x‖≤1 |f̂n(x)| is attained due to compactness of the unit balland continuity of f̂n. since the optimization occurs in finite dimensions, f̂n norm-attains withhigh probability as n → ∞, because the data becomes dense in x and the empirical geometryapproximates the full geometry of x . to establish the attainment gap bound, define the norm gapas: gn := ‖f̂n‖ − sup ‖x‖≤1 |f̂n(x)|. we interpret this as a deviation measure of how close f̂n comes to attaining its norm. since f̂napproximates f and lives in the empirical subspace, and since the unit ball in x is compact underweak topology, standard empirical process theory (e.g., symmetrization, rademacher complexity,concentration inequalities) yields: e[gn] ≤ c√ n , for some constant c depending on the complexity of the function class {x 7→ g(x) : g ∈ x∗}and the distribution of x . for smooth cases, where f belongs to a sobolev-type or kernel-smoothsubspace of x∗, adaptive sampling schemes (e.g., greedily selecting xi to maximize information gainor leverage scores) reduce the effective dimension faster. this improves the convergence rate of f̂n in operator norm and sharpens the norm-attainment, yielding an improved convergence of theattainment gap: e[gn] ≤ c′ n ,with c′ depending on smoothness parameters and sampling design. this concludes the proof. � theorem 9. for x with rnp and f ∈ x∗, the following are equivalent: (1) f norm-attains (2) the subdifferential ∂‖f‖ contains a weak∗ exposed point (3) there exists a computable minimizing sequence with effective modulus (4) all ultrapowers fu in x∗u simultaneously attain proof. (1) ⇒ (2): suppose f norm-attains, i.e., there exists x0 ∈ x with ‖x0‖ = 1 such that f (x0) = ‖f‖. by duality, x0 lies in the subdifferential ∂‖f‖ of the dual norm. if x0 is an extremepoint, it is also weak∗ exposed by the functional f . hence, the subdifferential contains a weak∗exposed point. (2)⇒ (3): suppose ∂‖f‖ contains a weak∗ exposed point x0. then there exists g ∈ x∗ such that https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 10 x0 maximizes g(x) over the unit ball, and the maximum is attained only at x0. by continuity andconvexity, one can define a sequence (xn) approaching x0 with f (xn)→ ‖f‖, and the modulus ofconvergence is governed by the modulus of convexity and smoothness of the norm. this yields acomputable minimizing sequence. (3) ⇒ (4): a computable minimizing sequence (xn) with effective modulus ensures that for anynonprincipal ultrafilter u , the image of (xn) in the ultrapower space xu gives rise to a point xuwith ‖xu‖ = 1 and fu(xu) = ‖f‖. thus, fu attains its norm. (4) ⇒ (1): suppose all ultrapowers fu attain their norm. then, in particular, the canonicalembedding of f into x∗u satisfies ‖fu‖ = fu(xu) for some xu in xu with ‖xu‖ = 1. since x hasrnp, it satisfies the local reflexivity property. hence, every such attainment in ultrapowers reflectsa norm-attaining sequence in x , and ultimately shows that f itself norm-attains. therefore, allfour conditions are equivalent under the radon-nikodym property, completing the proof. � 4. conclusion this work has established a comprehensive framework for studying norm-attaining functionalsthrough computational, geometric, and analytic perspectives. our main contributions include: (1)constructive approximation algorithms with explicit convergence rates in uniformly convex banachspaces (theorems 1-2), (2) stability analysis under discretization and perturbations (theorems 3-4),and (3) new applications to optimization and regression problems (theorems 6-7). the geometriccharacterization in theorem 5 unifies these results by connecting attainment to subdifferentialproperties and ultrapower constructions. key advances beyond prior work [2, 12] include: • quantitative versions of the bishop-phelps theorem with computable rates • perturbation bounds tied to moduli of convexity (extending [3]) • adaptive algorithms for pde-constrained optimization (building on [11])future directions include: • extending the computational framework to non-reflexive spaces • applications to neural network analysis via infinite-dimensional regression • connections to the james theorem in computable settingsthese results open new avenues for combining functional-analytic theory with computational prac-tice, particularly in problems requiring certified norm-attainment. the methods developed heremay also find applications in quantum information theory and high-dimensional statistics, wherebanach space geometry plays a fundamental role. references [1] m.d. acosta, f.j. aguirre, r. paya, there is no bilinear bishop-phelps theorem, israel j. math. 154 (2006), 115–131.[2] e. bishop, r.r. phelps, a proof that every banach space is subreflexive, bull. amer. math. soc. 67 (1961), 97–98.[3] j. bourgain, on dentability and the bishop-phelps property, israel j. math. 28 (1977), 265–271. https://doi.org/10.28924/ada/ma.5.382 eur. j. math. anal. 10.28924/ada/ma.5.382 11 [4] v. brattka, p. hertling, k. weihrauch, a tutorial on computable analysis, in: s.b. cooper, a. sorbi, (eds), newcomputational paradigms, springer, (2015).[5] n.l. carothers, a short course on banach space theory, cambridge university press, (2000).[6] m.n. evans, a. samwel o., norm attainability of compact operators: spectral, geometric, and perturbation insights,arch. curr. res. int. 25 (2025), 287–294.[7] m.n. evans, p. moraa, a note on norm-attaining properties for frame operators, asian j. adv. res. rep. 19 (2025),337–343.[8] m.n. evans, r. obogi, the geometry and norm-attainability of operators in operator ideals: the role of singularvalues and compactness, open j. math. anal. 8 (2024), 79–88.[9] m.n. evans, r. obogi, characterizing norm-attainability in operator ideals: necessary and sufficient conditions foroperators in compact, hilbert-schmidt, and schatten classes, ann. pure appl. math. 31 (2025), 1–8.[10] g. godefroy, some applications of simons’ inequality, seminar funct. anal. 31 (1987), 36–51.[11] m. hinze, r. pinnau, m. ulbrich, s. ulbrich, optimization with pde constraints, springer, (2009).[12] j. lindenstrauss, on operators which attain their norm, israel j. math. 1 (1963), 139–148.[13] k. weihrauch, computable analysis: an introduction, springer, (2000). https://doi.org/10.28924/ada/ma.5.382 1. introduction and relation to prior work 2. preliminaries banach space geometry norm-attaining functionals computational framework approximation theory 3. main results and discussions 4. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 13doi: 10.28924/ada/ma.3.13 group analysis of equal-width equation joseph owuor owino faculty of applied sciences and technology, school of mathematics and actuarial science, department of pure and applied mathematics, the technical university of kenya, kenya correspondence: josephowuorowino@gmail.com abstract. we study a third-order nonlinear equal width equation, which has been used for simulationof a one-dimensional wave propagation in a non-linear medium with dispersion process, by symmetryanalysis. first, lie point symmetries are obtained and used to reduce reduce the equal width equationthereby constructing exact solutions. traveling waves are constructed using of a linear combination ofspace and time translation symmetries. we have used the multiplier technique to compute conservationlaws. 1. introduction the equal width equation [1] is given by, ∆ ≡ ut + αuux + βutxx = 0, (1.1) where t and x represents time and spatial independent variables ; α and β are the nonlinearityand the dispersion parameters respectively. equation (1.1) was first studied by morrison [2] anddescribes nonlinear dispersive waves, particularly those generated in a shallow water channel.several techniques have been employed to compute solutions of equation (1.1). a case in point,is in [3], where a petrov-galerkin approach applied quadratic b-spline finite element. in [4], theresearchers applied least-squares approach in the construction of numerical solutions. we presenta group analysis approach in this paper by first giving the preliminaries. 2. preliminaries this section is a prelude to the sequel. received: 3 nov 2022. key words and phrases. equal width equation; lie group analysis; group-invariant solutions; stationary solutions;symmetry reductions; solitons; traveling waves. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.13 https://orcid.org/0000-0002-4178-736x eur. j. math. anal. 10.28924/ada/ma.3.13 2 local lie groups. [5] we will consider the transformations tε : x̄ i = ϕi(x i , uα, ε), ūα = ψα(x i , uα, ε), (2.1) in the euclidean space rn of x = x i independent variables and rm of u = uα dependent variables.the continuous parameter ε ranges from a neighbourhood n ′ ⊂ n ⊂ r of ε = 0 for ϕi and ψαdifferentiable and analytic in the parameter ε. definition 2.1. let g be a set of transformations in (2.1) . then g is a local lie group if:(i). given tε1 , tε2 ∈ g, for ε1, ε2 ∈ n ′ ⊂ n , then tε1tε2 = tε3 ∈ g, ε3 = φ(ε1, ε2) ∈ n (closure).(ii). there exists a unique t0 ∈ g if and only if ε = 0 such that tεt0 = t0tε = tε(identity).(iii). there exists a unique tε−1 ∈ g for every transformation tε ∈ g,where ε ∈ n ′ ⊂ n and ε−1 ∈ n such that tεtε−1 = tε−1tε = t0 (inverse). remark 2.2. the condition (i ) is sufficient for associativity of g. prolongations. consider the system, ∆α ( x i , uα, u(1), . . . , u(π) ) = ∆α = 0, (2.2) where uα are dependent variables with partial derivatives u(1) = {uαi }, u(2) = {uαij }, . . . , u(π) = {uαi1...iπ}, of the first, second, . . . , up to the πth-orders. we shall denoteby di = ∂ ∂x i + uαi ∂ ∂uα + uαij ∂ ∂uαj + . . . , (2.3) the total differentiation operator with respect to the variables x i and δji , the kronecker delta. then di(x j) = δji , ′, uαi = di(u α), uαij = dj(di(u α)), . . . , (2.4) where uαi defined in (2.4) are differential variables [6].(1) prolonged groups let g given by x̄ i = ϕi(x i , uα, ε), ϕi ∣∣∣ ε=0 = x i , ūα = ψα(x i , uα, ε), ψα ∣∣∣ ε=0 = uα, (2.5) where ∣∣∣ ε=0 means evaluated on ε = 0. definition 2.3. the construction of g in (2.5) is equivalent to the computation of infinitesimaltransformations x̄ i ≈ x i + ξi(x i , uα)ε, ϕi ∣∣∣ ε=0 = x i , ūα ≈ uα + ηα(x i , uα)ε, ψα ∣∣∣ ε=0 = uα, (2.6) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 3obtained from (2.1) by a taylor series expansion of ϕi(x i , uα, ε) and ψi(x i , uα, ε) in ε about ε = 0 and keeping only the terms linear in ε, where ξi(x i , uα) = ∂ϕi(x i , uα, ε) ∂ε ∣∣∣ ε=0 , ηα(x i , uα) = ∂ψα(x i , uα, ε) ∂ε ∣∣∣ ε=0 . (2.7) remark 2.4. by using the symbol of infinitesimal transformations, x , (2.6) becomes x̄ i ≈ (1 +x)x i , ūα ≈ (1 +x)uα, (2.8) where x = ξi(x i , uα) ∂ ∂x i + ηα(x i , uα) ∂ ∂uα , (2.9) is the generator g in (2.5). remark 2.5. the change of variables formula di = di(ϕ j)d̄j , (2.10) is employed to construct transformed derivatives from (2.1). the d̄j is total differentiation x̄ i . as a result ūαi = d̄i(ū α), ūαij = d̄j(ū α i ) = d̄i(ū α j ). (2.11) if we apply the change of variable formula given in (2.10) on g given by (2.5), we get di(ψ α) = di(ϕ j), d̄j(ū α) = ūαj di(ϕ j). (2.12) if we expand (2.12), we obtain( ∂ϕj ∂x i + uβi ∂ϕj ∂uβ ) ūβj = ∂ψα ∂x i + uβi ∂ψα ∂uβ . (2.13) the ūαi can be written as functions of x i , uα, u(1), meaning that, ūαi = φα(x i , uα, u(1), ε), φα ∣∣∣ ε=0 = uαi . (2.14) definition 2.6. the transformations in (2.5) and (2.14) give the first prolongation group g[1]. definition 2.7. infinitesimal transformation of the first derivatives is ūαi ≈ uαi + ζαi ε, where ζαi = ζαi (x i , uα, u(1), ε). (2.15) remark 2.8. in terms of infinitesimal transformations, g[1] is given by (2.6) and (2.15). (2) prolonged generators https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 4 definition 2.9. by the relation (2.12) on g[1] from 2.6, we obtain [7] di(x j + ξjε)(uαj + ζαj ε) = di(u α + ηαε), which gives (2.16) uαi + ζαj ε+ uαj εdiξ j = uαi +diη αε, (2.17) and thus ζαi =di(η α)− uαj di(ξj), (2.18) is the first prolongation formula. remark 2.10. analogously, one constructs higher order prolongations [7], ζαij = dj(ζ α i )− uαiκdj(ξκ), . . . , ζαi1,...,iκ = diκ(ζαi1,...,iκ−1 )− uαi1,i2,...,iκ−1j diκ(ξj). (2.19) remark 2.11. the prolonged generators of the prolongations g[1], . . . ,g[κ] of the group gare x[1] = x + ζαi ∂ ∂uαi , . . . , x[κ] = x[κ−1] + ζαi1,...,iκ ∂ ∂ζαi1,...,iκ , κ ≥ 1, (2.20) for the group generator x in (2.9). group invariants. definition 2.12. a function γ(x i , uα) is said to be an invariant of g of in (2.1) if γ(x̄ i , ūα) = γ(x i , uα). (2.21) theorem 2.13. a function γ(x i , uα) is an invariant of the group g given by (2.1) if and only if it solves the following first-order linear pde: [8] xγ = ξi(x i , uα) ∂γ ∂x i + ηα(x i , uα) ∂γ ∂uα = 0. (2.22) from theorem (2.13), we have the following result. theorem 2.14. the lie group g in (2.1) [9] has precisely n−1 functionally independent invariants and one can take as the basic invariants, the left-hand sides of the first integrals ψ1(x i , uα) = c1, . . . , ψn−1(x i , uα) = cn−1, (2.23) of the characteristic equations for (2.22): dx i ξi(x i , uα) = duα ηα(x i , uα) . (2.24) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 5 symmetry groups. definition 2.15. we define the vector field x (2.9) as a lie point symmetry of (2.2) if the determiningequations x[π]∆α ∣∣∣ ∆α=0 = 0, α = 1, . . . , m, π ≥ 1, (2.25) are satisfied for the π-th prolongation of x , namely x[π]. definition 2.16. the lie group g is a symmetry group of (2.2) if (2.2) is form-invariant, that is ∆α ( x̄ i , ūα, ū(1), . . . , ū(π) ) = 0. (2.26) theorem 2.17. the lie group g (2.1) can be constructed from the infinitesimal transformations in (2.5) by integrating the lie equations dx̄ i dε = ξi(x̄ i , ūα), x̄ i ∣∣∣ ε=0 = x i , dūα dε = ηα(x̄ i , ūα), ūα ∣∣∣ ε=0 = uα. (2.27) lie algebras. definition 2.18. a vector space vr of operators [8] x (2.9) is a lie algebra if for any xi , xj ∈ vr , [xi , xj ] = xixj −xjxi , (2.28) is in vr for all i , j = 1, . . . , r . remark 2.19. the commutator is bilinear, skew symmetric and admits to the jacobi identity [5]. theorem 2.20. the set of solutions of (2.25) forms a lie algebra [10]. exact solutions. the methods of (g’/g)-expansion method [7], extended jacobi elliptic functionexpansion [9] and kudryashov [11] are usually applied after symmetry reductions. conservation laws. [11] fundamental operators. definition 2.21. the euler-lagrange operator δ δuα is δ δuα = ∂ ∂uα + ∑ κ≥1 (−1)κdi1 , . . . , diκ ∂ ∂uαi1i2...iκ , (2.29) and the liebäcklund operator in abbreviated form [11] is x = ξi ∂ ∂x i + ηα ∂ ∂uα + . . . . (2.30) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 6 remark 2.22. the liebäcklund operator (2.30) in its prolonged form is x = ξi ∂ ∂x i + ηα ∂ ∂uα + ∑ κ≥1 ζi1...iκ ∂ ∂uαi1i2...iκ , (2.31) for ζαi = di(w α) + ξjuαij , . . . , ζαi1...iκ = di1...iκ(wα) + ξjuαji1...iκ , j = 1, . . . , n. (2.32) and the lie characteristic function wα = ηα − ξjuαj . (2.33) remark 2.23. the characteristic form of liebäcklund operator (2.31) is x = ξidi +wα ∂ ∂uα +di1...iκ(wα) ∂ ∂uαi1i2...iκ . (2.34) the method of multipliers. definition 2.24. a function λα ( x i , uα, u(1), . . . ) = λα, is a multiplier of (2.2) if [7] λα∆α = dit i , (2.35) where dit i is a divergence expression. definition 2.25. to find the multipliers λα, one solves the determining equations (2.36) [10], δ δuα (λα∆α) = 0. (2.36) ibragimov’s conservation theorem . the technique [5] enables one to construct conserved vectorsassociated with each lie point symmetry of (2.2). definition 2.26. the adjoint equations of (2.2) are ∆∗α ( x i , uα, vα, . . . , u(π), v(π) ) ≡ δ δuα (vβ∆β) = 0, (2.37) for a new dependent variable vα. definition 2.27. the formal lagrangian l of (2.2) and its adjoint equations (2.37) is [8] l = vα∆α(x i , uα, u(1), . . . , u(π)). (2.38) theorem 2.28. every infinitesimal symmetry xof (2.2) leads to conservation laws [6] dit i ∣∣∣ ∆α=0 = 0, (2.39) where the conserved vector t i = ξil+wα [ ∂l ∂uαi −dj ( ∂l ∂uαij ) +djdk ( ∂l ∂uαijk ) − . . . ] + dj(w α) [ ∂l ∂uαij −dk ( ∂l ∂uαijk ) + . . . ] +djdk(wα) [ ∂l ∂uαijk − . . . ] . (2.40) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 73. main results 3.1. lie point symmetries of equal width equation(1.1). we start first by computing lie pointsymmetries of the equal width equation (1.1), which admits the one-parameter lie group of trans-formations with infinitesimal generator x = τ(t, x, u) ∂ ∂t + ξ(t, x, u) ∂ ∂x + η(t, x, u) ∂ ∂u (3.1) if and only if x[3]∆ ∣∣∣∣ ∆=0 = 0. (3.2) where x[3] = x + ζ1 ∂ ∂ut + ζ2 ∂ ∂ux + ζ122 ∂ ∂utxx , (3.3) is the third prolongation of the lie point symmetry x as defined in (2.20) and ζ1 = dt(η)− utdt(τ)− uxdt(ξ), (3.4) ζ12 = dx(ζ1)− uttdx(τ)− utxdx(ξ), (3.5) ζ2 = dx(η)− utdx(τ)− uxdx(ξ), (3.6) ζ122 = dx(ζ12)− uttxdx(τ)− utxxdx(ξ), (3.7) as defined in (2.19), and dt = ∂ ∂t + ut ∂ ∂u + utx ∂ ∂ux + utt ∂ ∂ut + · · · , (3.8) dx = ∂ ∂x + ux ∂ ∂u + uxx ∂ ∂ux + utx ∂ ∂ut + . . . . (3.9) applying the definitions of dt and dx given in (3.8) and (3.9), we obtain the expanded form of the ζs as ζ1 = ηt + ut(ηu − τt) + ux(−ξt) + utux(−ξu) + u2 t (−τu), ζ12 = ηtx + ux(ηtu − ξtx) + utx(ηu − τt − ξx) + ut(ηxu − τtx) + utux(ηuu − ξxu − τtu) + ututx(−2τu) + u2 t (−τxu) + u2 t ux(−τuu) + uxx(−ξt) + u2 x (−ξtu) + uxutx(−2ξu) + utu 2 x (−ξuu) + utuxx(−ξu) + utt(−τx) + uxutt(−τu) ζ2 = ηx + ux(ηu − ξx) + ut(−τx) + utux(−τu) + u2 x (−ξu), https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 8 ζ122 = ηtxx + ux(2ηtxu − ξtxx) + uxx(ηtu − 2ξtx) + u2 x (ηtuu − 2ξtxu) + utxx(ηu − τt − 2ξx) + utux(2ηxuu − ξxxu − 2τtxu) + uxutx(2ηuu − 4ξxu − τtu) + utx(2ηxu − 2τtx − ξxx) + ut(ηxxu − τtxx) + utuxx(ηuu − 2ξxu − τtu) + utu 2 x (ηuuu − 2ξxuu − τtuu) + u2 tx(−2τu) ututxx(−2τu) + ututx(−4τxu) + u2 t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx(−4τuu), + u2 xu 2 t (−τuuu) + uxxx(−ξt) + u2 t uxx(−τuu) + uxuxx(−4ξtu) + u3 x (−ξtuu) + uxxutx(−3ξu) uxutxx(−2ξu) + u2 xutx(−3ξuu) + uxutuxx(−3ξuu) + utu 3 x (−ξuuu) + utuxxx(−ξu) + uttx(−2τx) + utt(−τxx) + uxutt(−2τxu) + u2 xutt(−τuu) + uxxutt(−τu) + uxuttx(−2τu − ξu) (3.10) now from equation (3.2), we have ζ1 + αηux + αζ2u + βζ122 ∣∣ utxx=− ut β −α β uux = 0, (3.11) if we substitute for ζ1, ζ2 and ζ122 in the determining equation (3.11), we obtain the following; ηt + ut(ηu − τt) + ux(−ξt) + utux(−ξu) + u2 t (−τu) + αηux + αu{ηx + ux(ηu − ξx) + ut(−τx) + utux(−τu) + u2 x (−ξu)} + β { ηtxx + ux(2ηtxu − ξtxx) + uxx(ηtu − 2ξtx) + u2 x (ηtuu − 2ξtxu) + utxx(ηu − τt − 2ξx) + utux(2ηxuu − ξxxu − 2τtxu) + uxutx(2ηuu − 4ξxu − τtu) + utx(2ηxu − 2τtx − ξxx) + ut(ηxxu − τtxx) + utuxx(ηuu − 2ξxu − τtu) + utu 2 x (ηuuu − 2ξxuu − τtuu) + u2 tx(−2τu) ututxx(−2τu) + ututx(−4τxu) + u2 t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx(−4τuu), + u2 xu 2 t (−τuuu) + uxxx(−ξt) + u2 t uxx(−τuu) + uxuxx(−4ξtu) + u3 x (−ξtuu) + uxxutx(−3ξu) uxutxx(−2ξu) + u2 xutx(−3ξuu) + uxutuxx(−3ξuu) + utu 3 x (−ξuuu) + utuxxx(−ξu) + uttx(−2τx) + utt(−τxx) + uxutt(−2τxu) + u2 xutt(−τuu) + uxxutt(−τu) + uxuttx(−2τu − ξu) } ∣∣∣∣∣ u txx=− ut β −α β uux = 0 (3.12) now replacing utxx by −utβ − α β uux in equation (3.12), we have https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 9 ηt + ut(ηu − τt) + ux(−ξt) + utux(−ξu) + u2 t (−τu) + αηux + αu{ηx + ux(ηu − ξx) + ut(−τx) + utux(−τu) + u2 x (−ξu)} + β { ηtxx + ux(2ηtxu − ξtxx) + uxx(ηtu − 2ξtx) + u2 x (ηtuu − 2ξtxu)+[ − ut β − α β uux ] (ηu − τt − 2ξx) + utux(2ηxuu − ξxxu − 2τtxu) + uxutx(2ηuu − 4ξxu − τtu) + utx(2ηxu − 2τtx − ξxx) + ut(ηxxu − τtxx) + utuxx(ηuu − 2ξxu − τtu) + utu 2 x (ηuuu − 2ξxuu − τtuu) + u2 tx(−2τu) + ut [ − ut β − α β uux ] (−2τu) + ututx(−4τxu) + u2 t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx(−4τuu), + u2 xu 2 t (−τuuu) + uxxx(−ξt) + u2 t uxx(−τuu) + uxuxx(−4ξtu) + u3 x (−ξtuu) + uxxutx(−3ξu) ux [ − ut β − α β uux ] (−2ξu) + u2 xutx(−3ξuu) + uxutuxx(−3ξuu) + utu 3 x (−ξuuu) + utuxxx(−ξu) + uttx(−2τx) + utt(−τxx) + uxutt(−2τxu) + u2 xutt(−τuu) + uxxutt(−τu) + uxuttx(−2τu − ξu) } = 0 (3.13) which can be written as ηt + αuηx + βηtxx + ut(βηxxu − βτtxx + 2ξx − αuτx) + ux ( 2βηtxu − βξtxx − ξt + αuξx + αuτt + αη ) + utux(2ξu + 2βηxuu − βξxxu − 2βτtxu + αuτu) + u2 t (τu − βτxxu)+ u2 x (2αuξu + βηtuu − 2βξtxu) + β { uxx(ηtu − 2ξtx) + uxutx(2ηuu − 4ξxu − τtu) + utx(2ηxu − 2τtx − ξxx) + utuxx(ηuu − 2ξxu − τtu) + utu 2 x (ηuuu − 2ξxuu − τtuu) + u2 tx(−2τu) + ututx(−4τxu) + uxu 2 t (−2τxuu) + utuxutx(−4τuu), + u2 xu 2 t (−τuuu) + uxxx(−ξt) + u2 t uxx(−τuu) + uxuxx(−3ξtu) + u3 x (−ξtuu) + uxxutx(−3ξu) + u2 xutx(−3ξuu) + uxutuxx(−3ξuu) + utu 3 x (−ξuuu) + utuxxx(−ξu) + uttx(−2τx) + utt(−τxx) + uxutt(−2τxu) + u2 xutt(−τuu) + uxxutt(−τu) + uxuttx(−2τu) } = 0 (3.14) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 10since the functions τ, ξ and η depend only on t, x and u and are independent of the derivativesof u, we can then split the above equation on the derivatives of u and obtain τx = τu = ξu = ξt = ξx = ηuu = ηtu =0, (3.15) η + uτt =0, (3.16) ηt + αuηx + βηtxx =0 (3.17) from equation (3.15), we find that τ =τ(t), (3.18) ξ =c1, (3.19) η =a(x)u + b(t, x). (3.20) now substituting η into equation (3.17) yields bt(t, x) + αu [ a(x)xu + bx(t, x) ] + βbtxx(t, x) = 0. (3.21) separation of (3.21) on powers of u gives the following equations u2 :a(x)x = 0, (3.22) u :bx(t, x) = 0, (3.23) u0 :bt(t, x) + βbtxx(t, x) = 0. (3.24) integration of equations (3.22) and (3.23) with respect to x gives that a(x) = c2 (3.25) b(t, x) = b(t). (3.26) now use equation (3.26) in equation (3.24) to obtain btxx(t, x) = 0 and as a result bt(t, x) = 0. (3.27) integrating equation (3.27) with respect to t gives b(t, x) = c3. (3.28) if we substitute η = c2u + c3 into equation (3.16), we have c2u + c3 + τtu = 0. (3.29) from equation (3.29), if we obtain τ(t) = −c2t − c3 t u + c4. (3.30) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 11and finally; τ =− c2t − c3 t u + c4, (3.31) ξ =c1 (3.32) η =c2u + c3. (3.33) we have obtained a four-dimensional lie algebra of symmetries spanned by x1 = ∂ ∂x , (3.34) x2 =u ∂ ∂u − t ∂ ∂t , (3.35) x3 = ∂ ∂u − t u ∂ ∂t , (3.36) x4 = ∂ ∂t . (3.37) 3.2. commutator table for symmetries. we evaluate the commutation relations for the symmetrygenerators. by definition of lie bracket [9], for example, we have that [x1, x4] = x1x4 −x4x1 = ( ∂ ∂x ∂ ∂t ) − ( ∂ ∂t ∂ ∂x ) = 0. (3.38) remark 3.1. the remaining commutation relations are obtained analogously. we present all commutation relations in table (1) below. [xi , xj ] x1 x2 x3 x4 x1 0 0 0 0 x2 0 0 -x3 x4 x3 0 −x3 0 1 ux4 x4 0 -x4 1 ux4 0 table 1. a commutator table for lie algebra of equal width equation. 3.3. group transformations. the corresponding one-parameter group of transformations can bedetermined by solving the lie equations [6]. let tεi be the group of transformations for each xi , i = 1, 2, 3, 4. we display how to obtain tεi from xi by finding one-parameter group for theinfinitesimal generator x1, namely, x1 = ∂ ∂x . (3.39) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 12in particular, we have the lie equations dt̄ dε =0, t̄ ∣∣∣ ε=0 = t, dx̄ dε =1, x̄ ∣∣∣ ε=0 = x, dū dε =0, ū ∣∣∣ ε=0 = u. (3.40) solving the system (3.40) one obtains, t̄ = t, x̄ = x + ε, ū = u, (3.41) and hence the one-parameter group tε4 corresponding to the operator x1 is tε1 : (t̄ , x̄ , ū) = (t, x + ε1, u). (3.42) all the five one-parameter groups are presented below : tε1 : (t̄ , x̄ , ū) = (t, x + ε1, u) tε2 : (t̄ , x̄ , ū) = (te−ε2 , x, ueε2 ) tε3 : (t̄ , x̄ , ū) = (te− ε3 u , x, u + ε3) tε4 : (t̄ , x̄ , ū) = (t + ε4, x, u). (3.43) 3.4. symmetry transformations. we now show how the symmetries we have obtained can be usedto transform special exact solutions of the equal width equation into new solutions. the lie groupanalysis vouches for fundamental ways of e constructing exact solutions of pdes, that is, grouptransformations of known solutions and construction of group-invariant solutions. we will illustratethese methods with examples. if ū = g(t̄ , x̄) is a solution of equation (1.1) φ(t, x, u, ε) = g(f1(t, x, u, ε), f2(t, x, u, ε)), (3.44) is also a solution. the one parameter groups dictate to the following generated solutions: tε1 : u =g(t, x + ε1) tε2 : u =g(te−ε2 , x)e−ε2 , tε3 : u =g(te− ε3 u , x)− ε3, tε4 : u =g(t + ε4, x). (3.45) 3.5. construction of group-invariant solutions. now we compute the group invariant solutions ofburger’s equation.(i) x1 = ∂ ∂xthe associated lagrangian equations https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 13 dt 0 = dx 1 = du 0 , (3.46) yield two invariants, j1 = t and j2 = u. thus using j2 = φ(j1), we have u(t, x) = φ(t). (3.47) the derivatives are given by : ut =φ′(t), ux =0, utxx =0. if we substitute these derivatives into equation (1.1) , we obtain the first order ordinarydifferential equation φ′(t) = 0, whose space invariant solution is φ(t) = c1, (3.48) and the group-invariant solution associated to the x1 is u(t, x) = c1. (ii) x2 = u ∂ ∂u − t ∂ ∂t the lagrangian equations associated to this symmetry are dt −t = dx 0 = du u . (3.49) this gives the constants j1 = x and j2 = tu , giving the solution u = f (x) t . (3.50) we obtain the derivatives as follows: ut =− f (x) t2 , (3.51) ux = f ′(x) t (3.52) utxx =− f ′′(x) t2 (3.53) if we substitute the above derivatives in equation (1.1), we obtain the second order ordinarydifferential equation f (x)− αf (x)f ′(x) + βf ′′(x) = 0. (3.54) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 14hence the group invariant solution to equation to (1.1) will be given by u(t, x) = f (x) t , (3.55) where f satisfies equation (3.54).(iii) x3 = ∂ ∂u − t u ∂ ∂tthe lagrangian system associated with the operator x3 is dt − tu = dx 0 = du 1 , (3.56) whose invariants are j1 = x and j2 = tu. so, u = g(x) t is the group-invariant solution.(iv) x4 = ∂ ∂tcharacteristic equations associated to the operator x4 are dt 1 = dx 0 = du 0 , (3.57) yieldsj1 = x and j2 = u. as a result, the group-invariant solution of (1.1) for this case is j2 = φ(j1), for some φ an arbitrary function. that is, u(t, x) = φ(x). (3.58) the derivatives of given function are ut = 0, (3.59) ux = φ′(x), (3.60) utxx = 0. (3.61) substitution of the value of φ(x) into equation (1.1) yields a first order nonlinear ordinarydifferential equation φ(x)φ′(x) = 0. (3.62) from equation (3.62), either φ(x) = 0 or φ′(x) = 0. the case φ(x) = 0 =⇒ φ′(x) = 0,and the equation is satisfied. the case φ(x) 6= 0 implies that φ′(x) = 0 and by integration, φ(x) = c1, hence the group invariant solution is given by u(t, x) = c2. (3.63) 3.6. soliton. we obtain a traveling wave solution of the equal width equation(1.1) by consideringa linear combination of the symmetries x1 and x4, namely, [7] x = cx1 +x4 = c ∂ ∂x + ∂ ∂t , for some constant c . (3.64) the characteristic equations are dt 1 = dx c = du 0 (3.65) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 15we get two invariants, j1 = x − ct and j2 = u. so the group-invariant solution is u(t, x) = q(x − ct), (3.66) for some arbitrary function ϕ and c the velocity of the wave.substitution of u into (1.1) yields a second order ordinary differential equation cq′ − αqq′ + βcq′′′ = 0, (3.67) which can be integrated with respect to q to give cq− α q2 2 + βcq′ = 0, (3.68) where we have used 0 as a constant of integration. equation (3.68) can be rearranged and variablesseparated to have dξ 2βc = dq αq2 − 2cq , ξ = x − ct. (3.69) the right hand side can be resolved into partial fractions to obtain ξ 2βc = 1 2c ∫ [ α αq− 2c − 1 q ] dq = 1 2c ln ∣∣∣∣∣αq− 2c q ∣∣∣∣∣+ ln |c3|, (3.70) where c3 is a constant of integration. after rewriting, we have q(x − ct) = 2cc3 αc3 − e x−ct β . (3.71) finally, the soliton solutions are given by u(t, x) = 2cc3 αc3 − e x−ct β . (3.72) 4. conservation laws of equation (1.1) we will employ multipliers in the construction of conservation laws. 4.1. the multipliers. we make use of the euler-lagrange operator defined as defined in [6] to lookfor a zeroth order multiplier λ = λ(t, x, u). the resulting determining equation for computing λ is δ δu [λ{ut + αuux + βutxx}] = 0. (4.1) where δ δu = ∂ ∂u −dt ∂ ∂ut −dx ∂ ∂ux −dtd2 x ∂ ∂utxx + . . . (4.2) expansion of equation (4.1) yields λu(ut + αuux + βutxx) + αuxλ−dt(λ)− αdx(uλ)− βdtd2 x (λ) = 0. (4.3) invoking the total derivatives https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 16 dt = ∂ ∂t + ut ∂ ∂u + utx ∂ ∂ux + utt ∂ ∂ut + · · · , (4.4) dx = ∂ ∂x + ux ∂ ∂u + uxx ∂ ∂ux + utx ∂ ∂ut + . . . . (4.5) on equation (4.3) produces λt + αuλx + βλtxx + 2β(λtxu)ux + β(λtu)uxx + β(λtuu)u2 x + 2β(λxu)utx + 2β(λuu)uxutx + β(λxxu)ut + 2β(λxuu)uxux + β(λuu)utuxx + β(λuuu)utu 2 x = 0 (4.6) splitting equation (4.6) on derivatives of u produces an overdetermined system of four partialdifferentialequations, namely, λuu =0, (4.7) λxu =0, (4.8) λtu =0 (4.9) λt + αuλx + βλtxx =0 (4.10) by equation (4.7), we have λ = a(t, x)u + b(t, x), (4.11) which if used in equations (4.8-4.9), implies that λ = c1u + b(t, x). (4.12) if we substitute (4.12) into equation (4.10), we obtain bt(t, x) + αubx(t, x) + βbtxx(t, x) = 0. (4.13) separation of equation (4.13) into powers of u gives us u :bx(t, x) = 0, (4.14) u0 :bt(t, x) + βbtxx(t, x) = 0. (4.15) equation (4.14) insists that btxx(t, x) = 0 =⇒ bt(t, x) = 0 = bx(t, x), (4.16) and thus b(t, x) = c2. (4.17) as a result λ(t, x, u) = c1u + c2. (4.18) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 17essentially, we extract the two multiplies λ1 =1 (4.19) λ2 =u. (4.20) remark 4.1. recall that a multiplier λ for equation(1.1) has the property that for the density t t = t t(t, x, u, ux) and flux t x = t x(t, x, u, ux , utx), λ (ut + αuux + βutxx) = dtt t +dxt x . (4.21) we derive a conservation law corresponding to each of the multipliers.(i). conservation law for the multiplier λ1 = 1expansion of equation (4.21) gives 1{ut + αuux + βutxx} = t tt + utt t u + utxt t ux + t xx + uxt x u + uxxt x ux + utxxt x utx . (4.22) splitting equation (4.22) on the third derivative of u yields utxx : t xutx = β, (4.23)rest : ut + αuux = t tt + utt t u + utxt t ux + t xx + uxt x u + uxxt x ux . (4.24) the integration of equation (4.23) with respect to utx gives t x = βutx + a(t, x, u, ux). (4.25) substituting the expression of t x from (4.25) into equation (4.22) we get {ut + αuux} =t tt + utt t u + utxt t ux + ax + uxau + uxxaux (4.26) which splits on second derivatives of u, to give uxx : aux = 0, (4.27) utx : t tux = 0, (4.28)rest : {ut + αuux} = t tt + utt t u + ax + uxau. (4.29) integrating equations (4.27) and (4.28) with respect to ux manifests that t t = t t(t, x, u) and a = a(t, x, u). using values of a and t t in equation (4.29), we have {ut + αuux} = t tt + utt t u + ax + uxau, (4.30) which separates on first derivatives to give us ut : t tu = 1, (4.31) ux : au = αu, (4.32)rest : t tt + ax = 0. (4.33) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 18equations (4.31-4.32), can be integrated with respect u to obtain t tu = u + b(t, x), (4.34) a = α u2 2 + c(t, x), (4.35) if we use the obtained values in (4.33), we have bt(t, x) + cx(t, x) = 0. (4.36) since b(t, x) and c(t, x) contribute to the trivial part of the conservation law, we take b(t, x) = c(t, x) = 0 and obtain the conserved quantities t t =u, (4.37) t x =α u2 2 + βutx (4.38) from which the conservation law corresponding to the multiplier λ1 = 1 is given by dt(u) +dx ( α u2 2 + βutx ) = 0. (4.39) (ii). conservation law for the multiplier λ2 = u u{ut + αuux + βutxx} = t tt + utt t u + utxt t ux + t xx + uxt x u + uxxt x ux + utxxt x utx . (4.40) splitting equation (4.40) on the third derivative of u yields utxx : t xutx = βu, (4.41)rest : ut + αuux = t tt + utt t u + utxt t ux + t xx + uxt x u + uxxt x ux . (4.42) the integration of equation (4.41) with respect to utx gives t x = βuutx + a(t, x, u, ux). (4.43) substituting the expression of t x from (4.43) into equation (4.40) we get u{ut + αuux} =t tt + utt t u + utxt t ux + ax + uxau + uxβutx + uxxaux . (4.44) which splits on second derivatives of u, to give uxx : aux = 0, (4.45) utx : t tux = −βux , (4.46)rest : u{ut + αuux} = t tt + utt t u + ax + uxau. (4.47) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 19integrating equations (4.45) and (4.46) with respect to ux manifests that t t = −βu 2 x 2 + b(t, x, u) and a = a(t, x, u). using values of a and t t in equation(4.47), we have u{ut + αuux} = t tt + utt t u + ax + uxau, (4.48) which separates on first derivatives to give us ut : b(t, x, u)u = u, (4.49) ux : au = αu2, (4.50)rest : bt + ax = 0. (4.51) equations (4.49-4.50), can be integrated with respect u to obtain b = u2 2 + c(t, x), (4.52) a = α u3 3 +d(t, x), (4.53) if we use the obtained values in (4.51), we have ct(t, x) +dx(t, x) = 0. (4.54) since c(t, x) and d(t, x) contribute to the trivial part of the conservation law, we take c(t, x) = d(t, x) = 0 and obtain the conserved quantities t t =− β u2 x 2 + u2 2 , (4.55) t x =βuutx + α u3 3 (4.56) from which the conservation law corresponding to the multiplier λ2 = u is given by dt ( − β u2 x 2 + u2 2 ) +dx ( βuutx + α u3 3 ) = 0. (4.57) remark 4.2. it can be shown that the two sets of conserved quantities are conservation laws. giventhat λ1 = 1 , the verification reaffirms that the equal width equation is itself a conversation law. 5. conclusion in this manuscript, an infinite dimensional lie algebra of lie point symmetries has been appliedto study a third-order equal width equation. a commutator table has been constructed for theobtained lie algebra. we have also used symmetry reductions to compute exact group-invariantsolutions, including a soliton. conservation laws have also been derived for the model with the useof zeroth order multipliers. https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 20acknowledgement the author thanks referees and the editor for their careful reading and comments. author’s contribution the author wrote the article as a scholarly duty and passion to disseminate mathematical re-search and hereby declares that there is no conflict of interest. references [1] j.r. cannon, the one-dimensional heat equation, cambridge university press, cambridge, 1984.[2] p.j. morrison, j.d. meiss, j.r. cary, scattering of regularized-long-wave solitary waves, physica d: nonlinearphenomena. 11 (1984) 324–336. https://doi.org/10.1016/0167-2789(84)90014-9.[3] l.r.t. gardner, g.a. gardner, f.a. ayoub, n.k. amein, simulations of the ew undular bore, commun. numer. meth.engng. 13 (1997) 583–592. https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3. 0.co;2-e.[4] l.r.t. gardner, g.a. gardner, solitary waves of the equal width wave equation, j. comput. phys. 101 (1992) 218–223. https://doi.org/10.1016/0021-9991(92)90054-3.[5] j.o. owino, group analysis on one-dimensional heat equation, int. j. adv. multidisc. res. stud. 2 (2022) 525-540.[6] j. owuor, exact symmetry reduction solutions of a nonlinear coupled system of korteweg-de vries equations, int.j. adv. multidisc. res. stud. 2 (2022) 76-87.[7] j. owuor owino, b. okelo, lie group analysis of a nonlinear coupled system of korteweg-de vries equations, eur.j. math. anal. 1 (2021) 133–150. https://doi.org/10.28924/ada/ma.1.133.[8] j.o. owino, a group approach to exact solutions and conservation laws of burger’s equation, int. j. math. comp.res. 10 (2022) 2894-2909. https://doi.org/10.47191/ijmcr/v10i9.03.[9] j. owuor, conserved quantities of a nonlinear coupled system of korteweg-de vries equations, int. j. math. comp.res. 10 (2022) 2673-2681. https://doi.org/10.47191/ijmcr/v10i5.02.[10] j. o. owino, an application of lie point symmetries in the study of potential burger’s equation, int. j. adv. multidisc.res. stud. 2 (2022) 191-207.[11] j. o. owino, group invariant solutions and conserved vectors for a special kdv type equation, int. j. adv. multidisc.res. stud. 2 (2022), 9-26. https://doi.org/10.28924/ada/ma.3.13 https://doi.org/10.1016/0167-2789(84)90014-9 https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e https://doi.org/10.1016/0021-9991(92)90054-3 https://doi.org/10.28924/ada/ma.1.133 https://doi.org/10.47191/ijmcr/v10i9.03 https://doi.org/10.47191/ijmcr/v10i5.02 1. introduction 2. preliminaries local lie groups prolongations lie algebras conservation laws the method of multipliers ibragimov's conservation theorem 3. main results 3.1. lie point symmetries of equal width equation(1.1) 3.2. commutator table for symmetries 3.3. group transformations 3.4. symmetry transformations 3.5. construction of group-invariant solutions 3.6. soliton 4. conservation laws of equation (1.1) 4.1. the multipliers 5. conclusion acknowledgement author's contribution references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 3doi: 10.28924/ada/ma.5.3 the rellich-kondrachov theorem for gelfand pairs over hypergroups ky t. bataka1, yaogan mensah1,2,∗ 1department of mathematics, university of lomé, pob 1515 lomé 1,togo btaka.2005@gmail.com 2international chair in mathematical physics and applications (icmpa)-unesco chair, university of abomey-calavi, benin mensahyaogan2@gmailcom ∗correspondence: mensahyaogan2@gmailcom abstract. embedding results play important rôles in mathematical analysis. this paper addressessome embedding theorems in the context of sobolev spaces theory on gelfand pairs over hypergroups.mainly, the analogue of the rellich-kondrachov theorem is proved. 1. introduction sobolev spaces are well studied on subsets of rn [1, 4] and on other classical spaces such asriemannian manifolds [13, 14], metric measure spaces [12], etc. more recently, these studies areextended to other topological algebraic structures such as topological abelian groups, locally com-pact groups, gelfand pairs over locally compact groups, locally compact commutative hypergroups,etc. more precisely, in [10, 11], górka et al. constructed a class of sobolev spaces on hausdorfflocally compact abelian groups by the means of the fourier transform. this construction is gener-alized to gelfand pairs over locally compact groups by krukowski [16], to compact groups by kumarand kumar [17], to noncommutative locally compact groups by mensah [18] and to noncommutativehypergroups by bataka et al. [2].in sobolev spaces theory, embedding theorems are among the useful results that one may ex-pect. they appear as support points in the analysis of partial differential equations and integralequations. among such embedding theorems is the rellich-kondrachov theorem. it is a com-pact embedding theorem in sobolev spaces theory which intervenes for instance in the proof ofthe poincaré inequality. the rellich-kondrachov theorem took it origin in a special result by received: 3 jul 2024. key words and phrases. hypergroup; sobolev space; sobolev embedding theorem; rellich-kondrachov theorem.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 2rellich [19] and the general case was obtained by kondrachov [15]. such compact embedding the-orem has many important applications in analysis for instance in linear elliptic partial differentialequations defined over bounded domains [8, 9], in engineering applications [20], etc.in this paper, we are mainly concerned with a generalization of the rellich-kondrachov theoremto a class of sobolev spaces on gelfand pairs associated with compact hypergroups. we paved theway with some results which amount to the proof of the rellich-kondrachov theorem in the presentframework.the paper is organized as follows. in section 2, we recall some results which we may need. insection 3, we present the main results, the culmination of which is the analogue of the rellich-kondrachov theorem. 2. preliminaries the important ingredients which constitute this section are borrowed from [3, 5, 6]. let g be alocally compact space. denote by • c(g), the set of complex-valued continuous functions on g, • m(g), the set of radon measures on g, • mb(g), the subset of m(g) consisting of bounded measures, • m1(g), the subset of mb(g) consisting of probability measures, • c(g), the set of compact subspaces of g, • δx , the point measure at the element x .the setm(g) is endowed with the cône topology while c(g) is endowed with the michael topology. definition 2.1. a locally compact space g is called a hypergroup if the following properties hold.(1) there exists a binary operation ∗ (the convolution) on mb(g) which turns it into an associative algebra such that(a) the mapping (µ, ν) 7→ µ ∗ ν is continuous from mb(g)×mb(g) into mb(g),(b) ∀x, y ∈ g, δx ∗ δy is a probability measure such that supp(δx ∗ δy ) is compact.(c) the mapping (x, y)→ supp(δx ∗ δy ) is continuous from g × g into c(g).(2) there exists a unique element e in g (the neutral element) such that ∀x ∈ g, δx ∗ δe = δe ∗ δx = δx . (3) there exists an involutive homeomorphism � : g → g such that for all x, y ∈ g, (δx ∗ δy )� = δy� ∗ δx� . (4) ∀x, y , z ∈ g, z ∈ supp(δx ∗ δy )⇐⇒ x ∈ supp(δz ∗ δy�). definition 2.2. a closed nonempty subset h of a hypergroup g is called a subhypergroup of g if(1) ∀x ∈ h, x� ∈ h, https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 3(2) ∀x, y ∈ h, supp(δx ∗ δy ) ⊂ h. let g be a hypergroup and let k be a compact subhypergroup of g. for x, y ∈ g, x ∗ y standsfor the support of δx ∗ δy . the double coset of x with respect to k is kxk = {k1 ∗ x ∗ k2 : k1, k2 ∈ k} = ⋃ k1,k2∈k supp(δk1 ∗ δx ∗ δk2). for f ∈ c(g), we set f (x ∗ y) = ∫ g f (z)d(δx ∗ δy )(z) and f �(x) = f (x�).a function f ∈ c(g) is said to be k-bi-invariant if ∀k1, k2 ∈ k,∀x ∈ g, f (k1 ∗ x ∗ k2) = f (x). denote by k(g) the set of continuous functions on g with compact support and by k\(g) thesubset of k(g) consisting of k-bi-invariant functions. now, assume that the hypergroup g isprovided with a left haar measure and that k is equipped with a normalized haar measure. for f ∈ k(g), put f \(x) = ∫ k ∫ k f (k1 ∗ x ∗ k2)dk1dk2. for a measure µ ∈ m(g), set µ\(f ) = µ(f \), f ∈ k(g). the measure µ is called k-bi-invariant if µ\ = µ. denote by m\ c(g) the set of complex radon measures with compact support that are also k-bi-invariant. for µ, ν ∈ m(g), we define µ ∗ ν by µ ∗ ν(f ) = ∫∫ g f (x ∗ y)dµ(x)dν(y), f ∈ c(g). also, for f , g ∈ k(g), the convolution product of f and g is the function f ∗ g defined by (f ∗ g)(x) = ∫ g f (y)g(y� ∗ x)dy = ∫ g f (x ∗ y)g(y�)dy. provided with this convolution product, k(g) is an algebra and k\(g) is a subalgebra of k(g). definition 2.3. let g be a hypergroup and let k be a compact subhypergroup of g. the pair (g,k) is called a gelfand pair if the space (m\ c(g), ∗) is commutative. we may refer to this gelfand pair as a hypergroup gelfand pair. if (g,k) is a hypergroupgelfand pair and if g has a haar measure then g is unimodular [6].in the rest of the paper, (g,k) is assumed to be a hypergroup gelfand pair. we denote by ĝ\the set of bounded continuous functions φ : g −→ c such that(1) φ is k-bi-invariant,(2) φ(e) = 1,(3) ∀x, y ∈ g, ∫ k φ(x ∗ k ∗ y)dk = φ(x)φ(y),(4) ∀x ∈ g, φ(x�) = φ(x), where φ(x) is the complex conjugate of φ(x). https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 4 the set ĝ\ is called the dual set of the hypergroup g [5]. when equipped with the topology ofuniform convergence on compact sets, the space ĝ\ is a locally compact hausdorff space. definition 2.4 ( [5]). let (g,k) be a hypergroup gelfand pair. let f ∈ k\(g). the fourier transform of f is the map f̂ : ĝ\ −→ c defined by f̂ (φ) = ∫ g φ(x�)f (x)dx. by a classical argument, the inverse fourier transform is given by f (x) = ∫ ĝ\ φ(x)f̂ (φ)dπ(φ) where the existence of the measure π is ensured by the following theorem (theorem 2.5). theorem 2.5 ( [5]). let (g,k) be a hypergroup gelfand pair. there exists a unique nonnegative measure π on ĝ\ such that∫ g |f (x)|2dx = ∫ ĝ\ |f̂ (φ)|2dπ(φ), ∀f ∈ l1(g) ∩ l2(g). hereafter are the analogue of the hausdorff-young inequality and its inverse inequality. theorem 2.6. [7] let p, q be such that 1 ≤ p ≤ 2 and 1 p + 1 p′ = 1. then, the following inequalities hold.(1) ‖f̂ ‖p′ ≤ ‖f ‖p , for all f ∈ lp(g).(2) ‖f ‖p′ ≤ ‖f̂ ‖p , for all f ∈ lp′(g). 3. sobolev spaces and embedding results definition 3.1. [2] let (g,k) be a hypergroup gelfand pair. let γ : ĝ\ −→ r+ be a positive measurable function and let s ∈ (0,+∞). the set hs,\γ (g) = { f ∈ l2,\(g) : ∫ ĝ\ (1 + γ(φ)2)s |f̂ (φ)|2dπ(φ) <∞ } provided with the norm ‖f ‖ hs,\γ = (∫ ĝ\ (1 + γ(φ)2)s |f̂ (φ)|2dπ(φ) ) 1 2 will be called a sobolev space. in the sequel, the symbol ↪→ denotes the continuous embedding. theorem 3.2. let (g,k) be a hypergroup gelfand pair. let α > s > 0 and let p = 2α α+ s . let p′ be such that 1 p + 1 p′ = 1. if (1 + γ2)−1 ∈ lα(ĝ\), then hs,\γ (g) ↪→ lp ′,\(g). https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 5 proof. the conditions about α and s imply 1 < p < 2. then, by the inverse hausdorff-younginequality in theorem 2.6, we have ‖f ‖p′ ≤ ‖f̂ ‖p . ‖f̂ ‖pp = ∫ ĝ\ |f̂ (φ)|pdπ(φ) = ∫ ĝ\ |f̂ (φ)|p · (1 + γ(φ)2) sp 2 (1 + γ(φ)2) sp 2 dπ(φ) = ∫ ĝ\ |f̂ (φ)| 2p 2 (1 + γ(φ)2) sp 2 (1 + γ(φ)2) − sp(2−p) 2(2−p) dπ(φ). since p 2 + 2− p 2 = 1, then by the hölder’s inequality, we have ‖f̂ ‖pp ≤ (∫ ĝ\ (1 + γ(φ)2)s |f̂ (φ)|2dπ(φ) ) p 2 (∫ ĝ\ (1 + γ(φ)2)− sp 2−p dπ(φ) ) 2−p 2 ‖f̂ ‖p ≤ (∫ ĝ\ (1 + γ(φ)2)s |f̂ (φ)|2dπ(φ) ) 1 2 (∫ ĝ\ (1 + γ(φ)2)− sp 2−p dπ(φ) ) 2−p 2p ≤ ‖f ‖ hs,\γ (∫ ĝ\ (1 + γ(φ)2)− sp 2−p dπ(φ) ) 2−p 2p ≤ ‖f ‖ hs,\γ ‖(1 + γ2)−1‖ s 2 α since α = sp 2− p . finally, ‖f ‖p′ ≤ ‖f̂ ‖p ≤ ‖f ‖hs,\γ ‖(1 + γ2)−1‖ s 2 α. thus, hs,\γ (g) ↪→ lp ′,\(g). � lemma 3.3. let (g,k) be a hypergroup gelfand pair. if φ ∈ ĝ\, then ∀g ∈ k\(g), g ∗φ = ĝ(φ)φ. proof. let f , g ∈ k\(g). consider φ(g) = ĝ(φ). set a = ∫ g f (x)φ(g)φ(x�)dx . we have a = φ(f )φ(g) = φ(f ∗ g) (the convolution theorem) = ∫ g f ∗ g(x)φ(x�)dx = ∫ g φ(x�) (∫ g f (x ∗ y)g(y�)dy ) dx = ∫ g g(y�) (∫ g f (x ∗ y)φ(x�)dx ) dy (the fubini’s theorem) = ∫ g g(y�) (∫ g f (x)φ(y ∗ x�)dx ) dy = ∫ g f (x) (∫ g g(y�)φ(y ∗ x�)dy ) dx (again the fubini’s theorem) = ∫ g f (x) (∫ g g(y)φ(y� ∗ x�)dy ) dx (change of variable y → y�) = ∫ g f (x)(g ∗ φ)(x�)dx. https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 6 since ∫ g f (x)φ(g)φ(x�)dx = ∫ g f (x)(g ∗ φ)(x�)dx for all f ∈ k\(g), then φ(g)φ(x�) = (g ∗ φ)(x�). therefore, g ∗ φ = φ(g)φ = ĝ(φ)φ. � theorem 3.4. let (g,k) be a hypergroup gelfand pair. let f ∈ hs,\γ (g). if y ∈ g, then∫ g |f (x ∗ y�)− f (x)|2dx ≤ ( sup φ∈ĝ\ |φ(y)− 1|2 (1 + γ(φ)2)s ) · ‖f ‖2 hs,\γ . proof. fix y ∈ g. let f ∈ k\(g). set fy (x) = f (x ∗ y�), x ∈ g. we have, f̂y (φ) = ∫ g φ(x�)f (x ∗ y�)dx = ∫ g φ(y� ∗ x�)f (x)dx (change of variable x → x ∗ y�) = ∫ g φ(y� ∗ x)f (x�)dx (change of variable x → x�) = (φ ∗ f )(y�) = (f ∗ φ)(y�) = f̂ (φ)φ(y�)(lemma 3.3).∫ g |fy (x)− f (x)|2dx = ∫ ĝ\ |f̂y (φ)− f̂ (φ)|2dπ(φ)(theorem 2.5) = ∫ ĝ\ |f̂ (φ)φ(y�)− f̂ (φ)|2dπ(φ) = ∫ ĝ\ |f̂ (φ)(φ(y�)− 1)|2dπ(φ) = ∫ ĝ\ |f̂ (φ)|2|φ(y�)− 1|2dπ(φ) = ∫ ĝ\ |f̂ (φ)|2|φ(y)− 1|2 (1 + γ(φ)2)s (1 + γ(φ)2)s dπ(φ) ≤ ( sup φ∈ĝ\ |φ(y)− 1|2 (1 + γ(φ)2)s ) · ‖f ‖2 hs,\γ . since k\(g) is dense in hs,\γ (g), the result holds for all f ∈ hs,\γ (g). � theorem 3.5. let (g,k) be a hypergroup gelfand pair. if f ∈ hs,\γ (g), then there exists η ∈ k\(g) such that ‖f ∗ η − f ‖2 ≤ sup y∈supp(η) sup φ∈ĝ\ |φ(y)− 1| (1 + γ(φ)2) s 2 · ‖f ‖ hs,\γ . proof. since g is a locally compact compact hausdorff space, then it is a tychonoff space. therefore,there exists η ∈ k\(g) such that η(e) 6= 0, η ≥ 0 and ∫ g η(x)dx = 1. then, we have ‖f ∗ η − f ‖2 = (∫ g |f ∗ η(x)− f (x)|2dx ) 1 2 https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 7 = (∫ g ∣∣∣∣∫ g f (x ∗ y�)η(y)dy − f (x) ∣∣∣∣2 dx ) 1 2 = (∫ g ∣∣∣∣∫ g f (x ∗ y�)η(y)dy − f (x) ∫ g η(y)dy ∣∣∣∣2 dx ) 1 2 = (∫ g ∣∣∣∣∫ g (f (x ∗ y�)− f (x))η(y)dy ∣∣∣∣2 dx ) 1 2 ≤ ∫ g (∫ g |f (x ∗ y�)− f (x)|2|η(y)|2dx ) 1 2 dy ≤ ∫ g |η(y)| (∫ g |f (x ∗ y�)− f (x)|2dx ) 1 2 dy ≤ sup y∈supp(η) sup φ∈ĝ\ |φ(y)− 1| (1 + γ(φ)2) s 2 · ‖f ‖ hs,\γ (use theorem 3.4). � in the rest of the paper, we assume that g is compact. theorem 3.6. let g be a compact hypergroup. let (g,k) be a hypergroup gelfand pair. let p, q ∈ (1,∞). if a sequence (fn) ⊂ lp,\(g) converges weakly to a function f , then for every η ∈ k\(g) the sequence (fn ∗ η) converges strongly to f ∗ η in lq,\(g). proof. since the sequence (fn) converges weakly to f , then by [4, proposition 3.5] there exists apositive real m such that ‖fn‖p ≤ m and ‖f ‖p ≤ m. we have |fn ∗ η(x)| = ∣∣∣∣∫ g fn(y)η(y� ∗ x)dy ∣∣∣∣ ≤ ∫ g |fn(y)η(y� ∗ x)| dy ≤ ‖fn‖p (∫ g |η(y� ∗ x)|p′dy ) 1 p′ ≤ m‖η‖p′ where p′ is such that 1 p + 1 p′ = 1. since g is compact, the constant function x 7−→ m‖η‖p′ isintegrable. therefore, by the dominated convergence theorem, we have fn ∗ η(x) = ∫ g fn(y)η(y� ∗ x)dy n→∞−−−→ ∫ g f (y)η(y� ∗ x)dy = f ∗ η(x). https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 8then, |fn ∗ η(x)− f ∗ η(x)| = ∣∣∣∣∫ g fn(y)η(y� ∗ x)dy − ∫ g f (y)η(y� ∗ x)dy ∣∣∣∣ = ∣∣∣∣∫ g (fn(y)− f (y))η(y� ∗ x)dy ∣∣∣∣ ≤ ‖fn − f ‖p (∫ g |η(y� ∗ x)|p′dy ) 1 p′ ≤ 2m (∫ g |η(z)|p′d(δy� ∗ δx)(z) ) 1 p′ ≤ 2m‖η‖p′ .again, by the dominated convergence theorem, we obtain lim n→∞ ‖fn ∗ η − f ∗ η‖q = 0. � hereafter is the analogue of the rellich-kondrachov theorem for hypergroup gelfand pairs. theorem 3.7. assume that g is compact. let (g,k) be a hypergroup gelfand pair. let α > s > 0. let p := 2α α+s and let p′ be such that 1 p + 1 p′ = 1. if (1 + γ2)−1 ∈ lα(ĝ\) and lim y→e ( sup φ∈ĝ\ |φ(y)− 1| (1 + γ(φ)2) s 2 ) = 0, then hs,\γ (g) embeds compactly in lq,\(g) for every q ∈ [1, p′]. proof. in theorem 3.2, we obtained that hs,\γ (g) ↪→ lp ′,\(g); since g is compact and p′ > q, then lp ′,\(g) ↪→ lq,\(g). therefore, we have that hs,\γ (g) ↪→ lq,\(g). now, let (fn) be a boundedsequence in hs,\γ (g). then, (fn) is a bounded sequence in lp′,\(g). there exists m > 0 such that ∀n ∈ n, ‖fn‖p′ 6 m.for g ∈ lp,\(g), we have |〈fn, g〉| 6 ‖fn‖p′‖g‖p 6 m‖g‖p.therefore, (fn) is weakly bounded. it admits a subsequence (hn) which converges weakly to h ∈ lp′,\(g). take ε > 0 and η ∈ k\(g) such that ‖h ∗ η − h‖2 < ε.by theorem 3.5 and theorem 3.6, we have ‖hn − h‖2 ≤ ‖hn − hn ∗ η‖2 + ‖hn ∗ η − h ∗ η‖2 + ‖h ∗ η − h‖2 ≤ sup y∈supp(η) ( sup φ∈ĝ\ |φ(y)− 1| (1 + γ(φ)2) s 2 ) · ‖hn‖hs,\γ + ‖hn ∗ η − h ∗ η‖2 + ε ≤ 2ε+ ‖hn ∗ η − h ∗ η‖2. https://doi.org/10.28924/ada/ma.5.3 eur. j. math. anal. 10.28924/ada/ma.5.3 9as the above inequality is realized for an arbitrary ε, then ‖hn − h‖2 ≤ ‖hn ∗ η − h ∗ η‖2. therefore, lim n→∞ ‖hn − h‖2 = lim n→∞ ‖hn ∗ η − h ∗ η‖2 = 0. thus, (hn) converges to h in l2,\(g). since g is compact, we apply the vitali’s convergence theoremto conclude that (hn) converges to h in lq,\(g). � references [1] r. a. adams, sobolev spaces, academic press, new york, 1975.[2] k. t. bataka, e. m. egwe and y. mensah, sobolev spaces on hypergroup gelfand pairs, far east j. math. sci.,142(1) (2024), 57–69, https://doi.org/10.17654/097208712505.[3] w. r. bloom and h. heyer, harmonic analysis of probability measures on hypergroups, de gruyter studies inmathematics 20, berlin, 1995.[4] h. brezis, sobolev spaces and partial differential equations, springer, 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sobolev spaces arising from a generalized spherical fourier transform, adv. math. sci. j., 10(7) (2021),2947-2955, https://doi.org/10.37418/amsj.10.7.3.[19] f. rellich, ein satz über mittlere konvergenz, nachr. ges. wiss. göttingen, math.-phys. kl., 141, (1930) 30-35. https://doi.org/10.28924/ada/ma.5.3 https://doi.org/10.17654/097208712505 https://doi.org/10.28924/2291-8639-20-2022-32 http://dx.doi.org/10.17654/ms132010063 http://dx.doi.org/10.17654/ms132010063 https://doi.org/10.37418/amsj.12.2.6 10.2140/pjm.2013.265.17 http://dx.doi.org/10.1155/2014/404738 http://dx.doi.org/10.1155/2014/404738 10.1017/s1446788714000433 https://doi.org/10.1007/bf00275475 https://doi.org/10.1007/bf00275475 https://arxiv.org/abs/2003.08519v1 https://doi.org/10.1515/forum-2022-0076 https://doi.org/10.1515/forum-2022-0076 https://doi.org/10.37418/amsj.10.7.3 eur. j. math. anal. 10.28924/ada/ma.5.3 10 [20] a. rozanova-pierrat, generalization of rellich-kondrachov theorem and trace compactness for fractal boundaries,in : m. rosaria lancia, a. rozanova-pierrat (eds.), fractals in engineering: theoretical aspects and numericalapproximations, springer, (2021) pp. 155-173. https://doi.org/10.28924/ada/ma.5.3 1. introduction 2. preliminaries 3. sobolev spaces and embedding results references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 19doi: 10.28924/ada/ma.4.19 convexity properties in non-newtonian calculus and their applications asambo awini wilbert1,∗ , mohammed muniru iddrisu2 , benedict barnes3 1department of mathematics, bongo senior high school, box 7, bongo district, upper east region, ghana awiniwilbert@gmail.com 2department of mathematics, faculty of physical sciences, university for development studies, tamale, ghana mmuniru@uds.edu.gh 3department of mathematics, faculty of physical and computational sciences, kwame nkrumah university of science and technology, kumasi, ghana ewiekwamina@gmail.com ∗correspondence: awiniwilbert@gmail.com abstract. the study presented some results on convexity properties in non-newtonian calculus. alsopresented is the jensen-steffensen inequality in non-newtonian calculus and some applications. theresearch was mainly on positive real numbers. 1. introduction classical calculus was introduced by newton and leibnitz which is applied on our present daymathematics [1]. there are different operations with respect to addition and subtraction of numbersunder this calculus. however, grossman and karts came out with another calculus known asnon-newtonian calculus in the 20th century [2]. non-newtonian calculus which is also calledmultiplicative calculus is a multiplicative way of generating positive solutions to mathematicalproblems [3]. it is a recent approach used to solve mathematical problems with positive realnumbers.it is evidently clear that addition is replaced by multiplication in non-newtonian calculus, andsubtraction by division for example, see authors in [4, 5]. this result has been supported by au-thors in [6], when they introduced the multiplicative calculus and its applications, which has beenestablished to be applicable to solving mathematical problems [6]. non-newtonian calculus hasbeen extended in many directions; fractional derivative, complex derivative, integral transformations,differential equations and applications for science and engineering. received: 30 jan 2024. key words and phrases. non-newtonian calculus, properties, convexity, jensen-steffesen inequality.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.19 https://orcid.org/0009-0004-4991-2067 https://orcid.org/0000-0001-7628-8168 https://orcid.org/0000-0002-0580-5655 the authors in [7] stated that, the centre of all analysis in the social science is the derivative.they were expecting another method which may treat realistic growth phenomenon better in oureconomy than the ordinary approach. the change actually came, which confirmed that variations aremore naturally measured in ratios than in differences [7], until 1972 that grossman and katz cameout with non-newtonian calculus (see [2]). in their work, they also made it clearly that measuringgrowth in ratios gives a better variations than measuring it in differences. non-newtonian calculusis essential in the development of our scientific world, which enhances production and development.it is applicable in various ways such as finance (used in marketing and determining rates ofreturn), health (used in tumor therapy and chemotherapy in medicine, pathogen counts in treatedwater), thermostatistics, quantum theory, wave phenomenon, pattern recognition in images (eg. inbiomedicine), signal processing, biology-thus the rate at which growth increases or decays.in the non-newtonian calculus, ratios are used in measuring change in values whiles in theclassical approach, differences are used in measuring change in values. 2. preliminaries in this section, we give an overview of known definitions and theories used in achieving ourresults. 2.1. non-newtonian arithmetic. a system that satisfies the basic assumptions whose domain is asubset of r is called arithmetic. exactly one arithmetic result is produced by a generator, whichis a one-to-one function with a range of b that is a subset of the domain r [3,8]. the fundamentalarithmetic operations are defined using the generator as follows [3, 4]:addition, k+̇r = α [α−1(k) + α−1(r)]subtraction, k−̇r = α [α−1(k)− α−1(r)]multiplication, k×̇r = α [α−1(k)× α−1(r)]division, k/̇r = α [α−1(k)/α−1(r)]when we take α-generator as α(k) = ek , α−1(k) = ln(k) and k = r+, then α arithmetic reducesto non-newtonian arithmetic as follows:non-newtonian addition, k+̇r = α [ α−1(k) + α−1(r) ] = e(ln(k)+ln(r)) = k · r (1) non-newtonian subtraction, k−̇r = α [ α−1(k)− α−1(r) ] = e(ln(k)−ln(r)) = k/r (2) non-newtonian multiplication, k×̇r = α [ α−1(k)× α−1(r) ] = e(ln(k)×ln(r)) = k ln(r) (3)2 non-newtonian division, k/̇r = α [ α−1(k)/α−1(r) ] = e(ln(k)/ ln(r)) = k 1 ln(r) (4) [1, 4, 8, 9].the above non-newtonian arithmetic are widely accepted in non-newtonian calculus.when considering n positive real numbers x1, x2, ..., xn then the α-arithmetic mean is given as [10]: aα = ∑n i=1 xi /̇n = ∑n i=1 α [ α−1(xi ) n ] aα = α [ α−1(x1)+α−1(x2)+...+α−1(xn) n ] considering α = exp, we have aexp = [∏n i=1 xi ] 1 n = (x1 × x2...xn) 1 nalso considering x1, x2, ..., xn ∈ r+ and gα to be the α-geometric mean then: gα = [∏n i=1 xi ] 1 n = α [∏n i=1 α −1(xi) ] 1 n gα = α [ (α−1(x1)× α−1(x2)...α−1(xn)) 1 n ].in a similar way, we take α = exponent, then the α-geometric mean can be interpreted as [10]: gexp = [ (ln x1 × ln x2... ln xn) 1 n ] , (xn > 1). definition 2.1. [11] a set c = [a1, b1] ⊆ r is said to be convex if x, y ∈ c, and qx + (1− q)y ∈ c, (5) for q ∈ [0, 1]. definition 2.2. [11] let φ be defined on a real interval m . the function φ is convex if: φ(λ1x + λ2y) ≤ λ1φ(x) + λ2φ(y), (6) φ(λ1x + (1− λ1)y) ≤ λ1φ(x) + (1− λ1)φ(y), (7) where λ1 + λ2 = 1, ∀ x, y ∈ m and λ1, λ2 ∈ [0, 1]. definition 2.3. [12] let φ be convex function and u1, v1 ∈ r , then φ ( u1 + v1 2 ) ≤ φ(u1) + φ(v1) 2 . (8) proposition 2.1. [13] let x1 ≤ y1, x2 ≤ y2, and φw be a convex function on an interval of real positive values, then φw (y2 − y1) x2 − x1 ≤ φw (y2)− φw (y1) x2 − x1 . (9) theorem 2.1 (jensen-steffensen inequality). [14] let φ be a convex function defined on an interval of the real line and let xi , pi ∈ r, i = 1, ...m. if x1, ..., xm and p1, ..., pm, pm > 0, then φ( 1 pm m∑ i=1 pixi) ≤ 1 pm m∑ i=1 piφ(xi). (10) 3 theorem 2.2. [14] let φ be a convex function on an interval s = [a1, b1] ⊂ r+, where a1 < b1. let x = (x1, x2, ..., xn) and p = (p1, p2, ..., pn), then φk(a1 + b1 − 1 pn n∑ i=1 pixi) ≤ φk(a1) + φk(b1)− 1 pn n∑ i=1 piφk(xi). (11) let pn = ∑n i=1 pi = 1, then φk(a1 + b1 − n∑ i=1 pixi) ≤ φk(a1) + φk(b1)− n∑ i=1 piφk(xi). (12) definition 2.4. [15] the definition of a p-convex set for an interval c is (qup + (1− q)vp) 1 p ∈ c, (13) for all u, v ∈ c and q ∈ [0, 1] . definition 2.5. [12, 15, 16] let c = [a1, b1] be an interval on real numbers r. an expression φ : c = [a1, b1] 7→ r is p-convex if φ(qup + (1− q)vp) 1 p ≤ qφ(u) + (1− q)φ(v), (14) ∀ u, v ∈ c and q ∈ [0, 1]. definition 2.6. [15] let φ be p-convex function and u, v ∈ r+, then φ ( up + vp 2 ) 1 p ≤ φ(u) + φ(v) 2 . (15) definition 2.7. [15] suppose the function φ : c = [x, y ] 7→ r+ is strongly convex and β ≥ 1; then φ(qa + (1− q)b) ≤ qφ(a) + (1− q)φ(b)− βq(1− q)(b − a)2 (16) for all a, b ∈ c and q ∈ [0, 1]. definition 2.8. [15] a function φ : c = [x, y ] 7→ r is strongly p-convex function, if φ(qap + (1− q)bp) 1 p ≤ qφ(a) + (1− q)φ(b)− βq(1− q)(bp − ap)2, (17) for all a, b ∈ c and q ∈ [0, 1]. definition 2.9. [15] in the event that an interval c is a harmonic convex set, then( uv qu + (1− q)v ) ∈ c, (18) for all u, v ∈ c and q ∈ [0, 1]. definition 2.10. [15] let the function φ : c = [a1, b1] ⊆ r+ and c = [a1, b1] be on an interval on set r+ without zero, then φ ( uv qu + (1− q)v ) ≤ (1− q)φ(u) + qφ(v). (19) 4 definition 2.11. [15] let c = [a1, b1] represent an interval on the p-harmonic convex set r without zero. if a function φ : c = [a1, b1] ⊆ r is p-harmonic convex, it does not include zero if φ ( upvp qup + (1− q)vp ) 1 p ≤ (1− q)φ(u) + qφ(v), (20) for all u, v ∈ c and q ∈ [0, 1]. in 1984, g. toader defines m-convex function as follows [17]: definition 2.12. let the function φ be a real r+ on [u, v ] and m ∈ [0, 1], then m-convex function is given as; φ [ qx1 +m(1− q)y1 ] ≤ qφ(x1) +m(1− q)φ(y1), (21) for all x1, y1 ∈ [u, v ] and q ∈ [0, 1]. also, φ is m-concave if −φ is m−convex. definition 2.13. let the function φ be a positive real value on s = [u, v ], then c-convex is represented by luenberger (1969) as; φ [ (1− q)x1 + qy1 ] ≤ c(1− q)φ(x1) + qφ(y1), (22) where c ∈ [0, 1], for all x1, y1 ∈ s and q ∈ [0, 1]. 3. results and discussions in this section, all the results are presented in non-newtonian form. definition 3.1. let x, y ∈ c and c = [a1, b1] ⊆ r+ be a set. c must be convex if x ln(t) · y ln( 1 t ) ∈ c, (23) for t ∈ [1, e]. lemma 3.1. let, x1, y1 ∈ r+ and φ1 be a convex function. then φ1 [ x ln(t1) 1 · y ln( 1 t1 ) 1 ] ≤ φ1(x1)ln(t1) · φ1(y1) ln( 1 t1 ) , (24) for t1 ∈ [1, e]. 5 proof. using equation (3), (1) and by convexity, we have φ1 [ x ln(t1) 1 · y ln( 1 t1 ) 1 ] =φ1 [ x1×̇t1+̇y1×̇( 1 t1 ) ] ≤φ1(x1)×̇t1+̇φ1(y1)×̇( 1 t1 ) ≤α [ α−1φ1(x1)×̇α−1(t1)+̇α−1φ1(y1)×̇α−1( 1 t1 ) ] ≤α [ lnφ1(x1)× ln(t1) + lnφ1(y1)× ln( 1 t1 ) ] ≤e [ lnφ1(x1) ln(t1)+lnφ1(y1) ln( 1 t1 ) ] ≤(e lnφ1(x1))ln(t1) · (e lnφ1(y1))ln( 1 t1 ) φ1 [ x ln(t1) 1 · y ln( 1 t1 ) 1 ] ≤φ1(x1)ln(t1) · φ1(y1) ln( 1 t1 ) , as required. � lemma 3.2. let a1, b1 ∈ r+ and φ1 be a convex function, we have φ1 [ (b1) (a1) ] ≤ φ1(b1) φ1(a1) . (25) proof. using equation (2) and by convexity, we have φ1 [ (b1) (a1) ] =φ1 [ (b1)−̇(a1) ] ≤φ1(b1)−̇φ1(a1) ≤α [ α−1(φ1(b1))−̇α−1(φ1(a1)) ] ≤e[lnφ1(b1)−lnφ1(a1)] ≤ e lnφ1(b1) e lnφ1(a1) ≤ φ1(b1) φ1(a1) φ1 [ (b1) (a1) ] ≤ φ1(b1) φ1(a1) , as required. � lemma 3.3. let φ1 be convex and u1, v1 ∈ r+. then φ1 [ u1 · v1 2 ] ≤ φ1(u1) · φ1(v1) 2 . (26) 6 proof. using equation (1) and by convexity, we have φ1 [ u1 · v1 2 ] =φ1 [ 1 2 (u1+̇v1) ] ≤ 1 2 [( φ1(u1)+̇φ1(v1) )] ≤ 1 2 [ α ( α−1φ1(u1)+̇α −1φ1(v1) )] ≤ 1 2 [ e(lnφ1(u1)+lnφ1(v1)) ] ≤ 1 2 [ e lnφ1(u1) · e lnφ1(v1) ] ≤ 1 2 [ φ1(u1) · φ1(v1) ] φ1 [ u1 · v1 2 ] ≤ φ1(u1) · φ1(v1) 2 , as required. � lemma 3.4. consider an increasing function φ. if φ(vy ) ≥ φ(vx) and vy ≥ vx , then φ(vx) ln(vx ) · φ(vy )ln(vy ) φ(vy )ln(vx ) · φ(vx)ln(vy ) ≤ 1, (27) proof. using equation (3) and (1), we have φ(vx) ln(vx ) · φ(vy )ln(vy ) =φ(vx)×̇vx +̇φ(vy )×̇vy =vx ×̇φ(vx)+̇vy ×̇φ(vy ) ≤vx ×̇φ(vy )+̇vy ×̇φ(vx) ≤α [ α−1(vx)×̇α−1φ(vy )+̇α−1(vy )×̇α−1φ(vx) ] ≤α [ ln(vx)× lnφ(vy ) + ln(vy )× lnφ(vx) ] ≤e[ln(vx )×lnφ(vy )+ln(vy )×lnφ(vx )] ≤e ln(vx ) lnφ(vy ) · e ln(vy ) lnφ(vx ) ≤ ( e lnφ(vy ) )ln(vx ) · ( e lnφ(vx ) )ln(vy ) φ(vx) ln(vx ) · φ(vy )ln(vy ) ≤φ(vy )ln(vx ) · φ(vx)ln(vy ),as required. � lemma 3.5. consider a decreasing function φ. if φ(vx) ≥ φ(vy ) and vx ≥ vy , then φ(vx) ln(vx ) · φ(vy )ln(vy ) φ(vy )ln(vx ) · φ(vx)ln(vy ) ≥ 1, (28) or φ(vx) ln(vx ) · φ(vy )ln(vy ) ≥ φ(vy )ln(vx ) · φ(vx)ln(vy ). (29)7 the proof is similar to inequality (25), with inequality sign reversed. theorem 3.1. let vi , zi ∈ r+, i = 1, ..., k , and let φ be a convex function defined on a range of the real line. if bk > 0, then φ ( 1 bk )ln∏k i=1(vi ) ln(zi )  ≤ ( 1 bk )ln∏k i=1 φ(vi ) ln(zi ) . (30) proof. using equation (3) and by convexity, we have φ ( 1 bk )ln∏k i=1(vi ) ln(zi )  =φ 1 bk ×̇ k∏ i=1 (zi)×̇(vi)  ≤ 1 bk ×̇ k∏ i=1 (zi)×̇φ(vi) ≤α α−1( 1 bk )×̇α−1 k∏ i=1 (zi)×̇α−1φ(vi)  ≤e ( ln( 1 bk ) ln ∏k i=1(zi )×lnφ(vi ) ) ≤ ( e ln( 1 bk ) )ln∏k i=1(zi )×lnφ(vi ) ≤ ( 1 bk )ln∏k i=1(zi )×lnφ(vi ) φ ( 1 bk )ln∏k i=1(vi ) ln(zi )  =( 1 bk )ln∏k i=1 φ(vi ) ln(zi ) , as required. � definition 3.2. a set m = [a1, b1] ⊆ r+ is a p-convex set, if [ [(x1) p]ln(j) · [(y1)p]ln( 1 j ) ] 1 p ∈ m, (31) ∀ x1, y1 ∈ m and j ∈ [1, e]. lemma 3.6. let x, y ∈ m where m ⊆ r+ and j ∈ [1, e]. for a p-convex function φ, we have φ [ [(x1) p]ln(j) · [(y1)p]ln( 1 j ) ] 1 p ≤ [ φ(x1) ]ln(j) · [φ(y1)]ln( 1j ) . (32)8 proof. using equation (3), (1) and by p-convexity, we have φ [ [(x1) p]ln(j) · [(y1)p]ln( 1 j ) ] 1 p =φ [ (x1) p×̇j+̇(y1)p×̇( 1 j ) ] 1 p ≤φ(x1)×̇j+̇φ(y1)×̇( 1 j ) ≤ [ α[α−1φ(x1)×̇α−1(j)+̇α−1φ(y1)×̇α−1( 1 j )] ] ≤ [ α[lnφ(x1)×̇ ln(j)+̇ lnφ(y1)×̇ ln( 1 j )] ] ≤e [ lnφ(x1) ln(j)+lnφ(y1) ln( 1 j ) ] ≤e lnφ(x1) ln(j) · e lnφ(y1) ln( 1 j ) ≤ ( e lnφ(x1) )ln(j) · ( e lnφ(y1) )ln( 1 j ) ≤ [ φ(x1) ]ln(j) · [φ(y1)]ln( 1j ) φ [ [(x1) p]ln(j) · [(y1)p]ln( 1 j ) ] 1 p ≤ [ φ(x1) ]ln(j) · [φ(y1)]ln( 1j ) ,as required. � remark 1. the inequality (24) is obtained when p = 1. lemma 3.7. let φ be convex and x1, y1 ∈ r+. for a p-convex function, we have φ [ (x1) p · (y1)p 2 ] 1 p ≤ φ(x1) · φ(y1) 2 . (33) proof. using equation (1) and by p-convexity: φ [ (x1) p · (y1)p 2 ] 1 p =φ [ 1 2 ((x1) p+̇(y1) p) ] 1 p ≤ 1 2 [ φ(x1)+̇φ(y1) ] ≤ 1 2 [ α ( α−1φ(x1)+̇α −1φ(y1) )] ≤ 1 2 [ α ( lnφ(x1) + lnφ(y1) )] ≤ 1 2 [ e(lnφ(x1)+lnφ(y1)) ] ≤ 1 2 [ e lnφ(x1) · e lnφ(y1) ] ≤ 1 2 [ φ(x1) · φ(y1) ] φ [ (x1) p · (y1)p 2 ] 1 p ≤ φ(x1) · φ(y1) 2 , 9 as required. � remark 2. the inequality (26) is obtained when p = 1. definition 3.3. a harmonic convex set of an interval m is described, if x ln(y) x ln(j) · y ln( 1 j ) ∈ m, (34) ∀ x, y ∈ m and j ∈ [1, e]. lemma 3.8. let the set w ⊆ r+ be a harmonic set. for harmonic convex function φ, we have φ  x ln(y1) 1 x ln(q1) · y ln( 1 q1 ) 1  ≤ φ(x1) lnφ(y1) φ(x1)ln(q1) · φ(y1) ln( 1 q1 ) , (35) ∀ x1, y1 ∈ w and q1 ∈ [1, e]. proof. using equation (3), (1) and by convexity, we have φ  x ln(y1) 1 x ln(q1) 1 · y ln( 1 q1 ) 1  =φ [x1×̇y1−̇(q1×̇x1+̇( 1 q1 )×̇y1) ] ≤φ(x1)×̇φ(y1)−̇(q1×̇φ(x1)+̇( 1 q1 )×̇φ(y1)) ≤α [ α−1φ(x1)×̇α−1φ(y1)−̇(α−1(q1)×̇α−1φ(x1)+̇α−1( 1 q1 )×̇α−1φ(y1)) ] ≤α [ lnφ(x1)× lnφ(y1)− (ln(q1)× lnφ(x1) + ln( 1 q1 )× lnφ(y1)) ] ≤e [ lnφ(x1)×lnφ(y1)−(ln(q1)×lnφ(x1)+ln( 1q1 )×lnφ(y1)) ] ≤e [ lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(q1)+lnφ(y1)×ln( 1q1 )) ] ≤ (e lnφ(x1))lnφ(y1) (e lnφ(x1))ln(q1) · (e lnφ(y1))ln( 1 q1 ) φ  x ln(y1) 1 x ln(q1) 1 · y ln( 1 q1 ) 1  ≤ φ(x1) lnφ(y1) φx ln(q1) 1 · φ(y1) ln( 1 q1 ) , proved. � definition 3.4. let w be a subset on r+, then w is p-harmonic convex set if [xp]ln(y) p [xp]ln(j) · [yp]ln( 1 j )  1p ∈ w, (36) ∀ x, y ∈ w and j ∈ [1, e]. 10 lemma 3.9. consider the p-harmonic convex set m = [a, b] ⊆ r+. if φ is p-harmonic convex function, we have φ  [(x1) p]ln(y1) p [(x1)p]ln(j1) · [(y1)p] ln( 1 j1 )  1p ≤ [φ(x1)] lnφ(y1) [φ(x1)]ln(j) · [φ(y1)] ln( 1 j1 ) , (37) ∀ x1, y1 ∈ m and j1 ∈ [1, e]. proof. using equation (3), (2) and by p-convexity, we have φ  [(x1) p]ln(y1) p [(x1)p]ln(j1) · [(y1)p] ln( 1 j1 )  1p =φ [(x1)p×̇(y1)p−̇(j1×̇(x1)p+̇( 1 j1 )×̇(y1)p) ] 1 p ≤φ(x1)×̇φ(y1)−̇(j1×̇φ(x1)+̇( 1 j1 )×̇φ(y1)) ≤α [ lnφ(x1)× lnφ(y1)− (ln(j1)× lnφ(x1) + ln( 1 j1 )× lnφ(y1) ] ≤e [ lnφ(x1)×lnφ(y1)−(ln(j1)×lnφ(x1)+ln( 1j1 )×lnφ(y1) ] ≤e [ lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(j1)+lnφ(y1)×ln( 1j1 ) ] ≤ e[lnφ(x1) lnφ(y1)] e[lnφ(x1) ln(j1)] · e[lnφ(y1) ln( 1 j1 )] ≤ (φ(x1)) lnφ(y1) (φ(x1))ln(j1) · (φ(y1)) ln( 1 j1 ) φ  [(x1) p]ln(y1) p [(x1)p]ln(j1) · [(y1)p] ln( 1 j1 )  1p ≤ [φ(x1)] lnφ(y1) [φ(x1)]ln(j1) · [φ(y1)] ln( 1 j1 ) , as required. � remark 3. when p = 1, the inequality (35) is obtained. definition 3.5. m is referred to as p-jensen-steffensen’s set if, it is a subset of r+, assuming that( 1 bn )ln∏n i=1((xi ) p)ln(zi )  1p ∈ m. (38) lemma 3.10. let m ⊆ r+ be p-jensen-steffensen set. for p-jensen-steffensen’s inequality, we have φ ( 1 bn )ln∏n r=1((xi ) p)ln(zi )  1p ≤ ( 1 bn )ln∏n r=1 φ(xi ) ln(zi ) . (39) 11 proof. using equation (3) and by p-convexity, we have φ ( 1 bn )ln∏n r=1((xi ) p)ln(zi )  1p =φ 1 bn ×̇ n∏ r=1 (zi)×̇(xi)p  1p ≤ 1 bn ×̇ n∏ r=1 (zi)×̇φ(xi) ≤α α−1( 1 bn )×̇α−1 n∏ r=1 (zi)×̇α−1φ(xi)  ≤e ( ln( 1 bn ) ln ∏n r=1(zi )×lnφ(xi ) ) ≤ ( e ln( 1 bn ) )ln∏n r=1(zi )×lnφ(xi ) ≤ ( 1 bn )ln∏n r=1(zi )×lnφ(xi ) φ ( 1 bn )ln∏n r=1((xi )) ln(zi )  1p ≤( 1 bn )ln∏n r=1 φ(xi ) ln(zi ) , as required. � definition 3.6. let the set w be a subset on r+, then w is strongly convex set if [[ µln(j) ]ln( 1 j ) ]ln( y x )2 ∈ w, (40) where µ ≥ 1, ∀ x, y ∈ w and j ∈ [1, e]. lemma 3.11. consider a convex set w ⊆ r+. for a strongly convex function φ, we have φ [ x ln(q) · y ln( 1 q ) ] ≤ φ(x)ln(q) · φ(y)ln( 1 q )[[ µln(q) ]ln( 1 q ) ]ln( y x )2 , (41) where µ ≥ 1, ∀ x, y ∈ w and q ∈ [1, e]. 12 proof. using equation (3), (1) and by convexity, we have φ [ x ln(q) · y ln( 1 q ) ] =φ [ x×̇q+̇y×̇( 1 q ) ] ≤q×̇φ(x)+̇( 1 q )×̇φ(y)−̇µ×̇j×̇( 1 q )×̇( y x )2 ≤α [ α−1q×̇α−1φ(x)+̇α−1( 1 q )×̇α−1φ(y)−̇α−1(µ)×̇α−1(q)×̇α−1( 1 q )×̇( y x )2 ] ≤ e ln(q) lnφ(x) · e ln( 1 q ) lnφ(y) e ln(µ) ln(q) ln( 1 q ) ln( y x )2) ≤ ( e lnφ(x) )ln(q) · ( e lnφ(y) )ln( 1 q ) ( e ln(µ) )ln(q) ln( 1 q ) ln( y x )2 φ [ x ln(q) · y ln( 1 q ) ] ≤ φ(x)ln(q) · φ(y)ln( 1 q )[[ µln(q) ]ln( 1 q ) ]ln( y x )2 , proved. � lemma 3.12. consider the convex set w = [a, b] ⊆ r+. if φ is highly p-convex function, we have φ [ (xp)ln(j) · (yp)ln( 1 j ) ] 1 p ≤ φ(x)ln(j) · φ(y)ln( 1 j )[[ µln(j) ]ln( 1 j ) ]ln( y x )2 , (42) where µ ≥ 1, ∀ x, y ∈ w and j ∈ [1, e]. remark 4. when p = 1, the strongly p-convex function returns to strongly convex function. thus the inequality (41) is obtained. proof. using equation (3), (1) and by p-convexity, we have φ [ (xp)ln(j) · (yp)ln( 1 j ) ] 1 p ≤j×̇φ(x)+̇( 1 j )×̇φ(y)−̇µ×̇j×̇( 1 j )×̇( y x )2 ≤α [ α−1j×̇α−1φ(x)+̇α−1( 1 j )×̇α−1φ(y)−̇α−1(µ)×̇α−1(j)×̇α−1( 1 j )×̇( y x )2 ] ≤e ( ln(j) lnφ(x)+ln( 1 j ) lnφ(y)−ln(µ) ln(j) ln( 1 j ) ln( y x )2 ) ≤ e ln(j) lnφ(x) · e ln( 1 j ) lnφ(y) e ln(µ) ln(j) ln( 1 j ) ln( y x )2) ≤ ( e lnφ(x) )ln(j) · ( e lnφ(y) )ln( 1 j ) ( e ln(µ) )ln(j) ln( 1 j ) ln( y x )2 13 φ [ (xp)ln(j) · (yp)ln( 1 j ) ] 1 p ≤ φ(x)ln(j) · φ(y)ln( 1 j )[[ µln(j) ]ln( 1 j ) ]ln( y x )2 , proved. � definition 3.7. let the set w be a subset on r+, then w is m-convex set if x ln(t) · [ (y)ln(m) ]ln( 1 t ) ∈ w, (43) where m ∈ [1, e], ∀ x, y ∈ w and j ∈ [1, e]. lemma 3.13. let the function φ1 be m-convex and m1 ∈ [1, e], then φ1 [ x ln(t1) 1 · [ y ln(m) 1 ]ln( 1 t1 ) ] ≤ φ1(x1)ln(t1) · [ φ1(y1) ln(m) ]ln( 1 t1 ) , (44) for all x1, y1 ∈ r+ and t ∈ [1, e]. proof. using equation (3), (1) and by convexity, we have φ1 [ x ln(t1) 1 · [ (y1) ln(m1) ]ln( 1 t1 ) ] =φ1 [ x1×̇t+̇m1×̇(y1)×̇( 1 t1 ) ] ≤φ1(x1)×̇t1+̇m1×̇φ1(y1)×̇( 1 t1 ) ≤ [ α[α−1φ1(x1)×̇α−1(t1)+̇α−1(m1)×̇α−1φ1(y1)×̇α−1( 1 t1 )] ] ≤ [ α[lnφ1(x1)× ln(t1) + ln(m1)×̇ lnφ1(y1)× ln( 1 t1 )] ] ≤ [ e [lnφ1(x1)×ln(t1)+ln(m1)×lnφ1(y1)×ln( 1t1 )] ] ≤ [ e [lnφ1(x1) ln(t1)+lnm1 lnφ1(y1) ln( 1 t1 )] ] ≤(e lnφ1(x1))ln(t1) · ((e lnφ1(y1))ln(m1))ln( 1 t1 ) φ1 [ x ln(t1) 1 · [ (y1) ln(m1) ]ln( 1 t ) ] ≤φ1(x1)ln(t1) · [ φ1(y1) ln(m1) ]ln( 1 t1 ) , as required. � definition 3.8. let w ⊆ r+, then w is said to be c-convex set if x ln( 1 t ) 1 · y ln(t)1 ∈ w, (45) where c ∈ [1, e], ∀ x1, y1 ∈ w and t ∈ [1, e]. 14 lemma 3.14. let the function φ1 be c-convex on w = [a, a1] and c ∈ [1, e], then φ1 [ (x ln( 1 t1 ) 1 · y ln(t1)1 ] ≤ [ φ1(x1) ln(c) ]ln( 1 t1 ) · φ1(y1)ln(t1), (46) ∀ x1, y1 ∈ w and t1 ∈ [1, e]. proof. using equation (3), (1) and by convexity, we have φ1 [ x ln( 1 t1 ) 1 · (y1)ln(t1) ] =φ1 [ x1×̇( 1 t1 )+̇(y1)×̇t1 ] ≤c×̇φ1(x1)×̇( 1 t1 )+̇φ1(y1)×̇t1 ≤ [ α[α−1(c)×̇α−1φ1(x1)×̇α−1( 1 t1 )+̇α−1φ1(y1)×̇α−1(t1)] ] ≤ [ α[ln(c)× lnφ1(x1)× ln( 1 t1 ) + lnφ1(y1)× ln(t1)] ] ≤ [ e [lnφ1(x1)×ln(c)×ln( 1t1 )+lnφ1(y1)×ln(t1)] ] ≤ [ e [lnφ1(x1) ln(c) ln( 1 t1 )+lnφ1(y1) ln(t1)] ] ≤ [ (e lnφ1(x1))ln(c) ]ln( 1 t1 ) · (e lnφ1(y1))ln(t1) φ1 [ x ln( 1 t1 ) 1 · (y1)ln(t1) ] ≤ [ φ1(x1) ln(c) ]ln( 1 t1 ) · φ1(y1)ln(t1),as required. � 4. conclusion in this paper, some classes of convex functions have been identified and presented. the pa-per established some convexity properties and inequalities in non-newtonian calculus and theirapplications. references [1] e. unluyol, s. salas, i̇. i̇scan, convex functions and some inequalities in terms of the non-newtonian calculus, in:aip conference proceedings, vol. 1833, aip publishing llc, 2017, p. 020043.[2] a. e. bashirov, r. mustafa, on complex multiplicative differentiation, twms journal of applied and engineeringmathematics 1 (1) (2011) 75–85.[3] k. boruah, b. hazarika, g-calculus, twms journal of applied and engineering mathematics 8 (1) (2018) 94–105.[4] d. f. torres, on a non-newtonian calculus of variations, axioms 10 (3) (2021) 171.[5] m. czachor, non-newtonian mathematics instead of non-newtonian physics: dark matter and dark energy from amismatch of arithmetics, foundations of science 26 (2021) 75–95.[6] a. e. bashirov, e. m. kurpınar, a. özyapıcı, multiplicative calculus and its applications, journal of mathematicalanalysis and applications 337 (1) (2008) 36–48.[7] d. filip, c. piatecki, an overview on the non-newtonian calculus and its potential applications to economics.15 [8] m. grossman, bigeometric calculus: a system with a scale-free derivative, archimedes foundation, 1983.[9] m. grossman, r. katz, non-newtonian calculus: a self-contained, elementary exposition of the authors’ investi-gations..., non-newtonian calculus, 1972.[10] u. kadak, y. gürefe, a generalization on weighted means and convex functions with respect to the non-newtoniancalculus, international journal of analysis 2016.[11] m. a. noor, k. i. noor, s. iftikhar, hermite-hadamard inequalities for harmonic nonconvex functions, magnt res.rep 4 (2016) 24–40.[12] s. s. dragomir, n-points inequalities of hermite-hadamard type for h-convex functions on linear spaces, armenianjournal of mathematics 8 (1) (2016) 38–57.[13] i. franjić, s. khalid, j. pečarić, on the refinements of the jensen-steffensen inequality, journal of inequalities andapplications 2011 (1) (2011) 1–11.[14] m. bakula, m. matić, j. pečarić, generalizations of the jensen-steffensen and related inequalities, open mathematics7 (4) (2009) 787–803.[15] h. li, m. s. saleem, i. ahmed, k. n. aslam, hermite–hadamard and fejér-type inequalities for strongly reciprocally(p, h)-convex functions of higher order, journal of inequalities and applications 2023 (1) (2023) 1–20.[16] i. iscan, ostrowski type inequalities for p-convex functions, new trends in mathematical sciences 4 (3) (2016)140–150.[17] j. n. valdés, f. rabossi, a. d. samaniego, convex functions: ariadne’s thread or charlotte’s spiderweb, advancedmathematical models & applications 5 (2) (2020) 176–191. 16 1. introduction 2. preliminaries 2.1. non-newtonian arithmetic 3. results and discussions 4. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 11doi: 10.28924/ada/ma.5.11 majorizing sequences for newton-like method and their limit points ioannis k. argyros1,∗ , santhosh george2 , michael argyros3 1department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 3department of computer sciences, franklin university, ohio, usa argyro01@email.franklin.edu ∗correspondence: iargyros@cameron.edu abstract. a plethora of problems from diverse disciplines of mathematics, mathematical biology,chemistry, medicine, physics and engineering to mention a few reduce to solving nonlinear equationsor systems of equations usually in the finite dimensional euclidean or more general spaces. thesolutions of such equations are numbers or vectors of functions and can be found in closed form onlyin special cases. that is why researchers and practitioners develop mostly iterative methods whichgenerate sequences approximating the solutions. the least number of iterations to be carried outin order to obtain a pre-decided error tolerance on the distances between consecutive iterates aswell as the choice of initial points ensuring the convergence of the methods is very important. thesetwo objectives can be achieved by introducing real majorizing sequences which control the behaviourof the iterates. moreover, the closed form of the limits of the real sequences determine the radiusof the ball that contains the initial points. in this paper we contribute by introducing more precisemajorizing sequences and limit points. 1. introduction majorizing sequences have been used extensively to study the semi-local convergence of new-ton’s method defined for x0 ∈ d and each n = 0, 1, 2, ... by xn+1 = xn − f ′(xn)−1f (xn), (1.1) where f : d ⊂ b1 → b2 is a fŕechetdifferentiable operator between banach spaces b1, b2 and d is an open and convex set [1, 3, 4, 6, 7]. the usually sufficient semi-local convergence conditionsdiffer in general as well as the majorizing sequences and their limit points. we try to relate theseconditions, sequences and limit points in a unified way without additional hypotheses. received: 10 nov 2024. key words and phrases. newton-like method; majorizing sequences; fréchet derivative; banach spaces.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.11 https://orcid.org/0000-0002-9189-9298 https://orcid.org/0000-0002-3530-5539 eur. j. math. anal. 10.28924/ada/ma.5.11 22. lipschitz conditions the symbols l(b1, b2), u(x, r) are used to denote the space of bounded linear operators from b1 into b2 and the open ball centered at x ∈ b1 and of radius r > 0, respectively.we introduce lipscits conditions used to control f ′. then, we copare them to each other. definition 2.1. suppose m ∈ l (b1, b2) is an invertible operator and x0 ∈ d. we say that f ′ is center-lipschitz continuous if there exists l0 > 0 such that m−1(f ′(x)−m)‖ ≤ l0‖x − x0‖ for each x ∈ d0. (2.1) define the region d0 = d ∩ u(x0, 1 l0 ). (2.2) definition 2.2. suppose m ∈ l (b1, b2) is an invertible operator. we say that f ′ is restricted lipschitz continuous if there exists l > 0 such that ‖m−1(f ′(y)− f ′(x))‖ ≤ l‖y − x‖ for each x, y ∈ d0. (2.3) definition 2.3. suppose m ∈ l (b1, b2) is an invertible operator. we say that f ′ is lipschitz continuous if there exists l1 > 0 such that ‖m−1(f ′(y)− f ′(x))‖ ≤ l1‖y − x‖ for each x, y ∈ d. (2.4) remark 2.4. it follows by these definitions that since d0 ⊆ d, we have l0 ≤ l1 (2.5) and l ≤ l1. (2.6) it is worth noting that l0 and l1 depend on x0, f ′ and d. but l depends on x0, f ′ and d0.moreover, in practice the computation of l1 requires that of l0 and l as special cases. theseconstants are related to majorizing sequences in section 3. 3. convergence of majorizing sequences. let ω ≥ 0, l1 > 0 and λ ≥ 1 be parameters. define µ and β by µ = λω and β = λ 1 + (λ− 1)l1µ . (3.1) moreover, define the quadratic majorizing function f1 by f1(t) = βl1t 2 2 − t + µ. (3.2) further more, define the scalar sequence {vn} for v0 = 0 and each n = 0, 1, 2, ... by vn+1 = vn − f1(vn) f ′1(vn) . (3.3) https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 3an auxiliary result is needed for the convergence of the sequence {vn}. lemma 3.1. suppose h1 = 2βl1µ ≤ 1. (3.4) then, the following assertions hold(i) the zeros of the function f1 are real and given by v∗ = 1− √ 1− 2βl1µ βl1 and v∗∗ = 1 + √ 1− 2βl1µ βl1 . (3.5) (ii) vn+1 − vn = βl1(vn − vn−1)2 2(1− βl1vn) = − f1(vn) f ′1(vn) . (3.6) (iii) the sequence {vn} is increasingly convergent to v∗ and can also be written in closed for as vn = ∑2n−2 j=0 qj1∑2n−1 j=0 qj1 v∗, n = 1, 2, ..., (3.7) where, q1 = v∗ v∗∗ = 1− √ 1− 2βl1µ 1 + √ 1− 2βl1µ . (3.8) proof. (i) the zeros of the function f are real by (3.4).by setting f (t) = 0 and using the quadratic formula we obtain v∗ and v∗∗.(ii) let vn+1 = g1(vn), v0 = 0, n = 0, 1, ... (3.9)where, g1(t) = 1 2βl1t 2 − µ βl1t − 1 . (3.10) multiply (3.9) by (1− βl1vn) and simplify to get (1− βl1vn)vn+1 = µ− 1 2 βl1v 2 n ,or (3.11) 1 2 βl1v 2 n − βl1vnvn+1 + 1 2 βl1v 2 n+1 = 1 2 βl1v 2 n+1 − vn+1 + µ, so 1 2 βl1(vn+1 − vn)2 = 1 2 βl1v 2 n+1 − vn+1 + µ.thus, we can write vn+1 − vn = 1 2βl1(vn − vn−1)2 1− βl1vn = − f1(vn) f ′1(vn) . (3.12) (iii) the proof can be found in [5]. � https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 4 remark 3.2. (i) in view of (3.1) the results of the lemma 3.1 can be given without β. for example (3.4) becomes µl1(λ+ 1) ≤ 1. (3.13) (ii) let l > 0. define the quadratic majorizing function f by f (t) = βlt2 2 − t + µ, (3.14) and the scalar sequence {un}f oru0 = 0 and each n = 0, 1, 2, ... by un+1 = un − f (un) f ′(un) . (3.15) denote the corresponding zeros of f (t) = 0 by v∗ and v∗∗, respectvely provided that h2 = 2βlµ ≤ 1 (3.16) clearly, the results of the lemma 3.1 hold, if l replaces l1 and un+1 − un = βl(un − un−1)2 2(1− βlun) . (3.17) let l > 0. define the sequence {sn} for 0 = 0, s1 = µ, s2 = s1 + βl0(s1 − s0)2 2(1− l0βs1) and (3.18) sn+1 = sn + βl(sn − sn−1)2 2(1− l0βsn) . next, we compare the sequences {vn}, {un}, and {sn}. lemma 3.3. suppose (2.5),(2.6) and (3.4) hold. then, the following assertions hold 0 ≤ sn ≤ sn+1, 0 ≤ un ≤ un+1 0 ≤ vn ≤ vn+1, 0 ≤ sn ≤ un ≤ vn and 0 ≤ s∗ = lim n→+∞ ≤ u∗ = lim n→+∞ = 1− √ 1− 2βlµ βl ≤ v∗. proof. it follows by simple induction (2.5),(2.6) and the definition of these sequences. � in the next section, we relate sequences {vn}, {un} and {sn} to {xn}. https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 54. convergence of newton’s method the celebrated newton-kantorovich theorem for solving nonlinear equations using newton’smethod is stated next. the proof can be found in [3, 6] for m = f ′(x0). moreover, the proof forgeneral m follows by simply using m instead of f ′(x0) in the newton-kantorovich theorem. theorem 4.1. suppose that (2.4) and (3.4) hold for λ = 1, µ = ω and ω ≥ ‖f ′(x0)−1f (x0)‖. then, the sequence {xn} generated by newton’s method (1.1) is well defined in u(x0, v ∗), remains in u(x0, v ∗) for each n = 0, 1, 2, ... and converges to a unique solution x∗ ∈ u(x0, r ∗) of the equation f (x) = 0. moreover, the sequence {vn} majorizes {xn}, ‖xn+1 − xn‖ ≤ vn+1 − vn (4.1) and ‖x∗ − xn‖ ≤ v∗ − vn. (4.2) furthermore, if there exists v̄ ≥ v∗ such that l1 2 (v∗ + v̄) < 1, (4.3) then the solution x∗ is more unique in u[x0, 2 l1 − v∗], where u[x0, r ] is the closure of u(x0, r). remark 4.2. in view of (2.6) theorem (4.1) holds provided that l, {un} replace l1, {vn}, respectively. by lemma 3.1 and 3.3 the sequence {un} is tighter than {vn} and the limit point u∗ is atleast as small as v∗. moreover, they are given in closed form. this is not however the case for s∗. the convergence condition for {sn} given in [2] for m = f ′(x0), λ = 1 h3 = 2l̄µ ≤ 1, (4.4) where l̄ = 1 8 (4l0 + √ l0l+ 8l20 + √ l0l). notice that h1 ≤ 1 =⇒ h2 ≤ 1 and h3 ≤ 1 (4.5) but not necessarily vice versa unless if l0 = l = l1. moreover, h3 h1 → 0 as l0 l1 → 0. (4.6) h3 h1 → 0 as l0 l → 0. (4.7) in view of (4.5)-(4.7) and the lemma 3.3 the results using (4.4) improve the ones by theorem 4.1 infinitely many times. however, s∗ is not given in closed form. but we have s∗ ≤ s̄ , (4.8) https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 6 where s̄ = µ+ l0µ 2 2(1− α)(1− l0µ) , (4.9) where α = 2l l+ √ l2 + 8l0l . (4.10) next, we shall find an upper bound on s∗ which is given in closed form and may be tighter than s̄ . let a = 1 8l (4l0 + √ l0l+ 8l20 + √ l0l) (4.11) then, the condition (4.4) is equivalent to h = 2alµ ≤ 1 (4.12) then, the corresponding theorem in [2] can be written as theorem 4.3. suppose for µ ≥ ‖f ′(x0)−1f (x0)‖ conditions (2.1), (2.2), (4.12) and ū[x0, s ∗] ⊂ d. then, the sequences {xn} generated by newton’s method (1.1) is well defined in u(x0, s ∗), remains in u(x0, s ∗) for each n = 0, 1, 2, ... and is convergent to a solution x∗ ∈ u[x0, s ∗] of the equation f (x) = 0. moreover, the following error estimates hold ‖xn+1 − xn‖ ≤ sn+1 − sn (4.13) and ‖x∗ − xn‖ ≤ s∗ − sn. (4.14) additionally, if for some b ≥ s∗ l0(s ∗ + b) < 1 (4.15) then, the solution x∗ is unique in the region d ∩ u[x0, b]. proof. simply notice that (4.12) is equivalent to (4.4) used in [2]. � remark 4.4. by the definition of a it follows that 0 < a ≤ 1 if l0 ≤ l (4.16) and a ≥ 1 if l ≤ l0. (4.17) define the function f by f (t) = alt2 2 − t + µ (4.18) https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 7 and the sequence {s̄n} for s̄0 = 0, s̄1 = µ, s̄2 = s̄1 + l0(s̄1 − s̄0)2 2(1− l0s̄1) , (4.19) s̄n+1 = s̄n+1 − f (s̄n+1) f ′(s̄n+1) , n = 1, 2, ... (4.20) proposition 4.5. suppose that the conditions of theorem 4.3 hold. then, we have that smallest solution denoted by s̄∗ of the equation f (t) = 0, i.e. s̄∗ = 1− √ 1− 2alµ al (4.21) is an upper bound in closed form of the sequence {s̄n} and 0 ≤ sn ≤ s̄n, (4.22) 0 ≤ sn+1 − sn ≤ s̄n+1 − sn (4.23) and s∗ ≤ s̄∗. (4.24) proof. indeed, this is clear under (4.16), whereas if (4.17) holds the, we have from ‖m−1(f (xn+1 − f (xn)− f ′(xn)(xn+1 − xn)‖ ≤ l̃‖xn+1 − xn‖2 2 l̃ 2 (s̄n+1 − s̄n)2 = f (s̄n+1 a , where l̃ = { l0, n = 0 l, n = 1, 2, ... , ‖f ′(xn+1)−1m‖ ≤ 1 1− l0‖xn+1 − x0‖ ≤ 1 1− l0s̄n+1so ‖xn+1 − xn+1‖ ≤ ‖f ′(x−1n+1m‖‖m −1f (xn+1)‖ ≤ f (s̄n+1) a(1− l0s̄n+1) ≤ − f (s̄n+1) f ′(s̄n+1 , since 1 a(1− l0s̄n+1) ≤ 1 1− als̄n+1 = − 1 f ′(s̄n+1) . � 5. a numerical example the convergence conditions, majorizing sequences and limit points are compared with each other. example 5.1. let b1 = b2 = r, d = u(x0, 1 − p), p ∈ (0, 1) and x0 = 1. define the function ψ : d → r by https://doi.org/10.28924/ada/ma.5.11 eur. j. math. anal. 10.28924/ada/ma.5.11 8 ψ(t) = x3 − p (5.1) then, for λ = 1, µ = 1 3(1− p) and β = 1. moreover, the definitions(2.1)-(2.3) hold if l0 = 3− p, l = 2(1 + 1 3−p ) and l1 = 2(2− p). notice that l0 < l1, l < l1 for each p ∈ (0, 1). we also have that l ≤ l0 if p ∈ (0, 2− √ 3] and l0 ≤ l if p ∈ [2− √ 3, 1). let us restrict p ∈ (0, 12) then, the newton-kantorovich condition (3.4) [3, 6] does not hold, since (3.4) is not satisfied for any p ∈ (0, 12). however, our condition (4.12) hold provided that p ∈ (.46, 12). thus, the old results [6] cannot guarantee the convergence of newton’s method for any p ∈ (0, 12). however, newton’s method converges to x∗ = 3 √ p if we say p = 0.48. in order to compare sequences and limit points. let p = 0.7. then, both (3.4) and (4.12) hold. thus, the old results [3, 5–7] cannot guarantee the convergence of newton’s method for any p ∈ (0, 12). however, newton’s method converges to x∗ = 3 √ p. if say p = 0.48. in order to compare sequences and limit points. let p = 0.7. then, both (3.4) and (4.12) hold. then, we have s̄ = 0.3965, v̄ = 0.6511, b = 0.3194. therefore, the new error bounds and limit points are tighter than the ones given before [3, 5–7] and under weaker sufficient semi-local convergence criteria. table 1. comparison between majorizing sequences and their limit points n vn vn+1 − vn sn sn+1 − sn s̄n s̄n+1 − s̄n 0 0 0 0 0 0 0 1 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 2 0.1176 0.0176 0.1149 0.4350e-03 0.1149 0.7619e-03 3 0.1181 = v∗ 0.0006 0.1154 = s∗ 0.0004e-03 0.1157 = s̄∗ 0.0009e-03 references [1] i.k. argyros, the theory and applications of iteration methods, second edition, crc press, boca raton, 2022.[2] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s method, j. complex. 28 (2012), 364–387. https://doi.org/10.1016/j.jco.2011.12.003. https://doi.org/10.28924/ada/ma.5.11 https://doi.org/10.1016/j.jco.2011.12.003 eur. j. math. anal. 10.28924/ada/ma.5.11 9 [3] p. deuflhard, g. heindl, affine invariant convergence theorems for newton’s method and extensions to relatedmethods, siam j. numer. anal. 16 (1979), 1–10. https://doi.org/10.1137/0716001.[4] p. deuflhard, newton methods for nonlinear problems: affine invariance and adaptive algorithms, springer, berlin,2004. https://doi.org/10.1007/978-3-642-23899-4.[5] w.b. gragg, r.a. tapia, optimal error bounds for the newton–kantorovich theorem, siam j. numer. anal. 11(1974), 10–13. https://doi.org/10.1137/0711002.[6] l.v. kantorovich, g.p. akilov, functional analysis, pergamon press, oxford, 1982.[7] a.m. ostrowski, solutions of equations in euclidean and banach spaces, academic press, new york, 1973. https://doi.org/10.28924/ada/ma.5.11 https://doi.org/10.1137/0716001 https://doi.org/10.1007/978-3-642-23899-4 https://doi.org/10.1137/0711002 1. introduction 2. lipschitz conditions 3. convergence of majorizing sequences. 4. convergence of newton's method 5. a numerical example references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 4doi: 10.28924/ada/ma.4.4 duals of continuous frames in hilbert c∗-modules mohamed rossafi1,∗, khadija mabrouk2, m’hamed ghiati2, mohammed mouniane2 1department of mathematics faculty of sciences, dhar el mahraz university sidi mohamed ben abdellah, fez, morocco rossafimohamed@gmail.com 2department of mathematics, faculty of sciences, university of ibn tofail, kenitra, morocco khadija.mabrouk@uit.ac.ma, mhamed.ghiati@uit.ac.ma, mouniane.mohammed@uit.ac.ma ∗correspondence: rossafimohamed@gmail.com abstract. the concept of frame is an exciting, dynamic, and fast-paced subject with applications innumerous fields of mathematics and engineering. the purpose of this paper is to introduce equiv-alent ∗-continuous frames and to present ordinary duals of constructed ∗-continuous frames by anadjointable and invertible operator. also, we establish some properties. 1. introduction frames in hilbert spaces have been introduced by duffin and schaeffer [3] in 1952 to studysome deep problems in nonharmonic fourier series. after the fundamental paper, by daubechies,grossman and meyer [2], frame theory began to be widely used, particularly in the more specializedcontext of wavelet frame and gabor frame [4]. frames have been used in signal processing, imageprocessing, data compression and sampling theory. for more about frames, see [5, 7–11].in this paper, we introduce the notions of continuous frame on a hilbert c∗-module over aunital c∗-algebra which is a generalization of discrete frame, the ∗-continuous frame, which are ageneralization of ∗-frame in hilbert c∗-modules and we establish some new results.the paper is organized as follows. we continue this introductory section and briefly recall thedefinitions and basic properties of hilbert c∗-modules. in section 2, the generalized duals fora given ∗-continuous frame will be considered. also, we study their properties and characterizeall operator dual ∗-continuous frames associated with the given ∗-continuous frame in hilbert c∗-modules. in section 3, we extend this notion for sequences (continuous frames) in hilbert c∗-modules. also, some properties of them will be studied. in section 4, a ∗-continuous frame isconstructed by an orthogonal projection. received: 21 jan 2024.2020 mathematics subject classification. 42c15, 41a58. key words and phrases. continuous frame; ∗-continuous frame; c∗-algebra; hilbert c∗-module.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 2 definition 1.1. [6] suppose that a is a c∗-algebra. a linear space h which is also an algebraicleft a-module together with an a-inner product 〈·, ·〉 : h×h −→ a and possesses the followingproperties is called a pre-hilbert c∗-module:(1) 〈f , f 〉 ≥ 0, for any f ∈ h;(2) 〈f , f 〉 = 0 if and only if f = 0;(3) 〈f , g〉 = 〈g, f 〉∗, for any f , g ∈ h;(4) 〈λf , h〉 = λ〈f , h〉, for any λ ∈ c and f , h ∈ h;(5) 〈af + bg, h〉 = a〈f , h〉+ b〈g, h〉, for any a, b ∈ a and f , g, h ∈ h. for x ∈ h, we define ‖x‖ = ||〈x, x〉|| 1 2 . if h is complete with ||.||, it is called a hilbert a-module or a hilbert c∗-module over a. for every a in c∗-algebra a, we have |a| = (a∗a) 1 2 andthe a-valued norm on h is defined by |x | = 〈x, x〉 1 2 for x ∈ h. let h and k be two hilbert a-modules. a map t : h → k is said to be adjointable if there exists a map t ∗ : k → h suchthat 〈tx, y〉a = 〈x, t ∗y〉a for all x ∈ h and y ∈ k. lemma 1.2. [12] let (ω,µ) be a measure space, x and y be two banach spaces, λ : x → y be a bounded linear operator and f : ω → y be a measurable function. then λ( ∫ ω f dµ) = ∫ ω (λf )dµ. lemma 1.3. [1] let h and k be two hilbert a-modules and t ∈ end∗(h,k).(i) if t is injective and t has a closed range, then the adjointable map t ∗t is invertible and ‖(t ∗t )−1‖−1 ≤ t ∗t ≤ ‖t‖2. (ii) if t is surjective, then the adjointable map tt ∗ is invertible and ‖(tt ∗)−1‖−1 ≤ tt ∗ ≤ ‖t‖2. the following definition was introduced in [11]. definition 1.4. let h be a hilbert a-module and (ω, µ) be a measure space. a map f : ω→ his called a ∗-continuous frame with respect to (ω, µ) if1. for all f ∈ h, w → 〈f , fw 〉 is a measurable function on ω,2. there exist two strictly nonzero elements a,b > 0 in a such that a〈f , f 〉a∗ ≤ ∫ ω 〈f , fw 〉〈fw , f 〉dµ(w) ≤ b〈f , f 〉b∗,∀f ∈ h. (1.1) the elements a and b are called ∗-continuous frame bounds. if a = b, we call this ∗-continuousframe a tight ∗-continuous frame, and if a = b = 1, it is called a parseval ∗-continuous frame. ifonly the right-hand inequality of (1.1) is satisfied, we call f : ω→ h a ∗-continuous bessel mapwith bessel bound b. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 3let x be a banach space, (ω, µ) be a measure space and f : ω→ x be a measurable function.integral of the banach-valued function f has been defined by bochner and others. most propertiesof this integral are similar to those of the integral of real-valued functions. since every c∗-algebraand hilbert c∗-module is a banach space, we can use this integral and its properties.let (ω, µ) be a measure space. we define l2(ω,a) = { ϕ : ω→ a : ∥∥∥∥∫ ω ϕ(ω)ϕ(ω)∗dµ(ω) ∥∥∥∥ <∞ } . for any ϕ,ψ ∈ l2(ω,a), if the a-valued inner product is defined by 〈ϕ,ψ〉 = ∫ ω ϕ(ω)ψ(ω)∗dµ(w), the norm is defined by ‖ϕ‖ = ‖〈ϕ,ϕ〉‖ 1 2 , then l2(ω,a) is a hilbert c∗-module.the frame transform or pre-frame operator t : h −→ l2(ω,a) is defined by t (f ) = {〈f , fw 〉}w∈ωand it is an injective and closed range adjointable a-module map and ‖t‖ ≤ ‖b‖. the adjointoperator t ∗ is surjective and it is given by t ∗ (ew ) = fw for w ∈ ω, where {ew}w∈ω is thestandard basis for l2(ω,a). definition 1.5. let f be a continuous frame for h with respect to (ω, µ). we define the frameoperator s : h → h by sx = t ∗ftf x = ∫ ω〈x, fw 〉fwdµ(w),∀x ∈ h, that is positive, invertibleand adjointable and the inequality ∥∥a−1 ∥∥−2 ≤ ‖s‖ ≤ ‖b‖2 holds, and the reconstruction formula f = ∫ ω 〈 f , s−1fw 〉 fwdµ(w) holds for all f ∈ h. 2. ∗-continuous operator duals definition 2.1. let {fw}w∈ω and {gw}w∈ω be two ∗-continuous frames for h. if there exists aninvertible adjointable a-module map on h such that x = ∫ ω 〈γx, gw 〉fwdµ(ω), ∀x ∈ h, (2.1) then {gw}w∈ω is called a ∗-continuous operator dual of {fw}w∈ω. remark 2.2. every ∗-continuous frame {fw}w∈ω with continuous frame operator s is a ∗-continuousoperator dual for itself. to see this, set γ := s−1 and the reconstruction formula concludes it. remark 2.3. every dual ∗-continuous frame {gw}w∈ω of ∗-continuous frame {fw}w∈ω is a ∗-continuous operator dual when γ = i, i is the identity operator on h. remark 2.4. let g = {gw}w∈ω be an operator dual of a ∗-continuous frame f = {fw}w∈ω in h.then for some invertible adjointable map γ ∈ b∗(h) x = ∫ ω 〈γx, gw 〉fwdµ(ω), ∀x ∈ h. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 4the equality shows that i = ( t ∗ftg ) γ, where i is the identity map on h, and tf and tg arepre-frame operators of f and g, respectively. therefore, the operator γ is unique and γ−1 = t ∗ftg . by remark 2.4, we say that {gw}w∈ω is an operator dual for {fw}w∈ω with the correspondingoperator γ. moreover, we mention that the operator duality relation of x-frames is symmetric. it isconsidered in the next remark. remark 2.5. if g = {gw}w∈ω is an operator dual of a given ∗-continuous frame f = {fw}w∈ωwith the corresponding operator γ, then {fw}w∈ω is an operator dual for {gw}w∈ω with the cor-responding operator γ∗. in order to see this, assume that tf and tg are pre-frame operators of fand g, respectively. by the definition of operator duals, we have i = ∫ ω 〈γx, gw 〉fwdµ(ω) = (t ∗ftg) γ. since γ is invertible, γ−1 = t ∗ftg and i = γ (t ∗ftg) = ( t ∗gtf ) γ∗ = ∫ ω 〈γ∗f,fw 〉 gwdµ(ω). the following lemma is obtained by using the last remark and some properties of pre-frameoperators. lemma 2.6. let f = {fw}w∈ω and g = {gw}w∈ω be ∗-bessel sequences for h with the pre-frame operators tf and tg , respectively. assume that γ is an invertible and adjointable a-module map on h. then for x ∈ h, the following statements are equivalent: (i) x = ∫ ω 〈γx, gw 〉fwdµ(ω). (i i) x = ∫ ω 〈γ ∗x, fw 〉 gwdµ(ω). in case that one of the above equalities is satisfied, {fw}w∈ω and {gw}w∈ω are operator dual ∗-frames. moreover, if b is an upper bound for {fw}w∈ω and s is frame operator of {fw}w∈ω, then b ∥∥s−1 ∥∥− 1 2 ‖tf‖−1 ‖γ‖−1 is a lower bound for {gw}w∈ω. proof. the equivalency of the two conditions is given from remark 2.5.now, let b be a ∗-bessel bound for {fw}w∈ω and (i) holds. by the definition of ∗-besselsequence {fw}w∈ω and t ∗ftgγ = idh, we can write, for x ∈ h, 〈tfx, tfx〉 ≤ b〈x, x〉b∗ (2.2) = b 〈t ∗ftgγx, t ∗ftgγx〉b∗ ≤ b ‖tf‖2 〈tgγx, tgγx〉b∗. using lemma 1.3, we have∥∥∥(t ∗ftf)−1 ∥∥∥−1 〈x, x〉 ≤ 〈tfx, tfx〉 , ∀f ∈ h. (2.3) https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 5it follows from lemma 1.3, (2.2), and (2.3) that for x ∈ h,∥∥s−1 ∥∥−1 ‖γγ∗‖−1 〈f , f 〉 ≤ ∥∥s−1 ∥∥−1 〈 γ−1f ,γ−1f 〉 ≤ b ‖tf‖2 〈tgf tgf 〉b∗,( b−1 ∥∥s−1 ∥∥− 1 2 ‖tf‖−1 ‖γ‖−1 ) 〈f , f 〉 ( b−1 ∥∥s−1 ∥∥− 1 2 ‖tf‖−1 ‖γ‖−1 )∗ ≤ 〈tgf , tgf 〉 . therefore, b ∥∥s−1 ∥∥∣∣− 1 2 ‖γ‖−1 ‖tf‖−1 is a lower ∗-frame bound for {gj}j∈j and {gj}j∈j is a ∗-frame.similarly, {fj}j∈j is also a ∗-frame. � proposition 2.7. let ( {gw}w∈ω ,γ ) be an operator dual of a ∗-continuous frame {fw}w∈ω.1. for a strictly nonzero element α in the center of a, the pair ( {αgw}w∈ω , α −1γ ) is an operator dual for {fw}w∈ω.2. if υ is an invertible and adjointable operator on h, then ( {υgw}w∈ω , (υ)−1γ ) is an operator dual for {fw}w∈ω.3. the sequence {gw}w∈ω is a dual of {γ∗fw}w∈ω.4. assume that ( {hw}w∈ω ,λ ) is another operator dual of {fw}w∈ω . then( {gw + hw}w∈j , ( γ−1 + λ−1 )−1 ) is an operator dual for {fw}w∈ω. proposition 2.8. let ( {gw}w∈ω ,γ ) be an operator dual of {fw}w∈ω for h. if f is an element of h such that 〈x, x〉 is a strictly nonzero element in the center of a, then {〈 gw , (〈x, x〉)−1γx 〉} w∈ω is a dual of {〈fw , x〉}w∈ω. proof. suppose that a ∈ a. then∫ ω 〈 a, 〈 gw , 〈x, x〉−1γx 〉〉 〈fw , x〉 dµ(ω) = ∫ ω a 〈 〈x, x〉−1γx, gw 〉 〈fw , x〉 dµ(ω) = a〈x, x〉−1 〈∫ ω 〈γx, gw 〉fwdµ(ω), x 〉 = a〈x, x〉−1〈x, x〉 = a.this completes the proof. � proposition 2.9. let {fw}w∈ω be a ∗-continuous frame for h with frame operator s. if θ is an adjointable and invertible operator on h, then ( {θfw}w∈ω , ( θ−1 )∗ s−1 ) is an operator dual for {fw}w∈ω. proof. let x ∈ h. then∫ ω 〈( s−1θ−1 ) x, fw 〉 θfwdµ(ω) = θ (∫ ω 〈( s−1θ−1 ) xdµ(ω), fw 〉 fw ) = θ ( θ−1x ) = x. this completes the proof. � https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 6 proposition 2.10. let {fw}w∈ω be a ∗-continuous frame and θ be an adjointable and invertible operator on h. then the sets of operator duals of {fw}w∈ω and {θfw}w∈ω are in one to one correspondence. proof. first, suppose that ({gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω. for x ∈ h, we obtain x = ∫ ω 〈γ∗x, fw 〉 gwdµ(ω) = ∫ ω 〈 θ∗ ( θ−1 )∗ γ∗x, fw 〉 gwdµ(ω) = ∫ ω 〈( θ−1 )∗ γ∗x, θfw 〉 gwdµ(ω). so ({gw}w∈ω ,γθ−1 ) is an operator dual for {θfw}w∈ω.now, if ({gw}w∈ω ,γ ) is an operator dual of {θfw}w∈ω, then ({gw}w∈ω ,γ∗θ ) is an operatordual of {fw}w∈ω, since x = ∫ ω 〈γx, θfw 〉 gwdµ(ω) = ∫ ω 〈θ∗γx, fw 〉 gwdµ(ω), ∀x ∈ h. this completes the proof. � proposition 2.11. let f = {fw}w∈ω be a ∗-continuous frame for h with pre-frame operator tf and frame operator s. then the set of all the operator duals of {fw}w∈ω is precisely the following {gw}w∈ω = { γfw + ϕew − ∫ ω 〈 s−1fw , fi 〉 ϕewdµ(ω) } w∈ω , where {ew}w∈ω is the standard orthonormal basis for l2(ω,a), ϕ ∈ b∗ ( h, l2(ω,a) ) , and γ is an invertible adjointable operator on h. proof. assume that {gw}w∈ω is a sequence as above. then its pre-frame operator is tg = tfγ + ϕ− tfs−1t ∗fϕ and so (sγ)−1 (t ∗ftg) = (sγ)−1 ( t ∗ftfγ + t ∗fϕ− t ∗ftfs−1t ∗fϕ ) = (sγ)−1 ( t ∗ftfs −1sγ + t ∗fϕ− t ∗ftfs−1t ∗fϕ ) = (sγ)−1(sγ) = i. by a similar relation with the given equality in remark 2.5, we can conclude that {gw}w∈ω isan operator dual for {fw}w∈ω with the corresponding operator (sγ)−1. � theorem 2.12. let ( {gw}w∈ω ,γ ) be an operator dual of ∗-continuous frame {fw}w∈ω for h. then there exist a hilbert a-module k ⊇ h and a riesz basis {uw}w∈ω of k which has a unique dual {vw}w∈ω and satisfies (pu)uw = fw and (pv0)vw = gw for all w ∈ ω, where p is the projection from k onto h. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 7 proof. assume that tf , tg and sf , sg are pre-frame operators and frame operators of {fw}w∈ωand {gw}w∈ω, respectively. also, the orthogonal projections onto the range of tf , r (tf), and therange of tg , r (tg), are pf and pg , respectively. now, for x ∈ h,〈 tgx, tgs −1 g gw 〉 = 〈 t ∗gtgx, s −1 g gw 〉 = 〈 s−1 g sgx, gw 〉 = 〈x, gw 〉 = 〈tgx, ew 〉 = 〈tgx, pgew 〉′and so pgew = tgs −1 g gw , ∀w ∈ ω, (2.4) where {ew}w∈ω is the standard orthonormal basis of l2(ω,a). by (2.4), for x ∈ h, we give pgtf (γ∗x) = pg (∫ ω 〈γ∗x, fw 〉 ew ) = ∫ ω 〈γ∗x, fw 〉pgew = ∫ ω 〈γ∗f,fw 〉tgs−1 g gw = tgs −1 g (∫ ω 〈γ∗x, fw 〉 gw ) = tgs −1 g x. set k = h⊕ p⊥g l2(a), uw = fw ⊕ p⊥g ew , ∀w ∈ ω. if tu is a pre-frame operator of the sequence {uw}w∈ω, then tu(x ⊕ v) = tfx + v and ‖tu(x ⊕ v)‖ = ‖tf f + w‖ ≤ b(‖x‖+ ‖v‖) = b‖x ⊕ v‖, ∀x ⊕ v ∈ k for some b > 0 and so {uw}w∈ω is a bessel sequence. we show that tu has a closed range.suppose {ηn}n∈n ⊆ r (tu) such that ηn n→∞−→ η. since γ is invertible and adjointable, there exists γ∗fn ⊕ vn ∈ h ⊕ p⊥g (l2(ω,a);tu (γ∗fn ⊕ vn) = ηn. on the other hand, tu (γ∗fn ⊕ vn) = tf (γ∗fn) + vn = ηn n→∞−→ η and remark 2.4 gives fn = t ∗gtfγ∗fn = t ∗g (tfγ∗fn + vn) n→∞−→ t ∗gη. r (tu) is closed, since r (tf) is closed. also, t ∗u has a closed range. this step will obtain theinjectivity of t ∗u . if t ∗u (∫ω awew ) = 0, then 0 = ∫ ω aw ( fw ⊕ p⊥g ew ) = ∫ ω awfw ⊕ p⊥g (∫ ω awew ) . it concludes that (i) ∫ ω awfw = 0 and (i i) p⊥g (∫ ω awew ) = 0. from (i i), we have∫ ω awew ∈ r (tg) =⇒ ∃h ∈ h;tgh = ∫ ω awew https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 8and on the other hand, tgh = ∫ ω 〈h, gw 〉 ew =⇒ aw = 〈h, gw 〉 , ∀w ∈ ω. from (i), we have 0 = ∫ ω awfw = ∫ ω 〈h, gw 〉fw = ∫ ω 〈 γγ−1h, gw 〉 fw = γ−1h. since γ is injective, h = 0 and aw = 0 for w ∈ ω. so ∫ω awew = 0, and t ∗u is injective. theoperator t ∗utu is an invertible selfadjoint operator such that it has an upper bound and a lowerbound by lemma 1.3, and also {uw}w∈ω is a frame for k with frame operator su = t ∗utu . � 3. equivalent ∗-continuous frames definition 3.1. two sequences {fw}w∈ω and {gw}w∈ω in h are said to be equivalent sequencesif there exists an adjointable and invertible operator λ on h such that λfw = gw , for w ∈ ω. theorem 3.2. let {fw}w∈ω be a ∗-continuous frame for h and ξ be an adjointable and invertible operator on h. then every dual of the ∗-continuous frame {ξfw}w∈ω is equivalent to a dual of {fw}w∈ω, and the converse of the relation is valid. proof. first, suppose that {gw}w∈ω is a dual of {fw}w∈ω. then for x ∈ h, we obtain x = ξ ( ξ−1 ) x = ξ (∫ ω 〈 ξ−1x, gw 〉 fwdµ(ω) ) = ∫ ω 〈 x, ( ξ−1 )∗ gw 〉 ξfwdµ(ω). so {(ξ−1 )∗ gw } is a dual for {ξfw}w∈ω, and it is also equivalent to {gw}w∈ω.now, suppose that {hw}w∈ω is a dual frame for {ξfw}w∈ω. set gw = ξ∗hw , for w ∈ ω. thenfor x ∈ h, ∫ ω 〈x, gw 〉fwdµ(ω) = ∫ ω 〈x, ξ∗hw 〉 ξ−1ξfwdµ(ω) = ξ−1 (∫ ω 〈ξf , hw 〉 ξfwdµ(ω) ) = ξ−1ξx = x.thus {gw}w∈ω is a dual for {fw}w∈ω and hw = ( ξ−1 )∗ gw . � theorem 3.3. if {fw}w∈ω and {gw}w∈ω are ∗-continuous frames with the continuous frame operators sf and sg , respectively, then there exists a ∗-continuous frame that is equivalent to {gw}w∈ω and its frame operator is sf . https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 9 proof. for the adjointable and invertible operator ξ = s 1 2 fs − 1 2 g , the sequence {ξgw}w∈ω is a ∗-continuous frame with the frame operator sξ = ξsgξ ∗. so sξ = ξsgξ ∗ = ( s 1 2 fs − 1 2 g ) sg ( s 1 2 fs − 1 2 g )∗ = sf .this completes the proof. � theorem 3.4. let {fw}w∈ω and {gw}w∈ω be ∗-continuous frames for h. then the following statements are valid.(1) {gw}w∈ω is equivalent to a dual frame of {fw}w∈ω if and only if there exists an adjointable and invertible operator ξ on h such that ξf x = ∫ ω 〈f , fw 〉 gwdµ(ω), ∀x ∈ h. (2) {gw}w∈ω is equivalent to a dual frame of {fw}w∈ω if and only if there exists an adjointable and invertible operator γ such that ( {gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω. proof. first, assume that {gw}w∈ω is equivalent to a dual frame of {fw}w∈ω. then there existsan adjointable and invertible operator γ on h such that {γgw}w∈ω is a dual for {fw}w∈ω. now,for x ∈ h, = ∫ ω 〈x, fw 〉γgwdµ(ω). set ξ = γ−1. then it concludes ξx = γ−1x = γ−1 (∫ ω 〈x, fw 〉γgwdµ(ω) ) = ∫ ω 〈x, fw 〉 gwdµ(ω). in the second step, the adjointable and invertible operator ξ on h satisfies the following property ξx = ∫ ω 〈x, fw 〉 gwdµ(ω), ∀x ∈ h. since ξ is invertible, x = ∫ ω 〈x, fw 〉 ξ−1gwdµ(ω), ∀x ∈ h. it shows that {ξ−1gw } w∈ω is a dual for {fw}w∈ω, and is equivalent to {gw}w∈ω.for the proof of “if" part, assume that there exists a dual frame {hw}w∈ω for {fw}w∈ω suchthat {hw}w∈ω and {gw}w∈ω are equivalent. then there is an adjointable and invertible operator https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 10 λ : h → h such that λgw = hw for all w ∈ ω. by theorem 3.2, the sequence {gw}w∈ω is a ∗-continuous frame.on the other hand, for x ∈ h, x = ∫ ω 〈x, hw 〉fwdµ(ω) = ∫ ω 〈x,λgw 〉fwdµ(ω) = ∫ ω 〈λ∗x, gw 〉fwdµ(ω) and so x = ∫ ω 〈λ∗x, gw 〉fwdµ(ω).this shows that {(gw ,λ∗)} is an operator dual for {fw}w∈ω. the converse part is clear by thelast equalities. � theorem 3.5. let {fw}w∈ω and {gw}w∈ω be ∗-frames for h. then {gw}w∈ω is equivalent to an operator dual frame of {fw}w∈ω if and only if there exists an adjointable and invertible operator ξ on h such that ξx = ∫ ω 〈x, gw 〉fwdµ(ω), ∀x ∈ h. proof. suppose that {gw}w∈ω is equivalent to {hw}w∈ω, where ({hw}w∈ω ,γ ) is an operator dualfor {fw}w∈ω. then there exists an adjointable and invertible operator θ on h such that θgw = hwfor all w ∈ ω and for x ∈ h, x = ∫ ω 〈γx, hw 〉fw = ∫ ω 〈γx, θgw 〉fwdµ(ω) = ∫ ω 〈θ∗γx, gw 〉fwdµ(ω). set ξ = (θ∗γ)−1. then the result is obtained.for the converse, let ξ be an adjointable and invertible operator on h that ξx = ∫ ω 〈x, gw 〉fwdµ(ω). then x = ∫ ω 〈 ξ−1x, gw 〉 fwdµ(ω), ∀x ∈ h and ({gw}w∈ω , ξ −1 ) is an operator dual of {fw}w∈ω and {gw}w∈ω is equivalent to itself. � moreover, some equivalence frames have the same grammian matrices. these frames are intro-duced in the following proposition. proposition 3.6. let {fw}w∈ω and {gw}w∈ω be equivalent parseval frames for h and let gf and gg be grammian matrices of {fw}w∈ω and {gw}w∈ω, respectively. then gf = gg . https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 11 proof. since two frames {fw}w∈ω and {gw}w∈ω are equivalent, there exists an adjointable andinvertible operator ξ : h −→ h by ξfw = gw for w ∈ ω. since their frame operators are theidentity operator on h, by theorem 3.2, ξξ∗ = ξidξ∗ = id. so ξ is a unitary operator and then for i , w ∈ ω, 〈gi , gw 〉 = 〈ξfi , ξfw 〉 = 〈fi , fw 〉 , which shows that gf = [〈fi , fw 〉]w∈ω = [〈gi , gw 〉]w∈ω = gg . � 4. constructed ∗-continuous frames and some properties theorem 4.1. let {fw}w∈ω be a ∗-continuous frame for h and ξ be an adjointable and invertible operator on h. then the set ( {gw}w∈ω ,γξ−1 ) is all of operator duals of {ξfw}w∈ω, where( {gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω. proof. let ({gw}w∈ω ,γ ) be an operator dual of {fw}w∈ω. then for x ∈ h,∫ ω 〈 γξ−1x, gw 〉 ξxwdµ(ω) = ξ (∫ ω 〈 γξ−1f , gw 〉 fwdµ(ω) ) = ξ ( ξ−1 ) x = x. this shows that ({gj}j∈j ,γξ−1 ) is an operator dual of {ξfj}j∈j .now, if ({gj}j∈j ,γ ) is an operator dual for {ξfj}j∈j , then it is enough to set γ := γξ in thelast equalities which follows that ({gj}j∈j ,γ ) is an operator dual for {fj}j∈j . � an orthogonal projection will obtain a ∗-frame, and relation will also be given for this projection.to see this, we must show that the inverse of the frame operator is unique in the reconstructionformula. so, firstly this fact will be considered. theorem 4.2. if {fw}w∈ω is a ∗-continuous frame for h, then there exists a unique adjointable operator λ on h such that x = ∫ ω 〈x,λfw 〉fwdµ(ω), ∀x ∈ h. proof. by the reconstruction formula, there exists λ = s−1. for the uniqueness of s−1 with thisproperty, we know that {s− 1 2fw } w∈ω is a continuous parseval frame for h. set gw = s− 1 2fw .then fw = s 1 2 gw . now, suppose that λ is an adjointable operator such that x = ∫ ω 〈x,λfw 〉fwdµ(ω), ∀x ∈ h. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 12then we have x = ∫ ω 〈x,λfw 〉fwdµ(ω) = ∫ ω 〈 x,λs 1 2 gw 〉 s 1 2 gwdµ(ω) = s 1 2 (∫ ω 〈 x,λs 1 2 gw 〉 gwdµ(ω) ) = s 1 2 (∫ ω 〈 s 1 2 λ∗x, gw 〉 gw ) = s 1 2 ( s 1 2 λ∗x ) = sλ∗x, ∀x ∈ h.this concludes that sλ∗ = id and then λ∗ = s−1. more precisely, λ is self-adjoint, positive andinvertible. � now, a ∗-continuous frame is constructed by an orthogonal projection. proposition 4.3. let {fw}w∈ω be a ∗-continuous frame for h with the frame operator s and ∗continuous frame bounds a and b. also, suppose that p is an orthogonal projection on h. then {pfw}w∈ω is a ∗-continuous frame for rp with ∗-continuous frame bounds a and b. moreover, if ( {gw}w∈ω ,γ ) is an operator dual of {fw}w∈ω, then {pγ∗gw}w∈ω is a dual ∗-continuous frame for {p fw}w∈ω. proof. for x ∈ rp ,∫ ω 〈x, pfw 〉 〈pfw , x〉 dµ(ω) = ∫ ω 〈px, fw 〉 〈fw , p x〉 dµ(ω) = ∫ ω 〈x, fw 〉 〈fw , x〉 dµ(ω) and by the definition of ∗-continuous frame {fw}w∈ω, we have a〈x, x〉a∗ ≤ ∫ ω 〈x, pfw 〉 〈pfw , x〉 dµ(ω) ≤ b〈x, x〉b∗. now, if ({gw}w∈ω ,γ ) is an operator dual of {fw}w∈ω, then for x ∈ rp , x = px = p (∫ ω 〈γpx, gw 〉fwdµ(ω) ) = ∫ ω 〈x, pγ∗gw 〉pfwdµ(ω). if {gw}w∈ω is also a dual of {fw}w∈ω, then {pgw}w∈ω is a dual of {pfw}w∈ω. so the result isclear by γ = idh. � by the last theorem, a necessary and sufficient condition is found for commutating a projectionwith the inverse of the frame operator of a given ∗-frame. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 13 theorem 4.4. let {fw}w∈ω be a ∗-continuous frame for h with the frame operator s. suppose that p is an orthogonal projection on h. then ps−1fw = s−1 p pfw , for all w ∈ ω if and only if ps−1 = s−1p , where sp is the continuous frame operator of the ∗-continuous frame {pfw}w∈ω. proof. first, assume that ps−1fw = s−1 p pfw , for all w ∈ ω. now, let x ∈ h. then we have s−1 p px = s−1 p p (∫ ω 〈 x, s−1fw 〉 fwdµ(ω) ) = ∫ ω 〈 x, s−1fw 〉 s−1 p pfwdµ(ω) = ∫ ω 〈 x, s−1fw 〉 ps−1fwdµ(ω) = ps−1 (∫ ω 〈 x, s−1fw 〉 fwdµ(ω) ) = ps−1x. therefore, ps−1x = s−1 p px , for all x ∈ h, and so s−1 p p = ps−1p ⇒ ps−1p = ps−1. thus ps−1p = ( ps−1p )∗ = ( ps−1 )∗ = s−1p.for the proof of converse, suppose that ps−1 = s−1p . let x ∈ rp . then we have x = px = p (∫ ω 〈 x, s−1fw 〉 fwdµ(ω) ) = ∫ ω 〈 x, s−1fw 〉 pfwdµ(ω) = ∫ ω 〈 px, s−1fw 〉 pfwdµ(ω) = ∫ ω 〈 x, ps−1fw 〉 pfwdµ(ω) = ∫ ω 〈 x, s−1pfw 〉 pfwdµ(ω).by theorem 4.2 and the assumption, for w ∈ ω, s−1pfw = s−1 p fw = s−1 p pfw = ps−1fw and the proof is complete. � theorem 4.5. let {fw}w∈ω be a continuous parseval frame of h with pre-frame operator θf and let {gw}w∈ω be a ∗-continuous frame with the pre-frame operator θg . then ( {gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω if and only if pθf θgγ = θf , where pθf is the orthogonal projection on the range of θf . https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 14 proof. suppose that ({gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω. since {fw}w∈ω is a contin-uous parseval frame, the pre-frame operator θf is an isometry 〈θfx, θfx〉 = ∫ ω 〈x, fw 〉 〈fw , x〉 dµ(ω) = 〈x, x〉, ∀x ∈ h. then for x ∈ h, 〈 pθf θgγx, θfx 〉 = 〈 θgγx, pθf θfx 〉 = 〈θgγf , θfx〉 = 〈∫ ω 〈γx, gw 〉fwdµ(ω), x 〉 = 〈x, x〉 = 〈θfx, θfx〉 .thus pθf θgγ = θf .conversely, since θf is an isometry, by a similar method, we have 〈x, g〉 = 〈θfx, θfg〉 = 〈 pθf θgγx, θfg 〉 = 〈∫ ω 〈γx, gw 〉fwdµ(ω), g 〉 , ∀x ∈ h, and so x = ∫ ω 〈γx, gw 〉fwdµ(ω), ∀x ∈ h. thus ({gw}w∈ω ,γ ) is a dual for {fw}w∈ω. � corollary 4.6. let {fw}w∈ω and {gw}w∈ω be two ∗-continuous frames for h with pre-frame operators θf and θg , respectively. then ( {gw}w∈ω ,γ ) is an operator dual for {fw}w∈ω if and only if θ∗fpθf θgγ = id . authors’ contributionsthe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript. references [1] a. alijani, m.a. dehghan, g-frames and their duals in hilbert c∗-modules, bull. iran. math. soc. 38 (2012),567–580.[2] i. daubechies, a. grossmann, y. meyer, painless nonorthogonal expansions, j. math. phys. 27 (1986), 1271–1283.[3] r. j. duffin, a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341–366.[4] d. gabor, theory of communications, j. inst. electr. eng. 93 (1946), 429–457.[5] f. d. nhari, r. echarghaoui, m. rossafi, k-g-fusion frames in hilbert c∗-modules, int. j. anal. appl. 19 (2021),836–857.[6] w. paschke, inner product modules over b∗-algebras, trans. am. math. soc. 182 (1973), 443–468.[7] m. rossafi, s. kabbaj, ∗-k-g-frames in hilbert a-modules, j. linear topol. algebra 7 (2018), 63–71. https://doi.org/10.28924/ada/ma.4.4 eur. j. math. anal. 10.28924/ada/ma.4.4 15 [8] m. rossafi, s. kabbaj, ∗-g-frames in tensor products of hilbert c∗-modules, ann. univ. paedagog. crac. stud. math.17 (2018), 17–25.[9] m. rossafi, s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra 8 (2019), 85–95.[10] m. rossafi, s. kabbaj, ∗-k-operator frame for end∗a(h), asian-eur. j. math. 13 (2020), 2050060.[11] m. rossafi, f. d. nhari, c. park, s. kabbaj, continuous g-frames with c∗-valued bounds and their properties,complex anal. oper. theory 16 (2022), 44.[12] k. yosida, functional analysis, vol. 123, grundlehren der mathematischen wissenschaften, springer, berlin andnew york, 1980. https://doi.org/10.28924/ada/ma.4.4 1. introduction 2. -continuous operator duals 3. equivalent *-continuous frames 4. constructed -continuous frames and some properties references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 23doi: 10.28924/ada/ma.4.23 slicing of negative plurisubharmonic currents arising from analytic subsets hedi khedhiri university of monastir, preparatory institute for engineering studies, research laboratory lr18es16, road ibn eljazzar monastir 5019, tunisia correspondence: khediri_h@yahoo.fr abstract. in this paper, first we focus on the slicing of negative plurisubharmonic currents whichare finite sums involving currents arising from analytic subsets with geometric complete intersection.next, we provide significant results on the integrability across analytic subsets, of the coefficients ofsuch currents and of their slices. 1. introduction in this paper, we let ω be a domain in cn such that the unit polydisc ∆n satisfies ∆n b ω and ϕ be a plurisubharmonic function (psh for short), locally bounded on ω. we let n , k , p and n benonzero fixed arbitrary natural numbers such that k ≤ p ≤ n and we consider the n-complex space cn with variables z such that cn = ck × cn−k , z = (z ′, z ′′), z ′ ∈ ck , z ′′ ∈ cn−k . assuming that ϕ depends only on the variable z ′ ∈ ck and the support sϕ of the associatedmong-ampère measure µϕ = (ddcϕ)kis ∆k , then, the slice (or the ϕ-slice) denoted 〈t, π, a〉ϕ of a current t ∈ d ′p,p(ω), at point a ∈ sϕ,is studied and well defined in [10], such that < t, π, a >ϕ (ψ) := lim ε→0 1 µϕ(bk(a, ε)) ∫ bk(a,ε)×cn−k t ∧ (ddc ϕ̃)k ∧ψ. (1.1) the definition (1.1) makes sense as well as the limit exists in c for any test form ψ ∈ d(p−k,p−k)(ω),where, ϕ̃ = ϕ ◦ π stands for the composite of ϕ with the projection map π : cn −→ ck (z ′, z ′′) 7−→ z ′ received: 6 jun 2024.2020 mathematics subject classification. 32u05, 31c10, 32u25, 32c30, 32u40. key words and phrases. plurisubharmonic function, plurisubharmonic current, slice, analytic set, complete intersection.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 2and bk(a, ε) is the open ball in ck centered at point a and of radius ε > 0.in the present work we are concerned by topics studied in [9–12]. particularly, we will studytopics based on the definition (1.1) recently investigated in [11], for negative psh currents of smallsupport. a fundamental result was provided showing that for any psh function v non identicallyequals −∞, there exists a pluripolar subset e in ck , such that the slice < v, π, a >ϕ of v at point a is well defined and is expressed explicitly by v(a, .) as well as the point a lies outside e.as a consequence, an existence slicing result was deduced for negative psh currents having theirsupports contained in a strip. in particular, for positive or negative closed currents with supportcontained in a strip, it was established that the slices are well defined and are vanishing everywhereas soon as they are vanishing outside a pluripolar subset.in this paper, we shall develop the work in [11] by providing others slicing results for negativepsh currents which are arising from analytic subsets having suitable intersections.among several currents, we are interested on those written as finite sums of the form∑ 1≤j≤n (log |fj |) [ xj ] , ∑ 1≤j≤n vj [xj ], ∑ 1≤j≤n u[xj ] ∧ [yj ], or ∑ 1≤j≤n [xj ] ∧ u[yj ], where (fj)1≤j≤n is a finite family of holomorphic functions non identically vanishing on ∆n, (xj)1≤j≤nis a finite family of analytic subsets all of pure dimension p, (yj)1≤j≤n is another finite family ofanalytic subsets such that each yj forms a complete intersection with xj in ∆n, (vj)1≤j≤n is afinite family of psh functions non identically −∞ on ∆n and u[xj ] is the lelong-skoda potentialassociated to the current [xj ], (1 ≤ j ≤ n). the first result describes properly the slices of the current ∑ 1≤j≤n (log |fj |) [ xj ]. theorem 1.1. assume we have in ∆n, a finite family (xj)1≤j≤n of analytic subsets all of the same pure dimension q > k and a finite family (fj)1≤j≤n of holomorphic functions non identically vanishing such that for all 1 ≤ j ≤ n the hypersurface yj = {fj = 0} yields a complete intersection near the origin with xj . if σj : xj ∩ π−1(a) −→ ∆n denotes the canonical injection, then, there exist, a closed pluripolar subset e of ∆k , an integer m ∈ n and homogenous polynomials qm,j on xj , 1 ≤ j ≤ n , such that for all a ∈ ∆k r e the slice 〈 ∑ 1≤j≤n (log|fj |)[xj ], π, a〉ϕ is defined such that(1) if σj∗fj 6≡ 0 on xj ∩ π−1(a), ∀ 1 ≤ j ≤ n, then 〈 ∑ 1≤j≤n (log|fj |)[xj ], π, a〉ϕ = ∑ 1≤j≤n ( log |σj∗fj | ) [ xj ∩ π−1(a) ] , (2) if σj∗fj ≡ 0 and qm,j |xj∩π−1(a) 6≡ 0, ∀ 1 ≤ j ≤ n, then 〈 ∑ 1≤j≤n (log|fj |)[xj ], π, a〉ϕ = ∑ 1≤j≤n ( log |σj∗qm,j | ) [ xj ∩ π−1(a) ] . https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 3the second result generalizes theorem 1.1 to psh functions. it provides the existence and theexpression of the slices of a negative psh current of the form ∑ 1≤j≤n vj [xj ]. theorem 1.2. assume we have in ∆n, a finite family (xj)1≤j≤n of analytic subsets all of pure dimension p and a finite family (vj)1≤j≤n of negative psh functions such that for all 1 ≤ j ≤ n the set of singular points of vj is contained in a hypersurface yj that yields a complete intersection with xj . then, there is a pluripolar subset e of ∆k , such that for all a ∈ ∆k r e, the current∑ 1≤j≤n vj [xj ] admits a slice expressed by 〈 ∑ 1≤j≤n vj . [ xj ] , π, a〉ϕ = ∑ 1≤j≤n vj |xj∩π−1(a). [ xj ∩ π−1(a) ] . next, we let n(x) = − 1 (n−1)(4π)n 1 |x |2n−2 be the newton kernel in cn and we let η be a positivesmooth function with compact support in ω such that 0 ≤ η ≤ 1, η ≡ 1 on a neighborhood of ∆n.for a given analytic subset x in ∆n, we take the lelong-skoda potential u[x] associated to thecurrent [x], as the (n − p − 1, n − p − 1)-negative current defined by the following integral u[x](z) = ∫ ξ∈cn η(ξ)n(z − ξ)[x](ξ) ∧ ( ddc |z − ξ|2 )n−1 . we prove the third result of this paper on the existence and the expression of the slices of thewedge product current ∑ 1≤j≤n u[xj ] ∧ [yj ] introduced and studied in [9]. theorem 1.3. assume we have in ∆n, finite families (xj)1≤j≤n and (yl)1≤l≤n of analytic subsets such that for all 1 ≤ j ≤ n,, xj is of pure dimension p and yj is of pure dimension q and xj yields a complete intersection with yj . then, there exists a subset e contained in a countable union of analytic subsets of ∆k of dimensions ≤ k − 1 such that for all a ∈ ∆k r e, the slice of∑ 1≤j≤n u[xj ] ∧ [yj ] at point a is given by 〈 ∑ 1≤j≤n u[xj ] ∧ [yj ], π, a〉ϕ = ∑ 1≤j≤n j∗au[xj ] ∧ [ yj ∩ π−1(a) ] . in the sequel, we show that the above results enjoy interesting properties like the integrability ofcoefficients of the above currents and their slices. for instance, the following theorem 1.4 providesan important integrability property for a given psh function v not identically −∞ on ∆n. in fact, itdescribes the integrability of the function exp(−v) across an analytic subset x . consequently, weget the following beautiful result in terms of slices, about the integrability of exp(−v) across theintersection π−1(a) ∩x . theorem 1.4. assume we have a finite family (vj)1≤j≤n of psh functions in ∆n such that for all 1 ≤ j ≤ n the set of singular points of vj is contained in a hypersurface x that yields a complete intersection with another analytic subset y in ∆n. then, there exist a constant α > 0 and a https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 4 pluripolar subset e of ∆k , such that for any point a ∈ ∆kre, the function ∑ 1≤j≤n exp (−αvj)|y ∩π−1(a) lies in l1 loc ( y ∩ π−1(a) ) . finally, as an other application we establish the following result valid for analytic subsets xand y yielding a complete intersection in ω. theorem 1.5. assume we have analytic subsets x and y of pure dimensions p and q respectively, yielding a complete intersection in ∆n. then, there exist a constant δ > 0 and a pluripolar subset e in ck , such that for all a ∈ ∆k re, the coefficients of the current j∗a (u[x])∧ [ y ∩ π−1(a) ] lie in l1+δ loc (y ∩ π−1(a)). 2. preliminaries let ω be a domain in cn, we use the standard notations for the operators d = ∂ + ∂̄ and dc = i(∂̄ − ∂); the operator ddc is then defined by ddc = 2i∂∂. the space d ′p,p(ω) of (n− p, n− p)-currents (of bidimension (p, p)) on ω, is the dual of the space dp,p(ω) of smooth compactlysupported (p, p)-forms on ω. we say that a current t ∈ d ′p,p(ω) is positive, if for all smooth (1, 0)-forms α1, . . . , αp on ω, the product t ∧ iα1 ∧ ᾱ1 ∧ . . . ∧ iαp ∧ ᾱp is a positive measure. the current t is said to be closed if dt = 0 and psh if ddct ≥ 0.in particular, if p = n, then t is a psh function u on ω, that is a distribution such that∑ 1≤j,k≤n ∂2u ∂zj∂z̄k idzj ∧ dzk is a positive current of bidimension (n − 1, n − 1), the derivatives are taken here in the sense ofdistributions.we denote psh(ω) the set of psh functions on ω and l∞loc(ω)∩ psh(ω) the subset of elementsin psh(ω) which are locally bounded. the kähler form on cn is denoted β(t) = ddc |t|2. it canbe decomposed into the sum β′(t ′) + β′′(t ′′) where β′ and β′′ are kähler forms on ck and cn−krespectively. for a point a ∈ ck , we denote ja the map defined by ja : cn−k −→ {a} × cn−k z ′′ 7−→ (a, z ′) 3. proof of theorem 1.1: slicing of the current (log |f |)[x] first of all, we require the following well known propositions: proposition 3.1. let x be an analytic subset of ∆n and let m be its complex dimension. then, the following statements hold: https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 5(1) if m < k , then π(x) is contained in a countable union of analytic subsets of ∆k of dimensions ≤ m.(2) if m ≥ k , then the set z = {a ∈ ∆k / dimc(x ∩ π−1(x)) ≥ m − k + 1} is contained in a countable union of analytic subsets of ∆k of dimensions ≤ k − 1. as a consequence of the expansion of holomorphic functions in power series we have: proposition 3.2. let f 6≡ 0 be a non vanishing holomorphic function on ∆n such that for all z ′′ ∈ ∆n−k , f (0, z ′′) = 0. then, there exist a natural number m ≥ 1 and holomorphic functions aµ(z ′′) on ∆n−k , such that, for all (z ′; z ′′) ∈ ∆k × ∆n−k , we have f (z ′, z ′′) = ∑∞ j=mqj(z ′; z ′′) where qm(z ′, z ′′) = ∑ |µ|=m z ′µaµ(z ′′) and |µ| = µ1 + · · ·+ µk . now, given a psh function v on ∆n such that v is locally integrable on the analytic subset x ,then, for all test form ψ, for all fixed ε > 0 and all a ∈ sϕ, we shall justify the well definition andthe finiteness of the quantity ∫ bk(a,ε)×cn−k v [x] ∧ (ddc ϕ̃)k ∧ψ. proposition 3.3. let x ⊂ ∆n be analytic of pure dimension p > k , let v ∈ psh(∆n) ∩ l1 loc(x) be negative and let ϕ ∈ psh(∆k)∩l∞loc(∆k) be given such that sϕ = ∆k . then, for any point a ∈ ∆k , for any fixed ε > 0 and for any positive test form ψ ∈ d(p−k,p−k)(∆n), we have 1 µϕ(bk(a, ε)) ∫ bk(a,ε)×cn−k −v [x] ∧ (ddc ϕ̃)k ∧ψ <∞. proof. the result is local. we may take a = 0, without loss of generality we may choose the testform ψ such that ψ = ψ(z)β′′n−k ∈ d(n−p,n−p)(∆n−k) where ψ is a positive test function on ∆n−k .let denote γε := 1 µϕ(bk(a, ε)) ∫ bk(a,ε)×cn−k −v [x] ∧ (ddc ϕ̃)k ∧ψ. (3.2) there exists a neighborhood u = u ′ × u ′′ ⊂ ∆k × ∆p−k × ∆n−p of 0 and a coordinate system (t, ζ, z ′′) such that the projection πx : x ∩ u → ∆p = ∆k × ∆p−k( (t, ζ), z ′′ ) 7→ (t, ζ)is a ramified covering of x . let z be the ramification locus of πx and set xz = x ∩ ((u ′ r z)× u ′′) ⊂ xreg. the restriction of πxz : xz → u ′ r z https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 6is then a covering with a finite sheet number. the expression of γε given by (3.2) will be transformedas γε = 1 µϕ(bk(0, ε)) ∫ xz∩bk(0,ε)×cn−k πxz ∗(− v(ddcϕ)k ∧ψ ) . (3.3) we may choose ε > 0 small enough, so that uε := xz ∩ bk(0, ε)× cn−kbk(0, ε)× ∆p−k × {0}cn−p .then, the expression given by (3.3) can be written as γε = 1 µϕ(bk(0, ε)) ∫ bk(0,ε) w(t)(ddcϕ)k , (3.4) where w(t) = ∫ ∆p−k π∗xz (−vψ)(t, ζ)dλp−k(ζ). furthermore, following [6] we have ||w(ddcϕ)k ||bk(0,ε) ≤ ||w ||l1(bk(0,2ε))||ϕ||kl∞(bk(0,2ε)), (3.5) the inequality (3.5) implies that w is a locally integrable function with respect to the positivemeasure µϕ and hence the integral given by (3.4) is finite. � now we are ready to give the proof of theorem 1.1. proof. the result is local. without loss of generalities we may suppose n = 1. furthermore, forsimplicity, we may consider the slice at point a = 0. put t = log |f |[y ] and γε(t ) = 1 µϕ(bk(0, ε)) ∫ bk(0,ε)×cn−k t ∧ (ddc ϕ̃)k ∧ψ, (3.6) where ψ is a test form such that ψ = h(z)β′′n−k ∈ d(n−q,n−q)(∆n−k) where h is a positive smoothfunction with compact support. we may find a neighborhood u = u ′×u ′′ of 0 in ∆k×∆q−k×∆n−qand a coordinate system (t, ζ, z ′′) such that the projection πy : y ∩ u → ∆q( (t, ζ), z ′′ ) 7→ (t, ζ)defines a ramified covering of y . let z be the ramification locus of π and yz = y ∩((u ′rz)×u ′′) ⊂ yreg . the restriction of πyz : yz → u ′ r z is then a covering with a finite sheet number. equality (3.6) can be written as γε(t ) = 1 µϕ(bk(0, ε)) ∫ yz∩bk(0,ε)×cn−k π∗ ( log |f |(ddcϕ)k ∧ψ ) . (3.7) we have to find the limit, as ε→ 0, of (6.20). for ε > 0 small enough, the set uε := yz∩bk(0, ε)× cn−k can be viewed as bk(0, ε) × ∆q−k × {0}cn−q . the integral in the right hand side of (6.20)can be written such that γε(t ) = 1 µϕ(bk(0, ε)) ∫ bk(0,ε) wh(t)(ddcϕ)k , (3.8) https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 7where wh(t) = ∫ ∆q−k log |f (t, ζ, 0)|h(ζ, 0)dλq−k(ζ) = ∫ yz∩π−1(0) log |i∗ yz∩π−1(0) f (t, ζ, z ′′)|i∗ yz∩π−1(0) h(ζ, z ′′),and iyz∩π−1(0) : yz ∩π−1(0)→ ∆n is the canonical injection. let e be the closed pluripolar subsetof ∆k defined by e = {a ∈ ∆k : w(a) = −∞ or w 6∈ l1 loc(µϕ) near a}. by theorem 1.1 in [11], for all a ∈ ∆k r e, we have:(1) if (iyz∩π−1(0))∗f 6≡ 0, then, as ε→ 0, the limit of the integral given by (3.8) will be limε→0 γε = wh(0) = ∫ ∆q−k log |f (0, ζ, 0)|h(ζ, 0)dλq−k(ζ) = ∫ y ∩π−1(0)(iyz∩π−1(0))∗[log |f | ∧ψ] = 〈 ( iyz∩π−1(0) )∗ (log |f |) [ y ∩ π−1(0) ] , (iyz∩π−1(0))∗(ψ)〉ϕ.(2) if (iyz∩π−1(0))∗f ≡ 0, then by proposition 3.2 applied with the function (iyz∩π−1(0))∗f on ∆q = ∆k × ∆q−k , there exist an integer m ∈ n, a non vanishing homogenous polynomial qm(t, ζ) and smooth functions (t, ζ) 7→ r(t, ζ) ∈ c∞(∆k × ∆q−k) such that f (t, ζ) = qm(t, ζ) + tr(t, ζ) on bk(0, ε)× ∆p−k . thanks to taylor’s formula, the rest r(t, ζ) can be chosen such that |r(t, ζ)| ≤ 1. thiswith the triangle inequality legitimate the following inequalities log ||qm| − |tr|| ≤ log ||f (t, ζ)|| = log |(iyz∩π−1(0))∗f (t, w, z ′′)| ≤ log(|qm|+ |tr|).since (t, ζ) ∈ bk(0, ε) × ∆p−k and |r(t, ζ)| ≤ 1, then, as ε → 0, |t| → 0 and |tr| → 0.hence, we get limε→0 γε = wh(0) = ∫ ∆q−k log |qm(0, ζ)|h(ζ, z ′′)dλq−k(ζ) = ∫ y ∩π−1(0)(iyz∩π−1(0))∗[log |qm(0, ζ| ∧ψ] = 〈(log |(iyz∩π−1(0))∗qm|) [ y ∩ π−1(0) ] , (iyz∩π−1(0))∗(ψ)〉ϕ.the proof is completed. � the following example illustrates theorem 1.1. example 3.4. in c3 we let f (z) = z2 1 + z2 2 − z3, y = {z1 = −z3}, k = 1 and ϕ(z ′) = |z ′|2 = |z1|2.it is clear that the subsets x = {f = 0} and y yield a complete intersection near 0. hence,according to [9], we deduce that log |f | ∈ l1 loc(y ). by theorem 1.1, we have 〈log |f |. [ y ] , π, 0〉 = log |z2 2 − z3|. [ z1 = z3 = 0 ] . https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 84. proof of theorem 1.2: slicing of the current v [x] proof. we use notations and arguments as in the proof of theorem 1.1 and we may suppose n = 1.we have γε(v [x]) = 1 µϕ(bk(0, ε)) ∫ bk(0,ε)×cn−k v [x] ∧ (ddc ϕ̃)k ∧ψ. (4.9) let xreg be the set of regular points of x. assume that 0 ∈ xreg and put z1 = {a ∈ ∆k : dimc(x ∩ π−1(a)) > p − k}. then, by proposition 3.1, z1 is contained in a countable union of analytic subsets of ∆k of dimen-sions ≤ k − 1. as the dimension m of the set xsing of singular points, satisfies m ≤ p − 1, then,by proposition 3.1, there exists a set z2 contained in a countable union of analytic subsets of ∆kof dimensions ≤ k − 1, such that for all a ∈ ∆k rz2, xsing ∩π−1(a) is an analytic subset of ∆k ofdimension m − k (otherwise is empty). put z = z1 ∪ z2 and denote π̄ := π|xreg . the expression (4.9) can be transformed to γε(v [x]) = 1 µϕ(bk(a, ε)) ∫ xregr(π̄−1(z))∩(bk(a,ε)×cn−k) π̄∗[v(ddcϕ)k ∧ψ]. (4.10) in a local chart of coordinates (z1, . . . , zk , w1, . . . , wn−k), such that xreg r (π̄−1(z)) ∩ (bk(a, ε)× cn−k) = bk(a, ε)× cp−k × {0}cn−p , when ε > 0 is small enough, the integral (4.10), can be written as γε = 1 µϕ(bk(a, ε)) ∫ ∆pr(π̄−1(z))∩(bk(a,ε)×cn−k) v(z ′, w)(ddcϕ)k ∧ψ(z ′, w). (4.11) according to [11], since z ′ 7→ v(z ′, w) is locally integrable on xreg r (π̄−1(z), then when ε→ 0,we get the limit of (4.11), as follows limε→0 γε(v [x]) = ∫ ∆p−k v(a, .)ψ(a, w)β′′p−k = ∫ xreg\(π̄−1(z))∩π−1{a} v(a, .)ψ(a, w)β′′p−k = 〈v|x∩π−1(a). [ x ∩ π−1(a) ] , (ix∩π−1(a))∗ψ〉.this finishes the proof. � we illustrate theorem 1.2 by the following example. example 4.1. in c4 = c× c3, we take v(z) = log(|z1|3 + |z3|5), y = {z4 = z2 1 z 4 2} , k = 1 and ϕ(z ′) = |z ′|2 = |z1|2. the subset x = {z1 = z3 = 0} that contains {v = −∞}, yields a complete intersection with thesubset y near 0. by theorem 1.2, up to a constant, we have 〈v [y ], π, 0〉 = log |z3| [ z1 = z2 = z4 = 0 ] . https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 9for the next section, we will be concerned with points a ∈ ∆k at which the slice 〈u[x]∧ [y ], π, a〉of the current u[x] ∧ [y ], is well defined and how it can be expressed.remember that the wedge product u[x] ∧ [y ] was defined in [10] as a current on y , as wellas, the analytic subsets x and y yield a complete intersection in ω. actually a necessary andsufficient condition was established, by expressing that the current u[x] has integrable coefficientswith respect to the trace measure of the current [y ], if and only if x and y yield a completeintersection in ω. the condition of complete intersection is optimal for the definition of this wedgeproduct in the weak sense of currents. recall that, x and y yield a complete intersection, if for anyirreducible components xj of x and yk of y , we have codim(xj ∩ yk) = codimxj + codimyk . 5. proof of theorem 1.3: slicing of the current u[x] ∧ [y ] in order to avoid complications, we assume here that the function ϕ is smooth and hence themeasure µϕ can be considered as the lebesgue measure on ck , with density a smooth functiondenoted µϕ(z ′). proof. let ψ ∈ d(p+q−n+1−k,p+q−n+1−k)(∆n) be a test form. we have to prove the existence of apluripolar subset e of ∆k such that for all a 6∈ e, we have lim ε→0 γε = ∫ ∆n j∗au[x] ∧ [ y ∩ π−1(a) ] ∧ j∗a (ψ) where γε = 1 µϕ(bk(a, ε)) ∫ bk(a,ε)×cn−k u[x] ∧ [y ] ∧ (ddc ϕ̃)k ∧ψ. (5.12) since the potential u[x] involved in (5.12), is a finite linear combination of forms with coefficients bi,j(s, z) given by the following expression bi,j(s, z) = ∫ ξ∈cn η(ξ)n(z − ξ)[x](ξ) ∧ βp−s(ξ) ∧ 2s i s 2 dξi ∧ dξj (|i| = |j| = s). (5.13) we need to find limε→0 γε(bi,j(s, z)), where γε(bi,j(s, z)) is given by γε(bi,j(s, z)) := 1 µϕ(bk(a, ε)) ∫ bk(a,ε)×cn−k bi,j(s, z) ∧ [y ] ∧ (ddc ϕ̃)k ∧ψ. (5.14) since the current u[x]∧ [y ] is well defined then, the functions z 7→ bi,j(s, z) are locally integrableon y , and hence γε(bi,j(s, z)) given by (5.14), can be written as γε := γε(bi,j(s, z)) = 1 µϕ(bk(a,ε)) ∫ (ξ,z)∈vε η(ξ)n(z − ξ) [ x × y ] (ξ, z) ∧ (ddcϕ)k ∧ f (ξ, z), where f (ξ, z) = βp−s(ξ) ∧ 2s i s 2 dξi ∧ dξj ∧ψ(z) and vε = cn × [ bk(a, ε)× cn−k ] . https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 10let [x × y ]reg be the set of regular points of x × y. we may assume that 0 ∈ [x × y ]reg . put z1 = {a ∈ ∆k : dimc((x × y ) ∩ (π−1(a)× π−1(a)) > p + q − k} = {a ∈ ∆k : dimc(x ∩ π−1(a))× (y ∩ π−1(a)) > p + q − k}. following proposition 3.1, z1 is contained in a countable union of analytic subsets of ∆k ofdimensions ≤ k − 1. as the dimension m of the subset (x × y )sing of singular points, satisfies m ≤ p + q − 1, then, again by proposition 3.1, there exists a subset z2 contained in a countableunion of analytic subsets of ∆k of dimensions ≤ k − 1, such that for any point a ∈ ∆k rz2, the set (x × y )sing ∩ (π−1(a) × π−1(a) is analytic in ∆k of dimension m − k (otherwise is empty). put z = z1 ∪ z2 and denote π̄ := π|(x×y )reg , we have γε = ∫ (x×y )regr(π̄−1(z))∩vε π̄∗ [ η(ξ)n(z − ξ)(ddcϕ)k ∧ f (ξ, z) ] . (5.15) we can find local coordinates (ξ, z) = (ξ, (z ′, w)) = ( ξ, (z1, . . . , zk , w1, . . . , wn−k) ) , so that, for all ε > 0 small enough, (x × y )reg r (π̄−1(z)) ∩ vε = cp × {0}cn−p × bk(a, ε)× cq−k × {0}cn−q . since µϕ(bk(a, ε)) ∼ ω2kε 2kµϕ(a) as ε → 0. then, by an application of fubini’s theorem andby the change of variables (ξ, z ′) ↔ (ξ, z ′−a ε ), when ε > 0 is small enough, the equality (5.15)can be transformed to the following γε = 1 ω2kµϕ(a) ∫ v µϕ(a + εt)η(ξ)n((a + εt, w)− ξ)f (ξ, (a + εt, w))dν(t, w), (5.16) where v = cp+q−k ×bk(0, 1) and dν(t, w) = dλk(t)⊗ dλp+q−k(w). by letting ε→ 0 in (5.16),we get limε→0 γε = ∫ cp+q−k η(ξ)n((a, w)− ξ)f (ξ, (a, w))dλp+q−k(w) = ∫ (x×y )reg∩(π−1(a)×π−1(a)) η(ξ)n((a, w)− ξ) ∧ f (ξ, (a, w)) = ∫ (x∩π−1(a))reg×(y ∩π−1(a))reg η(ξ)n((a, w)− ξ) ∧ f (ξ, (a, w)) = ∫ cn−k bi,j(s, (a, w)) ∧ [y ∩ π−1(a)] ∧ψ(a, z ′′) = 〈j∗a(bi,j(s, z)) ∧ [y ∩ π−1(a)], j∗a (ψ)〉.the last equality holds true since we have dim ( (x × y )sing ∩ (π−1(a)× π−1(a)) ) < p + q − k. this achieves the proof of theorem 1.3. � let illustrate theorem 1.3 with the following example. https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 11 example 5.1. consider in c5 = c × c4, x = {z2 = z3 = z4 = 0} and y = {z4 = z2 2 z 2 3} take k = 1 and ϕ(z ′) = |z ′|2 = |z1|2. since x is smooth, then a potential of [x] is u = log(|z2|2 + |z3|2 + |z4|2)ddc log(|z2|2 + |z3|2 + |z4|2) |z2|2 + |z3|2 + |z4|2 . as x and y yield a complete intersection, then by [9], the current u ∧ [y ] is well defined. thecurrent u ∧ [y ] is given by the following u ∧ [y ] = i∗y [ log(|z2|2 + |z3|2 + |z2z3|4)ddc log(|z2|2 + |z3|2 + |z2z3|4) |z2|2 + |z3|2 + |z2z3|4 ] . we have x ∩ π−1(0) = {z1 = z2 = z3 = z4 = 0} and y ∩ π−1(0) = {z1 = z2 4 − z2z3 = 0}. it is clear that x∩π−1(0) and y ∩π−1(0) yield a complete intersection in {0}×c4. if σ = iy ∩π−1(0),then, 〈u ∧ [y ], π, 0〉 = σ∗ [ log(|z2|2 + |z3|2 + |z2z3|4)ddc log(|z2|2 + |z3|2 + |z2z3|4) |z2|2 + |z3|2 + |z2z3|4 ] . 6. proofs of theorem 1.4 and theorem 1.5 with applications this section aims to give the proof of theorem 1.4 and the proof of theorem 1.5. we may suppose n = 1. indeed, for the proof of theorem 1.5, we use the fact that a finite union of pluripolar subsetsis pluripolar. in addition, theorem 1.4 is an immediate application of the following theorem 6.1.in fact, if for all 1 ≤ j ≤ n , there exists αj > 0 for which theorem 6.1 holds for the psh function vj , then for α = min1≤j≤n (αj), theorem 1.4 can be deduced from theorem 6.1 in terms of slicesand it works with the psh function v = ∑ 1≤j≤n exp(−αvj). theorem 6.1. assume we have a psh function v on ∆n such that its set of singular points is contained in a hypersurface x forming a complete intersection with the analytic subset y of ∆n. then, there exists α > 0, such that the function exp (−αv)|y lies in l1 loc(y ). proof. the result is local, so it’s enough to prove it near a singular point z0 = 0 ∈ {v = −∞} ⊂ x .we may suppose z0 is a regular point of x ∩y . let q be the dimension of y at z0 (1 ≤ q ≤ n−1),since x and y yield a complete intersection near z0, then we may find a neighborhood v (z0) of z0 and local coordinates (z1, . . . , zn) such that k ∩x ∩ y ⊂ k ∩ y ∩ {zq = zq+1 = · · · = zn = 0}, k = v (z0). (6.17) let (vδ)δ>0 be a decreasing sequence of continuous psh functions such that limδ→0 vδ = v pointwise.we choose a fixed δ0 > 0 small enough, such that vδ0 (z)− 1 ≤ v(z) ≤ vδ0 (z), ∀ z ∈ k rx ∩ y. (6.18) https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 12according to a classical result due to h. j. bremermann and p. lelong [4, 14], there exists (fj)j asequence of holomorphic functions on ∆n such that the function vδ0 is the regularized supremumlimit on k of (1 j log |fj | ) j . which means that vδ0 (z) = [ limj→∞ sup 1 j log |fj(z)| ]∗ = limε→0 supζ∈b(z,ε)[limj→∞ 1 j log |fj(ζ)], z ∈ k. (6.19) for the compact k, in view of (6.18) and (6.19), we can find a natural number n that depends on k, and analytic functions on ∆n, f1, . . . , fn , such that the function v and the function vn definedby vn(z) = max 1≤j≤n 1 j log |fj(z)|, (6.20) satisfy the following inequalities vn − 1 ≤ v ≤ vn ≤ 0 on k rx ∩ y . (6.21) we may suppose the compact k is sufficiently small so that k ∩x ∩ y ⊂ k ∩ y ∩ {fj = 0, j = 1, . . . , n}. by the weierstrass preparation theorem (see [6]), for each indice 1 ≤ j ≤ n , one can write fj suchthat fj(z) = hj(z)pj(z ′, zn), (6.22) where hj is an invertible holomorphic function on ∆n and pj(z ′, zn) is a weierstrass polynomial in zn, of the form pj(z ′, zn) = z mj n + a1,j(z ′)z mj−1 n + · · ·+ aν,j(z ′)z mj−ν n + · · ·+ amj ,j,(z ′), aν,j(0) = 0, with mj ≥ 1 is the vanishing order of fj at z0 = 0 and (aν,j,(z ′))1≤µ≤mj are holomorphic coefficientson the polydisc ∆n−1 in cn−1. furthermore, for all 1 ≤ j ≤ n and all 1 ≤ ν ≤ mj , there is apositive constant c1 such that for all z ′ ∈ ∆n−1, the following inequality holds |aν,j(z ′)| ≤ c1||z ′||ν . (6.23) therefore, by the triangle inequality, (6.23) provides the existence of a constant c2 = c3(k) > 0such that for all 1 ≤ j ≤ n and for all z = (z ′, zn) ∈ k, the following inequality holds |pj(z ′, zn)| ≤ c2||z ′||mj . (6.24) on the other hand, since {fj = 0, j = 1, . . . , n} = {pj = 0, j = 1, . . . , n} and since the polynomial pj vanishes at z0 = 0, then there is a constant c3 > 0, such that for any point z = (z ′, zn) ∈ ksufficiently close to 0, the following inequality holds |zn| ≤ c3||z ′||. (6.25) https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 13hence, using the triangle inequality with (6.25) and reasoning by induction on the degree mj of pj , 1 ≤ j ≤ n , we can find a constant c4 = c4(k) > 0, such that for all 1 ≤ j ≤ n and for all z = (z ′, zn) ∈ k, the polynomial pj(z ′, zn) satisfies the following inequality c4||z ′||mj ≤ |pj(z ′, zn)|. (6.26) in addition, since for all 1 ≤ j ≤ n , the holomorphic function hj is invertible and n depends onlyon k, then we can find a constant c5 = c5(k) > 0 such that for all z ∈ k, the function defined by v ′n(z) = max 1≤j≤n 1 j log |hj(z)| satisfies the following inequality − c5 ≤ v ′n(z) ≤ 0. (6.27)if we denote βn = min1≤j≤n mj j and wn(z) = wn(z ′, zn) = βn log ||z ′||, then following (6.22) thefunction vn defined by (6.20) is such that vn(z) = max1≤j≤n 1 j log |pj(z ′, z,n )|+ max1≤j≤n 1 j log |hj,δ>0(z)| ≤ βn log ||z ′||+ max1≤j≤n 1 j log |hj(z)| = wn(z) + v ′n(z) ≤ 0. (6.28) hence, in view of (6.21), (6.24), (6.26) and (6.28), we can find constants c6 = c6(k) > 0 and c7 = c7(k) > 0 such that for all z ∈ k r y ∩ x , the functions v and wn satisfy the followinginequalities c6wn(z)− c7 ≤ v(z) ≤ wn(z) ≤ 0. (6.29)it is clear that (βn)n is a positive and decreasing sequence. hence it has a limit β ≥ 0 as n → +∞.in view of (6.17) and (6.29) we have β > 0. indeed, if not, the function v will be bounded near 0and then the point z0 is not a singular point of v . therefore, by letting n → +∞ in (6.29), we canfind constants β > 0 and c8 = c8(k) > 0 such that for all z = (z ′, zn) ∈ k r y ∩x , the function v satisfies the following inequality exp (−v(z)) ≤ c8||z ′||−β. (6.30) taking α > 0 so that αβ ∈ (0, 2q − 2), hence z 7→ 1 ||z ||αβ ∈ l 1 loc(cq−1), and investigating (6.17)with (6.30), we get the following∫ z∈k\x∩y exp(−αv(z)) ≤ c8 ∫ z∈∆q−1 dλq−1(z) ||z ||αβ <∞. (6.31) consequently, our proof is achieved thanks to (6.31). � remark 6.2. theorem 6.1 generalizes a result in [9] showing that, if y = {f = 0} is a hyper-surface in ω given by a holomorphic function not identically vanishing and if y yields a completeintersection with another analytic subset x ⊂ ω, then there exists δ > 0 such that the coefficientsof the current log |f |[x] lies in l1+δ loc (x). moreover, there exists α > 0, such that the coefficients https://doi.org/10.28924/ada/ma.4.23 eur. j. math. anal. 10.28924/ada/ma.4.23 14of the current |f |−α[x] lie in l1 loc(x). if we replace the hypersurface y = {f = 0} by a positive (1, 1)-closed current of the form t = ddcv for some psh function v , then we provide theorem 1.4.note that we invite interested readers to proceed defining and investigating quaternionic hyper-surfaces and quaternionic analytic sets in the space hc recently studied in [13] and find similarresults as given in this paper. in the direction of theorem 1.3 and proceeding as in the proof of theorem 3.3 in [9], we finallygive the proof of theorem 1.5. proof. we consider the analytic subsets x ∩ π−1(a), y ∩ π−1(a) and e = {a ∈ ∆k : dimc(x ∩ π−1(a))× (y ∩ π−1(a)) > p + q − k}. by proposition 3.1, e is contained in a countable union of analytic subsets of ∆k of dimension ≤ k−1. therefore the set e is pluripolar. we work as in [9] ( proof of theorem 3.3) around a point a 6∈ e. � competing interests the author(s) declare(s) that there is no conflict of interest regarding the publication of thispaper. references [1] e. bedford, b.a. taylor, a new capacity for plurisubharmonic functions, acta math. 149 (1982) 1–41.[2] h. ben messaoud, h. el mir, operateur de monge-ampre et formule de tranchage pour un courant positifs fermé,c.r.a.s. paris, t316 i (1993), 1173–1176.[3] h. ben messaoud, h. el mir, tranchage et prolongement des courants positifs fermés, math. ann. 307 (1997),473–487.[4] h.-j. bremermann, on the conjecture of the equivalence of the plurisubharmonic functions and the hartogs functions,math. ann. 131 (1956), 76–86.[5] c.o. kiselman, sur la définition de l’opérateur de monge-ampère complexe, analyse complexe, proceedings ofjournées fermat (smf), toulouse, may, 1983.[6] j.p. demailly, complex analytic and differential geometry, e-book, 2007.[7] h. federer, geometric measure theory, springer, 1969.[8] r. harvey, b. shiffman, a characterization of holomorphic chains, ann. math. 28 (1974), 553–587.[9] h. khedhiri, wedge product of currents, lobachevskii j. math. 31 (2010), 224–231.[10] h. khedhiri, slicing of currents associated to a plurisubharmonic function, punj univ. j. math. 47 (2015), 21–34.[11] h. khedhiri, ϕ-slicing results for negative plurisubharmonic currents, uzbek j. math. 68 (2024), 102-112.[12] h. khedhiri, on construction of positive closed currents with prescribed lelong numbers, j. sib. fed. univ. math.phys. 13 (2020), 331-341.[13] h. khedhiri, t. mkademi, foundational aspects of a new matrix holomorphic structure, arab j. math. sci. (2024).[14] p. lelong, intégration sur un ensemble analytique complexe, bull. soc. math france. 853 (1957), 239–262. https://doi.org/10.28924/ada/ma.4.23 1. introduction 2. preliminaries 3. proof of theorem 1.1: slicing of the current (log|f|)[x] 4. proof of theorem 1.2: slicing of the current v[x] 5. proof of theorem 1.3: slicing of the current u[x][y] 6. proofs of theorem 1.4 and theorem 1.5 with applications competing interests references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 17doi: 10.28924/ada/ma.5.17 nonlinear geometry of norm-attaining functionals: variational principles, subdifferential calculus, and polynomial optimization in locally convex spaces mogoi n. evans1, priscah moraa2,∗ 1department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, kenya mogoievans4020@gmail.com 2department of mathematics and actuarial science, kisii university, kenya priscahmoraa@kisiiuniversity.ac.ke ∗correspondence: priscahmoraa@kisiiuniversity.ac.ke abstract. we develop a unified theory of norm-attainment for nonlinear functionals in locally con-vex spaces, extending classical results to sublinear, quasiconvex, and polynomial settings. our maincontributions include: (1) nonlinear bishop-phelps theorems establishing density of norm-attainingfunctionals, (2) a subdifferential characterization of attainment via interiority conditions, (3) a krein-milman principle for convex functionals on compact sets, and (4) a complete solution to the poly-nomial norm-attainment problem through tensor product geometry. the work combines innovativeapplications of choquet theory, variational analysis, and complex-geometric methods to reveal newconnections between functional analysis and optimization. key applications address stochastic vari-ational principles and reproducing kernel hilbert space optimization, with tools applicable to pdeconstraints and high-dimensional data science. these results collectively bridge fundamental gapsbetween linear and nonlinear functional analysis while providing fresh geometric insight into infinite-dimensional phenomena. introduction the study of norm-attaining functionals originated with the seminal bishop-phelps theorem [13],which established the density of norm-attaining linear functionals in banach spaces. while thisresult has been extended in various directions [10, 11], existing theories remain constrained bybanach space limitations and lack comprehensive frameworks for nonlinear functionals. our workovercomes these limitations by developing a unified theory in locally convex spaces, combininginnovative tools from variational analysis [5], convex geometry [13], and polynomial functional anal-ysis [8]. the classical bishop-phelps theorem has seen partial extensions to nonlinear settings, received: 16 apr 2025. key words and phrases. norm-attaining functionals; nonlinear functional analysis; locally convex spaces; variationalprinciples; subdifferential geometry; polynomial optimization; bishop-phelps theorem; infinite-dimensional convexity.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 2including sublinear functionals [3] and quasiconvex cases [6], but these advances have been re-stricted to either banach spaces or specific functional classes. our theorem 1 breaks new groundby characterizing the nonlinear bishop-phelps property in general locally convex spaces, revealingan essential connection with lattice norms and τ-lower semicontinuity that extends both the orig-inal results and their convex generalizations [7]. the proof introduces novel variational techniquesinspired by the borwein-preiss principle [5]. in subdifferential geometry, theorem 2 establishes thefirst complete characterization of norm-attainment for sublinear functionals through weak∗-exposedpoints and subdifferential monotonicity. this bridges classical subdifferential calculus [14] withmodern theories of barrelled spaces [9], while theorem 8 extends the brondsted-rockafellar theo-rem with new interiority conditions for norm-attainment. the quasilinear separation in theorem 4generalizes the hahn-banach theorem while preserving norm-attainment, with immediate applica-tions to game theory and economic equilibrium [4]. for polynomial functionals, theorem 10 solvesthe long-standing attainment problem through projective tensor products and radon-nikodym prop-erties [1, 8]. this complements theorem 6’s surprising link between operator norm-attainment andplurisubharmonic norms, combining operator theory with complex analysis [12]. the nonlinearkrein-milman theorem (theorem 9) and james-type characterization (theorem 7) complete thepicture, employing choquet theory [13] and geometric methods [2] to extend fundamental results togeneral locally convex spaces. collectively, these advances bridge critical gaps between linear andnonlinear functional analysis while providing powerful new tools for optimization, stochastic pdes,and high-dimensional statistics. the synthesis of variational principles, geometric methods, andcomplex-analytic techniques reveals previously unrecognized connections across these domains. notation x locally convex space ⊗̂nπ projective tensor product ∂p(x) subdifferential at x preliminaries throughout this work, we consider x to be a hausdorff locally convex space (lcs) over r or c,with topology τ generated by a separating family of seminorms {ρα}α∈i . we denote by x∗ thetopological dual space, equipped with the weak-∗ topology σ(x∗, x). fundamental references forthese concepts include [9] and [4]. functional analytic foundations. definition 1 (continuous sublinear functionals). a functional p : x → r is called sublinear if: • p(λx) = λp(x) for all λ ≥ 0 (positive homogeneity) • p(x + y) ≤ p(x) + p(y) (subadditivity) https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 3 such p is τ-continuous if and only if it is dominated by some continuous seminorm ρα [14]. definition 2 (norm-attainment). for a functional f : x → r, we say f attains its p-norm at x0 ∈ x if: f (x0) = ‖f ‖p := sup x∈x p(x)≤1 |f (x)| where p is a given continuous sublinear functional. when p is the minkowski functional of a bounded set b, we write ‖f ‖b . convex analysis tools. definition 3 (subdifferentials). for f : x → r ∪ {+∞}, the subdifferential at x ∈ dom(f ) is: ∂f (x) := {φ ∈ x∗ : f (y) ≥ f (x) + φ(y − x) ∀y ∈ x} when f is continuous and convex, ∂f (x) is nonempty and σ(x∗, x)-compact [14]. proposition 1 (borwein-preiss variational principle [5]). let (x, τ) be a complete lcs and f : x → r ∪ {+∞} proper, lower semicontinuous, and bounded below. then there exists a τ-dense gδ set g ⊂ x such that for all ξ ∈ g, the perturbed functional f + ξ attains its strong minimum on x . geometric properties. definition 4 (plurisubharmonic norms). a norm ‖ · ‖ on complex x is plurisubharmonic if for all x, y ∈ x , the function: λ 7→ ‖x + λy‖ is subharmonic on c. this generalizes the notion of complex convexity [8]. definition 5 (radon-nikodym property). a locally convex space x has the radon-nikodym prop-erty (rnp) if every continuous linear operator t : l1[0, 1]→ x is representable by an x-valued bochner integrable function [9]. polynomial mappings. definition 6 (n-homogeneous polynomials). a mapping p : x → c is a continuous n-homogeneouspolynomial if there exists a continuous n-linear form l : x × · · · ×x︸ ︷︷ ︸ n → c such that: p (x) = l(x, . . . , x) the space p(nx) carries the topology of uniform convergence on bounded sets [1]. https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 4 proposition 2 (projective tensor representation). for any n-homogeneous polynomial p , there exists a unique linear functional p̃ on the projective tensor product ⊗̂nπx such that: p (x) = p̃ (x ⊗ · · · ⊗ x︸ ︷︷ ︸ n ) norm-attainment of p corresponds to norm-attainment of p̃ [8]. key topological concepts. definition 7 (mackey-arens property). a lcs (x, τ) satisfies the mackey-arens property if τ coincides with the mackey topology τ(x,x∗) [9]. definition 8 (quasi-completeness). x is quasi-complete if every bounded cauchy net converges. this generalizes completeness for non-metrizable spaces [3]. definition 9 (barrelled spaces). x is barrelled if every closed, absolutely convex, absorbing set is a τ-neighborhood of 0. this ensures the uniform boundedness principle holds [9]. these foundational concepts will be essential throughout our analysis of nonlinear norm-attainment phenomena in the subsequent sections. main results and discussions theorem 1 (nonlinear bishop-phelps property). let x be a locally convex space and p : x → r a continuous sublinear functional. the following are equivalent: (1) every bounded below p-dominated convex functional attains its strong minimum (2) the set {f ∈ x∗ : f attains its p-norm} is dense in (x∗, β(x∗, x)) (3) x admits an equivalent τ-lower semicontinuous lattice norm proof of theorem 1 (nonlinear bishop-phelps property). we employ a nonlinear geometric ap-proach inspired by borwein’s variational techniques. step 1: (1) ⇒ (2). for any f ∈ x∗ and ε > 0, define g(x) := p(x) − f (x). by (1), there exists xε ∈ x attaining infx∈x g(x). the perturbed functional: fε := f + ε∂p(xε) attains its p-norm at xε since: ‖fε‖p = sup x [f (x) + εp(xε)− εp(x − xε)] = f (xε) + εp(xε) moreover, ‖fε − f ‖ ≤ 2ε by construction. step 2: (2)⇒ (3). using the density, construct a sequence (fn) ⊂ x∗ of norm-attaining functionalsseparating points. the lattice norm: ‖x‖ := sup n |fn(x)| ‖fn‖p + ρ(x) https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 5where ρ is the original τ-lsc seminorm, has the required properties by the nachbin-shirota theorem. step 3: (3)⇒ (1). for any p-dominated convex h, the sublevel sets {x : h(x) ≤ α} are τ-closed andbounded in the lattice norm, hence τ-compact by the generalized alaoglu theorem. the attainmentfollows from lower semicontinuity. � example 1 (non-attaining sublinear functional). let x = c[0, 1] with p(f ) = supx∈[0,1/2] |f (x)|. the functional φ(f ) = ∫ 1 0 f fails to attain its p-norm, illustrating theorem 1’s lattice condition necessity. theorem 2 (characterization of sublinear norm-attainment). for a sublinear p : x → r on a barrelled space, the following are equivalent: • p attains its norm at some x0 ∈ x • ∂p(0) ∩x∗ contains a weak∗-exposed point • the subdifferential map x 7→ ∂p(x) is not norm-decreasing proof of theorem 2 (characterization of sublinear norm-attainment). we develop a new subdif-ferential calculus approach: (i) ⇒ (ii): if p attains its norm at x0, then for any f ∈ ∂p(x0) we have: f (x0) = p(x0) = ‖p‖ thus f exposes ∂p(0) at x0 in the weak∗ topology. (ii) ⇒ (iii): let f be a weak∗-exposed point of ∂p(0). there exists x0 ∈ x such that: f (x0) > g(x0) ∀g ∈ ∂p(0) \ {f } this implies ‖∂p(x0)‖ = ‖f ‖ since any other subgradient would violate the exposing property. (iii)⇒ (i): by the barrelledness assumption, the subdifferential map is locally bounded. if ‖∂p(x)‖is non-decreasing, then for some x0 we must have: ‖∂p(x0)‖ = sup x∈x ‖∂p(x)‖ = ‖p‖ the attainment follows from the hahn-banach theorem applied to ∂p(x0). � theorem 3 (non-convex variational principle). let x be a quasi-complete locally convex space and f : x → r gateaux differentiable. if f is τ-lower semicontinuous and coercive, then there exists a dense gδ set g ⊂ x∗ such that for all ξ ∈ g, the perturbed functional f + ξ attains its exact norm on x . proof of theorem 3 (non-convex variational principle). we combine phelps’ perturbed minimiza-tion with christensen’s category methods: step 1: define the family: f := {ξ ∈ x∗ : f + ξ attains its norm} https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 6 step 2 : for each n ∈ n, consider the open sets: un := ⋃ x∈x ‖x‖>n {ξ ∈ x∗ : (f + ξ)(x) > ‖f + ξ‖ − 1 n } these are dense by the quasi-completeness and the ekeland variational principle. step 3: the set g := ⋂ n∈n un is a dense gδ by baire’s theorem. for any ξ ∈ g, take a sequence (xn) with ‖xn‖ → ∞ and: (f + ξ)(xn)→ ‖f + ξ‖the coercivity and lower semicontinuity ensure the existence of a norm-attaining point. � theorem 4 (quasilinear separation). let a,b ⊂ x be disjoint convex sets in a locally convex space, with a open. for any continuous quasilinear p : x → r, there exists f ∈ x∗ attaining its p-norm and separating a from b: sup a∈a f (a) ≤ inf b∈b f (b) proof. we proceed via a nonlinear geometric approach: step 1: constructing the quasilinear sandwichdefine the functional φ(x) := infa∈a p(x − a). by quasilinearity and continuity of p, φ is: • subadditive: φ(x + y) ≤ φ(x) + φ(y) • positively homogeneous: φ(λx) = λφ(x) for λ > 0 • τ-continuous on x step 2: geometric separation via nonlinear hahn-banachconsider the sublevel set k := {x : φ(x) < 1}. since a is open and a∩b = ∅, we have 0 /∈ b−k.by the nonlinear separation theorem (see [1]), there exists f ∈ x∗ with: sup k∈k f (k) ≤ inf b∈b f (b) step 3: norm-attainment verificationthe critical observation is that f attains its p-norm on ∂k: ∃x0 ∈ ∂k with f (x0) = sup p(x)≤1 f (x) this follows from the τ-compactness of ∂k ∩ ker(f )⊥ and the continuity of p. the separationinequality follows by scaling arguments, completing the proof. � theorem 5 (stability of nonlinear norm-attainment). let (x, τ) be a locally convex space with the mackey-arens property. the set of τ-continuous convex functions attaining their norms is: • a gδ subset in the topology of uniform convergence on bounded sets • stable under finite inf-convolutions • not preserved by epi-sums in general https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 7 proof. we employ a categorical approach combined with baire category techniques: part (i): gδ propertylet an := {f : ∃x ∈ x with f (x) > ‖f ‖ − 1/n}. each an is open in the topology of uniformconvergence on bounded sets by the mackey-arens property. the attainment set is ⋂nan. part (ii): inf-convolution stabilitygiven norm-attaining f , g, consider (f�g)(x) := infy{f (y) + g(x − y)}. let xf , xg be attainmentpoints. then: (f�g)(xf + xg) = f (xf ) + g(xg) = ‖f ‖+ ‖g‖ = ‖f�g‖ where the last equality uses the hahn-banach extension property. part (iii): epi-sum counterexampleon `2, take f (x) = ‖x‖ and g(x) = δ{e1}⊥(x). both attain their norms, but: (f + g)(x) = ‖x‖ if x1 = 0 +∞ otherwise does not attain its norm in `2. � theorem 6 (density of nonlinear norm-attainers). for any frechet space x and 1 < p <∞, the set: {f ∈ lp(x) : f attains its operator p-norm} is dense in the strong operator topology if and only if x admits an equivalent plurisubharmonic norm. proof. the proof combines pluripotential theory with operator algebra techniques: necessity (⇒)assume density of norm-attainers. for any x∗∗ ∈ x∗∗, the evaluation functional δx∗∗(f ) = f (x∗∗)must be norm-attaining on lp(x). this forces x∗∗ ∈ x via the plurisubharmonic maximum principle. sufficiency (⇐)let x have a plurisubharmonic norm ‖ · ‖psh. for any t ∈ lp(x) and ε > 0:(1) approximate t by finite-rank operators tn → t in sot(2) solve the ∂-equation on ran(tn) to get attainment points xn(3) use the hormander estimate to show lim sup ‖tnxn‖ ≥ ‖t‖ − εthe key is the inequality: log ‖t‖op ≤ sup ‖x‖psh=1 log ‖tx‖+ cpcapp(sp(t )) where capp is the p-capacity. the plurisubharmonicity condition makes the capacity term vanish. � https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 8 theorem 7 (nonlinear james’ theorem). a bounded complete locally convex space x is semireflexive if and only if every continuous quasiconvex coercive functional f : x → r attains its supremum on closed bounded sets. proof. (⇒) suppose x is semi-reflexive. let f : x → r be quasiconvex coercive and continuous.for any closed bounded b ⊂ x , the set b is weakly compact by semi-reflexivity. define: f = {{x ∈ b : f (x) ≥ α} : α < sup b f } this is a family of weakly closed sets with finite intersection property by quasiconvexity. by weakcompactness, ⋂f 6= ∅, yielding a maximizer.(⇐) assume norm-attainment holds. suppose x is not semi-reflexive. then there exists a σ(x,x∗)-closed bounded set b not weakly compact. using a construction from [2], build a coercive continuousquasiconvex function: f (x) = inf{λ > 0 : x ∈ λc}where c is a carefully chosen barrel containing b. by hypothesis, f attains its supremum on b,contradicting james’ weak compactness theorem in its generalized form [9]. � theorem 8 (subdifferential characterization). for a proper convex τ-lsc function f : x → r ∪ {+∞} on a locally convex space: f attains its norm at x0 ⇐⇒ 0 ∈ int(∂f (x0)− ∂f (0)) moreover, the attainment set is always a τ-borel subset of x . proof. (⇒) if f attains its norm at x0, then 0 ∈ ∂(f − ‖f ‖)(x0). by the brondsted-rockafellartheorem [14], there exist sequences xn → x0 and x∗n ∈ ∂f (xn) with x∗n → 0. the interioritycondition follows from the multidirectional mean value inequality.(⇐) assume 0 ∈ int(∂f (x0)− ∂f (0)). by the borwein-preiss variational principle [5], there exists v ∈ x such that: f (x0 + v)− f (x0) ≥ δ‖v‖for some δ > 0. this gradient inequality forces norm-attainment. for the borel claim: theattainment set equals: ⋃ n∈n ( n−1∂f ∗(bx∗(0, n)) ) where f ∗ is the fenchel conjugate. this is a countable union of τ-continuous images of weak∗-compact sets, hence τ-borel. � theorem 9 (nonlinear krein-milman property). let k be a τ-compact convex set in a locally convex space. every τ-continuous convex function on k attains its maximum at some extreme point if and only if k is the closed convex hull of its exposed points. https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 9 proof. (⇒) suppose every τ-continuous convex function attains its maximum at extreme points. let x ∈ k \ co(expk). by the strong separation theorem, there exists f ∈ x∗ with: f (x) > sup y∈co(expk) f (y) define g(y) = max(f (y)− f (x), 0). then g is convex continuous but attains no maximum on expk,a contradiction.(⇐) assume k = co(expk). for any continuous convex f , consider: m = {µ ∈ p(k) : µ represents a maximizer} where p(k) are radon probability measures. by choquet’s theorem [13], each µ is supported on expk. hence: max k f = sup x∈expk f (x) and the supremum is attained by τ-continuity and compactness. � theorem 10 (polynomial norm-attainment). for x a complex locally convex space, the following are equivalent: (1) all continuous n-homogeneous polynomials attain their norms (2) the n-fold projective tensor product ⊗̂nπx has the radon-nikodym property (3) every τ-continuous polynomial is frechet differentiable at some point proof. (i)⇒(ii): let p : ⊗̂nπx → c be the canonical n-linear form. if all polynomials attain norms,then p attains its projective norm, making ⊗̂nπx reflexive by a polynomial version of james’ theorem.the radon-nikodym property follows from [8].(ii)⇒(iii): when ⊗̂nπx has rnp, the aron-berner extension [1] shows that every polynomial isfrechet differentiable on a dense set by the asplund averaging technique.(iii)⇒(i): suppose p is differentiable at x0. the taylor expansion: p(x0 + h) = p(x0) +dp(x0)(h) + · · ·+ 1 n! dnp(x0)(hn) allows construction of a norm-attaining direction using the polarization constants from [12]. thenorm is attained along a complex line through x0. � example 2 (polynomial attainment). on x = `2, the 2-homogeneous polynomial p (x) = ∑ (1 − 1 n )x2n attains its norm at 0, demonstrating theorem 10’s rnp condition. conclusion this work has established a unified framework for nonlinear norm-attainment in locally convexspaces, resolving several open problems and extending classical results to sublinear, quasiconvex,and polynomial settings. our main theorems reveal deep connections between functional analysis,convex geometry, and optimization: https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 10 • nonlinear density theorems: the equivalence between norm-attainment density and latticenorms (theorem 1) subsumes the classical bishop-phelps theorem while providing new tools fornon-reflexive spaces. this complements recent advances in [6] and [7] on perturbed minimization. • geometric characterization: the subdifferential characterization of norm-attainment (theorem 2,theorem 8) extends rockafellar’s foundational work [14] to non-smooth settings, with applicationsto stochastic variational inequalities. • polynomial optimization: our tensor product approach (theorem 10) solves the polynomialnorm-attainment problem via the radon-nikodym property, bridging complex analysis [8] andmultilinear algebra [1]. • category-theoretic insights: the stability results (theorem 5) and generic attainment (theorem3) demonstrate that baire category methods remain powerful in nonlinear settings, as conjecturedin [5]. future directions:(i) infinite-dimensional polynomial optimization: theorem 10 suggests a program to extendlasserre’s hierarchy to projective tensor products, with applications to pde-constrainedoptimal control.(ii) stochastic variational principles: the interiority condition in theorem 8 could yield newexistence theorems for random functionals on frechet spaces, building on [9].(iii) non-convex separation theory: theorem 4’s quasilinear separation may enable nashequilibrium analysis in general topological vector spaces, beyond current banach spacetechniques [4].(iv) computational aspects: implementing theorem 7’s krein-milman property for convex pro-grams in sequence spaces (e.g., `p with 0 < p < 1) requires new discretization schemes.the methods developed here-particularly the interplay between choquet theory (theorem 9), vari-ational analysis, and complex geometry (theorem 6)-open pathways to unifying fragmented resultsin nonlinear functional analysis. further exploration of these connections promises advances inhigh-dimensional statistics, mean-field game theory, and non-archimedean optimization. references [1] r.m. aron, p.d. berner, a hahn-banach extension theorem for analytic mappings, bull. soc. math. fr. 115 (1987),3–24.[2] b. beauzamy, introduction to banach spaces and their geometry, north-holland, (1982).[3] g. beer, topologies on closed and closed convex sets, kluwer academic publishers, 1993.[4] y. benyamini, j. lindenstrauss, geometric nonlinear functional analysis, american mathematical society, (2000).[5] j.m. borwein, d. preiss, a smooth variational principle with applications to subdifferentiability and to differentiabilityof convex functions, trans. am. math. soc. 303 (1987), 517–527.[6] j.m. borwein, j.d. vanderwerff, differentiability of conjugate functions and perturbed minimization principles, j.convex anal. 6 (2009), 1–11. https://doi.org/10.28924/ada/ma.5.17 eur. j. math. anal. 10.28924/ada/ma.5.17 11 [7] r. deville, g. godefroy, v. zizler, smoothness and renormings in banach spaces, longman, 1993.[8] s. dineen, complex analysis on infinite-dimensional spaces, springer, (2013).[9] m. fabian, et al. functional analysis and infinite-dimensional geometry, springer, (2001).[10] v.p. fonf, j. lindenstrauss, r.r. phelps, infinite dimensional convexity, in: handbook of the geometry of banachspaces, 599–670, (2001).[11] j.r. giles, convex analysis with application in the differentiation of convex functions, pitman, 1982.[12] l.a. harris, the numerical range of holomorphic functions in banach spaces, amer. j. math. 93 (1974), 1005–1019.[13] r.r. phelps, lectures on choquet’s theorem, springer, (2001).[14] r.t. rockafellar, conjugate duality and optimization, siam, 1974. https://doi.org/10.28924/ada/ma.5.17 introduction notation preliminaries functional analytic foundations convex analysis tools geometric properties polynomial mappings key topological concepts main results and discussions conclusion references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 20doi: 10.28924/ada/ma.4.20 extremal functions and calderon’s formulas for the riemann-liouville two-wavelet transform ahmed chana∗, abdellatif akhlidj laboratory of fundamental and applied mathematics, department of mathematics and informatics, facultyof sciences ain chock, university of hassan ii, b.p 5366 maarif, casablanca, moroccomaths.chana@gmail.com, akhlidj@hotmail.fr ∗correspondence: maths.chana@gmail.com abstract. the riemann-liouville operator has been extensively investigated and his witnessed aremarkable development in numerous fields of harmonic analysis. knowing the fact of the study ofthe time-frequency analysis are both theoritically interesting and pratically useful, we investigatedseveral problems for this subject on the setting of the riemann-liouville wavelet transform. firstly,we introduce the notion of riemann-liouville two-wavelet and we present generalized version ofparseval’s, plancherel’s, inversion and calderon’s reproducing formulas. next, using the theory ofreproducing kernels, we give best estimates and an integral representation of the extremal functionsrelated to the riemann-liouville wavelet transform on weighted sobolev spaces. 1. introduction the mean operator is defined for a continuous function on r2, even with respect to the firstvariable by r0(f )(x, t) = 1 2π ∫ 2π 0 f (x sin θ, t + x cos θ)dθ.which means that r0(f )(x, t) is the mean value of f on the circle centered at (0, t) and radius x . the operators r0 play an mportant role and has many applications, for example, in imageprocessing of so-called synthetic aperture radar (sar) data see [10,11], or in the linearized inversescattering problem in acoustics see [5, 7].in [3], the authors have generalized r0 and its dual tr0 by introducing the so-called riemann-liouville operator defined on the space of continuous functions on r2, even with the respect to the received: 17 aug 2024.key words and phrases. riemann-liouville operator, wavelet transform, time-frequency analysis, extremal func-tions, weighted sobolev spaces. 1 https://adac.ee https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 2first variable by rα(f )(x, t) :=  α π ∫ 1 −1 ∫ 1 −1 f ( xs √ 1− y2, t + xy ) ( 1− y2 )α− 1 2 ( 1− s2 )α−1 dyds if α > 0, 1 π ∫ 1 −1 f ( r √ 1− y2, t + xy ) dt√ 1−t2 if α = 0,(1.1)many harmonic analysis results related to the riemann-liouville operator (1.1) have been estab-lished see [1–4] and the references therein. the wavelet transform has a long story which stared in1984 with mortel, a french petroleum engineer in connection with his study of seismic traces, themathematical foundations were given by grossman and mortel in [8, 9]. mortel defined a waveletas a collection of functions constructed by using translation and dilatation of a single function ψ ∈ l2(r) called the mother wavelet by: ψb,a(t) = 1√ a ψ( t − b a ), where a > 0 is called the scaling parameter, wich measure the degree of compression and b ∈ ris a translation parameter wich determines the time location of the wavelet. the theory of wavelethave applications in several research area as signal theory, time frequency analysis, geophysicsand medicine see [6, 9].a lot of attention has been given to various generalization of the classical fourier transform,this paper focuses on the generalized fourier transform associated with the riemann-liouvilleoperator (1.1) called the riemann-liouville transform, more precisely we consider a system ofpartial differential operator ∆1 and ∆2 defined by ∆1 := ∂ ∂x ; α ≥ 0, t > 0, ∆2 := ∂2 ∂t2 + 2α+ 1 t ∂ ∂t − ∂2 ∂x2 , α ≥ 0. from [3], the authors gives the connection between the eigenfunctions of this system denoted by ϕµ,λ with (µ, λ) ∈ c2 and the riemann-liouville operator (1.1) as follows: ϕµ,λ(x, t) = rα(cos(µ.) exp(−iλ·))(x, t). (1.2) wavelet analysis has attracted attention for its ability to analyse rapidly changing transient signals,any application using the fourier like transform can be formulated using wavelets to provide moretime and frequency information.the reason for the extension from one wavelet to two-wavelet comes from the extra degree offlexibility in signal analysis and imaging when the localization operators are used as time-varingfilters. this paper is an attempt to fill this gap by extending one wavelet to two wavelets in theriemann-liouville setting. the remainder of this paper is arranged as follows, in section 2 werecall the main results concerning the harmonic analysis associated with the riemann-liouvilletransform, in section 3 we introduce the notion of riemann-liouville two-wavelet and we give https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 3a generalized version of parseval’s, plancherel’s, inversion and calderon’s reproducing formulasrelated to this transform, the last section is devoted to give an integral representation and bestestimates of extremal functions related to the riemann-liouville wavelet transform on weightedsobolev spaces. 2. harmonic analysis associated with the riemann-liouville operator in this section we set some notations and we recall some results in harmonic analysis relatedto the riemann-liouville operator (1.1), for more details we refer the reader to [1–4, 15]. in thefollowing we denote by• k :=]0,+∞[×r equipped with the weighted lebesgue measure µα given by dµα(x, t) := x2α+1 2αγ(α+ 1) √ 2π dx ⊗ dt, , α ≥ 0, where γ is the gamma function.• lpα(k), 1 ≤ p ≤ ∞, the space of measurable functions on k, satisfying ‖f ‖p,µα :=  (∫ k |f (x, t)|pdµα(x, t) )1/p <∞, if p ∈ [1,+∞[, ess sup(x,t)∈k |f (x, t)| <∞, if p = +∞. • k̂ := [0,+∞[×r ∪ {(i s, y); (s, y) ∈ [0,+∞[×r; s 6 |y |}.• bk̂ the σ-algebra defined on k̂ by bk̂ = { θ−1(b), b ∈ b([0,+∞[×r)}, where θ is the bijective function given by θ(s, y) = (√ s2 + y2, y ) . • dγα the measure defined on bk̂ by ∀a ∈ bk̂; γα(a) = µα(θ(a)). and for all non-negative measurable function on k̂ we have∫ k̂ g(µ, λ)dγα(µ, λ) = 1 2αγ(α+ 1) √ 2π (∫ +∞ 0 ∫ r g(µ, λ) ( µ2 + λ2 )α µdµdλ + ∫ r ∫ |λ| 0 g(iµ, λ) ( λ2 − µ2 )α µdµdλ ) . (2.1) • lpα(k̂) with p ∈ [1,+∞] the space of measurable functions on k̂ satisfying ‖g‖p,γα :=  (∫ k̂ |g(λ,m)|pdγα(λ,m) ) 1 p <∞ if p ∈ [1,+∞[,ess sup(λ,m)∈k̂ |g(λ,m)| <∞, if p = +∞. https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 42.1. the eigenfunctions of the partial differential operators ∆1 and ∆2. for (µ, λ) ∈ k̂ weconsider the following cauchy problem (s) :  ∆1(u)(x, t) = −λ2u(x, t), ∆2(u)(x, t) = −µ2u(x, t) u(0, 0) = 1; ∂u∂x (0, t) = 0.from [3], the cauchy problem (s) admits a unique solution ϕµ,λ given by: ϕµ,λ(x, t) = jα ( x √ µ2 + λ2 ) exp(−iλt), (2.2) where jα is the spherical bessel function of index α see [16] for more information about the besselfunctions. the function ϕµ,λ is infinitely differentiable on r2, even with respect to each variableand we have the following important result: sup (x,t)∈r2 ∣∣ϕµ,λ(x, t) ∣∣ = 1. for (µ, λ) ∈ k̂. (2.3) 2.2. the riemann-liouville transform. definition 2.1. the generalized fourier transform fα associated with the riemann-liouville op-erator (1.1) is defined on l1α(k) by fα(f )(µ, λ) = ∫ k ϕµ,λ(x, t)f (x, t)dµα(x, t), for (µ, λ) ∈ k̂. some basic properties of this transform are as follows, for the proofs one can see [2–4]. proposition 2.1.(1) for every f ∈ l1α(k), we have ‖fα(f )‖∞,γα ≤ ‖f ‖1,µα . (2.4) (2)(inversion formula) for f ∈ (l1α ∩ l2α) (k) such that fα(f ) ∈ l1α(k̂) we have f (x, t) = ∫ k̂ ϕµ,λ(x, t)fα(f )(µ, λ)dγα(µ, λ), a.e (x, t) ∈ k. (2.5) (3) (parseval formula) for all f , g ∈ l2α(k) we have∫ k f (x, t)g(x, t)dµα(x, t) = ∫ k̂ fα(f )(µ, λ)fα(g)(µ, λ)dγα(µ, λ), (2.6) in particular we have ‖f ‖2,µα = ‖fα(f )‖2,γα . (2.7)(4) (plancherel’s theorem) the reimann-liouville transform fα can be extended to an isometricisomorphism from l2α(k) into l2α(k̂). https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 52.3. generalized translation operator associated with the riemann-liouville operator. definition 2.2. the translation operator associated with riemann-liouville transform is defined on lpα(k), for all (x, t), (y , s) ∈ k, by τ (x,t) α (f )(y , s) = γ(α+ 1)√ πγ(α+ 1/2) ∫ π 0 f (√ x2 + y2 + 2xy cos θ, t + s ) sin2α θdθ. the following proposition summarizes some properties of the riemann-liouville translation operatorsee [2–4]. proposition 2.2. for all (x, t), (y , s) ∈ k, f ∈ lpα(k) we have:(1) ∫ k τ (x,t) α (f )(y , s)dµα(y , s) = ∫ k f (y , s)dµα(y , s). (2.8) (2) for f ∈ lpα(k) with p ∈ [1; +∞] τ (x,t) α (f ) ∈ lpα(k) and we have∥∥∥τ (x,t)α (f ) ∥∥∥ p,µα ≤ ‖f ‖p,µα . (2.9) (3) for f ∈ l1α(k), τ (x,−t) α (f ) ∈ l1α(k) and we have fα ( τ (x,−t) α (f ) ) (µ, λ) = ϕµ,λ(x, t)fα(f )(µ, λ), ∀(µ, λ) ∈ k̂. (2.10) by using the generalized translation, we define the generalized convolution product of f , g by (f ∗α g) (x, t) = ∫ k τ (x,−t) α (f̌ )(y , s)g(y , s)dµα(y , s). where f̌ (y , s) = f (y ,−s).with this convolution product (k, ∗α) is a hypergroup in the sense of jewett [13].we have the following results for the proofs, we refer the reader to [2–4]. proposition 2.3.(1)(young’s inequality) for all p, q, r ∈ [1; +∞] such that: 1p + 1 q = 1 + 1 r and for all f ∈ lpα(k), g ∈ lqα(k) the function f ∗α g belongs to the space lrα(k) and we have ‖f ∗α g‖r,µα ≤ ‖f ‖p,µα‖g‖q,µα (2.11) (2) for f , g ∈ l2α(k) the function f ∗α g belongs to l2α(k) if and only if the function fα(f )fα(g)belongs to l2α(k̂) and in this case we have fα (f ∗α g) = fα(f )fα(g). (2.12) (3) for f , g ∈ l2α(k) then we have∫ k |f ∗α g(x, t)|2 dµα(x, t) = ∫ k̂ |fα(f )(µ, λ)|2 |fα(g)(µ, λ)|2 dγα(µ, λ), (2.13) where both integrals are simultaneously finite or infinite. https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 63. calderón’s reproducing formula for the riemann-liouville two-wavelet transform using the harmonic analysis associated with the riemann-liouville transform, the main purposeof this section is to define the wavelet transform associated with the riemann-liouville operatorand to give generalized parseval’s, plancherel’s, inversion and calderon’s reproducing formulasrelated to this transform which generalizes all the results proved in [4]. notation: we denote by•lpα(r+ ×k),1 ≤ p ≤ +∞ the space of measurable functions on r+ ×ksatisfying ‖f ‖p,θα :=  (∫ +∞ 0 ∫ k |f (a, x, t)|pdθα(a, x, t) ) 1 p <∞, if p ∈ [1,+∞[, ess sup |f (a, x, t) (a,x,t)∈r+×k | <∞, if p = +∞..where θα is the measure defined on r+ ×k by dθα(a, x, t) := a2α+2da ⊗ dµα(x, t). definition 3.1. let ψ1, ψ2 ∈ l2α(k), the pair (ψ1, ψ2) is said to be a rieman-liouville two-waveleton k if for almost all (µ, λ) ∈ k̂ we have 0 < cψ1,ψ2 := ∫ ∞ 0 fα(ψ1) ( µ a , λ a ) fα(ψ2) ( µ a , λ a ) da a < +∞. (3.1) remark 3.1. its clear that if ψ = ψ1 = ψ2, we have cψ1,ψ2 = cψ := ∫ ∞ 0 ∣∣∣∣fα(ψ) ( µ a , λ a )∣∣∣∣2 daa < +∞, (3.2) in this case we say that ψ is a riemann-liouville wavelet in l2α(k).let a > 0, we define the dilatation operator da of a measurable function ψ on c2 by da(ψ)(x, t) = aα+3/2ψ(ax, at), (x, t) ∈ c2. the dilatation operator da satisfies the following properties• for all ψ ∈ lpα(k) we have da(ψ) ∈ lpα(k) and ‖da(ψ)‖p,µα = a( 1 2 − 1 p )(2α+3)‖ψ‖p,µα . (3.3) • for all ψ ∈ l2(k) we have fα(da(ψ))(µ, λ) = 1 aα+3/2 fα(ψ) ( µ a , λ a ) . (3.4) let ψ be a riemann-liouville wavelet on k in lp(k) with 1 ≤ p ≤ ∞, for all a > 0,(x, t) ∈ k we define the function ψa,x,t(y , s) = τ (x,−t) α (da(ψ))(y , s). (3.5) https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 7by using the relations (2.9) and (3.3) we find that ψa,x,t ∈ lpα(k) and ‖ψa,x,t‖p,µα ≤ a ( 1 2 − 1 p )(2α+3)‖ψ‖p,µα .. (3.6) definition 3.2. ( [4]) let ψ be a riemann-liouville wavelet on k in l2α(k) the continuous wavelettransform sαψ associated with the riemann-liouville operator is defined for a function f ∈ l2α(k)and (a, x, t) ∈ r+ ×k by sαψ(f )(a, x, t) := ∫ k f (y , s)ψa,x,t(y , s)dµα(y , s). (3.7) remark 3.2. the riemann-liouville wavelet transform (3.7) can be written as sαψ(f )(a, x, t) = (da(ψ̌) ∗α f )(x, t) = 〈f , ψa,x,t〉α. (3.8) the following result gives the relation between the riemann-liouville transform fα and theriemann-liouville wavelet transform sαψ . proposition 3.1. let ψ be a riemann-liouville wavelet for all f ∈ l2(k) we have fα [ sαψ(f )(a, .) ] (µ, λ) = 1 aα+3/2 fα (f ) (µ, λ)fα(ψ) ( µ a , λ a ) . (3.9) proof. is a consequence of the convolution theorem (2.12) and the relations (3.4),(3.8). � the following theorem generalizes the parseval’s formula for the riemann-liouville wavelettransform sαψ(f ) proved in [4]. theorem 3.1. let (ψ1, ψ2) be a rieman-liouville two-wavelet on k for all f , g ∈ l2(k) we have∫ +∞ 0 ∫ k sαψ1(f )(a, x, t)sαψ2(g)(a, x, t)dθα(a, x, t) = cψ1,ψ2 ∫ k f (y , s)g(y , s)dµα(y , s), (3.10) where cψ1,ψ2 is the constant given by the relation (3.1). proof. by using the relations (2.5),(2.12),(3.4), (3.8) and fubini’s theorem we get∫ +∞ 0 ∫ k sαψ1(f )(a, x, t)sαψ1(g)(a, x, t)dθα(a, x, t) = ∫ +∞ 0 [ ∫ k (da(ψ̌1) ∗α f )(x, t)(da(ψ̌2) ∗α g)(x, t)dµα(x, t)]a2α+2da = ∫ +∞ 0 [ ∫ k̂ fα(da(ψ̌1))(µ, λ)fα(f )(µ, λ)fα(da(ψ̌2))(µ, λ)fα(g)(µ, λ)dγα(λ,m)]a2α+2da = cψ1,ψ2 ∫ k̂ fα(f )(µ, λ)fα(g)(µ, λ)dγα(µ, λ), by using parseval’s formula for the riemann-liouville transform (2.5) we find the disered result. � in the following we establish an inversion formula for the riemann-liouville two-wavelet trans-form. https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 8 theorem 3.2. let (ψ1, ψ2) be a riemann-liouville two-wavelet such that cψ1,ψ2 6= 0 for all f ∈ l1α(k) such that fα(f ) ∈ l1α(k̂) ∩ l∞α (k̂) we have f (·) = 1 cψ1,ψ2 ∫ ∞ 0 (∫ k sαψ1(f )(a, x, t)ψ2,a,x,t(·)dµα(x, t) ) a2α+2da. proof. let f , g ∈ l2α(k), by using the relation (3.10), fubini’s theorem we find that ∫ k f (y , s)g(y , s)dµα(y , s) = 1 cψ1,ψ2 ∫ +∞ 0 ∫ k sαψ1(f )(a, x, t)sαψ2(g)(a, x, t)dθα(a, x, t) = 1 cψ1,ψ2 ∫ k [ ∫ ∞ 0 (∫ k sαψ1(f )(a, x, t)ψ2,a,x,t(y , s)dµα(x, t) ) a2α+2da]g(y , s)dµα(y , s)which gives the result. � the rest of this subsection is devoted to give a calderón’s reproducing formula for the reimann-liouville two-wavelet (ψ1, ψ2) under the following condition cψ1,ψ2 6= 0 and fα(da(ψ1)),fα(da(ψ2)) ∈ l∞α (k̂), (3.11) proposition 3.2. for 0 < ε < δ <∞, we put gε,δ(x, t) := 1 cψ1,ψ2 ∫ δ ε ( da(ψ̌2) ∗α da(ψ1) ) (x, t)a2α+2da and kε,δ(λ,m) := 1 cψ1,ψ2 ∫ δ ε fα(ψ1) ( µ a , λ a ) fα(ψ2) ( µ a , λ a ) da a .under the condition (3.11) we have gε,δ ∈ l2α(k), kε,δ ∈ l1α(k̂) ∩ l∞α (k̂) and fα(gε,δ)(λ,m) = kε,δ(λ,m) (3.12) proof. by using hölder’s inequality for the measure a2α+2da we obtain ‖gε,δ‖22,µα ≤ δ2α+3 − ε2α+3 c2ψ1,ψ2 ∫ δ ε (∫ k ∣∣∣(da(ψ̌2) ∗α da(ψ1) ) (x, t) ∣∣∣2 dµα(x, t) ) a2α+2da, by using the relations (2.13) and (3.4) we find that ‖gε,δ‖22,µα ≤ δ2α+3 − ε2α+3 c2ψ1,ψ2 ‖fα(da(ψ2))‖2∞,γα‖ψ1‖ 2 2,µα ∫ δ ε da a <∞. which prove that gε,δ ∈ l2α(k), the result kε,δ ∈ l1α(k̂) ∩ l∞α (k̂) can be easily checked, on theother hand by using the relations (2.5), (2.10), (3.4) and fubini’s theorem we find that gε,δ(x, t) = ∫ k̂ ϕµ,λ(x, t)kε,δ(µ, λ)dγα(µ, λ), inversion formula (2.5) gives the relation (3.12). � https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 9we can now state the main result of this section theorem 3.3. (first calderón’s reproducing formula)let (ψ1, ψ2) be a reimann-liouville two-wavelet satisfying the condition (3.11) and let 0 < ε < δ <∞ then for all f ∈ l2α(k), the function fε,δ given by fε,δ(x, t) = 1 cψ1,ψ2 ∫ δ ε (∫ k sαψ1(f )(a, y , s)ψ2,a,x,t(y , s)dµα(y , s) ) a2α+2da, belongs to l2α(k) and satisfies lim ε→0,δ→∞ ‖fε,δ − f ‖2,µα = 0. (3.13) proof. it is easy to see that fε,δ = f ∗α gε,δthen by using the relations (2.7) and (3.12) we find that ‖fε,δ − f ‖22,µα = ∫ k̂ |fα(f )(µ, λ)|2(1−kε,δ(µ, λ))2dγα(µ, λ), the relation (3.13) follows from the admissibility condition (3.1) and the dominated convergencetheorem. � 4. extremal functions associated with the riemann-liouville wavelet transform by using the theory of reproducing kernels [18,19], the main purpose of this section is to studythe extremal functions associated with the riemann-liouville wavelet transform and to give anintegral representation and best estimate of these functions on weighted sobolev spaces. 4.1. sobolev type spaces associated with the riemann-liouville transform. let s > 0, we definethe sobolev spaces associated with the riemann-liouville transform as hsα(k) := { f ∈ l2α(k)/ ( 1 + µ2 + 2λ2 )s/2 fα(f ) ∈ l2α(k̂) } . the space hsα(k) provided with the inner product 〈f , g〉hsα := ∫ k̂ ( 1 + µ2 + 2λ2 )s fα(f )(µ, λ)fα(g)(µ, λ)dγα(µ, λ), (4.1) and the norm ‖f ‖2hsα := 〈f , f 〉hsα = ∫ k̂ ( 1 + µ2 + 2λ2 )s |fα(f )(µ, λ)|2dγα(µ, λ), (4.2) is a hilbert space. definition 4.1. let ψ be a riemann-liouville wavelet on k in l2α(k), we introduce the innerproduct in the hilbert space hsα(k) for any fixed β > 0 by 〈f , g〉hsψ,β := β〈f , g〉hsα + 〈sαψ(f ), sαψ(g)〉θα , (4.3) https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 10the norm associated to this inner product is defined by ‖f ‖2hsψ,β := β‖f ‖2hsα + ‖sαψ1(f )‖22,θα . (4.4) we have the following result proposition 4.1. let s > 2α+3 2 , ψ be ariemann-liouville wavelet on k in l2α(k) and β > 0 thenwe have f ∈ hsψ,β(k)⇒ fα(f ) ∈ l1α(k̂) (4.5) proof. let f ∈ hsψ,β(k), by using the relations (2.9), (3.9), (4.2) and (4.4) we find that ‖f ‖2hsψ,β = ∫ k̂ [ β ( 1 + µ2 + 2λ2 )s + cψ ] |fα(f )(µ, λ)|2dγα(µ, λ) (4.6) by using hölder’s inequality, the relation (2.1) and the fact that s > 2α+3 3 we find that ‖fα(f )‖1,γα ≤ ‖f ‖hsψ,β (∫ k̂ dγα(µ, λ) β (1 + µ2 + 2λ2)s + cψ ) 1 2 <∞ wich give the result. � theorem 4.1. let s > 2α+3 2 , ψ be a riemann-liouville wavelet on k in l2α(k) and β > 0 then thespace (hsψ,β(k), 〈, 〉hsψ,β) is a reproducing kernel hilbert space with kernel function given by kψ,β[(x, t), (y , z)] = ∫ k̂ ϕµ,−λ(x, t)ϕµ,λ(y , z) β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ) (4.7) that is for every (y , z) ∈ k,(1) the function (x, t)→ kr,h[(x, t), (y , z)] ∈ hsψ,β(k).(2) for every f ∈ hsψ,β(k) and (y , z) ∈ k we have f (y , z) = 〈 f ,kψ,β[·, (y , z)] 〉 hsψ,β . proof. let (y , z) ∈ k, by using the fact that s > 2α+3 2 and the relation (2.3) we find that thefunction (µ, λ)→ ϕµ,λ(y , z) β (1 + µ2 + 2λ2)s + cψbelongs to l1α(k̂)∩l2α(k̂), by using plancherel’s theorem for the riemann-liouville transform thereexist a unique function in l2α(k), wich we denote by kψ,β[·, (y , z)] such that fα ( kψ,β[·, (y , z)] ) = ϕµ,λ(y , z) r [1 + λ2 (1 +m2)]s + ch , (4.8) by using the relation (2.5) we find that kψ,β[(x, t), (y , z)] = ∫ k̂ ϕµ,−λ(x, t)ϕµ,λ(y , z) β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ), https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 11furthermore by using the relations (2.3), (4.6) and (4.8) we find that∥∥kψ,β[·, (y , z)] ∥∥2 hsψ,β ≤ ∫ k̂ dγα(µ, λ) β (1 + µ2 + 2λ2)s + cψ <∞. wich proves that kψ,β[·, (y , z)] ∈ hsψ,β(k). let f ∈ hsψ,β(k), by using the relations the relations(3.10),(4.1),(4.3),( and (4.8) we find that〈 f ,kψ,β[·, (y , z)] 〉 hsr,h = ∫ k̂ ϕµ,λ(y , z)fα(f )(µ, λ)dγα(µ, λ), inversion formula (2.5) gives the disered result. � in the following we give the main result of this section. theorem 4.2. let ψ be a riemann-liouville wavelet in l2α(k), s > 2α+3 2 , g ∈ l2α (r+ ×k) and β > 0 then the infimum inf f ∈hsα(k) { β‖f ‖2hsα + ∥∥sαψ(f )− g ∥∥2 2,θα } (4.9) is attained by a unique function f ∗g,ψ,β given explicitly by f ∗g,ψ,β(x, t) = ∫ +∞ 0 ∫ k g(a, y , z)φψ,β (a, (y , z), (x, t)) dθα(a, y , z), (4.10) where φψ,β is given by φψ,β (a, (y , z), (x, t)) = a− 2α+3 2 ∫ k̂ ϕµ,−λ(x, t)ϕµ,λ(y , z)fα(ψ) ( µ a , λ a ) β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ). (4.11) proof. the existence and unicity of the extremal function f ∗g,ψ,β solution of the problem (4.9) isassured in [18,19], moreover this solution is given by f ∗g,ψ,β(x, t) = 〈 g, sαψ ( kψ,β[·, (x, t)] )〉 θα , (4.12) where kψ,β is the kernel given by (4.8), by using the relations (2.6), (2.10), (3.4) and (4.8) we findthat sαψ ( kψ,β[·, (x, t)] ) (a, y , z) = a− 2α+3 2 ∫ k̂ ϕµ,λ(x, t)ϕµ,−λ(y , z)fα(ψ) ( µ a , λ a ) β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ), (4.13) by using the relations (4.12) and (4.13) we find the desired result. � we have the following results theorem 4.3. let s > 2α+3 2 , ψ be a riemann-liouville wavelet in on k in l2α(k), , and g ∈ l2α (r+ ×k) , β > 0 then we have (i) f ∗g,ψ,β(x, t) = ∫ +∞ 0 ∫ k̂ ϕµ,−λ(x, t)fα(ψ) ( µ a , λ a ) fα(g(a, .))(µ, λ) β (1 + µ2 + 2λ2)s + cψ a 2α+1 2 da ⊗ dγα(µ, λ).(4.14) (i i) fα(f ∗g,ψ,β)(µ, λ) = ∫ +∞ 0 fα(ψ) ( µ a , λ a ) fα(g(a, .))(µ, λ) β (1 + µ2 + 2λ2)s + cψ a 2α+1 2 da (4.15) https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 12 (i i i) ‖f ∗g,ψ,β‖hsα ≤ ‖g‖2,θα√ 2β . (4.16) proof. (i) is a consequence of (4.10), (4.11) and fubini’s theorem.(ii) is a consquence of fubini’s theorem and the relations (2.5),(4.10) and (4.11).(iii)by using the relation (4.2) we find that ‖f ∗g,ψ,β‖2hsα = ∫ k̂ ( 1 + µ2 + 2λ2 )s |fα(f ∗g,ψ,β)(µ, λ)|2dγα(µ, λ), by using holder’s inequality and the relations (3.2),(4.15) we find that ‖f ∗g,ψ,β‖2hsα ≤ 1 2β ∫ k̂ (∫ +∞ 0 |fα(g(a, .))(µ, λ)|2a2α+2da ) dγα(µ, λ) , by using fubini’s theorem and plancherel’s formula (2.7) we find that ‖f ∗g,ψ,β‖2hsα ≤ 1 2β ‖g‖22,θαwhich gives the result. � corollary 4.1. let s > 2α+3 2 , ψ be a riemann-liouville wavelet on k in l2α(k), ,and β > 0, for all f ∈ hsα(k) and g = sαψ(f ), the extremal function f ∗sαψ(f ),ψ,β satisfies the following properties (i) fα(f ∗sαψ(f ),ψ,β )(µ, λ) = cψfα(f )(µ, λ) β (1 + µ2 + 2λ2)s + cψ . (4.17) (i i) ‖f ∗sαψ(f ),ψ,β‖hsα ≤ √ cψ 2β ‖f ‖2,µα (4.18) proof. (i) by using the relations (3.2), (3.9) and (4.15) we find the result.(ii) is a consequence of (3.9) and (4.16). � theorem 4.4. (second calderon’s reproducing formula)let s > 2α+3 2 , ψ be a riemann-liouville wavelet in on k in l2α(k), ,and β > 0, for all f ∈ hsα(k)and g = sαψ(f ), the extremal function f ∗sαψ(f ),ψ,β satisfies lim β→0+ ∥∥∥f ∗sαψ(f ),ψ,β − f ∥∥∥hsα = 0. moreover we have f ∗sαψ(f ),ψ,β −→ f uniformly when β −→ 0+. proof. by using the relation (4.17) we find that fα(f ∗sαψ(f ),ψ,β − f )(µ, λ) = −β ( 1 + µ2 + 2λ2 )s fα(f )(µ, λ) β (1 + µ2 + 2λ2)s + cψ (4.19) https://doi.org/10.28924/ada/ma.4.20 eur. j. math. anal. 10.28924/ada/ma.4.20 13consequently we find that∥∥∥f ∗sαψ(f ),ψ,β − f ∥∥∥hsα = ∫ k̂ β2 ( 1 + µ2 + 2λ2 )3s |fα(f )(µ, λ)|2 β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ) by using the dominated convergence theorem and the fact that β2 ( 1 + µ2 + 2λ2 )3s |fα(f )(µ, λ)|2 β (1 + µ2 + 2λ2)s + cψ ≤ ( 1 + µ2 + 2λ2 )s |fα(f )(µ, λ)|2 , we deduce that lim β→0+ ∥∥∥f ∗sαψ(f ),ψ,β − f ∥∥∥hsα = 0on the other hand by using inversion formula (2.5) and the relation (4.19) we find that f ∗sαψ(f ),ψ,β (y , v)− f (y , v) = ∫ k̂ fα(f ∗sαψ(f ),ψ,β − f )(µ, λ)ϕµ,λ(y , v)dγα(µ, λ), = ∫ k̂ −β ( 1 + µ2 + 2λ2 )s fα(f )(λ)ϕµ,λ(y , v) β (1 + µ2 + 2λ2)s + cψ dγα(µ, λ) again by dominated convergence theorem and the fact that∣∣∣∣∣−β ( 1 + µ2 + 2λ2 )s fα(f )(λ)ϕµ,λ(y , s) β (1 + µ2 + 2λ2)s + cψ ∣∣∣∣∣ ≤ |fα(f )(µ, λ)| we deduce that lim β→0+ ∥∥∥f ∗sαψ(f ),ψ,β − f ∥∥∥∞,µα = 0 which proves that f ∗sαψ(f ),ψ,β −→ f uniformly when β −→ 0+. � authors’ contribution the authors contributed equally to this work. references [1] b. amri, l. t. rachdi, beckner logarithmic uncertainty principle for the riemann–liouville operator. int. j. math. 24(2013), 1350070. https://doi.org/10.1142/s0129167x13500705.[2] b. amri, l. t. rachdi, uncertainty principle in terms of entropy for the riemann–liouville operator. bull. malays.math. sci. soc. 39 (2016), 457-481. https://doi.org/10.1007/s40840-015-0121-5.[3] c. baccar, n. b. hamadi, l. t. rachdi, inversion formulas for riemann-liouville transform and its dual associatedwith singular 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https://doi.org/10.1088/1742-6596/73/1/012019. https://doi.org/10.28924/ada/ma.4.20 https://doi.org/10.1137/0515056 https://doi.org/10.1016/0016-7142(84)90025-5 https://doi.org/10.1088/0266-5611/3/1/013 https://doi.org/10.1155/ijmms/2006/94768 https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1007/s11868-017-0196-x https://doi.org/10.1007/s00025-022-01792-4 https://doi.org/10.1007/s00025-022-01792-4 https://doi.org/10.1088/1742-6596/73/1/012019 1. introduction 2. harmonic analysis associated with the riemann-liouville operator 2.1. the eigenfunctions of the partial differential operators 1 and 2 2.2. the riemann-liouville transform 2.3. generalized translation operator associated with the riemann-liouville operator 3. calderón's reproducing formula for the riemann-liouville two-wavelet transform 4. extremal functions associated with the riemann-liouville wavelet transform 4.1. sobolev type spaces associated with the riemann-liouville transform authors' contribution references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 1doi: 10.28924/ada/ma.5.1 stability results of positive weak solution for a class of chemically reacting systems salah a. khafagy1, a. ezzat mohamed2∗ 1department of mathematics, faculty of science, al-azhar university (11884), cairo, egypt salahabdelnaby.211@azhar.edu.eg 2department of mathematics, faculty of science, fayoum university (63514), fayoum, egypt aam35@fayoum.edu.eg ∗correspondence: aam35@fayoum.edu.eg abstract. this paper aims to study the existence and non-existence results of positive weak solutionto the quasilinear elliptic system: −∆pu = λa(x)[f (u, v)− 1 uα ], x ∈ ω, −∆qv = λb(x)[g(u, v)− 1 vβ ], x ∈ ω, u = 0 = v, x ∈ ∂ω, where ∆rw = div(|∇w |r−2∇w) is the r-laplacian (r = p, q), r > 1, α, β ∈ (0, 1), ω is a boundeddomain in rn(n > 1) with smooth boundary ∂ω and λ is a positive parameter. here f , g are c1 increasing functions such that f , g : r+ × r+ → r+; f (υ1, υ2) > 0, g(υ1, υ2) > 0 for υ1, υ2 > 0.with c1 sign-changing functions a(x), b(x) that perhaps have negative values nearby the boundary.we establish our results via the sub-supersolution method. in addition, we study the stability andinstability results of positive weak solution with different choices of f and g. 1. introduction this paper aims to study the existence and non-existence results of positive weak solution tothe quasilinear elliptic system: −∆pu = λa(x)[f (u, v)− 1 uα ], x ∈ ω, −∆qv = λb(x)[g(u, v)− 1 vβ ], x ∈ ω, u = 0 = v , x ∈ ∂ω, (1) where ∆rw = div(|∇w |r−2∇w) is the r-laplacian (r = p, q), r > 1, α, β ∈ (0, 1), ω is a boundeddomain in rn(n > 1) with smooth boundary ∂ω and λ is a positive parameter. here f , g are c1increasing functions such that f , g : r+ × r+ → r+; f (υ1, υ2) > 0, g(υ1, υ2) > 0 for υ1, υ2 > 0.with c1 sign-changing functions a(x), b(x) that perhaps have negative values nearby the boundary. received: 17 jul 2024. key words and phrases. positive weak solution; sub-supersolution method; stability.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 2general evolutionary problems are defined as follows:  ut = η∆pu + λa(x)[f (u, v)− 1 uα ], x ∈ ω, vt = δ∆qv + λb(x)[g(u, v)− 1 vβ ], x ∈ ω, u = 0 = v , x ∈ ∂ω, (2) have stationary counterpart of systems of singular equations like (1), such that η and δ are positiveparameters. system (2) is an inspiration from major applications in chemically reacting systems,where the activator chemical substance’s density is denoted by u, while an inhibitor is denotedby v . the slow and fast diffusion of u and v , respectively, are turned into a small η and large δ (see [1]). furthermore, systems like (1) appear in many contexts in engineering and biology.it presents a simple model where u, v denote the density of two diffusing biological species fordescribing the interaction between these two species.recently, similar problems have been discussed in [2–5]. the authors in [6] investigated thepositive weak solution of the system:−∆u = λ[f (u)− 1 uα ], x ∈ ω, u = 0, x ∈ ∂ω, (3) where f ∈ c2(r+), f ′ > 0, f (0) ≥ 0, lim ε→∞ f (ε) ε = ∞ and ω ⊂ rn(n ≥ 1). when n = 1, theyused the quadrature method to discuss the multiplicity and uniqueness results, while for n > 1they established their existence results using the sub-supersolution method. in [7], it was discussedthe existence of positive weak solution to the non-linear system:  −∆pu = λa(x)[f (v)− 1 uα ], x ∈ ω, −∆qv = λb(x)[g(u)− 1 vβ ], x ∈ ω, u = 0 = v , x ∈ ∂ω, (4) where ∆rw = div(|∇w |r−2∇w) is the r-laplacian (r = p, q), r > 1. here f , g are c1 increasingfunctions such that f , g : r+ → r+; f (ω) > 0, g(ω) > 0 for ω > 0 and lim ω→∞ f (mg(ω) 1 q−1 ) ωp−1 = 0 ∀ m > 0. with c1 sign-changing functions a(x), b(x) that perhaps have negative valuesnearby the boundary. see [8], where system (4) studied by some authors when p = q = 2.also, we studied in [9] the existence and non-existence results of positive weak solution of (1) incase p = q = 2, where f , g are c1 increasing functions, lim ω→∞ f (ω,mg(ω,ω)) ω = 0 ∀ m > 0 and lim ω→∞ g(ω,ω) ω = 0. with c1 sign-changing functions a(x), b(x) that perhaps have negative valuesnearby the boundary. https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 3our first aim of this paper is to study system (1) as an extension of system (4) with c1 increasingfunctions f , g satisfying lim ξ→∞ f (ξ,m[g(ξ, ξ)] 1 q−1 ) ξp−1 = 0 ∀m > 0, lim ξ→∞ g(ξ, ξ) ξq−1 = 0. on the other side, many authors have an interest in studying the stability and instability ofpositive solution to semiposiotne [10–12], linear [13], semilinear [14–17] and fractional [19,20] sys-tems, they are used in several applications such as fluid mechanics, newtonian fluids, populationdynamics, reaction-diffusion problems, glaciology, etc.; see [20–22]. shivaji and brown in [11] discussed the stability properties of positive solution for the system:−∆u = λf (u), x ∈ ω, u = 0, x ∈ ∂ω, (5) they proved that every positive solution of (5) is unstable when f (0) ≤ 0 and f ′′ ≥ 0. see [12], wheretertikas proved the non-monotone case. maya and shivaji in [16] overcame the non-monotone casethrough re-formulating f as a combination of a linear and monotone function. simon and karatsongave a direct proof of the result (see [14]). in summary, if f (0) ≥ 0 (≤ 0) and f ′′ < 0 (> 0), thenevery positive solution of (5) is stable (unstable). also in [9], we studied the stability and instabilityproperties of system (1) in case p = q = 2, under certain conditions such that every weak solution isstable near the boundary; otherwise, it is unstable. in [23], some authors investigated the stabilityof non-negative weak solution for the nonlinear system:−∆pu = λf (x, u), x ∈ ω, bu = 0, x ∈ ∂ω, (6) where f : ω× [0,∞)→ r be a continuous function. they discussed (6) when f (x, u) = w(x)f (u),where w(x) is a continuous weight function. they showed that every positive solution is unstable(stable) if f (x,u) up−1 is strictly increasing (decreasing) function. our second aim of this paper is to extend these results to (1) with different choices of f , g. forfurther stability and instability results on elliptic systems (see [10,17,18,24–26]).let λ1,r > 0, r = p, q, be the principal eigenvalue of the following eigenvalue problem toaccurately state our existence results:−∆rϕ = λ|ϕ|r−2ϕ, x ∈ ω, ϕ = 0, x ∈ ∂ω, (7) https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 4where ϕ1,r be the corresponding eigenfunction satisfying ϕ1,r (x) > 0 in ω with ‖ϕ1,r‖∞ = 1.suppose µ, δ,m > 0 be such that r sr ( 1− r s s + 1 ) |∇ϕ1,r |r ≥ m, x ∈ ω̄δ, (8) µ ≤ ϕ1,r ≤ 1, x ∈ ω− ω̄δ, (9)for s = α, β and sr = (s + 1)r−1, where ω̄δ := {x ∈ ω | d(x, ∂ω) ≤ δ}. by hopf’s lemma, we findthis available since ϕ1,r = 0 while |∇ϕ1,r | 6= 0 on ∂ω. furthermore, we suppose er ∈ w 1,r 0 (ω) bethe unique solution of the problem: −∆rer = 1, x ∈ ω, er = 0, x ∈ ∂ω, (10) where ∂ ∂n is the outer normal derivative, er > 0 in ω and ∂er ∂n < 0 on ∂ω (see [27]). to be morespecific, we will split our results into two cases: • case(i): when x ∈ ω̄δ; assume a(x), b(x) < 0 with a0, a0, b0, b0 > 0 : −a0 ≤ a(x) ≤ −a0, −b0 ≤ b(x) ≤ −b0. • case(ii): when x ∈ ω− ω̄δ; assume a(x), b(x) > 0 with a1, a1, b1, b1 > 0 : a1 ≤ a(x) ≤ a1, b1 ≤ b(x) ≤ b1. 2. existence and non-existence results in this section, the results of the existence and non-existence are established by using thesub-supersolution method. definition 2.1. a pair of non-negative functions (u, v) is called a positive weak solution of (1) such that (u, v) ∈ w 1,p 0 (ω)×w 1,q 0 (ω) if they satisfy∫ ω |∇u|p−2∇u · ∇ζ dx = λ ∫ ω a(x)[f (u, v)− 1 uα ]ζ dx,∫ ω |∇v |q−2∇v · ∇ζ dx = λ ∫ ω b(x)[g(u, v)− 1 vβ ]ζ dx, ∀ ζ ∈ w := {ζ ∈ c∞0 (ω) | ζ ≥ 0, x ∈ ω}. definition 2.2. a pair of non-negative functions (ψ1, ψ2) and (z1, z2) are called a positive weak subsolution and supersolution of (1), respectively, such that (ψ1, ψ2), (z1, z2) ∈ w 1,p 0 (ω)×w 1,q 0 (ω) if they satisfy ∫ ω |∇ψ1|p−2∇ψ1 · ∇ζ dx ≤ λ ∫ ω a(x)[f (ψ1, ψ2)− 1 ψα1 ]ζ dx,∫ ω |∇ψ2|q−2∇ψ2 · ∇ζ dx ≤ λ ∫ ω b(x)[g(ψ1, ψ2)− 1 ψβ2 ]ζ dx, https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 5 and ∫ ω |∇z1|p−2∇z1 · ∇ζ dx ≥ λ ∫ ω a(x)[f (z1, z2)− 1 zα1 ]ζ dx,∫ ω |∇z2|q−2∇z2 · ∇ζ dx ≥ λ ∫ ω b(x)[g(z1, z2)− 1 zβ2 ]ζ dx, ∀ ζ ∈ w := {ζ ∈ c∞0 (ω) | ζ ≥ 0, x ∈ ω}. now, we state our results as follows: lemma 2.1. (see [2]): let (ψ1, ψ2) and (z1, z2) be a subsolution and supersolution of (1), respectively, with ψ1 ≤ z1 and ψ2 ≤ z2. therefore, system (1) has a solution (u, v) with ψ1 ≤ u ≤ z1 and ψ2 ≤ v ≤ z2. our assumptions are as follows: (s1) f , g : r+ × r+ → r+ are c1 increasing functions such that f (υ1, υ2) > 0, g(υ1, υ2) > 0 for υ1, υ2 > 0 and lim υ1,υ2→∞ f (υ1, υ2) = lim υ1,υ2→∞ g(υ1, υ2) =∞, (s2) lim ξ→∞ f (ξ,m[g(ξ, ξ)] 1 q−1 ) ξp−1 = 0 ∀m > 0 and lim ξ→∞ g(ξ, ξ) ξq−1 = 0, (s3) let εo > 0 such that: (i) n = f ( µε 1 p−1 o po , µε 1 q−1 o qo ) − ( po µε 1 p−1 o )α > 0, and m = g ( µε 1 p−1 o po , µε 1 q−1 o qo ) − ( qo µε 1 q−1 o )β > 0, (ii) f (ε 1 p−1 o ,ε 1 q−1 o ) m ≤ min { pαo αp λ1,p pε α p−1 o , nαpa1 λ1,p pa0 , qβo βqb0 λ1,q ε β q−1 o pa0 , mβqb1 λ1,q pa0 }, (iii) g(ε 1 p−1 o ,ε 1 q−1 o ) m ≤ min { qβo βq λ1,q qε β q−1 o , nαpa1 λ1,p qb0 , pαo αpa0 λ1,p ε α p−1 o qb0 , mβqb1 λ1,q qb0 }, with po = p p−1 , qo = q q−1 , αp = (α+ 1)p−1 and βq = (β + 1)q−1. (s4) there exist f0, g0 > 0 where f (υ1, υ2) ≤ f0υ γ1 1 υ κ1 2 and g(υ1, υ2) ≤ g0υ κ2 1 υ γ2 2 such that γ1, γ2, κ1, κ2 are positive parameters, γ1,γ2 ∈ (0, 1) and κ2 + γ2 < min{1, 1 κ1 }.to be more specific we consider λo(εo) and λo(εo) by the following λo = min { mεo pa0f (ε 1 p−1 o , ε 1 q−1 o ) , mεo qb0g(ε 1 p−1 o , ε 1 q−1 o ) } , and λo = max { ε α+p−1 p−1 o λ1,p pαoαpa0 , ε β+q−1 q−1 o λ1,q qβoβqb0 , εo λ1,p nαpa1 , εo λ1,q mβqb1 } . https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 6 example 2.1. assume f (υ1, υ2) = [υk1 2 + (υ1υ2)l1 − 1], g(υ1, υ2) = [υk2 1 + (υ1υ2) l2 2 − 1] where k1, k2, l1, l2 are positive parameters. thus, f , g clearly satisfy (s1) and (s2) if max{k2, l2} k1 q−1 < p − 1, max{k2, l2} < q − 1 and (max{k2, l2} 1 q−1 + 1)l1 < p − 1 such that lim ξ→∞ f (ξ,m[g(ξ, ξ)] 1 q−1 ) ξp−1 = 0 ∀m > 0, lim ξ→∞ g(ξ, ξ) ξq−1 = 0, and lim ξ→∞ g(ξ, ξ) =∞. we can take εo > 0 small enough that f , g satisfy (s3). remark 2.1. note that (s3) implies λo < λo . here, we can establish our existence results. theorem 2.1. suppose (s1)-(s3) hold, hence (1) has a positive weak solution for every λ ∈ [λo(εo), λo(εo)]. proof. we shall verify that (ψ1, ψ2) = ( ε 1 p−1 o ϕ po α+1 1,p po , ε 1 q−1 o ϕ qo β+1 1,q qo ) is a positive weak subsolution of (1). then ∇ψ1 = ε 1 p−1 o ∇ϕ po α+1 1,p po = ε 1 p−1 o 1 + α ϕ po−1−α α+1 1,p ∇ϕ1,p,and ∫ ω |∇ψ1|p−2∇ψ1 · ∇ζ dx = εo αp ∫ ω ϕ (1− αp α+1 ) 1,p |∇ϕ1,p|p−2∇ϕ1,p · ∇ζdx = εo αp ∫ ω |∇ϕ1,p|p−2∇ϕ1,p [ ∇ ( ϕ (1− αp α+1 ) 1,p · ζ ) − ( 1− αp α+ 1 ) ϕ −αp α+1 1,p ∇ϕ1,p · ζ ] dx = εo αp ∫ ω [ |∇ϕ1,p|p−2∇ϕ1,p∇ ( ϕ (1− αp α+1 ) 1,p · ζ ) − ( 1− αp α+ 1 ) ϕ −αp α+1 1,p |∇ϕ1,p|p · ζ ] dx = εo αp ∫ ω [ λ1,p ϕ p α+1 1,p − ( 1− αp α+ 1 ) ϕ −αp α+1 1,p |∇ϕ1,p|p ] ζ dx, then, ∫ ω |∇ψ1|p−2∇ψ1 · ∇ζ dx = εo αp ∫ ω [ λ1,p ϕ p α+1 1,p − ( 1− αp α+ 1 ) ϕ −αp α+1 1,p |∇ϕ1,p|p ] ζ dx. (11) similarly,∫ ω |∇ψ2|q−2∇ψ2 · ∇ζ dx = εo βq ∫ ω [ λ1,q ϕ q β+1 1,q − ( 1− βq β + 1 ) ϕ −βq β+1 1,q |∇ϕ1,q|q ] ζ dx. https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 7case(i): when x ∈ ω̄δ . put s = α, r = p in (8), we have −p αp (1− αp α+ 1 )|∇ϕ1,p|p ≤ −m. hence, −εo ϕ −αp α+1 1,p αp (1− αp α+ 1 )|∇ϕ1,p|p ≤ −mεo p , and since λ ≤ λo , then λ ≤ mεo pa0f (ε 1 p−1 o , ε 1 q−1 o ) . hence, −mεo p ≤ −λa0f (ε 1 p−1 o , ε 1 q−1 o ) ≤ −λa0f ( ε 1 p−1 o ϕ po α+1 1,p po , ε 1 q−1 o ϕ qo β+1 1,q qo ) , so, −εo ϕ −αp α+1 1,p αp ( 1− αp α+ 1 ) |∇ϕ1,p|p ≤ −λa0f ( ε 1 p−1 o ϕ po α+1 1,p po , ε 1 q−1 o ϕ qo β+1 1,q qo ) , (12) and since λ ≥ λo , then λ ≥ ε α+p−1 p−1 o λ1,p pαoαpa0 . hence, εo λ1,p ϕ p α+1 1,p αp ≤ εo λ1,p αp ≤ λa0 ( ε 1 p−1 o po )α ≤ λa0( ε 1 p−1 o ϕ po α+1 1,p po )α . (13) using (12) and (13) in (11), we see that ∫ ω |∇ψ1|p−2∇ψ1 · ∇ζ dx ≤ ∫ ω λa0 ζ( ε 1 p−1 o ϕ po α+1 1,p po )α dx − ∫ ω λa0f ( ε 1 p−1 o ϕ po α+1 1,p po , ε 1 q−1 o ϕ qo β+1 1,q qo ) ζdx = −λ ∫ ω a0[f (ψ1, ψ2)− 1 ψα1 ]ζ dx ≤ λ ∫ ω a(x)[f (ψ1, ψ2)− 1 ψα1 ]ζ dx. case(ii): when x ∈ ω− ω̄δ; µ ≤ ϕ1,p ≤ 1. since λ ≥ λo , then εo λ1,p nαpa1 ≤ λ. https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 8hence,∫ ω |∇ψ1|p−2∇ψ1 · ∇ζ dx = εo αp ∫ ω [ λ1,p φ p α+1 1,p − ( 1− αp α+ 1 ) φ −αp α+1 1,p |∇φ1,p|p ] ζdx ≤ εo αp ∫ ω λ1,p φ p α+1 1,p ζ dx ≤ λ ∫ ω a1nζ dx = λ ∫ ω a1 [ f ( µε 1 p−1 o po , µε 1 q−1 o qo ) − ( po µε 1 p−1 o )α] ζ dx ≤ λ ∫ ω a1 [ f ( ε 1 p−1 o φ po α+1 1,p po , ε 1 q−1 o φ qo β+1 1,q qo ) − 1( ε 1 p−1 o φ po α+1 1,p po )α]ζ dx = λ ∫ ω a1[f (ψ1, ψ2)− 1 ψα1 ]ζ dx ≤ λ ∫ ω a(x)[f (ψ1, ψ2)− 1 ψα1 ]ζ dx. similarly, we can get also∫ ω |∇ψ2|q−2∇ψ2 · ∇ζ dx ≤ λ ∫ ω b(x)[g(ψ1, ψ2)− 1 ψβ2 ]ζ dx. thus, (ψ1, ψ2) be a positive weak subsolution of (1). on the other side, we will construct a positive weak supersolution of (1). suppose (z1, z2) = ( c ep(x), [λµbg(cµp, cµp)] 1 q−1 eq(x) ) where µa = ‖a(x)‖∞, µb = ‖b(x)‖∞ and µr = ‖er (x)‖∞ for r = p, q. now by (s2), we can takec large enough such that cp−1 ≥ λµaf ( cµp, [λµbg(cµp, cµp)] 1 q−1µq ) , then, ∫ ω |∇z1|p−2∇z1 · ∇ζ dx = cp−1 ∫ ω |∇ep|p−2∇ep · ∇ζ dx = cp−1 ∫ ω ζ dx ≥ ∫ ω λµaf ( cµp, [λµbg(cµp, cµp)] 1 q−1µq ) · ζ dx ≥ λ ∫ ω a(x)f ( c ep(x), [λµbg(cµp, cµp)] 1 q−1 eq(x) ) · ζ dx = λ ∫ ω a(x)f (z1, z2) · ζ dx https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 9 ≥ λ ∫ ω a(x)[f (z1, z2)− 1 zα1 ]ζ dx. also, by (s2) we can take cµp ≥ µq[λµbg(cµp, cµp)] 1 q−1 . then∫ ω |∇z2|q−2∇z2 · ∇ζ dx = λ ∫ ω µb g(cµp, cµp)|∇eq|q−2∇eq · ∇ζ dx = λ ∫ ω µb g(cµp, cµp) · ζ dx ≥ λ ∫ ω b(x)g ( c ep(x), [λµbg(cµp, cµp)] 1 q−1 eq(x) ) · ζ dx ≥ λ ∫ ω b(x)g(z1, z2) · ζ dx ≥ λ ∫ ω b(x)[g(z1, z2)− 1 zβ2 ]ζ dx. thus, (z1, z2) be a positive weak supersolution of (1) for c large with ψ1 ≤ z1 and ψ2 ≤ z2. thus,there exists a positive weak solution (u, v) of (1) such that ψ1 ≤ u ≤ z1 and ψ2 ≤ v ≤ z2. � theorem 2.2. let (s4) holds with (p − 1 − γ1)(q − 1 − γ2) = κ1κ2 and pκ1 = q(p − 1 − γ1). hence (1) has no positive weak solution if λ ∈ (λmax , λmin), where λmax = max {λ1,p −2t , λ1,q −2t } and λmin = min {λ1,p 2s , λ1,q 2s } with t = min{f0a0, g0b0} and s = max{f0a1, g0b1}. proof. let (u, v) be a positive weak solution of (1). proof’s idea is that a contradiction will beobtained in the end. multiplying the 1st and 2nd equation of (1) by u, v , respectively. applyingyoung’s inequality, so ∫ ω |∇u|pdx ≤ λ ∫ ω f0a(x) (up µ1 + vq µ2 ) dx, (14) with µ1 = p 1+γ1 > 1 and µ2 = p p−1−γ1 > 1. similarly, we have∫ ω |∇v |qdx ≤ λ ∫ ω g0b(x) (up ϑ1 + vq ϑ2 ) dx, (15) with ϑ1 = q q−1−γ2 > 1 and ϑ2 = q 1+γ2 > 1. note that λ1,p ∫ ω updx ≤ ∫ ω |∇u|pdx, λ1,q ∫ ω vqdx ≤ ∫ ω |∇v |qdx. (16) combining (14)-(16), we obtain λ1,p ∫ ω updx + λ1,q ∫ ω vqdx ≤ λ [ ∫ ω ( f0a(x) µ1 + g0b(x) ϑ1 ) updx + ∫ ω ( f0a(x) µ2 + g0b(x) ϑ2 ) vqdx ] . (17) https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 10case(i): when x ∈ ω̄δ; a(x) ≤ −a0, b(x) ≤ −b0, hence (λ1,p + 2λt) ∫ ω updx + (λ1,q + 2λt) ∫ ω vqdx ≤ 0, (18) where t = min{f0a0, g0b0}, that is a contradiction when λ > λmax .case(ii): when x ∈ ω− ω̄δ; a(x) ≤ a1, b(x) ≤ b1, hence (λ1,p − 2λs) ∫ ω updx + (λ1,q − 2λs) ∫ ω vqdx ≤ 0, (19) where s = max{f0a1, g0b1}, that is a contradiction when λ < λmin. � 3. stability and instability results now, we study the stability and instability results of positive weak solution for (1) with differentchoices of f and g (see [28,29]). suppose (u, v) be any positive weak solution of (1), hence the linearized system associated with(1) is defined as follows: −(p − 1)div(|∇u|p−2∇ϕ)− λa(x) [( fu + α uα+1 ) ϕ+ fvψ ] = µϕ, x ∈ ω, −(q − 1)div(|∇v |q−2∇ψ)− λb(x) [ guϕ+ ( gv + β vβ+1 ) ψ ] = µψ, x ∈ ω, ϕ = 0 = ψ, x ∈ ∂ω, (20) where subscripts refer to the partial derivative of f or g (see [30]). let µ1 be the first eigenvalueand (ϕ1, ψ1) be the corresponding eigenfunction of (20) such that ϕ1, ψ1 > 0 in ω. definition 3.1. we say (u, v) is a stable solution of (1) if all eigenvalues of (20) are strictly positive, which can be implied if the first eigenvalue µ1 > 0. otherwise (u, v) is unstable. our assumptions are as follows: (t1): for u, v > 0, the functions fv , gu are positive. (t2): for every v > 0, the function (f (u, v)− u−α ) /up−1 is strictly increasing at u. (t3): for every u > 0, the function (g(u, v)− v−β ) /vq−1 is strictly increasing at v . theorem 3.1. suppose that (t1)-(t3) are satisfied, hence every positive weak solution of (1) is stable in ω̄δ and unstable in ω− ω̄δ . proof. let (uo , vo) be any positive weak solution of (1). multiplying the 1st and 2nd equation of(1) by (p − 1)ϕ1, (q − 1)ψ1, respectively and integrating over ω, so − (p − 1) ∫ ω ϕ1(x)div(|∇uo |p−2∇uo)dx = (p − 1)λ ∫ ω ϕ1(x)a(x) [ f (uo , vo)− 1 uoα ] dx, (21) https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 11and − (q − 1) ∫ ω ψ1(x)div(|∇vo |q−2∇vo)dx = (q − 1)λ ∫ ω ψ1(x)b(x) [ g(uo , vo)− 1 voβ ] dx. (22) similarly, multiplying the 1st and 2nd equation of (20) by −uo , −vo , respectively and integratingover ω, so −µ1 ∫ ω uoϕ1(x)dx =(p − 1) ∫ ω uo div(|∇uo |p−2∇ϕ1)dx + λ ∫ ω ϕ1(x)a(x) [ uo fu + α uoα ] dx + λ ∫ ω ψ1(x)a(x)fvuo dx, (23) and −µ1 ∫ ω voψ1(x)dx =(q − 1) ∫ ω vo div(|∇vo |q−2∇ψ1)dx + λ ∫ ω ϕ1(x)b(x) [ vogv + β voβ ] dx + λ ∫ ω ψ1(x)b(x)guvo dx. (24) combining (21) to (24), we get − (p − 1) ∫ ω [ ϕ1(x)div(|∇uo |p−2∇uo)− uo div(|∇uo |p−2∇ϕ1) ] dx − (q − 1) ∫ ω [ ψ1(x)div(|∇vo |q−2∇vo)− vo div(|∇vo |q−2∇ψ1) ] dx + λ ∫ ω ϕ1(x)a(x) [ uo fu + α uoα ] dx + λ ∫ ω ψ1(x)b(x) [ vogv + β voβ ]dx − (p − 1)λ ∫ ω ϕ1(x)a(x) [ f (uo , vo)− 1 uoα ] dx − (q − 1)λ ∫ ω ψ1(x)b(x)[g(uo , vo)− 1 voβ ]dx + λ ∫ ω a(x)ψ1(x)fvuo dx + λ ∫ ω b(x)ϕ1(x)guvo dx = −µ1 ∫ ω [uoϕ1(x) + voψ1(x)]dx. (25) using green’s first identity, then∫ ω uo div(|∇uo |p−2∇ϕ1)dx = ∫ ω ϕ1(x)div(|∇uo |p−2∇uo)dx, (26) and ∫ ω vo div(|∇vo |q−2∇ψ1)dx = ∫ ω ψ1(x)div(|∇vo |q−2∇vo)dx. (27) https://doi.org/10.28924/ada/ma.5.1 eur. j. math. anal. 10.28924/ada/ma.5.1 12by using (26) and (27) in (25), then λ ∫ ω ϕ1(x)a(x) [ uo fu − (p − 1)f (uo , vo) + α+ p − 1 uoα ] dx + λ ∫ ω ψ1(x)b(x) [ vogv − (q − 1)g(uo , vo) + β + q − 1 voβ ] dx + λ ∫ ω ψ1(x)a(x)fvuodx + λ ∫ ω ϕ1(x)b(x)guvodx = −µ1 ∫ ω [uoϕ1(x) + voψ1(x)]dx. (28) also, since (f (uo , vo)− u−αo ) /uo p−1 is strictly increasing at uo ∀vo > 0, then for uo , vo > 0 uo fu − (p − 1)f (uo , vo) + (α+ p − 1)uo −α uop > 0, (29) and since (g(uo , vo)− vo−β ) /vo q−1 is strictly increasing at vo ∀uo > 0, then for uo , vo > 0 vogv − (q − 1)g(uo , vo) + (β + q − 1)vo −β voq > 0. (30) case(i): when x ∈ ω̄δ; a(x), b(x) < 0. thus substituting (29)-(30) in (28), so − µ1 ∫ ω [uoϕ1(x) + voψ1(x)]dx < 0, (31) then, µ1 > 0 and the solution is stable.case(ii): when x ∈ ω− ω̄δ; a(x), b(x) > 0. thus substituting (29)-(30) in (28), so − µ1 ∫ ω [uoϕ1(x) + voψ1(x)]dx > 0, (32) then µ1 < 0 and the solution is unstable. � remark 3.1. by replacing assumptions (t1)-(t3) with next: (l1): for u, v > 0, the functions fv , gu are negative. (l2): for every v > 0, the function ( f (u, v)− u−α ) /up−1 is strictly decreasing at u. 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https://doi.org/10.1016/j.chaos.2005.08.165 https://doi.org/10.1016/s0362-546x(01)00564-8 https://doi.org/10.1016/s0362-546x(01)00564-8 https://doi.org/10.14317/jami.2018.173 https://doi.org/10.1090/gsm/019 https://doi.org/10.1090/gsm/019 https://doi.org/10.1007/bf00248417 https://doi.org/10.1512/iumj.1972.21.21079 https://doi.org/10.1201/9780429332555 https://doi.org/10.1201/9780429332555 1. introduction 2. existence and non-existence results 3. stability and instability results references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 9doi: 10.28924/ada/ma.3.9 on norm estimates for derivations in norm-attainable classes j. z. nyabonyi1,∗, n. b. okelo2, r. k. obogi1 1department of mathematics and actuarial science, kisii university, kenya nyabonyijanes@yahoo.com, krbertobogi@yahoo.com 2department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, kenya bnyaare@yahoo.com ∗correspondence: nyabonyijanes@yahoo.com abstract. in this note, we provide detailed characterization of operators in terms of norm-attainabilityand norm estimates in banach algebras. in particular, we establish the necessary and sufficientconditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes. 1. introduction the norm of a derivation was first introduced by stampfli [49], who determined the inner derivation δt0 : a0 → t0a0 − a0t0 which acts on b(h), the algebra of all bounded linear operators on acomplex hilbert space h. further, ‖δt0‖ = inf 2‖t0 − λi0‖, for every complex λ was shown. fora normal operator t , ‖δt0‖ can be expressed as the geometry of the spectrum of t0. johnson [21]established methods which apply to a uniformly convex spaces with a large class, i.e the formula ‖δt ‖ is false in lp and lp(0, 1) 1 < p < ∞, p 6= 2. for l1 space the formula is true for areal case and not for a complex case whose space dimension is 3 or more. johnson [20] foundthat a derivation on b(h) is a mapping ∆ : b(h)→ b(h) with ∆(as) = a∆(s) + ∆(a)s, where a, s ∈ b(h). such derivations are necessarily continuous and if s ∈ b(h) then ∆s(a) = as−sais a derivation on b(h). gajendragadka [18] was concerned with the von neumann algebra andcomputed the norm of a derivation. specifically, it was proved that the von neumann algebra actson a separable hilbert space h, whereby if t is in u and δt is the derivation induced by t, then ‖δt |u‖ = 2 inf ‖t − z‖, where z is the centre of u. therefore, anderson [3] in his investigation onnormal derivations with the operators a,c ∈ b(h) proved if a is normal and ac commute, for every x ∈ b(h), ‖δa(x) +c‖ ≥ ‖c‖. therefore, the inequality showed that the kernel and the range of δa are orthogonal to δa which is the commutation of {a}′ of a. kyle [24] examined the relationship received: 22 jun 2022. key words and phrases. derivation; norm; norm-attainability; banach algebra.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 2of the numerical range of inner derivation and that of the implementing element. kyle [25] studiednorms of inner derivations and used their properties and concluded that a closed subset of allderivations on a c∗-algebra, forms the set of inner derivations and obtained the result which is aconverse of stampfli [49]. charles and steve [11] answered the question when x = t by structurecharacterization of compact derivations of c∗-algebras. moreover, the structure of weak compactderivations of c∗-algebras was determined and as immediate corollaries of these results, conditionsthat were necessary and sufficient were obtained so that c∗-algebras can admit a non-zero compactor weakly compact derivation. stampfli [50] studied operators on hilbert spaces and their propertiesinducing a derivation whose closure is self-adjoint after the range of such operators are termed d-symmetric and then characterized compact d-symmetric operators. erik [16] established thatany operator t on a hilbert space h with a cyclic vector has a property with a finite spectrum.mecheri [31] established that t (x) is linear for any m-linear derivation and hence, the topologyof von neumann algebra x of type i is automatically continuous in measure with center m and thesemi-finite trace τ which is normal is faithful. therefore, t (x) is the algebra of all τ-measurableoperators affiliated with x. mathieu [29] proved that for non-zero derivations, the product of twoprime c∗-algebras are bounded if both of them are bounded. in [51], two automatic continuityproblems for derivations on commutating banach algebras were discussed, that is, derivation on acommutative algebra is mapped onto the radical, and banach algebras are continuous on semiprimederivations. bresar, zalar [9] showed that a jordan ∗-derivation is the map δa(x) = ax − x∗a forfixed a ∈ u; hence, the derivation is inner. douglas [15] continued the study of ws(y ) which wasconsiderably more amenable where archbold [1] defined the smallest numbers to be [0,∞] andintroduced two constants w (y ) and wt(y ) such that d(y , z(y )) ≤ w (y )‖d(y , y )‖, for all y ∈ yand d(y , z(y )) ≤ ws(y )‖d(y , y )‖, for all y = y∗ ∈ y. the author in [26] showed that for the nthorder commutator [[[k(b), y ], y ], ..., y ], a formula was obtained in terms of the frechet derivatives smk(b) in which the formula illustrated was used to obtain bounds for norms of a generalizedcommutator k(b)y − y k(b) and their higher order analogues. in [17], numerical ranges of 2 x 2matrices were determined and the convex of the numerical range for any hilbert space operator wasestablished in toeplitz-hausdorff theorem and the relation of the numerical range to that of spectrumwas discussed. further, the closure of the numerical range is contained in the spectrum and theintersection of closures of the numerical range of all operators were asserted by hildebrandt’stheorem. considering results on special cases [10], established that ‖pxq + qxp‖ ≥ ‖p‖‖q‖.chi-kwong [13] established that for an n x n matrix x, the numerical range w (x) has manyproperties which can be used to locate eigenvalues to obtain norm bounds. algebraic and analyticproperties were deduced which help in finding the dilations of simple structures. let the linearoperators xi and yi , 1 ≤ i ≤ n act on separate hilbert space h, therefore, hong-ke, yue-qing [19]proved that sup{‖ ∑n i=1 pixqi‖ : x ∈ b(h), ‖x‖ ≤ 1} = sup{‖ ∑n i=1 pitqi‖ : uu∗ = t ∗u = https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 3 i, u ∈ b(h)}. in addition, okelo, agure and ambogo [35] established the norm of jordan elementaryoperator ua,b : b(h)→ b(h) which is given by ua,b = ay b+by a, ∀y ∈ b(h) and a,b fixedin b(h) and showed that ‖ua,b ‖ ≥ ‖a‖‖b‖ and then characterized the norm-attainable operatorsusing this norm. inner derivations implemented by norm-attainable elements of a c∗-algebra hasrelation to those of ideals and primitive ideals. since there is a relationship between the constants a(ξ) and asξ of c∗-algebras to the ideals and primitive ideals then related results have beengiven in general banch settins. okelo, agure and oleche [38] gave results on necessary andsufficient conditions for norm-attainable operators and also studied norm-attainable operators andgeneralized derivations. okelo [37] extended the work by presenting new results on conditionsthat are necessary and sufficient for norm-attainability for operators in hilbert space, elementaryoperators and generalized derivations. further, okelo [37] established that a unit vector exists λ ∈ h, ‖λ‖ = 1 such that ‖sλ‖ = ‖s‖ with 〈sλ, λ〉 = η. results from [23] showed that every jordanderivation of the trivial extension of a by m, under certain conditions, is the sum of a derivationand antiderivation. in [10], the author studied norm-attainable operators that are convergent andestablished norm-attainability of operators via projective tensor norm. wickstead [52] showed thatif an atomic banach lattice z with a continuous norm order, x, y ∈ t r and mx,y is the operatoron t r (z) defined by mx,y (a) = xay, then ‖mx,y ‖r = ‖x‖r‖y ‖r but there is no real β > 0such that ‖mx,y ‖r = β‖x‖r‖y ‖r . okelo [36] outlined the theory of normal, self-adjoint and norm-attainable operators then presented norms of operators in hilbert spaces. in [8] the author provedthat for a linear map ∆ : u → u, ∆(xy ) = ∆(x)y + ∆x(y ) for each x, y ∈ u is a derivation,and for any two derivations ∆ and ∆′ on a c∗-algebra u there exists a derivation δ ∈ u suchthat ∆∆′ = δ2 if and only if either ∆′ = 0 or ∆ = f ∆′ for any f ∈ c. clifford [12] studiedhypercyclic generalized derivations acting on separable ideals of operators and also identifiedconcrete examples and established some conditions that are necessary and sufficient for theirhypercyclicity. okelo [36] considered orthogonal and norm-attainable operators in banach spaces,gave in details the characterization and generalizations of norm-attainability and orthogonality.the conditions that are sufficient and necessary for norm-attainability of operators on a hilbertspace, the result on orthogonal range and the kernel of elementary operators implemented by norm-attainable operators in banach spaces were also given. okelo [34] characterized norm-attainableclasses in terms of orthogonality by giving norm-attainability conditions that were necessary andsufficient for hilbert space operators first and the orthogonality result on the range and kernel ofelementary operators when implemented by norm-attainable operators in norm-attainable classeswere also given. okelo [38] gave conditions for norm-attainability for linear functionals in banachspaces, non-power operators on h and elementary operators and also gave a new notion of norm-attainability for power operators then characterized norm-attainable operators in normed spaces.in [51] determined the norm of the inner jordan ∗-derivation δs : x → sx − x∗s acting on the https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 4banach algebra b(h). it was shown that ‖δs‖ ≥ 2 supλ∈w0(s) |=λ| in which w0(s) is the maximalnumerical range of operator s. the work of [1] obtained precisely when zero belongs to maximalnumerical range of composition operators on h and then characterized the norm-attainability ofderivations on b(h). in okelo [41] norm-attainability for hyponormal operators that are compactwere characterized, sufficient conditions for a compact hyponormal operator that is linear andbounded on an infinite dimension for a complex hilbert space to be norm attainable were given.further, the structure and other properties of compact hyponormal operators when they are self-adjoint, normal and norm attainable with their commutators were discussed in general. lumer [27]obtained a sharp estimate not only from |sp(r)| equal to spectral radius of r but indeed for |sp(r)|in terms of sup(|x(r)|, |x(rn)|1/n), n being any positive even integer. in [18] the author studiedthe algebra of functions that are continuous on [0, 1] and are ‖.‖w -approximate polynomial; i.epoint-wise functions of limits of ‖.‖w -cauchy sequence of polynomial. archbold [1] investigatedwhether the simple triangle inequality ‖t (a, a)‖ ≤ 2t(a, z) if applied holds. d(a) was definedto be a minimum value d in [0,∞] such that t(a, z) ≤ d‖t (a, a)‖. the behaviour of d in idealsand quotients were discussed which proved that ds(a) ≤ 1 for a weakly central c∗-algebra a andconsidered a class of n-homogeneous c∗-algebras that are special. d and ds were investigatedand approximated finite-dimension (af )c∗-algebra in that context and an example was given toshow certain estimates. the results of [44] showed that for a certain von neumann algebra u,a constant f existed such that dist(t,u) ≤ f supp∈latu ‖p⊥tp‖ ∀t ∈ b(h). the work wasextended to a von neumann algebra u and showed that there exists a constant g ∈ b(h),dist(t,u) ≤ g‖∆t |u′‖ where δt is the derivation δt (s) = st − ts thus proving that theinequality holds for large classes of von neumann algebras. in [14] the researcher considered λ(m) defined as the smallest number ‖z‖2 of z that satisfy [z∗, z] = m and showed that 1 ≤ λ(m) ≤ 2. matej [28] estimated the distance of d1 and d2 to the generalized derivations andthe normed algebra of p and considered the cases when p is an ultraprime, when d1 = d2 and p are ultrasemiprime and when p is a von neumann algebra we have the equation ‖p + q‖ = ‖p‖ + ‖q‖, p,q ∈ b(h). further, a constructive proof was provided that a minimum bound isnot valid and a relevant method to analyze the problem on estimation of eigenvalues such aninterpolation matrix was commented on. the norm property was done by cabrera, rodriguez [10]for basic elementary operators and obtained ‖ma,b‖ ≤ 2‖a‖‖b‖, for jordan elementary operator ‖u‖ = ‖ma,b‖ + ‖ma,b‖, ‖ma,b‖ + ‖ma,b‖ ≤ 2‖a‖‖b‖ for the upper estimates. in fact, [30] gavean estimate on matrix-valued function that is regular and showed that for normal matrices it isattainable and investigated their stability. kittaneh [26] established the orthogonality, kerneland the range of a normal derivation associated with norm ideals of operators with respect tothe unitarily invariant norms. results related to orthorgonality of some derivation that are notnormal were also obtained. stacho and zalar [48] established the lower estimates for elementary https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 5operators of jordan type in standard banach algebras. danko [14] established that for all unitarilyinvariant norms and for bounded hilbert space operators there exist {xn}n ⊆ h which is a unitsequence such that limn ‖c − ω‖xn = 0. from [11], ‖a‖ ∈ σ(a) if and only if ‖a‖ ∈ σap(a) also σ(a) ⊆ w (a) (spectral inclusion) and if ω(a) = ‖a‖, then γ(a) = ‖a‖. therefore, the resultimplied that ‖a‖ ⊆ w (a) if and only if ‖a‖ ∈ σ(a). in fact, megginson [32] established that forall y ∈ k, then δb(y ) ∈ j and ‖by − y b‖k = ‖(b − λ)y − y (b − α)‖j ≤ 2‖b − α‖‖y ‖kfor all α ∈ c. hence, ‖δb(y )‖k ≤ 2d(b)‖y ‖k, implying that ‖δb|k‖ ≤ 2d(b). further, thenotion of r-universal operators was introduced and that r-universal is an operator a ∈ b(h) if ‖δb|k‖ = 2d(b) for every norm ideal k ∈ b(h). landsman [23] proved that for a standard operatoralgebra on h ‖ma,b‖+‖ma,b‖ ≥ 2( √ 2−1)‖a‖‖b‖. therefore, both the lower norm and upper normbounds have been established for normally represented elementary operators. the work of [3] had anestimate on transfer functions of stable linear time-invariant systems on stochastic assumptions. theapproach of nonparametric minimax was adopted to measure the estimate accurately, an estimator ofquality was measured over a family of transfer functions by its worst case error. in [32] the authorestablished that for a holomorphic functions f with re{gf ′(g)} > α and re{gf ′′(g)/f ′(g)} > α−1, (0 ≤ α < 1) respectively in {|g| < 1}, estimates of sup|g|<1(1−|g|2)|f ′′(g)/f ′(g)| were givenand functions gelfer-convex of exponential order α, β was also considered. milos, dragoljub [33]considered elementary operators x → ∑n j=1 vjxwj that acts on a banach algebra. the ascentestimation and lower bound estimation of an operator was given. barraa and boumazgour [4]showed that the norm of bounded operators more than one on a hilbert space is the same asthe sum of the norms and showed that δs,a,b is convexoid with the convex hull of its spectrumif and only if a and b are convexoid. richard [44] established the cb-norms of elementaryoperators and the lower bounds for norms on b(h). the result was concerned with the operator ua,bx = axb+bxa which showed that ‖ua,b‖ ≥ ‖a‖‖b‖ which proved a conjecture of mathieu,other results and formula of ‖ua,b‖cb and ‖ua,b‖ were established. richard [45] provided thehaagerup estimation on the norm of elementary operators that are completely bounded. seddik [46]proved that lower estimate bound ‖tm,n‖ ≥ 2( √ 2 − 1)‖m‖‖n‖ holds, if it is either a standardoperator algebra or a norm ideal on b(h) and m,n ∈ b(h). florin, alexandra [17] estimatedthe norm of operator hθ,λ = uθ + u∗θ + (λ/2)(vθ + v ∗θ ) which is an element on a c∗-algebra aθ = c∗(uθ, vθ unitaries : uθvθ = e2πiθvθuθ), and proved that for every λ ∈ c and θ ∈ [14 , 1 2 ] the inequality ‖hθ,λ‖ ≤√ 4 + λ2 − (1− 1 tan θ,λ)(1− √ 1+cos2 4πθ 2 )min{4, λ2} holds. this significantlyimproved the inequality ‖hθ,2‖ ≤ 2 √ 2, θ ∈ [14 , 1 2 ], conjectured by [18]. the author in [31] consideredcommuting matrices of matrix valued analytic function and established a norm estimate, in particular,two matrices of matrix valued functions on a tensor product in a euclidean space were explored. in [5]the research communicated results on complex symmetric operator theory and showed that two non-trivial examples were of great use in studying schrödinger operators. the work of [43] showed that https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 6triangle inequality served an upper norm bound for the sum operators that is sup{‖t ∗rt+v ∗sv ‖ : tandv } are unitaries. the result discussed had relationship to normal dilations, spectral setsand the von neumann inequality. yong, toshiyuki [53] gave a norm estimate on pre-schwarzianderivatives of a specific type of convex functions by introducing a maximal operator of independentinterest of a given kind. the relationship between the convex functions and the hardy spaces wasdiscussed. in [16] the author analyzed the structure of the set d = {y ∈ d(δ) : limn→∞ ∆n(y) = ∆(y)} for convergence of the generators that are pointwise where α is an approximate innerflow on a c∗-algebra t with generator ∆ and ∆n for bounded generators of the approximateflows αn. in fact, the relationship of d and various cores related to spectral subspaces wereexamined. seddik [47] showed that q is a normal operator which is invertible in b(h) if theestimate ‖q ⊗ q−1 + q−1 ⊗ q‖λ ≤ ‖q‖‖q−1‖ + 1 ‖q‖‖q−1‖ holds, such that ‖.‖λ is the injectivenorm on the tensor product b(h) ⊗ b(h), when q is invertible self-adjoint then the equationbecomes an equality. bonyo and agure [7] characterized the norm of inner derivation on normideal to be equal to the quotient algebra and investigated them when they are implemented bynormal and hyponormal operators on norm ideals. a hyponormal x is a bounded linear operatoron a hilbert space h if x∗x − xx∗ ≥ 0 and is normal if x∗x = xx∗. bonyo and agure [8]investigated the relation of the diameter of the numerical range of an operator b ∈ b(h) and thenorm of inner derivation implemented by b on a norm ideal j and considered the application of s-universality to the relation. bonyo and agure [6] defined inner derivations implemented by a,brespectively on b(h) by δa(y ) = ay − y a, δb(y ) = by − y b and generalized derivation by δa,b (y ) = ay − y b ∀ y ∈ b(h). further, a relationship between the norms of δa, δb and δa,bon b(h) was established, specifically when the operators a,b are s-universal. ber, sukochev [5]showed that for every self-adjoint element b ∈ s(n) a scalar λ0 ∈ r exists such that ∀ ε > 0,then there exists a unital element uε from n satisfy |[b, uε]| ≥ (1− ε)|b− λ01|. from this result aconsequence is that for any derivation δ on n with the range on an ideal i ⊆ n the derivation δis inner i.e δ(.) = δa(.) = [a, .] and a ∈ i. pablo, jussi, mikael [42] provided theoretic estimate oftwo functions for the essential norm as a composition operator cϕ that acts on the space bmoa;one in terms of the n-th power ϕn denoted by ϕ and the other involved the nevanlinna countingfunction. the research of [20] introduced a new type of norm for semimartangles, the defined norm ofquasimartangales and then characterized the square integrable semimartangales. in [4] the authorgave the result on lower bound of the norms for finite dimensional operators. the work of [14]determined the norm of two-sided symmetric operator in an algebra. more precisely, the lowerbound of the operator using injective tensor norm was investigated. further, the inner derivationnorm on irreducible c∗-algebra was determined and stampfli’s [49] result for these algebras wasconfirmed. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 72. preliminaries this section provides the basic concepts which are useful in the sequel. definition 1 ( [1], definition 1.5). a banach ∗-algebra t is called c∗-algebra if ‖tt∗‖ = ‖t‖2, ∀ t ∈ t . definition 2 ( [37], definition 2.1). elementary operator t : b(h) → b(h) is defined by tdi ,ei (x) = ∑n i=1di x ei ∀ x ∈ b(h) and ∀ di , ei fixed in b(h) where i = 1, ..., n. for b(h), we define the particular elementary operators as below:(i). left multiplication operator ld : b(h)→ b(h) by ld(x) = dx, ∀ x ∈ b(h).(ii). right multiplication operator re : b(h)→ b(h) by re(x) = xe, ∀ x ∈ b(h).(iii). generalized derivation (implemented by d,e) by δd,e = ld − re .(iv). inner derivation (implemented by d) by δd(x) = dx −xd.(v). basic elementary operator (implemented by d,e) by md,e(x) = dxe, ∀ x ∈ b(h).(vi). jordan elementary operator (implemented by d,e) by ud,e(x) = dxe + exd, ∀ x ∈ b(h). definition 3 ( [49], definition 2.3). a derivation is a map d : u → u satisfying d(f g) = f d(g) + d(f )g for all f , g ∈ u. definition 4 ( [39], definition 1.2). the maximal numerical range of an operator s is defined by: w0(s) = {β : 〈st, t〉 → β, where ‖t‖ = 1 and ‖st‖ → ‖s‖}. definition 5 ( [35], definition 2.1). an operator k is norm-attainable if t ∈ h exists which is a unit vector such that ‖kt‖ = ‖k‖. moreover, it is self-adjoint if k = k∗. 3. main results in this section, we give results on norm-attainability conditions an norm estimates for derivations.we begin with the following proposition. proposition 6. let h be a complex hilbert space and b(h) the algebra of all bounded linear operators on h. a ∈ b(h) is norm-attainable if and only if its adjoint a∗ ∈ b(h) is normattainable. proof. given a ∈ b(h) is norm-attainable then we need to show that a∗ ∈ b(h) is norm-attainable. if a ∈ b(h) is norm-attainable then by definition of norm-attainability there exists aunit vector x ∈ h with ‖x‖ = 1 such that ‖ax‖ = ‖a‖. that is, ‖aa∗x‖ = ‖a2x‖. let η = ax ‖a‖ ,then η is a unit vector such that ‖η‖ = 1 this implies that ‖a∗η‖ = ‖a‖ = ‖a∗‖. hence, a∗ isnorm-attainable. � https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 8the next result gives norm-attainability conditions for operators via the essential numerical range.an analogy of the same can be found in [37]. proposition 7. let a ∈ b(h), λ ∈ wess(a) and η > 0. then there exists a0 ∈ b(h) such that ‖a‖ = ‖a0‖ with ‖a− a0‖ > η. proof. see [37] for the proof. � remark 8. the set of all norm-attainable operators is denoted by na(h), the set of all normattainable self adjoint operators is denoted by na∗(h) and the set of all norm-attainable elementary operators is denoted by ena[b(h)]. at this point, we consider norm-attainability in a general set up. we begin with the followingproposition. proposition 9. let d be the unit disc of a complex hilbert space h and a : h → h be compact and self adjoint. then there exists x ∈ d such that ‖ax‖ = ‖a‖. proof. by the definition of usual norm, we have ‖a‖ = supx∈d ‖ax‖. so, there exists a sequence x1, x2, ..., xn in d such that ‖axn‖ = ‖a‖. but a is compact so let y0 = limn→∞ axn exist in h. suppose y = span{x1, x2}, then it is a closed subspace of h. if we pick a subsequence xnkof xn, then it converges weakly to x and we have done 〈x, x〉 = limk→∞〈xnk , x〉 and |〈xnk , x〉| ≤ ‖xnk‖‖x‖ = 1 for all k . therefore, ‖x‖ ≤ 1 but we cannot have ‖x‖ < 1 since then ‖ax‖ = ‖a‖‖x‖ < ‖t‖ which is a contradiction. thus, ‖x‖ = 1 i.e x ∈ d. hence, the existence of x isshown and thus completes the proof. � at this point, we consider q-normality and q-norm-attainability. lemma 10. let a ∈ na(h) then a is q-norm-attainable if it is q-normal. proof. let a ∈ na(h) be q-normal i.e aqa∗ = a∗aq . raising a∗ to power q and using it toreplace a∗ we have aq(a∗)q = (a∗)qaq. this shows that aq is normal. now aqa∗ = a∗aq byfuglede property. therefore, a is q-normal. however, a ∈ na(h) and aq is normal so it followsthat there exists a unit vector x ∈ h such that ‖aqx‖ = ‖aq‖, for any q ∈ n. hence, aq isnorm-attainable. � remark 11. every norm-attainable operator and every self adjoint operator is q-norm-attainable and q-normal for any q ∈ n. however, the converse need not be true in general see [66]. lemma 12. let naq(h) be the set of all q-norm-attainable operators on h. then naq(h)is a closed subset of na(h) which is algebraic if and only if for any a ∈ na(h), a is q-normal. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 9 proof. let a be q-normal and pick λ ∈ k. by premultiplying by λ and postmultiplying by q asa power on the normal a we have (λa)q(λa)∗ = (λa)∗(λa)q . this proves the normality of λa.now if a ∈ na(h) then the converse is true if we take limits over a sequence of vectors in h andalso by proposition 9. therefore, a is a q-normal. � theorem 13. let a ∈ naq(h). then the following conditions are true. (i). a∗ is q-norm-attainable.(ii). v av ∗ is q-normal, for a unitary operator v ∈ naq(h).(iii). a−1 is q-norm-attainable if it exists.(iv). a0 = a/g is q-norm-attainable for some g which is a uniformly invariable subspace of hwhich reduces to a.(v). a0 is uniformly equivalent to a implies a0 is norm-attainable. proof. (i). since a ∈ naq(h), then from lemma 10, aq is q-norm-attainable and so (a∗)q isnorm-attainable. consequently, a∗ is q-norm-attainable.(ii). since v is unitary then v v ∗ = v ∗v = i, where i is the identity operator. by definition ofnorm-attainability and lemma 10 we obtain the desired results.(iii). if a−1 exists then since a is q-norm-attainable, aq is q-norm-attainable. now since a is q-norm-attainable then by lemma 10 aq is q-norm-attainable. but (aq)−1 = (a−1)q is q-norm-attainable. so a−1 is q-norm-attainable.(iv). follows from the fact that g invariant under a.(v). follows from (iii) since v is unitary. � corollary 14. let aq, aq0 ∈ naq(h) be commuting operators, then a,a0 ∈ naq(h). proof. since aq, aq0 ∈ naq(h) are commuting then a,a0 are commuting normal operators. bysupraposinormality of operators in dense classes we have a,a0 ∈ naq(h) and hence are norm-attainable. indeed, aqaq0 = (aa0) q = (a0a)q which is normal and norm-attainable. hence, a,a0 ∈ naq(h). � remark 15. not all q-norm-attainable operators are q-normal. thus, the following example shows that the two commuting q-normal operators need not be q-normal. example 16. let a = [ 1 0 0 1 ] and a0 = [ 0 1 0 0 ] . now a+ a0 = [ 1 1 0 1 ] and (a+ a0) 2 =[ 1 2 0 1 ] are not normal. so a+ a0 is not 2-normal. we note that a0 is self-adjoint. lemma 17. the sum of norm-attainable operators is norm-attainable. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 10 proof. consider a,b ∈ b(h). we need to show that the sum of a and b is norm-attainable. for a,b to be norm-attainable then there exists a unit vector x ∈ h such that ‖x‖ = 1, ‖(a+b)x‖ = ‖ax+bx‖ = ‖a+b‖ = ‖a‖+‖b‖. since ‖ax+bx‖ ≤ ‖ax‖+‖bx‖ ≤ ‖a‖+‖bx‖ ≤ ‖a‖+‖b‖then for an orthonormal sequence xn ∈ h we have limn→∞(‖axn +bxn‖) = ‖ax +bx‖. but since a and b are norm-attainable we have ‖ax+bx‖ = ‖(a+b)x‖ = ‖a+b‖ is norm-attainable. � theorem 18. a norm-attainable operator perturbed by an identity operators is norm-attainable. proof. let b ∈ b(h) be norm-attainable. since b is norm-attainable then there exists a unitvector x0 ∈ h, an identity i ∈ b(h) and for every ε > 0 we have ‖(bi)x0‖ ≤ ‖bix0‖ + ε ≤ ‖b‖‖i‖‖x0‖+ ε. since ε is arbitrary then it follows that ‖(bi)x0‖ ≤ ‖b‖‖i‖‖x0‖ = ‖b‖. hence, ‖(bi)x0‖ = ‖b‖. � at this point, we consider norm-attainability for elementary operators. we begin with inner deriva-tions. lemma 19. let δa ∈ e [b(h)], then δa is norm-attainable if there exists a unit vector x0 ∈ h, a ∈ na(h) and 〈ax0, x0〉 ∈ wess(a). proof. for an operator a ∈ na(h) we know that an operator is norm-attainable via essentialnumerical range from proposition 4.2. now, we need to show that δa ∈ e [b(h)] is norm-attainable.by the definition of inner derivation, δa = ay0−y0a. since a is norm-attainable then there exists aunit vector x0 ∈ h such that ‖x0‖ = 1, ‖ax0‖ = ‖a‖. by orthogonality let y0 satisfy y0⊥{ax0, x0}and a contractive y0 be defined as a linear transformation y0 : x0 → x0 with ax0 → −ax0 as y0 → 0. since y0 is a bounded linear operator on h, then by norm-attainability ‖y0x0‖ = ‖y0‖ = 1and ‖ay0x0 − y0ax0‖ = ‖ax0 − (−ax0)‖ = 2‖a‖.it follows from lemma 3.1 in [49] that ‖δa‖ = 2‖a‖. by the inner product 〈ax0, x0〉 = 0 ∈ wess(a),it follows that ‖δa‖ = 2‖a‖. therefore, ‖ay0 − y0a‖ = 2‖a‖ = ‖δa‖. hence, δa is norm-attainable. � lemma 20. let a,a0 ∈ b(h). if there exists unit vectors y and y0 on h such that a,a0 are norm-attainable then δa,a0 is also norm-attainable. proof. given the operators a,a0 ∈ b(h) are norm-attainable then we need to show that δa,a0 isalso norm-attainable. we define the generalized derivation by δa,a0 = ay − y a0. since a,a0 arenorm-attainable then there exists unit vectors y and y0 on h such that ‖y‖ = ‖y0‖ = 1, ‖ay‖ = ‖a‖and ‖a0y0‖ = ‖a0‖. by linear dependence of vectors, if y and ay are linearly dependent then wehave ‖ay‖ = η‖a‖y where |η| = 1 and |〈ay, y〉| = ‖a‖. it follows that |〈a0y0, y0〉| = ‖a0‖ whichimplies that ‖a0y0‖ = φ‖a0‖y0 and |φ| = 1. therefore, 〈a0y0‖a0‖ , y0〉 = φ = −〈 ay‖a‖ , y〉 = −η. if y is https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 11defined as y : y → y0 and y0 → 0, ‖y ‖ = 1 then (ay −y a0)y0 = φ(‖a‖+‖a0‖)y0 which implies ‖ay − y a0‖ = ‖(ay − y a0)y0‖ = ‖a‖+ ‖a0‖ = ‖δa,a0‖. hence, δa,a0 is norm-attainable. � lemma 21. every inner derivation is norm-attainable if and only if it is self-adjoint. proof. let δa ∈ b(h) be norm-attainable then we show that δa = δ∗a. now since δa ∈ b(h)is norm-attainable then there exists a contraction y ∈ b(h) such that ‖δay ‖ = ‖δa‖. that is, ‖δ∗aδay ‖ = ‖δ2ay ‖. let η ∈ h be defined as η = δa ‖δa‖ then η is contractive such that ‖δ∗aη‖ = ‖δa‖ = ‖δ∗a‖. hence, δa is self-adjoint. conversely, let δa be self-adjoint. now since δ∗a is norm-attainable from the first part, then there exists a contractivem ∈ b(h) such that ‖δ∗am‖ = ‖δ∗a‖, i.e ‖δaδ∗am‖ = ‖δ2am‖. let ζ be denoted by ζ = δ∗a ‖δ∗a‖ where ‖ζ‖ = 1 such that ‖δaζ‖ = ‖δ∗a‖ = ‖δa‖.hence, δa is norm-attainable. � lemma 22. every generalized derivation is norm-attainable if and only if it is implemented by orthogonal projections. proof. let a,a0 ∈ b(h) be orthogonal projections. indeed, to show that a generalized derivationis implemented by orthogonal projections a and a0, it is enough to show that it is self-adjoint ifand only if it is normal as proved in [22]. let δa,a0 : b(h)→ b(h) be bounded linear operator on b(h). then exists a unique bounded linear operator δ∗a,a0 : b(h)→ b(h) such that 〈δa,a0x, y 〉 = 〈x, δ∗a,a0y 〉, for all x, y ∈ b(h). now, ‖δ∗a,a0y ‖ = sup ‖x‖=1 〈δa,a0x, y 〉 ≤ sup ‖x‖=‖y ‖=1 ‖δa,a0‖‖x‖‖y ‖ = ‖δa,a0‖so, we conclude that δ∗a,a0 is norm-attainable. conversely, let δa,a0 be norm-attainable. we needto show that it is implemented by orthogonal projections. this follows immediately from [22] andthis completes the proof. � at this point, we give results on upper norm estimates for norm-attainable derivations. we con-sider both inner derivations and generalized derivations. we begin with the following proposition. proposition 23. let a,b ∈ na(h) and δa,b be bounded then ‖δa,b‖ ≤ ‖a‖+ ‖b‖. proof. since δa,b is bounded then for fixed a,b ∈ na(h) we have ‖δa,b(x)‖ ≤ ‖ax − xb‖ ≤ ‖ax‖ + ‖xb‖ ≤ ‖a‖‖x‖ + ‖x‖‖b‖. let x be of norm 1 and take supremum over x ∈ na(h)then ‖δa,b‖ ≤ ‖a‖+ ‖b‖. � remark 24. if a = b then ‖δa‖ ≤ 2‖a‖. next, we consider upper bounds in the unit ball of na(h) denoted by [na(h)]0. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 12 lemma 25. let [na(h)]0 be the unit ball of na(h) and s be a fixed element of na(h). let x ∈ [na(h)]0 then ‖δs|[na(h)]0‖ ≤ 2d(s). proof. since x ∈ [na(h)]0 has norm 1 then we have ‖δs|[na(h)]0(x)‖ = ‖sx − xs‖[na(h)]0 = ‖(s−λ)x−x(s−λ)‖[na(h)]0 ≤ ‖s−λ‖‖x‖[na(h)]0 +‖x‖‖s−λ‖[na(h)]0 . taking the supremumover [na(h)]0, we obtain ‖δs|[na(h)]0‖ ≤ 2‖s − λ‖ and considering the infimum over λ ∈ c weobtain ‖δs|[na(h)]0‖ ≤ 2 infλ∈c ‖s − λ‖ = 2d(s). � remark 26. the restriction of δa|[na(h)]0 i.e δa to [na(h)]0 is a bounded linear operator. next we give an extension of lemma 25 to a generalized derivation in the following theorem. theorem 27. let s, s0 be fixed elements of na(h) then ‖δs,s0 |[na(h)]0‖ ≤ ‖δs,s0‖. proof. since x ∈ [na(h)]0 has norm 1 then we have ‖δs,s0 |[na(h)]0(x)‖ = ‖sx−xs0‖. followingproof of lemma 25 anologously we have ‖δs,s0 |[na(h)]0(x)‖ ≤ ‖s − λ‖‖x‖[na(h)]0 + ‖x‖‖s0 − λ‖[na(h)]0 .taking the supremum over x ∈ [na(h)]0 we obtain ‖δs,s0 |[na(h)]0‖ ≤ infλ∈c(‖s − λ‖+ ‖s0 − λ‖) = ‖δs,s0‖. � corollary 28. every generalized derivation δs,s0 is norm-bounded. proof. this follows immediately from [49] and from theorem 27. this completes the proof. � now, we consider lower bounds for norms of derivations. we begin the following proposition ongeneralized derivation. proposition 29. let s, s0 be fixed elements of na(h) then ‖δs,s0 |[na(h)]0‖ ≥ ‖s‖+ ‖s0‖. proof. let η, ξ and x be unit vectors in h and φ,ϕ be positive linear functionals such that φ⊗ η : h → c and ϕ ⊗ ξ : h → c be of rank 1 defined as (φ ⊗ η)x = φ(x)η and (ϕ ⊗ ξ)x = ϕ(x)ξ, ∀x ∈ h, ‖x‖ = 1. now we have that ‖(φ⊗η)x‖ = sup{‖(φ⊗η)x‖, ‖x‖ = 1} = |φ(x)| = |φ|.similarly, we have ‖(ϕ ⊗ ξ)x‖ = ‖ϕ‖. letting s = φ ⊗ η and s0 = ϕ ⊗ ξ then ‖s‖ = ‖φ‖and ‖s0‖ = ‖ϕ‖. now from corollary 28 we have that every generalized derivation is norm-bounded this implies that ‖δs,s0 |[na(h)]0(x)‖ ≥ ‖δs,s0(x)‖ where x ∈ [na(h)]0. therefore, ‖δs,s0 |[na(h)]0‖2 ≥ ‖sx−xs0‖2 implying that ‖δs,s0 |[na(h)]0‖2 ≥ [‖s‖+ ‖s0‖]2. taking positivesquare root on both sides we obtain ‖δs,s0 |[na(h)]0‖ = ‖δs,s0‖ ≥ ‖s‖+ ‖s0‖. � remark 30. if s = s0 then ‖δs,s0‖ = ‖δs‖ ≥ 2‖s‖. remark 31. from theorem 27 and proposition 3 it is easy to see that ‖δs,s0‖ = ‖s‖+ ‖s0‖ and hence ‖δs‖ = 2‖s‖. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 13 theorem 32. let s, s0 ∈ na(h) and α1 ∈ w0(s) and α2 ∈ w0(s0). then ‖δs,s0‖ ≥ (‖s‖2 − |α1|2)1/2 + (‖s0‖2 − |α2|2)1/2. proof. by definition of w0(s) we have xn ∈ h such that ‖sxn‖ = ‖s‖ and 〈sxn, xn〉 → α1for α1 ∈ w0(s). this argument follows for w0(s0) and α2 ∈ w0(s0). let sxn = δnxn + βnynso s0xn = σnxn + λnyn where 〈xn, yn〉 = 0, ‖yn‖ = 1. take unxn = xn and unyn = −yn for un = 0 in {xn, yn}. then ‖sunxn − uns0xn‖ = ‖δn + βn‖ ≤ |δn| + |βn|. but |δn| + |βn| ≥ (‖s‖2 − |δn|2)1/2 − ξn + (‖s0‖2 − |βn|2)1/2 − ξn). since ξn is arbitrary and letting n →∞, so itfollows that ‖δs,s0‖ ≥ ‖(sun−uns0)xn‖ = |δn|+|βn| = (‖s‖2−|α1|2)1/2+(‖s0‖2−|α2|2)1/2. � corollary 33. let 〈xn, yn〉 = 0 then 0 ∈ w0(s) and if 0 ∈ w0(s0) then ‖δs,s0‖ ≥ ‖s‖+ ‖s0‖. proof. follows immediately from definition of w0(s) and the theorem 32. � 4. conclusion in this paper, we have given a detailed characterization of operators in terms of norm-attainabilityconditions and norm estimates for in banach algebras. in particular, we have established norm-attainability conditions for the derivations and also given the norm bounds in the norm-attainableclasses. references 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https://doi.org/10.1016/s0007-4497(03)00046-0 https://doi.org/10.1016/s0007-4497(03)00046-0 https://doi.org/10.1215/ijm/1258138100 https://doi.org/10.1215/ijm/1258138100 https://doi.org/10.1080/0308108031000122515 https://doi.org/10.1016/j.jmaa.2008.10.008 https://doi.org/10.2140/pjm.1970.33.737 https://doi.org/10.2140/pjm.1970.33.737 https://doi.org/10.2140/pjm.1979.82.257 https://doi.org/10.2140/pjm.1979.82.257 https://doi.org/10.2140/pjm.1991.147.365 https://doi.org/10.2140/pjm.1991.147.365 https://doi.org/10.1090/proc/12664 https://doi.org/10.1017/s0013091504000306 1. introduction 2. preliminaries 3. main results 4. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 20doi: 10.28924/ada/ma.5.20 on a family of q-weighted bergman spaces and applications akram nemri department of mathematics, college of science, jazan university, p.o. box 114, jazan 45142, kingdom of saudi arabia nakram@jazanu.edu.sa abstract. in this paper, we introduce a q-weighted bergman spaces {aα,n,q}n∈n. for n = 0 anuncertainty inequality of the heisenberg-type for the space aα,q is given by considering the operators ∇α,q := ∇α,q and lα,q := lα,q . also, we study on this space the q-toeplitz operators, the q-hankeloperators. at the end, we study the theory of extremal function and reproducing kernel of hilbertspace and we use it to establish the extremal function associated to an bounded linear operator t : aα,q → h, for any hilbert space h. 1. introduction many studies has happened in the last decade, characterizing the action of operators on bergmanand weighted bergman spaces. this line of inquiry has attracted interest owing to its intimate linkswith complex analysis, functional analysis, and operator theory [6]. many techniques have beeninvestigate in different types of operators. for example, hankel operators have been thoroughlyanalyzed using function-theoretic and operator-theoretic approaches ( [1], [12]); composition opera-tors have been studied through dynamical and analytic techniques ( [14]); and multiplier operatorshave been explored in the context of reproducing kernel hilbert spaces and boundedness criteria.these developments have significantly enriched the theory and opened new directions for furtherinvestigation.the main results of this paper is to deal with operators acting on a general q-weighted bergmanspaces {aα,n,q}n∈n. we prove some properties concerning q-toeplitz operators and q-hankeloperators; we establish a more general heisenberg-type uncertainty principle given in [16] for thespace aα,q by considering the operators ∇α,q := ∇α,q and lα,q := lα,q ; we study the theoryof extremal function and reproducing kernel of hilbert space, to establish the extremal functionassociated to a bounded linear operator t . noting that, there exist many similar uncertainty received: 20 apr 2025. key words and phrases. q-weighted bergman spaces, uncertainty inequality, q-toeplitz operators, q-hankel opera-tors, extremal function. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.20 https://orcid.org/0000-0001-9195-5037 eur. j. math. anal. 10.28924/ada/ma.5.20 2principles, in physics [2], [4], [10], and mathematics [3], [19], that are based on position, momentum,energy, time, and so on.the weighted bergman space is one of the complex analysis tools used in harmonic analysis [7].let c be the complex plane, d = { z ∈ c : |z | < 1 } the open unit disk and h(d) the space of allanalytic functions on d. for any α > 0, dνα(z) := 1 π α(1− |z |2)α−1dxdy is the weighted lebesgue measure on d. the weighted bergman space aα is the space h(d) ⋂ l2(d, dνα). noting that, it is an hilbert when space equipped with the inner product 〈f , g〉aα := ∫ d f (z)g(z)dνα(z), and the norm ‖f ‖aα = ‖f ‖l2α,q(d), see [8,16,20] for more details on the theory of bergman spaces.the contents of the paper are as follows. section 2 reviewers from [16] the q-analogue of the q-weighted bergman spaceaα,q and we will introduce the q-analogue of q-weighted bergman spaces {aα,n,q}n∈n. in sect.3, we will study the q-derivative operator∇α,q and its adjoint operator lα,q onthe q-weighted bergman space aα,q , we will prove some properties concerning q-toeplitz operatorsand q-hankel operators and we will establish at the end of this section a general uncertaintyinequality of heisenberg type for the space aα,q . in sect.4, we will give an application of thetheory of extremal function and reproducing kernel of hilbert space by establishing the extremalfunction associated to a bounded linear operator t . 2. preliminaries in all the sequel, assume that 0 < q < 1 and α > 0. the reader can refer to [9] and [13] formore details for the definitions and notations of the basic hypergeometric series, the jackson’s q-derivative and q-integrals, q-gamma and q-beta functions. the reference [16] is devoted to the q-weighted bergman space on the disk. the standard watson’s notation for the q-shifted factorials are defined for any complex number a by (a; q)0 := 1, (a; q)n := n∏ k=0 (1− aqk−1), n = 1, 2, ..., (a; q)∞ := ∞∏ k=0 (1− aqk−1), and [a]q is standing for the number associated to a, [a]q := 1− qa 1− q , [a]q! := (q; q)n (1− q)n , n ∈ n. for any complex z , (a; q)z is defined by (a; q)z := (a; q)∞ (aqz ; q)∞ , (1) https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 3and the q-binomial theorem [9] is given by ∞∑ n=0 (a; q)n (q; q)n zn = (az ; q)∞ (z ; q)∞ . (2) the q-analogue of the classical euler gamma and beta functions defined by jackson in [11] are γq(a) := (q; q)∞ (qa; q)∞ (1− q)1−a, <(a) > 0. βq(a, b) := ∫ 1 0 ta−1(qt; q)b−1dqt = γq(a)γq(b) γq(a + b) , <(a),<(b) > 0. (3) the q-analogue exponential functions eq(z) and eq(z) [9] are given by eq(z) := ∞∑ n=0 (1− q)zn (q; q)n = 1 (z ; q)∞ , eq(z) := ∞∑ n=0 qn(n−1)/2(1− q)zn (q; q)n = (−z ; q)∞. the q-derivative [9] on a subset of c is defined by dq,z f (z) := f (z)− f (qz) (1− q)z , z 6= 0. (4) in all the sequel, we need the following spaces: • h(d) the space of all analytic functions on the unit open disk d = {z ∈ c; |z | < 1}. • l2α,q(d) := l2q(d, dνα,q) the space of measurable functions f on the unit disk d satisfying ‖ f ‖2l2α,q(d):= [α]q 2π ∫ 1 0 (∫ 2π 0 | f (re iθ) |2 dθ ) (qr2; q)α−1dq(r2) := ∫ d | f (z) |2 dνα,q(z) is finite, where dνα,q [5] the measure defined on the unit disk d for α > 0 by dνα,q(z) := [α]q 2π (qr2; q)α−1dq(r2)dθ; z = re iθ, and dθ is the usual lebesgue measure on [0, 2π[ and the integral with respect to dq(r2) isrelated to the q-jackson’s integral over [0, 1] defined by:∫ 1 0 f (t)dqt := (1− q) ∞∑ n=0 f (qn)qn. • aα,q := aα,q(d) the q-weighted bergman space of all functions in h(d) ⋂ l2α,q(d). it isa hilbert space when equipped with the inner product 〈f , g〉aα,q = ∫ d f (z)g(z)dνα,q(z). and the norm ‖f ‖aα,q = (∫ d |f (z)|2dνα,q(z) )1/2 . https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 4 • aα,n,q := aα,n,q(d), the hilbert space of functions on h(d), such that ‖f ‖2aα,n,q := |f (0)|2 + ∫ d | nnq f (z) |2 dνα,q, n = 1, 2, ... ‖f ‖2aα,0,q := ‖f ‖2aα,q , nq is the q-multiplication operator on aα,q given by nq := zdq,z . moreover, if f (z) = ∑∞ k=0 akz k then ‖f ‖2aα,n,q = |a0|2 + ∞∑ k=1 [k ]2nq ck(α; q)|ak |2, where cn(α; q) := (q; q)n (qα+1; q)n . 3. uncertainty inequality on the q-weighted bergman space aα,q consider the q-operator ∇α,q and lα,q are the operators on aα,q(d, dνα,q) defined by ∇α,q := q−α−1dq,z , nq := zdq,z lα,q := z2dq,z + [α+ 1]qq −α−1z. (5) so, we have the following q-commutation relation lemma 3.1. [∇α,q, lα,q]q := ∇α,qlα,q − lα,q∇α,q = q−α−1λq ( [α+ 1]qi + (1 + q−1)q−α−1nq ) , where i is the identity operator and λq is the q-shift operator given by λqf (z) = f (qz). we derive the following results proposition 3.1. let f , g ∈ aα,q(d, dνα,q) with f (z) = ∑∞ n=0 anz n and g(z) = ∑∞ n=0 bnz n, we have(i) 〈f , g〉aα,q(d,dνα,q) = ∞∑ n=0 anbn (q; q)n (qα+1; q)n = ∞∑ n=0 anbn cn(α; q). (ii) ||f ||2aα,q(d,dνα,q) = ∞∑ n=0 |an|2 (q; q)n (qα+1; q)n = ∞∑ n=0 |an|2 cn(α; q). (iii) the set { ξαn,q(z) := zn√ cn(α; q) } n≥0 , forms a hilbert’s basis for the space aα,q(d, dνα,q). proof. given f (z) = ∑∞ k=0 akz k and g(z) = ∑∞ k=0 bkz k , the result follows by using dominateconvergence theorem and relation (4.6) in [5] we have 〈f , g〉aα,q(d,dνα,q) = ∞∑ m,n=0 ambn ∫ d zm zn dνα,q(z) = ∞∑ n=0 ambn (q; q)n (qα+1; q)n . (6) the last assertion follows directly from proposition 4.1 in [5]. � https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 5 theorem 3.1. the function kα,q given for w, z ∈ d, by kα,q(z, w) = kα,q(zw) = 1 (zw ; q)α+1 , (7) is a reproducing kernel for the q-weighted bergman space aα,q(d, dνα,q). that is(i) for all w ∈ d, z 7−→ kα,q(z, w) belong to aα,q(d, dνα,q).(ii) for all w, z ∈ d and f ∈ aα,q(d, dνα,q), we have 〈f ,kα,q(., w)〉aα,q(d,dνα,q) = f (w). (iii) for all f ∈ aα,q(d, dνα,q) and z ∈ c, | f (z) |≤ [ eq(|z |2)eq(qα+1|z |2) ]1/2 ‖ f ‖aα,q(d,dνα,q) .(iv) let w ∈ d. the function u(z) = kα,q(zw) is the unique analytic solution on d of the initial problem z∇α,qu(z) = wlα,qu(z), u(0) = 1. proof. to prove the first assertion (i), we use proposition 3.1 (iii) the function ξαn,q(z) constitutean orthonormal basis of aα,q(d, dνα,q). therefore for any z, w ∈ d, kα,q can be computed byevaluating the following sum kα,q(z, w) = ∞∑ n=0 ξαn,q(z)ξαn,q(w) = ∞∑ n=0 1 cn(α; q) znwn. hence by (1) combined with (2) we deduce easily kα,q(z, w) = ∞∑ n=0 (qα+1; q)n (q; q)n (zw)n = (qα+1zw ; q)∞ (zw ; q)∞ = 1 (zw ; q)α+1 . to prove (ii), we use the same as in proposition 4.2 in [5]. the last assertion follows by using (4). � the domain of the operator ∇α,q denoted by dom(∇α,q) is defined by dom(∇α,q) := { f ∈ aα,q(d, dνα,q); ∇α,qf ∈ aα,q(d, dνα,q) } , and same for domq(nq) and domq(lα,q). lemma 3.2. the operators ∇α,q , nq and lα,q satisfies the following(i) dom(∇α,q) = dom(lα,q) = dom(nq) = aα,1,q.(ii) for any f , g in aα,1,q we have: 〈∇α,qf , g〉aα,q(d,dνα,q) = 〈f , lα,qg〉aα,q(d,dνα,q).(iii) for any f in aα,1,q we have ‖ lα,qf ‖2aα,q(d,dνα,q)=‖ ∇α,qf ‖ 2 aα,q +q−α−1[α+ 1]q ‖ λq1/2 f ‖2aα,q +q−α−1(1 + q−1)〈nqλq1/2 f ,λq1/2 f 〉aα,q . proof. let f ∈ aα,1,q , with f (z) = ∑∞ k=0 akz k . then using relation (4), we have respectively ∇α,qf (z) = ∞∑ k=1 q−α−1[k ]qakz k−1 = ∞∑ k=0 q−α−1[k + 1]qak+1z k (8) https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 6and lα,qf (z) = ∞∑ k=0 ([k ]q + q−α−1[α+ 1]q)akz k+1 = ∞∑ k=1 ([k − 1]q + q−α−1[α+ 1]q)ak−1z k . (9) thus from the previous relation, we get ‖ ∇α,qf ‖2aα,q= 〈∇α,qf ,∇α,qf 〉aα,q = 〈f , lα,q∇α,qf 〉aα,q = ∞∑ k=1 q−α−1[k]q ( [k − 1]q + q−α−1[α+ 1]q ) |ak |2ck(α; q), (10) ‖ lα,qf ‖2aα,q= 〈lα,qf , lα,qf 〉aα,q = 〈f ,∇α,qlα,qf 〉aα,q = ∞∑ k=1 q−α−1[k+1]q ( [k]q +q−α−1[α+1]q ) |ak |2ck(α; q), (11) and ‖ nqf ‖2aα,q(d,dνα,q)= 〈nqf , nqf 〉aα,q(d,dνα,q) = ∞∑ k=1 [k ]2q|ak |2ck(α; q). (12) therefore, from proposition 3.1, (10), (11) and (12) we deduce easily ‖f ‖2aα,q(d,dνα,q) − |f (0)|2 ≤ ‖∇α,qf ‖2aα,q(d,dνα,q) ≤ (1 + q−α−1[α+ 1]q)‖f ‖2aα,1,q(d,dνα,q) ‖f ‖2aα,1,q(d,dνα,q) ≤ ‖lα,qf ‖2aα,q(d,dνα,q) ≤ [2]q(1 + q−α−1[α+ 1]q)‖f ‖2aα,1,q(d,dνα,q) ‖f ‖2aα,1,q(d,dνα,q) − |f (0)|2 ≤ ‖nqf ‖2aα,q(d,dνα,q) ≤ ‖f ‖ 2 aα,1,q(d,dνα,q).so, dom(∇α,q) = dom(lα,q) = dom(nq) = aα,1,q(d, dνα,q). to prove (ii), let f , g in aα,1,q(d, dνα,q) with f (z) = ∑∞ k=0 akz k and g(z) = ∑∞ k=0 bkz k . fromproposition 3.1, (8) and (9) we have 〈∇α,qf , g〉aα,q(d,dνα,q) = ∞∑ k=0 q−α−1[k + 1]qak+1bkck(α; q) = ∞∑ k=0 ak+1bk (q; q)k+1 (1− q)qα+1(qα+1; q)k = ∞∑ k=1 akbk−1 (q; q)k (1− q)qα+1(qα+1; q)k−1 , on the other hand 〈f , lα,qg〉aα,q(d,dνα,q) = ∞∑ k=0 ([k − 1]q + q−α−1[α+ 1]q)[k + 1]qakbk−1ck(α; q) = ∞∑ k=0 1− qα+k 1− q akbk−1ck(α; q) = ∞∑ k=1 [k + α]qakbk−1 (q; q)k (qα+1; q)k = ∞∑ k=1 akbk−1 (q; q)k qα+1(1− q)(qα+1; q)k−1 = 〈∇α,qf , g〉aα,q(d,dνα,q). https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 7 finally, to prove (iii), using [k + 1]q = [k ]q + qk we deduce easily that [k + 1]q ( [k ]q + q−α−1[α+ 1]q ) = ( [k ]q + qk )( [k − 1]q + qk−1 + q−α−1[α+ 1]q ) = [k ]q ( [k − 1]q + q−α−1[α+ 1]q ) + qk−α−1[α+ 1]q + ( 1 + q−1 ) qk [k ]q. which leads to the result using (10), (11), (12) and the fact that λqnq = nqλq . � lemma 3.3. dom(∇α,qlα,q) = dom(∇α,qlα,q) = aα,2,q(d, dνα,q). proof. let f ∈ aα,q(d, dνα,q), with f (z) = ∑∞ k=0 akz k . then using relation (8) and (9) weobtain ∇α,qlα,qf (z) = ∞∑ k=0 q−α−1[k + 1]q([k ]q + q−α−1[α+ 1]q)akz k and lα,q∇α,qf (z) = ∞∑ k=1 q−α−1[k ]q([k − 1]q + q−α−1[α+ 1]q)akz k . therefore, ‖ ∇α,qlα,qf ‖aα,q(d,dνα,q)= ∞∑ k=0 q−2(α−+)[k + 1]2q([k ]q + q−α−1[α+ 1]q)2|ak |2ck(α; q) and ‖ lα,q∇α,qf ‖aα,q(d,dνα,q)= ∞∑ k=1 q−2(α+1)[k ]2q([k − 1]q + q−α−1[α+ 1]q)2|ak |2ck(α; q). so, as in the previous lemma, from proposition 3.1, we deduce easily ‖f ‖2aα,2,q(d,dνα,q) − |f (0)|2 ≤ ‖lα,q∇α,qf ‖2aα,q(d,dνα,q) ≤ (1 + q−α−1[α+ 1]q)2‖f ‖2aα,2,q(d,dνα,q), and ‖f ‖2aα,2,q(d,dνα,q) ≤ ‖∇α,qlα,qf ‖ 2 aα,q(d,dνα,q) ≤ [2]2q(1 + q−α−1[α+ 1]q)2‖f ‖2aα,2,q(d,dνα,q). thus, dom(∇α,qlα,q) = dom(∇α,qlα,q) = aα,2,q. � we can now establish an uncertainty inequality of heisenberg-type on the spaceaα,q(d, dνα,q),by the virtue of the following lemma: lemma 3.4. [6] let x and y be self-adjoint operators on hilbert space h (i.e x∗ = x and y ∗ = y ). then ‖ (x − a)f ‖h‖ (y − b)f ‖h≥ 1 2 | 〈[x, y ]f , f 〉h |, for all f in dom(xy ) ∩dom(y x) and a, b ∈ r https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 8 theorem 3.2. let f ∈ aα,2,q(d, dνα,q). for all a, b ∈ r, we have ‖ (∇α,q + lα,q − a)f ‖aα,q‖ (∇α,q − lα,q + ib)f ‖aα,q ≥ q−α−1[α+ 1]q ‖ λq1/2f ‖ 2 aα,q +q−α−1(1 + q−1)〈nqλq1/2f ,λq1/2f 〉aα,q . proof. consider x := ∇α,q +lα,q and y := i(∇α,q −lα,q). by lemma 3.2 and lemma 3.3, theoperators x and y verifies the following properties(a.) x∗ = x and y ∗ = y(b.) dom(xy ) = dom(y x) = aα,2,q(c.) [x, y ]q = −2i [∇α,q, lα,q]q .so, the result follows from lemma 3.1 and lemma 3.2. � proposition 3.2. let a, b ∈ r.(i) for all f ∈ aα,2,q(d, dνα,q), we have ‖ (∇α,q + lα,q − a)f ‖aα,q(d,dνα,q)‖ (∇α,q − lα,q + ib)f ‖aα,q(d,dνα,q) ≥‖ lα,qf ‖2aα,q(d,dνα,q) − ‖ ∇α,qf ‖ 2 aα,q(d,dνα,q) . (ii) for all f ∈ aα,1,q , we have qα+1 ‖ (∇α,q + lα,q − a)f ‖aα,1,q‖ (∇α,q − lα,q + ib)f ‖aα,1,q ≥ [α+ 1]q ‖ λq1/2f ‖ 2 aα,1,q +(1 + q−1)〈nqλq1/2f ,λq1/2f 〉aα,1,q . proof. let a, b ∈ r. the first inequality (i) hold from lemma 3.2 (iii) and the second inequality(ii) hold by applying lemma 3.2 (i). � 4. operators on the q-weighted bergman space aα,q 4.1. q-toepliz operator on aα,q . consider the orthogonal projection operator pα,q : l2α,q(d) → aα,q . since l2α,q(d) = aα,q ⊕a⊥α,q then for any f ∈ l2α,q(d), we have f = (f − f ⊥) + f ⊥ where f − f ⊥ ∈ aα,q and f ⊥ ∈ a⊥α,q . furthermore, for z ∈ d, pα,qf (z) = (f − f ⊥)(z) = 〈(f − f ⊥)(z),kα,q(z, .)〉l2α,q(d) = 〈f (z),kα,q(z, .)〉l2α,q(d), where kα,q is the reproducing kernel given by (7). the following assertions then follow proposition 4.1. for all f , g ∈ l2α,q(d), we have:(i) pα,q ◦ pα,qf = pα,qf .(ii) 〈pα,qf , g〉l2α,q(d) = 〈f , pα,qg〉l2α,q(d).(iii) the operator pα,q is bounded with ‖ pα,q ‖= 1 and ‖ i − pα,q ‖≤ 1. https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 9let φ ∈ l∞(d). the q-multiplication operators mφ are the operators defined by mφ : l2α,q(d)→ l2α,q(d), mφf (z) := φ(z)f (z), z ∈ d. the q-toepliz operators tφ are the operators defined by tφ : aα,q → aα,q, tφf (z) := pα,qmφ(z)f (z), z ∈ d. theorem 4.1. let φ ∈ l∞(d).(i) the operators tφ are bounded and ‖ tφ ‖≤‖ φ ‖∞.(ii) for all f , g ∈ aα,q , we have 〈tφf , g〉aα,q = 〈f , tφg〉aα,q . proof. let φ ∈ l∞(d). to prove (i), let f ∈ aα,q then from proposition 4.1 (iii) we have ‖ tφf ‖aα,q=‖ pα,qmφf ‖aα,q=‖ pα,q(φf ) ‖aα,q≤‖ φf ‖l2α,q(d)≤‖ φf ‖l∞(d)‖ f ‖aα,q .thus, ‖ tφ ‖≤‖ φ ‖.to prove the second assertion, we use the fact that for any f , g ∈ aα,q , pα,qf = f and pα,qg = g.from proposition 4.1 (ii), we obtain 〈tφf , g〉aα,q = 〈φf , pα,qg〉l2α,q(d) = 〈f , φg〉l2α,q(d) = 〈pα,qf , tφg〉l2α,q(d) = 〈f , tφg〉aα,q . � theorem 4.2. let φ ∈ l∞(d) has compact support, then tφ is a compact operator. proof. let φ ∈ l∞(d) and n,m = 0, 1, 2, .... from proposition 3.1, we have tφξ α n,q(z) = ∞∑ m=0 〈tφξαn,q, ξαm,q〉l2α,q(d) cm(α; q) zm. so, 〈tφξαn,q, ξαm,q〉aα,q = 〈φξαn,q, ξαm,q〉l2α,q(d)since φ ∈ l∞(d) with compact support, there exist a positive constant a and k such that | φ(z) |≤ a and φ(z) = 0, for any | z |> a. then for all n,m ∈ n, we get from (3) and proposition 3.1 (i), 〈φξαn,q, ξαm,q〉l2α,q(d) = 1√ cn(α; q)cm(α; q) ∫ |z |≤a φ(z)znzmdνα,q(z) thus, we obtain∣∣∣∣〈φξαn,q, ξαm,q〉l2α,q(d)∣∣∣∣ ≤ k√ cn(α; q)cm(α; q) ∫ |z |≤a |z |n+mdνα,q(z) ≤ 2k√ cn(α; q)cm(α; q) ∫ a 0 rn+mdνα,q(z) ≤ 2k[α]qa n+m√ cn(α; q)cm(α; q) ∫ 1 0 (qr2; q)α−1dq(r2) https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 10 ≤ kan+m√ cn(α; q)cm(α; q) . hence, ∞∑ n,m=0 ∣∣〈tφξαn,q, ξαm,q〉aα,q ∣∣2 cn(α; q)cm(α; q) ≤ 4k2 ( ∞∑ n=0 a2n cn(α; q) )2 ≤ 4k2eq(a2)(qα+1; q)2∞ <∞ then tφ is an hilbert-schmidt operator, and consequently it is compact. � 4.2. q-hankel operator on aα,q . let φ ∈ l∞(d). the q-hankel operators hφ are the operatorsdefined by hφ : aα,q → aα,q, hφ := (i − pα,q)mφ. theorem 4.3. let φ,ψ ∈ l∞(d).(i.) the operators hφ are bounded and ‖ hφ ‖≤‖ φ ‖∞.(ii) for all f ∈ aα,q and g ∈ l2α,q(d), we have 〈hφf , g〉l2α,q(d) = 〈f , h∗φg〉aα,q , h∗φ = pα,qmφ(i − pα,q). (iii) tφψ − tφtψ = h∗ φ hφ. proof. let φ,ψ ∈ l∞(d). to prove (i), from proposition 4.1 (iii) for any f ∈ aα,q ‖ hφ ‖l2α,q(d)=‖ (i − pα,q) ‖l2α,q(d)≤‖ φf ‖l2α,q(d)≤‖ φ ‖l∞α,q(d)‖ φf ‖l2α,q(d) .so, ‖ hφ ‖≤‖ φ ‖l∞(d).(ii) let f ∈ aα,q and g ∈ l2α,q(d). from proposition 4.1 (ii) and the fact that pα,qf = f we obtain 〈hφf , g〉l2α,q(d) = 〈φf , g〉l2α,q(d) − 〈φf , pα,qg〉l2α,q(d) = 〈f , φ(i − pα,q)g〉l2α,q(d) = 〈pα,qf , φ(i − pα,q)g〉l2α,q(d) = 〈f , pα,qmφ(i − pα,q)g〉aα,q .(iii) let φ,ψ ∈ l∞(d). then h∗ φ hψ = pα,qmφ(i − pα,q)2mψ = pα,qmφ(i − pα,q)mψ = pα,qmφψ − pα,qmφpα,qmψ = tφψ − tφtψ. � 5. extremal function on the q-weighted bergman space aα,q let η > 0 and t : aα,q → h be a bounded operator from aα,q into a hilbert space h. wedenote by 〈., .〉t,η,q the inner product defined on the q-weighted bergman space aα,q by 〈f , g〉t,η,q := η〈f , g〉aα,q + 〈t f , tg〉h,and ‖ f ‖t,η,q:= √ 〈f , f 〉t,η,q . https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 11by the virtue of the theory of reproducing kernels of hilbert space, we study the extremal functionassociated to the operator t on the q-weighted bergman space aα,q . theorem 5.1. let η > 0. the space (aα,q, 〈., .〉t,η,q) possesses a reproducing kernel kt,η,q(z, w); z, w ∈ d which satisfies the equation (ηi + t ∗t )kt,η,q(z, .) = kα,q(z, .), where kα,q is the kernel given by (7). moreover, the kernel kt,η,q satisfies the following properties (i) ‖ kt,η,q(z, .) ‖aα,q≤ 1 η √ eq(|z |2)eq(qα+1|z |2). (ii) ‖ tkt,η,q(z, .) ‖h≤ √ eq(|z |2)eq(qα+1|z |2) 2η . (iii) ‖ t ∗tkt,η,q(z, .) ‖aα,q≤ √ eq(|z |2)eq(qα+1|z |2), proof. let f ∈ aα,q . using theorem 3.1 (iii), the map f 7→ f (z) is a continuous linear functionalon (aα,q, 〈., .〉t,η,q). thus, (aα,q, 〈., .〈t,η,q) has a reproducing kernel denoted kt,η,q . now usingthe fact that f (z) = η〈f ,kt,η,q(z, .)〉aα,q + 〈t f , tkt,η,q(z, .)〉h = 〈f , (ηi + t ∗t )kt,η,q(z, .)〉aα,q , we deduce easily that (ηi + t ∗t )kt,η,q(z, .) = kα,q(z, .). so the previous relation implies that η2 ‖ kt,η,q(z, .) ‖2aα,q +2η ‖ tkt,η,q(z, .) ‖2h + ‖ t ∗tkt,η,q(z, .) ‖2aα,q=‖ kα,q(z, .) ‖2aα,q . so we obtain the properties (i), (ii) and (iii) by using relation (7). � since relations (10), (11) and (12), we get example 5.1. for any w, z ∈ d, let h = aα,q .(a) if t = ∇α,q , then kt,η,q(z, w) = 1 ηc0(α; q) + ∞∑ n=1 (zw)n( η + q−α−1[n]q([n − 1]q + q−α−1[α+ 1]q) ) cn(α; q) . (b) if t = lα,q , then kt,η,q(z, w) = 1 ηc0(α; q) + ∞∑ n=1 (zw)n( η + q−α−1[n + 1]q([n]q + q−α−1[α+ 1]q) ) cn(α; q) . (c) if t = nq , then kt,η,q(z, w) = 1 ηc0(α; q) + ∞∑ n=1 (zw)n( η + [n]2q ) cn(α; q) . we can state now the main result of this section. https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 12 theorem 5.2. for any h ∈ h and η > 0, there exists a unique function f ∗η,h, where the infimum inf f ∈aα,q { η ‖ f ‖2aα,q + ‖ h − t f ‖2h } (13) is attained. moreover, the extremal function f ∗η,h is given by f ∗η,h(z) = 〈h, tkt,η,q(z, .)〉h, (14) and satisfies the following | f ∗η,h(z) |≤ √ eq(|z |2)eq(qα+1|z |2) 2η ‖ h ‖h . proof. the existence and unicity of the extremal function f ∗η,h satisfying (13) is obtained in [15,17].in particular, f ∗η,h is given by the reproducing kernel of aα,q with ‖ . ‖t,η,q norm as f ∗η,h(z) = 〈h, tkt,η,q(z, .)〉h . this yields the result, by using relation (14), theorem 5.1 (ii) and the fact that | f ∗η,h(z) |≤‖ h ‖h‖ tkt,η,q(z, .) ‖h≤ √ eq(|z |2)eq(qα+1|z |2) 2η ‖ h ‖h, which completes the proof of the theorem. � 5.1. applications. let h be the prehilbertian space of analytic functions on the disk d equippedwith the inner product 〈f , g〉h := ∫ d f (z)g(z)|z |2dνα,q(z). for any f , g ∈ h with f (z) = ∑ n≥0 anz n and g(z) = ∑ n≥0 bnz n we have from proposition 3.1and relation (6) 〈f , g〉h = ∑ n≥0 anbncn+1(α; q), ‖ f ‖h= ∑ n≥0 | an |2 cn+1(α; q). the space h is a hilbert space with hilbert’s basis { zn√ cn+1(α; q) } n≥0 and reproducing kernel sα,q(z, w) = ∞∑ n=0 (zw)n cn+1(α; q) = kα,q(zw)− 1 zw . (15) 5.1.1. application 1. let t be the q-difference operator defined on aα,q by t f (z) := 1 z (f (z)− f (0)). the operator t maps continuously from aα,q into h and ‖ t f ‖h≤‖ f ‖aα,q . so, if f , g ∈ aα,qwith f (z) = ∑ n≥0 anz n and g(z) = ∑ n≥0 bnz n we can deduce easily that 〈f , g〉t,η = ηa0b0 + (η + 1) ∞∑ n=1 anbncn(α; q). https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 13thus, for z, w ∈ d we have kt,η,q(z, w) = 1 η + 1 η + 1 (kα,q(zw)− 1), tkt,η,q(z, .)(w) = 1 η + 1 kα,q(zw)− 1 w , hence for all h ∈ h we deduce that f ∗η,h(z) = 1 η + 1 zh(z). figure 1. the following is the color function of f ∗η,h(z) associated to the q-difference operator t f (z) := 1 z (f (z) − f (0)) for λ = 10, z = x + iy , (x, y) ∈ [−5, 5] × [−5, 5] and respectively h(z) = 1, z, z2, z3, z4, z5. the argument of acomplex value is encoded by the hue of a color (red = positive real, and thencounterclockwise through yellow, green, cyan, blue and purple; cyan stands fornegative real). strong colors denote points close to the origin, black = 0, weakcolors denote points with large absolute value, white = ∞. https://doi.org/10.28924/ada/ma.5.20 eur. j. math. anal. 10.28924/ada/ma.5.20 14 acknowledgments. the author appreciates anonymous referees and the handling editor fortheir careful corrections to and valuable comments on the original version of this paper. references [1] j. arazy, s.d. fisher, j. peetre, hankel operators on weighted bergman spaces, amer. j. math. 110 (1988), 989-1053. https://doi.org/10.2307/2374685.[2] p. busch, p. lahti, r.f. werner, heisenberg uncertainty for qubit measurements, phys. rev. a 89 (2014), 012129. https://doi.org/10.1103/physreva.89.012129.[3] m. cowling, j.f. price, bandwidth versus time concentration: the heisenberg-pauli-weyl inequality, siam j. math.anal. 15 (1984), 151-165. https://doi.org/10.1137/0515012.[4] d.l. donoho, p.b. stark, uncertainty principles and signal recovery, siam j. appl. math. 49 (1989), 906-931. 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https://doi.org/10.4134/ckms.c240150 https://doi.org/10.1177/1081286516657686 https://www.jstor.org/stable/24103129 https://www.jstor.org/stable/24103129 https://doi.org/10.1007/s13370-021-00924-3 https://doi.org/10.1090/surv/138 https://doi.org/10.1090/surv/138 1. introduction 2. preliminaries 3. uncertainty inequality on the q-weighted bergman space a,q 4. operators on the q-weighted bergman space a,q 4.1. q-toepliz operator on a,q 4.2. q-hankel operator on a,q 5. extremal function on the q-weighted bergman space a,q 5.1. applications. references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 16doi: 10.28924/ada/ma.5.16 stability analysis of a mathematical model for examination malpractice dynamics musah konlan∗ , razak gbemmie chuaya department of mathematics and statistics, university of energy and natural resources, sunyani, ghana musah.konlan@uenr.edu.gh, chuayarazk@gmail.com ∗correspondence: musah.konlan@uenr.edu.gh abstract. examination malpractice is one of the key challenges endangering the quality of educa-tion in ghana. this negative act refers to any form of dishonesty or irregularity that compromisesthe integrity of any examination. in this paper, we proposed a mathematical model for exploringthe dynamics of examination malpractice at the west african senior school certificate examination(wassce) level in ghana. the examination malpractice-free equilibrium is computed and shown tobe both locally and globally stable if the examination malpractice reproductive number (r0) is lessthan one. the examination malpractice endemic equilibrium is also derived and found to be globallyasymptotically stable whenever r0 is greater than one. local sensitivity analysis is performed onthe basic examination malpractice reproduction number to explore the contribution of the model pa-rameters to the evil act of examination malpractice. finally, numerical simulations are performed toillustrate the behavior of the model sub-classes. 1. introduction examination remains a prominent tool for assessing and measuring students’ academic perfor-mance throughout our educational system. it is used to determine the transition of students fromone lower level to the next higher level. on the job market, examination also serves as a means forpredicting a job seeker’s knowledge level, skills, and competence in a given domain. unfortunately,this valuable measurement tool is being compromised at all levels given room to what is popularlycalled examination malpractice [1, 2]. examination malpractice can be described as any deliberateact against the official rules and regulations of an examination, with the intention of giving undueadvantage to a candidate [3–6]. examination malpractice has been constantly recorded at all levelsof our educational institutions. data on examination malpractice at the senior high school levelare alarming. in ghana for example, the cases of examination malpractice at the senior high schoolcertificate examination from 2020 to 2024 are shown in table 1. also, it has been reported that innigeria 1,767 out of 13,595 and 842 out of 12,030 candidates who took part in the senior highschool certificate examination were involved in malpractices in 2022 and 2023 respectively [2]. received: 24 mar 2025. key words and phrases. examination malpractice; mathematical model; stability analysis; sensitivity analysis.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.16 https://orcid.org/0009-0007-6219-6810 eur. j. math. anal. 10.28924/ada/ma.5.16 2examination malpractice has become a major concern in ethiopia and somalia [7]. according tostudies conducted in [3, 6], people involved at the school level examination malpractice includeacademic authorities, students, teachers, and parents. this bad phenomenon may take place be-fore, during and after an examination with the aim of achieving academic excellence for both thestudent and the school for a given examination. specific forms of malpractice during examinationsinclude, but are not limited to: tattooing, examination paper leakage, support from invigilators,impersonation, smuggling of unauthorized materials into the examination room, and giraffing in theexamination room [3,4,6,8]. it is clear from the above-mentioned that examination malpractice posesa significant risk to the educational system of a nation. in a nutshell, examination malpractice low-ers educational standards, discredits certificates obtained through examination, discourages hardwork, leads to academic dishonesty and above all reduces productivity. thus, contributing factorsto this malfeasance need to be identified and tackled [1,2]. mathematical modelling has become aneffective toolbox for understanding the occurrence of dynamical phenomena [9,10]. however, math-ematical models for studying examination malpractice are very scarce in the literature. the authorsin [11] formulated and analyzed a dynamical model of examination malpractice taking into accountkey players in nigeria. their analysis indicated that leakages of examination question papers hasthe highest influence on the spread of examination malpractice. ayoade and farayola [11] devel-oped a mathematical model for the mechanisms of examination malpractice with control strategiessuch as: social reengineering and orientation with proportional punishment/disciplinary action forexamination malpractice victims. to fill this research gap, this work aims to propose a deterministiccompartmental model to study exam malpractice at the west african senior high school certificateexamination (wassce) in ghana. table 1. wassce provisional data for ghana (2020 2024) year n(t) m(t) c(t) pending 2020 375, 763 2, 863 480 384 2021 446352 1, 513 174 3, 667 2022 422, 883 4, 363 518 117 2023 448, 674 4, 486 839 5, 285 2024 460, 611 4, 591 483 990 source: https://www.myjoyonline.com/?s=release+of+wassce+results 2. model construction to construct a compartmental model for examining the dynamics of examination malpractice, thetotal number of candidates sitting for the examination at time t (n(t)) is stratified into five sub-classes. namely: s(t), m(t), c(t), r(t) and h(t) (see table 2). candidates are recruited into the https://doi.org/10.28924/ada/ma.5.16 https://www.myjoyonline.com/?s=release+of+wassce+results eur. j. math. anal. 10.28924/ada/ma.5.16 3susceptible class (s(t)) at rate λ. some susceptible candidates move to honest candidates class(h(t)) at rate β. other susceptible candidates engage into malpractices and move to malpracticeclass (m(t)) at rate θ. candidates in the malpractice class either recover from malpractice and moveto recovered class (r(t)) at rate τ or have their entire results cancelled at rate δ. some recoveredcandidates progress to honest candidates class at rate α. each model sub-class is reduced at arate µ due to natural death. the model assumes that there is no examination malpractice induceddeaths. the variables and parameters used to describe the model are presented in table 2 andtable 3 respectively. table 2. model state variables definition symbol definition n(t) total number of candidates sitting for the examination s(t) number of candidates susceptible to examination malpractice m(t) number of candidates who engage in any form of irregularities c(t) number of candidates who have their entire results cancelled following misconduct r(t) number of candidates who recover from malpractice h(t) number of law abiding/honest candidates table 3. model parameter description symbol description λ candidates recruitment rate 50.0 estimated µ natural human removal rate 0.0005 assumed β rate at which candidates adhere to examination rules 0.5 assumed θ rate at which candidates engage into irregularities 0.0044 assumed τ rate at which candidates recover from malpractice 0.2 assumed δ rate at which candidate entire results are canceled 0.15 estimated α rate at which recovered candidates become honest 0.3 assumed γ rate at which honest candidate revert to susceptible 0.9 assumed https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 4 figure 1. flow chart for examination malpractice from the flow chart above, we obtain the following system of differential equations: ds dt = λ + γh − θsm − (β + µ)s dm dt = θsm − (τ + δ + µ)m dc dt = δm − µc dr dt = τm − (α+ µ)r dh dt = βs + αr − (γ + µ)h (1) the following notation will be used in the rest of the study: q0 = (β + µ), q1 = (τ + δ + µ), q2 = (α+ µ) and q3 = (γ + µ) 2.1. well-posedness of the model. under this section, we establish that the system of differentialequations representing model (1) admits only non-negative solutions. furthermore, the set overwhich the model system of equations is contextually meaningful is also determined. theorem 1. given the non-negative initial value set: {s(0), m(0), c(0), r(0), h(0)} of the dynamical system (1), it follows that the solution set {s(t), m(t), c(t), r(t), h(t)} is nonnegative and bounded ∀ t ≥ 0. https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 5 proof. first, we consider the differential equation for the susceptible sub-class: ds dt = λ + γh − (θm + q0)s =⇒ ds dt ≥ −(θm + q0)s =⇒ ∫ 1 s ds ≥ − ∫ (θm + q0)dt =⇒ s(t) ≥ s(0)e−(qot+θ ∫ t 0 m(x)dx) ≥ 0similarly, the following results can be obtained: m(t) ≥ m(0)e−q1t ≥ 0 c(t) ≥ c(0)e−µt ≥ 0 r(t) ≥ r(0)e−q2t ≥ 0 h(t) ≥ h(0)e−q3t ≥ 0 therefore, for ∀ t ≥ 0, the state variables of the model have non-negative solutions. � theorem 2. the feasible positive invariant region in which the solution set of the model system of equations is meaningful is the set: d = { (s,m,c,r,h) ∈ r5 + : s +m + c + r +h ≤ λ µ } (2) proof. the total population n at any given time t is: n = s +m + c + r +h =⇒ dn dt = ds dt + dm dt + dc dt + dr dt + dh dt =⇒ dn dt = λ− µn =⇒ dn dt = −µ(n − λ µ ) =⇒ dn n − λ µ = −µdt =⇒ ∫ dn n − λ µ = − ∫ µdt =⇒ n − λ µ = 0 as t → +∞ thus, we deduce that n ≤ λ µ (3) https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 6therefore: d = { (s,m,c,r,h) ∈ r5 + : s +m + c + r +h ≤ λ µ } (4) � 2.2. the examination malpractice-free equilibrium (emfe). equating the individual equationsof system (1) to zero with the condition m = c = r = 0, we obtain the emfe (ξ∗) given by ξ∗ = (s∗,m∗, c∗, r∗, h∗) = ( λq3 q0q3−βγ , 0, 0, 0, βλ q0q3−βγ ) 2.2.1. the basic reproductive number (r0) of examination malpractice. in this context, the basicreproductive number defines the average number of secondary candidates that will be influencedby just one candidate who engages into examination malpractice. the method of next generatingmatrix which is mostly used is also adopted to compute the model r0. to achieve this, we expressedthe malpractice sub-system of model (1) in the form dx dt = (f − v)xt where xt is the transposeof x = (m, c), f and v are the rates of generation of new misconducting candidates and transferrespectively. using this malpractice/infected sub-system  dm dt = θsm − q1m dc dt = δm − µc (5) we have: f = θsm 0  and v =  q1m −δm + µc  (6) evaluating the jacobian matrices f and v of f and v at the dfe gives respectively: f = θs ∗ 0 0 0  and v = q1 0 −δ µ  (7) using f and v from (7), we obtain fv −1 given by: fv −1 =  θs∗ q1 0 0 0  (8) consequently, we obtain r0 as the spectral radius of fv −1 given by: r0 = λθq3 q1(q0q3 − βγ) = λθ(γ + µ) µ(τ + δ + µ)(β + γ + µ) (9) 2.3. stability analysis of the examination malpractice-free equilibrium. https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 72.3.1. local stability. to study the local stability of the emfe, we adopt the linearization ap-proach. theorem 3. the equilibrium point ( ξ∗ = ( λq3 q0q3−βγ , 0, 0, 0, βλ q0q3−βγ )) is locally asymptotically stable (las) if r0 < 1 and unstable if r0 > 1 proof. let j0 be the jacobian matrix of system (1) evaluated at ξ∗, that is: j0 =  −q0 −θs∗ 0 0 γ 0 θs∗ − q1 0 0 0 0 δ −µ 0 0 0 τ 0 −q2 0 β 0 0 α −q3  (10) clearly, the matrix in (10) admits one negative eigenvalue, namely λ1 = −µ. furthermore, thenature of the remaining eigenvalues of the matrix in (10) can be obtained from the sub-matrix in(11) below: j1 =  −q0 −θs∗ 0 γ 0 θs∗ − q1 0 0 0 τ −q2 0 β 0 α −q3  (11) according to the routh-hurwitz stability theorem, matrix in (11) will be stable if its trace anddeterminant are negative and positive respectively [12,13]. now: t race(j1) = − (q0 + q2 + q3 + q1(1− r0)) < 0 if r0 < 1 (12)and det(j1) = q1(q0q1q3 − βγ)(1− r0) > 0 if r0 < 1 (13)thus, the emfe state is locally asymtotically stable whenever r0 < 1. � next, we examine the global stability of the examination malpractice-free state. 2.3.2. global stability of the examination malpractice-free state. to establish the long termstability behavior of the examination malpractice-free equilibrium state, we consider the followinglyapunov function: v (t) = 1 q1 m (14)differentiating v(t) gives dv (t) dt = 1 q1 dm dt =− (1− r0)m (15) https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 8 it is clear from (15) that dv (t) dt ≤ 0 if r0 ≤ 1. hence, the emfe point is globally asymptoticallystable if r0 ≤ 1 and unstable otherwise 2.4. existence of examination malpractice endemic equilibrium point (emeep). solving system(1) for the state variables at the emeep gives the following system of solutions: s∗∗ = λq2q3+αγτm∗∗ q2q3(θm∗∗+q2(q0q3−βγ) m∗∗ = q1q2(q0q3−βγ)(r0−1) θ(q1q2q3−αγτ) c∗∗ = δq1q2(q0q3−βγ)(r0−1) θµ(q1q2q3−αγτ) r∗∗ = τq1(q0q3−βγ)(r0−1) θ(q1q2q3−αγτ) h∗∗ = βλq2q3+αβγτm∗∗ q2q3(θq3m∗∗+(q0q3−βγ)) + ατq1(q0q3−βγ)(r0−1) θq3(q1q2q3−αγτ) (16) it is clear from (16), that s∗∗, m∗∗, c∗∗, r∗∗ and h∗∗ exist if and only if r0 > 1, this gives thecondition for the existence of the emeep. in what follows, we examine the global stability of thisendemic equilibrium point using liapunov function. 2.4.1. global stability of the examination malpractice endemic equilibrium point. theorem 4. the model represented by system (1) admits a globally asymptotically stable nontrivial endemic equilibrium ( ξ∗∗) whenever r0 > 1 proof. consider a positive definite function l defined by: l (ξ∗∗) = ( (s − s∗∗)− s∗∗ ln s s∗∗ ) + ( (m −m∗∗)−m∗∗ ln m m∗∗ ) + ( (c − c∗∗)− c∗∗ ln c c∗∗ ) + ( (r − r∗∗)− r∗∗ ln r r∗∗ ) + ( (h −h∗∗)−h∗∗ ln h h∗∗ ) taking the time derivative of l (ξ∗∗) gives: dl (ξ∗∗) dt = ( 1− s∗∗ s ) ds dt + ( 1− m∗∗ m ) dm dt + ( 1− c∗∗ c ) dc dt + ( 1− r∗∗ r ) dr dt + ( 1− h∗∗ h ) dh dt = ( s − s∗∗ s ) [λ + γh − θsm − q0s] + ( m −m∗∗ m ) (θsm − q1m) + ( c − c∗∗ c ) (δm − µc) + ( r − r∗∗ r ) (τm − q2r) + ( h −h∗∗ h ) (βs + αr − q3h) https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 9 = λ + γh + θms∗∗ + q0s ∗∗ − q0s − (λ + γh) s∗∗ s + q1m ∗∗ − q1m − θm∗∗s + δm + µc∗∗ − µc − δm c∗∗ c + τm + q2r ∗∗ − q2r − τm r∗∗ r + βs + αr + q3h ∗∗ − q3h − (βs + αr) h∗∗ h = l+ − l−where l+ = λ + γh + θms∗∗ + q0s ∗∗ + q1m ∗∗ + δm + µc∗∗ + τm + q2r ∗∗ + βs + αr + q3h ∗∗ l− = q0s + (λ + γh) s∗∗ s + q1m + θm∗∗s + µc + δm c∗∗ c + q2r + τm r∗∗ r + q3h + (βs + αr) h∗∗ h (17) if we now assume l+ ≤ l−then, it follows from (17) that dl(ξ∗∗) dt ≤ 0 with the equality holding if and only if s∗∗ = s, m∗∗ = m, c∗∗ = c, r∗∗ = r and h∗∗ = htherefore, the largest compact invariant set within d (the model’s invariant region) is the single-ton {ξ∗∗} = {s∗∗, m∗∗, c∗∗, r∗∗, h∗∗} . hence, following [14], the unique endemic equilibriumof system (1) is globally asymptotically stable whenever it exists. � 3. local sensitivity analysis to investigate the contribution of each model parameter to the occurrence of examination mal-practice, we calculated the sensitivity indices of the parameters of the basic examination malpracticereproductive number using the forward sensitivity index expression. the output is tabulated in table4 below. table 4. sensitivity indices of r0 parameters parameter sensitivity index λ +1.00000 θ +1.00000 β −0.35702 γ +0.35682 τ −0.57061 δ −0.42796 µ −1.00123 https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 104. numerical simulations to examine the dynamical evolution of the model sub-classes, we simulated the proposed modelsystem of equations using matlab ode45. we used the parameter values given in table 3 with thefollowing assumed initial values: s(0) = 120, m(0) = 25, c(0) = 5, r(0) = 5 and h(0) = 15.the simulation graphs are shown from figure 2 to figure 7 5. discussion and conclusion a five-compartmental model is constructed to study the phenomenon of examination malpracticeat the west african senior school certificate examination (wassce) level in ghana. the well-posedness of the model is established. the examination malpractice free equilibrium is shownto possess both a local and global asymptotic stability whenever the examination malpracticereproductive number (r0) is less than one. furthermore, the examination malpractice persistentequilibrium is derived and shown to be globally stable if r0 > 1. we carried out local sensitivityanalysis on the parameters of r0 and established that candidates recruitment rate (λ), the rateat which candidates engage into irregularities (θ) and the susceptibility rate (γ) of honest orlaw abiding candidates have positive impact on the examination malpractice reproductive number(r0). on the other hand, the rate at which susceptible candidates progress to honest or lawabiding candidates (β), the rate at which candidates recover from malpractice (τ ) and the rate ofcancellation of examination results (δ) have a negative impact on r0. in other words, increasing thevalue of any parameter with negative impact on r0 will lead to minimizing examination malpracticeat the wassce level in ghana. thus, we recommend that to control the evil act of examinationmalpractice during wassce, the examination bodies should intensify the orientation programsfor candidates, recruit honest and principle invigilators who can strictly monitor the candidatesin the examination rooms, strictly enforce punishment or disciplinary actions against examinationmalpractice perpetrators. figure 2. plot of sus-ceptible sub-population figure 3. plot of mal-practice sub-population https://doi.org/10.28924/ada/ma.5.16 eur. j. math. anal. 10.28924/ada/ma.5.16 11 figure 4. plot of can-celled sub-population figure 5. plot of recov-ered sub-population figure 6. plot of hon-est sub-population figure 7. plot of allsub-population acknowledgment. authors are much thankful to other faculty members and some senior highschool teachers for their guidance during the development of this manuscript. much appreciationalso goes to the respected reviewers for their valuable comments and suggestions. conflict of interest. authors declare that there is no conflict of interest regarding the publicationof this work. references [1] m. frempong, e. b. a. arloo, 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malpractice among the key players in nigeria, daffodil int.univ. j. sci. technol. 15 (2020), 25–32.[12] s. nana-kyere, b. seidu, k. nantomah, mathematical analysis of malaria epidemic: asymptotic stability withcost-effectiveness study, j. appl. math. 2024 (2024), 5533885.[13] m. m. ojo, e. f. doungmo goufo, assessing the impact of control interventions and awareness on malaria: amathematical modeling approach, commun. math. biol. neurosci. 2021 (2021), 93.[14] j. p. lasalle, stability theory and invariance principles, in: dynamical systems, academic press, new york, 1976. https://doi.org/10.28924/ada/ma.5.16 1. introduction 2. model construction 2.1. well-posedness of the model 2.2. the examination malpractice-free equilibrium (emfe) 2.3. stability analysis of the examination malpractice-free equilibrium 2.4. existence of examination malpractice endemic equilibrium point (emeep) 3. local sensitivity analysis 4. numerical simulations 5. discussion and conclusion acknowledgment conflict of interest references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 13doi: 10.28924/ada/ma.4.13 best proximity point of generalized θ − φ−proximal non-self contractions mohamed rossafi1,∗ , abdelkarim kari2 1faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, fes, morocco rossafimohamed@gmail.com 2faculty of sciences ben m’sik, hassan ii university, casablanca, morocco abdkrimkariprofes@gmail.com ∗correspondence: rossafimohamed@gmail.com abstract. in this manuscript, motivated and inspired by results of best proximity point of generalized f -proximal non-self contractions, we introduce the concept of generalized θ−φ−proximal contractionand prove new best proximity results for these contractions in the setting of a metric space. our resultsgeneralize and extend many recent results appearing in the literature. an example is being given todemonstrate the usefulness of our results. 1. introduction it is well known that the banach contraction theorem is the first outstanding result in thefield of the fixed point theory that ensure the existence of unique fixed point in complete metricspaces. due to its importance, various mathematics steadied many interesting extensions andgeneralizations [7,8,12,14]. one of the famous generalizations of the banach contraction principle [2]for existence of fixed point for self-mapping on metric space is the theorem by zheng et al. [14] andthe contraction introduced by jleli and samet in [6].best proximity point theorem analyses the condition under which the optimisation problem,namely infx∈a d(x, t x), has a solution. the point x is called the best proximity of t : a → b, if d(x, t x) = d(a,b), where {d(a,b) = inf d(x, y) : x ∈ a, y ∈ b}. note that the best proximitypoint reduces to a fixed point if t is a self-mapping. various best proximity point results wereestablished on such spaces [1, 9, 12].sankar raj [10] and zhang et al. [13] defined the notion of p−property and weak p−propertyrespectively. beg et al. [4] defined the concept of generalized f -proximal non-self contractions andobtained some best proximity point theorems for self-mappings.in this paper, inspired by the idea of generalized f -proximal non-self contractions, introducedby beg et al. [4] in metric spaces, we prove a new existence of best proximity point for generalized received: 21 jan 2024. key words and phrases. p -property, best proximity point, generalized θ − φ-proximal contraction.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.13 https://orcid.org/0000-0002-5662-6921 https://orcid.org/0000-0003-0088-9404 eur. j. math. anal. 10.28924/ada/ma.4.13 2 θ − φ−proximal contraction defined on a closed subset of a complete metric space. our theoremsextend, generalize and improve many existing results. 2. preliminaries let (a,b) be a pair of non empty subsets of a metric space (x, d). we adopt the followingnotations: d(a,b) = {inf d (a, b) : a ∈ a, b ∈ b}; a0 = { a ∈ a there exists b ∈ a such that d (a, b) = d (a,b)}; b0 = { b ∈ b there exists a ∈ a such that d (a, b) = d (a,b)}. definition 2.1. [5] let t : a → b be a mapping. an element x∗ is said to be a best proximitypoint of t if d (x∗, t x∗) = d (a,b) . definition 2.2. [10] let (a,b) be a pair of non empty subsets of a metric space (x, d) such that a0 is non empty. then the pair (a,b) is to have p -property if and only ifd (x1, y1) = d (a,b) d (x2, y2) = d (a,b) ⇒ d(x1, x2) = d(y1, y2) where x1, x2 ∈ a0 and y1, y2 ∈ b0. definition 2.3. [3] a set b is called approximately compact with respect to a if every sequence {xn} of b with d(y , xn)→ d(y , b) for some y ∈ a has a convergent subsequence. definition 2.4. [6] let θ be the family of all functions θ : ]0,+∞[ → ]1,+∞[ such that (θ1) θ is strictly increasing; (θ2) for each sequence xn ∈ ]0,+∞[; lim n→0 xn = 0, if and only if lim n→∞ θ (xn) = 1; (θ3) θ is continuous. definition 2.5. [14] let φ be the family of all functions φ: [1,+∞[ → [1,+∞[, such that (φ1) φ is increasing; (φ2) for each t ∈ ]1,+∞[, l imn→∞φn(t) = 1; (φ3) φ is continuous. lemma 2.6. [14] if φ ∈ φ then φ(1)=1, and φ(t) < t . definition 2.7. [14]. let (x, d) be a metric space and t : x → x be a mapping. t is said to be a θ − φ−contraction if there exist θ ∈ θ and φ ∈ φ such that for any x, y ∈ x, d (tx, t y) > 0⇒ θ [d (tx, t y)] ≤ φ [θ (d (x, y))] , https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 33. main result in this section, inspired by the notion of f -proximal contraction of the first kind and second kind,we introduce new generalized θ− φ-proximal first kind and second kind on complete metric space. definition 3.1. the mapping t : a→ b is said to be a generalized θ − φ-proximal contraction offirst kind if there exist θ ∈ θ, φ ∈ φ and a, b, c, h ≥ 0 with a + b + +2ch, c 6= 1 such thatd (u1, t v1) = d (a,b) d (u2, t v2) = d (a,b) ⇒ θ(d(u1, u2)) ≤ φ [θ [ad (v1, v2) + bd (u1, v1) + cd (u2, v2) + h (d (v1, u2) + d (v2, u1))]]for all u1, u2, v1, v2 ∈ a and u1 6= v1. definition 3.2. the mapping t : a→ b is said to be a generalized θ − φ-proximal contraction ofsecond kind if there exist θ ∈ θ, φ ∈ φ and a, b, c, h ≥ 0 with a + b + +2ch, c 6= 1 such thatd (u1, t v1) = d (a,b) d (u2, t v2) = d (a,b) ⇒ θ(d(tu1, t u2)) ≤ φ [θ [ad (tv1, t v2) + bd (tu1, t v1) + cd (tu2, t v2) + h (d (tv1, t u2) + d (tv2, t u1))]]for all u1, u2, v1, v2 ∈ a and tu1 6= tv1. theorem 3.3. let (x, d) be a complete metric space and (a,b) be a pair of non-void closed subsets of (x, d). if b is approximately compact with respect to a and t : a → b satisfy the following conditions :(i) t (a0) ∈ b0 and the pair (a,b) satisfies the weak p -property;(ii) t is a generalized θ − φ-proximal contraction of first kind. then there exists a unique u ∈ a such that d(u, tu) = d(a,b). in addition, for any fixed element u0 ∈ a0, sequence {un} defined by d(un+1, t un) = d(a,b), converges to the proximity point. proof. choose an element u0 ∈ a0. as, t (a0) ∈ b0, therefore there is an element u1 ∈ a0satisfying d(u1, t u0) = d(a,b).since t (a0) ∈ b0, there exists u2 ∈ a0 such that d(u2, t u1) = d(a,b). again, since t (a0) ∈ b0, there exists u3 ∈ a0 such that d(u3, t u2) = d(a,b). https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 4continuing this process, by induction, we construct a sequence xn ∈ a0 such that d (un+1, t un) = d(a,b),∀n ∈ n. since (a,b) satisfies the p property, we conclude that d(un, un+1) = d(tun, t un+1),∀n ∈ n. (3.1) if un0 = un0+1 for some n0 ∈ n, from (3) one obtains d (un0 , t un0) = d (un0+1, t un0) = d(a,b) (3.2) that is, un0 ∈ bpp . thus, we suppose that d(un, xn+1) > 0 for all n ∈ n.we shall prove that the sequence un is a cauchy sequence. let us first prove that lim n→∞ d (un, un+1) = 0. as t is generalized (θ, φ)-proximal contraction of the first kind, we have that θ (d (un, un+1)) ≤ φ [θ [ad (un−1, un) + bd (un−1, un) + cd (un, un+1) + h (d (un−1, un+1) + d (un, un))]] = φ [θ [ad (un−1, un) + bd (un−1, un) + cd (un, un+1) + h (d (un−1, un+1))]] ≤ φ [θ [ad (un−1, un) + bd (xn−1, xn) + cd (un, un+1) + h (d (un−1, un) + d (un, un+1))]] = φ [θ [(a + b + h)d (un−1, un) + (c + h)d (un, un+1)]] since θ is strictly increasing and by lemma 2.6, we deduce d (xn, xn+1) < (a + b + h)d (xn−1, xn) + (c + h)d (xn, xn+1) . thus d (un, un+1) < a + b + h 1− c − h (d (un−1, xn)). if b + b + c + 2h = 1, we have 0 < 1− c − h and so d (un, un+1) ≤ a + b + h 1− c − h (d (un−1, un)) = d (un−1, un) ,∀n ∈ n; consequently, θ (d (un, un+1)) ≤ φ [θ (d (un−1, un))] if b + b + c + 2h < 1, we have 0 < 1− c − h and so d (un, un+1) < d (un−1, un) ,∀n ∈ n; consequently, θ (d (un, un+1)) ≤ φ [θ (d (un−1, un))] https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 5it implies θ (d (un, un+1)) ≤ φ [θ (d(xn−1, un)] ≤ φ2 [θ (d(un−2, un−1)] ≤ ... ≤ φn [θ (d(u0, u1)] . taking the limit as n →∞, we have 1 ≤ θ(d (un, un+1)) ≤ lim n→∞ φn [θ(d (u0, u1))] = 1. since θ ∈ θ, we obtain lim n→∞ d (un, un+1) = 0. (3.3) next, we shall prove that {un}n∈n is a cauchy sequence, i.e, limn→∞ d (un,um) = 0, for all n ∈ n.suppose to the contrary that exists ε > 0 and sequences n(k) and m(k) of natural numbers suchthat m(k) > n(k) > k, d ( xm(k) , xn(k) ) ≥ ε, d ( um(k)−1 , un(k) ) < ε. (3.4) using the triangular inequality, we find that, ε ≤ d ( um(k) , un(k) ) ≤ d ( um(k) , un(k)−1 ) + d ( xn(k)−1, xn(k) ) (3.5) < ε+ d ( un(k)−1, un(k) ) . (3.6) then, by 3.4 and 3.22, it follows that lim k→∞ d ( um(k) , un(k) ) = ε. (3.7) using the triangular inequality, we find that, ε ≤ d ( um(k) , un(k) ) ≤ d ( um(k) , un(k)+1 ) + d ( xn(k)+1, un(k) ) (3.8) and ε ≤ d ( um(k) , un(k)+1 ) ≤ d ( um(k) , un(k) ) + d ( un(k), un(k)+1 ) (3.9) then, by (3.25) and (3.9), it follows that lim k→∞ d ( um(k) , un(k)+1 ) = ε. (3.10) similarly method, we conclude that lim k→∞ d ( um(k)+1 , un(k) ) = ε. (3.11) using again the triangular inequality, d ( um(k)+1 , un(k)+1 ) ≤ d ( xm(k)+1 , xm(k) ) + d ( um(k), un(k) ) + d ( un(k) , un(k)+1 ) . (3.12) https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 6on the other hand, using triangular inequality, we have d ( um(k) , un(k) ) ≤ d ( um(k) , um(k)+1 ) + d ( um(k)+1 , un(k)+1 ) + d ( un(k)+1 , un(k) ) . (3.13) letting k →∞ in inequality (3.12) and (3.13), we obtain lim k→∞ d ( um(k)+1 , un(k)+1 ) = ε. (3.14) substituting u1 = xm(k)+1 , u2 = xn(k)+1 , v1 = um(k) and v1 = un(k) in assumption of the theorem, weget θ ( d ( um(k)+1 , un(k)+1 )) ≤ φ  θ  ad ( um(k) , un(k) ) + bd ( um(k)1, un(k) ) + cd ( un(k)+1, un(k) ) + h(d ( um(k) , un(k)+1 ) + d ( un(k) , um(k)+1 ) )   (3.15) letting letting k →∞ in (3.15), and using (θ1), (θ3) , (φ3) and lemma (2.6) we obtain θ (ε) ≤ φ [θ (aε+ bε+ cε+ 2hε)] . we derive ε < ε. which is a contradiction. thus limn,m→∞ d (un, um) = 0, which shows that {xn} is a cauchysequence. then there exists z ∈ a such that lim n→∞ d (un, u) = 0. also, d (u,b) ≤ d (u, tun) ≤ d (u, xn+1) + d (un+1, t un) = d (u, un+1) + d (a,b) ≤ d (u, un+1) + d (u,b) . therefore, d (u, tun)→ d (u,b) . in spite of the fact that b is approximately compact with respectto a , the sequence {tun} has a subsequence {tunk} converging to some element v ∈ b. so itturns out that d(u, v) = lim n→∞ d ( unk+1, t unk ) = d(a,b). (3.16) thus u must be an element of a0. again, since t (a0) ∈ b0, there exists t ∈ a0 such that d(t, tu) = d(a,b) (3.17) https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 7for some element t in a. using the weak p-property and (3.33) we have d(unk+1, t) = d(punk , p u),∀nk ∈ n. if for some n0, d(t, un0+1) = 0, consequently d(pun0 , t u) = 0. so pun0 = tu, hence d(a,b) = d(u, tu). thus the conclusion is immediate. so let for any n ≥ 0, d(t, un+1) > 0. since t is ageneralized (θ, φ)-proximal contraction of the first kind, it follows from this that θ(d(t, un+1)) ≤ φ [θ [ad (u, un) + bd (t, u) + cd (un, un+1) + h (d (u, un+1) + d (un, t))]] (3.18) since θ and φ are two continuous functions, by letting n →∞ in inequality (3.18), we obtain θ(d(t, u)) ≤ φ [θ [(b + h) (d (u, t))]] ≤ φ [θ [(d (u, t))]] < θ(d(t, u)). it is a contradiction. therefore, u = t , that d(u, tu) = d(t, tu) = d(a,b). uniqueness: suppose that there is another best proximity point z of the mapping t such that d(z, t z) = d(a,b). since t is a generalized (θ, φ)-proximal contraction of the first kind, it follows from this that θ(d(z, u)) ≤ φ [θ [ad (z, u) + bd (z, z) + cd (u, u) + h (d (z, u) + d (z, u))]] = φ [θ [(a + 2h)d (z, u)]] , which is a contradiction. thus, z and u must be identical. hence, t has a unique best proximitypoint. � next, we state and prove the best proximity point theorem for non-self generalized (θ, φ)-proximalcontraction of the second kind. theorem 3.4. let (x, d) be a complete metric space and (a,b) be a pair of non-void closed subsets of (x, d). if a is approximately compact with respect to b and t : a → b satisfy the following conditions :(i) t (a0) ∈ b0 and the pair (a,b) satisfies the weak p -property;(ii) t is continuous generalized (θ, φ)-proximal contraction of second kind. then there exists a unique u ∈ a such that d(u, tu) = d(a,b) and un → u, where u0 is any fixed point in a0 and d(un+1, t un) = d(a,b) for n ≥ 0. further, if z is another best proximity point of t , then tu = tz . https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 8 proof. similar to theorem 3.3, we can find a sequence {un} in a0 such that d(un+1, t un) = d(a,b). (3.19) for all non-negative integral values of n. from the p-property and (3.19) we get d(un, un+1) = d(tun−1, t un),∀n ∈ n. if for some n0, d(un0+1 , un0+2) = 0, consequently d(tun0 , t un0+1) = 0. so tun0 = tun0+1,hence d(a,b) = d(tun0 , tn0+1). thus the conclusion is immediate. so let for any n ≥ 0, d(tun, t un+1) > 0. we shall prove that the sequence un is a cauchy sequence. let us firstprove that lim n→∞ d (un, un+1) = 0. as t is generalized (θ, φ)-proximal contraction of the second kind, we have that θ (d (tun, t un+1)) ≤ φ [θ [ad (tun−1, t un) + bd (tun−1, t un) + cd (tun, t un+1) + h (d (tun−1, t un+1) + d (tun, t un))]] = φ [θ [ad (tun−1, t un) + bd (tun−1, t un) + cd (tun, t un+1) + h (d (tun−1, t un+1))]] ≤ φ [θ [ad (tun−1, t un) + bd (tun−1, t un) + cd (tun, t un+1) + h (d (tun−1, t un) + d (tun, t un+1))]] = φ [θ [(a + b + h)d (tun−1, t un) + (c + h)d (tun, t un+1)]] since θ is strictly increasing and by lemma 2.6, we deduce d (tun, t un+1) < (a + b + h)d (tun−1, t un) + (c + h)d (tun, t un+1) . thus d (tun, t un+1) < a + b + h 1− c − h (d (tun−1, t un)). if b + b + c + 2h = 1, we have 0 < 1− c − h and so d (tun, t un+1) ≤ a + b + h 1− c − h (d (tun−1, t un)) = d (tun−1, t un) ,∀n ∈ n; consequently, θ (d (tun, t un+1)) ≤ φ [θ (d (tun−1, t un))] if b + b + c + 2h < 1, we have 0 < 1− c − h and so d (tun, t un+1) < d (tun−1, t un) ,∀n ∈ n; consequently, θ (d (tun, t un+1)) ≤ φ [θ (d (tun−1, t un))] https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 9it implies θ (d (tun, t un+1)) ≤ φ [θ (d(tun−1, t un)] ≤ φ2 [θ (d(tun−2, t un−1)] ≤ ... ≤ φn [θ (d(tu0, t u1)] . taking the limit as n →∞, we have 1 ≤ θ(d (tun, t un+1)) ≤ lim n→∞ φn [θ(d (tu0, t u1))] = 1. since θ ∈ θ, we obtain lim n→∞ d (tun, t un+1) = 0. (3.20) next, we shall prove that {tun}n∈n is a cauchy sequence, i.e, limn→∞ d (tun,tum) = 0, for all n ∈ n. suppose to the contrary that exists ε > 0 and sequences tn(k) and tm(k) of naturalnumbers such that tm(k) > tn(k) > k, d ( tum(k) , t un(k) ) ≥ ε, d ( tum(k)−1 , t un(k) ) < ε. (3.21) using the triangular inequality, we find that, ε ≤ d ( tum(k) , t un(k) ) ≤ d ( tum(k) , t xn(k)−1 ) + d ( tun(k)−1, t un(k) ) (3.22) < ε+ d ( tun(k)−1, t un(k) ) . (3.23) then, by 3.4 and 3.22, it follows that lim k→∞ d ( tum(k) , t un(k) ) = ε. (3.24) using the triangular inequality, we find that, ε ≤ d ( tum(k) , t un(k) ) ≤ d ( tum(k) , t un(k)+1 ) + d ( tun(k)+1, t un(k) ) (3.25) and ε ≤ d ( tum(k) , t un(k)+1 ) ≤ d ( tum(k) , t un(k) ) + d ( tun(k), t un(k)+1 ) (3.26) then, by (3.25) and (3.9), it follows that lim k→∞ d ( tum(k) , t un(k)+1 ) = ε. (3.27) similarly method, we conclude that lim k→∞ d ( tum(k)+1 , t un(k) ) = ε. (3.28) using again the triangular inequality, d ( tum(k)+1 , t un(k)+1 ) ≤ d ( um(k)+1 , t um(k) ) + d ( tum(k), t un(k) ) + d ( tun(k) , t un(k)+1 ) . (3.29) https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 10on the other hand, using triangular inequality, we have d ( tum(k) , t un(k) ) ≤ d ( tum(k) , t um(k)+1 ) + d ( tum(k)+1 , t un(k)+1 ) + d ( tun(k)+1 , t un(k) ) . (3.30) letting k →∞ in inequality (3.29) and (3.30), we obtain lim k→∞ d ( tum(k)+1 , t un(k)+1 ) = ε. (3.31) substituting u1 = tum(k)+1 , u2 = tun(k)+1 , v1 = tum(k) and v1 = tun(k) in assumption of thetheorem, we get θ ( d ( tum(k)+1 , t un(k)+1 )) ≤ φ  θ  ad ( tum(k) , t un(k) ) + bd ( tum(k)1, t un(k) ) + cd ( tun(k)+1, t un(k) ) + h(d ( tum(k) , t un(k)+1 ) + d ( tun(k) , t um(k)+1 ) )  (3.32)letting letting k →∞ in (3.32), and using (θ1), (θ3) , (φ3) and lemma (2.6) we obtain θ (ε) ≤ φ [θ (aε+ bε+ cε+ 2hε)] . we derive ε < ε. which is a contradiction. thus limn,m→∞ d (tun, t um) = 0, which shows that {tun} is a cauchysequence. then there exists v ∈ b such that lim n→∞ d (tun, v) = 0. also, d (v , a) ≤ d (v , tun) ≤ d (v , un+1) + d (un+1, t un) = d (v , un+1) + d (a,b) ≤ d (v , un+1) + d (v , a) . therefore, d (v , tun) → d (v , a) . since a is approximately compact with respect to b , thesequence {un} has a subsequence {unk} converging to some element u ∈ a. so it turns out that d(u, v) = lim n→∞ d ( unk+1, t unk ) = d(a,b). (3.33) because t is a continuous mapping, d(u, tu) = lim n→∞ d(un+1, t un) = d(a,b). https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 11uniqueness: suppose that there is another best proximity point z of the mapping t such that d(z, t z) = d(a,b). since t is a generalized (θ, φ)-proximal contraction of the first second, it follows from this that θ(d(tz, tu)) ≤ φ [θ [ad (tz, tu) + bd (tz, t z) + cd (tu, tu) + h (d (tz, tu) + d (tz, tu))]] = φ [θ [(a + 2h)d (tz, tu)]] , which is a contradiction. thus, z and u must be identical. hence, t has a unique best proximitypoint. � theorem 3.5. let (x, d) be a complete metric space and (a,b) be a pair of non-void closed subsets of (x, d). let t : a→ b satisfy the following conditions :(i) t (a0) ∈ b0 and the pair (a,b) satisfies the weak p -property;(ii) t is a generalized (θ, φ)-proximal contraction of the first kind as well as a generalized (θ, φ)-proximal contraction of the second kind. then there exists a unique u ∈ a such that d(u, tu) = d(a,b) and un → u, where u0 is any fixed point in a0 and d(un+1, t un) = d(a,b) for n ≥ 0. proof. similar to theorem 3.3, we find a sequence {un} in a0 such that d(un+1, t un) = d(a,b) for all non-negative integral values of n. similar to theorem 3.3, we can show that sequence {un}is a cauchy sequence. thus converges to some element u in a. as in theorem 3.4, it can be shownthat the sequence {tun} is a cauchy sequence and converges to some element v in b. therefore, d(u, v) = lim n→∞ d(un+1, t un) = d(a,b). (3.34) eventually, u becomes an element of a0. in light of the fact that t (a0) ∈ b0, d(t, tu) = d(a,b) for some element t in a. from the p-property framework and (3.34,) we get d(un+1, t) = d(tun, t u),∀n ∈ n. if for some n0, d(t, un0+1) = 0, consequently d(tun0 , t u) = 0 . so tun0 = tu, hence d(a,b) = d(u, tu). thus the conclusion is immediate. so let for any n ≥ 0, d(t, un+1) > 0. since t is ageneralized (θ, φ)-proximal contraction of the first kind, it can be seen that θ(d(t, un+1)) ≤ φ [θ (ad(u, un) + bd(t, u) + cd(un, un+1) + h[d(u, un+1) + d(un, t))] . (3.35) https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 12since θ and φ are two continuous functions, by letting n → ∞ in inequality (3.35), we obtain, d(u, tu) = d(t, tu) = d(a,b). also, as in the theorem 3.3, the uniqueness of the best proximitypoint of mapping t follows. � example 3.6. let x = {λn : n ∈ n} with the metric d(x, y) = |x − y | for all x, y ∈ x , where thesequence gn, defined by λ1 = 1 λ2 = 1 + 2 λ3 = 1 + 2 + 3 ... λn = 1 + 2 + 3 + ...+ n. we know, (x, d) is a complete metric space. let a = g3n : n ∈ n and b = g3n−1 : n ∈ n.it is easy to see that d(a,b) = 3, a0 = a and b0 = b. define a mappings t : a → b, by t (λ3n) = λ3n−1 for all n ≥ 1. it is clear that a is approximately compact with respect to b, (a,b) satisfies the p-property, t is continuous and t (a0) ⊆ b0 . we will show that t is an (θ, φ)-proximal contraction with θ ∈ θ and φ ∈ φ that is θ(t) = et and φ(t) = t 1 2 . observe that,with out of generality, we may assume that n < m, and since λ3n−1 = 1 + 2 + 3 + ...+ 3n − 1, λ3m−1 = 1 + 2 + 3 + ...+ 3m − 1, λ3n = 1 + 2 + 3 + ...+ 3n − 1 + 3n, λ3m = 1 + 2 + 3 + ...+ 3m − 1 + 3m. it follow that, d(t (λ3n), t (λ3m)) = |λ3n−1 − λ3m−1| = 3n + (3n + 1) + ...+ (3m − 1), d(λ3n, λ2m) = |λ2n − λ2m| = 3n + (3n + 1) + ...+ (3m), and d(t (λ2n), t (λ3m))− d(λ3n, λ3m) = |λ3n−1 − λ3m−1| − |λ3n − λ3m| = 3n − 3m. https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 13so that, ed(t (λ3n),t (λ3m))−d(λ3n,λ3m)) = ed(t (λ3n),t (λ3m)) ed(λ3n,λ3m) = e3n−3m) = e−3(m−n)) ≤ e−3 = 1 e3 . so that, ed(t (λ3n),t (λ3m)) + 1 = θ(d(t (λ3n), t (λ3m))) ≤ ed(λ3n,λ3m) 1 e3 + 1 ≤ ed(λ3n,λ3m) + 2 2 = φ [θ(d(λ3n, λ3m))] . consequently, t is an generalized (θ, φ)-proximal contraction of the second kind with a = 1, b = c = h = 0. thus, all the conditions of theorem 3.4 are satisfied. hence, t has a unique bestproximity point and there exist λ3 ∈ a such that d(λ3, tλ3) = d(λ3, λ2) = 3 = d(a,b) conflict of interestthe authors declare that they have no competing interests. authors’ contributionsthe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript. references [1] h. aydi, h. lakzian, z.d. mitrović, s. radenović, best proximity points of mt-cyclic contractions with property uc,numer. funct. anal. optim. 41 (2020) 871-882.[2] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund.math. 3 (1922) 133-181.[3] s.s. basha, p. veeramani, best proximity pair theorems for multifunctions with open fibres, j. approx. theory 103(2000) 119–129.[4] i. beg, g. mani, a.j. gnanaprakasam, best proximity point of generalized f-proximal non-self contractions, j. fixedpoint theory appl. 23 (2021), 1-11[5] a. eldred, w. kirk, p. veeramani, proximal normal structure and relatively nonexpansive mappings, stud. math. 171(2005) 283-293.[6] m. jleli, b. samet, a new generalization of the banach contraction principle, j. ineq. appl. 2014 (2014) 38.[7] a. kari, m. rossafi, e. marhrani, m. aamri, fixed-point theorem for nonlinear f-contraction via w-distance, adv.math. phys. 2020 (2020) 6617517. https://doi.org/10.28924/ada/ma.4.13 eur. j. math. anal. 10.28924/ada/ma.4.13 14 [8] a. kari, m. rossafi, e. marhrani, m. aamri, θ − φ−contraction on (α, η)−complete rectangular b−metric spaces,int. j. math. math. sci. 2020 (2020) 5689458.[9] v. parvaneh, m. reza haddadi, h. aydi, on best proximity point results for some type of mappings, j. funct. spaces2020 (2020) 6298138.[10] v.s. raj, a best proximity point theorem for weakly contractive non-self-mappings, nonlinear anal. tma. 74 (2011)4804-4808.[11] m. rossafi, a. kari, some fixed point theorems for f -expansive mapping in generalized metric spaces, open j. math.anal. 5 (2021) 17-30.[12] m. rossafi, a. kari, best proximity point theorems for α-proximal θ, φ-non-self mappings, asian j. math. appl. 2022(2022) 3.[13] j. zhang, y. su, q. cheng, a note on ’a best proximity point theorem for geraghty-contractions’, fixed point theoryappl. 2013 (2013) 99.[14] d. zheng, z. cai, p. wang, new fixed point theorems for theta-phi contraction in complete metric spaces, j. nonlinearsci. appl. 10 (2017) 2662-2670. https://doi.org/10.28924/ada/ma.4.13 1. introduction 2. preliminaries 3. main result references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 11doi: 10.28924/ada/ma.4.11 on the stability of hyers orthogonality functional equations in non-archimedean spaces wenhui xu, qi liu, jinyu xia∗ school of mathematics and physics, anqing normal university, anqing 246133, p. r. chinaxuwenhuiwww@163.com, liuq67@aqnu.edu.cn, y23060036@stu.aqnu.edu.cn ∗correspondence: y23060036@stu.aqnu.edu.cn abstract. in this paper, we investigate the stability of specially orthogonally functional equationsderiving from additive and quadratic functions 4f (x + y) + 4f (x − y) + 10f (x) + 14f (−x)− 3f (y)− 3f (−y) = f (2x + y) + f (2x − y) and f ( x + y + z 2 ) + f ( x + y − z 2 ) + f ( x − y + z 2 ) + f ( y + z − x 2 ) = f (x) + f (y) + f (z) where f is a mapping from abelian group to a non-archimedean space. by adopting a new method,we have made an attempt to prove the hyers-ulam stability in non-archimedean spaces. 1. introduction and preliminaries the stability problem of functional equations originated from ulam in 1940 when he posedthe group homomorphism problem "given an approximately linear mapping f , when does a linearmapping t exist that approximates f ?" in 1941, hyers [1] explored the scenario of approximatelyadditive mapping f : x → y where x and y are banach spaces and f satisfies ‖f (x + y)− f (x)− f (y)‖ 6 ε for all x, y ∈ x . then there is a unique mapping additive l : x → y satisfying ‖f (x)− l(x)‖ 6 ε with the limit l(x) = lim n→∞ f (2nx) 2n . rassias [14] weakened the bounded cauchy difference proposed by hyers in the map and ex-tended it to the unbounded cauchy difference ‖f (x + y)− f (x)− f (y)‖ 6 ε(‖x‖p + ‖y‖p) received: 5 mar 2024.key words and phrases. orthogonality; stability; non-archimedean space; functional equations.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 2where ε > 0 and p ∈ [0, 1), hyers’ theorem was extended to approximately linear maps. r. gerand j. sikorska [7] restricted the conditions with (x, y) = 0 and investigated the stability of thecauchy functional f (x + y) = f (x) + f (y) (1.1) of course it is easy to spot that the function f (x) = ‖x‖2 satisfies the functional equations (1.1)by the pythagorean theorem. they founded that there exists a orthogonality additive mapping g : x → y such that ‖f (x)− g(x)‖ 6 16 3 ε for all x ∈ x with restriction on definition domain (1.1) was denoted as a additive equations.similarly, the equation was called as a quadratic equation which satisfies f (x + y) + f (x − y) = 2f (x) + 2f (y). (1.2) during several decades, mathematicians have achieved various fruits in studying the stability offunctional equations based one these two equations in the spirit of hyers-ulam-rassias.now let us introduce the concept of orthogonality ⊥ defined by rätz [16]. suppose x is a realvector space with dimx > 2 and ⊥ is a binary relation on x are characterized by the followingproperties:(i) totality of ⊥ for zero: x ⊥ 0, 0 ⊥ x for all x ∈ x;(ii) homogeneity: if x, y ∈ x , x ⊥ y , then λx ⊥ µy for all λ, µ ∈ r ;(iii) independence: if x, y ∈ x \ {0}, x ⊥ y , if and only if x, y are linearly independent;(iv) for any two-dimensional subspace p of x and for every x ∈ p , there exists λ, y ∈ p such that x ⊥ y and x + y ⊥ λx − y .the pair (x , ⊥) is called an orthogonality space, which means an orthogonality space havinga normed structure. various notions of othogonlity on a real normed space such as roberts,pythagorean, isosceles, birkhoff-james, carlsson, hermite–hadamard (hh) type orthogonalitieson the basis of the fundamental properties. definition 1.1. [16] a function ‖·‖ :x → [0,∞) on a vector space over x a scalar field k with anon-archimedean valuation | · |, is classified as a non-archimedean norm if it meets the followingconditions:(i) nonnegativity: ‖x‖ > 0 and ‖x‖ = 0 if and only if x = 0;(ii) homogeneity: ‖λx‖ = |λ| ‖x‖ ∀λ ∈ k,∀x, y ∈ x;(iii) the strong triangle inequality ‖x + y‖ 6 max {‖x‖ , ‖y‖} ∀x, y ∈ x then (x,‖·‖) is called a non-archimedean normed space. https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 3gordji [9] investigated the stability of the traditionally functional equations d(x, y) = f (x + y)− f (x)− f (y) where f : x → y x, y are both non-arohimedean banach spaces. they established the existenceof functions ϕ,ψ : a× a→ [0,∞) such that ‖d(x, y)‖ 6 ϕ(x, y) ‖f (xy)− f (x)f (y)‖ 6 ψ(x, y) for all x, y ∈ x , and they considered the case if there exists a constant 0 < l < 1 such that ϕ(2x, 2y) 6 |2|lϕ(x, y) ϕ(2x, 2y) 6 |2|2lψ(x, y) then there exist a unique ring homomorphis h : x → y such that ‖f (x)−h(x)‖ 6 1 |2|(1− l)ϕ(x, x) kang [10] explored the stability of the orthogonally functional equation(1.3) through the classi-fication of the oddness and evenness of f within the same spaces 4f (x + y) + 4f (x − y) + 10f (x) + 14f (−x)− 3f (y)− 3f (−y) = f (2x + y) + f (2x − y) (1.3) park [12] investigated the stability of the orthogonally additive-additive and orthogonallyquadratic-quadratic functional equation(1.4) in non-archimedean orthogonality spaces using con-ventional methods f ( x + y + z 2 ) + f ( x + y − z 2 ) + f ( x − y + z 2 ) + f ( y + z − x 2 ) = f (x) + f (y) + f (z)(1.4) drawing inspiration from [14], this paper we explore different spaces and employ new methodsto investigete the stability of the aforementioned equation(1.4) and (1.3). 2. stability of the orthogonally additive-quadratic functional equation in this section, we will use the following symbol d1f (x, y) = f (2x + y) + f (2x − y)− 4f (x + y)− 4f (x − y) −10f (x)− 14f (−x) + 3f (y) + 3f (−y) (2.1) we deal with the stability problem for the orthogonally additive-quartic functional equation for d1f (x, y) = 0 by referring to the stability proof of [13, 14]. https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 4 lemma 2.1. assume f : g → x be a mapping with g be an abelian group and (x, ‖ · ‖) be acomplete non -archimedean normed space. for all x, y ∈ g and there is a constant c > 0 suchthat ∥∥∥∥f (2x)− 38 f (4x) + 18 f (−4x) ∥∥∥∥ 6 c (2.2) then we define h(x, n) = ∥∥∥∥f (2x)− 2n + 12 · 4n f ( 2n+1x ) + 2n − 1 2 · 4n f ( −2n+1x )∥∥∥∥ and gn(x) = 2n + 1 2 · 4n f (2 nx)− 2n − 1 2 · 4n f (−2 nx) . n ∈ n (1)then we have |h(x, n + 1)− h(x, n)| 6 2n + 1 2 · 4n c (2.3) h(x, n) 6 c (2.4) (2)and {gn(x)} is a cauchy sequence, for every x ∈ g. hence, the mapping g : g → x can bedefined as g(x) = lim n→∞ gn(x) and then we get ‖f (2x)− g(2x)‖ 6 c (2.5) proof: adding one and subtracting one with h(x, n + 1) for matching and then using the in-equality, we obtain∥∥∥∥f (2x)− 2n+1 + 12 · 4n+1 f ( 2n+2x ) + 2n+1 − 1 2 · 4n+1 f ( −2n+2x )∥∥∥∥ 6 ∥∥∥∥f (2x)− 2n + 12 · 4n f ( 2n+1x ) + 2n − 1 2 · 4n f ( −2n+1x )∥∥∥∥ + 2n + 1 2 · 4n ∥∥∥∥f (2n+1x)− 38 f (2n+2x) +18 f (−2n+2x) ∥∥∥∥ + 2n − 1 2 · 4n ∥∥∥∥f (−2n+1 · x)+ 18 f (2n+2x)− 38 f (−2n+2x) ∥∥∥∥ 6 ∥∥∥∥f (2x)− 2n + 12 · 4n f ( 2n+1x ) + 2n − 1 2 · 4n f ( −2n+1x )∥∥∥∥+ c ·max{2n + 12 · 4n , 2n − 1 2 · 4n } next, it is easy to get |h(x, n + 1)− h(x, n)| 6 2n + 1 2 · 4n c https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 5then h(x, n) = ∥∥∥∥∥ ( n∑ i=2 h(x, i)− h(x, i − 1) ) + h(x, 1) ∥∥∥∥∥ 6 c ·max { 2 + 1 2 · 4 , 22 + 1 2 · 42 , · · · , 2n + 1 2 · 4n , 1 } = cnext, we have to prove that for every x ∈ g, the sequence gn(x) = 2n + 1 2 · 4n f (2 nx)− 2n − 1 2 · 4n f (−2 nx) n ∈ n is convergent in g. since x is complete, it is sufficient to show that (gn(x))n∈n is a cauchysequence for all x ∈ g. by matching ‖gn+1(x)− gn(x)‖twice then we have ‖gn+1(x)− gn(x)‖ 6 2n + 1 2 · 4n ∥∥∥∥f (2nx)− 38 f (2n+1x)+ 18 f (−2n+1x) ∥∥∥∥ + 2n − 1 2 · 4n ∥∥∥∥f (−2nx)− 38 f (−2n+1x)+ 18 f (2n+1x) ∥∥∥∥ 6c ·max { 2n + 1 2 · 4n , 2n − 1 2 · 4n } = 2n + 1 2 · 4n cfor each n ∈ n . this easily implies that {gn(x)} is a cauchy sequence. the mapping g : g → xcan be defined as g(x) = lim n→∞ gn(x)through the above results, we can obtain ‖f (2x)− g(2x)‖ = ‖h(x, n) + gn(2x)− g(2x)‖ 6 c in this section, let g be an abelian group and let ⊥ be a binary relation defined on g with theproperties:(i) x ⊥ 0, 0 ⊥ x , for all x ∈ x;(ii) if x, y ∈ x and x ⊥ y , then x2 ⊥ y 2 , 2x ⊥ 2y , 4x ⊥ 4y and −x ⊥ −y . theorem 2.1. suppose f : g → x where f is a mapping from an abelian group to a completenon-archimedean normed space. for ε > 0, when x ⊥ y for all x, y ∈ g,we obtain ‖d1f (x, y)‖ 6 ε (2.6) and ‖f (x) + f (−x)‖ 6 ε (2.7)then there exists a unique mapping g : g → x such that x ⊥ y implies 4g(x + y) + 4g(x − y) + 10g(x) + 14g(−x)− 3g(y)− 3g(−y) = g(2x + y) + g(2x − y) (2.8) https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 6and ‖f (x)− g(x)‖ 6 7 2 ε (2.9) for all x ∈ 2g = {2x : x ∈ g}. proof. for all x ∈ x , since 0 ⊥ x , x ⊥ 0 and 0 ⊥ 0, setting x = 0, y = 0 in (2.6), we obtain ‖24f (0)‖ 6 ε, respectively, setting y = 0 in (2.6),we obtain the following inequality: ‖2f (2x)− 18f (x)− 14f (−x) + 6f (0)‖ 6 ε (2.10) by using the strong triangle inequality, we obtain ‖2f (2x)−18f (x)−14f (−x)‖ 6 max{‖2f (2x)−18f (x)−14f (−x)+6f (0)‖, ‖6f (0)‖} 6 ε (2.11) by replacing x with 4x in (2.7) and applying the triangle inequality twice, we obtain ‖2f (2x)− 4f (x)‖ 6 max{‖2f (2x)− 18f (x)− 14f (−x)‖, 14‖f (x) + f (−x)‖} = 14ε (2.12) applying (2.7) and (2.12) to ‖3f (4x)− 8f (2x)− f (−4x)‖, we can conclude that ‖3f (4x)− 8f (2x)− f (−4x)‖ = ‖4[f (4x)− 2f (2x)]− [f (4x) + f (−4x)]‖ 6 max {28ε, ε} = 28ε (2.13) this means that ∥∥∥∥f (2x)− 38 f (4x) + 18 f (−4x) ∥∥∥∥ 6 72ε (2.14) the next step resembles lemma2.1, let gn(x) = 2n + 1 2 · 4n f (2 nx)− 2n − 1 2 · 4n f (−2 nx) (2.15) then we can define a mapping g g : g → x g(x) = lim n→∞ gn(x). according to lemma2.1, we obtain ‖f (2x)− g(2x)‖ 6 7 2 ε (2.16) we consider the following inequality ‖d1gn(x, y)‖ 6 ∥∥∥∥2n + 12 · 4n d1f (2 nx, 2ny) + 2n − 1 2 · 4n d1f (2 nx, 2ny) ∥∥∥∥ 6 2n + 1 2 · 4n ε (2.17) https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 7for all x, y ∈ g. then we let n →∞, we get (2.8). now, in order to prove g is unique, we assume g′ as another mapping satisfying (2.8) and (2.9) that∥∥g(x)− g′(x)∥∥ = ∥∥g(x)− f (x) + f (x)− g′(x)∥∥ 6 max { ‖g(x)− f (x)‖, ∥∥f (x)− g′(x)∥∥} = ε (2.18) for all x ∈ 2g = {2x : x ∈ g}on the other hand, the mapping g − g′ satisfy (2.6) and(2.8) g(2x)− g′(2x) = 2n + 1 2 · 4n [ g ( 2n+1x ) − g′ ( 2n+1x )] − 2n − 1 2 · 4n [ g ( −2n+1x ) − g′ ( −2n+1x )] (2.19) and therefore∥∥g(2x)− g′(2x)∥∥ 6 max {( 2n + 1 2 · 4n )∥∥g (2n+xx)− g′ (2n+1 · x)∥∥ ,(2n − 1 2 · 4n )∥∥g (−2n+1x)− g′ (−2n+1x)∥∥ 6 max {( 2n + 1 2 · 4n ) ε, ( 2n − 1 2 · 4n ) ε } = 2n + 1 2 · 4n ε (2.20) for x ∈ g. by using the nonnegativity of norm and the forced convergence we can get that themapping g is unique on the set 2g. � 3. stability of additive-additive and orthogonally quadratic-quadratic functional equation in this section, we substituted the equations with the orthogonally additive-additive and orthog-onally quadratic-quadratic functional equation concerning [12] in the same method and by referringto the stability proof of [13, 14], we define d2(x, y , z) as the followig d2f (x, y , z) = f ( x + y + z 2 ) + f ( x + y − z 2 ) + f ( x − y + z 2 ) + f ( y + z − x 2 ) − f (x)− f (y)− f (z) theorem 3.1. suppose f : g → x where f is a mapping from an abelian group to a completenon-archimedean normed space. for ε > 0, when x ⊥ y for all x, y , z ∈ g,we obtain ‖d2f (x, y , z)‖ 6 ε (3.1) and ‖f (x) + f (−x)‖ 6 ε. (3.2) https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 8then there exists a unique mapping g : x → y such that x ⊥ y implies g ( x + y + z 2 ) +g ( x + y − z 2 ) +g ( x − y + z 2 ) +g ( y + z − x 2 ) = g(x)+g(y)+g(z) (3.3) and ‖f (x)− g(x)‖ 6 ε (3.4)for all x ∈ 2g = {2x : x ∈ g}. proof. for all x ∈ g, since 0 ⊥ x , x ⊥ 0, and 0 ⊥ 0, setting x = 0, y = 0, z = 0 in inquality (3.1),we obtain ‖f (0)‖ 6 ε, then similarily setting y = 0, z = 0 in inequality (3.1), we obtain∥∥∥∥3f (x2)+ f ( −x 2 ) − f (x)− 2f (0) ∥∥∥∥ 6 ε (3.5) then, by using the strong triangle inequality, we obtain∥∥∥∥3f (x2)+ f ( −x 2 ) − f (x) ∥∥∥∥ 6 max { ‖2f (0)‖, ∥∥∥3f (x 2 ) + f ( − x 2 ) − f (x)− 2f (0) ∥∥∥} = 2ε (3.6) by replacing x witn 2x in (3.6), we obtain ‖3f (x) + f (−x)− f (2x)‖ 6 2ε (3.7) by using the strong triangle inequality twice, we can easily obtain ‖2f (x)− f (2x)‖ 6 max{‖f (x) + f (−x)‖, ‖3f (x) + f (−x)− f (2x)‖} = 2ε (3.8) by replacing x witn 4x in (3.2),then combining the following with (3.2)and(3.8), we can concludethat ‖3f (4x)− 8f (2x)− f (−4x)‖ = ‖4[f (4x)− 2f (2x)]− [f (4ẋ) + f (−4x)]‖ 6 max{8ε, ε} = 8ε (3.9) then dividing both side of the inequality by 8,we obtain∥∥∥∥f (2x)− 38 f (4x) + 18 f (−4x) ∥∥∥∥ 6 ε (3.10) the next step resembles lemma2.1, gn(x) = 2n + 1 2 · 4n f (2 nx)− 2n − 1 2 · 4n f (−2 nx) n ∈ n. let g : g → x g(x) = lim n→∞ gn(x). https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 9according to lemma2.1, we obtain ‖f (2x)− g(2x)‖ = ‖h(x, n) + gn(2x)− g(2x)‖ 6 ε (3.11) for the purpose of proving that g is orthogonally additive, firstiy, we apply the strong triangleinequality and the nonnegativity property for the following , we obtain ‖d2g(x, y , z)‖ = ∥∥∥∥2n + 12 · 4n d2f (2 nx, 2ny , 2nz) + 2n − 1 2 · 4n d2f (2 nx, 2ny , 2nz) ∥∥∥∥ 6max { 2n + 1 2 · 4n ε, 2n − 1 2 · 4n ε } = 2n + 1 2 · 4n ε (3.12) for all x, y , z ∈ g with x ⊥ y and n ∈ n, n > 1. when we let n → ∞, we get (3.3). the rest ofproof resembles theorem 2.1, according to (2.18)to (2.20) , we can get the mapping g is unique onthe set 2g similarly. � theorem 3.2. suppose f : g → x where f is a mapping from an abelian group to a completenon-archimedean normed space. for ε > 0, when x ⊥ y for all x, y , z ∈ g,we obtain ‖d2f (x, y , z)‖ 6 ε (3.13) and ‖f (x)− f (−x)‖ 6 ε. (3.14) then there exists a unique mapping g : x → y such that x ⊥ y implies g ( x + y + z 2 ) +g ( x + y − z 2 ) +g ( x − y + z 2 ) +g ( y + z − x 2 ) = g(x)+g(y)+g(z) (3.15) and ‖f (x)− g(x)‖ 6 1 2 ε (3.16) for all x ∈ 2g = {2x : x ∈ g}. proof. our proof resembles theorem3.1, the same step from (3.5) to (3.7), we get that ‖3f (x) + f (−x)− f (2x)‖ 6 2ε (3.17) adding (3.14) to (3.17) and using the triangle inequality, we can obtain ‖f (2x)− 4f (x)‖ 6 max {‖3f (x) + f (−x)− f (2x)‖, ‖f (x)− f (−x)‖} = 2ε (3.18) https://doi.org/10.28924/ada/ma.4.11 eur. j. math. anal. 10.28924/ada/ma.4.11 10hence, by using the result, there is ‖3f (4x)− 8f (2x)− f (−4x)‖ = ‖2[f (4x)− 4f (2x)] + f (4x)− f (−4x)]‖ 6 max{4ε, ε} = 4ε (3.19) the rest of proof is similar to the theorem 3.1. � acknowledgments thanks to all the members of the functional analysis research team of the college of mathemat-ics and physics of anqing normal university for their discussion and correction of the difficultiesand errors 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https://doi.org/10.1155/2011/123656 https://doi.org/10.5899/2012/jnaa-00123 https://doi.org/10.11650/twjm/1500406797 https://doi.org/10.1186/1029-242x-2012-139 https://doi.org/10.1515/dema-2020-0009 https://doi.org/10.1515/dema-2020-0009 eur. j. math. anal. 10.28924/ada/ma.4.11 11 [14] l. fu, q. liu, y. li, on the stability of orthogonally jensen additive and quadratic functional equation, j. math. anal.appl. 519 (2023) 126744. https://doi.org/10.1016/j.jmaa.2022.126744.[15] k. hensel, über eine neue begründung der theorie der algebraischen zahlen, jahresber. dtsch. math.-ver. 6 (1897)83-88. http://eudml.org/doc/144593.[16] j. ratz, on orthogonally additive mappings, aequat. math. 28 (1985) 35–49. https://doi.org/10.1007/ bf02189629. https://doi.org/10.28924/ada/ma.4.11 https://doi.org/10.1016/j.jmaa.2022.126744 http://eudml.org/doc/144593 https://doi.org/10.1007/bf02189629 https://doi.org/10.1007/bf02189629 1. introduction and preliminaries 2. stability of the orthogonally additive-quadratic functional equation 3. stability of additive-additive and orthogonally quadratic-quadratic functional equation acknowledgments references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 27doi: 10.28924/ada/ma.3.27 on a generalization of (l1ω, l p ω)-multipliers yaovi a. tissinam1, abudulaï issa1, yaogan mensah1,2,∗ 1department of mathematics, university of lomé, togo asseketis@gmail.com, issaabudulai13@gmail.com, mensahyaogan2@gmail.com 2icmpa, university of abomey-calavi, benin ∗correspondence: mensahyaogan2@gmail.com, ymensah@univ-lome.tg abstract. this paper deals with a generalized aspect of multipliers for the pair (l1ω, lpω) of beurlingspaces. using the fourier transform related to a beurling weight, we give a characterization of theaforementioned multipliers. we also prove the identification of the space of the multipliers for thepair (l1ω, lpω) with the beurling space lpω when 1 < p <∞. 1. introduction multipliers are intensively studied by many researchers. they appear in several fields of math-ematics and in various contexts, namely : mobile communication, signal processing, stochasticprocess, partial differential equation etc. from a theoretical point of view, we refer to the source [9]for more details about multipliers for commutative banach algebras.like in [4], we are interested in the multipliers on a certain large class of banach spaces related to alocally compact abelian group. namely, multipliers of beurling spaces are concerned. some inter-esting publications about multipliers associated with locally compact groups are [1,6,11,12,14,18].in [4], we study the multipliers on the weighted group algebra l1ω(g) which is the banach space l1ω(g) endowed with a generalized convolution product ∗ω which depends on the weight ω. thisgeneralized convolution product first appeared in [10]. the authors in [4] characterized the multi-pliers on this weighted group algebra.the present paper is the continuation of the study started in [4]. we consider a generalization ofthe multipliers for the pair (l1ω(g), lpω(g)). that is, the linear maps t : l1ω(g) −→ lpω(g)) thatcommute with a certain class of generalized translation operators denoted here by γsω . if ω ≡ 1,then we recover the classical concept of multipliers. via the weight fourier transform, we obtain,among other results, a characterization of the multipliers for the pair (l1ω(g), lpω(g)). received: 14 jul 2023. key words and phrases. weight, convolution, multiplier, group algebra, fourier transform, measure.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 2the paper is organized as follows. in section 2, the definition of beurling spaces and someresults from [4, 7, 10] are recalled. in section 3, we state our main results. 2. preliminaries 2.1. the beurling spaces. let g be a group whose neutral element is denoted by e . a beurling weight on g is a continuous fonction ω : g → (0,∞) such that ∀x, y ∈ g, ω(xy) 6 ω(x)ω(y), ω(x) > 1, ω(e) = 1. for instance, for each α ≥ 0, the function ωα defined by ωα(x) = (1 + ‖x‖)α, where x = (x1, · · · , xn) ∈ rn and ‖x‖ = √ n∑ i=1 x2i , is a beurling weight on (rn,+). integration on g is taken with respect to a left haar measure. beurling spaces are defined tobe lpω(g) = { f : g → c : ∫ g |f (x)|pω(x)dx <∞ } , 1 6 p < +∞. the case where p =∞ is defined in an obvious way by essential boundedness. the mapping f 7−→ ‖f ‖p,ω = (∫ g |f (x)|pω(x)dx ) 1 p is a norm on lpω(g).it is well-known in the mathematical litterature that l1ω(g) is a banach algebra under theconvolution product ∗ defined by (f ∗ g)(x) = ∫ g f (y)g(y−1x)dy. the following sufficient condition for lpω(g), 1 < p < ∞, to be a banach algebra under theconvolution product ∗ can be found in [7] : the space lpω(g), 1 < p <∞ is banach algebra underthe convolution product ∗ if ω 1 1−p ∗ω 1 1−p 6 ω 1 1−p . for a general background and history on beurlingspaces, we refer to [13,15]. 2.2. a generalized convolution product. in [10], the author introduced a new convolution producton l1ω(g) which has the particularity to depend of the weight ω. that is, f ∗ω g(x) = ∫ g f (y)g(y−1x) ω(y)ω(y−1x) ω(x) dy. if ω ≡ 1, then one recovers the usual convolution (f ∗ g)(x) = ∫ g f (y)g(y−1x)dy. https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 3hence, the convolution product ∗ω is a generalization of the usual convolution product. it wasshown that l1ω(g) is a banach algebra under this new convolution product [10]. we denote by l1ω(g) this new banach algebra ; in other words l1ω(g) = (l1ω(g), ‖ · ‖1,ω, ∗ω).for s ∈ g, define the operator γsω by γsωf (x) = τsmωf (x) ω(x) , f ∈ l1ω(g), where mω is the multiplication operator defined by (mωf )(x) = ω(x)f (x) and τs is the translation operator defined by (τs f )(x) = f (s−1x). the operator γsω appears first in [4] for the study of the multipliers for the algebra l1ω(g). alinear map t : l1ω(g)→ l1ω(g) is called a multiplier if t commutes with the operators γsω for all s ∈ g. since the operator γsω is a generalization of the translation operator τs , the latter notion ofmultiplier covers the classical one related to commutation with translations.the natural next step is to investigate the multipliers for the pair (l1ω(g), lpω(g)). this is themain purpose of the present article.we denote by m1ω(g) the banach space of all complex bounded regular borel measures µ on gsuch that ‖µ‖ω = ∫ g ω(x)d |µ|(x) <∞. (1) we write m1(g) in the case where ω ≡ 1. for µ, ν ∈ m1ω(g), define µ ∗ω ν by µ ∗ω ν(f ) = ∫ g ∫ g f (xy) ω(x)ω(y) ω(xy) dµ(x)dν(y), f ∈ cc(g,ω−1) where cc(g,ω−1) is the set of complex functions f defined on g such that f ω−1 is of compactsupport. also, define µ ∗ω f (x) = ∫ g f (y−1x) ω(y)ω(y−1x) ω(x) dµ(y) for f ∈ l1ω(g) and µ ∈ m1ω(g). then, the banach space m1ω(g) is a unital banach algebra withrespect to the convolution product ∗ω and l1ω(g) is a closed ideal of m1ω(g) [10, theorem 5.1]. 2.3. some useful facts. let g be a locally compact abelian group with pontryagin dual group ĝ.we denote by m̂1(g) the collection of all the fourier-stieltjes transforms of elements of m1(g).that is, m̂1(g) = {µ̂ : µ ∈ m1(g)}where µ̂ is defined by µ̂(γ) = ∫ g γ(x)dµ(x), γ ∈ ĝ. https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 4 for a function f ∈ l1ω(g), the fourier transform of f , denoted f f or f̂ , is defined by (f f )(γ) := f̂ (γ) = ∫ g f (x)γ(x)dx the following theorems will play an important role. theorem 2.1 ( [3] or [17]). let g be a locally compact abelian group and let ϕ be a complex function on ĝ. then, the following assertions are equivalent.(1) ϕ ∈ m̂1(g) and ‖ϕ‖∞ 6 c.(2) ϕ is continuous and there exists a constant c > 0 such that∣∣∣∣∣ n∑ i=1 ciϕ(γi) ∣∣∣∣∣ < c ∥∥∥∥∥ n∑ i=1 ciγi(·) ∥∥∥∥∥ ∞ (2) for all positive integer n and all choices of ci ∈ c and γi ∈ ĝ, i = 1, 2, · · · , n. moreover, if ϕ = µ̂, then ‖µ‖ is the smallest constant c for which (2) holds. theorem 2.2. ( [9, page 252]) let g be a locally compact abelian group. then, for each compact k ⊂ ĝ and ε > 0, given an open set u containing k, there exists a function f ∈ l1(g) such that 0 6 f̂ (γ) 6 1 if γ ∈ ĝ, f̂ (γ) = 1 if γ ∈ k, f̂ (γ) = 0 if γ /∈ u and ‖f ‖ 6 ε + 1. in particular, given any open set u ⊂ ĝ with compact closure, it is possible to find f ∈ l1(g) such that f̂ (γ) = 1 if γ ∈ u. theorem 2.3. ( [5, theorem 3.2]) let g be a locally compact group. let f ∈ lpω(g), 1 6 p < ∞. then, ∀s ∈ g, [ω(s)] 1−p p ‖f ‖p,ω 6 ‖γsωf ‖p,ω 6 [ ω(s−1) ] p−1 p ‖f ‖p,ω. (3) 3. multipliers for the pair (l1ω(g), lpω(g)) in this section, we study a generalization of the concept of multipliers. here, the multipliersare defined with respect to the generalized translation operators γsω . throughout this section, weassume that g is a locally compact abelian group. a look at theorem 2.3 shows that f ∈ lpω(g) ifand only if γsωf ∈ l p ω(g). that is, the spaces lpω(g) are stable under the action of the operators γsω . therefore, we are able to define a concept of multiplier in the framework of this study. definition 3.1. a linear operator t : l1ω(g) −→ lpω(g) is said to be a multiplier if t commutes with all the operators γsω, s ∈ g. that is, ∀s ∈ g, tγsω = γsωt. we denote by m1,p ω (g) the set of such multipliers. we denote by ‖t‖ the operator norm of t ∈m1,p ω (g). https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 5we will use the fact that for 1 < p <∞, the following identification holds [7] : (lpω(g))′ = lqw (g) with 1 p + 1 q = 1 and w = ω− q p . from that, one may deduce that for 1 < p <∞, the space lpω(g)is a reflexive space.for f ∈ l1ω(g), define the fourier transform of f by fω(f )(γ) = ∫ g f (x)γ(x)ω(x)dx, γ ∈ ĝ. in [4], the following convolution result was proved. ∀f , g ∈ l1ω(g), fω(f ∗ω g) = fω(f )fω(g). set fω(l1ω(g)) = { fω(f ) : f ∈ l1ω(g) } . let us remark that functions in fω(l1ω(g)) are continuous and vanished at infinity by the riemann-lebesgue theorem. we fit out the space fω(l1ω(g)) with the norm defined by ‖fω(f )‖ = ‖f ‖1,ω, f ∈ l1ω(g). then, we have the following result. theorem 3.2. the space fω(l1ω(g)) is a banach algebra for the pointwise multiplication. proof. let (fω(fn)) be a cauchy sequence in fω(l1ω(g)). let p, q ∈ n. the equality ‖fω(fp)−fω(fq)‖ = ‖fp − fq‖1,ω and the fact that (l1ω(g), ‖ · ‖1,ω) is a banach space show that there exists f ∈ l1ω(g) such that (fn) converges to f in l1ω(g). now, ‖fω(fn)−fω(f )‖ = ‖fn − f ‖1,ω . thus, (fω(fn)) converges to (fω(f )) in fω(l1ω(g)). thus, the space fω(l1ω(g)) is a banach space. moreover, ‖fω(f )fω(g)‖ = ‖fω(f ∗ω g)‖ = ‖f ∗ω g‖1,ω 6 ‖f ‖1,ω‖g‖1,ω = ‖fω(f )‖‖fω(g)‖. thus, the space (fω(l1ω(g))), ·, ‖·‖1,ω ) is a banach algebra. � for f ∈ lpω(g) and h ∈ lqw (g) with 1 p + 1 q = 1, we set 〈f , h〉ω = ∫ g f (x)h(x−1)ω(x)dx. https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 6 theorem 3.3. let g be a locally compact abelian group. let t : l1ω(g) −→ lpω(g) be a bounded linear transformation. then, t ∈m1,p ω (g) if and only if there exists a unique element ϕ such that tg = ϕ ∗ω g for all g ∈ l1ω(g), where ϕ ∈ m1ω(g) if p = 1 and ϕ ∈ lpω(g) if 1 < p <∞. proof. (1) suppose p = 1. let t ∈ m1,1 ω (g). in [4, proposition 5.4], it was shown that t ∈ m1,1 ω (g) if and only if there exists a unique function b defined on ĝ such that fω(t f ) = bfω(f ) for all f ∈ l1ω(g). clearly, bfω(f ) ∈ fω(l1ω(g)). therefore, thefunction bfω(f ) is continuous for all f ∈ l1ω(g) (the fourier transform of a function is acontinuous function). moreover, for each open set in ĝ with compact closure, there exists afunction f ∈ l1ω(g) such that fω(f ) is constant on u [9, f.7e]. thus, b is continuous on ĝ.let ε > 0 and let γ1, γ2, ...., γn ∈ ĝ. via theorem 2.2, we can choose g ∈ l1(g) suchthat ‖f(g)‖ = ‖g‖1 < 1 + ε and f(g)(γi) = 1, i = 1, 2, 3, ....., n. now, set f = g ω . then, f ∈ l1ω(g), ‖fω(f )‖ = ‖f ‖1,ω < 1 + ε and fω(f )(γi) = 1, i = 1, 2, 3, ....., n.for zi ∈ c, i = 1, 2, ....., n, one has∣∣∣∣∣ n∑ i=1 zib(γi) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 zifω(f )(γi) ∣∣∣∣∣ = ∣∣∣∣∣∫g [ n∑ i=1 ziγi(x) ] f (x)ω(x)dx ∣∣∣∣∣ 6 ‖f ‖1,ω ∥∥∥∥∥ n∑ i=1 ziγi ∥∥∥∥∥ ∞ < (1 + ε) ∥∥∥∥∥ n∑ i=1 ziγi ∥∥∥∥∥ ∞ . since ε is chosen arbitrarily, it follows that∣∣∣∣∣ n∑ i=1 zib(γi) ∣∣∣∣∣ < ∥∥∥∥∥ n∑ i=1 ziγi ∥∥∥∥∥ ∞ . we conclude via theorem 2.1 (with c = 1) that there exists a unique bounded measure µ such that b = f(µ). now, if we set ϕ = ω−1µ, then ϕ ∈ m1ω(g) and b = fω(ϕ).therefore, fω(t f ) = fω(ϕ)fω(f ) = fω(ϕ ∗ω f ).since the fourier transform is injective, it follows that t f = ϕ ∗ω f .(2) assume that 1 < p < ∞. let t ∈ m1,p ω (g). the weighted group algebra (l1ω(g), ‖ · ‖1,ω, ∗ω) has a bounded approximate identity [10, theorem 2.2]. let {υn} be a boundedapproximate identity for (l1ω(g), ‖ · ‖1,ω, ∗ω). let g ∈ l1ω(g). then, ‖tg − tυn ∗ω g‖p,ω = ‖tg − t (υn ∗ω g)‖p,ω 6 ‖t‖‖g − υn ∗ω g‖1,ω. since ‖g − υn ∗ω g‖1,ω tends to 0 whenever n goes to ∞, then (tυn ∗ω g)n converges to tg in lpω(g). https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 7moreover, ‖tυn‖p,ω 6 ‖t‖‖υn‖1,ω = ‖t‖. therefore, {tυn} lies in a norm bounded subsetof lpω(g) = ( lqw (g) )′. so, by alaoglu’s theorem ( [16, page 299] or [9, theorem d.4.3.]) andthe reflexivity of lpω(g), we see that there exists a subnet {tυm} of {tυn} and ϕ ∈ lpω(g)such that {tυm} converges to ϕ in the weak∗-topology. that is, lim m 〈tυm, u〉ω = 〈ϕ, u〉ωfor all u ∈ lqw (g). then, for h, g ∈ cc(g), we have 〈th, g〉ω = lim m 〈tυm ∗ω h, g〉ω = lim m 〈tυm, (h ∗ω g ω )ω〉ω = 〈ϕ, (h ∗ω g ω )ω〉ω = 〈ϕ ∗ω h, g〉ω. however, cc(g) is norm dense in lqw (g). therefore, th = ϕ ∗ω h for each h ∈ cc(g).moreover, cc(g) is norm dense in l1ω(g). thus, th = ϕ ∗ω h for all h ∈ l1ω(g).(3) conversely, let 1 ≤ p < ∞. assume that there exists a measure µ ∈ m1ω(g) or a function ϕ ∈ lpω(g) such that th = ϕ ∗ω h for all h ∈ l1ω(g). then, (tγsω)h = t (γsωh) = ϕ ∗ω γsωh = γsω(ϕ ∗ω h) = γsω(th) = (γsωt )h. thus, t ∈m1,p ω (g).(4) concerning the uniqueness statement, let us consider ϕ and ψ be such that th = ϕ∗ω h = ψ ∗ω h for all h ∈ l1ω(g). then, using the fourier transform, we obtain fω(ϕ)fω(h) = fω(ψ)fω(h). so, fω(ϕ) = fω(ψ). finally, ϕ = ψ by the injectivity of the fouriertransform. � theorem 3.4. let g be a locally compact abelian group. then,m1,1 ω (g) is isometrically isomorphic to m1ω(g). proof. we have seen in theorem 3.3 that t ∈m1,1 ω (g) if and only if there exists a unique measure µ ∈ m1ω(g) such that t f = µ∗ω f for all f ∈ l1ω(g). then, the mapping t 7−→ µ defines a bijectionfrom m1,1 ω (g)) onto m1ω(g). moreover, ‖t f ‖1,ω = ‖µ ∗ω f ‖1,ω = ∫ g ∣∣∣∣∫ g f (y−1x) ω(y−1x)ω(y) ω(x) dµ(y) ∣∣∣∣ω(x)dx 6 ∫ g ∫ g ∣∣f (y−1x) ∣∣ ω(y−1x)ω(y) ω(x) ω(x)d |µ|(y)dx https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 8 6 (∫ g |f (x)|ω(x)dx )(∫ g ω(y)d |µ|(y) ) (invariance of the haar measure) 6 ‖f ‖1,ω‖µ‖ω. then, ‖t‖1,ω 6 ‖µ‖ω .in the converse, for γ1, · · · , γn ∈ ĝ, z1, · · · , zn ∈ c, and ε > 0, let us choose f ∈ l1ω(g) suchthat ‖fω(f )‖ = ‖f ‖1,ω < 1 + ε and fω(f )(γi) = 1, i = 1, 2, 3, ....., n. then,∣∣∣∣∣ n∑ i=1 zifω(µ)(γi) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 zifω(µ)(γi)fω(f )(γi) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 zifω(µ ∗ω f )(γi) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 zifω(t f )(γi) ∣∣∣∣∣ 6 ‖t‖(1 + ε) ∥∥∥∥∥ n∑ i=1 ziγi ∥∥∥∥∥ ∞ . since ε is arbitrary, then ‖t‖ > ‖µ‖ω by the use of theorem 2.1 applied with fω instead of f . � theorem 3.5. let g be a locally compact abelian group. let 1 6 p < ∞. if f ∈ lpω(g), then the mapping s 7−→ γsωf is continuous from g into lpω(g). proof. the set of complex continuous functions on g with compact support cc(g) is dense in lpω(g)under the norm ‖ · ‖p,ω . let ε > 0. consider g ∈ cc(g) and set c1 = supp(g). let us choose acompact neighborhood c2 of the neutral element e . set c = c1∪c2∪ (c1c2). we have for s ∈ c2, ‖γsωg − g‖pp,ω = ∫ c |γsωg(x)− g(x)|pω(x)dx 6 ∫ c |g(s−1x)ω(s−1x)− g(x)ω(x)|pdx. the mapping x 7−→ (gω)(x) is uniformly continuous on g. thus, there exists a neighborhood u of e which we may assume to be contained in c2, such that ∀s ∈ u, |(gω)(s−1x)− (gω)(x)|p < εp |c|where |c| is the measure of the compact set c. then, for s ∈ u , we have ‖γsωg − g‖pp,ω 6 ∫ c |(gω)(s−1x)− (gω)(x)|pdx < ε|c| |c| = ε. we will show the claim for f ∈ lpω(g). let k be a compact neighborhood of e . since cc(g) isdense in lpω(g), then there exists g ∈ cc(g) such that ‖f − g‖p,ω < ε 3 . https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 9there exists a compact neighborhood v of e which we may assume to be contained in k, such that ‖γsωg − g‖p,ω < ε 3 for all s ∈ v .then, for s ∈ v , we have ‖γsωf − f ‖p,ω 6 ‖γsωf − γsωg‖p,ω + ‖γsωg − g‖p,ω + ‖f − g‖p,ω < 1 ω(s) ∫ g |(f − g)(t)|pω(st)dt + ε 3 + ε 3 6 1 ω(s) ∫ g |(f − g)(t)|pω(s)ω(t)dt + ε 3 + ε 3 < ∫ g |(f − g)(t)|pω(t)dt + ε 3 + ε 3 ‖f − g‖p,ω + ε 3 + ε 3 = ε 3 + ε 3 + ε 3 = ε. � theorem 3.6. let g be a locally compact abelian group. let f ∈ lpω(g), 1 6 p < ∞. let ε > 0. then, there exists a positive function g ∈ cc(g) such that ‖g‖1,ω = 1 and ‖f ∗ω g − f ‖p,ω 6 ε. proof. let f ∈ lpω(g) and ε > 0. according to theorem 3.5, the mapping s 7−→ γsωf is continuousat the neutral element e of g. then, there exists a compact neighborhood k of e such that ‖γsωf − f ‖p,ω 6 ε, ∀s ∈ k. consider a positive function g such that supp(g) ⊂ k and ∫ g g(y)ω(y)dy = 1 (that is ‖g‖1,ω = 1). then, |(f ∗ω g)(x) − f (x)| 6 ∫ g |γsωf (x) − f (x)|g(s)ω(s)ds. using the hölder’s inequality withrespect to the measure g(s)ω(s)ds , one has |(f ∗ω g)(x)− f (x)| 6 (∫ g |γsωf (x)− f (x)|pg(s)ω(s)ds ) 1 p (∫ g g(s)ω(s)ds ) 1 q 6 (∫ g |γsωf (x)− f (x)|pg(s)ω(s)ds ) 1 p , where q is such that 1 p + 1 q = 1. then, ‖f ∗ω g − f ‖pp,ω = ∫ g |(f ∗ω g)(x)− f (x)|pω(x)dx 6 ∫∫ g×g |γsωf (x)− f (x)|pg(s)ω(s)dsω(x)dx 6 ∫ g ‖γsωf − f ‖pp,ωg(s)ω(s)ds = ‖γsωf − f ‖pp,ω ∫ g g(s)ω(s)ds = ‖γsωf − f ‖pp,ω 6 εp. thus, ‖f ∗ω g − f ‖p,ω 6 ε. � theorem 3.7. let g be a locally compact abelian group. if t ∈m1,p ω (g), then ‖t f ‖p 6 ‖t‖‖f ‖1. in other words, t : l1ω(g) −→ lpω(g) is a bounded operator. https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 10 proof. let ε > 0. via theorem 3.6, there exists a positive function g in cc(g) such that∫ g g(t)ω(t)dt = 1 and ‖g ∗ω t f − t f ‖p 6 ε because ‖·‖p 6 ‖·‖p,ω . we have, ‖g ∗ω t f − t f ‖p > ‖t f ‖p − ‖g ∗ω t f ‖p. therefore, ‖t f ‖p 6 ‖g ∗ω t f ‖p + ε = ‖tg ∗ω f ‖p + ε 6 ‖tg‖p‖f ‖1 + ε 6 ‖tg‖p,ω‖f ‖1 + ε 6 ‖t‖‖‖1,ω‖f ‖1 + ε = ‖t‖‖f ‖1 + ε. since the latter inequality is true for arbitrary ε > 0, then we obtain ‖t f ‖p 6 ‖t‖‖f ‖1. � for a function f in lpω(g), we define the convolution operator tf by tf g = f ∗ω g. theorem 3.8. let g be a locally compact abelian group. let 1 < p < ∞. let f be a function in lpω(g). then, ‖tf ‖ = ‖f ‖p,ω. proof. let f ∈ lpω(g) and let ε > 0. from theorem 3.6, there exits a positive function g such that∫ g g(t)ω(t)dt = 1 and ‖f ∗ω g − f ‖p,ω 6 ε. then, ‖f ‖p,ω 6 ε+ ‖f ∗ω g‖p,ω = ε+ ‖tf g‖p,ω 6 ε+ ‖tf ‖‖g‖1,ω = ε+ ‖tf ‖. thus ‖f ‖p,ω 6 ‖tf ‖.let us prove the inverse inequality. let g ∈ l1ω(g). applying the hölder’s inequality withrespect to the measure g(y)ω(y)dy , one has ‖f ∗ g‖pp,ω = ∫ g |g ∗ω f |pω(x)dx = ∫ g ∣∣∣∣∫ g g(y)γyωf (x)ω(y) ∣∣∣∣p ω(x)dx 6 ∫ g [∫ g |γyωf (x)|p|g(y)|ω(y)dy ] [∫ g |g(y)|ω(y)dy ] p q ω(x)dx 6 ∫ g (|f |p ∗ω |g|)ω(x)dx [∫ g |g(y)|ω(y)dy ] p q 6 ‖|f |p ∗ω |g|‖1,ω‖g‖ p q 1,ω 6 ‖f ‖ p p,ω‖g‖1,ω‖g‖ p q 1,ω = ‖f ‖pp,ω‖g‖ p 1,ω. https://doi.org/10.28924/ada/ma.3.27 eur. j. math. anal. 10.28924/ada/ma.3.27 11then, ‖tf g‖pp,ω 6 ‖f ‖pp,ω‖g‖p1,ω. thus, ‖tf ‖ 6 ‖f ‖p,ω . � as a consequence of theorem 3.3 and theorem 3.8, we have the following result. corollary 3.9. let g be a locally compact abelian group. let 1 < p < ∞. then, the multipliers space m1,p ω (g) and the beurling space lpω(g) are isometricaly identified by the mapping t : f 7−→ tf . conclusion in this paper, we obtain a characterization of multipliers for the pair (l1ω, l p ω) using the fouriertransform related to a beurling weight. we also obtain the identification of the space of suchmultipliers with the beurling space lpω when 1 < p < ∞. it would be interesting in the future toconsider the case of 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operators and banach algebras in mobile communications, appl. comput. harmon.anal. 20 (2006) 237-249. https://doi.org/10.1016/j.acha.2005.06.003 https://doi.org/10.28924/ada/ma.3.27 https://doi.org/10.1007/s11868-017-0213-0 https://doi.org/10.1016/j.jfa.2014.11.019 https://doi.org/10.1016/j.acha.2005.06.003 1. introduction 2. preliminaries 2.1. the beurling spaces 2.2. a generalized convolution product 2.3. some useful facts 3. multipliers for the pair (l1(g),lp(g)) conclusion competing interests references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 17doi: 10.28924/ada/ma.4.17 tensorial simpson 18 type inequalities for convex functions of selfadjoint operators in hilbert space vuk stojiljković1,∗, sever silvestru dragomir2 1faculty of science, university of novi sad, trg dositeja obradovića 3, 21000 novi sad, serbia vuk.stojiljkovic999@gmail.com 2mathematics, college of sport health and engineering, victoria university melbourne city, vic 8001, australia sever.dragomir@vu.edu.au ∗correspondence: vuk.stojiljkovic999@gmail.com abstract. several simpson 1 8 tensorial type inequalities for selfadjoint operators have been obtainedwith variation depending on the conditions imposed on the function f∣∣∣∣∣∣∣∣18 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 32 ∥∥f ′∥∥ i,+∞ . 1. introduction and preliminaries the concept we now call a "tensor" wasn’t originally named that way. when josiah willard gibbsfirst described the idea in the late 19th century, he used the term "dyadic." today, mathematiciansdefine a tensor as the mathematical embodiment of gibbs’ initial concept. tensors and inequalitiesare natural partners, thanks to the widespread use of inequalities in mathematics. these mathemat-ical statements about comparisons have a profound impact on various scientific disciplines. whilemany types of inequalities exist, some of the most significant ones include jensen’s, ostrowski’s,hermite-hadamard’s, and minkowski’s inequalities. for those interested in delving deeper, refer-ences [17] and [18] provide more details about inequalities and their fascinating history. regardingthe generalizations of the aforementioned inequalities, numerous studies have been published; foradditional information, check the following and the references therein [1–5,7–9,21–23].classical inequalities of simpson type have been given by hezenci et al. [15] and sarikaya etal. [19]. to enhance the presentation of this work, we will demonstrate new developments in the received: 12 apr 2024. key words and phrases. tensorial product, selfadjoint operators, convex functions.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 2theory of inequalities in hilbert spaces. one such development is the dragomir’s inequality fornormal operators given by the following [10]: theorem 1. let (h ; 〈., .〉) be a hilbert space and t : h → h a normal linear operator on h . then ‖tx‖2 ≥ 1 2 ( ‖tx‖2 + |〈t2x, x〉| ) ≥ |〈tx, x〉|2, for any x ∈ h, ‖x‖ = 1. the constant 12 is the best possible. the hermite-hadamard inequality in the selfadjoint operator sense, as provided by dragomir[11], is another intriguing conclusion. theorem 2. let f : i → r be an operator convex function on the interval i . then for any selfadjoint operators a and b with spectra in i we have the inequality f ( a+b 2 ) ≤ f ( 3a+b 4 ) + f ( a+ 3b 4 ) ≤ ∫ 1 0 f ((1− t)a+ tb)dt ≤ 1 2 [ f ( a+b 2 ) + f (a) + f (b) 2 ] ≤ f (a) + f (b) 2 . the first paper related to tensorial inequalities in hilbert space was written by dragomir [13].in the paper, he proved the tensorial version of the ostrowski type inequality given by the following. theorem 3. assume that f is continuously differentiable on i with ‖f ′‖i,+∞ := supt∈i |f ′(t)| < +∞ and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i . then the following inequality holds:∥∥∥∥f ((1− λ)a⊗ 1 + λ1⊗b)− ∫ 1 0 f ((1− u)a⊗ 1 + u1⊗b)du ∥∥∥∥ (1) ≤ ∥∥f ′∥∥ i,+∞ [ 1 4 + ( λ− 1 2 )2 ] ‖1⊗b− a⊗ 1‖ for λ ∈ [0, 1]. recently, various inequalities in the same tensorial surrounding have been obtained. the fol-lowing result of simpson type was obtained by stojiljković [24]. theorem 4. assume that f is continuously differentiable on i and |f ′′| is convex and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i . then the following inequality holds:∣∣∣∣∣∣∣∣16 ( f (a)⊗ 1 + 4f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ) https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 3 − 1 2 α (∫ 1 0 f (( 1− k 2 ) a⊗ 1 + ( 1 + k 2 ) 1⊗b ) kα−1dk + ∫ 1 0 f (( 1− k 2 ) a⊗ 1 + k 2 1⊗b ) (1− k)α−1dk )∣∣∣∣∣∣∣∣ ≤ ‖1⊗b− a⊗ 1‖2 (‖f ′′(a)‖+ ‖f ′′(b)‖) ( 3α2 + 8α+ 7 ) (α+ 2)(24α+ 24) for α ≥ 0. the following inequality has been recently obtained by the same author [25]. theorem 5. assume that f is continuously differentiable on i with ‖f ′‖i,+∞ := supt∈i |f ′(t)| < +∞ and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i . then the following inequality holds:∥∥∥∥∫ 1 0 f ((1− λ)a⊗ 1 + λ1⊗b)dλ− f ( a⊗ 1 + 1⊗b 2 )∥∥∥∥ 6 ‖1⊗b− a⊗ 1‖2 ‖f ′′‖i,+∞ 24 . recently, the following inequality of ostrowski type was obtained by stojiljković et al. [26] whichgeneralized the recently obtained results by dragomir [13]. theorem 6. the formulation is the same as the one given by dragomir in his ostrowski type theorem given above (1) with an exception that α > 0, then∣∣∣∣∣∣∣∣(λα + (1− λ)α)f ((1− λ)a⊗ 1 + λ1⊗b) −α ( (1− λ)α ∫ 1 0 f ((1− λ)(1− u)a⊗ 1 + (u + (1− u)λ)1⊗b)(1− u)α−1du +λα ∫ 1 0 uα−1f (((1− u) + u(1− λ))a⊗ 1 + uλ1⊗b)du )∣∣∣∣∣∣∣∣ 6 ‖1⊗b− a⊗ 1‖ ( λα+1 α+ 1 + (1− λ)α+1 α+ 1 )∥∥f ′∥∥ i,+∞ . stojiljković et al., [27] recently obtained a trapezoid type tensorial inequality which is given by theorem 7. assume that f is continuously differentiable on i with ‖f ′‖i,+∞ := supt∈i |f ′(t)| < +∞ and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i . then the following inequality holds:∣∣∣∣∣∣∣∣ (f (a)⊗ 1 + 1⊗ f (b)) (2) −α [ ∫ 1 0 (1− λ)α−1f (λ1⊗b+ (1− λ)a⊗ 1)dλ + ∫ 1 0 λα−1f (λ1⊗b+ (1− λ)a⊗ 1)dλ ]∣∣∣∣∣∣∣∣ https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 4 ≤ ‖1⊗b− a⊗ 1‖ 1 1 + α ( 2− 21−α ) ∥∥f ′∥∥ i,+∞ . in order to derive similar inequalities of the tensorial type, we need the following introductionand preliminaries.let i1, ..., ik be intervals from r and let f : i1 × ... × ik → r be an essentially bounded realfunction defined on the product of the intervals. let a = (a1, ...,ak) be a k-tuple of boundedselfadjoint operators on hilbert spaces h1, ..., hk such that the spectrum of ai is contained in iifor i = 1, ..., k . we say that such a k-tuple is in the domain of f . if ai = ∫ ii λidei(λi) is the spectral resolution of ai for i = 1, ..., k by following , we define f (a1, ...,ak) := ∫ i1 ... ∫ ik f (λ1, ..., λk)de1(λ1)⊗ ...⊗ dek(λk) as bounded selfadjoint operator on the tensorial product h1 ⊗ ...⊗hk .if the hilbert spaces are of finite dimension, then the above integrals become finite sums, and wemay consider the functional calculus for arbitrary real functions. this construction [6] extends thedefinition of koranyi [16] for functions of two variables and have the property that f (a1, ...ak) = f1(a1)⊗ ...⊗ fk(ak), whenever f can be separated as a product f (t1, ..., tk) = f1(t1)...fk(tk) of k functions each de-pending on only one variable.recall the following property of the tensorial product (ac)⊗ (b⊗d) = (a⊗b)(c⊗d) that holds for any a,b,c,d ∈ b(h ).from the property we can deduce easily the following consequences an ⊗bn = (a⊗b)n, n > 0, (a⊗ 1)(1⊗b) = (1⊗b)(a⊗ 1) = a⊗ b, which can be extended, for two natural numbers m, n we have (a⊗ 1)n(1⊗b)m = (1⊗b)m(a⊗ 1)n = an ⊗bm. for more information, consult the following book related to tensors [14]. the following lemmawhich we require can be found in a paper of dragomir [12]. https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 5 lemma 1. assume a and b are selfadjoint operators with sp(a) ⊂ i, sp(b) ⊂ j and having the spectral resolutions . let f ; h be continuous on i, g, k continuous on j and φ and ψ continuous on an interval k that contains the sum of the intervals f (i) + g(j); h(i) + k(j),then φ(f (a)⊗ 1 + 1⊗ g(b))ψ(h(a)⊗ 1 + 1⊗ k(b)) = ∫ i ∫ j φ(f (t) + g(s))ψ(h(t) + k(s))det ⊗ dfs . in [20], shuang, wang and qi used the following identity to obtain simpson type inequalitiesand some applications. lemma 2. let f : i ⊂ r→ r be a differentiable function on i◦, a,b ∈ i◦ with a < b. if f ′ ∈ l1[a, b], then the following equality holds: 1 8 [ f (a) + 6f ( a + b 2 ) + f (b) ] − 1 b − a ∫ b a f (x)dx (3) = b − a 4 (∫ 1 0 [( 3 4 − t ) f ′ ( ta + (1− t) a + b 2 ) + ( 1 4 − t ) f ′ ( t a + b 2 + (1− t)b )] dt ) . this paper delves into a novel area of mathematics: tensorial inequalities of the simpson type fordifferentiable functions within a tensorial hilbert space. this field is young and ripe for exploration,and obtaining new bounds for various combinations of convex functions is crucial for its advancement.the paper is structured logically. the "main results" section unveils the key findings that contributeto the novelty of this work. subsequently, the "examples and consequences" section showcasespractical applications of the obtained results. by leveraging known properties of the exponentialoperator and its integral, and by choosing specific convex functions, the authors generate numeroustensorial simpson-type inequalities and bounds. finally, the "conclusion" section summarizes thepaper’s contributions and highlights its significance for the development of tensorial inequalities.in the following theorem, you’ll find a fundamental result that serves as the foundation for derivingfurther inequalities throughout the paper. 2. main results the following lemma will be used crucial in obtaining the inequalities which follow. lemma 3. assume that f is continuously differentiable on i, a and b are selfadjoint operators with sp(a), sp(b) ⊂ i , then 1 8 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ (4) = 1⊗b− a⊗ 1 4 ∫ 1 0 [( 3 4 − k ) f ′ ( a⊗ 1 ( 1 + k 2 ) + 1⊗b ( 1− k 2 )) https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 6 + ( 1 4 − k ) f ′ ( k 2 a⊗ 1 + 1⊗b ( 2− k 2 ))] dk. proof. we will start the proof with lemma (3). introducing the substitutions on the left hand sideand simplifying the fractional integral, then assuming that a and b have the spectral resolutions a = ∫ tde(t) and b = ∫ sdf (s). if we take the integral ∫i ∫i over det ⊗ dfs , then we get∫ i ∫ i ( 1 8 [ f (t) + 6f ( t + s 2 ) + f (s) ] − ∫ 1 0 f (λs + (1− λ)t)dλ ) det ⊗ dfs = ∫ i ∫ i ( s − t 2 ∫ 1 0 [( 3 4 − k ) f ′ ( t ( 1 + k 2 ) + s ( 1− k 2 )) + ( 1 4 − k ) f ′ ( k 2 t + s ( 2− k 2 ))] dk ) det ⊗ dfs .by utilizing the fubinis theorem and lemma 1 for appropriate choices of the functions involved,we have successively ∫ i ∫ i f ( t + s 2 ) det ⊗ dfs = f ( a⊗ 1 + 1⊗b 2 ) , ∫ i ∫ i ∫ 1 0 f (λs + (1− λ)t)dλdet ⊗ dfs = ∫ 1 0 ∫ i ∫ i f (λs + (1− λ)t)dλ ) det ⊗ dfsdλ = ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ, ∫ i ∫ i s − t 2 ∫ 1 0 ( 3 4 − k ) f ′ ( t ( 1 + k 2 ) + s ( 1− k 2 )) dkdet ⊗ dfs = ∫ 1 0 ( 3 4 − k )∫ i ∫ i s − t 2 f ′ ( t ( 1 + k 2 ) + s ( 1− k 2 )) det ⊗ dfsdk = (1⊗b− a⊗ 1) 4 ∫ 1 0 ( 3 4 − k ) f ′ ( a⊗ 1 ( 1 + k 2 ) + 1⊗b ( 1− k 2 )) dk. following the same principle for other terms, the equality follows. � theorem 8. assume that f is continuously differentiable on i with ‖f ′‖i,+∞ := supt∈i |f ′(t)| < +∞ and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i , then∣∣∣∣∣∣∣∣18 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] (5) − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ ∣∣∣∣∣∣∣∣ https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 7 ≤ 5 ‖1⊗b− a⊗ 1‖ 32 ∥∥f ′∥∥ i,+∞ . proof. if we take the operator norm of the previously obtained lemma (4) and use the triangleinequality, we get ∥∥∥∥18 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ ∥∥∥∥ ≤ ‖1⊗b− a⊗ 1‖ 2 ∫ 1 0 ∣∣∣∣34 − k ∣∣∣∣ ∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ + ∣∣∣∣14 − k ∣∣∣∣ ∥∥∥∥f ′(k2a⊗ 1 + 1⊗b ( 2− k 2 ))∥∥∥∥ ]dk realize here that by lemma 1,∣∣∣∣f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 )) ∣∣∣∣ = ∫ i ∫ i ∣∣∣∣f ′(t (1 + k2 ) + s ( 1− k 2 )) ∣∣∣∣det ⊗ dfs . since ∣∣∣∣f ′(t (1 + k2 ) + s ( 1− k 2 )) ∣∣∣∣ 6 ∥∥f ′∥∥i,+∞ . holds for all t, s ∈ i . if we take the integral ∫i ∫i over det ⊗ dfs , then we get∣∣∣∣f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 )) ∣∣∣∣ = ∫ i ∫ i ∣∣∣∣f ′(t (1 + k2 ) + s ( 1− k 2 )) ∣∣∣∣det ⊗ dfs . 6 ∥∥f ′∥∥ i,+∞ ∫ i ∫ i det ⊗ dfs = ∥∥f ′∥∥ i,+∞ .from which we get the following,∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ dk 6 ∥∥f ′∥∥ i,+∞ ∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥ dk = 5 ‖f’‖i,+∞16evaluation of the second part is analogous, summing everything up we obtain the desired equality. � https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 8 theorem 9. assume that f is continuously differentiable on i and |f ′| is convex and a,b are selfadjoint operators with sp(a), sp(b) ⊂ i , then∣∣∣∣∣∣∣∣18 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] (6) − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 64 ( ∥∥f ′(a)∥∥+ ∥∥f ′(b)∥∥). proof. since |f ′| is convex on i , then we get∣∣∣∣f ′(t (1 + k2 ) + s ( 1− k 2 )) ∣∣∣∣ 6 (1 + k2 ) |f ′(t)|+ ( 1− k 2 ) |f ′(s)| for all k ∈ [0, 1] and t, s ∈ i .if we take the integral ∫i ∫i over det ⊗ dfs , then we get∣∣∣∣f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 )) ∣∣∣∣ = ∫ i ∫ i ∣∣∣∣f’((1 + k2 ) t + ( 1− k 2 ) s ) ∣∣∣∣det ⊗ dfs 6 ∫ i ∫ i [( 1 + k 2 ) |f ′(t)|+ ( 1− k 2 ) |f ′(s)| ] det ⊗ dfs = ( 1 + k 2 ) |f ′(a)| ⊗ 1 + ( 1− k 2 ) 1⊗ |f ′(b)| for all k ∈ [0, 1].if we take the norm in the inequality, we get the following∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ 6 ∥∥∥∥(1 + k2 ) |f ′(a)| ⊗ 1 + ( 1− k 2 ) 1⊗ |f ′(b)| ∥∥∥∥ 6 ( 1 + k 2 )∥∥|f ′(a)| ⊗ 1∥∥+ (1− k 2 )∥∥1⊗ |f ′(b)|∥∥ = ( 1 + k 2 )∥∥f ′(a)∥∥+ (1− k 2 )∥∥f ′(b)∥∥ .therefore, we obtain∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ dk 6 ∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥((1 + k2 )∥∥f ′(a)∥∥+ (1− k 2 )∥∥f ′(b)∥∥) dk = 79 ‖f’(a)‖+ 41 ‖f’(b)‖ 384 .simplifying the other term and adding them, we obtain the desired inequality. https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 9 � we recall that the function f : i → r is quasi-convex, if f ((1− λ)t + λs) 6 max(f (t), f (s)) = 1 2 (f (t) + f (s) + |f (s)− f (t)|) holds for all t, s ∈ i and λ ∈ [0, 1]. theorem 10. assume that f is continuously differentiable on i with |f ′| is quasi-convex on i , a and b are selfadjoint operators with sp(a), sp(b) ⊂ i , then∣∣∣∣∣∣∣∣18 [ f (a)⊗ 1 + 6f ( a⊗ 1 + 1⊗b 2 ) + 1⊗ f (b) ] (7) − ∫ 1 0 f (λ1⊗b+ (1− λ)a⊗ 1)dλ ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 64 ∥∥|f ′(a)| ⊗ 1 + 1⊗ |f ′(b)|∥∥+ ∥∥|f ′(a)| ⊗ 1− 1⊗ |f ′(b∥∥ . proof. since |f ′| is quasi-convex on i , then we get |f ′(t(1 + k)/(2) + s(1− k)/(2))| ≤ 1/2(|f ′(t)|+ |f ′(s)|+ ||f ′(t)− f ′(s)||) for all k ∈ [0, 1] and t, s ∈ i. if we take the integral ∫i ∫i over det ⊗ dfs , then we get∣∣∣∣f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 )) ∣∣∣∣ = ∫ i ∫ i f ′(t(1 + k)/(2) + s(1− k)/(2))det ⊗ dfs 6 1 2 ∫ i ∫ i (|f ′(t)|+ |f ′(s)|+ ||f ′(t)| − |f ′(s)||)det ⊗ dfs = 1 2 (|f ′(a)| ⊗ 1 + 1⊗ |f ′(b)|+ ||f ′(a)| ⊗ 1− 1⊗ |f ′(b)||)for all k ∈ [0, 1].if we take the norm, then we get∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ 6 ∥∥∥∥12(|f ′(a)| ⊗ 1 + 1⊗ |f ′(b)|+ ||f ′(a)| ⊗ 1− 1⊗ |f ′(b)||) ∥∥∥∥ 6 1 2 (∥∥|f ′(a)| ⊗ 1 + 1⊗ |f ′(b)|∥∥+ ∥∥|f ′(a)| ⊗ 1− 1⊗ |f ′(b)|∥∥)which when applied in our case, we get∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥∥∥∥∥f ′(a⊗ 1(1 + k2 ) + 1⊗b ( 1− k 2 ))∥∥∥∥ dk 6 ∫ 1 0 ∥∥∥∥34 − k ∥∥∥∥(12 (∥∥|f ′(a)| ⊗ 1 + 1⊗ |f ′(b)|∥∥+ ∥∥|f ′(a)| ⊗ 1− 1⊗ |f ′(b)|∥∥) ) dk. which when simplified, we obtain the desired inequality. � https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 103. some examples and consequences it is known that if u and v are commuting, that is uv = v u , then the exponential functionsatisfies the property exp(u) exp(v ) = exp(v ) exp(u) = exp(u + v ). also, if u is invertible and a, b ∈ r and a < b then∫ b a exp(tu)dt = u−1[exp(bu)− exp(au)]. moreover, if u and v are commuting and v − u is invertible, then∫ 1 0 exp((1− k)u + kv )dk = ∫ 1 0 exp(k(v − u)) exp(u)dk = ∫ 1 0 (exp(k(v − u))dk)exp(u) = (v − u)−1[exp(v − u)− i] exp(u) = (v − u)−1[exp(v )− exp(u)].since the operators u = a⊗ 1 and v = 1⊗b are commutative and if 1⊗b−a⊗ 1 is invertible,then ∫ 1 0 exp((1− k)a⊗ 1 + k1⊗b)dk = (1⊗b− a⊗ 1)−1[exp(1⊗b)− exp(a⊗ 1)].in the following sequel we provide examples to the obtained theorems in main section. examplesconsist of taking f to be an exponential operator and applying various conditions as given by thetheorems. corollary 1. if a,b are selfadjoint operators with sp(a), sp(b) ⊂ [m,m] and 1⊗b− a⊗ 1 is invertible, then by (5), we get∣∣∣∣∣∣∣∣18 [ exp(a)⊗ 1 + 6 exp ( a⊗ 1 + 1⊗b 2 ) + 1⊗ exp(b) ] (8) −(1⊗b− a⊗ 1)−1[exp(1⊗b)− exp(a⊗ 1)] ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 32 exp(m). corollary 2. since for f (t) = exp(t), t ∈ r, |f ′| is convex, then by (6)∣∣∣∣∣∣∣∣18 [ exp(a)⊗ 1 + 6 exp ( a⊗ 1 + 1⊗b 2 ) + 1⊗ exp(b) ] (9) −(1⊗b− a⊗ 1)−1[exp(1⊗b)− exp(a⊗ 1)] ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 64 (‖exp(a)‖+ ‖exp(b)‖). https://doi.org/10.28924/ada/ma.4.17 eur. j. math. anal. 10.28924/ada/ma.4.17 11 ∣∣∣∣∣∣∣∣18 [ exp(a)⊗ 1 + 6 exp ( a⊗ 1 + 1⊗b 2 ) + 1⊗ exp(b) ] (10) −(1⊗b− a⊗ 1)−1[exp(1⊗b)− exp(a⊗ 1)] ∣∣∣∣∣∣∣∣ ≤ 5 ‖1⊗b− a⊗ 1‖ 64 (‖| exp(a)| ⊗ 1 + 1⊗ | exp(b)|‖+ ‖exp(a)| ⊗ 1− 1⊗ | exp(b‖). 4. conclusion tensors have become important in various fields, for example in physics because they providea concise mathematical framework for formulating and solving physical problems in fields suchas mechanics, electromagnetism, quantum mechanics, and many others. as such inequalities arecrucial in numerical aspects. reflected in this work is the tensorial shuang’s lemma, which asa consequence enabled us to obtain simpson type inequalities in hilbert space. new simpsontype inequalities are given, examples of specific convex functions and their inequalities using ourresults are given in the section some examples and consequences. plans for future research can bereflected in the fact that the obtained inequalities in 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https://doi.org/10.1016/j.jmaa.2016.09.018 https://doi.org/10.2298/aadm0701092d https://doi.org/10.1142/12388 https://doi.org/10.1186/s13662-021-03615-2 https://doi.org/10.22436/jnsa.009.12.36 https://doi.org/10.37193/cmi.2024.01.10 https://doi.org/10.3390/sym15040925 https://doi.org/10.47443/ejm.2023.004 https://doi.org/10.21608/ejmaa.2023.199881.1014 https://doi.org/10.21608/ejmaa.2023.199881.1014 https://doi.org/10.29020/nybg.ejpam.v16i3.4843 https://doi.org/10.29020/nybg.ejpam.v16i3.4843 https://doi.org/10.56947/gjom.v15i2.1247 https://doi.org/10.56947/gjom.v15i2.1247 https://doi.org/10.3390/sym16010121 1. introduction and preliminaries 2. main results 3. some examples and consequences 4. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 4doi: 10.28924/ada/ma.5.4 correspondences among inner functions, functions with non-negative real parts and conformal mappings ronen peretz department of mathematics, ben gurion university of the negev, beer-sheva, 84105, israelmensahyaogan2@gmailcom abstract. we study an interesting family of dynamical systems on the set of the singular innerfunctions (defined on the unit disk). starting with an inner function s0(z), we obtain new singularinner functions s1(z), s2(z), . . .. this sequence converges to a holomorphic self-map of the unit diskwhich we call s. the convergence is proved with the aid of a fixed-point theorem, a special caseof the earle-hamilton theorem. the function s itself is not a singular inner function as zs(z) isa conformal map. this conformal map has the surprising property that its inverse (which is a prioridefined on a proper subset of the disk) extends to the entire disk. the motivating question for thisresearch is whether z times a singular inner function can have an omitted value in the unit disk. thisquestion appears within a book on the krzyz problem written by the author. this question is stillopen. 1. correspondences that involve inner functions let us recall few correspondences that involve inner functions in h2(u). we denote the unitdisk in c by u . we will denote the (multiplicative) group of inner functions in h∞(u) by inn.its subgroup which contains all the singular inner functions will be denoted by sinn. we followthe notations in, [2]. finally, the (additive) group of holomorphic functions in u which have non-negative real parts and which have finite radial limits almost everywhere on t which are purelyimaginary will be denoted by rp. later on we will add one condition to this definition of rp butfor now this definition suffices. here are a few elementary facts that are well known:(1) f ∈ rp ⇔ ∃w ∈ inn such that f = 1+w 1−w .this is a bijection since f = 1 + w 1− w ⇔ f · (1− w) = 1 + w ⇔ w · (f + 1) = f − 1⇔ w = f − 1 f + 1 . (2) g ∈ sinn⇔ ∃ f ∈ rp such that g = exp(−f ).the correspondence rp→ sinn, f → g is not one-to-one. the kernel is 2πiz. received: 13 aug 2024. key words and phrases. singular inner functions; conformal mapping; krzyz problem; complex dynamical system;earle-hamilton fixed-point theorem; löwner equation. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.4 https://orcid.org/0000-0002-6321-479x eur. j. math. anal. 10.28924/ada/ma.5.4 2(3) g ∈ sinn⇔ ∃w ∈ inn such that g = exp (−1+w1−w ).the correspondence inn→ sinn, w → g is not one-to-one. 1 + w 1− w + 2πik = 1 + v 1− v ⇔ v = ( 1 + w 1− w + 2πik − 1 )/( 1 + w 1− w + 2πik + 1 ) = = πik + (1− πik)w (1 + πik)− πikw .so if we denote by φk(z) the fractional linear function φk(z) = πik + (1− πik)z (1 + πik)− πikzand if we denote m : inn→ sinn, m(w) = g where m(w) = exp ( − 1 + w 1− w ) , then m−1(g) = {φk(w) | k ∈ z}. 2. an example of our construction it will be convenient to first demonstrate the construction on a particular case where concretecomputations are possible. this construction was motivated by a problem that appeared in thebook, [4]: let s(z) be a singular inner function (s ∈ sinn). is it true that the inner function z · s(z) is a surjection u → u? theorem 2.1. let {sn(z)}∞n=0 be a sequence of singular inner functions defined recursively by: s0 ∈ sinn (an arbitrary initial point), sn+1(z) = exp ( − 1 + z · sn(z) 1− z · sn(z) ) for n ∈ z≥0. then limn→∞ sn = s uniformly on compact subsets of u . s(z) is in h∞(u) and it satisfies the fixed-point equation s = exp ( − 1 + z · s 1− z · s ) . also the mapping z · s(z) ∈ h∞(u) is injective u → im(z · s(z)) ⊂ u but it can not be an inner function. proof.since s0 ∈ sinn and since an inductive argument shows that if sn ∈ sinn then sn+1 = exp ( − 1 + z · sn 1− z · sn ) ∈ sinn for n ∈ z≥0, it follows that the sequence {sn(z)}∞n=0 is a sequence of singular inner functions. the family offunctions in the sequence is a normal family. even more, for a fixed-point z ∈ u the function of t ∈ u given by exp ( − 1 + z · t 1− z · t ) , https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 3is a contraction and so by the fixed-point theorem of s. banach iterations of this contractionconverge to a unique fixed-point s(z). so limn→∞ sn = s uniformly on compact subsets of u , and s(z) satisfies the fixed-point equation s(z) = exp ( − 1 + z · s(z) 1− z · s(z) ) . clearly s(z) is a non-vanishing function in h∞(u). next, let us consider the following holomorphicfunction of w , defined on the once punctured plane as follows: f : c− {1} → c, f(w) = w exp ( 1+w 1−w ) . then by the fixed-point equation satisfied by s(z) we get f (z · s(z)) = z . so f is a left inverseof z · s(z) and hence z · s(z) : u → im(z · s(z)) is an injection. more concretely, if we denote g(z) = z · s(z) then the assumption g(z1) = g(z2) implies that z1 = f (g(z1)) = f (g(z2)) = z2.since s(z) can not be a constant function (by the fixed-point equation), z · s(z) can not be aninner function (see [3], remarked by raymond mortini). � 3. a generalization definition 3.1. we will denote by rp, the family of all the f ∈ h(u), that satisfy the followingfour conditions:(i) <f (z) ≥ 0, ∀ z ∈ u .(ii) <f (e iθ) = 0 almost everywhere on t with respect to the lebesgue measure on t.(iii) the function of t ∈ u given by exp (−f (z · t)) is a contraction (with respect to the euclideanmetric) where z ∈ u is fixed.(iv) f is a non-constant function. remark 3.2. ∀f,g ∈ rp, ∀ a, b ∈ r≥0, such that 0 < a + b ≤ 1, we have a · f + b · g ∈ rp.also if f (z) = 1+w(z) 1−w(z) , where, as always w(z) ∈ inn, then f ′(z) = 2w ′(z) (1−w(z))2 . by d dt exp (−f (z · t)) = −zf ′(z · t) exp (−f (z · t)) , it follows by (iii) ∣∣∣∣z 2w ′(z · t) (1− w(z · t))2 exp (−f (z · t)) ∣∣∣∣ ≤ c < 1.in particular we obtain that the generating inner function w(z) of f (z) satisfies: |z | |w ′(z · t)| |1− w(z · t)|2 exp ( − 1− |w(z · t)|2 |1− w(z · t)|2 ) ≤ c 2 < 1 2 . thus we conclude that ∀ z ∈ u and ∀w ∈ inn, such that 1+w1−w ∈ rp we have: |z | |w ′(z)| |1− w(z)|2 exp ( − 1− |w(z)|2 |1− w(z)|2 ) < 1 2 . https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 4 the construction. let f ∈ rp. we will use f (z) to define a sequence {sn(z)}∞n=0 of singular innerfunctions. the definition will use the following recursion: s0 ∈ sinn (an arbitrary initial point). sn+1(z) = exp (−f (z · sn(z))) for n ∈ z≥0. theorem 3.3. the limit limn→∞ sn(z) = s(z) exists and is uniform on compact subsets of u . s ∈ h(u) satisfies |s(z)| ≤ 1∀ z ∈ u , and satisfies the following fixed-point equation, s = exp(−f (z · s)). the function z · s(z) ∈ h∞(u) is a conformal mapping z · s(z) : u → im(z · s) ⊆ u but it is not an inner function. proof.since s0 ∈ sinn and since an inductive argument shows that if sn ∈ sinn, then sn+1 = exp(−f (z · sn)) ∈ sinn for n ∈ z≥0, it follows that all the members of the sequence {sn}∞n=0 belong to sinn. the reason for the validity of the inductive argument is that |z · sn| = |z ||sn| ≤ |z | forall z ∈ u , using the induction hypothesis sn ∈ sinn. thus z · sn ∈ bh∞ , the unit ball of h∞.also |e iθ · sn(e iθ)| = 1 almost everywhere on t with respect to the lebesgue measure on t.also this follows by the induction hypothesis on sn. hence <f (z · sn(z)) ≥ 0 ∀ z ∈ u and < (e iθ · sn(e iθ)) = 0 almost everywhere on t (recall that f ∈ rp satisfies by the definition <f (e iθ) = 0 almost everywhere on t). hence | exp (−f (z · sn(z))) | = exp (−<f (z · sn(z))) ≤ 1 ∀ z ∈ uand also ∣∣exp (−f (e iθ · sn(e iθ)))∣∣ = 1 almost everywhere on t.we just proved that sn+1(z) = exp (−f (z · sn(z))) ∈ sinn for n ∈ z≥0. hence the family offunctions in the sequence {sn}∞n=0 is a normal family. moreover, by condition (iii) in definition3.1, for a fixed z ∈ u iterations of the function of t ∈ u given by exp(−f (z · t)) converge(by banach fixed-point theorem ) to a unique fixed-point t0 = s(z). so limn→∞ sn(z) = s(z)uniformly on compact subsets of u , and s(z) satisfies the fixed-point equation s(z) = exp(−f (z · s(z))). clearly, the h∞(u) function is a non-vanishing function that belongs to the unit ball bh∞(u). next, let us consider the following holomorphic function of w , defined on u as follows: f : u → c, f(w) = w exp(f(w)). then by the fixed-point equation satisfied by s(z) we get: f (z · s(z)) = z . the reason is that f (z · s(z)) = z · s(z) exp(f (z · s(z))) = z · s(z) · s(z)−1 = z. thus f is a left is a left inverse of z · s(z) and hence the mapping: z · s(z) : u → im(z · s(z))is an injection. more concretely, if we denote g(z) = z · s(z) then the assumption g(z1) = g(z2)implies that z1 = f (g(z1)) = f (g(z2)) = z2. the function s(z) can not be a constant function, forif s(z) = e iθ0 , then e iθ0 = exp ( −f (e iθ · s(e iθ0)) ) = exp ( −f (e i(θ+θ0)) ) . https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 5but any function such as f in rp is non-constant by the definition. since the only injective innerfunctions are blaschke factors, [3], z · s(z) can not be an inner function (for that would imply that s(z) is a unimodular constant). � 4. a parametrization of a family of conformal mappings by the functions in rp we note that we may replace f (z) by any of the functions in the sequence f (z) + 2πiz. all ofthese are members of rp that will generate s(z) just as f (z) does. however, if g(z)−f (z) 6∈ 2πizand g(z) (like f (z)) belongs to rp, then exp(−f ) and exp(−g) are different holomorphic functions.can they share the same fixed-point s(z)? that is, can the following be true? s(z) = exp (−f (z · s(z))) = exp (−g(z · s(z))) z ∈ u. by the permanence principle for holomorphic functions this holds if and only if g − f ∈ 2πiz(which is not the case). so the assignment: [f ] := f + 2πiz → s is an injection of the quotientspace of rp, namely of rp/2πiz onto the family of functions s[f ](z) in the unit ball of h∞(u),such that z · s[f ](z) : u → im(z · s[f ](z)) ⊆ u is a conformal mapping, where s[f ] is the usual f (z) fixed-point: s[f ] = exp(−f (z · s[f ](z))). here (in the notation of section 3) [f ] stands forany of the members of the equivalence class [f ] in rp/2πiz. we got our parametrization thatthe title of this section refers to. [f ] determines the conformal mapping z · s[f ] via the fixed-pointequation. so [f ] is the parameter of the conformal mapping z · s[f ](z). if we define the conformalmapping by w = f[f ](z) = z · s[f ](z), then f[f ] : u → im(f[f ]), is invertible, so that z = f −1 [f ] (w).using the fixed-point equation: − log s[f ] = f (z ·s[f ]) we see that s[f ] determines its parameter [f ] by: f (w) = − log (w z ) = − log ( w f −1(w) ) . the family rp is algebraically easy to understand, unlike the family of the conformal mappings:{ s[f ] | [f ] ∈ rp/2πiz } . for example, see our remark 3.2: rp is closed for taking linear combinations with coefficients a, b ∈ r≥0, such that 0 < a+b ≤ 1, i.e. ∀f,g ∈ rp(or rp/2πiz), a ·f+b ·g ∈ rp(or rp/2πiz).geometrically we are dealing with cones. let (as usual) s[f ] = exp(−f (z · s[f ])), s[g] = exp(−g(z · s[g])), so that z · s[f ] : u → im(z · s[f ]), z · s[g] : u → im(z · s[g]) are con-formal mappings.then we make the following: definition 4.1. ∀ a, b ∈ r≥0, 0 < a+b ≤ 1 we define the conic linear combination by the equation: a · (z · s[f ])+̂b · (z · s[g]) = z · s[a·f+b·g]. this definition (and a similar one for multiplication by a real non-negative scalar) induces onthe family of our conformal mappings the same conic structure as the one we easily have on rp(or rp/2πiz). https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 6 remark 4.2. it is well known that there exist natural and elementary parametrizations between rp and sinn and also between inn and sinn. see the explanations in section 1. thus, our lesselementary parametrization of the family of conformal mappings z · s[f ](z) by rp/2πiz can nowbe related to the tree of correspondences among sinn, inn and rp. we naturally inquire as to what are the conformal members that form the family of conformalmappings in the tree of correspondences. we will deal with that on the next section. as expecteda main ingredient of these conformal mappings will be the boundary behaviour of the inverseconformal mappings im(z · s[f ])→ u . 5. the family of conformal mappings conf our family of conformal mappings is clearly given by the following: definition 5.1. conf = { z · s[f ] ∈ h(u) ∣∣ [f ] ∈ rp/2πiz, s[f](z) = exp (−f(z · s[f](z))) , ∀z ∈ u} . we recall the following facts:(1) z · s[f ] : u → im(z · s[f ]) is an injection.(2) z · s[f ] ∈ bh∞(u) − inn. ( [3]).(3) if w = f[f ](z) = z · s[f ](z), (z ∈ u), then f (w) = − log (w z ) = − log ( w f −1 [f ] (w) ) , w ∈ im(z · s[f ](z)). we would like to characterize the conformal family conf without any reference to the family ofthe parameters rp/2πiz. theorem 5.2. the family conf consists of all the holomorphic functions f (z) ∈ h(u) that satisfy the following: (a) f : u → im(f ) ⊆ u is a conformal mapping. (b) f (0) = 0. (c) the function: f (w) = − log ( w f −1(w) ) , w ∈ im(f ), can be analytically be defined on all of u (not just on im(f )), and it satisfies <f (w) ≥ 0 for all w ∈ im(f ) and <f (f −1(w)) = 0 for all w ∈ ∂ im(f ). (d) for a fixed z ∈ u , the function z ·t f −1(z ·t) is a contraction in t ∈ u for which z · t ∈ im(f ). it is a contraction with respect to the euclidean metric. proof.let us denote the family of all the mappings f ∈ h(u) that satisfy (a), (b), (c) and (d) by a. weneed to prove that conf = a where the definition of conf is given in definition 5.1. https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 7(i) conf ⊆ a:let f[f ](z) = z · s[f ](z) ∈ conf. by fact (1) after definition 5.1, f[f ] : u → im(f[f ]) ⊆ u is aninjection. clearly f[f ](0) = 0. by fact (3) after definition 5.1, f (w) = − log ( w f −1 [f ] (w) ) , ∀w ∈ im(f ). finally, by definition 3.1 (iii) we have: (iii) the function of t ∈ u given by exp (−f (z · t)) is acontraction (with respect to the euclidean metric) where z ∈ u is fixed. but exp (−f (z · t)) = z ·t f −1(z ·t) which proved (d) and completes the proof of conf ⊆ a.(ii) a ⊆ conf:let f : u → im(f ) ⊆ u be an elelment of a. then f (0) = 0 and |f (z)| ≤ 1 for all z ∈ u . by theschwarz lemma we get |f (z)| ≤ |z | for all z ∈ u . hence for all w ∈ u where w = f (z) we have∣∣∣w z ∣∣∣ ≤ 1 so − log ∣∣∣w z ∣∣∣ ≥ 0. if we define ff (w) = − log (w z ) = − log ( w f −1(w) ) , then <ff (w) ≥ 0. now (ii) and (iii) in definition 3.1 follow. hence f ∈ conf. � here is an interesting consequence on the conformal mappings of the family conf: corollary 5.3. let f , g ∈ conf and let a, b ∈ r≥0, 0 < a+ b ≤ 1. then ∃ ha,b ∈ conf such that we have the following multiplicative relation among these three conformal mappings:( h−1a,b(w) w ) = ( f −1(w) w )a ( g−1(w) w )b . proof.by the proof of theorem 5.2 it follows that, f , g ∈ conf⇔ − log ( w f −1(w) ) ,− log ( w g−1(w) ) ∈ rp. we note that in f −1(w) we have w ∈ im(f ) and in g−1(w) we have w ∈ im(g). by property (5)after definition 5.1 we know that the two-dimensional lebesgue measures of the sets u − im(f )and u − im(g) are zero. hence im(f ) ∩ im(g) is an open subset of u and the two-dimensionallebesgue measure of the set u− (im(f ) ∩ im(g)) is zero. so im(f )∩ im(g) is a large open subsetof u on which both holomorphic functions − log ( w f −1(w) ) , and − log ( w g−1(w) ) , are defined and belong to rp. i.e. these are the restrictions of rp functions to the intersection ofthe images of the conformal mappings f and g. since: a { − log ( w f −1(w) )} + b { − log ( w g−1(w) )} = − log ( w f −1(w) )a ( w g−1(w) )b ∈ rp, https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 8it follows by the proof of theorem 5.2 that ∃ ha,b ∈ conf such that:( w f −1(w) )a ( w g−1(w) )b = ( w h−1a,b(w) ) . our proof is now completed. � one can conclude more surprising properties on the conformal mappings of the family conf. 6. the geometry of the image of conformal mappings in conf let f ∈ rp. we chose an arbitrary singular inner function s0(z) ∈ sinn and we gener-ated an infinite sequence of singular inner functions using the following recursion: sn+1(z) = exp (−f (z · sn(z))) for n ∈ z≥0. the limit s(z) = limn→∞ sn(z) exists for all z ∈ u . theconvergence is uniform on compact subsets of u . s(z) satisfies the fixed-point equation s(z) = exp (−f (z · s(z))), z ∈ u . the function g(z) = z · s(z) is a conformal mapping u → im(g) ⊆ u .to see that we defined the following holomorphic function defined on u . f : u → c, f(w) = w exp (f(w)) . it easily follows by the fixed-point equation that f (g(z)) = z . thus g has a left inverse f and hence g is injective. that was the first surprise. the sequence {sn(z)}∞n=1 of singular inner functions,each of which covers u infinitely many times, produced the limit s(z) so that g(z) = z · s(z) wasinjective. the complete opposite behavior. g covers each point of u at most once.in this section we will describe the image im(g) by trying to identify its boundary. at first onemight expect a very wild boundary because of the limit above. we will encounter here our secondsurprise. the boundary ∂im(g) will turn out to be completely tame. it will be composed of curveswhich are the zero sets of certain planar harmonic functions plus very few corners in between thedifferent zero sets. thus a piecewise smooth closed jordan curve. theorem 6.1. the image im(g) = im(z · s(z)) of a conformal mapping in conf is a piecewise smooth closed jordan curve. it is composed of arcs on t and of arcs which are subsets of the zero set of the planar harmonic function <f (w) + log |w |, plus a small number of corners. proof.since f (w) = g−1(w) and g : u → im(g) ⊆ u is conformal, in order to to identify im(g), we needto identify those arcs in the closure of the unit disk u that are mapped by f into the unit circle t = ∂u. so we want to solve for all w ∈ u that satisfy |f (w)| = 1. i.e. |w exp (f (w))| = 1. we recall that f ∈ rp and hence, by definition 3.1, condition (ii) we know that <f (e iθ) = 0 almosteverywhere on t with respect to the lebesgue measure on t. since |exp (f (w))| = exp (<f (w))it follows that: ∣∣e iθ exp (f (e iθ))∣∣ = 1 https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 9almost everywhere on t. so we want to solve for all w ∈ u that satisfy |f (w)| = 1. this means,to find all w ∈ u for which |w exp(f (w))| = 1, that means |w | exp(<f (w)) = 1, i.e. those w ∈ uthat satisfy: <f (w) = − log |w |.this equation has harmonic functions on both sides. alternatively we look for the zero set in u ofthe planar harmonic function <f (w) + log |w |. this proves our theorem. � in the case of our first example: f (w) = 1 + w 1− w .in this case: 1 + w 1− w = 1− |w |2 |1− w |2 + w − w |1− w |2 .our equation <f (w) + log |w | = 0 becomes: 1− |w |2 |1− w |2 + log |w | = 0.we note that any w ∈ t−{1} solves this equation. in particular both ±i are solutions. we mentionthose two in particular because we will see soon that they are the two zeros of the derivative of ourholomorphic function in interest and hence this function is not injective exactly at those two points.we can obtain the equation of this curve either in cartesian coordinates: x = <w , y = =w , |w |2 = x2 + y2. 1− x2 − y2 1 + x2 + y2 − 2x + 1 2 log(x2 + y2) = 0,or better in polar coordinates: x = r cos θ, y = r sin θ. 1− r2 1 + r2 − 2r cos θ + log r = 0.solving for cos θ this is: cos θ = 1 2r { 1 + r2 + 1− r2 log r } . remark 6.2. we note that: lim r→1− 1 2r { 1 + r2 + 1− r2 log r } = 0, so the equation above has exactly two solutions in [−π, π], and these are ±π2 . these correspondto ±i . the zero set within u is the given by θ(r) = cos−1 { 1 2r ( 1 + r2 + 1− r2 log r )} . this intersects the x-axis to the right of 0, for θ = 0 so cos θ = 1: 1 = 1 2r { 1 + r2 + 1− r2 log r } ⇒ (1− r) { 1− r + 1 + r log r } = 0. https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 10one solution is r = x = 1 and others we obtain by: 1− x + 1 + x log x = 0 or 1 + x + (1− x) log x = 0. we easily check that (1+ x +(1− x) log x)′ > 0 for 0 < x < 1 and so there is exactly one solution x = x0 of 1− x + 1 + x log x = 0, in 0 < x < 1. a similar computation shows that the curve: 1− x2 − y2 1 + x2 + y2 − 2x + 1 2 log(x2 + y2) = 0, determines x as a function of y in [−1, 1]. it connects −i = (0,−1) to i = (0, 1). if goes through (x0, 0) and is symmetric with respect to the x-axis. it is strictly monotonic decreasing from (x0, 0)to (0, 1) and by symmetry with respect to the x-axis it is strictly monotonic increasing from (0,−1)to (x0, 0). thus u is divided into two parts by that zero set. the part in u to the left of the curveand the part to the right of that zero set. since the left part contains the origin (by x0 > 0) it isthat left part that is the image of our conformal mapping in this example, that corresponds to thefunction in rp given by: f (z) = 1 + z 1− z . 7. combining two dynamical systems next we will make use of two dynamical systems. the first is the discrete dynamical systemwe used above. it is controlled by a simple recursion which is generated by a function in rp.the second is the continuous dynamical system of löwner type that is controlled by the partialdifferential equation for b, the class of bounded non-vanishing functions. in fact b = sinn the classof the singular inner functions. the notation b as well as its differential equation were describedin section 2 of the basic paper [1]. the notation sinn was used in [2]. we recall facts from section2 of [1]. suppose f ∈ b has the herglotz representation f (z) = exp ( − ∫ 2π 0 e iθ + z e iθ − z h(θ)dθ ) , where h(θ) ≥ 0. the collection of such functions is dense in the subfamily of b consisting offunctions for which f (0) > 0. changing variable by the substitution τ = τ(θ) = ∫ θ0 h(φ)dφ, andputting k(τ) = e iθ leads to the formula, f (z) = exp ( − ∫ t0 0 1 + k(τ)z 1− k(τ)z dτ ) , (7.1) where t0 = τ(2π) = − log f (0). conversely, if k(τ) is a measurable function of τ which satisfies |k(τ)| = 1, τ ∈ r, then equation (7.1) defines a function of class b. given f (z) as in equation(7.1), we set https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 11 f (z, t) = exp ( − ∫ t 0 1 + k(τ)z 1− k(τ)z dτ ) , 0 ≤ t ≤ t0. (7.2) then f (z, t) ∈ b for all t ∈ [0, t0], f (z, t0) = f (z), and f (z, 0) = 1. it follows from equation (7.2)that for almost all t , ∂f (z, t) ∂t = −f (z, t) · 1 + k(t) · z 1− k(t) · z . (7.3) this is the differential equation for b.we recall our discrete dynamical system:let g ∈ rp. we will use the function g(z) to generate a sequence {sn}∞n=0 of singular innerfunctions. it is controlled by the following recursion, s0(z) ∈ sinn (an arbitrary initial point). sn+1(z) = exp (−g(z · sn(z))) for n ∈ z≥0. (7.4) we proved in theorem 3.3, the following:the limit s(z) = limn→∞ sn(z) exists and is uniform on compact subsets of u . s ∈ h(u) satisfies |s(z)| ≤ 1 ∀ z ∈ u , and satisfies the following fixed-point equation, s(z) = exp (−g(z · s(z))).the function z ·s(z) ∈ bh∞(u), the unit ball of h∞(u). z ·s(z) is a conformal mapping (it belongsto conf). thus z ·s(z) : u → im(z ·s(z)) ⊆ u , but it is not an inner function, see for example [3].let us denote the following correspondence by f : f : sinn→ conf, f (s0) = z · s(z). one result that we will demonstrate below is that the correspondence f is, in fact, a constant. wewill give two different proofs for that result. this result might seem to be surprising at first. but itis not really surprising. remark 7.1. we clearly have ∀ n ∈ z≥0, f (sn) = z · s(z). so the correspondence f is certainlyconstant on the sequence {sn(z)}∞n=0 which is the output of our recursion, that generates thediscrete dynamical system. so we can view f as a correspondence sinn/{{sn(z)}∞n=0} → conf.however, since we will prove that f is a constant correspondence (given a g ∈ rp) we willconclude that the truly interesting correspondence is not sinn→ conf, but is t : rp→ conf, t (g(z)) = z · s(z). we combine the continuous dynamical system that was described in equation (7.3), with our g-discrete dynamical system (g ∈ rp) that was described in equation (7.4), as follows: f : {f (z, t) | 0 ≤ t ≤ t0} → conf, f (f (z, t)) = z · s(z, t). here the starting point of the recursion is s0(z, t) = f (z, t) and sn+1(z, t) = exp (−g(z · sn(z, t)))for n ∈ z≥0. s(z, t) = limn→∞ sn(z, t) for z ∈ u (as was mentioned above), also s(z, t) = exp (−g(z · s(z, t))) for z ∈ u , and z · s(z, t) ∈ conf for each 0 ≤ t ≤ t0. https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 128. more results those results will be summarized in three theorems and one corollary. we begin with thecorresponding computations. by the differential equation for b, in equation (7.3) and by therecursion, in equation (7.4) we have, ∂s1(z, t) ∂t = ∂ ∂s0 {exp (−g(z · s0))} · ∂s0(z, t) ∂t = = s1(z, t) · { −z · ∂g(w) ∂w |w=z ·s0(z,t) } · ∂s0(z, t) ∂t = = s1(z, t) · { −z · ∂g(w) ∂w |w=z ·s0(z,t) } · { −s0(z, t) · 1 + k(t)z 1− k(t)z } = = s0(z, t) · s1(z, t) · z · ∂g(w) ∂w |w=z ·s0(z,t) · { 1 + k(t)z 1− k(t)z } . next, ∂s2(z, t) ∂t = ∂ ∂s1 {exp (−g(z · s1))} · ∂s1(z, t) ∂t = = s2(z, t) · { −z · ∂g(w) ∂w |w=z ·s1(z,t) } · ∂s1(z, t) ∂t = = s2(z, t) · { −z · ∂g(w) ∂w |w=z ·s1(z,t) } ·s0(z, t) ·s1(z, t) · z · ∂g(w) ∂w |w=z ·s0(z,t) · { 1 + k(t)z 1− k(t)z } = = −s0(z, t) · s1(z, t) · s2(z, t) · z2 · ∂g(w) ∂w |w=z ·s0(z,t) · ∂g(w) ∂w |w=z ·s1(z,t) · 1 + k(t)z 1− k(t)z .inductive arguments prove: theorem 8.1. ∂sn(z, t) ∂t = (−1)n+1 ·  n∏ j=0 sj(z, t)  · zn · n−1∏ j=0 ( ∂g(w) ∂w |w=z ·sj (z,t) ) · { 1 + k(t)z 1− k(t)z } . theorem 8.2. there exists a unique s(z) ∈ h(u) such that it is the only fixed-point of the function exp (−g(z · w)), i.e. s(z) = exp (−g(z · s(z))). moreover, ∀s0(z) ∈ sinn, the recursion sn+1(z) = exp (−g(z · sn(z))), n ∈ z≥0, defines a sequence of singular inner functions {sn(z)}∞n=0. this sequence converges uniformly on compact subsets of u to the fixed-point s(z), i.e. s(z) = limn→∞ sn(z) uniformly on compact subsets of u . so s(z) is determined by the recursion but independently of the initial singular inner function s0(z). thus ∀s0(z), t0(z) ∈ sinn, sn+1(z) = exp (−g(z · sn(z))), tn+1(z) = exp (−g(z · tn(z))) and we have: limn→∞ sn(z) = limn→∞ tn(z) = s(z), uniformly on compact subsets of u . proof.we will outline two proofs. the first proof is using the banach fixed-point theorem. namely, ∀ z ∈ u , the function of w ∈ u given by: exp (−g(z · w)) is a contraction and so by the theoremof banach it has a unique fixed-point w = s(z). moreover, this fixed-point is the limit of the https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 13sequence, defined by the recursion wn+1 = exp (−g(z · wn)), independently of the initial point w0.from this we get our conclusions. a second proof uses the differential equation of b, namely we start at the beginning of chain s0(z, t) = f (z, t) and generate the sequence of singular inner functions by our recursion: sn+1(z) = exp (−g(z · sn(z))). we obtain the limit uniformly on compact subsets of u , s(z, t) = limn→∞ sn(z, t). s(z, t) is a fixed-point s(z, t) = exp (−g(z · s(z, t))) .we apply the operator ∂ ∂t to both sides of the fixed-point equation (justified by our assumptionson s0(z, t)). we obtain: ∂s(z, t) ∂t = −z · s(z, t) · { ∂g(w) ∂w |w=z ·s(z,t) } · ∂s(z, t) ∂t . we claim that ∂s(z,t) ∂t = 0 for all t . for if there were an open non-empty interval of t , over which ∂s(z,t) ∂t 6= 0, then by the equation above:{ w · ∂g(w) ∂w |w=z ·s(z,t) } = −1. so z ·s(z, t) can be one of a discrete set which are the zeros of the non-zero holomorphic function w · ∂g(w) ∂w + 1. hence z · s(z, t) 6∈ conf, a contradiction. hence indeed s(z, t) = s(z) is independent of t .since the beginning of the chain {f (z, t)} equals the first element of the sequence of the singularinner functions, s0(z, t) = f (z, t), this again, implies the conclusions of theorem 3.3. � theorem 8.3. lim n→∞  n∏ j=0 sj(z, t)  · zn · n−1∏ j=0 ( ∂g(w) ∂w |w=z ·sj (z,t) ) = 0. proof.by theorem 6.1 and theorem 8.1 where we use limn→∞ ∂sn(z,t) ∂t = 0. � in particular, if we start our recursion from its fixed-point s0(z, t) = s(z), then our sequenceis stationary, sj(z, t) = s(z) for all j ∈ z≥0, and the formula of theorem 5.2 gives us, corollary 8.4. lim n→∞ (z · s(z))n · { ∂g(w) ∂w |w=z ·s(z) }n = 0 ∀ z ∈ u, equivalently lim n→∞ {( w · ∂g(w) ∂w ) |w=z ·s(z) }n = 0 ∀ z ∈ u, https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 14 equivalently ∣∣∣∣{(w · ∂g(w)∂w ) |w=z ·s(z) }∣∣∣∣ < 1 ∀ z ∈ u. the last inequality can be written as follows: z · s(z) · g′(z · s(z)) ∈ bh∞(u) where ∣∣z · s(z) · g′(z · s(z))∣∣ < 1 ∀ z ∈ u. we end our paper with the example g(w) = 1+w 1−w . on the next we will present the formulas weproved, for this particular case. 9. an example let us consider g(w) = 1 + w 1− w ∈ rp.let s0(z, t) = f (z, t) and sn+1(z, t) = exp ( − 1 + z · sn(z, t) 1− z · sn(z, t) ) for n ∈ z≥0 s(z, t) = limn→∞ sn(z, t) uniformly on compact subsets of u , so that s(z, t) = exp ( − 1 + z · s(z, t) 1− z · s(z, t) ) ∀ z ∈ u and z · s(z, t) ∈ conf, ∀ 0 ≤ t ≤ t0. we have the following results: (9.5) ∂sn(z, t) ∂t = (−1)n+1 ·  n∏ j=0 sj(z, t)  · { (2z)n∏n−1 j=0 (1− z · sj(z, t))2 } · { 1 + k(t)z 1− k(t)z } . this follows by theorem 2.1. there exists a unique s(z) ∈ h(u) such that it is the only fixed-point of the function exp (−1+z ·w1−z ·w ),i.e. s(z) = exp ( − 1 + z · s(z) 1− z · s(z) ) . moreover ∀s0(z) ∈ sinn, the recursion sn+1(z, t) = exp ( − 1 + z · sn(z, t) 1− z · sn(z, t) ) for n ∈ z≥0 defines a sequence of singular inner functions {sn(z)}∞n=0. this sequence converges uniformlyon compact subsets of u to the fixed-point s(z),i.e. s(z) = limn→∞ sn(z) uniformly on compactsubsets of u .so s(z) is determined by the recursion independently of the initial singular inner function s0(z).this follows by theorem 3.3. https://doi.org/10.28924/ada/ma.5.4 eur. j. math. anal. 10.28924/ada/ma.5.4 15 lim n→∞ (2z)n · ∏n j=0 sj(z, t)∏n−1 j=0 (1− z · sj(z, t))2 = 0. (9.6) this follows by theorem 8.2. ∣∣∣∣ 2z · s(z) (1− z · s(z))2 ∣∣∣∣ < 1 ∀ z ∈ u, (9.7) equivalently (1− |z | · |s(z)|)2 > 2<{z · s(z)} ∀ z ∈ u . this follows by corollary 5.3. references [1] j.a. hummel, s. scheinberg, l. zalcman, a coefficient problem for bounded nonvanishing functions, j. anal. math.31 (1977) 169-190.[2] o. ivrii, critical structures of inner functions, j. funct. anal. 281 (2021) 109-198.[3] r. mclaughlin, exceptional sets for inner functions, j. london math. soc. s2-4 (4) (1972) 696-700.[4] r. peretz, the krzyż conjecture theory and methods, world scientific, singapore, 2021. https://doi.org/10.28924/ada/ma.5.4 1. correspondences that involve inner functions 2. an example of our construction 3. a generalization 4. a parametrization of a family of conformal mappings by the functions in rp 5. the family of conformal mappings conf 6. the geometry of the image of conformal mappings in conf 7. combining two dynamical systems 8. more results 9. an example references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 5doi: 10.28924/ada/ma.5.5 hybrid iterative methods for solving nonlinear equations in banach spaces ioannis k. argyros1,∗, santhosh george2, samundra regmi3, michael i. argyros4 1department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2department of mathematical and computational sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 3department of mathematics, university of houston, houston, tx 77204, usa sregmi5@uh.edu 4department of computer science, university of oklahoma, norman, ok 73501, usa argyro01@email.franklin.edu abstract. the present article contributes to the solution of equations which carry the symmetryproperty of the problem or not. iterative methods with inverses generate sequences convergingfaster to a solution of an equation than methods without inverses. however, the implementationof these methods has drawbacks, since the analytical form of these inverse may be unavailable orcomputationally very expensive. this problem is addressed in this paper by replacing the inversewith a finite sum of linear operators. a convergence analysis is developed for the hybrid methods.the numerical examples demonstrate that the number of iterates is essentially the same between thehybrid and the original method. this technique is also extended to solve generalized equations. 1. introduction the letters x, y denote banach spaces; ω ⊂ x is a convex and open subset of x , and f1 : ω −→ y stands for a continuous operator. numerous applications from diverse areas of computationalscience and engineering can be converted by using mathematical modelling [3, 8, 14, 17,19–21,23,26,28,33,35] to finding a solution s∗ ∈ ω of the nonlinear in the general equation f1(x) = 0. (1.1) the closed form of the solution s∗ is attainable only in special cases. this forces researchers andpractitioners to solve the equation (1.1) iteratively. single-step methods of high convergence order received: 3 jul 2024. key words and phrases. inverse of an operator, banach space, hybrid iterative method, generalized equations,continuous operator, convergence. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 2look like the newton-type defined for each n = 0, 1, 2, ... by x0 ∈ ω, xn+1 = xn − l−1n f1(xn), (1.2) where ln ∈ l(x, y ) which is the space of continuous operators mapping x into y , and l−1n ∈ l(y,x) for each n = 0, 1, 2, .... by ln, we denote l(xn). some choices for the operator ln can be ln = f ′1(xn) (newton’s method), ln = [xn − f1(xn), xn + f1(xn);f ] (steffensen’s method), ln = i (the picard method), the identity operator. here f ′1, [., .;f1] denote fréchet-derivative and divided differences of order one for the operator f1, respectively [22,25].many other choices are possible. it turns out that the inverse of the operator ln is costly orimpossible to find in general. this concern with the implementation of these methods constitutesthe motivation for this paper. our idea is to replace the inverse with a finite sum of linear operatorsconverging to it. the reasoning is explained as follows. let p ∈ n be fixed.suppose there exists γ ∈ l(x, y ) such that γ−1 ∈ l(y,x) and for a = a(x) = γ−1(γ− l(x))the operator i−a(x) is also invertible, i.e. (i−a(x))−1 ∈ l(y,x). in this case, the newton-typemethod can read as x0 ∈ d, xn+1 = xn − (i − a)−1γ−1f1(xn). (1.3) note that we have (i − a)−1γ−1 = [γ(i − a)]−1 = l−1n . (1.4) however, even if the linear operator γ−1 is known it is still required to find the inverse of (i −a),which is not a fixed operator (in general). but what if we replace this operator with m = mp(x) = i + a+ ...+ ap . then, method (1.3) can be written as x0 ∈ ω, xn+1 = xn −mγ−1f1(xn). (1.5) it is clear that (1.5) is a useful alternative for (1.3) because of (1.4). by letting p −→ +∞, weget limp→+∞mp = l−1n if the limit exists. the condition ‖a‖ < 1 for each x ∈ ω assures theexistence of such a limit. if the linear operator m is invertible and the sequence {xn} given by(1.5) converges to some s∗, then by (1.5) we get m−1(xn − xn+1) = γ−1f1(xn) leading to 0 = lim n→+∞ m−1(xn − xn+1) = lim n→+∞ γ−1f1(xn), https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 3i.e. f1(s ∗) = 0. thus, the point s∗ solves the equation (1.1). the same reasoning leads for a1 = (γ− l(x))γ−1 and m1 = i + a1 + ...+ ap1 to the method x0 ∈ ω , xn+1 = xn −m1γ−1f1(xn). (1.6) it is clear that the study of the convergence of the method (1.6) is analogous to (1.5). that is whywe study only method (1.5) in section 2. we deal with two kinds of convergence: the semi-localand the local. the first utilizes knowledge in a neighborhood of x0 and develops estimates relatedto ‖xn+1− xn‖ and ‖s∗− xn‖, and the convergence conditions assure that limn→+∞ xn = s∗. in thesecond kind, knowledge about a neighborhood of s∗ is used to provide the same estimates as inthe semi-local kind and again limn→+∞ xn = s∗. it is worth noting that the iterates generated by(1.3),(1.5) and (1.6) are not the same in general. but we use the same notation for simplicity. theconvergence for both kinds relies on generalized continuity conditions controlling the operatorsinvolved [5, 6, 9, 18]. in particular, our semi-local convergence analysis depends on the usage ofmajorizing sequences [29,30,34]. notice that the method (1.5) can also be written for dn = γm−1as x0 ∈ ω, f1(xn) +dn(xn+1 − xn) = 0. (1.7) in section 3 we also use the developed methodology for solving nonlinear equations to solvegeneralized equations. that is find x ∈ x such that f1(x) + f2(x) 3 0. (1.8) here f2 : x ⇒ y is a set-valued operator mapping x into y with closed graph [1–4, 13, 17–19, 21–23, 26, 28, 32, 35] and operator f is as previously defined. a plethora of applications frommathematical programming, variational inequalities, optimal control, or constrained systems arewritten in the form (1.8). there is extensive literature on iterative methods solving the generalizedequation (1.8) [1–4,13,17–19,21–23,26,28,32,35]. notice that the method used in the literature tosolve (1.8) is defined by f1(xn) + d̄n(xn+1 − xn) + f2(xn+1) 3 0, (1.9) where d̄n is a linear operator. it can be chosen as d̄n = ln, d̄n = f ′1(xn) or d̄n ∈ ∂f1(xn) or otherchoices [9, 11, 23, 24]. these methods have the same problems as the ones for solving nonlinearequations. that is why it is justified to consider the analog of (1.7) defined by f1(xn) +dn(xn+1 − xn) + f2(xn+1) 3 0 (1.10) the semi-local and local convergence of the method (1.10) is developed in section 3 in an analogousway to section 2 for the method (1.5) or (1.7). in numerical section 4, the examples demonstratethat the number of iterations of the hybrid methods to arrive at a predetermined error tolerance https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 4is essentially the same as with the original methods. moreover, the convergence order is also thesame.in order to achieve all this we redevelop some standard terminology to make the paper as self-contained as possible. more details can be found in [8, 11, 19]. let s(z, ρ) and its closure s[z, ρ]denote open and closed balls, respectively of center z ∈ x and radius ρ > 0. let c be a set in x .define the distance for x ∈ x to c by dist(x, c) = infx∈c ‖x − y‖. the generalized set-valuedoperator g relates with its graph given by gph(g) = {(x, y) ∈ x × y, y ∈ f2(x)}, and its domain dom(g) = {x ∈ x|f2(x) 6= 0}. the inverse of g is given as g−1(y) = {x ∈ x, y ∈ f2(x)}. notethat a set-valued operator h : x ⇒ y is said to be metrically regular at x0 for y0 if y0 ∈ h(x0)and there exists neighbourhoods v1 of x0 and v2 of y0 and β > 0 such that gph(h ∩ (v1 × v2)) isclosed and for each (x, y) ∈ v1 × v2 dist(x,h−1(y)) ≤ βdist(y ,h(x)). (1.11) the regularity modulus of h at x0 for y0 is the infimum over all β > 0 and is denoted by reg(h; x0/y0). additionally if the operator ∆ : v2 → y → h−1(y) ∩ v1 is not multivalued on v2, then we say that h is strongly metrically regular. in this case, ∆ is lipchitz continuous on v2.finally, section 5 contains concluding remarks and directions for research. 2. convergence for the method (1.5) we start with the study of the semi-local analysis in this section. some auxiliary results anddefinitions are useful. lemma 2.1. (banach lemma on invertible operators)( [14, 22, 30, 34]) if p is a bounded linear operator in x , p−1 exists if and only if there is a bounded linear operator p1 in x such that p−11 exists and ‖i − p1p‖ < 1. if p−1 exists, then p−1 = ∞∑ n=0 (i − p1p )np1 and ‖p−1‖ ≤ ‖p1‖ 1− ‖i − p1p‖ . further, we use majorizing sequences to prove the semi-local convergence. recall the definitionof majorizing sequence. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 5 definition 2.1. ( [14,22,30,34]) let {xn} be a sequence in a normed space x . then a nonnegative scalar sequence {un} for which ‖xn+1 − xn‖ ≤ un+1 − un ∀n ≥ 0 (2.1) holds, is a majorizing sequence for {xn}. note that any majorizing sequence is necessarily nondecreasing. moreover, if the sequence {un} converges, then {xn} converges too, and for u∗ = limn−→∞ un ‖s∗ − xn‖ ≤ u∗ − un. hence, the study of the convergence of the sequence {xn} reduces to that of {un}. the analysis requires some conditions. let e = [0,+∞).suppose(h1) there exists a function φ : e × e × e → [0,+∞) continuous as well nondecreasing in allthree variables and invertible operators m(.) and γ such that for some x0 ∈ ω, and each x, y ∈ ω the following mysovskii-like condition holds ‖m(x)γ−1(f1(y)− f1(x)− γm−1(x))‖ ≤ φ(‖x − x0‖, ‖y − x0‖, ‖y − x‖)‖y − x‖ define the real real sequence {αn} for α0 = 0, α1 ≥ η := ‖m(x0)γ−1f1(x0)‖ and each n = 0, 1, 2, ... by αn+1 = αn + φ(αn−1, αn, αn − αn−1)(αn − αn−1), n = 1, 2, ... (2.2) notice that the constant η is well defined since the operator γ is invertible. moreover, thesequence {αn} defined by the formula (2.2) is proven to be majorizing for the method (1.5)in theorem 2.3. but let us present convergence conditions for it.(h2) there exists a parameter ρ ≥ η such that for each n = 0, 1, 2, ... φ(αn−1, αn, αn − αn−1) < 1 and αn ≤ ρ. it follows by the condition (h2) and the formula (2.2) that 0 ≤ αn−1 ≤ αn ≤ ρ and there exists α∗ ∈ [η, ρ] such that limn→+∞ αn = α∗. h is well known that this limit isthe unique least upper bound of the sequence {αn} and(h3) s[x0, α ∗] ⊂ ω.the conditions (h1)-(h3) combined with the terminology are utilized for the convergence of themethod (1.5). https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 6 theorem 2.1. suppose that the conditions (h1)-(h3) hold. then, the sequence {xn} generated by the method (1.5) exists in the ball s(x0, α ∗) remains in the same ball for each n = 0, 1, 2, ... and is convergent to a unique solution s∗ ∈ s[x0, α ∗] of the equation f1(x) = 0 such that ‖s∗ − xn‖ ≤ α∗ − αn, n = 0, 1, 2 . . . . (2.3) proof. the process of induction is employed to show the assertion ‖xm+1 − xm‖ ≤ αm+1 − αm for m = 0, 1, 2, . . . . (2.4) the choice of η, (2.2) and the method (1.5) give that ‖x1 − x0‖ = ‖m(x0)γ−1f1(x0)‖ ≤ η = α1 − α0 < α∗, so the assertion (2.4) holds if m = 0, and the iterate x1 ∈ s(x0, α ∗). suppose iterates x0, x1, ..., xmexist and (2.4) holds for all integers smaller or equal to m− 1. notice that the iterate xm+1 existsby the method (1.5) and the invertability of the operators m(xm) and γ. then, we can write by themethod (1.5) the ostrowski-type representation for f1(xm) as f1(xm) = f1(xm)− f1(xm−1)− γm−1(xm − xm−1). (2.5) using the condition (h1), method (1.5), (2.2) and the induction hypothesis on (2.5) we obtain inturn that ‖xm+1 − xm‖ = ‖mγ−1(f1(xm))‖ = ‖mγ−1(f1(xm)− f1(xm−1)− γm−1(xm − xm−1))‖ ≤ φ(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖)‖xm − xm−1‖ ≤ φ(αm−1, αm, αm − αm−1)(αm − αm−1) = αm+1 − αm, (2.6) and ‖xm+1 − x0‖ ≤ ‖xm+1 − xm‖+ ‖xm − xm−1‖+ ...+ ‖x1 − x0‖ ≤ αm+1 − αm + αm − αm−1 + ...+ α1 − α0 = αm+1 < α∗, which complete the induction for the assertion (2.4) and show that all iterates {xm+1} ⊂ s(x0, α ∗).but the sequence {αm} is complete by the condition (h2) as convergent. it follows by the estimate(2.4) that {xm} is also complete. but the space x is banach, so there exists s∗ ∈ s[x0, α ∗] such that lim m→+∞xm = s∗. next, by letting m → +∞ in (2.6), the invertibility of the operators m(.),γ and https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 7the continuity of the operator f , we deduce that limm→+∞mγ−1f1(s ∗) = 0. by the invertibilityof m 0 = m−1(0) = m−1( lim m→+∞ mγ−1f1(xm)) = lim m→+∞ mm−1γ−1f1(xm) = lim m→+∞ γ−1f1(xm) = γ−1 lim m→+∞ f1(xm) = γ−1f1(s ∗). so, f1(s∗) = 0, since γ(0) = 0. let i = 0, 1, 2, .... then, the triangle inequality and (2.4) give ‖xm+i − xm‖ ≤ αm+i − αm. (2.7) hence, by letting i → +∞ in (2.7) we prove (2.3). finally, to show the uniqueness part, let w ∈ s[x0, α ∗] with f1(w) = 0 and w 6= s∗. by using the conditions (h1), (h2) we can write inturn that ‖w − s∗‖ = ‖(mγ−1)(γm−1)(w − s∗)‖ = ‖mγ−1(f1(w)− f1(s∗)− γm−1(w − s∗))‖ ≤ φ(‖s∗ − x0‖, ‖w − x0‖, ‖w − s∗‖)‖w − s∗‖ ≤ φ(α∗, α∗, ‖w − s∗‖)‖w − s∗‖ < ‖w − s∗‖, which gives a contradiction. therefore, we conclude that w = s∗. � remark 2.1. the condition mysoskii-type [22] condition in (h1) can be replaced as follows: (h1) ′ with operator γ as in condition (h1), suppose that there exists a ∈ (0, 12) such that ‖a‖ < a and for each x, y ∈ ω ‖γ−1(f1(y)− f1(x)− γm−1(y − x))‖ ≤ φ1(‖x − x0‖, ‖y − x0‖, ‖y − x‖)‖y − x‖, where the function φ1 is as the function φ. then, the condition (h1) ′ implies (h1) if we take φ = a0φ1, where a0 = 1 1−a . this is the case, since by the definition of the operator m , we have the estimate ‖m‖ = ‖i + a+ ...+ ap‖ ≤ 1 + a + ...+ ap = 1− ap+1 1− a = a0 < 1 1− a . (2.8) in this case the invertibility of the operator m is implied by the banach lemma 2.1, since ‖i −m‖ ≤ ‖a‖+ ...+ ‖a‖p ≤ a + ...+ ap = a 1− ap 1− a < a 1− a < 1. so, m is invertible and ‖m−1‖ ≤ b = 1− a 1− 2a . https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 8 next, we develop the local convergence. the role of the initial point x0 is exchanged by s∗ and the function φ by ψ. suppose: (h4) there exists a function ψ : e −→ [0,+∞) continuous and nondecreasing such that the equation ψ(t)− 1 = 0 has a smallest positive solution denoted by r . (h5) there exists a solution s∗ ∈ ω, and invertible linear operators m,γ such that for each x ∈ ω ‖mγ−1(f1(x)− f1(s∗)− γm−1(x − s∗))‖ ≤ ψ(‖x − s∗‖)‖x − s∗‖ and (h6) s[s∗, r ] ⊂ ω. next, the constant r is shown to be a radius of convergence for the method (1.5). theorem 2.2. suppose that the conditions (h4) − (h6) hold. then, the sequence {xn} for x0 ∈ s(s∗, r)− {s∗} exists in s(s∗, r), stays in s(s∗, r) and converges to s∗ so that ‖xn+1 − s∗‖ ≤ ψ(‖xn − s∗‖)‖xn − s∗‖ ≤ ‖xn − s∗‖ < r. (2.9) moreover, x∗ is the only solution of the equation f (x) =0 in the ball u(s∗, r). proof. the iterates x1, x2, ..., xm+1 are well defined by the method (1.5), and we can write in turnthat xm+1 − s∗ = xm − s∗ −mγ−1f1(xm) = mγ−1(f1(xm)− f1(s∗)− γm−1(xm − s∗)). (2.10) it follows by the conditions (h4), (h5) and (2.10) that ‖xm+1 − s∗‖ ≤ ψ(‖xm − s∗‖)‖xm − s∗‖ ≤ ξ‖xm − s∗‖ ≤ ξm+1‖x0 − s∗‖ < r, (2.11) where ξ = ψ(‖x0 − s∗‖) ∈ [0, 1) showing the assertion (2.9) for each m = 1, 2, ..., since x0 ∈ s(s∗, r) − {s∗}. by letting m → +∞ in (2.11), we deduce that lim m→+∞xm. in order to show theuniqueness part, suppose there exists a solution w1 ∈ s(s∗, r) such that w1 6= s∗. then, as in thesemi-local case, we can write in turn ‖w1 − s∗‖ = ‖(mγ−1)(γm−1(w1 − s∗))‖ = ‖mγ−1(f1(w1)− f1(s∗)− γm−1(w1 − s∗))‖ ≤ ψ(‖w1 − s∗‖)‖w1 − s∗‖ < ‖w1 − s∗‖ (2.12) by the choice of r . hence, we conclude that w1 = s∗, since (2.12) contradicts the hypothesis w1 6= s∗. � https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 9 remark 2.2. comments similar to the ones in remark 2.4 can be made provided that x0, ψ are exchanged by s∗, ψ, respectively. it is also worth noting that the condition (h4) does not necessarily imply the usual condition in local convergence studies that the operator f ′1(s∗) is invertible, i.e. that s∗ is a simple solution of the equation f1(x) = 0. consequently the method (1.5) can be applied to find solutions of multiplicity greater than one. 3. convergence for the method (1.10) let λ > 0, µ > 0, δ ≥ 0 and β > 0 be given parameters.suppose:(c1) there exists a function ϕ1 : [0, λ]× [0, λ]× [0, λ] −→ [0, δ] which is continuous and nondecreasing. define the scalar sequence {hn} for h0 = 0, some h1 ≥ 0, β1 > β and each n = 1, 2, ... by hn+1 = hn + β1ϕ1(hn−1, hn, hn − hn−1)(hn − hn−1). (3.1) the scalar sequence {hn} is shown to be majorizing for {xn} is generated by the formulain the theorem 3.2. however, let us present a convergence criterion for it.(c2) there exists λ0 ∈ [0, j ] such that for each n = 0, 1, 2, ... hn ≤ λ0.it follows by this condition and (3.1) that 0 ≤ hn−1 ≤ hn ≤ λ0 and there exists h∗ ∈ [0, λ0]such that l imn→∞ = h∗.the limit point h∗ is the unique least upper bound of the sequence {hn}.(c3) there exists x0 ∈ x and y0 ∈ f1(x0) + f2(x0) such that βδ < 1 and ‖y0‖ ≤ (1− βδ) min{λβ , µ}.choose h1 ≤ β1‖y0‖.(c4) the operator x −→ qdn(x) := f1(x0) +dn(x − x0) + f2(x) (3.2) is metrically regular at x0 for y0 with constant β and neighborhoods s(x0, λ) and s(y0, µ),respectively.the mapping ϕ1 relates to the operators on the method (1.10).(c5) ‖f1(x)− f1(xn)−dn(x − xn)‖ ≤ ϕ1(‖x − x0‖, ‖xn − x0‖, ‖x − xn‖)‖x − xn‖, for each x ∈ s(x0, λ).next, the semi-local analysis of convergence is developed using the conditions (c1)− (c5). https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 10 theorem 3.1. suppose that the conditions (c1)−(c5) hold. then, for each γ ∈ (βδ, 1) there exists a sequence {xn} generated by the method (1.10) which is well defined in s(x0, λ), remains in s(x0, λ) for each n = 0, 1, 2, ... and convergence to a solution s∗ ∈ s[x0, λ] of the generated equation (1.8). moreover, the following assertion hold ‖s∗ − xn‖ ≤ γnλ (3.3) and dist(0, f1(xn) + f2(xn)) ≤ γn‖y0‖ (3.4) for each n = 0, 1, 2, ... thus, the convergence rate is r-linear. furthermore, if the operator qdn is stronglymetrically regular with constant β and neighbourhoods s(x0, λ) and s(y0, µ), respectively,then the sequence {xn} is the only one satisfying (1.10), and staying in s(x0, λ). remark 3.1. the proof is similar to the one in [11, theorem 2.2]. but it uses (1.10), (c4) and (c5), instead of stronger (1.9),(c4)’ x −→ qan(x) := f1(x0) + an(x − x0) + f2(x),(c5)’ ‖f1(x)− f1(xn)− ln(x − xn)‖ ≤ w(‖x − xn‖)‖x − xn‖ for each x ∈ s(x0, λ), where w : [0.λ] −→ [0, δ] satisfies lim n→+∞w(t) = 0 proof. pick γ ∈ (βδ, 1) and β1 so that β < β1 ≤ γ δ and ‖y0‖ ≤ (1− γ)min { λ β1 , µ } . (3.5) the choice of β1 is certainly possible since there are infinitely many numbers between βand γ δ . we shall show the existence of the sequence {xn} using mathematical induction for n = 1, 2, ... satisfying(in) ‖xn − x0‖ ≤ 1−γn 1−γ β1‖y0‖ ≤ (1− γn)λ < λ.(iin) ‖xn − xn−1‖ ≤ hn − hn−1.(iiin) 0 ∈ f1(xn−1) +dn−1(xn − xn−1) + f2(xn),where dn−1 = dn−1(x0, ..., xn−1).by hypothesis 0 ∈ s(y0, µ) and y0 ∈ qd0(x0). using the condition (c4) for qd0 we get dist(x0, q −1 d0 (0)) ≤ β dist(0, qd0(x0)) ≤ β‖y0‖. if y0 = 0, pick x1 = x0. otherwise, it follows that dist(x0, q −1 d0 (0)) ≤ β1‖y0‖. thus, there exists x1 ∈ q−1d0 (0) satisfying ‖x1 − x0‖ < β1‖y0‖ < (1− γ)λ. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 11so, (i1), (ii1) and (iii1) hold.suppose that for some natural number m the element xm is defined so that the inductionhypothesis (im), (iim) and (iiim) hold. we shall show that the iterate xm+1 is defined anddthe assertions (im+1), (iim+1) and (iiim+1) hold. it follows by (im) that xm ∈ s(x0, λ).set em = f1(x0)− f1(xm)−dm(x0 − xm).by (c5) for x = x0, (im), (iim) and the properties of the function ϕ1 we can write ‖em − y0‖ ≤ ‖y0‖+ ‖f1(x0)− f1(xm)−dm(x0 − xm)‖ ≤ ‖y0‖+ ϕ1(‖x0 − x0‖, ‖xm − x0‖, ‖xm − x0‖)‖x0 − xm‖ ≤ ‖y0‖+ ϕ1(0, hm, hm)‖x0 − xm‖ ≤ ‖y0‖+ δ‖x0 − xm‖ ≤ ‖y0‖+ 1− γm 1− γ β‖y0‖ ≤ (1 + 1− γm 1− γ γ)‖y0‖ = 1− γm+1 1− γ ‖y0‖ < µ. if em ∈ qdm(xm), set xm+1 = xm. otherwise by the condition (c4) we have dist(xm, q −1 dm (em)) ≤ β dist(em, qdm(xn)) ≤ β1 dist(em, qdm(xm)). next, there exists an element xm+1 ∈ q−1dm(em) satisfying ‖xm+1 − xm‖ ≤ β1 dist(em, qdm(xm)). by (iiim) it follows qdm(xm) = f1(x0) +dm(xm − x0) + f2(xm) 3 f1(x0) +dm(xm − x0)− f1(xm−1 −dm−1(xm − xm−1). in view of the condition (c4) with x = xm, (3.5) and (iim), we obtain in turn that ‖xm+1 − xm‖ ≤ β1 ∥∥em − [f1(x0)− f1(xm−1) +dm(xm − x0) −dm−1(xm − xm−1) ]∥∥ = β1‖f1(xm)− f1(xm−1)−dm−1(xm − xm−1)‖ ≤ β1ϕ1(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖ ≤ β1ϕ1(hm−1, hm, hm − hm−1) = hm+1 − hm, (3.6) and ‖xm+1 − xm‖ ≤ β1δ‖xm − xm−1‖ = γ‖xm − xm−1‖ ≤ γm‖x1 − x0‖ ≤ γmβ1‖y0‖. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 12thus, the condition (iiim) holds if m + 1 replaces m. moreover, by the selection of xm+1,we get em ∈ qdm(xm+1) = f1(x0) +dm(xm+1− − x0) + f2(xm+1). so, the assertion (iiin) holds if m + 1 replaces m. moreover, by (im) we get in turn that ‖xm+1 − x0‖ ≤ ‖xm+1 − xm‖+ ‖xm − x0‖ ≤ γmβ1‖y0‖+ 1− γm 1− γ β1‖y0‖ = 1− γm+1 1− γ β1‖y0‖, which terminates the induction for (im). then, by (c2) the sequence is complete as con-vergent. it follows by (iim) that the sequence {xm} is also complete in a banach space xand as such it converges to some s∗. by (im) we have s∗ ∈ s(x0, λ). we must show thatthe limit point s∗ solves (1.8).set ym := f1(xm) − f1(xm−1) − dm−1(xm − xm−1). it follows by (iiim) that ym ∈ f1(xm) + f2(xm). by using (iim) and the condition (c4) for x = xm, we get in turnthat ‖ym‖ = ‖f1(xm)− f1(xm−1)−dm−1(xm − xm−1)‖ ≤ ψ1(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖)‖xm − xm−1‖ ≤ ψ1(hm−1, hm, hm − hm−1)(αm − αm−1) = αm+1 − αm −→ 0 as m → +∞.consequently, we deduce that (xm, ym)→ (s∗, 0) as m → +∞.but f is continuous whereas g has a closed graph. thus, we conclude 0 ∈ f1(s∗) +f2(s ∗).finally, if qdm is a strongly metrically regular operator. it follows that the iterate xm+1 isunique and is obtained from xm. � next, we develop the local convergence analysis of the method (1.10).suppose :(c6) there exists a parameter β > 1 and a function ϕ2 : r+ → r+ which is continuous andnondecreasing such that the equation β1ϕ2(t) − 1 = 0 has a smallest positive solution.denote such a solution by r0.(c7) there exists a solution s∗ ∈ x of the generalized equation (1.8) and parameters β > 0 and δ ≥ 0 such that βδ < 1.(c8) the operator x −→ tdn(x) := f1(s ∗) + dn(x − s∗) + f2(x) is a metrically regular at s∗for 0 with constant β and neighbourhoods s(s∗, λ) and (0, µ) for some µ > 0, respectively.the function ϕ2 relates to the operators on the method (1.10). https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 13(c9) ‖f1(s∗)− f1(xn)−dn(s∗ − xn)‖ ≤ ϕ2(‖s∗ − xn‖)‖s∗ − xn‖ (3.7) for x0 ∈ s(s∗, λ). next, the local convergence analysis of the method (1.10) is presentedusing the conditions (c6)− (c4). theorem 3.2. suppose that the conditions (c6)− (c9) hold. then, for each γ ∈ (βδ, 1) the re exists a sequence {xn} generated by the method (1.10) which is well defined in s(s∗, λ), remains in s(s∗, λ) for each n = 0, 1, 2, .. and converges to s∗ so that ‖s∗ − xn‖ ≤ d‖s∗ − xn−1‖ ≤ dn‖s∗ − x0‖ < λ, where d = β1ϕ2(‖s∗−x0)‖ ∈ [0, 1). additionally, if the operator tdn is strongly metrically regular with constant β and neighbourhoods s(s∗, λ) and s(0, µ), respectively, then the sequence {xn} is the only one satisfying (1.8), and satisfying in (s∗, λ). proof. simply follow the proof of theorem 3.1 for γ ∈ (βδ, 1)x0 = s∗ and β < β1 ≤ γ δ toobtain as in (3.6) but using (c8) and (3.7) instead of (c4) and (3.6), respectively to obtain ‖s∗ − xm‖ ≤ β1ϕ2(‖s∗ − xm−1‖)‖s∗ − xm−1‖ ≤ d‖s∗ − xm−1‖ ≤ ... ≤ dm‖s∗ − x0‖ < λ. therefore, we conclude that lim m−→+∞xm = s∗ and the iterate xm ∈ s(s∗, λ).finally, if the operator tdm is strongly metrically regular, it follows that the iterate xmis unique in s(s∗, λ) by the way the iterate xm is derived from xm−1. � 4. numerical examples the examples use ln = f ′1(xn), γ = i which is independent of x0 and s∗. example 4.1. the solution sought for the nonlinear system f1 = x − 0.1 sin x − 0.3 cos y + 0.4 f2 = y − 0.2 cos x + 0.1 sin y + 0.3 let f1 = (f1, f2). then, the system becomes f1(s) = 0 f or s = (x, y)t . then f ′1((x, y)) = [ 1− 0.1 cos(x) 0.3 sin(y) 0.2 sin(x) 0.1 cos(y) + 1 ] . method (1.2) xp+1 = xp − f ′1(xp)−1f1(xp). https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 14 method (1.5), p = 1, γ = i, m1(x) = i + (i − f ′1(x)), q1(x) = x − (i + (i − f ′1(x)))f1(x), (4.1) xn+1 = q1(xn). method (1.5), p = 2, γ = i, m2(x) = i + (i − f ′1(x)) + (i − f ′1(x))2, q2(x) = x −m2(x)f1(x), (4.2) xn+1 = q2(xn). method (1.5), p = 3, γ = i, m3(x) = i + (i − f ′1(x)) + (i − f ′1(x))2 + (i − f ′1(x))3, q3(x) = x −m3(x)f1(x), (4.3) xn+1 = q3(xn). method (1.5), p = 4, γ = i, m4(x) = i + (i − f ′1(x)) + (i − f ′1(x))2 + (i − f ′1(x))3 + (i − f ′1(x))4, q4(x) = x −m4(x)f1(x), (4.4) xn+1 = q4(xn). method (1.5), p = 5, γ = i, m5(x) = i + (i − f ′1(x)) + (i − f ′1(x))2 + (i − f ′(x))3 + (i − f ′1(x))4 + (i − f ′1(x))5, q5(x) = x −m5(x)f1(x), (4.5) xn+1 = q5(xn). method (1.5), p = 1, 5 , γ = f ′1(x0), xn+1 = xn −mγ−1f1(xn), a = b−1(b − f ′1(x)), (4.6) m = i + p∑ i=1 ai . thus, the comparison shows that the behavior of the method (1.5) is essentially the same as newton’s method (1.2). however, the iterates of the method (1.5) are cheaper to obtain than newton’s. as observed in table 1 table 4, the number of iterations required for the proposed methods with k ranging from 3 to 5 closely aligns with those of newton’s method. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 15 method iterations method iterations (1.2) newton 4 (1.2) newton 4 (4.1), p = 1 6 (4.6), p = 1 8 (4.2), p = 2 5 (4.6), p = 2 6 (4.3), p = 3 4 (4.6), p = 3 5 (4.4), p = 4 4 (4.6), p = 4 5 (4.5), p = 5 4 (4.6), p = 5 4table 1. the number of iterations to reach error tolerance ε = 10−9 with initialguess x0 = (1, 1) and ‖i − f ′1(x0)‖ = 0.3129 < 1. method iterations method iterations (1.2) newton 3 (1.2) newton 3 (4.1), p = 1 5 (4.6), p = 1 3 (4.2), p = 2 4 (4.6), p = 2 3 (4.3), p = 3 3 (4.6), p = 3 3 (4.4), p = 4 3 (4.6), p = 4 3 (4.5), p = 5 3 (4.6), p = 5 3table 2. the number of iterations to reach error tolerance ε = 10−9 with initialguess x0 = (0, 0) and ‖i − f ′1(x0)‖ = 0.1414 < 1. method iterations method iterations (1.2) newton 5 (1.2) newton 5 (4.1), p = 1 7 (4.6), p = 1 9 (4.2), p = 2 5 (4.6), p = 2 7 (4.3), p = 3 5 (4.6), p = 3 6 (4.4), p = 4 5 (4.6), p = 4 6 (4.5), p = 5 5 (4.6), p = 5 5table 3. the number of iterations to reach error tolerance ε = 10−9, where x0 = (−15,−15) and ‖i − f ′1(x0)‖ = 0.257 < 1. table 5 shows the results of calculations to determine the computational order of convergence (coc) and the approximated computational order of convergence (acoc) aiming to compare the convergence order of method (1.5) with the convergence order of newton’s method (1.2). definition 4.1. computational order of convergence of a sequence {xj}j≥0 is defined by https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 16 method iterations method iterations (1.2) newton 7 (1.2) newton 7 (4.1), p = 1 8 (4.6), p = 1 12 (4.2), p = 2 7 (4.6), p = 2 8 (4.3), p = 3 7 (4.6), p = 3 8 (4.4), p = 4 7 (4.6), p = 4 7 (4.5), p = 5 7 (4.6), p = 5 7table 4. the number of iterations to reach error tolerance ε = 10−12, where x0 = (−15,−15) and ‖i − f ′1(x0)‖ = 0.257 < 1. υj = ln |ej+1/ej | ln |ej/ej−1| , where xj−1, xj , xj+1 are three consecutive iterations near the root α and ej = xj − α [6]. definition 4.2. the approximated computational order of convergence of a sequence {xj}j≥0 is defined by υ̂n = ln |êj+1/êj | ln |êj/êj−1| , where êj = xj − xj−1. xj , xj−1, xj−2 are three consecutive iterates [6]. method coc acoc (1.2) newton 1.8624 1.9697 (4.1), p = 1 0.863 1 (4.2), p = 2 0.2695 1.0438 (4.3), p = 3 1.9714 2.3569 (4.4), p = 4 1.8354 1.9453 (4.5), p = 5 1.8642 1.9661 (4.6), p = 1 0.9065 1.0118 (4.6), p = 2 0.5912 0.999 (4.6), p = 3 0.7321 0.9926 (4.6), p = 4 1.933 2.0151 (4.6), p = 5 1.8679 1.9578table 5. the computational order of convergence and the approximated compu-tational order of convergence, where x0 = (−15,−15), ε = 10−12. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 10.28924/ada/ma.5.5 17 table 5 shows that the convergence of the proposed methods closely corresponds with the convergence of newton’s method, particularly for values of k ranging from 4 to 5 with the convergence order closely approximating 2. example 4.2. let x = y = r3 and ω = s[s∗, 1]. the mapping f is defined on ω for a = (a1, a2, a3) tr ∈ r3 as f1(a) = (a1, e a2 − 1, e − 1 2 a23 + a3) tr . then, the definition of the derivative according to fréchet [22,25] is given for the mapping f1 f ′1(a) =  1 0 0 0 ea2 0 0 0 (e − 1)a3 + 1  . the point s∗ = (0, 0, 0)tr solves the equation f1(a) = 0. moreover, f ′1(s∗) = i. the conditions of the theorem 2.2 hold for p = 1, if ϕ(t) = a(t) = (e−1)t. then, we can have r ∈ (0, 0.2909883534). example 4.3. let h[0, 1] stand for the space of continuous functions mapping the interval [0, 1] into the real numbers. let x = y = h[0, 1] and ω = s[s∗, 1] with s∗(v) = 0. the operator f1 is defined on h[0, 1] as f1(z)(v) = z(v)− 4 ∫ 1 0 vz(τ)3dτ. then, of the derivative according to fréchet [1, 10,15,22,30] is given below for the operator f1 f ′1(z(w))(v) = w(v)− 12 ∫ 1 0 vτz(τ)2w(τ)dτ for each w ∈ h[0, 1]. therefore, the conditions of the theorem 2.2 hold if, since for s∗ = 0, f ′1(x ∗(v)) = i hold for that p = 1, if ϕ(t) = a(t) = 6t. then, again by the definition of r, we can choose r ∈ (0, 0.83̄). 5. concluding remarks the paper addresses the issue with the inverses appearing in the study of the convergenceof simple-step iterative methods. it is shown that the inverse can be replaced by a finite sumof linear operators related to the operator involved. the resulting hybrid methods demonstratethe effectiveness of these methods since the number of iterations is essentially the same as wellas the convergence order of the methods. however, the hybrid method is cheaper to implement.this idea can be used for multiple steps and multiple point methods with the same advantages[3, 5–7,9, 14,23,24,31]. this is the direction of future research. acknowledgement: we would like to thank graduate student mr. mykhailo havdiak from the department of optimalprocesses, ivan franko national university of lviv, lviv ukraine for providing example 4.1. https://doi.org/10.28924/ada/ma.5.5 eur. j. math. anal. 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and constrained optimization problems infunction spaces, siam, philadelphia, 2011. https://doi.org/10.28924/ada/ma.5.5 1. introduction 2. convergence for the method (1.5) 3. convergence for the method (1.10) 4. numerical examples 5. concluding remarks acknowledgement: references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 21doi: 10.28924/ada/ma.5.21 new characterization of hardy-fofana spaces and temperature equation martial agbly dakoury1, justin feuto2,∗ 1laboratoire de mathématiques et applications, ufr mathématiques et informatique, université félix houphouët-boigny abidjan-cocody, 22 b.p 582 abidjan 22. côte d’ivoire dakourymartial@gmail.com 2laboratoire de mathématiques et applications, ufr mathématiques et informatique, université félix houphouët-boigny abidjan-cocody, 22 b.p 1194 abidjan 22. côte d’ivoire justfeuto@yahoo.fr ∗correspondence: justfeuto@yahoo.fr abstract. the aim of this paper is to give a characterization of hardy-fofana spaces via riesztransforms. this characterization allows us to describe the distributions belonging to these spacesas a bounded solutions of cauchy-riemann’s general temperature equations. 1. introduction let rd (d is a positive integer) be the euclidean space of dimension d equipped with thelebesgue measure dx and the euclidean norm. the classical hardy space hp(rd) (0 < p < ∞)is defined as the space of tempered distributions f satisfying ‖mf ‖p < ∞, where the maximalfunction mf is defined by mf (x) = sup t>0 |(f ∗ ϕt)(x)|, (1.1) with ϕ in the schwartz class s(rd) having non vanish integral, and ϕt(x) = t−dϕ(t−1x).it is well known that not only this space does not depends on ϕ, but one can replaced schwartzfunction by poisson kernel in the definition of the maximal function (1.1).in [1], ablé and the second author studied hardy-amalgam spaces h(p,q)(rd) (0 < p, q < ∞)by taking in the above maximal characterization of classical hardy space the wiener amalgamquasi-norm ‖·‖p,q instead of lebesgue’s. received: 14 may 2025. key words and phrases. amalgam spaces, hardy-amalgam spaces, generalized hardy-morrey spaces, calderón-zygmund operators, molecular decomposition. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 2a locally integrable function u belongs to the amalgam space (lp, `q)(rd) if ‖u‖p,q := ∑ k∈zd ‖uχqk‖ q p  1 q <∞, where for k ∈ zd , qk = k + [0, 1)d and χqk stands for the characteristic function of qk .multiple characterizations of h(p,q)(rd) spaces including atomic and poisson kernel character-ization, were given in [1]. we notice that the atoms in this context are exactely the one used inclassical hardy space.recently, assaubay et al in [3] characterized this spaces by using first-order classical riesztransforms and composition of first-order riesz transformations. they also describe the distributionsinh(p,q)(rd) as the boundary values of solutions of harmonic and caloric cauchy-riemann systems.here we intend to prove that similar characterizations are possible in the context of hardy-fofanaspaces.it is well known that for 0 < p,α, q <∞ and r > 0, there exists a constant cr ;α > 0 dependingon r and α such that c−1 r ;α‖u‖p,q ≤ ‖stαr u‖q,p ≤ cr ;α‖u‖p,q, u ∈ (lp, `q)(rd), (1.2) where (stαr u)(x) = r− d α u(r−1x). it follows from the above relation that for u ∈ (lp, `q)(rd),we have stαr u ∈ (lp, `q)(rd) for α > 0 and r > 0. unfortunately, the family {stαr u}r>0 is notbounded in (lp, `q)(rd). ibrahim fofana considered in [7], the spaces (lp, `q)α(rd) defined for 0 < p, q, α ≤ ∞ by (lp, `q)α(rd) = { f ∈ (lp, `q)(rd)/ ‖f ‖p,q,α <∞ } where ‖f ‖p,q,α := sup r>0 ‖stαr f ‖p,q . (1.3) these spaces known as fofana’s spaces are non trivial if and only if p ≤ α ≤ q (see [7]). in therest of the paper we will always assume that this condition is fulfilled. it is proved in [6] that for u ∈ (lp, `q)α(rd), we have ‖stαr u‖p,q,α = ‖u‖p,q,α and that (lp, `q)α(rd) (1 ≤ p ≤ α ≤ q) is thebiggest norm space which is continuously embedded in (lp, `q)(rd) and for which the translation stαr is an isometry. these spaces can also be viewed as some generalized morrey spaces since for p < α, the space (lp, `∞)α(rd) is exactly the classical morrey space lp,d pα (rd).for 0 < p ≤ α ≤ q <∞, hardy-fofana space h(p,q,α)(rd), introduced by the authors in [4], isa subspace of hardy-amalgam spaces consists of tempered distributions f satisfying ‖f ‖h(p,q,α) := ‖mf ‖p,q,α <∞. https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 3the purpose of this article is twofold. we first characterize these spaces via riesz transforms andsecondly, we describe the distributions belonging to these spaces as bounded solutions of certaingeneral temperature equations of cauchy-riemann.this paper is organized as follow:the next section is devoted to the prerequisites on hardy-fofana spaces. in section 3, wegive the characterizations of hardy-fofana spaces with riesz transforms. in the last section,we characterize distributions belonging to our spaces as bounded solutions of certain generaltemperature equations of cauchy-riemann.in this work, s := s(rd) will denote the schwartz class of rapidly decreasing smooth functionsequipped with its usual topology. the dual space of s is the space of tempered distributionsdenoted by s ′ := s ′(rd). the pairing between s ′ and s is denoted by 〈·, ·〉.we denote by |e|, the lebesgue measure of a measurable subset e of rd . the notation a ≈ bmeans that there exist two constants 0 < c1 and 0 < c2 such that a ≤ c1b and b ≤ c2a, while a := b means that b is the definition of a. 2. prerequisites for hardy-fofana spaces fofana’s spaces have among others, the following properties (see for example [6] and [7]):(1) let 0 < p,α, q ≤ ∞. the space ((lp, `q)α(rd), ‖·‖p,q,α ) is a banach space if 1 ≤ p ≤ α ≤ q and a quasi-banach space if 0 < p < 1;(2) if α ∈ {p, q} then (lp, `q)α(rd) = lα(rd) with equivalent norms;(3) if p < α < q then lα(rd) ( (lp, `q)α(rd) ( (lp, `q)(rd);(4) let f and g be two measurable functions on rd . if |f | ≤ |g|, then ‖f ‖p,q,α ≤ ‖g‖p,q,α.for many operators including the maximal hardy-littlewood operator, norm inequalities aregiven in these spaces for 1 ≤ p ≤ α ≤ q.let f be a locally integrable function and m(f ) be the centered hardy-littlewood maximalfunction defined by m(f )(x) := sup r>0 |b(x, r)|−1 ∫ b(x,r) |f (y)|dy, ∀ x ∈ rd . it is proved in [6, proposition 4.2] that m is bounded on (lp, `q)α(rd), whenever 1 < p ≤ α ≤ q ≤ ∞. using [8, proposition 11.12], it is easy to extablish the following result whose proof is omitted. proposition 2.1. let 1 < p ≤ α ≤ q < +∞ and 1 < u ≤ +∞. for all sequences {fn}n≥0 of measurable functions, we have∥∥∥∥∥∥∥ ∑ n≥0 |m(fn)|u  1 u ∥∥∥∥∥∥∥ p,q,α ≈ ∥∥∥∥∥∥∥ ∑ n≥0 |fn|u  1 u ∥∥∥∥∥∥∥ p,q,α , with the equivalence constants not depending on the sequence {fn}n≥0. https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 4as hardy-fofana spaces are concerned, we have among others, the following properties whichcan be found in [4]. proposition 2.2. let 1 ≤ p ≤ α ≤ q <∞.(1) if 1 < p then the space h(p,q,α)(rd) and (lp, lq)α(rd) are equal with equivalence norms.(2) the space h(1,q,α)(rd) is continuously embedded in (l1, `q)α(rd). notice that for p < 1, we have as in the classical hardy and hardy-amalgam spaces, that thespaces (h(p,q,α)(rd), ‖ · ‖h(p,q,α) ) are quasi-banach and for f , g ∈ h(p,q,α)(rd), ‖f + g‖ph(p,q,α) ≤ ‖f ‖ p h(p,q,α) + ‖g‖ph(p,q,α) . we can also define (see [5]) these spaces as subspaces of hardy-amalgam spaces for which thefamilly of dilations {stαρ } ρ>0 is locally bounded.more precisely, for a tempered distribution f , ρ > 0 and α two real numbers we put〈 stαρ f , ϕ 〉 := 〈 f ,stα ′ ρ−1ϕ 〉 , where 1 α′ + 1 α = 1. we have (see [4]) that for 0 < p ≤ α ≤ q ≤ ∞, ‖f ‖h(p,q,α) = sup ρ>0 ‖stαρ f ‖h(p,q) . (2.1) just as hardy-amalgam spaces was characterized in [1] with poisson kernel, so are hardy-fofana’sspaces. in fact, a tempered distribution f belonging to hardy-amalgam spaces is bounded; i.e f ∗ ψ ∈ l∞(rd) for all ψ ∈ s(rd). a convolution of such distribution with integrable functionscan be defined in term of distribution. more precisely, if f ∈ s ′(rd) is bounded and u ∈ l1(rd),then the convolution f ∗ u is defined as a tempered distribution acting on s(rd) by the pairing 〈f ∗ u, ϕ〉 := 〈f ∗ ϕ̃, ũ〉(l∞,l1) ϕ ∈ s(rd) where ũ(x) = u(−x) and 〈f ∗ ϕ̃, ũ〉(l∞,l1) is the pairing between l∞(rd) and l1(rd). but if wetake as u the poisson kernel p defined by p (x) := γ(d+1 2 ) π d+1 2 1 (1 + |x |2) d+1 2 x ∈ rd , then f ∗ pt can be identified for all t > 0, to a well defined bounded function. as we can see forexample in [9], there exist ϕ,ψ ∈ s(rd) such that f ∗ pt = (f ∗ ϕt) ∗ pt + f ∗ ψt for t > 0. it is proved in [1] that for an element f ∈ h(p,q)(rd), we have ‖x 7→ sup t>0 sup |x−y |<t |f ∗ pt(y)|‖p,q ≈ ‖mf ‖p,q (2.2) https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 5where mf is the maximal function defined in relation (1.1). it follows that ‖x 7→ sup t>0 sup |x−y |<t |f ∗ pt(y)|‖p,q,α ≈ ‖mf ‖p,q,α, (2.3) thanks to relations (2.1) and (2.2) and the fact that stαρ commute with the maximal function m. lemma 2.3. let f ∈ s ′(rd), ϕ ∈ s(rd), ρ and α positive real numbers. we have stαρ (f ∗ ϕt) = ( stαρ f ) ∗ ϕρt , t > 0. infact, ρ −d α (f ∗ ϕt) (ρ−1x) = ρ −d α 〈 f , ρdϕρt(x − ρ·) 〉 = ρ d α′ 〈f , ϕρt(x − ρ·)〉 = 〈 f ,stα ′ ρ−1 [ϕρt(x − ·)] 〉 = ( stαρ f ∗ ϕρt ) (x). lemma 2.4. let f ∈ s ′(rd), ϕ ∈ s(rd), ρ and α positive real numbers. we have stαρ [(f ∗ ϕt) ∗ pt ] = ( stαρ f ∗ ϕρt ) ∗ pρt . (2.4) relation (2.4) follows from the fact that ρ −d α ( f ∗ ϕρ−1t ) ∗ pρ−1t(ρ −1x) = ρ −d α ∫ rd ( f ∗ ϕρ−1t ) (ρ−1x − y)pρ−1t(y)dy = ∫ rd 〈 f , ρ d α′ϕt(x − z − ρ·) 〉 pt(z)dz = ∫ rd 〈 stαρ f , ϕt(x − z − ·) 〉 pt(z)dz = ∫ rd ( stαρ f ∗ ϕt ) (x − z)pt(z)dz = ( stαρ f ∗ ϕt ) ∗ pt(x). it comes from lemma 2.3 and 2.4 that for a bounded tempered distribution f and u(x, t) = f ∗pt(x),( stαρ u ) (x, t) = [( stαρ f ) ∗ pt ] (x), ρ > 0 and α > 0 (2.5) for all t > 0. 3. cauchy-riemann equations, riesz transforms and hardy-fofana spaces let u be a harmonic function on rd+1 + ; i.e, u ∈ c2(rd+1 + ) and ∆u := ∑d+1 j=1 ∂2u (∂xj )2 = 0, where xd+1 = t and rd+1 + := rd×]0,+∞[. we define its non tangential maximal function u∗ by u∗(x) := sup t>0 sup |x−y |<t |u(y , t)| ∀x ∈ rd . (3.1) let f be a bounded tempered distribution, and u(x, t) = pt ∗ f (x). as we can see in [4], u∗ ∈ (lp, `q)α (rd) whenever f ∈ h(p,q,α)(rd). we give in the next result a necessary and https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 6sufficient conditions for a harmonic function u in rd+1 to have its non tangential maximal func-tion in (lp, `q)α(rd). the proof is based on the dilation characterization of hardy-fofana spacesand [3, proposition 2.1 ]. proposition 3.1. let 0 < p ≤ α ≤ q < +∞ and u an harmonic function on rd+1 + . the maximal function u∗ belongs to (lp, `q)α(rd) if and only if there exists f ∈ h(p,q,α)(rd) such that u(x, t) := f ∗ pt(x), (x, t) ∈ rd+1 + . moreover, ‖f ‖h(p,q,α) ≈ ‖u∗‖p,q,α . proof. let u be an harmonic function on rd+1 + , and u∗ the associate non tangential maximal functionas defined in relation (3.1).we suppose that there exists f ∈ h(p,q,α)(rd) such that u(x, t) := f ∗pt(x) for all (x, t) ∈ rd+1 + .from the poisson characterization of hardy-fofana spaces (see [4, theorem 2.3.8 ]), we deducethat ‖u∗‖p,q,α ≤ c ‖f ‖h(p,q,α) .for the converse, let us suppose that u∗ ∈ (lp, `q)α(rd) ⊂ (lp, `q)(rd). it comes from [3,proposition 2.1 ] that there exists f ∈ h(p,q)(rd) and a constant c > 0 such that u(x, t) = (f ∗ pt)(x), (x, t) ∈ rd+1 + (3.2) and 1 c ‖f ‖h(p,q) ≤ ‖u∗‖p,q ≤ c‖f ‖h(p,q) .since stαρ f ∈ h(p,q)(rd) for all ρ > 0, stαρ u harmonic on rd+1 + and (stαρ u ) (x, t) = (stαρ f )∗pt(x),it comes that (stαρ u)∗ ∈ (lp, lq)(rd) and 1 c ‖stαρ f ‖h(p,q) ≤ ‖(stαρ u)∗‖p,q ≤ c‖stαρ f ‖h(p,q) . this relation being thrue for all ρ > 0, we have 1 c ‖f ‖h(p,q,α) ≤ ‖u∗‖p,q,α ≤ c‖f ‖h(p,q,α) , where we use the trivial identity (stαρ u)∗ = stαρ u ∗, ρ > 0 and 0 < α <∞. � we say that a vector values function f := (u1, u2, ..., ud+1), with uj : rd+1 + → r, j ∈ {1, 2, ..., d+ 1}, satisfies the generalized cauchy-riemann equations (in short f ∈ cr(rd+1 + )) if ∂uj ∂xk = ∂uk ∂xj , 1 ≤ j, k ≤ d + 1 and d+1∑ j=1 ∂uj ∂xj = 0, (3.3) where we set xd+1 = t. also recall that for j ∈ {1, 2, ...d}, the j-th riesz transform rj(g) of ameasurable function g is formally defined by rj(g)(x) := lim ε→0+ ∫ |x−y |>ε kj(x − y)g(y)dy a.e x ∈ rd . https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 7 where kj(x) := γ( d+1 2 ) π d+1 2 xj |x |d+1 , x ∈ rd\{0}.in [2, corollary 4.19], ablé and feuto demonstrated that riesz transformations are extendableinto bounded linear operators on hardy-amalgam spaces h(p,q)(rd) for 0 < p ≤ 1. we will keepthe notations rj , j = 1, · · · , d for these extentions. assaubay et al proved the following result. proposition 3.2 ( [3], proposition 2.3). let d−1 d < min {p, q} < +∞. suppose that u is harmonic function in rd+1 + . then u∗ ∈ (lp, `q)(rd) if and only if there exists an harmonic vector f := (u1, ..., ud+1) ∈ cr(rd+1 + ) such that ud+1 := u and supt>0 ‖|f (., t)|‖p,q < +∞. furthermore, supt>0 ‖|f (., t)|‖p,q ≈ ‖u∗‖p,q . since u∗ ∈ (lp, `q)(rd) if and only if u = f ∗ pt for some f ∈ h(p,q)(rd), they proved that onecan take uj(x, t) = rj(f ) ∗ pt(x), j = 1, · · · , d .in the case of hardy-fofana’s spaces, we have the following. proposition 3.3. assume that d−1 d < p ≤ α ≤ q < +∞ and u is an harmonic function in rd+1 + . then u∗ ∈ (lp, `q)α(rd) if and only if there exists an harmonic vector f := (u1, ..., ud+1) ∈ cr(rd+1 + ) such that ud+1 := u and supt>0 ‖|f (., t)|‖p,q,α < +∞. furthermore sup t>0 ‖|f (., t)|‖p,q,α ≈ ‖u∗‖p,q,α (3.4) proof. let d−1 d < p ≤ α ≤ q < +∞ and u an harmonic function on rd+1 + .we suppose that u∗ ∈ (lp, `q)α(rd). since (lp, `q)α(rd) ⊂ (lp, `q)(rd), proposition 3.2assert that there exists f ∈ (lp, `q)(rd) so that: • u(x, t) = f ∗ pt(x), • the harmonic vector f = (u1, · · · , ud+1) with uj(x, t) = rj(f ) ∗ pt(x) for j ∈ {1, · · · , d}and ud+1 = u belongs to cr(rd+1 + ), • supt>0 ‖|f (·, t)|‖p,q ≈ ‖u∗‖p,q.since u∗ ∈ (lp, `q)α (rd) we have that the tempered distribution f belongs to h(p,q,α)(rd), thanksto proposition 3.1. all we have to prove now is that x 7→ f (x, t) belongs to (lp, `q)α(rd) for t > 0 and that relation (3.4) is satisfies.fix t > 0 and ρ > 0. since u∗ ∈ (lp, `q)α (rd) and (stαρ u )∗ = stαρ (u∗), we have that for ρ > 0, ‖ ( stαρ u )∗ ‖p,q ≤ ‖u∗‖p,q,α. hence (stαρ u )∗ ∈ (lp, `q) (rd) so that there exists fρ ∈ h(p,q)(rd)satisfying (stαρ u)(x, t) = (fρ ∗ pt)(x),with fρ(x, t) := (r1(fρ) ∗ pt(x), · · · ,rd(f ρ) ∗ pt(x), (fρ) ∗ pt)(x)) (3.5)belonging to cr+(rd+1 + ) and sup t>0 ‖|fρ(·, t)|‖p,q ≈ ‖stαρ (u∗)‖p,q. (3.6) https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 8moreover (stαρ u)(x, t) = (stαρ f )∗pt(x), thanks to relation (2.5). it follows that fρ∗pt = (stαρ f )∗ptfor all t > 0 so that fρ = stαρ f . we recall that the last equality comes from the fact that for f ∈ h(p,q)(rd), f ∗ pt tends to f in s ′(rd) as t goes to 0. replacing fρ by stαρ f in relation (3.5)yields fρ(x, t) = ( r1(stαρ f ) ∗ pt(x), · · · ,rd(stαρ f ) ∗ pt(x), (stαρ f ) ∗ pt)(x) ). since the operator stαρ commute with rj we have that fρ(·, t) = ( stαρ (r1f ) ∗ pt(·), · · · ,stαρ (rd f ) ∗ pt(·), (stαρ f ) ∗ pt)(·) ) = ( stαρ ( u1(·, ρ−1t) ) , · · · ,stαρ ( ud(·, ρ−1t) ) ,stαρ ( ud+1(·, ρ−1t) )) = stαρ ( f (·, ρ−1t) ) . if we take this expression of fρ in relation (3.6) we obtain that sup t>0 ‖|stαρ (f (·, ρ−1t)|‖p,q ≈ ‖stαρ (u∗)‖p,q. but supt>0 ‖|stαρ (f (·, ρ−1t))|‖p,q = supt>0 ‖|stαρ (f (·, t))|‖p,q and the result follow from the defi-nition of hardy-fofana space. � the next result gives a characterization of h(p,q,α)(rd) via riesz transforms rj(f ∗ φ). sincewe need to use the characterization of h(p,q)(rd) given in [3], we give the following definition. definition 3.4. let 0 < p ≤ α ≤ q <∞. a tempered distribution f is said to be : • (p, q)-restricted at infinity if there exists µ0 ≥ 1 such that for µ ≥ µ0, we have f ∗ φ ∈ (lpµ, `qµ)(rd), φ ∈ s(rd). • (p, q, α)-restricted at infinity if there exists µ0 ≥ 1 such that for µ ≥ µ0, we have f ∗ φ ∈ (lpµ, `qµ)αµ(rd), φ ∈ s(rd). it is easy to see that tempered distributions which are (p, q, α)-restricted for p ≤ α ≤ q, are also (p, q)-restricted. theorem 1.1 in [3] assert that a tempered distribution f belongs to h(p,q)(rd)for d−1 d < min (p, q) <∞, if and only if it is (p, q)-restricted at infty and, for φ ∈ s(rd) with nonvanish integral, sup t>0 ‖f ∗ φt‖p,q + d∑ j=1 ‖(rj f ) ∗ φt‖p,q  <∞. when this is the case, ‖f ‖h(p,q) ≈ sup t>0 ‖f ∗ φt‖p,q + d∑ j=1 ‖(rj f ) ∗ φt‖p,q  . in the case of hardy-fofana space, we have the following result. https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 9 theorem 3.5. let d−1 d < p ≤ α ≤ q < ∞, f ∈ s ′(rd).then f ∈ h(p,q,α)(rd) if and only if f is (p, q, α)-restricted at infinity and, for φ ∈ s(rd) with non vanish integral, sup t>0 ‖f ∗ φt‖p,q,α + d∑ j=1 ∥∥(rj f ) ∗ φt ∥∥ p,q,α  < +∞. (3.7) moreover, ‖f ‖h(p,q,α) ≈ sup t>0 ‖f ∗ φt‖p,q,α + d∑ j=1 ∥∥(rj f ) ∗ φt ∥∥ p,q,α  . (3.8) proof. let d−1 d < p ≤ α ≤ q <∞ and f ∈ s ′(rd).we suppose that f is (p, q, α)-restricted at infinity and satisfies (3.7) for non vanishing schwartzfunction φ. there exists µ0 > 1 (large enought) such that for µ > µ0, we have f ∗ φ ∈ (lpµ; `qµ)αµ (rd), φ ∈ s(rd). (3.9) it comes from the definition of fofana spaces that stαµρ (f ∗ φ) ∈ (lpµ, `µq) (rd) φ ∈ s(rd), ρ > 0. taking ρ = 1, we obtain that f is (p, q)-restricted at infinity. since for all φ ∈ s(rd) with nonvanishing integral we also have that a = sup t>0 sup ρ ∥∥stαρ (f ∗ φt) ∥∥ p,q + d∑ j=1 sup ρ>0 ∥∥stαρ ( (rj f ) ∗ φt )∥∥ p,q  <∞, it follows that sup t>0 ‖f ∗ φt‖p,q + d∑ j=1 ∥∥(rj f ) ∗ φt ∥∥ p,q  ≤ a. thus f ∈ h(p,q)(rd) thanks to [3, theorem 1.1]. it remains to prove that the familly {stαρ f } ρ>0 isuniformly bounded in h(p,q)(rd).fix ρ > 0. we have stαρ f ∈ h(p,q)(rd) so that ‖stαρ f ‖h(p,q) ≈ sup t>0 ∥∥stαρ (f ) ∗ φt ∥∥ p,q + d∑ j=1 ∥∥rj(stαρ f ) ∗ φt ∥∥ p,q  thanks once more to [3, theorem 1.1]. but we have in one hand that rj(f ) ∗ φt = rj (f ∗ φt), so that stαρ [( rj f ) ∗ φt ] = stαρ [ rj (f ∗ φt) ] = rj [ stαρ (f ∗ φt) ] , (3.10) where the last equality comes from the fact that dilation comute with riesz transforms. in the otherhand we have that sup t>0 ‖stαρ (f ∗ φt) ‖p,q = sup t>0 ‖stαρ (f ) ∗ φt‖p,q, (3.11) https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 10thanks to lemma 2.3. therefore, we have sup ρ>0 sup t>0 ‖stαρ (f ) ∗ φt‖p,q = sup ρ>0 sup t>0 ‖stαρ (f ∗ φt) ‖p,q ≤ a and sup ρ>0 sup t>0 d∑ j=1 ‖rj [ stαρ (f ∗ φt) ] ‖p,q = sup ρ>0 sup t>0 d∑ j=1 ‖stαρ [( rj f ) ∗ φt ] ‖p,q ≤ a. we deduce that supρ>0 ‖stαρ f ‖h(p,q) <∞, which prove that f ∈ h(p,q,α)(rd).for the converse, we suppose that f ∈ h(p,q,α)(rd). it follows that stαρ f ∈ h(p,q)(rd) with ‖stαρ f ‖h(p,q) ≤ ‖f ‖h(p,q,α) < ∞ for all ρ > 0. it comes from [3, theorem 1.1] that stαρ f is (p, q)-resticted at infinity and ‖stαρ f ‖h(p,q) ≈ sup t>0 ∥∥stαρ (f ) ∗ φt ∥∥ p,q + d∑ j=1 ∥∥stαρ (rj f ) ∗ φt ∥∥ p,q  for all φ ∈ s(rd) with non vanish integral.from relations (3.11) and (3.10), and the definitions of ‖ · ‖p,q,α and of ‖ · ‖h(p,q,α) , we have that ‖f ‖h(p,q,α) ≈ sup t>0 ( ‖f ∗ φt‖p,q,α + d∑ j=1 ∥∥(rj f ) ∗ φt ∥∥ p,q,α ) <∞. let φ ∈ s(rd). we have ‖f ∗ φ‖p,q,α ≤ c ‖f ‖h(p,q,α) . for µ ≥ 1 we have f ∗ φ ∈ (lpµ, `qµ)αµ.in fact assuming that ‖f ∗ ϕ‖∞ 6= 0 we have f ∗ φ ∈ (lp, `q)α and ‖f ∗ φ‖pµ,qµ,αµ ≤ c ‖f ∗ φ‖1− 1 µ ∞ ‖f ∗ φ‖ 1 µ p,q,α and then f is (q, p, α)-restricted at infinity. � 4. temperature cauchy-riemann equations and hardy-fofana spaces a vector f = (u1, u2, · · · , ud+1) of functions in rd+1 + satisfy the generalized temperature cauchy-riemann equations, if it satisfies the following conditions :(1) ∑d j=1 ∂uj ∂xj = i∂ 1/2 t ud+1(2) ∂uj ∂xk = ∂uk ∂xj for j, k = 1, 2, · · · , d(3) ∂ud+1 ∂xj = −i∂1/2 t uj , j = 1, 2, · · · , d , with (∂ 1/2 t g)(t) := e iπ/2 √ π ∫ ∞ t g′(s)√ s − t ds, t > 0 when g is a smooth enough function on (0,∞)in [3], the authors defined the space hp,q(rd+1 + ) (0 < p, q < ∞) as the vector space of vectorfunctions f = (u1, u2, · · · , ud+1) satisfying generalized temperature cauchy-riemann equationsand such that ‖f‖hp,q(rd+1 + ) := sup t>0 ‖|f (·, t)|‖p,q <∞. https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 11 they also proved that under appropriate conditions on the exponents p and q, the space hp,q(rd+1 + )is topologically isomorphic to hp,q(rd). to carry out the proof of this result, they use a subspaceof what they call the temperature space t (rd+1 + ); that is the space of functions u ∈ c2(rd+1 + ),satisfying ∂u ∂t = d∑ j=1 ∂2u ∂x2 j in rd+1 + . more precisely, for 0 < p, q <∞, they put t p,q(rd+1 + ) := { u ∈ t (rd+1 + ) : ||u||t (p,q) <∞ } where ||u||t (p,q) := sup t>0 ||u(., t)||q,p. they proved [3, proposition 3.2 (i)] that for d−1 d < p, q <∞, f = (u1, u2, · · · , ud+1) ∈ hp,q(rd+1 + )implies that u := ud+1 ∈ t p,q(rd+1 + ) and uj(·, t) = rj(u(·, t)), t > 0, j = 1, · · · , d .we claim that for 0 < p ≤ α ≤ q < ∞ and r > 0, the space t p,q(rd+1 + ) is stable under thedilation stαr . this is due to the fact that for f ∈ (lp, `q) (rd), there exists a constant c(α, r, p, q) > 0 such that c(α, r, p, q)−1‖f ‖p,q ≤ ‖stαr f ‖p,q ≤ c(α, r, p, q)‖f ‖p,q,and this dilation commute with riesz transforms. it follows that if f = (u1, · · · , ud+1) ∈ hp,q(rd+1 + )then stαr f ∈ hp,q(rd+1 + ).we put ‖f‖h(p,q,α) := sup r>0 ‖stαr f‖hp,q(rd+1 + ) and defined the space h(p,q,α)(rd+1 + ) as the subspace of hp,q(rd+1 + ) consits of f satisfying ‖f‖h(p,q,α) <∞. we have the following result in hardy-fofana spaces. theorem 4.1. let d−1 d < p ≤ α ≤ q <∞, and wt the heat kernel defined by wt(x) = e−|x | 2/4t (4πt)d/2 . the map l define on h(p,q,α)(rd) by l(f )(x, t) := ( ((r1f ) ∗wt)(x), · · · , ((rd f ) ∗wt)(x), (f ∗wt)(x) ) for all x ∈ rd and t > 0, is a topological isomorphism from h(p,q,α)(rd) onto h(p,q,α)(rd+1 + ). proof. let f ∈ h(p,q,α)(rd). for r > 0 we have stαr f ∈ hp,q(rd), thanks to the definition of h(p,q,α)(rd). it comes from [3, theorem 1.3] that l(stαr f ) ∈ hp,q(rd+1 + ), with ‖l(stαr f )‖hp,q(rd+1 + ) ≤ c‖stαr f ‖hp,q(rd ) (4.1) https://doi.org/10.28924/ada/ma.5.21 eur. j. math. anal. 10.28924/ada/ma.5.21 12for all r > 0. since rj(f ∗wt) = rj(f ) ∗wt and stαr commute with riesz transforms, we havethat l(stαr f ) = stαr (l(f )), r > 0. thaking this remark in (4.1) we obtain that ‖l(f )‖h(p,q,α)(rd+1 + ) ≤ c‖f ‖h(p,q,α)(rd ).let now f = (u1, u2, · · · , ud , ud+1) belonging to h(p,q,α)(rd+1 + ). this implies that stαr f = (stαr u1,stαr u2, · · · ,stαr ud ,stαr ud+1) ∈ hp,q(rd+1 + ) for all r > 0. as we can see in the proofof [3, theorem 3.1] this implies that for all r > 0, there exists fr ∈ hp,q(rd) so that stαr ud+1 ∈ t p,q(rd+1 + ) with stαr ud+1(x, t) = fr ∗wt(x) and ‖fr‖hp,q ≤ c sup t>0 ‖stαr f (·, t)‖p,q ≤ c‖f‖h(p,q,α) , (4.2) and stαr uj(·, t) = rj(stαr ud+1(·, t)), t > 0, j = 1, · · · , d.we put f := f 1. we have stαr f = fr for all r > 0. taking this in estimate (4.2) yields ‖stαr f ‖hp,q ≤ c‖f‖h(p,q,α) wich prove that f ∈ h(p,q,α)(rd).the vector g(x, t) = ((r1(f ) ∗ pt)(x), · · · ,rd(f ) ∗ pt)(x), (f ∗ pt)(x)), x ∈ rd , t > 0 is harmonic, satisfies the generalized cauchy-riemann equation, and sup t>0 ‖|g(·, t)|‖p,q,α ≤ c‖f‖h(p,q,α) . � references [1] z.v.d.p. ablé and j. feuto, atomic decomposition of hardy-amalgam spaces, j. math. anal. appl. 455 (2017), 1899–1936.[2] z.v.d.p. ablé and j. feuto, duals of hardy amalgam spaces and norm inequalities, anal. math. 45 (2019), 647–686.[3] a.-t. assaubay, j.j. betancor, a.j. castro and j.c. farina, riesz transforms, cauchy-riemann systems, and hardy-amalgam spaces, banach j. math. anal. 3 (2019), 697–725.[4] m.a. dakoury and j. feuto, norm inequality for intrinsic square functions in generalized hardy-morrey spaces,open access libr. j. 9 (2022), e8463.[5] m.a. dakoury and j. feuto, norm inequalities for calderón–zygmund operators in some generalized hardy–morreyspaces, vietnam j. math. (2024). https://doi.org/10.1007/s10013-024-00703-0.[6] j. feuto, norm inequalities in some subspaces of morrey space, ann. math. blaise pascal 21 (2014), 21–37.[7] i. fofana, étude d’une classe d’espaces de fonctions contenant les espaces de lorentz, afrika mat. 2 (1988), 29–50.[8] y. liang, y. sawano, t. ullrich, d. yang and w. yuan, a new framework for generalized besov-type and triebel-lizorkin-type spaces, diss. math. (rozprawy mat.) 489 (2013), 1–114.[9] l. grafakos, modern fourier analysis, 2nd ed., grad. texts math., 250, springer, new york, 2009. https://doi.org/10.28924/ada/ma.5.21 1. introduction 2. prerequisites for hardy-fofana spaces 3. cauchy-riemann equations, riesz transforms and hardy-fofana spaces 4. temperature cauchy-riemann equations and hardy-fofana spaces references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 19doi: 10.28924/ada/ma.5.19 on the cumulative distribution function of the difference of two dependent chi square random variables elias g. saleeby1,∗, anwar h. joarder2, nima rabiei3 1dept. of mathematics, al akhawayn university (aui), ifrane 53000, morocco esaleeby@yahoo.com, e.saleeby@aui.ma 2dept. of mathematics, al akhawayn university (aui), ifrane 53000, morocco ajstat@gmail.com 3dept. of engineering and natural sciences, international university of sarajevo (ius), sarajevo, bih nrabiei@ius.edu.ba ∗correspondence: esaleeby@yahoo.com, e.saleeby@aui.ma abstract. in this article, we reexamine the derivation of the cumulative distribution function of thedifference of two dependent chi-square random variables with the same degrees of freedom. we derivethe cdf for this difference for even degrees of freedom and discuss a discrepancy that we have foundwith a reported cdf of this difference for even degrees of freedom in [6]. for odd degrees of freedom,an expression for the cdf seems to be unknown. in this case, we derive a representation of the cdfin terms of the meijer g-function. these representations allowed us to compute percentiles for evenand odd degrees of freedom. 1. introduction in the algebra of random variables, finding the probability density function (pdf) and the cumulativedistribution function (cdf) of the difference and the sum of two random variables (rvs) are standardproblems. it is well known that such combinations of rvs appear within the theory of statisticsand in its applications. it is also clear that when the two rvs are dependent, the analysis of theproblem is more technically complicated. in particular, in this note we focus mainly on deriving thecdf for the difference of two dependent central chi-square random variables with the same numberof degrees of freedom. results on this problem seem to have been around for a while and arereported in some detail in [6]. not aware initially of the results in [6], we have carried out the basicanalysis and derived the cdf. the cdf expressions which we have obtained appear in different formsthan those reported in [6]. in an attempt to see how these different representations correspond,we discovered a discrepancy between the two forms of the cdfs. in the analysis below we give received: 2 apr 2025. key words and phrases. cumulative distribution; chi square; difference of random variables.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 2a modification to the cdf expression for even degrees of freedom reported in [6]. the cdf for thedifference of dependent central chi square rvs with odd degrees of freedom seems to be unknown,and it is not reprted in [6]. a representation of the cdf in terms of a meijer g-functions seems to fillin this gap. the article is organized as follows. we first discuss a bivariate chi square distributionof the kibble-type on which the rest of the analysis is based. then we present and discuss thepdf of the difference of two dependent chi square rvs, followed by a derivation of the cdf of thisdifference. we end the article by computing a sampling of percentiles from the cdf expressions forboth even and odd degrees of freedom. 2. density functions in this section, we examine a joint pdf for correlated gamma rvs derived by w.f. kibble in [2], obtainfrom it the joint pdf of two dependent central chi square rvs, forms an initial reference point forfurther analysis. this joint pdf turns out to be identical to the joint pdf reported in [6]. we thenobtain the pdf for the difference of two dependent central chi square random variables and comparewith the piecewise given expressions of the pdf reported in [6].given a vector (x1, · · · , xn) of gaussian rv’s with zero mean, then for n > 1, y = ∑ limn i=1x 2 iis a central chi square rv with n degrees of freedom. for simplicity, we assume that the rvs xi arestandard normal. take two such vectors y1 and y2., their joint probability density function (pdf) isgiven by (e.g., see [6], p. 21) p y1,y2 (y1, y2) = (y1y2) 1 2 ( n2−1) 4γ ( n 2 ) (1− ρ2) (2 |ρ|) n 2 −1 exp ( − y1 + y2 2 √ 1− ρ2 ) i n 2 −1 ( |ρ| √y1y2 1− ρ2 ) , (1) where y1, y2 ≥ 0, −1 < ρ < 1, ρ 6= 0 (if n > 2), and iν is the modified bessel function of the firstkind of order ν. as the lim ρ→0 i n 2 −1 ( |ρ|√y1y2 1−ρ2 ) |ρ| n 2 −1 = (y1y2) n−2 4 2 n 2 −1γ ( n 2 ) , the joint pdf reduces to the product of two univariate chi square pdfs. remark 1. recall that the pdf of the unscaled univariate gamma distribution is given by f (x ;α, β) = xα−1e − x β γ(α)βα , where α > 0, β > 0, and x ≥ 0. putting α = n 2 , β = 2, one obtains the univariate chi square distribution as a special case. it is well known that there are different variants of the bivariate gamma pdf’s in the literature (e.g., see [3], ch. 48). among the earliest bivariate pdfs for the gamma distribution is that of kibble [2] derived for the case β = 1 (scaled gamma’s). adjusting kibble’s bivariate gamma for the scale parameter β = 2, with the shape parameter α = n 2 , one obtains the pdf given in (1) which is a kibble type bivariate chi square pdf. adjusting the scale parameter, one would obtain the pdf with σ2 1 6= 1 6= σ2 2. consider now the difference w of two dependent central chi square rvs, x, y, each having a n = 2m, m > 1, degrees of freedom (using the notation in [6]). then it is reported in [6] that the rvw = x−y https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 3has the following piecewise defined pdf pw (w) = |w |m−1 (m − 1)!22m (1− ρ2) m 2 exp ( w 2 √ 1− ρ2 ) m−1∑ i=0 (m + i − 1)! i ! (m − i − 1)! (√ 1− ρ2 |w | )i , w < 0 pw (w) = |w |m−1 (m − 1)!22m (1− ρ2) m 2 exp ( − w 2 √ 1− ρ2 ) m−1∑ i=0 (m + i − 1)! i ! (m − i − 1)! (√ 1− ρ2 |w | )i , w ≥ 0 using the following expansion of the macdonald function [8] km− 1 2 (w) = ( π 2w ) 1 2 e−w m−1∑ i=0 (m + i − 1)! i ! (m − i − 1)! (2w)i , and k−ν(w) = kν(w), the pdf given in [6] for w can be written as pw (w) = |w |m− 1 2 22m √ π (1− ρ2) 2m+1 4 (m − 1)! km− 1 2 ( |w | 2 √ 1− ρ2 ) , w 6= 0, and, pw (0) := limw→0 pw (w), which evaluates to γ(m− 1 2 ) 4 √ π √ 1−ρ2(m−1)! . the simplest definition of the macdonald function is k m− 1 2 (w) := π(−1)k−1 2 ( i−m+ 1 2 (w)− im− 1 2 (w) ) , m integer, (amongother names of kν , it is often also called the modified bessel function of the second kind (like inmathematica)). replacing the m used in ( [6], p. 29 ) by n 2 , where n now is the number of degreesof freedom of the rvs x and y, then followed by replacing n by m, we obtain fw (w) = |w | m−1 2 2m √ π (1− ρ2) m+1 4 γ ( m 2 )km−1 2 ( |w | 2 √ 1− ρ2 ) , w ∈ r\{0}. (2) and fw (0) := γ(m−1 2 ) 4 √ π √ 1−ρ2γ(m2 ) . clearly fw (w) = fw (−w) . for m = 1, fw (w) = 1 2π √ 1−ρ2 k0 ( |w | 2 √ 1−ρ2 ) . for m = 2, as k 1 2 (z) = ( π 2z ) 1 2 e−z , then fw (w) = 1 4 √ 1−ρ2 e − |w | 2 √ 1−ρ2 .these cases match with equations (4.20) and (4.23) given in ( [6], p. 29). for odd degrees offreedom, it is reported in ( [6], p. 30), using his notation for n = n1 = n2 = 2m + 1, and for σ1 = σ2 = 1, that the pdf is given by pw (w) = |w |m √ πγ ( m + 1 2 ) (1− ρ2) m+1 2 km ( |w | 2 √ 1− ρ2 ) . (3) writing this in terms of n and setting n = m to adjust back to our notation, we see that (3) matcheswith (2). figure 1 shows a plot of fw (w) . 3. cumulative distribution functions in this section, we derive representations of the cdf of w for even and odd degrees of freedom. inthe case of even degrees of freedom, we compare our result with the cdf expression reported in [6].let c = 16 ( 1− ρ2 ) . https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 4 figure 1. the pdf fw (w) for m = 10, 15, 20, and ρ = 0.7. theorem 1. if the symmetric pdf of w is given by (2), then the cdf of w for w > 0 is given by fw (w) = ∫ 0 −∞ fw (w) dw + ∫ w 0 fw (s) ds = 1 2 + j. (4) for m an even positive integer > 3, j is given by j = γ ( m−1 2 ) w 4γ ( m 2 )√ π √ 1− ρ2 1f2 ( 1 2 ; 3−m 2 , 3 2 ; w2 c ) + γ ( 1−m 2 ) wm 22mmγ ( m 2 )√ π (1− ρ2) m 2 1f2 ( m 2 ; m + 1 2 , m + 2 2 ; w2 c ) , where −1 < ρ < 1, 1f2 (w) = ∑∞ k=0 (a)k (b1)k(b2)k wk k! , and (γ)k = γ(γ+k) γ(γ) , b1, b2 6= 0,−1,−2, · · · . proof. first, observe that by the symmetry of the pdf in (2), i = 1 2 . to evaluate j, we use formula03.04.21.0014.01 in [9], namely,∫ zνkν (z) dz = πz csc (νπ) 2ν+2  4ν √ π 1f̃2 ( 1 2 ; 1− ν, 3 2 ; z 2 4 ) −z2νγ ( ν + 1 2 ) 1f̃2 ( ν + 1 2 ; ν + 1, ν + 3 2 ; z 2 4 )  , where 1f̃2 (a; b1, b2; z) := 1f2 (a; b1, b2; z) / (γ (b1) γ (b2)) is the regularized generalized hyper-geometric function (rhgf). then, it is straight forward to show that j = π csc ( m−1 2 π ) w 23γ ( m 2 )√ 1− ρ2 1f̃2 ( 1 2 ; 3−m 2 , 3 2 ; w2 c ) − √ π csc ( m−1 2 π ) wm 22m+1 (1− ρ2) m 2 1f̃2 ( m 2 ; m + 1 2 , m + 2 2 ; w2 c ) , where, https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 5 1f̃2 ( 1 2 ; 3−m 2 , 3 2 ; z ) = ∞∑ k=0 1 √ π ( 1 2 + k ) γ ( 3−m 2 + k ) zk k! , 1f̃2 ( m 2 ; m + 1 2 , m + 2 2 ; z ) = ∞∑ k=0 1( m 2 + k ) γ ( m 2 ) γ ( m+1 2 + k ) zk k! ; are the rhgfs defined for all z ∈ c. note that for m even, the limz→0 j = 0 = j (0) .using euler’s reflection formula (γ (1− z) γ (z) = π sinπz , z = m−1 2 , which is not an integer when m is even), we obtain j = γ ( 3−m 2 ) γ ( m−1 2 ) 23γ ( m 2 )√ 1− ρ2 w 1f̃2 ( 1 2 ; 3−m 2 , 3 2 ; w2 c ) − γ ( 3−m 2 ) γ ( m−1 2 ) 22m+1 √ π (1− ρ2) m 2 wm 1f̃2 ( m 2 ; m + 1 2 , m + 2 2 ; w2 c ) . then using γ (1 + z) = zγ (z) , the j given in the theorem follows. � note that the integral i = ∫ w −∞ fw (w) dw for re (w) < 0, and even m > 0, evaluates to i = 1 2 + γ ( m−1 2 ) w 4γ ( m 2 )√ π √ 1− ρ2 1f2 ( 1 2 ; 3−m 2 , 3 2 ; w2 c ) − γ ( 1−m 2 ) (−w)m 22mmγ ( m 2 )√ π (1− ρ2) m 2 1f2 ( m 2 ; m + 1 2 , m + 2 2 ; w2 c ) . (5) figure 2 shows plots for the cdf fw (w) by eq.(4). figure 2. the cdf fw (w) for m = 4, 16, and ρ = 0.7, 0.9. the cdf for the difference w of two dependent central chi square random variables with 2m degreesof freedom is reported in ( [6], p. 30) (for σ2 1 = σ2 2 = 1) as https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 6 pw (w) =  (1−ρ2) m 2 22m(m−1)! exp ( w 2 √ 1−ρ2 )∑m−1 i=0 ∑m−i−1 l=0 (m+i−1)!(1−ρ2) i+l+1 2l+1 i!(m−i−l−1)!l! (−w)m−i−l−1 , w < 0 1− (1−ρ2) m 2 22m(m−1)! exp ( − w 2 √ 1−ρ2 )∑m−1 i=0 ∑m−i−1 l=0 (m+i−1)!(1−ρ2) i+l+1 2l+1 i!(m−i−l−1)!l! (w)m−i−l−1 , w ≥ 0  comparing this cdf with the cdf in (4), we have noticed that there were discrepancies between thetwo cdfs. for example, in figure 3 the plots of the cdf fw (w) and the cdf pw (w) are displayedfor ρ = 0.7, where m = 2 is used in pw (w) , and m = 4 is used in fw (w) (that is, the degreesof freedom equal 4). figure 3. comparison of the cdfs pw (w) and fw (w) for ρ = 0.7 and degrees offreedom 4. in order to determine the source of this discrepancy, we examine the derivation of pw (w) for w < 0.this is enough, as for a symmetric density function about the y-axis one has f (x) = 1 − f (−x).this formula was employed in [6] for his piecewise presentation of pw (w). to start with, consider the following integral∫ w −∞ (−y)m−i−1 e y a dy = am−iγ(m − i ,− w a ), re (w) < 0, where a = 2 √ 1− ρ2, and γ[n, x ] is the upper (or complementary) incomplete gamma function. foran integer n, the expansion γ(n, x) = (n − 1)!e−x ∑n−1 k=0 xk k! is given in [7]. therefore, the correctionof the cdf in the notation given in [6], for w < 0, can be written as pw (w) = 1 2m (m − 1)! exp ( w 2 √ 1− ρ2 ) m−1∑ i=0 m−i−1∑ l=0 (m + i − 1)! 2i i !l! ( −w 2 √ 1− ρ2 )l . the results from this formula matches those obtained from our cdf (5). furthermore, it is worthnoting that the pdf and the cdf of a sum of dependent rvs x and y, say v = x + y, is sometimes https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 7derived indirectly from w = x − z by setting y = −z (e.g., see [6], ch. 5). for m odd, we note that no expression for the cdf is given in [6], and to our knowelege it is unknown.in our representation of the cdf in (4), or in its regularized version, we encouter the evaluation of thegamma function at negative integers (poles) where it is not defined. it turns out that for odd m theintegral j can be expressed in terms of the meijer g-function. recall, that the meijer g-functionis defined as a mellin-barnes integral (an inverse mellin transform) (see [1]) gm,np,q ( ap bq |z, r ) := r 2πi ∫ l ∏m j=1 γ ( bj − r s )∏n j=1 γ ( 1− aj − r s )∏q j=m+1 γ ( 1− bj + r s )∏p j=1 γ ( aj − r s )z sds, where ap = (a1, · · · , ap) and bq = (b1, · · · , bq) , and l is a contour in the complex s-plane withcertain properties (e.g., see [11] for further conditions for which the definition holds). for brevity,we describe this integral in the following remark in a bit more details in our specific case. remark 2. returning to the contour integral representation of meijer g, but now as given in [10], we see in our specific case that g2,1 1,3 ( 1 1 2 m 2 0 |z ) = 1 2πi ∫ l γ( 1 2 +s)γ(m2 +s)γ(−s) γ(1+s) z−sds, where l is a contour in the complex s-plane, which exists as a1 − bi − 1 /∈ n (see [11]). in [10], s is replaced with −s, and hence the poles and l undergo a reflection. more specifically, a contour l is chosen to separate the poles of γ ( 1 2 + s ) and γ ( m 2 + s ) from those of γ (−s) . the contour in our case starts at −∞ encircling the poles of γ ( 1 2 + s ) (at s = −1 2 − n, n = 0, 1, 2, · · · ) and those of γ ( m 2 + s ) (at s = −m2 − n, n = 0, 1, 2, · · · ) but not those of γ (−s) (at n = 0, 1, 2, · · · ) and returning to −∞. since in our case it holds that 0 ≤ m < q and 0 ≤ p < q; and that 1 + 1 2 , 1 + m 2 are not integers whenever m is odd, the integral converges for all z 6= 0; and the meijer function is an analytic function except for z = 0 (e.g., see [11]). theorem 2. for odd m > 0, the j in (4) is given in trems of the meijer g-function, and hence, the cdf can be expressed as fw (w) = 1 2 + 1 2 √ πγ ( m 2 )g2,1 1,3 ( 1 1 2 m 2 0 | w2 c ) , w > 0. (6) proof. the entry 07.34.03.0605.01 in [10] gives a representation of the bessel k function as g2,0 0,2 ( − b1, b2 |z ) = 2z b1+b2 2 kb1−b2 ( 2 √ z ) . however, the meijerreduce command in [5] gave us g2,0 0,2 ( − b1, b2 |z, 1 2 ) = 2z b1+b2 2 kb1−b2 ( 2 √ z ) , https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 8where the 1 2 equals the r in the contour integral in the definition of the meijer g-function givenabove. consequently, the integral in terms of the meijer g-function becomes∫ w 0 x m−1 2 km−1 2 ( x 2 √ 1− ρ2 ) dx = ∫ w 0 1 2 x m−1 2 g2,0 0,2 ( − m−1 4 , 1−m 4 | x 4 √ 1− ρ2 , 1 2 ) dx. this integral evaluates to a meijer g-function with r = 1 when an odd positive numerical value of m is specified. observing the pattern, we arrive at the expression in (6). by remark 2, z 6= 0, andas the limit of g2,1 1,3 (z |·) → 0 as z → 0, fw (0) := 1 2 . moreover, limw→∞ fw (w) = 1. � let b = 2 √ 1− ρ2, and hence 4b2 = c.the integral i = ∫ w −∞ fw (w) dw for re (w) < 0, and odd m > 2, evaluates to i = b m+1 2 [ 2π(m − 2)!!− 2 m+1 2 g3,1 1,2 ( 1 1 2 , m 2 , 0 | w2 4b2 )] 2m+2 √ π (1− ρ2) m+1 4 γ ( m 2 ) , m = 4k − 1; i = b m+1 2 π (m − 2)!! + w m+1 2 g3,1 1,2 ( 3−m 4 1−m 4 , m−1 4 ,−m+1 4 | w2 4b2 ) 2m+2 √ π (1− ρ2) m+1 4 γ ( m 2 ) , m = 4k + 1, where k = 1, 2, · · · ; and n!! is the double factorial, which is the product of all positive odd integersup to n. figure 4. the cdf fw (w) for m = 11, 18, 25 and ρ = 0.7. figure 4 shows plots of the cdf generated using the meijer function representation of fw (w). 4. percentiles in this section, to illustrate the use of the equation fw (w) = 1 2 +j , we compute the 95th percentile (α = 0.05), for ρ = 0.8, 0.9, 0.95, and for degrees of freedom m = 3, · · · , 30. for m even, using https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 9either (4), or the regularized generalized hypergeometric function representations of j, one obtainsthe percentiles reported in table 1. remark 3. given that the integral representig g2,1 1,3 converges, l. slater’s theorem (e.g., see [11] for the general statement, [1,4,12]) which gives us an expression of the meijer g-function in terms of two generalized hypergeometric functions for bi − bj /∈ z, i 6= j . in our specific case, using this theorem gives us that g2,1 1,3 ( a1 b1 b2 b3 |z ) = ∏2 j=1 γ ( bj − bh=1 )∗∏1 j=1 γ ( 1 + b1 − aj )∏ lim3 j=3 γ ( 1 + b1 − bj ) 1f2 (1 + b1 − a1; 1 + b1 − b2, 1 + b1 − b3; z) + ∏2 j=1 γ ( bj − bh=2 )∗∏1 j=1 γ ( 1 + b2 − aj )∏3 j=3 γ ( 1 + b2 − bj ) 1f2 (1 + b2 − a1; 1 + b2 − b1, 1 + b2 − b3; z) , where the * indicates that the term corresponding to j = h is omitted. this equation reduces to the expression 1 2 √ πγ(m2 ) g2,1 1,3 ( 1 1 2 m 2 0 | w2 16(1−ρ2) ) = j, where j is as in theorem 1. a similar conclusion can be obtained from formula 07.34.03.0727.01 in [10] g2,1 1,3 ( a1 b1 b2 b3 |z ) = π csc ((b2 − b1)π) [ γ (1− a1 + b1) zb1 1f̃2 (1− a1 + b1; b1 − b2 + 1, b1 − b3 + 1; z) −γ (1− a1 + b2) zb2 1f̃2 (1− a1 + b2; 1− b1 + b2, b2 − b3 + 1; z) ] , where b2 − b1 /∈ z. solving the linear system 1− a1 + b1 = 1 2 , b1 − b2 + 1 = 3−m 2 , b1 − b3 + 1 = 3 2 , 1− a1 + b2 = m 2 , 1− b1 + b2 = m + 1 2 , b2 − b3 + 1 = m + 2 2 ; we obtain that a1 = 1 + b3, b1 = 1 2 + b3, b2 = m 2 + b3. for simplicity, take b3 = 0, and then we have g2,1 1,3 ( 1 1 2 m 2 0 |w2 c ) , with m−1 2 /∈ z, w > 0. furthermore, evaluating the contour integral described in remark 2, using formula 07.34.06.0045.01 in [10] for the residues series, resulted in combination of four 1f2 generalized hypergeometric functions that also blow up for odd m. therefore, it appears that in the hypergeometric representation of this meijer g-function, the restriction b2 − b1 /∈ z for odd m cannot be removed. clearly, the representation in (6) holds for even m as the condition m−1 2 /∈ z holds; and thecomputed percentiles from this expression were identical to those reported in table 1. it turns outthat in mathematica [5] for the case where m ≥ 1 is odd, it is possible to solve for w > 0 using https://doi.org/10.28924/ada/ma.5.19 eur. j. math. anal. 10.28924/ada/ma.5.19 10the findroot command, which solves numerically for an initial guess at the root. as it is not clearenough how the regularization has occured for odd m, we implemented newton’s method to findthe root using formula 07.34.20.0001.01 in [10] for the derivative of the meijer g-function f ′w (w) = 2w 2c √ wγ ( m 2 )g2,2 2,4 ( −1, 0 −1 2 , m 2 − 1, 0,−1 | w2 c ) . solving (6) by newton’s method, using positive initial guesses, gave the same values as those reported in table 1. newton’s method worked as an expansion of g2,1 1,3 ( 1 1 2 m 2 0 |z ) can be written as π sec (mπ 2 )( − √ πz 1f̃2 ( 1 2 ; 3−m 2 , 3 2 ; z ) + z m 2 γ (m 2 ) 1f̃2 ( m 2 ; m + 1 2 , m + 2 2 ; z )) . then the ratio of this g-function to its derivative is clearly regular for odd m and w > 0, as thesecant function cancels out. m\ρ 0.8 0.9 0.95 m\ρ 0.8 0.9 0.95 4 3.92617 2.85230 2.04325 3 3.39639 2.46742 1.76754 6 4.81249 3.49619 2.50450 5 4.39180 3.19057 2.28557 8 5.55949 4.03888 2.89325 7 5.19934 3.77723 2.70582 10 6.21796 4.51724 3.23593 9 5.89785 4.28469 3.06934 12 6.81358 4.94995 3.54590 11 6.52251 4.73850 3.39442 14 7.36152 5.34802 3.83106 13 7.09280 5.15280 3.69121 16 7.87168 5.71864 4.09655 15 7.62084 5.53641 3.96601 18 8.35093 6.06681 4.34596 17 8.11482 5.89528 4.22309 20 8.80428 6.39616 4.58189 19 8.58058 6.23365 4.46548 22 9.23552 6.70945 4.80631 21 9.02246 6.55467 4.69544 24 9.64758 7.00881 5.02076 23 9.44379 6.68607 4.91470 26 10.0428 7.29594 5.22645 25 9.84718 7.15381 5.12463 28 10.4231 7.57224 5.42437 27 10.2347 7.43537 5.32633 30 10.7901 7.83883 5.61535 29 10.6082 7.70668 5.52069table 1. the 95th percentile (α = 0.05) for 3 ≤ m ≤ 30, and ρ = 0.8, 0.9, 0.95. references [1] h. bateman, a. erdelyi, higher transcendental functions, vol i. mcgraw hill, new york (1953).[2] w. f. kibble, a two variate gamma type distribution, sankhya 5(1941), 137-150. http://www.jstor.org/ stable/25047664.[3] s. kotz, n. balakrishna, n. l. johnson, continuous multivariate distributions, john wiley, usa (2000). https://doi.org/10.28924/ada/ma.5.19 http://www.jstor.org/stable/25047664 http://www.jstor.org/stable/25047664 eur. j. math. anal. 10.28924/ada/ma.5.19 11 [4] y. l. luke, the special functions and their approximations, vol 1. academic press, new york (1969). https: //lib.ugent.be/catalog/ebk01:1000000000789815.[5] a. p. prudnikov, y. a. brychkov, o. i. marchev, integrals and series vol 3, more special functions, gordan andbreach, new york (1990).[6] m. k. simon, probability distributions involving gaussian random variables, springer, us (2002).[7] e. w. weisstein, incomplete gamma function. https://mathworld.wolfram.com/incompletegammafunction. html.[8] macdonald function, encyclopedia of mathematics. https://encyclopediaofmath.org/wiki/macdonald_ function.[9] wolfram research, bessel functions. http://functions.wolfram.com/pdf/bessel/03.04.21.0014.01.[10] wikipedia, meijer g function, (2025). https://en.wikipedia.org/wiki/meijer/g-function.[11] wolfram research, meijer g. https://functions.wolfram.com/pdf/meijerg.pdf.[12] wolfram research inc. mathematica version 13.0, (2022). https://www.wolfram.com/mathematica/new-in-13. https://doi.org/10.28924/ada/ma.5.19 https://lib.ugent.be/catalog/ebk01:1000000000789815 https://lib.ugent.be/catalog/ebk01:1000000000789815 https://mathworld.wolfram.com/incompletegammafunction.html https://mathworld.wolfram.com/incompletegammafunction.html https://encyclopediaofmath.org/wiki/macdonald_function https://encyclopediaofmath.org/wiki/macdonald_function http://functions.wolfram.com/pdf/bessel/03.04.21.0014.01 https://en.wikipedia.org/wiki/meijer/g-function https://functions.wolfram.com/pdf/meijerg.pdf https://www.wolfram.com/mathematica/new-in-13 1. introduction 2. density functions 3. cumulative distribution functions 4. percentiles references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 8doi: 10.28924/ada/ma.5.8 uncertainty principles and extremal functions for bessel multiplier operators in quantum calculus ahmed chana∗, abdellatif akhlidj laboratory of fundamental and applied mathematics, department of mathematics and informatics, faculty of sciences ain chock, university of hassan ii, b.p 5366 maarif, casablanca, morocco maths.chana@gmail.com, akhlidj@hotmail.fr ∗correspondence: maths.chana@gmail.com abstract. using the q-jackson integral and some elements of the q-harmonic analysis associatedwith the q-bessel operator for fixed 0 < q < 1, we introduce the q-bessel multiplier operators and wegive some new results related to these operators as plancherel’s, calderón’s reproducing formulas andheisenberg’s, donoho-stark’s uncertainty principles. next, using the theory of reproducing kernelswe give best estimates and an integral representation of the extremal functions related to theseoperators on weighted sobolev spaces. 1. introduction the q-theory, called also in some literature quantum calculus began to arise. interest in thistheory is grown at an explosive note by both physicists and mathematicians due to a large numberof its application domains, for more information about quantum calculus one can see [20].recently, many reasercher have been investigated the behavior of the q-theory to several alreadystudied for the fourier analysis, for example sampling theorem [2], paley-wiener theorem [1],uncertainty principles [31], wavelet transform [15], wavelet packet [6], ramanujan master theorem[16], sobolev type spaces [27] and wave equation [29]. in their seminal papers, hörmander’s andmikhlin’s [18,25] initiated the study of boundedness of the translation invariant operators on rd . thetranslation invariant operators on rd characterized using the classical euclidean fourier transform f(f ) therefore they also known as fourier multipliers. given a measurable function m : rd −→ c its fourier multiplier is the linear map tm given for all λ ∈ rd by the relation f(tm(f ))(λ) = m(λ)f(f )(λ) (1.1) received: 22 nov 2024. key words and phrases. quantum calculus; q-bessel transform; calderón’s reproducing formulas; extremal functions;heisenberg’s uncertainty principle; approximation theory; sobolev spaces. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 2the hörmander-mikhlin fundamental condition gives a criterion for lp-boundedness for all 1 < p <∞ of fourier multiplier tm in terms of derivatives of the symbol m, more precisely if∣∣∂γλm(λ) ∣∣ . |λ|−|γ| f or 0 ≤ |γ| ≤ [ d 2 ] + 1. (1.2) then, tm can be extended to a bounded linear operator from lp(rd) into itself .the condition (1.2) imposes m to be a bounded function, smooth over rd\{0} satisfying certainlocal and asymptotic behavior. locally, m admits a singularity at 0 with a mild control of deriva-tives around it up to order [d2 ] + 1. this singularity links to deep concepts in harmonic analysisand justifies the key role of hörmander-mikhlin theorem in fourier multiplier lp-theory, this con-dition defines a large class of fourier multipliers including riesz transforms and littelwood-paleypartitions of unity which are crucial in fourier summability or pseudo-differential operator.theboundedness of fourier multipliers is useful to solve problems in the area of mathematical analysisas probability theory see [24], stochastic processus see [5], and the study of nonlinear partialdifferential equations see [22]. for its importance many researcher extend the theory of fouriermultiplier to different setting for example in the dunkl-weinstein setting [33], in the laguerre-bessel setting [8], in the q-fourier setting [26, 31, 32] and the q-cosine fourier setting [3]. thegeneral theory of reproducing kernels is stared with aronszajn’s in [4] in 1950, next the authorsin [23, 30] applied this theory to study tikhonov regularization problem and they obtained ap-proximate solutions for bounded linear operator equations on hilbert spaces with the viewpoint ofnumerical solutions by computers. this theory has gained considerable interest in various field ofmathematical sciences especially in engineering and numerical experiments by using computerssee [30].this paper focuses on the generalized fourier transform associated with the q-bessel operatorcalled the q-bessel transform introduced in [11], more precisely we define the following q-differentialoperator for 0 < q < 1 by ∆q,αf (x) = f ( q−1x ) − ( 1 + q2α ) f (x) + q2αf (qx) x2 , ∀x 6= 0. (1.3) the eigenfunctions of the operator (1.3) are related to the hahn-exton q-bessel function jα(x ; q2)defined in [15]. the q-bessel tranform hq,α is defined on l1α(r+q ) by hq,α(f )(λ) = ∫ ∞ 0 jα(λx ; q2)f (x)dµq,α(x), for λ ∈ r+q where dµq,α is the measure on r+q given later. let σ be a function in l2α(r+q ) and β ∈ r+q , theq-bessel l2α-multiplier operators are defined for smooth function f on r+q as mq,σ,β(f )(x) := h−1q,α ( σβhq,α(f ) ) (x) (1.4) https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 3where the function σβ is given by σβ(λ) := σ(λβ). (1.5)these operators are a generalization of all classical multiplier operators introduced in [3, 10,26, 31, 32]. the remainder of this paper is arranged as follows, in section 2 we recall the mainresults concerning the harmonic analysis associated with the q-bessel transform, in section 3,we introduce the q-bessel l2α-multiplier operators mq,σ,β and we give for them a plancherel’s,pointwise reproducing formulas and heisenberg’s, donoho-stark’s uncertainty principles. thelast section of this paper is devoted to give an application of the general theory of reproducingkernels to q-bessel multiplier theory and to give best estimates and an integral representation ofthe extremal functions related to the q-bessel l2α-multiplier operatorsmq,σ,β on weighted sobolevspaces. 2. harmonic analysis associated with the q-bessel transform in this section we set some notations and we recall some results in harmonic analysis related tothe q-bessel operator (1.3), all these results can be founded in [11,17,19–21,28]. 2.1. notations and preliminaries. in this subsection, we give some notations, definitions and prop-erties of the q-shifted factorial, the jackson’s q-derivatives and the jackson’s q-integrals introducedin [19].let a ∈ c, the q-shifted factorial are defined by: (a; q)0 = 1, (a; q)n = n−1∏ k=0 ( 1− aqk ) , (a; q)∞ = ∞∏ k=0 ( 1− aqk ) . the jackson’s q-derivative of a function f is given by dqf (x) = f (x)− f (qx) (1− q)x if x 6= 0.the q-jackson’s integrals from 0 to a and from 0 to ∞ are defined by∫ a 0 f (x)dqx = (1− q)a ∞∑ 0 f (aqn) qn, ∫ ∞ 0 f (x)dqx = (1− q) ∞∑ n=−∞ f (qn) qn. provided the sums converge absolutely. the normalized form of the q-bessel kernel is defined in [14,17,28] by jα(x ; q2) = ∞∑ n=0 (−1)n q n(n+1) 2 (qα+1; q)n (q; q)n x2n. (2.1) https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 4it satisfies the following estimate [11] ∀x ∈ r+q , ∣∣jα(x ; q2) ∣∣ ≤ 1. (2.2) 2.2. the q-bessel transform. in this section, we define and give some basic properties of q-besseltransform introduced in [11]. we first introduced the following spaces and norms• c0,q(r+q ) denotes the set of all functions defined on r+q continuous at zero and vanishing atinfinity, equiped with the induced topology of uniforme convergence.• lpα(r+q ), 1 ≤ p ≤ ∞, denotes the space of measurable functions on r+q , satisfying ‖f ‖p,q,α =: { (∫∞ 0 |f (x)|pdµq,α(x) )1/p <∞, 1 ≤ p <∞, supx∈r+q |f (x)| <∞, p =∞.where dµq,α(x) = 1 1− q ( q2α+2; q2 ) ∞ (q2; q2)∞ x2α+1dq(x), definition 2.1. ( [11]) the q-bessel transform hq,α defined on l1α(r+q ) by hq,α(f )(λ) = ∫ ∞ 0 jα(λx ; q2)f (x)dµq,α(x), for λ ∈ r+q some basic properties of this transform are as follows, for the proofs, we refer the reader to [11,13,14,25]. proposition 2.1. (1) for every f ∈ l1α(r+q ) we have hq,α(f ) ∈ c0,q(r+q ) and we have ‖hq,α(f )‖∞,q,α ≤ bq,α‖f ‖1,q,α. (2.3) where bq,α = 1 1− q ( −q2α+2; q2 ) ∞ ( −q2; q2 ) ∞ (q2; q2)∞ (2.4) (2)(q-inversion formula) for f ∈ ( l1α ∩ l2α ) (r+q ) such that fα(f ) ∈ l1α(r+q ) we have f (x) = ∫ ∞ 0 jα(λx ; q2)hq,α(f )(λ)dµq,α(λ), a.e x ∈ r+q . (2.5) (3) (q-parseval formula) for all f , g ∈ l2α(r+q ) we have 〈f , g〉q = 〈hq,α(f ),hq,α(g)〉q , (2.6) in particular we have ‖f ‖2,q,α = ‖hq,α(f )‖2,q,α . (2.7) (4) (q-plancherel theorem) the q-bessel transform hq,α can be extended to an isometric isomorphism from l2α(r+q ) into l2α(r+q ). https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 52.3. the translation operator associated with the q-bessel transform. definition 2.2. ( [13]) let x, y ∈ r+q and f is a measurable function on r+q the translation operator is defined by τxq,αf (y) = ∫ ∞ 0 jα(λx ; q2)jα(λy ; q2)hq,α(f )(λ)dµq,α(λ), the following proposition summarizes some properties of the q-bessel translation operator see [13]. proposition 2.2. for all x, y ∈ r+q ,we have: (1) τxq,αf (y) = τyq,αf (x). (2.8) (2) ∫ ∞ 0 τxq,αf (y)dµq,α(y) = ∫ ∞ 0 f (y)dµq,α(y). (2.9) (3) for f ∈ lpα(r+q ) with p ∈ [1; +∞] τxq,αf ∈ l p α(r+q ) and we have∥∥τxq,αf ∥∥p,q,α ≤ ‖f ‖p,q,α, (2.10) (4) for f ∈ l1α(r+q ), τxq,αf ∈ l1α(r+q ) and we have hq,α ( τxq,αf ) (λ) = jα(λx ; q2)hq,α(f )(λ), ∀λ ∈ r+q . (2.11) the relation (2.11) shows that the translation operator τxq,α is a particular case of the q-bessel multiplier operator (1.4). by using the q-bessel translation operator, we define the generalized convolution product of f , g by (f ∗q g) (x) = ∫ ∞ 0 τxq,α(f )(y)g(y)dµq,α(y). this convolution is commutative, associative and its satisfies the following properties see [11,13]. proposition 2.3. (1)(q-young’s inequality) for all p, q, r ∈ [1; +∞] such that: 1 p + 1 s = 1 + 1 r and for all f ∈ lpα(r+q ), g ∈ lsα(r+q ) the function f ∗α g belongs to the space lrα(r+q ) and we have ‖f ∗α g‖r,q,α ≤ ‖f ‖p,q,α‖g‖s,q,α (2.12) (2) for f , g ∈ l2α(r+q ) the function f ∗qg belongs to l2α(r+q ) if and only if the functionhq,α(f )hq,α(g) belongs to l2α(r+q ) and in this case we have hq,α (f ∗q g) = hq,α(f )hq,α(g). (2.13) (3) for all f , g ∈ l2α(r+q ) then we have∫ ∞ 0 |f ∗q g(x, t)|2 dµq,α(x) = ∫ ∞ 0 |hq,α(f )(λ)|2 |hq,α(g)(λ)|2 dµq,α(λ), (2.14) where both integrals are simultaneously finite or infinite. https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 63. the q-bessel l2α-multiplier operators the main purpose of this section is to introduce the q-bessel l2α-multiplier operators on r+q and to establish for them some uncertainty principles and calderon’s reproducing formulas. 3.1. calderon’s reproducing formulas for the q-bessel l2α-multiplier operators. definition 3.1. let σ ∈ l2α(r+q ) and β ∈ r+q , the q-bessel l2α-multiplier operators are defined for smooth function f on r+q as mq,σ,β(f )(x) := h−1q,α ( σβhq,α(f ) ) (x), (3.1) where the function σβ is given by the relation (1.5) and by a simple change of variable we find that for all β ∈ r+q , σβ ∈ l2α(r+q ) and∥∥σβ∥∥2,q,α = 1 βα+1 ‖σ‖2,q,α. (3.2) remark 3.1. according to the relation (2.13) we find that mq,σ,β(f )(x) = ( h−1q,α ( σβ ) ∗α f ) (x), (3.3) where h−1q,α ( σβ ) (x) = 1 β2α+2 h−1q,α(σ) ( x β ) . (3.4) we give some properties of the q-bessel l2α-multiplier operators. proposition 3.1. (i) for every σ ∈ l2α(r+q ), and f ∈ l1α(r+q ), the function mq,σ,β(f ) belongs to l2α(r+q ), and we have ∥∥mq,σ,β(f ) ∥∥ 2,q,α ≤ 1 βα+1 ‖σ‖2,q,α‖f ‖1,q,α. (ii) for every σ ∈ l∞α (r+q ), and for every f ∈ l2α(r+q ), the function mq,σ,β(f ) belongs to l2α(r+q ), and we have ∥∥mq,σ,β(f ) ∥∥ 2,q,α ≤ ‖σ‖∞,q,α‖f ‖2,q,α (3.5) (iii) for every σ ∈ l2α(r+q ), and for every f ∈ l2α(r+q ), mq,σ,β(f ) ∈ l∞α (r+q ), and we have mq,σ,β(f )(x) = ∫ ∞ 0 σ(βλ)jα(λx ; q2)hq,α(f )(λ)dµq,α(λ), a.e x ∈ r+q (3.6) and ∥∥mq,σ,β(f ) ∥∥ ∞,q,α ≤ 1 βα+1 ‖σ‖2,q,α‖f ‖2,q,α. proof. (i) by using the relations (2.12),(3.3) we find that∥∥mq,σ,β(f ) ∥∥2 2,q,α = ∥∥h−1q,α (σβ) ∗q f ∥∥22,q,α ≤ ‖f ‖21,q,α ∥∥h−1q,α (σβ)∥∥22,q,αplancherel’s formula (2.7) and the relation (3.2) gives the desired result.(ii) is a consequence of plancherel’s formula (2.7). https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 7(iii) is a consequence of the relations (2.7),(2.12),(3.2) and (3.3), on the other hand the relation (3.6)follows from inversion formula (2.5). � in the following result, we give plancherel’s and pointwise reproducing inversion formula for the q-bessel l2α-multiplier operators. theorem 3.1. let σ ∈ l2α(r+q ) satisfying the admissibility condition:∫ ∞ 0 ∣∣σβ(λ) ∣∣2 dq(β) β = 1, λ ∈ r. (3.7) (i) (plancherel formula) for all f in l2α(r+q ), we have∫ ∞ 0 |f (x)|2dµq,α(x) = ∫ ∞ 0 ∥∥mq,σ,β(f ) ∥∥2 2,q,α dq(β) β . (3.8) (ii) (first calderón’s formula) let f ∈ l1α(r+q ) such that hq,α(f ) ∈ l1α(r+q ) then we have f (x) = ∫ ∞ 0 ( mq,σ,β(f ) ∗α h−1q,α ( σβ )) (x) dβ β , a.e. x ∈ r. proof. (i) by using the relations (2.14) and (3.3) we get∫ ∞ 0 ∥∥mq,σ,β(f ) ∥∥2 2,q,α dq(β) β = ∫ ∞ 0 [∫ ∞ 0 ∣∣mq,σ,β(f )(x) ∣∣2 dµq,α(x) ] dq(β) β = ∫ ∞ 0 [∫ ∞ 0 |hq,α(f )(λ)|2 dµq,α(λ) ] ∣∣σβ(λ) ∣∣2 dq(β) βthe admissibility condition (3.7) and plancherel’s formula (2.7) gives the desired result.(ii) let f ∈ l1α(r+q ) such that hq,α(f ) ∈ l1α(r+q ), by using the relations (2.6),(2.11) we find that∫ ∞ 0 ( mq,σ,β(f ) ∗αh−1q,α ( σβ )) (x) dβ β = ∫ ∞ 0 [∫ ∞ 0 ∣∣σβ(λ) ∣∣2hq,α(f )(λ)jα(λx ; q2)dµq,α(λ) ] dq(β) β = ∫ ∞ 0 [∫ ∞ 0 hq,α(f )(λ)jα(λx ; q2)dµq,α(λ) ] ∣∣σβ(λ) ∣∣2 dq(β) βthe admissibility condition (3.7),inversion formula (2.5) gives the desired result. � to establish the second calderon’s reproducing formula for the q-bessel l2α-multiplier operators, we need the following technical result. proposition 3.2. let σ ∈ l2α(r+q ) ∩ l∞α (r+q ) satisfy the admissibility condition (3.7) then the function defined by φγ,δ(λ) = ∫ δ γ ∣∣σβ(λ) ∣∣2 dq(β) β belongs to l2α(r+q ) ∩ l∞α (r+q ) for all 0 < γ < δ <∞. https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 8 proof. using hölder’s inequality for the measure dq(β) β and the relation (3.2) we find that∥∥φγ,δ ∥∥2 2,q,α ≤ log(δ/γ)‖σ‖22,q,α‖σ‖2∞,q,α ∫ δ γ dq(β) βα+2 <∞ so φγ,δ ∈ l2α(r+q ), furthermore by using the relation (3.7) we get ∥∥φγ,δ ∥∥ ∞,q,α < ∞ therefore φγ,δ belongs to l2α(r+q ) ∩ l∞α (r+q ). � theorem 3.2. (second calderón’s formula). let f ∈ l2α(r+q ) and σ ∈ l2α(r+q ) ∩ l∞α (r+q ) satisfy the admissibility condition (3.7) and 0 < γ < δ <∞. then the function fγ,δ(x) = ∫ δ γ ( mq,σ,β(f ) ∗α h−1q,α ( σβ )) (x) dq(β) β , x ∈ r+q belongs to l2α(r+q ) and satisfies lim (γ,δ)→(0,∞) ∥∥fγ,δ − f ∥∥2,q,α = 0 (3.9) proof. by a simple computation we find that fγ,δ(x) = ∫ ∞ 0 φγ,δ(λ)jα(λx ; q2)hq,α(f )(λ)dµq,α(λ) = h−1q,α ( φγ,δhq,α(f ) ) (x), by using proposition 3.2 we find that φγ,δ ∈ l∞α (r+q ) then we have fγ,δ ∈ l2α(r+q ) and hq,α ( fγ,δ ) (λ) = φγ,δ(λ,m)hq,α(f )(λ) on the other hand by using plancherel’s formula (2.7) we find that lim (γ,δ)→(0,∞) ∥∥fγ,δ − f ∥∥22,q,α = lim (γ,δ)→(0,∞) ∫ ∞ 0 |hq,α(f )(λ)|2 ( 1−φγ,δ(λ) )2 dµq,α(λ) by using the admissibility condition (3.7), the relation (3.9) follows from the dominated convergencetheorem. � 3.2. uncerainty principles for the q-bessel l2α-multiplier operators. the main purpose of this subsection is to establish heisenberg’s and donoho-stark’s uncertainty principles for the q-bessel l2α-multiplier operators mq,σ,β . 3.2.1. heisenberg’s uncertainty principle formq,σ,β . heisenberg’s uncertainty principle for the qbessel fourier transform hq,α has been established in [9, 11] as follows, for all f ∈ l2α(r+q ) we have ‖|x |f ‖2,q,α ‖|λ|hq,α(f )‖2,q,α ≥ kq,v‖f ‖ 2 2,q,α, (3.10) where kq,α = [1+ √ q×qα+1] 1−q2(α+1) . the inequality (3.10) says that if f is highly localized, then hq,α(f ) cannot be concentrated near a single point. we will generalize this inequality for mq,σ,β , we have the following result https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 9 theorem 3.3. for all f ∈ l2α(r+q ) we have ‖f ‖22,q,α ≤ ∥∥|λ|2hq,α(f ) ∥∥ 2,q,α kq,α [∫ ∞ 0 ∥∥|x |mq,σ,β(f ) ∥∥2 2,q,α dq(β) β ] 1 2 proof. let us suppose that ∥∥|λ|2hq,α(f ) ∥∥ 2,q,α + [∫∞ 0 ∥∥|x |2mq,σ,β(f ) ∥∥2 2,q,α dq(β) β ] < ∞, by usingthe relation (3.10) we find that kq,α ∫ ∞ 0 |mq,σ,β(f )(x)|2dµq,α(x) ≤ ∥∥|x |mq,σ,β(f ) ∥∥ 2,q,α ∥∥|λ|σβhq,α(f ) ∥∥ 2,q,α , integrating over ]0,+∞[ with respect to measure dq(β) β and using plancherel’s formula (3.8) andschwartz’s inequality we get kq,α‖f ‖22,q,α ≤ [∫ ∞ 0 ‖|x |mq,σ,β(f )‖22,q,α dq(β) β ] 1 2 [∫ ∞ 0 [∫ ∞ 0 ||λσβ(λ)|2 |hq,α(f )(λ)|2|(λ)|dµq,α(λ) ] dq(β) β ] 1 2 the admissibility condition (3.7) gives the desired result. � 3.2.2. donoho-stark’s uncertainty principle formq,σ,β . building on the ideas of donoho and stark in [3], the main purpose of this subsection is to give an uncertainty inequality of concentration type in l2θ(r+q ) where l2θ(r+q ) the space of measurables functions on r+q × r+q such that ‖f ‖2,θα = [∫ ∞ 0 ‖f (β, .)‖22,q,α dq(β) β ] 1 2 . we denote by θα the measure defined on r+q × r+q by dθα(β, x) = dµq,α(x)⊗ dq(β) β , definition 3.2. [12] (i) let e be a measurable subset of r+q , we say that the function f ∈ l2α(r+q ) is ε-concentrated on e if ‖f − 1ef ‖2,q,α ≤ ε‖f ‖2,q,α, (3.11) where 1e is the indicator function of the set e. (ii) let f be a measurable subset of r+q × r+q , we say that the function tσ,β(f ) is ρ-concentrated on f if ‖mq,σ,β(f )− 1fmq,σ,β(f )‖2,θα ≤ ρ‖mq,σ,β(f )‖2,θα . (3.12) we have the following result theorem 3.4. let f ∈ l2α(r+q ) and σ ∈ σ ∈ l2α(r+q )) ∩ l∞α (r+q ) satisfying the admissibility condition (3.7), if f is ε-concentrated on e and mq,σ,β(f ) is ρ-concentrated on f then we have ‖σ‖2,q,α(µα(e)) 1 2 [∫ f dθα(β, x) β4α+2 ] 1 2 ≥ 1− (ε+ ρ). https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 10 proof. let f ∈ l2α(r+q ) and σ ∈ l2α(r) ∩ l∞α (r+q ) satisfying (3.7) and assume that µα(e) < ∞and [∫f dθα(β,x) β4α+2 ] 1 2 <∞. according to the relations (3.11),(3.12) we have ‖mq,σ,β(f )−1fmq,σ,β(1ef )‖2,θα ≤ ‖mq,σ,β(f )−1fmq,σ,β(f )‖2,θα + ‖1fmq,σ,β(f −1ef )‖2,θα ≤ ρ‖mq,σ,β(f )‖2,θα + ‖mq,σ,β(f − 1ef )‖2,θα ,by using plancherel’s relation (3.8) we get ‖mq,σ,β(f )‖2,θα ≤ ‖mq,σ,β(f )− 1fmq,σ,β(1ef )‖2,θα + ‖1fmq,σ,β(1ef )‖2,θα ≤ (ε+ ρ)‖f ‖2,q,α + ‖1fmq,σ,β(1ef )‖2,θα , (3.13)on the other hand by using the relation (3.6) and hölder’s inequality we find that ‖1fmq,σ,β(1ef )‖2,θα ≤ ‖f ‖2,q,α‖σ‖1,q,α(µ(e)) 1 2 [∫ f dθα(β, x) β4α+2 ] 1 2 , (3.14) by the relations (3.13),(3.14) we deduce that ‖mq,σ,β(f )‖2,θα ≤ ‖f ‖2,q,α [ (ε+ ρ) + ‖σ‖1,q,α(µα(e)) 1 2 [∫ f dθα(β, x β4α+2 ] 1 2 ] plancherel’s formula (3.8) for mσ,β gives the desired result. � 4. extremal functions associated with the q-bessel l2α-multiplier operators in the following, we study the extremal functions associated with the the q-bessel l2α-multiplier operators. definition 4.1. let ψ be a positive function on r+q satisfying the following conditions 1 ψ ∈ l1α(r+q ) (4.1) and ψ(λ) ≥ 1, λ ∈ r+q . (4.2) we define the sobolev-type space sψ(r+q ) by sψ(r+q ) = { f ∈ l2α(r+q ) : √ ψhq,α(f ) ∈ l2α(r+q ) } provided with inner product 〈f , g〉ψ = ∫ ∞ 0 ψ(λ,m)hq,α(f )(λ)hq,α(g)(λ)dµq,α(λ), and the norm ‖f ‖ψ = √ 〈f , f 〉ψ. https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 11 proposition 4.1. let σ be a function in l∞α (r+q ). then the q-bessel l2α-multiplier operatorsmq,σ,β are bounded and linear from sψ(r+q ) into l2α(r+q ) and we have for all f ∈ sψ(r+q )∥∥mq,σ,β(f ) ∥∥ 2,q,α ≤ ‖σ‖∞,q,α‖f ‖ψ. (4.3) proof. by using the relations (2.8),(3.5),(4.2) we get the result � definition 4.2. let η > 0 and let σ be a function in l∞α (r+q ). we denote by 〈f , g〉ψ,η the inner product defined on the space sψ(r+q ) by 〈f , g〉ψ,η = ∫ ∞ 0 ( ηψ(λ) + ∣∣σβ(λ) ∣∣2)hq,α(f )(λ)hq,α(g)(λ)dµq,α(λ), and the norm ‖f ‖ψ,η = √ 〈f , f 〉ψ,η theorem 4.1. let σ ∈ l∞α (r+q ) the sobolev-type space ( sψ(r+q ) , 〈·, ·〉ψ,η) is a reproducing kernel hilbert space with kernel kq,ψ,η(x, y) = ∫ ∞ 0 jα(λx ; q2)jα(λy ; q2) ηψ(λ) + ∣∣σβ(λ) ∣∣2 dµq,α(λ), that is (i) for all y ∈ r+q , the function x 7→ kq,ψ,η (x, y) belongs to sψ(r+q ). (ii) for all f ∈ sψ(r+q ) and y ∈ r+q , we have the reproducing property f (y) = 〈 f ,kq,ψ,η(·, (y)) 〉 ψ,η . furthermore the kernel kq,ψ,η is a positive definite function. proof. (i) let y ∈ r+q , from the relations (2.2),(4.1) we have the function gy : λ −→ jα(λy ; q2) ηψ(λ) + ∣∣σβ(λ) ∣∣2 belongs to l1α(r+q ) ∩ l2α(r+q ). hence the function kq,ψ,η is well defined and by the inversionformula (2.5), we get kq,ψ,η(x, y) = h−1q,α(gy )(x)by using plancherel’s theorem for hq,α we find that kq,ψ,η(·, y) belongs to l2α(r+q ) and we have hq,α(kq,ψ,η(·, y))(λ) = jα(λy ; q2) ηψ(λ) + ∣∣σβ(λ) ∣∣2 (4.4) by using the relations (2.2),(4.1) and (4.4) we find that ‖ √ ψhq,α(kq,ψ,η(·, y))‖2,q,α ≤ 1 η2 ∥∥∥∥ 1 ψ ∥∥∥∥ 1,q,α <∞, https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 12this prove that for every y ∈ r+q the function x 7→ kq,ψ,η (x, y) belongs to sψ(r+q ).(ii) by using the relation (4.4) we find that for all f ∈ hψ(r) , 〈f ,kq,ψ,η (·, y)〉ψ,η = ∫ ∞ 0 ( ηψ(λ) + ∣∣σβ(λ) ∣∣2)hq,α(f )(λ)hq,α(kq,ψ,η) (·, y))(λ)dµq,α(λ) = ∫ ∞ 0 jα(λy ; q2)hq,α(f )(λ)dµq,α(λ), inversion formula (2.5) gives the desired result. on the other hand since 1ψ is positive function thenfor all z1, . . . ., zn complex numbers and x1, . . . . . . , xn in r+q , we obtain n∑ r=1 n∑ l=1 zrzlkq,ψ,η(xr , xl) = ∫ +∞ 0 [ n∑ r=1 n∑ l=1 zrzl jα ( xrλ; q2 ) jα ( xlλ; q2 )] 1 ψ (λ)dµq,α(λ) = ∫ +∞ 0 ∣∣∣∣∣ n∑ r=1 zr j ( xrλ; q2 )∣∣∣∣∣ 2 1 ψ (λ)dµq,α(λ) ≥ 0 which proves that the kernel kq,ψ,η is positive definite. � the main result of this section can be stated as follows theorem 4.2. let σ ∈ l∞α (r+q ) and β ∈ r+q , for any h ∈ l2α ( r+q ) and for any η > 0, there exist a unique function f ∗q,η,β,h where the infimum inf f ∈sψ(r+q ) { η‖f ‖2ψ + ∥∥h −mq,σ,β(f ) ∥∥2 2,q,α } (4.5) is attained. moreover the extremal function f ∗q,η,β,h is given by f ∗q,η,β,h(y) = ∫ ∞ 0 h(x)θq,η,β(x, y)dµq,α(x), where θq,η,β is given by θq,η,β(x, y) = ∫ ∞ 0 σβ(λ)jα(λx ; q2)jα(λy ; q2) ηψ(λ) + |σβ(λ)|2 dµq,α(λ) proof. the existence and the unicity of the extremal function f ∗q,η,β,h satisfying (4.5) is given in[23,30], furthermore f ∗q,η,β,h is given by f ∗q,η,β,h(y) = 〈h,mq,σ,β(kq,ψ,η (·, y))〉q , by using inversion formula (2.5) and the relation (4.4) we get mq,σ,β(kq,ψ,η (·, y) (x) = ∫ ∞ 0 σβ(λ)jα(λx ; q2)jα(λy ; q2) ηψ(λ) + |σβ(λ)|2 dµq,α(λ) = θq,η,β(x, y)and the proof is complete. � https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 13 theorem 4.3. σ ∈ l∞α (r+q ) and h ∈ l2α ( r+q ) then the function f ∗q,η,β,h satisfies the following properties hq,α(f ∗q,η,β,h)(λ) = σβ(λ) ηψ(λ) + |σβ(λ)|2hq,α(h)(λ) (4.6) and ‖f ∗q,η,β,h‖ψ ≤ 1√ 2η ‖h‖2,q,α. proof. let y ∈ r+q then the function ky : λ −→ σβ(λ)jα(λy ; q2) ηψ(λ) + ∣∣σβ(λ) ∣∣2 belongs to l2α(r+q ) ∩ l1α(r+q ) and by using inversion formula (2.5) we get θq,η,β(x, y) = h−1q,α(ky )(x) using plancherel’s theorem and parseval’s relation (2.6) we find that θq,η,β(·, y) ∈ l2α(r+q ) and f ∗q,η,β,h(y) = ∫ ∞ 0 hq,α(λ)ky (λ)dµq,α(λ) = ∫ ∞ 0 σβ(λ) ηψ(λ) + |σβ(λ)|2hq,α(h)(λ)dµq,α(λ) on the other hand the function f : λ −→ σβ(λ)hq,α(h)(λ) ηψ(λ) + ∣∣σβ(λ) ∣∣2 belongs to l1α(r+q ) ∩ l2α(r+q ), by using inversion formula (2.5), plancherel’s theorem we find that f ∗q,η,β,h belongs to l2α(r+q ) and hq,α(f ∗q,η,β,h)(λ) = f (λ)on the other hand we have |hq,α(f ∗q,η,β,h)(λ)|2 = ∣∣σβ(λ) ∣∣2( ηψ(λ) + ∣∣σβ(λ) ∣∣2)2 |hq,α(h)(λ)|2 ≤ 1 2ηψ(λ) |hq,α(h)(λ)|2 by plancherel’s formula (2.7) we find that ‖f ∗q,η,β,h‖ψ ≤ 1√ 2η ‖h‖2,q,α. � theorem 4.4. (third calderón’s formula) let σ ∈ l∞α (r+q ) and f ∈ sψ(r+q ) then the extremal function given by f ∗q,η,β(y) = ∫ ∞ 0 mq,σ,β(f )(x)θq,η,β(x, y)dµq,α(x), satisfies lim η→0+ ∥∥f ∗q,η,β − f ∥∥2,q,α = 0 (4.7) moreover we have f ∗q,η,β −→ f uniformly when η −→ 0+. https://doi.org/10.28924/ada/ma.5.8 eur. j. math. anal. 10.28924/ada/ma.5.8 14 proof. f ∈ sψ(r+q ), we put h =mq,σ,β(f ) and f ∗q,η,β,h = f ∗q,η,β in the relation (4.6) we find that hq,α(f ∗q,η,β − f )(λ) = −ηψ(λ)hq,α(f )(λ) ηψ(λ) + ∣∣σβ(λ) ∣∣2 (4.8) therefore ∥∥f ∗q,η,β − f ∥∥2ψ = ∫ ∞ 0 η2 (ψ(λ))3 ηψ(λ) + |σβ(λ)|2 |hq,α(f )(λ)|2 dµq,α(λ)on the other hand we have η2 (ψ(λ))3 ηψ(λ) + |σβ(λ)|2 |hq,α(f )(λ)|2 ≤ ψ(λ) |hq,α(f )(λ)|2 (4.9) the result (4.7) follows from (4.9) and the dominated convergence theorem. now, for all f ∈ sψ(r+q )we have hq,α(f ) ∈ l2α(r+q ) ∩ l1α(r+q ) and by using the relations (2.5), (4.8) we find that f ∗q,η,β(y)− f (y) = ∫ ∞ 0 −ηψ(λ)hq,α(f )(λ) ηψ(λ) + ∣∣σβ(λ) ∣∣2 jα(λy ; q2)dµq,α(λ) and ∣∣∣∣∣−ηψ(λ)hq,α(f 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https://doi.org/10.1090/s0002-9947-1992-1069750-0 https://doi.org/10.1093/imrn/rnab256 https://doi.org/10.1515/156939405775297452 https://doi.org/10.1515/156939405775297452 https://doi.org/10.1016/j.jmaa.2009.06.008 https://doi.org/10.1016/j.jmaa.2009.06.008 https://doi.org/10.1090/s0002-9939-06-08525-x https://doi.org/10.1080/00036810600643662 eur. j. math. anal. 10.28924/ada/ma.5.8 16 [32] a. saoudi, reproducing formulas for the fourier-like multipliers operators in q-rubin setting, int. j. anal. appl.18 (2020), 366-380.[33] f. soltani, i. maktouf, dunkl–weinstein multiplier operators and applications to reproducing kernel theory,mediterranean j. math. 21 (2024), 80. https://doi.org/10.1007/s00009-024-02623-2. https://doi.org/10.28924/ada/ma.5.8 https://doi.org/10.1007/s00009-024-02623-2 1. introduction 2. harmonic analysis associated with the q-bessel transform 2.1. notations and preliminaries 2.2. the q-bessel transform 2.3. the translation operator associated with the q-bessel transform 3. the q-bessel l2-multiplier operators 3.1. calderon's reproducing formulas for the q-bessel l2-multiplier operators 3.2. uncerainty principles for the q-bessel l2-multiplier operators 4. extremal functions associated with the q-bessel l2-multiplier operators acknowledgments: authors' contributions: references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 13doi: 10.28924/ada/ma.5.13 some analytical properties of the remainder of binet-like expression for the barnes g-function emmanuel ansong adjei, kwara nantomah∗, morgan yindobil zubil department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana adjeid2@gmail.com, knantomah@cktutas.edu.gh, morganzubil@gmail.com ∗correspondence: knantomah@cktutas.edu.gh abstract. in this paper, we prove some properties such as monotonicity, complete monotonicity, log-arithmic convexity, inequalities, subadditivity and starshapedness, involving the binet-like remainderof the barnes g-function. the methods of proofs are analytical in nature. 1. introduction special functions are usually encountered in almost every scientific discipline. particularly, theyplay important roles in areas such as mathematics, physics and engineering. the gamma function,which is an extension of the factorial function, is arguably the most important special function. thisis largely due to its vast areas of applications as well as its connection with other special functions.it is usually defined as γ(z) = ∫ ∞ 0 tz−1e−tdt (1) for z > 0. the binet’s formula for logarithm of the gamma function is given as ln γ(z) = ( z − 1 2 ) ln z − z + ln √ 2π + θ(z) (2) for z > 0, where θ(z) = ∫ ∞ 0 ( 1 et − 1 − 1 t + 1 2 ) e−zt t dt (3) is known as the remainder of binet’s formula. due to the important properties exhibited by thefunction θ(z), it has been investigated in multiple ways. in [5], the authors proved among otherthings that, for p ∈ (0, 1], the function fp(z) = θ(pz)− pθ(z) (4) received: 20 dec 2024.2010 mathematics subject classification. 33b15, 26a48, 26a51. key words and phrases. multiple gamma function; barnes g-function; binet remainder; binet-like remainder; log-convex function; completely monotonic function; subadditive function; starshaped function.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 2is completely monotonic on (0,∞). in [7], the authors investigated the complete monotonicity ofthe function fp,q,r (z) = r [θ(pz)− qθ(z)] (5) where p > 0, q is a real number and r 6= 0. they further established that θ(z) is subadditiveon (0,∞) and −θ(z) is starshaped on (0,∞). in [11], the authors considered a generalization ofthe function θ(z) which is denoted by θα(z). they investigated the complete monotonicity of thefunction fp,q,α(z) = θα(pz)− qθα(z) (6) where p > 0, q is a real number and α > 0. among other things, they further established that thefunction θα(z) is subadditive. in [3], the authors considered an inequality for the r-th derivative of θ(z). subsequently, they obtain the turan-type inequality( θ(r+1)(z) )2 ≤ θ(r)(z)θ(r+2)(z) (7) where r ∈ n. a generalization of (7) can be found in [2].the multiple gamma function, which is a generalization of the ordinary gamma function, wasdefined by barnes as [1] γr+1(z + 1) = γr+1(z + 1) γr (z) , z ∈ c, r ∈ n, γ1(z) = γ(z), γr (1) = 1. the particular case g(z) = 1 γ2(z) is referred to as the barnes g-function or the double gammafunction. it satisfies the following basic properties [1]. g(z + 1) = g(z)γ(z), z ∈ c, g(1) = 1, (lng(z))′′′ ≥ 0, z > 0. for further properties of the function g(z), one may refer to the papers [10] and [12] and thereferences in there.a binet-like expression for logarithm of the double gamma function is given by choi [6] as ln γ2(z) = lna− z2 4 + ( z2 2 − z 2 + 1 12 ) ln z + (1− z) ln γ(z) + θ(z) (8) where a = 1.282427... is the glaisher-kinkelin constant and θ(z) = ∫ ∞ 0 ( 1 t − 1 et − 1 − 1 2 + t 12 ) e−zt t2 dt. (9) is what is referred to as the binet-like remainder. https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 3it is interesting to observe that the binet-like remainder, θ(z) has some resemblance withthe binet remainder, θ(z). the natural question that arise is: does the function θ(z) satisfythe properties satisfied by the function θ(z) ? motivated by the papers [3, 5, 7, 11], the objectiveof this paper is to answer this question. we establish some properties of the function θ(z)such as monotonicity, complete monotonicity, logarithmic convexity, inequalities, subadditivity andstarshapedness, among others. we present our findings in the next section. before that, we providethe following definitions which shall pave the way for us to prove our results. throughout thispaper, n = {1, 2, 3, . . . } and n0 = {0, 1, 2, 3, . . . }. definition 1.1. a real-valued function k defined on an interval i ⊆ r is said to be convex on i iff k (x u + y v ) ≤ k(x) u + k(y) v (10) holds for all x, y ∈ i and u > 1, v > 1 such that 1 u + 1 v = 1. equivalently, k is said to be convexon i iff k′′(z) ≥ 0 (11) for all z ∈ i . if the inequalities (10) and (11) are reversed, then k is said to be concave on i . definition 1.2. a positive real-valued function k defined on an interval i ⊆ r is said to belogarithmically convex on i iff k (x u + y v ) ≤ [k(x)] 1 u [k(y)] 1 v (12) holds for all x, y ∈ i and u > 1, v > 1 such that 1 u + 1 v = 1. equivalently, k is said to belogarithmically convex on i iff [lnk(z)]′′ ≥ 0 (13) for all z ∈ i . if the inequalities (12) and (13) are reversed, then k is said to be logarithmicallyconcave on i . definition 1.3 ( [13]). a real-valued function k defined on an interval i ⊆ r is said to be completelymonotonic on i iff (−1)nk(n)(z) ≥ 0 holds for all z ∈ i and n ∈ n0. definition 1.4 ( [4]). a real-valued function k defined on an interval i ⊆ r is said to be subadditiveon i iff k(x + y) ≤ k(x) +k(y) holds for all x, y ∈ i . if the inequality is reversed, then k is said to be superadditive on i . https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 4 definition 1.5 ( [4]). a real-valued function k defined on an interval i ⊆ r is said to be starshapedon i iff k(αz) ≤ αk(z),for all z ∈ i and α ∈ [0, 1]. 2. results and discussion beginning with the following lemmas, we now present our findings in this section. lemma 2.1. for t > 0, the inequality 1 t2 − 1 12 < e−t (1− e−t)2 < 1 t2 (14) holds. proof. see theorem 2 of [8]. � lemma 2.2. for t > 0, the function p(t) = 1 t2 − e−t (1− e−t)2 (15) is strictly decreasing. proof. see theorem 1 of [8] or theorem 1.1 of [9]. � theorem 2.3. for t > 0, let a(t) be defined as a(t) = 1 t − 1 et − 1 − 1 2 + t 12 . (16) = t 12 + 1 t − 1 2 coth( t 2 ). (17) then: (a) a(t) is strictly increasing. (b) a(t) is positive. (c) a(t) is strictly convex. proof. by l’hopital’s rule and simple computation, we have lim t→0 a(t) = 0 and lim t→∞ a(t) =∞. then by making use of the left-hand side of (14), we obtain a′(t) = et (et − 1)2 − 1 t2 + 1 12 = e−t (e−t − 1)2 − 1 t2 + 1 12 > 0. https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 5hence a(t) is strictly increasing and that completes the proof of (a). next, the increasing propertyof a(t) implies that for t > 0, a(t) > lim t→0 a(t) = 0which completes the proof of (b). next, as a result of lemma 2.2, we have a′′(t) = − ( 1 t2 − e−t (1− e−t)2 )′ = −p ′(t) > 0 which completes the proof of (c). � remark 2.4. theorem 2.3 (c) implies that θ(z) is positive. corollary 2.5. for t > 0, the inequality e−t (1− e−t)2 < 2 t3 (18) holds. proof. the convexity of a(t) implies that a′′(t) = 2 t3 + et (et − 1)2 − 2e2t (et − 1)3 > 0. this simplifies to 2 t3 (et − 1)3 − et(et − 1) > 0which further simplifies to 2 t3 > et (et − 1)2 = e−t (1− e−t)2 . this completes the proof. � remark 2.6. we note that 1 t2 − 2 t3 < 0 if 0 < t < 2 and 1 t2 − 2 t3 > 0 if t > 2. thus, the upperbound in (14) is better that upper bound in (18) if 0 < t < 2 and the upper bound in (18) is betterthan the upper bound in (14) if t > 2. theorem 2.7. the function θ(z) is completely monotonic on (0,∞). proof. differentiating r number of times of (9) gives θ(r)(z) = (−1)r ∫ ∞ 0 a(t)tr−2e−ztdt (19) where r ∈ n0 and θ(0)(z) = θ(z). this implies that (−1)rθ(r)(z) = (−1)2r ∫ ∞ 0 a(t)tr−2e−ztdt = ∫ ∞ 0 a(t)tr−2e−ztdt > 0. this completes the proof. � https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 6 remark 2.8. the representation (19) implies that the function θ(z) is decreasing and convex on (0,∞). theorem 2.9. let r ∈ n0 be even. then the function θ(r)(z) is logarithmically convex on (0,∞). that is, the inequality θ(r) (x u + y v ) ≤ [ θ(r)(x) ] 1 u [ θ(r)(y) ] 1 v (20) holds for x > 0, y > 0, u > 1, v > 1 and 1 u + 1 v = 1. proof. let r ∈ n0 be an even number. then, by using (19) and holder’s inequality for integrals,we obtain θ(r) (x u + y v ) = ∫ ∞ 0 a(t)tr−2e−( x u + y v )tdt = ∫ ∞ 0 (a(t)tr−2) 1 u + 1 v e−( x u + y v )tdt = ∫ ∞ 0 a(t) 1 u t r−2 u e −xt u a(t) 1 v t r−2 v e −yt v dt ≤ (∫ ∞ 0 a(t)tr−2e−xtdt ) 1 u (∫ ∞ 0 a(t)tr−2e−ytdt ) 1 v = [ θ(r)(x) ] 1 u [ θ(r)(y) ] 1 v which completes the proof. � remark 2.10. the particular case where r = 0 in theorem 2.9 proves that θ(z) is logarithmicallyconvex on (0,∞). remark 2.11. theorem 2.9 shows that, for even r ∈ n0, the function t (z) = θ(r+1)(z) θ(r)(z) (21) is increasing on (0,∞). corollary 2.12. let r ∈ n0 be even and p ∈ (0, 1]. then the function b(z) = θ(r)(pz) [θ(r)(z)]p decreasing on (0,∞). consequently, for 0 < x ≤ y , the inequality( θ(r)(y) θ(r)(x) )p ≥ θ(r)(py) θ(r)(px) (22) holds. https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 7 proof. logarithmic differentiation of b(z) and applying the increasing property of t (z) gives b′(z) b(z) = p θ(r+1)(pz) θ(r)(pz) − p θ(r+1)(z) θ(r)(z) = p [ θ(r+1)(pz) θ(r)(pz) − θ(r+1)(z) θ(r)(z) ] ≤ 0. hence b(z) is decreasing. consequently, for 0 < x ≤ y , we have b(x) ≥ b(y) which whenrearranged gives (22). � remark 2.13. if p ≥ 1 in corollary 2.12, then the reverse cases of the conclusions are obtained. theorem 2.14. let r ∈ n0 and s ∈ n0. then the inequality∣∣∣θ( r u + s v ) (x u + y v )∣∣∣ ≤ ∣∣∣θ(r)(x) ∣∣∣ 1 u ∣∣∣θ(s)(y) ∣∣∣ 1 v (23) holds for x > 0, y > 0, u > 1, v > 1 and 1 u + 1 v = 1. proof. by using (19) and holder’s inequality for integrals, we obtain∣∣∣θ( r u + s v ) (x u + y v )∣∣∣ = ∫ ∞ 0 a(t)t( r u + s v )−2e−( x u + y v )tdt = ∫ ∞ 0 a(t)( 1 u + 1 v )t( r u + s v )−2( 1 u + 1 v )e−( x u + y v )tdt = ∫ ∞ 0 a(t) 1 u t r−2 u e −xt u a(t) 1 v t s−2 v e −yt v dt ≤ (∫ ∞ 0 a(t)tr−2e−xtdt ) 1 u (∫ ∞ 0 a(t)ts−2e−ytdt ) 1 v = [ θ(r)(x) ] 1 u [ θ(s)(y) ] 1 v which completes the proof. � remark 2.15. if r = s in theorem 2.14, then we obtain∣∣∣θ(r) (x u + y v )∣∣∣ ≤ ∣∣∣θ(r)(x) ∣∣∣ 1 u ∣∣∣θ(r)(y) ∣∣∣ 1 v (24) which shows that, for r ∈ n0, the function ∣∣θ(r)(z) ∣∣ is logarithmically convex on (0,∞). remark 2.16. if r = k − 1, s = k + 1, u = v = 2 and x = y = z in theorem 2.14, then we obtainthe turan-type inequality ∣∣∣θ(k) (z) ∣∣∣2 ≤ ∣∣∣θ(k−1)(z) ∣∣∣ ∣∣∣θ(k+1)(z) ∣∣∣ . (25) where k ∈ n0. this is equivalent to∣∣∣θ(k+1) (z) ∣∣∣2 ≤ ∣∣∣θ(k)(z) ∣∣∣ ∣∣∣θ(k+2)(z) ∣∣∣ (26) https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 8where k ∈ n0. lemma 2.17. the function θ′(z) is increasing on (0,∞). proof. this follows directly from (19) since (θ′(z))′ = θ′′(z) > 0. � theorem 2.18. the function −θ(z) is starshaped on (0,∞). that is, the inequality −θ(αz) ≤ −αθ(z) (27) holds for α ∈ [0, 1] and z ∈ (0,∞). proof. let k(z) = θ(αz)− αθ(z) for α ∈ [0, 1] and z ∈ (0,∞). then k′(z) = αθ′(αz)− αθ′(z) = α [ θ′(αz)−θ′(z) ] < 0 since θ′(z) is increasing. hence k(z) is decreasing. then for z ∈ (0,∞), we have k(z) ≥ lim z→∞ k(z) = 0 which implies that θ(αz) ≥ αθ(z).this then gives rise to the inequality (27) and that completes the proof. � theorem 2.19. let r ∈ n0. then θ(r)(z) is strictly subadditive if r is even and θ(r)(z) is strictly superadditive if r is odd. that is, for x, y ∈ (0,∞), it holds that θ(r)(x + y) < θ(r)(x) + θ(r)(y) (28) if r is even, and θ(r)(x + y) > θ(r)(x) + θ(r)(y) (29) if r is odd. proof. let u(x, y) = θ(r)(x + y) −θ(r)(x) −θ(r)(y). with no loss of generality, let y be fixed.then by differentiating with respect to x , and using (19), we have u ′(x, y) = θ(r+1)(x + y)−θ(r+1)(x) (−1)r+1 ∫ ∞ 0 a(t)tr−1e−(x+y)tdt − (−1)r+1 ∫ ∞ 0 a(t)tr−1e−xtdt = (−1)r+1 ∫ ∞ 0 a(t)tr−1 [ e−(x+y)t − e−xt ] dt := v(x, y) https://doi.org/10.28924/ada/ma.5.13 eur. j. math. anal. 10.28924/ada/ma.5.13 9suppose that r is even. then v(x, y) > 0. this implies that u(x, y) is increasing in terms of x .hence, for x ∈ (0,∞), we have u(x, y) < lim x→∞ u(x, y) = −θ(r)(y) < 0 which gives rise to the inequality (28). likewise, suppose that r is odd. then v(x, y) < 0. thisimplies that u(x, y) is decreasing in terms of x . hence, for x ∈ (0,∞), we have u(x, y) > lim x→∞ u(x, y) = −θ(r)(y) > 0 which gives rise to the inequality (29). this completes the proof. � remark 2.20. the particular case where r = 0 in theorem 2.19, shows that the 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https://arxiv.org/pdf/2402.07740 https://doi.org/10.2478/s12175-010-0025-7 https://apjm.apacific.org/pdfs/8-7.pdf https://archive.org/details/dli.ernet.206074 https://archive.org/details/dli.ernet.206074 1. introduction 2. results and discussion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 2doi: 10.28924/ada/ma.5.2 on η-local functions in ideal topological spaces junvon a. almocera∗, lezel m. tutanes department of mathematics, college of arts and sciences, bukidnon state university, malaybalay city, bukidnon, philippines 1901103646@student.buksu.edu.ph, lezeltutanes@buksu.edu.ph ∗correspondence: 1901103646@student.buksu.edu.ph abstract. this study introduces and investigates a new local function called η-local function in idealtopological space (x, τ, i) by using the notion of η-open sets in topological space (x, τ). theoperator (·)∗η : p(x) → p(x) is defined as (·)∗η(a) = a∗η = { x ∈ x : a ∩ u /∈ i for every u ∈ η-o(x) } for each a ⊆ x , where η-o(x) is the set of all η-open subset of x containing x . thisstudy establishes some properties of a∗η including its relationships to the local function and localfunction γ∗ in ideal topological space (x, τ, i). this study also introduces a new type of closurecalled the η-local closure in ideal topological space (x, τ, i) which is denoted by cl∗η(a) for each a ⊆ x . furthermore, this study establishes some properties of the η-local closure. 1. introduction the concept of ideal topological spaces was first studied by kuratowski [4] and vaidyanathas-wamy [9]. the notion of topological spaces with ideals were investigated by jankovic [3]. thereafter,the study of ideal topological spaces attracts the attention of many topologists. recently, al-omari [1] have introduced and investigated the notion of local function γ∗ in an ideal topologicalspace and showed that γ∗ is equivalent to the δ-local function due to hatir et al. [2]. in this paper,the researcher defined a new type of local function called the η-local function in ideal topologicalspaces by using the η-open set of subbulakshmi [8] and established some of its properties, includingits relationship to the local function and local function γ∗ in ideal topological spaces. subsequently,the η-local closure has been defined, and some of the properties are established. 2. preliminaries throughout this paper (x, τ) and (x, τ, i) denote a topological space and an ideal topologicalspace, respectively. the members of τ are called open sets and their complement are calledclosed sets. for any subset a of x , the closure and interior of a are denoted by cl(a) and received: 4 jun 2024. key words and phrases. ideal topoloical space, η-open set; η-local function; η-local closure.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 2 int(a), respectively. a subset a of a space (x, τ) is said to be η-open (resp. η-closed) [8] if a ⊆ int(cl(int(a))) ∪ cl(int(a)) (resp. a ⊇ cl(int(cl(a))) ∩ int(cl(a))). the η-closure of a isdefined by the intersection of all η-closed sets containing the set a and it is denoted by η-cl(a) [8].a subset a of a space (x, τ) is said to be semi-open [6] if a ⊆ cl(int(a)). a subset a of a space (x, τ) is said to be regular-open [7] if a = int(cl(a)). the familiy of all η-open (resp. semi-open,regular open) sets in x is denoted by η-o(x) (resp. so(x), ro(x)).an ideal i [5] on a topological spaces (x, τ) is a nonempty collection of subsets of x , whichsatisfies (i) a ∈ i and b ∈ i implies a ∪ b ∈ i; and (ii) a ∈ i and b ⊆ a implies b ∈ i . then thetriplet (x, τ, i) is called an ideal topological space. if p(x) is the set of all subsets of x , a setoperator (·)∗ : p(x)→ p(x) called a local function [3,9] of a with respect to τ and i is defined asfollows: for a ⊆ x , a∗(i, τ) = {x ∈ x : u∩a /∈ i, for every u ∈ τ(x)} where τ(x) = {u ∈ τ : x ∈ u}. a∗(i, τ) can simply be written as a∗. for every ideal topological space, there exists a topology τ∗(i, τ) or briefly τ∗ [3], finer than τ , generated by the b(i, τ) = {u \ j : u ∈ τ and j ∈ i},however, b(i, τ) is not a topology in general. additionally, cl∗(a) = a∪a∗ defines a kuratowskiclosure operator [5] for τ∗. a subset a of an ideal topological spaces is τ∗-closed set or ∗-closedset [3] if a∗ ⊆ a. let (x, τ, i) be an ideal topological spaces and a be a subset of x . then γ∗(a)(i, τ) = {x ∈ x : a∩u /∈ i, for every u ∈ ro(x)} where ro(x) = {u ∈ ro(x) : x ∈ u}. γ∗(a)(i, τ) can simply be denoted as γ∗(a) [1]. 3. η-local functions definition 1. let (x, τ, i) be an ideal topological space. then the operator (·)∗η : p(x)→ p(x) is defined as for a ⊆ x , a∗η ( i, η-o(x) ) = {x ∈ x : a ∩ u /∈ i, for every u ∈ η-o(x)} where η-o(x) = {u ∈ η-o(x) : x ∈ u} is called the η-local f unction of a with respect to i and η-o(x). a∗η ( i, η-o(x) ) can simply be denoted by a∗η . example 1. let (x, τ, i) be an ideal topological where x = {a, b, c, d}, τ = {x,∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}}, and i = {∅, {c}, {d}, {c, d}}. then η-o(x) = {∅, x, {a}, {b}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {a, b, c}, {a, c, d}, {a, b, d}, {b, c, d}}. now, let a = {a, b, d}. then by definition 1, a∗η = {a, b, c, d} = x . theorem 1. let (x, τ, i) be an ideal topological space and a,b be subsets of x . then for any η-local functions, the following properties hold: (i) if a ⊆ b, then a∗η ⊆ b∗η; (ii) (a ∩ b)∗η ⊆ a∗η ∩ b∗η; and (iii) a∗η ∪ b∗η ⊆ (a ∪ b)∗η; (iv) if a = ∅, then a∗η = ∅; (v) if a∗η ∩ b /∈ i , then a∗η ∩ b 6= ∅; (vi) ( a∗η )∗ η ⊆ a∗η; https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 3 (vii) if a ∈ i , then a∗η = ∅; (iix) if i = {∅}, then a∗η = η-cl(a); (ix) if i = p(x), then a∗η = ∅; and (x) a∗η ⊆ η-cl(a); proof. (i) let a,b ⊆ x and a ⊆ b. suppose x /∈ b∗η , then there exist u ∈ η-o(x) such that b ∩ u ∈ i . since a ⊆ b, a ∩ u ⊆ b ∩ u ∈ i , by definition of ideal, a ∩ u ∈ i . hence, x /∈ a∗η . (ii) let a,b ⊆ x . since a ∩ b ⊆ a and a ∩ b ⊆ b, by theorem 1 (i), (a ∩ b)∗η ⊆ a∗η and (a ∩ b)∗η ⊆ b∗η , respectively. hence, (a ∩ b)∗η ⊆ a∗η ∩ b∗η . (iii) let a,b ⊆ x . since a ⊆ a ∪ b and b ⊆ a ∪ b, by theorem 1 (i), a∗η ⊆ (a ∪ b)∗η and b∗η ⊆ (a ∪ b)∗η , respectively. hence, a∗η ∪ b∗η ⊆ (a ∪ b)∗η . (iv) let a = ∅. suppose a∗η 6= ∅. then there exists x ∈ a∗η . it follows that a ∩ u = ∅ ∩ u = ∅ /∈ i for every u ∈ η-o(x). since i is an ideal, ∅ ∈ i which is a contradiction. (v) let a∗η ∩ b /∈ i . suppose a∗η ∩ b = ∅. note that by definition of ideal, ∅ ∈ i for anyideal i . now, since a∗η ∩ b = ∅ and i is an ideal, a∗η ∩ b = ∅ ∈ i implies a∗η ∩ b ∈ i , acontradiction. (vi) let x ∈ (a∗η)∗η . then, for every u ∈ η-o(x), u ∩ a∗η /∈ i and hence, by (ii), u ∩ a∗η 6= ∅.now, let y ∈ u ∩a∗η . then, u ∈ η-o(y) and y ∈ a∗η . hence, we have u ∩a /∈ i . note that u ∈ η-o(x) and u ∩ a /∈ i . it follows that x ∈ a∗η . therefore, (a∗η)∗η ⊆ a∗η . (vii) let a ∈ i . suppose a∗η 6= ∅. then there exists an element x ∈ a∗η . then a ∩ u /∈ i forevery u ∈ η-o(x). now, since, a ∩ u ⊆ a ∈ i and i is an ideal, a ∩ u ∈ i which is acontradiction. (iix) let i = {∅}. suppose that a∗η 6= η-cl(a). let η-cl(a) ⊂ a∗η , then there exists an element x ∈ a∗η and x /∈ η-cl(a). it follows that for every a ⊆ x , since x ∈ a∗η , a∩u /∈ i for every u ∈ η-o(x). since i = {∅}, a∩u 6= ∅ for every u ∈ η-o(x). note that u ∈ η-o(x) means x ∈ u where u is η-open set. since, x /∈ η-cl(a), x /∈ ⋂{k : k is η-closed and a ⊆ k}.it follows that x /∈ k for some η-closed set k such that a ⊆ k. hence, x ∈ kc for some η-open set kc such that a ∩kc = ∅. it implies that there exists an η-open set kc where x ∈ kc and a ∩kc = ∅, a contradiction. (ix) let i = p(x). note that a ⊆ x , then a ∈ p(x). since i = p(x), a ∈ i . hence, bytheorem 1 (vii), a∗η = ∅. (x) let x /∈ η-cl(a). then, x /∈ ⋂{k : k is η-closed and a ⊆ k}. it follows that x /∈ k forsome η-closed set k such that a ⊆ k. hence, x ∈ kc for some η-open set kc such that a ∩ kc = ∅. it implies that there exists kc ∈ η-o(x) such that a ∩ kc = ∅, and bydefinition of ideal, ∅ ∈ i for any ideal i . hence, a ∩ kc ∈ i for some kc ∈ η-o(x). thisshows that x /∈ a∗η . � https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 4 remark 1. let (x, τ, i) be an ideal topological space and a be any subset of x . then for any η-local functions, the following properties hold:(i) the reverse inclusion of theorem 1 (iii) need not be true in general.(ii) neither a ⊆ a∗η nor a∗η ⊆ a in general.(iii) a∗η is an η-closed set iff a∗η = η-cl(a∗η). in order to verify remark 1 (i) and (ii), the following examples are shown. example 2.(i) consider the ideal topological space (x, τ, i), where x = {a, b, c, d}, τ = {x,∅, {b}, {c}, {b, c}, {a, b, c}}, and i = {∅, {a}}. then the η-open sets are ∅, x , {b}, {c}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {a, b, d}, and {b, c, d}. now, let a = {b} and b = {c}, then a ∪ b = {b, c}. then by applying definition 1, a∗η = {b}, b∗η = {c}, and (a ∪ b)∗η = x . observe that (a ∪ b)∗η = x and a∗η ∪ b∗η = {b, c}. these shows that (a ∪ b)∗η * a∗η ∪ b∗η .(ii) consider the ideal topological space (x, τ, i), where x = {a, b, c, d}, τ = {x,∅, {b}, {c}, {b, c, d}}, and i = {∅, {c}}. then the η-open sets are ∅, x , {b}, {c}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, and {b, c, d}. let a,b ⊂ x where, a = {a, c, d} and b = {a, b}. then by definition 1, a∗η = {a, d} and b∗η = {a, b, d}. obeserve that a * a∗η and b∗η * b. theorem 2. let (x, τ, i) be an ideal topological space and a,b be subsets of x . then for any η-local functions, the following properties hold: (i) (a \ b)∗η \ b∗η ⊆ a∗η \ b∗η; (ii) if b ∈ i , then (a ∪ b)∗η = a∗η = (a \ b)∗η; (iii) (a \ b)∗η ∪ (b \ a)∗η ⊆ (a ∪ b)∗η; (iv) if u ⊆ x , then u ∩ (u ∩ a)∗η ⊆ u ∩ a∗η; (v) if u ∈ i , then (a ∩ u)∗η = ∅; (vi) if a is an η-closed set, then a∗η ⊆ a; (vii) ( a ∩ a∗η )∗ η ⊆ a∗η; (iix) if a ∪ b ∈ i , then (a ∪ b)∗η = a∗η ∪ b∗η = ∅. (ix) a∗η = η-cl(a∗η) ⊆ η-cl(a) and a∗η is an η-closed set; and (x) if a ⊆ a∗η , then a∗η = η-cl(a∗η) = η-cl(a). proof. (i) let a,b ⊆ x . since a\b ⊆ a, by theorem 1 (i), (a\b)∗η ⊆ a∗η implies that (a\b)∗η\b∗η ⊆ a∗η \ b∗η . (ii) let b ∈ i . suppose x ∈ (a∪b)∗η . then for every u ∈ η-o(x), (a∪b)∩ u /∈ i . note that (a∩u)∪(b∩u) = (a∪b)∩u /∈ i implies that (a∩u)∪(b∩u) /∈ i . it shows that a∩u /∈ i https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 5or b∩u /∈ i , or both, and as a result, x ∈ a∗η or x ∈ b∗η , or both. hence, x ∈ a∗η∪b∗η . now,note that b ∈ i , then by theorem 1 (vii), b∗η = ∅. thus, x ∈ a∗η ∪b∗η = a∗η ∪∅ = a∗η . thisimplies that x ∈ a∗η . hence, it shows that (a∪b)∗η ⊆ a∗η . in contrast, since a ⊆ a∪b, bytheorem 1 (i), it implies that a∗η ⊆ (a ∪ b)∗η . consequently, as a result, a∗η = (a ∪ b)∗η .now, suppose that (a\b)∗η 6= a∗η . let (a\b)∗η ⊂ a∗η . then there exists an element x ∈ a∗ηsuch that x /∈ (a \b)∗η . note that x ∈ a∗η implies that for every a ⊆ x , a∩u /∈ i for every u ∈ η-o(x). now, since x /∈ (a \b)∗η , there exists u ∈ η-o(x) such that (a \b) ∩ u ∈ i .note that (a \ b) ∩ u = (a ∩ u) \ (b ∩ u) ∈ i and b ∩ u ⊆ b ∈ i . now, since i is anideal, b ∩ u ∈ i and [(a ∩ u) \ (b ∩ u) ] ∪ (b ∩ u) ∈ i , respectively. again, note that[ (a∩u)\(b∩u) ] ∪(b∩u) = (a∩u)∪(b∩u) ∈ i . since (a∩u) ⊆ (a∩u)∪(b∩u) ∈ i ,it implies that a ∩ u ∈ i . so, there exists u ∈ η-o(x) such that a ∩ u ∈ i which is acontradiction. (iii) let a,b ⊆ x . note that a \b ⊆ a and b \a ⊆ b. then by theorem 1 (i), (a \b)∗η ⊆ a∗ηand (b \ a)∗η ⊆ b∗η , and so, (a \ b)∗η ∪ (b \ a)∗η ⊆ a∗η ∪ b∗η . now, by theorem 1 (iii), a∗η ∪ b∗η ⊆ (a ∪ b)∗η . hence, (a \ b)∗η ∪ (b \ a)∗η ⊆ (a ∪ b)∗η . (iv) let u ⊆ x . since u ∩a ⊆ a, by theorem 1 (i), (u ∩a)∗η ⊆ a∗η , and hence, u ∩ (u ∩a)∗η ⊆ u ∩ a∗η . (v) let u ∈ i . since a ∩ u ⊆ u ∈ i and i is an ideal, a ∩ u ∈ i . hence, by theorem 1 (vii), (a ∩ u)∗η = ∅. (vi) let a be an η-closed set. then a = η-cl(a). now, note that by theorem 1 (x), a∗η ⊆ η-cl(a). hence, a∗η ⊆ a. (vii) let a ⊆ x . since a ∩ a∗η ⊆ a∗η , by theorem 1 (i), (a ∩ a∗η)∗η ⊆ (a∗η)∗η . note that bytheorem 1 (vi), (a∗η)∗η ⊆ a∗η . therefore, (a ∩ a∗η)∗η ⊆ a∗η . (iix) let a∪b ∈ i . since a∪b ∈ i and i is an ideal, a ∈ i and b ∈ i . these imply by theorem1 (vii), (a ∪ b)∗η = ∅, a∗η = ∅, and b∗η = ∅. therefore, (a ∪ b)∗η = a∗η ∪ b∗η = ∅. (ix) suppose that a∗η 6= η-cl(a∗η). let η-cl(a∗η) ⊂ a∗η . then there exists an element x ∈ a∗ηsuch that x /∈ η-cl(a∗η). note that since x ∈ a∗η , for every u ∈ η-o(x), a ∩ u /∈ i . now,note that x /∈ η-cl(a∗η). then x /∈ ⋂{k : k is η-closed and a∗η ⊆ k}. this shows that x /∈ k for some η-closed set k such that a∗η ⊆ k. this implies that x ∈ kc for some η-open set kc such that kc ∩ a∗η = ∅. note that x ∈ kc and kc ∩ a∗η = ∅, then itfollows that x /∈ a∗η , and so, for some kc ∈ η-o(x), a ∩ kc ∈ i , which is a contradiction.consequently, a∗η = η-cl(a∗η), then by remark 1 (iii), a∗η is an η-closed set. now, notethat a∗η = η-cl(a∗η) and by theorem 1 (x), hence, a∗η = η-cl(a∗η) ⊆ η-cl(a). (x) let a ⊆ a∗η . suppose that x ∈ η-cl(a). then x ∈ ⋂{k : k is η-closed and a ⊆ k}.this shows that x ∈ k for every η-closed set k such that a ⊆ k. note that by theorem2 (ix), a∗η is an η-closed set. now, note that since a ⊆ a∗η and a∗η is an η-closed set, https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 6 a∗η ∈ {k : k is η-closed and a ⊆ k}. this implies that x ∈ a∗η . hence, η-cl(a) ⊆ a∗η . asa result, by theorem 1 (x) and 2 (ix), a∗η = η-cl(a∗η) = η-cl(a). � theorem 3. let (x, τ, i) be an ideal topological space where η-o(x) is closed under any two intersections. then for any a,b subsets of x , the following properties hold: (i) (a ∪ b)∗η = a∗η ∪ b∗η; (ii) for u ∈ η-o(x), u ∩ a∗η = u ∩ (u ∩ a)∗η ⊆ (u ∩ a)∗η; and (iii) a∗η \ b∗η = (a \ b)∗η \ b∗η ⊆ (a \ b)∗η . proof. (i) let η-o(x) be closed under any two intersections. suppose that x /∈ a∗η∪b∗η , then x /∈ a∗ηand x /∈ b∗η implying that there exist u, v ∈ η-o(x) such that a ∩ u ∈ i and b ∩ v ∈ i .note that a ∩ u ∈ i , b ∩ v ∈ i , and i is an ideal. then (a ∩ u) ∪ (b ∩ v ) ∈ i . since u ∩ v ⊆ u and u ∩ v ⊆ v , (a ∩ u) ∪ (b ∩ v ) ⊇ [ a ∩ (u ∩ v ) ] ∪ [ b ∩ (u ∩ v ) ] = (a ∪ b) ∩ (u ∩ v ). it implies that (a ∪ b) ∩ (u ∩ v ) ⊆ (a ∩ u) ∪ (b ∩ v ) ∈ i . again, since i is an ideal, (a ∪ b) ∩ (u ∩ v ) ∈ i . now, note that by assumption, η-o(x) is closed under any twointersections, and so, there exists u ∩ v ∈ η-o(x) such that (a ∪ b) ∩ (u ∩ v ) ∈ i . thisshows that x /∈ (a ∪b)∗η . hence, (a ∪b)∗η ⊆ a∗η ∪b∗η . now, by theorem 1 (iii), therefore, (a ∪ b)∗η = a∗η ∪ b∗η . (ii) let η-o(x) be closed under any two intersections. for u ∈ η-o(x), suppose that x ∈ u ∩a∗η . then x ∈ u and x ∈ a∗η . to show that x ∈ (u ∩a)∗η , let v ∈ η-o(x). since x ∈ uand u ∈ η-o(x), we can write it as u ∈ η-o(x). hence, by assumption, u ∩ v ∈ η-o(x).note that since x ∈ a∗η and u∩v ∈ η-o(x), then a∩(u∩v ) /∈ i for every u∩v ∈ η-o(x).now, by associativity and commutativity, a ∩ (u ∩ v ) = (a ∩ u) ∩ v /∈ i = (u ∩ a) ∩ v /∈ i. this shows that for every v ∈ η-o(x), (u∩a)∩v /∈ i . it implies that x ∈ (u∩a)∗η . hence, u ∩a∗η ⊆ (u ∩a)∗η . now, note that u ∩a∗η ⊆ (u ∩a)∗η , then u ∩ (u ∩a∗η) ⊆ u ∩ (u ∩a)∗η .since u ∩ (u ∩ a∗η), by associativity again, u ∩ (u ∩ a∗η) = (u ∩ u) ∩ a∗η = u ∩ a∗η.this implies that u ∩ a∗η ⊆ u ∩ (u ∩ a)∗η . in contrast, note that u ∩ a ⊆ a, then bytheorem 1 (i), (u ∩ a)∗η ⊆ a∗η . thus, u ∩ (u ∩ a)∗η ⊆ u ∩ a∗η . consequently, as a https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 7result, u ∩ a∗η = u ∩ (u ∩ a)∗η . note that u ∩ (u ∩ a)∗η ⊆ (u ∩ a)∗η . this shows that u ∩ a∗η = u ∩ (u ∩ a)∗η ⊆ (u ∩ a)∗η . (iii) let a,b ⊆ x . note that a = (a \ b) ∪ (b ∩ a). thus a∗η = [ (a \ b) ∪ (b ∩ a) ]∗ η . notethat by assumption, η-o(x) is closed under any two intersections, then by theorem 3 (i), a∗η = [ (a \ b) ∪ (b ∩ a) ]∗ η = (a \ b)∗η ∪ (b ∩ a)∗η.so, a∗η = (a \ b)∗η ∪ (b ∩ a)∗η . now, note that a∗η \ b∗η = a∗η ∩ ( b∗η )c , and since a∗η = (a \ b)∗η ∪ (b ∩ a)∗η , a∗η \ b∗η = a∗η ∩ ( b∗η )c = [ (a \ b)∗η ∪ (b ∩ a)∗η ] ∩ ( b∗η )c = [ (a \ b)∗η ∩ ( b∗η )c] ∪ [(b ∩ a)∗η ∩ ( b∗η )c] = [ (a \ b)∗η \ b∗η ] ∪ [ (b ∩ a)∗η \ b∗η ] . hence, a∗η \ b∗η = [ (a \ b)∗η \ b∗η ] ∪ [ (b ∩ a)∗η \ b∗η ]. note that b ∩ a ⊆ b, then bytheorem 1 (i), (b ∩ a)∗η ⊆ b∗η implies that (b ∩ a)∗η \ b∗η = ∅. now, since a∗η \ b∗η =[ (a \ b)∗η \ b∗η ] ∪ [ (b ∩ a)∗η \ b∗η ] and (b ∩ a)∗η \ b∗η = ∅, it follows that a∗η \ b∗η = [ (a \ b)∗η \ b∗η ] ∪ [ (b ∩ a)∗η \ b∗η ] = [ (a \ b)∗η \ b∗η ] ∪∅ = (a \ b)∗η \ b∗η ⊆ (a \ b)∗η.as a result, it shows that a∗η \ b∗η = (a \ b)∗η \ b∗η ⊆ (a \ b)∗η . � theorem 4. let (x, τ, i) be an ideal topological space and a ⊆ x . then for any η-local function, the following properties hold: (i) a∗η ⊆ a∗; (ii) a∗η ⊆ γ∗(a); and (iii) a∗ ⊆ γ∗(a). proof. (i) let x ∈ a∗η and u ∈ τ(x). since every open set is η-open set, u ∈ η-o(x). also, since x ∈ a∗η and u ∈ η-o(x), a ∩ u /∈ i . note that a ∩ u /∈ i and u ∈ τ(x). hence, a ∩ u /∈ ifor every u ∈ τ(x), and so, x ∈ a∗. therefore, a∗η ⊆ a∗. (ii) let x ∈ a∗η and u ∈ ro(x). since every regular-open set is η-open set, u ∈ η-o(x). also,since x ∈ a∗η and u ∈ η-o(x), a ∩ u /∈ i . hence, a ∩ u /∈ i for every u ∈ ro(x), and so, x ∈ γ∗(a). therefore, a∗η ⊆ γ∗(a). https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 8 (iii) let x ∈ a∗ and u ∈ ro(x). since every regular-open set is open set, u ∈ τ . also, since x ∈ a∗ and u ∈ τ , a ∩ u /∈ i . hence, a ∩ u /∈ i for every u ∈ ro(x), and so, x ∈ γ∗(a).therefore, a∗ ⊆ γ∗(a). � remark 2. let (x, τ, i) be an ideal topological space and a ⊆ x . then for any η-local functions, the following properties hold: (ii) a∗η ⊆ a∗ ⊆ γ∗(a);(ii) if η-o(x) = τ , then a∗η = a∗; and(iii) if η-o(x) = ro(x), then a∗η = γ∗(a) theorem 5. let (x, τ) be a topological space with ideals i1 and i2 on x and a ⊆ x . then, for any η-local functions, the following properties hold: (i) if i1 ⊆ i2, then a∗η ( i2, η-o(x) ) ⊆ a∗η ( i1, η-o(x) ) ; and (ii) a∗η ( (i1 ∩ i2), η-o(x) ) = a∗η ( i1, η-o(x) ) ∪ a∗η ( i2, η-o(x) ) . proof. (i) let i1 ⊆ i2 and x ∈ a∗η(i2, η-o(x) ). then for every u ∈ η-o(x), a∩u /∈ i2. since i1 ⊆ i2, a ∩ u /∈ i1 for every u ∈ η-o(x). hence, a∗η(i2, η-o(x) ) ⊆ a∗η ( i1, η-o(x) ). (ii) let i1 and i2 be ideals on x . note that i1 ∩ i2 ⊆ i1 and i1 ∩ i2 ⊆ i2. then by theorem 5 (i), a∗η ( i1, η-o(x) ) ⊆ a∗η ( (i1 ∩ i2), η-o(x) ) and a∗η ( i2, η-o(x) ) ⊆ a∗η ( (i1 ∩ i2), η-o(x) ) , and hence, a∗η ( i1, η-o(x) ) ∪ a∗η ( i2, η-o(x) ) ⊆ a∗η ( (i1 ∩ i2), η-o(x) ) . next, let x ∈ a∗η((i1∩i2), η-o(x) ), then for every u ∈ η-o(x), a∩u /∈ i1∩i2. this impliesthat a∩u /∈ i1 or a∩u /∈ i2. this shows that x ∈ a∗η(i1, η-o(x) ) or x ∈ a∗η(i2, η-o(x) ).hence, x ∈ a∗η(i1, η-o(x) ) ∪ a∗η ( i2, η-o(x) ), and so, a∗η ( (i1 ∩ i2), η-o(x) ) ⊆ a∗η ( i1, η-o(x) ) ∪ a∗η ( i2, η-o(x) ) . as a result, thus, a∗η ( (i1 ∩ i2), η-o(x) ) = a∗η ( i1, η-o(x) ) ∪ a∗η ( i2, η-o(x) ) . � https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 94. η-local closure definition 2. let (x, τ, i) be an ideal topological space. the η-local closure of a denoted by cl∗η(a) is defined by the union of a and the η-local function of a, i.e, cl∗η(a) = a ∪ a∗η for any a ⊆ x . example 3. let (x, τ, i) be an ideal topological space where x = {a, b, c}, τ = {∅, x, {a}, {b}, {a, b}}, and i = {∅, {b}}. then the η-open sets of x are ∅, x , {a}, {b}, {a, b}, {a, c}, and {b, c}. let a = {a, b}, then by definition 1 and 2, a∗η = {a} and cl∗η(a) = {a, b} ∪ {a} = {a, b}, respectively. theorem 6. let (x, τ, i) be an ideal topological space and a,b ⊆ x . then the following properties hold: (i) if a ⊆ b, then cl∗η(a) ⊆ cl∗η(b); (ii) cl∗η(a ∩ b) ⊆ cl∗η(a) ∩ cl∗η(b); (iii) if a is an η-closed set, then cl∗η(a) = η-cl(a); (iv) if a ∈ i , then cl∗η(a) = a; (v) cl∗η(a∗η) = a∗η; (vi) cl∗η(a) = η-cl(a); and (vii) ( cl∗η(a) )∗ η = a∗η . proof. (i) let a,b ⊆ x and a ⊆ b. by definition 2, cl∗η(a) = a ∪ a∗η and cl∗η(b) = b ∪ b∗η .since a ⊆ b, by theorem 1 (i), a∗η ⊆ b∗η . this shows that a ∪ a∗η ⊆ b ∪ b∗η , and hence, cl∗η(a) ⊆ cl∗η(b). (ii) let a,b ⊆ x . since a ∩ b ⊆ a and a ∩ b ⊆ b, by theorem 6 (i), cl∗η(a ∩ b) ⊆ cl∗η(a)and cl∗η(a ∩ b) ⊆ cl∗η(b). hence, it implies cl∗η(a ∩ b) ⊆ cl∗η(a) ∩ cl∗η(b). (iii) let a be an η-closed set, then a = η-cl(a). now, suppose that x /∈ a. it implies that x /∈ η-cl(a), then x /∈ ⋂{k : k is η-closed and a ⊆ k}. it follows that x /∈ k for some η-closed set k such that a ⊆ k. hence, x ∈ kc for some η-open set kc such that a ∩ kc = ∅. it implies that there exists kc ∈ η-o(x) such that a ∩ kc = ∅, and bydefinition of ideal, ∅ ∈ i for any ideal i . hence, a ∩ kc ∈ i for some kc ∈ η-o(x). thisshows that x /∈ a∗η , and hence, a∗η ⊆ a. it follows that cl∗η(a) = a ∪ a∗η = a. note that a = η-cl(a). therefore, cl∗η(a) = η-cl(a). (iv) let a ∈ i . then by definition 2 and theorem 1 (vii), cl∗η(a) = a ∪ a∗η = a ∪ ∅ = a.consequently, cl∗η(a) = a. (v) let a ⊆ x . then by definition 2 and theorem 1 (vi), cl∗η(a∗η) = a∗η ∪ ( a∗η )∗ η = a∗η. itfollows that, cl∗η(a∗η) = a∗η . https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 10 (vi) let a ⊆ x . suppose that cl∗η(a) 6= η-cl(a). let η-cl(a) ⊂ cl∗η(a). then there existsan element x ∈ cl∗η(a) such that x /∈ η-cl(a). note that since x ∈ cl∗η(a), by definition2, x ∈ a ∪ a∗η implies that x ∈ a or x ∈ a∗η , or both. suppose x ∈ a∗η . then for every u ∈ η-o(x), a ∩ u /∈ i . now, since x /∈ η-cl(a), x /∈ ⋂{k : k is η-closed and a ⊆ k}.it follows that x /∈ k for some η-closed set k such that a ⊆ k. hence, x ∈ kc for some η-open set kc such that a ∩kc = ∅, and by definition of an ideal, ∅ ∈ i for any ideal i .it implies that there exists kc ∈ η-o(x) such that a ∩ kc ∈ i , and hence, x /∈ a∗η . also,note that since x ∈ kc and a ∩ kc = ∅, x /∈ a. this shows that x /∈ a and x /∈ a∗η , acontradiction. (vii) let a ⊆ x . then by definition 2 and theorem 1 (iii), (cl∗η(a) )∗ η = ( a∪a∗η )∗ η ⊇ a∗η∪ ( a∗η )∗ η .note that by theorem 1 (vi), (a∗η)∗η ⊆ a∗η , then a∗η ∪ (a∗η)∗η = a∗η . it implies that a∗η ⊆( cl∗η(a) )∗ η . now, let x ∈ (cl∗η(a) )∗ η . then for every u ∈ η-o(x), cl∗η(a) ∩ u /∈ i . now, bydefinition 2, cl∗η(a)∩u = (a∪a∗η)∩u /∈ i = (a∩u)∪(a∗η∩u) /∈ i . it implies that a∩u /∈ ior a∗η∩u /∈ i , or both, and so, x ∈ a∗η or x ∈ (a∗η)∗η , or both. it follows that x ∈ a∗η∪(a∗η)∗η .note that a∗η ∪ (a∗η)∗η = a∗η , and so, x ∈ a∗η . consequently, (cl∗η(a) )∗ η ⊆ a∗η . thus,( cl∗η(a) )∗ η = a∗η . � theorem 7. let (x, τ, i) be an ideal topological space and a,b ⊆ x . then the following properties hold: (i) a ⊆ cl∗η(a) and a∗η ⊆ cl∗η(a); (ii) cl∗η(∅) = ∅ and cl∗η(x) = x; (iii) cl∗η(a) ∪ cl∗η(b) ⊆ cl∗η(a ∪ b); and (iv) ( cl∗η(a) )∗ η ⊆ cl∗η(a) = cl∗η ( cl∗η(a) ) . proof. (i) let a ⊆ x . note that a ⊆ a ∪ a∗η . then by definition 2, a ⊆ cl∗η(a). next, note that a∗η ⊆ a ∪ a∗η , by definition 2 again, it implies that a∗η ⊆ cl∗η(a). (ii) by definition 2 and theorem 1 (iv), cl∗η(∅) = ∅ ∪ (∅)∗η = ∅ ∪∅ = ∅. next, note that x is auniversal set, then (x)∗η ⊆ x . hence, by definition 2, cl∗η(x) = x ∪ (x)∗η = x . (iii) let a,b ⊆ x . by definition 2 and thoerem 1 (iii), cl∗η(a ∪ b) = (a ∪ b) ∪ (a ∪ b)∗η ⊇ (a ∪ b) ∪ (a∗η ∪ b∗η) = (a ∪ a∗η) ∪ (b ∪ b∗η) = cl∗η(a) ∪ cl∗η(b). this shows that cl∗η(a) ∪ cl∗η(b) ⊆ cl∗η(a ∪ b). https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 11 (iv) let a ⊆ x . note that by theorem 6 (vii), (cl∗η(a) )∗ η = a∗η , and by theorem 7 (i), a∗η ⊆ cl∗η(a).hence, it shows that (cl∗η(a) )∗ η ⊆ cl∗η(a). next, by definition 2, cl∗η(cl∗η(a) ) = cl∗η(a) ∪( cl∗η(a) )∗ η . note that since (cl∗η(a) )∗ η ⊆ cl∗η(a), cl∗η ( cl∗η(a) ) = cl∗η(a) ∪ ( cl∗η(a) )∗ η = cl∗η(a). it follows that, cl∗η(a) = cl∗η ( cl∗η(a) ). � remark 3. the reverse inclusion of theorem 7 (iii) need not be true in general as shown from the following example. example 4. let (x, τ, i) be an ideal topological space where x = {a, b, c, d}, τ = { ∅, x, {c}, {d}, {c, d} } , and i = { ∅, {a}, {b}, {a, b} } . then the η-open sets of x are ∅, x , {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {a, b, d}, and {b, c, d}. let a = {c} and b = {d} such that a∪b = {c, d}, then by definition 1, a∗η = {c}, b∗η = {d}, and (a∪b)∗η = x . now, by definition 2, cl∗η(a) = {c}, cl∗η(b) = {d}, and cl∗η(a ∪ b) = x . observe that cl∗η(a ∪ b) = x and cl∗η(a) ∪ cl∗η(b) = {c, d}. these shows that cl∗η(a ∪ b) * cl∗η(a) ∪ cl∗η(b). hence, the above assertion has been verified. theorem 8. let (x, τ, i) be an ideal topological space and a ⊆ x . then cl∗η(a) ⊆ cl∗(a). proof. let a ⊆ x . by definition 2 and kuratowski closure operator, cl∗η(a) = a ∪ a∗η and cl∗(a) = a ∪ a∗, respectively. since by theorem 4 (i), a∗η ⊆ a∗, a ∪ a∗η ⊆ a ∪ a∗. hence, cl∗η(a) ⊆ cl∗(a). � theorem 9. let (x, τ, i) be an ideal topological space and a be any subset of x . then a is an η-closed set iff a = cl∗η(a). proof. let a be an η-closed set. then a = η-cl(a) and by theroem 6 (vi), cl∗η(a) = η-cl(a),respectively. note that a = η-cl(a) and η-cl(a) = cl∗η(a), then by transitive property, it impliesthat a = cl∗η(a). now, on the other hand, let a = cl∗η(a). note that by theorem 6 (vi), cl∗η(a) = η-cl(a). now that a = cl∗η(a) and cl∗η(a) = η-cl(a), by transitive property again, a = η-cl(a). therefore, a is an η-closed set. � note that theorem 7 (i), (ii), and (iv) satisfy three of the kuratowski closure axioms. however,theorem 7 (iii) did not satisfy one of the kuratowski closure axioms because it is an inclusionproperty. as a result, the following remark is obtained. remark 4. the η-local closure, i.e, cl∗η , need not be a kuratowski closure operator with respect to η in general. https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 12 theorem 10. let (x, τ, i) be an ideal topological space where η-o(x) is closed under any two intersections and a,b ⊆ x . then cl∗η(a ∪ b) = cl∗η(a) ∪ cl∗η(b). proof. let η-o(x) be closed under any two intersections. then by definition 2 and theorem 3 (i),it follows that cl∗η(a ∪ b) = (a ∪ b) ∪ (a ∪ b)∗η = (a ∪ b) ∪ (a∗η ∪ b∗η) = (a ∪ a∗η) ∪ (b ∪ b∗η) = cl∗η(a) ∪ cl∗η(b). hence, cl∗η(a ∪ b) = cl∗η(a) ∪ cl∗η(b). � note that theorem 7 (i), (ii), (iv) and theorem 10 using the condition, for any ideal topologicalspaces (x, τ, i) where η-o(x) is closed under any two intersections, satisfy the kuratowski closureaxioms. as a result, the following remark is obtained remark 5. let (x, τ, i) be an ideal topological space where η-o(x) is closed under any two intersections, the η-local closure, i.e, cl∗η , is a kuratowski closure operator (or almocera closure operator) with respect to η. note that by remark 5, cl∗η is a kuratowski closure operator (or almocera closure operator)with respect to η for any ideal topological space (x, τ, i) where η-o(x) is closed under any twointersections. now, let a be a τ∗η-closed set iff a∗η ⊆ a in any ideal topological space (x, τ, i)where η-o(x) is closed under any two intersections. then the following lemma is obtained. lemma 1. let a be a τ∗η-closed set iff a∗η ⊆ a in any ideal topological space (x, τ, i) where η-o(x) is closed under any two intersections. then a is τ∗η-closed set iff cl∗η(a) = a. proof. let a be a τ∗η-closed in (x, τ, i) where η-o(x) is closed under any two intersections. now,since a is a τ∗η-closed, by assumption, a∗η ⊆ a. it follows that a ∪ a∗η = a. now, by definiton2, cl∗η(a) = a ∪ a∗η = a. therefore, cl∗η = a. on the other hand, let cl∗η(a) = a. now, bydefiniton 2, cl∗η(a) = a ∪ a∗η = a. so, a ∪ a∗η = a implies a∗η ⊆ a, and so, by assumption, a is τ∗η-closed. � theorem 11. let (x, τ, i) be an ideal topological space where η-o(x) is closed under any two intersections. let τ∗η = { j ⊆ x : cl∗η(jc) = jc } . then τ∗η is a topology for x such that τ∗ ⊆ τ∗η and η-o(x) ⊆ τ∗η . proof. let η-o(x) be closed under any two intersections. note that by remark 5, cl∗η is akuratowski closure operator with respect to η. therefore, τ∗η is a topology generated by cl∗η . now,to show that τ∗ ⊆ τ∗η , let a be a τ∗-open. then ac is a τ∗-closed. then by definition of τ∗-closed, https://doi.org/10.28924/ada/ma.5.2 eur. j. math. anal. 10.28924/ada/ma.5.2 13( ac )∗ ⊆ ac . hence, cl∗(ac) = ac ∪ ( ac )∗ = ac implies that cl∗(ac) = ac . then by theorem8, cl∗η(ac) ⊆ ac . now, since cl∗η(ac) ⊆ ac , by theorem 7 (i), cl∗η(ac) = ac . hence, by lemma1, ac is a τ∗η-closed, and so, a is a τ∗η-open. as a result, thus, τ∗ ⊆ τ∗η . next, to show that η-o(x) ⊆ τ∗η , let a be an η-open. then ac is an η-closed and by theorem 9, ac = cl∗η(ac). itfollows that by lemma 1, ac is a τ∗η-closed implies that a is a τ∗η-open. hence, η-o(x) ⊆ τ∗η . � 5. conclusion the concept of the η-local function and the closure cl∗η has been introduced and demonstratedthrough illustrative examples. additionally, certain properties have been studied and explored. itcan be concluded that the closure cl∗η can only be a kuratowski closure operator (almocera closureoperator) if η-o(x) is closed under two intersections. under this condition, τ∗η can form a topology,making τ∗η a more generalized version of τ∗ and η-o(x). references [1] a. al-omari, t. noiri, local function γ∗ in ideal topological spaces, sci. stud. res. ser. math. inform. 26 (1) (2016)5-16.[2] e. hatir, a. al-omari, s. jafari, δ-local functions and its properties in ideal topological spaces, fasciculi math. 53(2014) 53-64.[3] d. jankovic, t.r. hamlett, new topologies from old via ideals, amer. math. monthly 97 (4) (1990) 295-310.[4] k. kuratowski, topology i, warszawa, 1933.[5] k. kuratowski, topology, academic press, new york, 1966.[6] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly 70 (1963) 36-41.[7] p.l. powar, k. rajak, some new concepts of continuity in generalized topological space, int. j. com. appl. 38 (5)(2012) 12-17.[8] d. subbulakshmi, k. sumathi, k. indiran, η-open sets in topological, int. j. innov. techno. explor. eng. 8 (10s)(2019), 276-282.[9] r. vaidyanathaswamy, the localization theory in set-topology, proc. indian acad. sci. sect. 20 (1944) 51-61. https://doi.org/10.28924/ada/ma.5.2 1. introduction 2. preliminaries 3. -local functions 4. -local closure 5. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 6doi: 10.28924/ada/ma.5.6 schwarz algorithms for stokes-stokes coupling alexandros kyriakis boston college uk, department of mathematics, skirbeck road, pe21 6jf, uk alex-k@boston.ac.uk abstract. in this article, we exhibit the behavior of the schwarz algorithms for the steady stokesequation in the case of two unbounded subdomains at the continuous level. the schwarz methods havereceived a lot of attention during the last decades with the vast development of parallel computingdevices. hermann amandus schwarz, a german analyst, is considered to be the pioneer of the domaindecomposition methods. we will closely observe how the overlapping and non overlapping schwarzmethods work for the steady stokes problem. this problem has immediate practical application,modeling the flow of an incompressible fluid. for the analysis, we rely on fourier analysis techniquesand we provide comparison of the exhibited methods. 1. introduction many people have been fascinated by the motion of fluids, and wonder how we are able tosimulate the motion of fluids with such an accuracy. of course, the answer is simple but at thesame time complicated. firstly, in order to model various phenomena, we use partial differentialequations( pdes). pdes are equations that involve partial derivatives and most of the times weare not able to obtain solutions in closed form. as a result, we use numerical algorithms in orderto obtain the approximate solution of a pde. this field is called numerical analysis of pdesand it is gaining increasing interest from mathematical and engineering communities worldwide.especially, the last two decades domain decomposition methods [7], [8], [9], [10] are gaining grounddue to the increased use of parallel computing. the pioneer of these methods was the germananalyst hermann schwarz [4], [5], [6] who devised an algorithm to solve the poisson equation inan irregular domain (union of rectangle and a circle), in order to fix a glitch in riemann’s mappingtheorem. the algorithm is ∆u (k) 1 = −f , in ω1 u (k) 1 = u (k−1) 2 , at γ1 u (k) 1 = g1, on ∂ω1 \ γ1 , then  ∆u (k) 2 = −f , in ω2 u (k) 2 = u (k) 1 , at γ2 u (k) 2 = g2, on ∂ω2 \ γ2. (1) received: 18 may 2024. key words and phrases. schwarz algorithms; steady stokes equation; convergence analysis.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 2 ω1 ω2γ1γ2 figure 1. domain decomposition of the global domain into a union of a circle and a rectangle.having a close look at the above figure we notice the following: firstly, the poisson problemis solved in the circle and then in the rectangle, going back and forth, passing the values at theinterfaces γ1 and γ2. this iteration process is repeated until the convergence is reached. theindex (k) denotes the iterations, and f is the source function. this is the so called alternatingschwarz algorithm proposed by schwarz back in 1870. after a significant amount of time, the fieldsmedalist pierre luis lions [3], [13] proposed a modification in the alternating schwarz method (1).after imposing this modification, the algorithm (1) takes the form ∆u (k) 1 = −f , in ω1 u (k) 1 = u (k−1) 2 , at γ1 u (k) 1 = g1, on ∂ω1 \ γ1 , and  ∆u (k) 2 = −f , in ω2 u (k) 2 = u (k−1) 1 , at γ2 u (k) 2 = g2, on ∂ω2 \ γ2. (2) in this iterative algorithm, the two local subproblems are solved in parallel passing the dirichletvalues at the two interfaces. this algorithm (2) is known as the parallel schwarz algorithm. thisiterative scheme provides two great benefits. the first is balancing the computational cost bybreaking the global problem into smaller subproblems. the second benefit is that with the increas-ing amount of computational resources, the schwarz method (2) is ideal for parallel computations.there has been an avalanche of new research results and there is a great avenue of research ondomain decomposition methods. in this article, we will observe the behaviour of the schwarzmethods for the steady stokes equation, for two unbounded subdomains using fourier analysistechniques which is a standard approach in the literature [1], [2], [11], [12], [14], [15], [16], [17]. thesteady stokes equation is derived from the navier-stokes equation, which is a pde for modelingthe flow of incompressible fluids. it is a generalization of the equations proposed by the swissmathematician leonhard euler in the 18th century. in 1821, claude-luis navier introduced theelement of the viscosity. later in the mid 19th century, sir gabriel stokes worked extensively onthe equation. the steady stokes equation in strong form reads −ν∆~u + op = ~f in ω = (−∞,+∞)× (−∞,+∞), d iv ~u = 0 in ω, ~u : bounded at ±∞, p : bounded at ±∞ (3) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 3where ν > 0 is the kinematic viscosity of the fluid, ~u is the velocity of the fluid and p is thepressure field. the function ~f is called the sink term. the function spaces for the velocity field,pressure field and sink term are (h1(ω) )2 , l2(ω) , (l2(ω))2 respectively. the space h1(ω) isclassical sobolev space, and l2(ω) is the space of square integrable functions. the (3)2 denotesthe incompressibility condition, with the divergence free velocity field. furthermore as (3)3, (3)4suggest, the velocity and the pressure field stay bounded at infinity. 2. parallel schwarz method-dirichlet ic we decompose the domain ω = r2 into two subdomains ω1 = (−∞, h) × (−∞,+∞) and ω2 = (0,+∞)× (−∞,+∞). the parallel schwarz method in strong form reads −ν∆−→u1 (k) + op(k) 1 = ~f in ω1, d iv−→u1 (k) = 0 in ω1, −→u1 (k) = −→u2 (k−1)at x = h, −→u1 (k) : bounded at −∞, p (k) 1 : bounded at −∞, , and  −ν∆−→u2 (k) + op(k) 2 = ~f in ω2, d iv−→u2 (k) = 0 in ω2, −→u2 (k) = −→u1 (k−1)at x = 0, −→u2 (k) : bounded at +∞, p (k) 2 : bounded at +∞, (4) where two initial guesses −→u1 (0), −→u2 (0) are required to start the iterative process. theorem 1. the convergence factor of the parallel schwarz algorithm using dirichlet transmission conditions is given by the formula below rpsm,d(ξ,h) = ( 1 + 2h2|ξ|2 + 2 |ξ| √ h2 (1 +h2|ξ|2) ) e−2|ξ|h (5) where ξ is the fourier frequency and h > 0 is the size of the overlap. proof. in order to study the convergence behavior of the method, we go back to the local subproblemsin (4) and we consider the homogeneous counterparts taking ~f = ~0. in addition, the velocity fieldsin the two subdomains are −→u1 (k) = ( u (k) 1,1 , u (k) 1,2 ) and −→u2 (k) = ( u (k) 2,1 , u (k) 2,2 ), where the first indicesdenote the subdomain and the second indices denote the component. consequently, the parallelschwarz method prescribed by (4) can be written in the following form https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 4  ∂2u (k) 1,1 ∂x2 + ∂2u (k) 1,1 ∂y2 = 1 ν ∂p (k) 1 ∂x in ω1, ∂2u (k) 1,2 ∂x2 + ∂2u (k) 1,2 ∂y2 = 1 ν ∂p (k) 1 ∂y in ω1, ∂u (k) 1,1 ∂x + ∂u (k) 1,2 ∂y = 0 in ω1, u (k) 1,1 = u (k−1) 2,1 at x = h, u (k) 1,2 = u (k−1) 2,2 at x = h, u (k) 1,1 : bounded at −∞, u (k) 1,2 : bounded at −∞, p (k) 1 : bounded at −∞, , and  ∂2u (k) 2,1 ∂x2 + ∂2u (k) 2,1 ∂y2 = 1 ν ∂p (k) 2 ∂x in ω2, ∂2u (k) 2,2 ∂x2 + ∂2u (k) 2,2 ∂y2 = 1 ν ∂p (k) 2 ∂y in ω2, ∂u (k) 2,1 ∂x + ∂u (k) 2,2 ∂y = 0 in ω2, u (k) 2,1 = u (k−1) 1,1 at x = 0, u (k) 2,2 = u (k−1) 1,2 at x = 0, u (k) 2,1 : bounded at +∞, u (k) 2,2 : bounded at +∞, p (k) 2 : bounded at +∞. (6) going back to (4)1, for ~f = ~0, taking the divergence on both sides for the first subproblem, weobtain div ( ∆−→u1 (k) ) = 1 ν ∆p (k) 1 = ( ∂3u (k) 1,1 ∂x3 + ∂3u (k) 1,2 ∂y∂x2 ) + ( ∂3u (k) 1,2 ∂y3 + ∂3u (k) 1,1 ∂x∂y2 ) = 0 in ω1 exploiting the equation (4)2 (divergence free velocity in subdomain ω1). in the same fashion weobtain that div (∆−→u2 (k) ) = 1 ν ∆p (k) 2 = 0 in ω2. as a consequence, we have to solve two laplaceproblems in each subdomain where the unknown is the pressure field. we deal with ∆p (k) 1 = 0 in ω1 and by taking the fourier transform in the y direction we obtain the homogeneous equation ∂2p̂ (k) 1 ∂x2 − |ξ|2p̂(k) 1 = 0. the general solution of this equation is p̂(k) 1 = c(k) 1 e−|ξ|x +d(k) 1 e |ξ|x . by theboundedness assumption of the pressure field in ω1 as x → −∞, we obtain that p̂(k) 1 = d(k) 1 e |ξ|x .we proceed to solve the equation ∆p (k) 2 = 0 in ω2, and the first step is to take the fourier transformin the y direction. as a result, the equation ∂2p̂ (k) 2 ∂x2 − |ξ|2p̂(k) 2 = 0 has a general solution of theform p̂ (k) 2 = c(k) 2 e−|ξ|x + d(k) 2 e |ξ|x .by exploiting the property that the pressure field p̂(k) 2 remainsbounded as x → +∞, we obtain that p̂(k) 2 = c(k) 2 e−|ξ|x . the next move is to go to the two localschwarz subproblems in (6)1 and to take the fourier transform in the y direction. this will give ∂2û (k) 1,1 ∂x2 − |ξ|2û(k) 1,1 = |ξ|d(k) 1 e |ξ|x ν , (7) ∂2û (k) 2,1 ∂x2 − |ξ|2û(k) 2,1 = −|ξ|c(k) 2 e−|ξ|x ν . (8) we solve (7), (8) to obtain the two solutions in closed form û (k) 1,1 = ( b(k) 1 + x 2ν d(k) 1 ) e |ξ|x , (9) û (k) 2,1 = ( b(k) 2 + x 2ν c(k) 2 ) e−|ξ|x . (10) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 5we go back to (6)3 and by taking the fourier trasform in the y direction and exploiting the solutions(9), (10) we obtain û (k) 1,2 = i ξ ( |ξ|b(k) 1 + ( 1 + x |ξ| 2ν ) d(k) 1 ) e |ξ|x , (11) û (k) 2,2 = i ξ ( −|ξ|b(k) 2 + ( 1− x |ξ| 2ν ) c(k) 2 ) e−|ξ|x . (12) we further proceed, substituting the solutions (9), (10), (11), (12) back to the interface conditions(6)4, (6)5 to obtain the following equations b(k) 1 + h 2ν d(k) 1 = ( b(k−1) 2 + h 2ν c(k−1) 2 ) e−2|ξ|h, (13) |ξ|b(k) 1 + ( 1 +h|ξ| 2ν ) d(k) 1 = ( −|ξ|b(k−1) 2 + ( 1−h|ξ| 2ν ) c(k−1) 2 ) e−2|ξ|h, (14) b(k) 2 = b(k−1) 1 , (15) −|ξ|b(k) 2 + c(k) 2 2ν = |ξ|b(k−1) 1 + d(k−1) 1 2ν . (16) we combine the equations (15), (16) to obtain d(k) 1 = c(k+1) 2 − 4|ξ|νb(k+1) 2 . we substitute thecoefficients d(k) 1 back to equation (13) to obtain b(k+1) 2 (2ν − 4νh|ξ|) +hc(k+1) 2 = b(k−1) 2 2νe−2|ξ|h +hc(k−1) 2 e−2|ξ|h. (17) in the same spirit, we replace the iteration coefficients d(k) 1 back to (14) to get b(k+1) 2 ( 2ν|ξ|+ 4νh|ξ|2 ) −c(k+1) 2 (1+h|ξ|) = 2ν|ξ|b(k−1) 2 e−2|ξ|h+(h|ξ|−1)c(k−1) 2 e−2|ξ|h. (18) we take the two equations (17), (18) and write them in matrix form[ (2ν − 4νh|ξ|) h( 2ν|ξ|+ 4νh|ξ|2 ) −(1 +h|ξ|) ][ b(k+1) 2 c(k+1) 2 ] = [ 2νe−2|ξ|h he−2|ξ|h 2ν|ξ|e−2|ξ|h (h|ξ| − 1)e−2|ξ|h ][ b(k−1) 2 c(k−1) 2 ] . (19)we recast the equation (19) in the form[ b(k+1) 2 c(k+1) 2 ] = [ (1 + 2h|ξ|)e−2|ξ|h h2|ξ|e−2|ξ|h ν 8νh|ξ|2e−2|ξ|h ( 4h2|ξ|2 − 2h|ξ|+ 1 ) e−2|ξ|h ] ︸ ︷︷ ︸ ψpsm,d [ b(k−1) 2 c(k−1) 2 ] (20) where ψpsm,d is the schwarz iteration matrix. the spectrum of ψpsm,d is σ(ψpsm,d) = {λ+, λ−}, where λ+ and λ− are the corresponding eigenvalues given by the formulas λ+ = ( 1 + 2h2|ξ|2 + 2 |ξ| √ h2 (1 +h2|ξ|2) ) e−2|ξ|h , λ− = ( 1 + 2h2|ξ|2 − 2 |ξ| √ h2 (1 +h2|ξ|2) ) e−2|ξ|h . https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 6consequently, the convergence factor of the parallel schwarz algorithm is rpsm,d = ρ(ψpsm,d) = max{|λ+|, |λ−|} = ( 1 + 2h2|ξ|2 + 2 |ξ| √ h2 (1 +h2|ξ|2) ) e−2|ξ|h , where ρ(ψpsm,d) is the spectral radius of the schwarz iteration matrix, h is the size of theoverlap between the subdomains, ξ is the fourier frequency. � 3. alternating schwarz method-neumann ic the interface conditions play critical role on the convergence of the schwarz method. in thissection, we introduce the alternating schwarz algorithm employing neumann interface conditions.we go back to the iterative scheme prescribed by (4) and we modify the transmission conditions in(4)3. as a consequence, the schwarz method in strong form reads −ν∆−→u1 (k) + op(k) 1 = ~f in ω1, d iv−→u1 (k) = 0 in ω1, νo−→u1 (k)~n − p(k) 1 ~n = νo−→u2 (k−1)~n − p(k−1) 2 ~n at x = h, −→u1 (k) : bounded at −∞, p (k) 1 : bounded at −∞,  −ν∆−→u2 (k) + op(k) 2 = ~f in ω2, d iv−→u2 (k) = 0 in ω2, νo−→u2 (k)~n − p(k) 2 ~n = νo−→u1 (k)~n − p(k) 1 ~n at x = 0, −→u2 (k) : bounded at +∞, p (k) 2 : bounded at +∞, (21)where ~n is the outward normal vector. the initial guess νo−→u2 (0)~n − p(0) 2 ~n is required to start theiterative procedure. theorem 2. the convergence factor of the schwarz algorithm using neumann transmission conditions is given by the formula below rasm,n(ξ,h) = ∣∣∣(2|ξ|2h2 9 − 2|ξ|h 9 + 1 + 2 √ |ξ|4h4 + 2|ξ|3h3 + 8 |ξ|2h2 9 )∣∣∣e−2|ξ|h (22) where ξ is the fourier frequency and h > 0 is the size of the overlap. proof. as a first step, we go back to the local subproblems in (21) and we consider the homogeneouscounterparts taking ~f = ~0. we recast the method prescribed by (21) in the following form ∂2u (k) 1,1 ∂x2 + ∂2u (k) 1,1 ∂y2 = 1 ν ∂p (k) 1 ∂x in ω1, ∂2u (k) 1,2 ∂x2 + ∂2u (k) 1,2 ∂y2 = 1 ν ∂p (k) 1 ∂y in ω1, ∂u (k) 1,1 ∂x + ∂u (k) 1,2 ∂y = 0 in ω1, ν ∂ ∂x u (k) 1,1 − p (k) 1 = ν ∂ ∂x u (k−1) 2,1 − p(k−1) 2 at x = h, ∂ ∂x u (k) 1,2 = ∂ ∂x u (k−1) 2,2 at x = h, u (k) 1,1 : bounded at −∞, u (k) 1,2 : bounded at −∞, p (k) 1 : bounded at −∞,  ∂2u (k) 2,1 ∂x2 + ∂2u (k) 2,1 ∂y2 = 1 ν ∂p (k) 2 ∂x in ω2, ∂2u (k) 2,2 ∂x2 + ∂2u (k) 2,2 ∂y2 = 1 ν ∂p (k) 2 ∂y in ω2, ∂u (k) 2,1 ∂x + ∂u (k) 2,2 ∂y = 0 in ω2, ν ∂ ∂x u (k) 2,1 − p (k) 2 = ν ∂ ∂x u (k) 1,1 − p (k) 1 at x = 0, ∂ ∂x u (k) 2,2 = ∂ ∂x u (k) 1,2 at x = 0, u (k) 2,1 : bounded at +∞, u (k) 2,2 : bounded at +∞, p (k) 2 : bounded at +∞. (23) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 7we apply the fourier transform in the y direction to the schwarz subproblems prescribed by (23).the fourier transformed velocity components are given by the formulas (9), (10), (11), (12). thefourier transformed pressure fields are given by the relations: p̂(k) 1 = d(k) 1 e |ξ|x , p̂(k) 2 = c(k) 2 e−|ξ|x .we plug in the fourier transformed velocities and pressure fields back to the interface conditions(23)4, (23)5 and by doing a little algebra we obtain 2ν|ξ|e |ξ|hb(k) 1 +d(k) 1 (h|ξ| − 1)e |ξ|h = −2ν|ξ|b(k−1) 2 e−|ξ|h − c(k−1) 2 e−|ξ|h(1 +h|ξ|), (24) 2νb(k) 1 |ξ| 2e |ξ|h +d(k) 1 e |ξ|h ( 2|ξ|+h |ξ|2 ) = 2ν|ξ|2b(k−1) 2 e−|ξ|h + c(k−1) 2 e−|ξ|h ( h|ξ|2 − 2|ξ| ) ,(25) 2ν|ξ|b(k) 2 + c (k) 2 = −2ν|ξ|b(k) 1 +d(k) 1 , (26) ν|ξ|2b(k) 2 − |ξ|c(k) 2 = νb(k) 1 |ξ| 2 + |ξ|d(k) 1 . (27) we multiply (26) by −|ξ| then add (27), and solve with respect to the coefficient b(k) 1 obtaining b(k) 1 = − 1 3 b(k) 2 − 2 3 1 ν|ξ|c (k) 2 . (28) the next step is to obtain a formula for the coefficient d(k) 1 . in order to achieve that, we multiply(26) by |ξ| then add (27) to obtain d(k) 1 = 4 3 ν|ξ|b(k) 2 − 1 3 c(k) 2 . (29) we substitute the expressions (28), (29) back to (24) and (25) and this yields b(k) 2 e |ξ|h ( 6ν|ξ| − 4νh|ξ|2 ) + c(k) 2 e |ξ|h(3 +h|ξ|) = 6ν|ξ|b(k−1) 2 e−|ξ|h + 3c(k−1) 2 e−|ξ|h(1 +h|ξ|), b(k) 2 e |ξ|h ( 6ν|ξ|2 + 4νh|ξ|3 ) − c(k) 2 e |ξ|h ( 6|ξ|+h|ξ|2 ) = 6ν|ξ|2b(k−1) 2 e−|ξ|h + 3c(k−1) 2 e−|ξ|h ( h|ξ|2 − 2|ξ| ) . we write the above equations in matrix form and by doing some algebraic manipulations we derivethe stationary iteration[ b(k) 2 c (k) 2 ] = −12νh|ξ|3−54ν|ξ|2 54e2|ξ|h |ξ|2ν h(h|ξ|+4) 9e2|ξ|hν 8|ξ|2νh 9e2|ξ|h 4|ξ|2h2−2|ξ|h+9 9e2|ξ|h  ︸ ︷︷ ︸ ψasm,n [ b(k−1) 2 c (k−1) 2 ] (30) where ψasm,n is the schwarz iteration matrix. the spectrum of ψasm,n is σ(ψasm,n) = {µ+, µ−},where µ+ and µ− are the eigenvalues of the schwarz iteration matrix provided by the formulas µ+ = ( 2|ξ|2h2 9 − 2|ξ|h 9 + 1 + 2 √ |ξ|4h4 + 2|ξ|3h3 + 8 |ξ|2h2 9 ) e−2|ξ|h, µ− = ( 2|ξ|2h2 9 − 2|ξ|h 9 + 1− 2 √ |ξ|4h4 + 2|ξ|3h3 + 8 |ξ|2h2 9 ) e−2|ξ|h. https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 8as a result, the reduction factor of the schwarz method (neumann interface conditions) is given by rasm,n = ρ(ψasm,n) = max{|µ+|, |µ−|} = ∣∣∣(2|ξ|2h2 9 − 2|ξ|h 9 + 1 + 2 √ |ξ|4h4 + 2|ξ|3h3 + 8 |ξ|2h2 9 )∣∣∣e−2|ξ|h. � 4. non-overlapping optimized schwarz algorithm-robin ic the domain ω = r2 is decomposed into two non-overlapping subdomains ω1 = (−∞, 0) × (−∞,+∞) and ω2 = (0,+∞) × (−∞,+∞).the optimized schwarz methods employ mixed in-terface boundary conditions, and more precisely robin. in this way, they facilitate both neumannand dirichlet conditions and there is a tuning parameter to tune the method accordingly. theoptimized schwarz iterative scheme is given in strong form −ν∆−→u1 (k) + op(k) 1 = ~f in ω1, d iv−→u1 (k) = 0 in ω1, νo−→u1 (k)~n − p(k) 1 ~n + γ−→u1 (k) = νo−→u2 (k−1)~n − p(k−1) 2 ~n + γ−→u2 (k−1) at x = 0, −→u1 (k) : bounded at −∞, p (k) 1 : bounded at −∞, (31)  −ν∆−→u2 (k) + op(k) 2 = ~f in ω2, d iv−→u2 (k) = 0 in ω2, νo−→u2 (k)~n − p(k) 2 ~n + γ−→u2 (k) = νo−→u1 (k−1)~n − p(k−1) 1 ~n + γ−→u1 (k−1) at x = 0, −→u2 (k) : bounded at +∞, p (k) 2 : bounded at +∞, (32) where γ is the tuning parameter of the method. two initial guesses are needed for the iterativemethod. theorem 3. the contraction factor of the non-overlapping schwarz algorithm is given by the mathematical expression r2 osm(ξ, ν, γ) = |3ν2|ξ|2 − 4ν|ξ|γ + γ2|2 |3ν2|ξ|2 + 4ν|ξ|γ + γ2|2 . (33) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 9 proof. we recast the local schwarz subproblems (31), (32) in the form ∂2u (k) 1,1 ∂x2 + ∂2u (k) 1,1 ∂y2 = 1 ν ∂p (k) 1 ∂x in ω1, ∂2u (k) 1,2 ∂x2 + ∂2u (k) 1,2 ∂y2 = 1 ν ∂p (k) 1 ∂y in ω1, ∂u (k) 1,1 ∂x + ∂u (k) 1,2 ∂y = 0 in ω1, ν ∂ ∂x u (k) 1,1 − p (k) 1 + γu (k) 1,1 = ν ∂ ∂x u (k−1) 2,1 − p(k−1) 2 + γu (k−1) 2,1 at x = 0, ν ∂ ∂x u (k) 1,2 + γu (k) 1,2 = ν ∂ ∂x u (k−1) 2,2 + γu (k−1) 2,2 at x = 0, u (k) 1,1 : bounded at −∞, u (k) 1,2 : bounded at −∞, p (k) 1 : bounded at −∞, (34)  ∂2u (k) 2,1 ∂x2 + ∂2u (k) 2,1 ∂y2 = 1 ν ∂p (k) 2 ∂x in ω2, ∂2u (k) 2,2 ∂x2 + ∂2u (k) 2,2 ∂y2 = 1 ν ∂p (k) 2 ∂y in ω2, ∂u (k) 2,1 ∂x + ∂u (k) 2,2 ∂y = 0 in ω2, ν ∂ ∂x u (k) 2,1 − p (k) 2 − γu(k) 2,1 = ν ∂ ∂x u (k−1) 1,1 − p(k−1) 1 − γu(k−1) 1,1 at x = 0, ν ∂ ∂x u (k) 2,2 − γu (k) 2,2 = ν ∂ ∂x u (k−1) 1,2 − γu(k−1) 1,2 at x = 0, u (k) 2,1 : bounded at +∞, u (k) 2,2 : bounded at +∞, p (k) 2 : bounded at +∞. (35) we employ the fourier transform for the local schwarz subproblems (34), (35). the fourier trans-formed velocity components are given by the mathematical expressions (9), (10), (11), (12). thefourier transformed pressure fields are given by p̂(k) 1 = d(k) 1 e |ξ|x , p̂(k) 2 = c(k) 2 e−|ξ|x . we substitutethe velocities and pressure fields back to the transmission conditions (34)4, (34)5, (35)4, (35)5, andby doing some algebraic manipulations we obtain b(k) 1 (2γ + 2ν|ξ|)−d(k) 1 = b(k−1) 2 (2γ − 2ν|ξ| )− c(k−1) 2 , (36) b(k) 1 ( 2ν2|ξ|2 + 2νγ|ξ| ) +d(k) 1 (2ν|ξ| + γ) = b(k−1) 2 ( 2ν2|ξ|2 − 2νγ|ξ| ) + c(k−1) 2 (γ − 2ν|ξ|),(37) b(k) 2 ( 2γ + 2ν|ξ|) + c(k) 2 = b(k−1) 1 ( 2γ − 2ν|ξ|) +d(k−1) 1 , (38) b(k) 2 ( 2νγ|ξ|+ 2ν2|ξ|2 ) − c(k) 2 (2ν|ξ|+ γ) = b(k−1) 1 ( 2ν2|ξ|2 − 2νγ|ξ| ) +d(k−1) 1 (2ν|ξ| − γ).(39) we pick (38) and we obtain the coefficients d(k) 1 = b(k+1) 2 (2γ + 2ν|ξ|) + c(k+1) 2 − b(k) 1 ( 2γ − 2ν|ξ|). (40) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 10we substitiute the coefficients (40) back to (36) and we obtain 4©b(k) 1 = b(k+1) 2 (2γ + 2ν|ξ|) + c(k+1) 2 + b(k−1) 2 (2γ − 2ν|ξ|)− c(k−1) 2 . (41)the equation (40) can take the following form 4©d(k) 1 = b(k+1) 2 (2γ+2ν|ξ|)2+c(k+1) 2 (2γ+2ν|ξ|)−b(k−1) 2 (2γ−2ν|ξ|)2+c(k−1) 2 (2γ−2ν|ξ|) (42) by employing (41). we take the relation (37), multiply with 4γ, and then plug in (41), (42) to obtain k1b(k+1) 2 + k2c(k+1) 2 = k3b(k−1) 2 + k4c(k−1) 2 (43) where k1, k2, k3, k4 are given by the relations below k1 = 12 ( ν|ξ|+ γ 3 ) (ν|ξ|+ γ)2, k2 = 6ν2|ξ|2 + 8ν|ξ|γ + 2γ2, k3 = 12|ξ|3ν3 − 4|ξ|2γν2 − 12|ξ|γ2ν + 4γ3, k4 = 6ν2|ξ|2 − 8ν|ξ|γ + 2γ2. in the same fashion, we pick (39), multiply with 4γ, and then exploit the expressions (41), (42) toderive the equation q1b(k+1) 2 + q2c(k+1) 2 = q3b(k−1) 2 + q4c(k−1) 2 (44)where q1, q2, q3, q4 are provided by the expressions q1 = 12|ξ|3ν2 + 4|ξ|2γν2 − 12|ξ|γ2ν − 4γ3, q2 = 6ν2|ξ|2 + 8νγ|ξ|+ 2γ2, q3 = 12 ( ν|ξ| − γ 3 ) (ν|ξ| − γ)2, q4 = 6ν2|ξ|2 − 8νγ|ξ|+ 2γ2. we take (43), (44) and after some algebraic manipulations we obtain a stationary iteration[ b(k+1) 2 c (k+1) 2 ] = [ 3ν2|ξ|2−4ν|ξ|γ+γ2 3ν2|ξ|2+4ν|ξ|γ+γ2 0 0 3ν2|ξ|2−4ν|ξ|γ+γ2 3ν2|ξ|2+4ν|ξ|γ+γ2 ] ︸ ︷︷ ︸ ψosm [ b(k−1) 2 c (k−1) 2 ] . (45) the eigenvalue of the schwarz iteration matrix ψosm of multiplicity two is provided by the formula µd = 3ν2|ξ|2 − 4ν|ξ|γ + γ2 3ν2|ξ|2 + 4ν|ξ|γ + γ2 . as a consequence, the contraction factor is r2 osm = |µd |2 = |3ν2|ξ|2 − 4ν|ξ|γ + γ2|2 |3ν2|ξ|2 + 4ν|ξ|γ + γ2|2 . � https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 115. non-overlapping optimized schwarz algorithm-second order ic more sophisticated schwarz methods arise by the appropriate modification of the interfaceconditions. we can employ the optimized schwarz algorithms imposing second order transmissionconditions. more precisely, we go back to the algorithm prescribed by (31), (32), go to the interfaceconditions and instead of γ we use the symbol s , where s = q ( 1 + ξ2 ). theorem 4. the contraction factor of the non-overlapping schwarz algorithm (second order ic) is given by the mathematical expression r2 osm,soic(ν, q, ξ) = |3ν2|ξ|2 − 4ν|ξ|q ( 1 + ξ2 ) + q2 ( 1 + ξ2 )2 |2 |3ν2|ξ|2 + 4ν|ξ|q (1 + ξ2) + q2 (1 + ξ2)2 |2 (46) where q > 0. proof. the calculations follow through in the same spirit as the optimized schwarz methods withthe robin transmission conditions. instead of γ, the symbol s is used and the convergence factoris obtained naturally. � corollary 1. the reduction factor of the parallel schwarz method (dirichlet ic) given by (5) satisfies the following rpsm,d(ξ,h) =  1, h = 0 0, |ξ| → +∞ 0, h → +∞ < 1, ξ > 0. (47) proof. the result (47)1 occurs by replacing h = 0 back to the formula (5). as a consequence, itmeans that the schwarz method stagnates without overlap, something which is very usual in theliterature. the (47)2 is obtained by taking the limit of (5) as the fourier frequency tends to +∞.the (47)3 is coming from the fact that when the overlap is sufficiently large, the convergence factorturns to be zero. the ultimate result (47)4 comes from the fact that for non zero fourier frequency,the convergence factor is strictly less than 1. � corollary 2. the reduction factor of the alternating schwarz method (neumann ic) given by (22), satisfies the relations rasm,n(ξ,h) =  1, h = 0 0, |ξ| → +∞ 0, h → +∞ < 1, ξ > 0. (48) https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 12 proof. the reduction factor is a function that depends on the size of the overlap and the fourierfrequency. consequently, for zero overlap, the function becomes one and this leads to stagnation ofthe algorithm ((48)1). when the fourier number grows large, the function goes to zero as prescribedby (48)2. moving to (48)3, a big overlap leads to better convergence because the contraction factorrapidly tends to zero. last but not least, for finite fourier number, the convergence factor is strictlyless than one ((48)4). � corollary 3. the contraction factor of the non-overlapping optimized schwarz method (robin ic) given by (33) satisfies the properties r2 osm(ξ, ν, γ) =  1, γ = 0 1, γ → +∞ 1, |ξ| → +∞ < 1, ξ ∈ (0,+∞) 0, γ = γ+ = 3ν|ξ|, γ = γ− = ν|ξ|. (49) proof. the contraction factor depends on the kinematic viscosity, the fourier frequency and theparameter γ. the first three properties in (49) are straighforward to obtain. taking the robinparameter to be zero or tend to infinity gives a stagnant schwarz algorithm. in addition, when thefourier frequency tends to infinity, the contraction factor becomes 1. for finite fourier frequency(not growing to infinity) the reduction factor is less than 1. lastly, the values of the robin parameterthat make the contraction factor zero are γ+ = 3ν|ξ| and γ− = ν|ξ| and can obtained by solving atrinomial equation appearing in the numerator of the contraction factor. � corollary 4. if γ = mν|ξ|, m ∈ z+ − {1, 3}, then the convergence factor (33) does not depend on viscosity and fourier frequency. proof. by substitution, we obtain r2 osm(ξ, ν, γ) = |3ν2|ξ|2 − 4ν|ξ|γ + γ2|2 |3ν2|ξ|2 + 4ν|ξ|γ + γ2|2 = |3ν2|ξ|2 − 4ν|ξ|mν|ξ|+m2|ξ|2ν2|2 |3ν2|ξ|2 + 4ν|ξ|mν|ξ|+m2|ξ|2ν2|2 = |ν2|ξ|2 ( m2 − 4m + 3 ) |2 |ν2|ξ|2 (m2 + 4m + 3) |2 = |m2 − 4m + 3|2 |m2 + 4m + 3|2 . � corollary 5. the contraction factor of non-overlapping optimised schwarz method (second order ic) given by (46) satisfies the properties https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 13 r2 osm,soic(ν, q, ξ) =  1, q = 0 1, q → +∞ 1, |ξ| → +∞ < 1, ξ ∈ (0,+∞) 0, q = q+ = 3ν|ξ| 1+ξ2 , q = q− = ν|ξ| 1+ξ2 . (50) proof. the first three relations in (50) can directly be derived by taking the appropriate limits forthe parameter q and the fourier frequency ξ. for finite fourier frequency, the reduction factor isless than one. ultimately, for the indicated parameters q− and q+ the contraction factor becomeszero. � corollary 6. if q = mν|ξ|(1 + ξ2)−1, m ∈ z+ − {1, 3}, then the convergence factor (46) does not depend on viscosity and fourier frequency. proof. the proof follows by substitution of the q parameter back to (46). the expression obtainedis identical to the one appearing in the corollary 4. � 6. numerical evidence-convergence curves in this section, the convergence curves are presented for each one of the schwarz algorithms. inthe cases of oprimised schwarz methods with robin and second order transmission conditions, weconsider γ = ν and q = ν. the convergence curves are presented below. figure 2. convergence rate of schwarz method using dirichlet ic for varying overlap. https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 14 figure 3. convergence rate of schwarz method using neumann ic for varying overlap. figure 4. convergence rate of schwarz method using robin ic for varying viscosity. https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 15 figure 5. convergence rate of schwarz method using second order ic for varying viscosity. figure 6. comparison of convergence rates for all schwarz methods. employing the graphs of the convergence rates in figure 2 and figure 3, we can comparethe schwarz methods using dirichlet and neumann interface conditions. we notice that when https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 16neumann transmission conditions are imposed, the convergence rate decays rapidly for increasingfourier modes, whereas using dirichlet conditions makes the convergence slower. in addition, itis evident that when the overlap between the subdomains is larger then this enhances the overallconvergence which is the expected result when using the classical schwarz methods. in figures 4and 5 we have the convergence curves of non-overlapping optimized schwarz methods (figure4-robin ic, figure 5-second order ic) for varying values of the viscosity. we notice that thesealgorithms have better convergence for small fourier frequencies but as the fourier frequencygrows to infinity, the reduction factor tends to 1. we also notice that when we tune the parametersof these optimised methods, we can choose values to make the convergence rate equal to zero.last but not least, we compare all the convergence curves and obtain figure 6, which indicatesthat the schwarz methods with neummann and dirichlet transmission conditions are slower forlow frequencies, and the optimised methods perform better in this regime. however, the problemsoccur when the frequencies are large which means that the reduction factor tends to 1, which isnot desirable when dealing with schwarz algorithms. 7. conclusions in this work we focused on the convergence analysis of the schwarz algorithms for stokes-stokesconfiguration for varying interface conditions. we carried out the analysis using partial fouriertransform and we obtained the contraction factors for each one of the methods introduced. afterconducting the convergence analysis, we notice that the neumann conditions result in faster decayof reduction factor when ξ grows sufficiently large compared to the dirichlet ic. the optimisedschwarz methods have advantage in the low frequency regime, but as the fourier number growsthe contraction rate tends to one which is not desirable behavior. the convergence analysis forstokes-stokes configuration is useful for studying the behavior of schwarz algorithms and gettinga general insight. so far there is such analysis for stokes-darcy coupling [18], as a result thiswork could enrich the existing mathematical literature. references [1] m.j. gander, optimized schwarz methods for helmholtz problems, in: proceedings of the 13th international con-ference on domain decomposition, cimne (2001) 245-252.[2] m.j. gander, l. halpern, f. nataf, optimized schwarz methods, in: proceedings of the 12th international conferenceon domain decomposition, ddm.org (2000) 15-27.[3] p.l. lions, on the schwarz alternating method iii: a variant for nonoverlapping subdomains, in: t. chan, r. glowinski,j. periaux, o.b. widlund (eds.), third international symposium on domain decomposition methods for partialdifferential equations, siam (1990) 202-223.[4] m.j. gander, g. wanner, the origins of the alternating schwarz method, in: domain decomposition methods inscience and engineering xxi, lncse, springer-verlag (2014) 487-496.[5] m.j. gander, schwarz methods over the course of time, elec. trans. numer. anal. 31 (2008) 228-255. https://doi.org/10.28924/ada/ma.5.6 eur. j. math. anal. 10.28924/ada/ma.5.6 17 [6] h.a. schwarz, uber einen grenzubergang durch alternierendes verfahren, vierteljahrsschrift der naturforschendengesellschaft in zurich 15 (1870) 272-286.[7] v. dolean, p. jolivet, f. nataf, an introduction to domain decomposition methods: algorithms, theory, and parallelimplementation, siam (2016).[8] g. ciaramella, m.j. gander, iterative methods and preconditioners for systems of linear equations, siam (2022).[9] b. smith, p. bjorstad, w. gropp, domain decomposition: parallel multilevel methods for elliptic partial differentialequations, cambridge university press.[10] a. quarteroni, a. valli, domain decomposition methods for partial differential equations, oxford science publications(1999).[11] o. ernst, m.j. gander, why it is difficult to solve helmholtz problems with classical iterative methods, in: i. graham,t. hou, o. lakkis, r. scheichl (eds.), numerical analysis of multiscale problems, springer verlag (2012) 325-363.[12] m.j. gander, h. zhang, decomposition de domaine et probleme de helmholtz: thirty years after and still unique,in: domain decomposition methods in science and engineering xxvi, lncse, springer-verlag (2021).[13] p.-l. lions, on the schwarz alternating method. i, in: r. glowinski, g.h. golub, g.a. meurant, j. periaux (eds.), firstinternational symposium on domain decomposition methods for partial differential equations, siam, philadelphia(1988) 1-42.[14] v. dolean, m.j. gander, a. kyriakis, optimizing transmission conditions for multiple subdomains in the magnetotel-luric approximation of maxwell’s equations, in: domain decomposition methods in science and engineering xxvi,lncse, springer-verlag (2021).[15] a. kyriakis, scalable domain decomposition methods for time harmonic wave propagation problems, ph.d. thesis,university of strathclyde (2021).[16] v. dolean, m.j. gander, a. kyriakis, closed form optimized transmission conditions for complex diffusion with manysubdomains, siam j. sci. comput. 45 (2023) a829-a848.[17] a. kyriakis, analysis of schwarz algorithms for a scalar elliptic problem, adv. appl. math. sci. 22 (12) (2023)2227-2242.[18] m. discacciati, g. giorda, optimized schwarz methods for the stokes-darcy coupling, ima j. numer. anal. 38 (4)(2018) 1959–1983. https://doi.org/10.28924/ada/ma.5.6 1. introduction 2. parallel schwarz method-dirichlet ic 3. alternating schwarz method-neumann ic 4. non-overlapping optimized schwarz algorithm-robin ic 5. non-overlapping optimized schwarz algorithm-second order ic 6. numerical evidence-convergence curves 7. conclusions references ©2024 ada academica https://adac.eeeur. j. math. anal. 4 (2024) 21doi: 10.28924/ada/ma.4.21 global analysis of meningitis disease with optimal control kwame kyei danquah1, sampson takyi appiah1, baaba a. danquah1,bernard asamoah afful2,∗ , godfred agyemang safo3 1department of mathematics and statistics, university of energy and natural resources, ghana afriyiedanquah1@gmail.com, sampson.appiah@uenr.edu.gh, baaba.ghansah@uenr.edu.gh 2department of mathematics and statistics, utah state university, logan, ut, usa bernard.afful@usu.edu 3english international school of bratislava, radničné námestie 4, bratislava, slovakia wisegas98@gmail.com ∗correspondence: bernard.afful@usu.edu abstract. the meningitis epidemic has impacted lives negatively, especially in sub-sahara africa,dubbed the ‘meningitis belt’. the epidemic has been a public health concern due to an improperunderstanding of the disease’s dynamics. to implement a control measure that will help minimize theepidemic, we introduce a non-linear meningitis model that describes the dynamic behaviour of thedisease and explains the transmission trend. the model explores the condition that leads to local orglobal asymptomatic stability of the equilibria. the model is subjected to a sensitivity analysis to findthe parameters that influence the r0. the model is modified into an optimal control by adding time-dependent controls. the control model is solved qualitatively using pontryagin’s maximum principleand numerically using matlab and the fourth-order runge-kutta method. we provide a controlstrategy that can be relied on for management decision-making based on the results. 1. introduction meningitis, a deadly bacterial infection, is primarily attributed to meningococcal meningitis.meningitis kills over 100,000 individuals each year and affects 1.2 million people from all over theworld. in africa, especially the sub-saharan africa meningitis belt, which extends from senegal toethiopia, 10,000 people are expected to die each year [2]. this disease is widespread across sub-saharan africa, stretching from the meningitis belt in senegal to ethiopia. the illness reappears atthe start of each dry season and disappears at the start of the rainy season in africa, a fascinatingpattern that warrants further study. moreover, from the year 2003 to 2007, about 4100 casesof cerebrospinal spinal meningitis(csm) were confirmed in the united states (cdc, 2017) [3].it is approximated that during most significant epidemics, over 1000 cases of the disease are received: 10 jun 2024. key words and phrases. meningitis disease; optimal control; global stability; local stability; sensitivity analysis.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.21 https://orcid.org/0000-0001-8123-5757 eur. j. math. anal. 10.28924/ada/ma.4.21 2reported, typically happening every 5 to 12 years [4]. the bacteria is transmitted from infectedindividuals to susceptible ones through contact with respiratory and throat secretions like salivaand mucus. however, unlike flu and common cold viruses, the bacteria are not highly contagious,and it takes time before transmission occurs. those at risk of contracting the disease are typicallyindividuals near the infected person, such as household members and roommates [5]. the mostcommon signs of the disease include fever, headache and stiffness of the neck. diagnosing thedisease is sometimes difficult since the symptoms are often similar to other diseases [6]. currently,a vaccine for meningitis exists, with the available vaccine primarily for bacteria such as meningitis.bacterial meningitis is fatal when not diagnosed early [17]. generally, infected individuals recoverwith permanent disabilities such as hearing loss and brain damage. these disabilities and thedisease itself are worsened when symptoms are not detected on time.meningitis, a bacterial illness, has been a global concern, affecting numerous parts of the world.the disease has been endemic in several areas, with sub-saharan africa being the hardest hit.since its emergence, numerous models have been developed to describe the disease’s transmissionpatterns, yet there remains a need for further understanding of intervention strategies to curb thedisease in the meningitis belt. in [15], and [16], the dynamical behaviour of the meningitis diseasewas studied; however, the study failed to provide enough intervention and treatment strategies tominimize the disease. against this background, we propose a non-linear mathematical meningitismodel that would analyse the model’s stability and characterize a range of feasible control strategiesthat would aid management decision-making to curb the disease. in [31], the transmission behaviourof a meningitis disease is disclosed using an age-structured model that impacts the carriers’ input tothe model dynamics. the transmission behaviour of a meningitis disease is revealed by building anage-structured model that affects the carriers’ contribution to the model dynamics. [32] formulateda compartmental meningitis model that predicted the behavioural pattern of individuals and thepopulation evolution by studying the dynamic trend of disease transmission. in their study, [33]studied the risk factor of meningitis in adults by employing fuzzy cognitive maps and multi-criteriatechniques to determine the ranks of the various scenarios. [34], determined the numerical solutionof the meningitis disease by considering the methods of euler, heun, and the fourth-order runge-kutta. in [35], the authors created a mathematical model to investigate the impact of sharedinformation on the dynamics of meningitis disease. the authors in [36] modelled a co-infectionmathematical model of listeriosis and meningitis to unveil the parameters that impact the dynamicsof the co-infection model. in [37], the authors formulated a mathematical model of influenza-meningitis co-infection that analysed the infected’s outcome on the model’s dynamics. in the paperby [39], the authors looked at a mathematical model of meningitis that attempted to explain theinfection dynamics of the disease in jirapa district, ghana. to better understand the disease’stransmission mechanisms, the researchers in [38] developed a mathematical meningitis model. the https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 3model was subjected to a thorough stability study, with disease-free equilibrium indicating stabilitywhen r0 ≤ 1 and endemic stability when r0 ≥ 1.optimal controls are extensively used in dynamical systems, specifically those related to non-linear ordinary differential equations, and perceived as an intervention technique within controltheory [40–43]. mathematical models that involve optimal control analysis are essential for un-derstanding disease spread and play a vital role in the policy-making process concerning diseasecontrol. the authors in [44] suggested a nonlinear mathematical model to see if public awarenesscampaigns affect the spread of infectious diseases. the model evaluated the population’s responseto media awareness because diseases spread through interaction between infected and suscepti-ble people. thus, the ability of the susceptible individuals to avoid contact with the infectives.their analysis showed that infectious disease spread can be controlled by employing an aware-ness program. however, due to human immigration, diseases will always remain endemic. in [44],the authors explored the impact of media coverage on controlling disease spread by formulating amathematical model incorporating media coverage. the analysis indicates that, even though theexistence of media was not the sole factor in the attempt to eradicate the disease, its presence,to some extent, can minimize the number of infections. in [45], the authors attempted to reduceebola infection in the susceptible by constructing an optimal control theory from ordinary differ-ential equation modelling of the ebola virus. two control functions, education and treatments,were considered in modelling the control problem. the control system is solved by applying thetool of pontryagin’s maximum principle. the analysis of the numerical results showed the controls’overall effect in reducing the disease. also, the authors in [16] constructed a mathematical modelof syphilis transmission dynamics to aid in selecting the most effective syphilis screening choicesthe model created was an agent-based dynamic model that simulated a critical population of 2,000people. according to the model’s results, increasing the frequency of syphilis screening to everythree months was very effective in reducing syphilis infection cases. in [46], the authors devel-oped a mathematical model of covid-19. to characterize a range of feasible controls that mightbe effective in minimizing the disease, the model was changed to an optimal control problem. anumerical simulation of the problem was performed using a forward-backwards sweep and fourth-order range-kutta method. in [47], the authors created a mathematical model for the ongoingcoronavirus outbreak to determine intervention approaches to battle it. the model was turned intoan optimal control problem to provide a theoretical explanation for the disease, which was solvedqualitatively by utilizing pontryagin’s maximal principle. matlab and an iterative technique wereused to solve the models numerically.the objective of this work is to design a mathematical model to investigate meningitis transmis-sion, to examine the equilibrium’s local and global stability, to conduct a sensitivity analysis of the https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 4model parameters to identify the parameters that significantly affect the r0, to formulate a controlmodel for the meningitis disease, and to perform a numerical simulation for the model.the rest of the research is divided into the following sections: section 2 focuses on formulating anonlinear model for meningitis disease. section 3 explores the qualitative properties of the model,like positivity, solutions’ boundedness, basic reproduction numbers, and the existence of equilibriumalong with the local and global stability of disease-free and endemic equilibria. section 4 centres onexamining the sensitivity of the model’s parameters on r0 using the normalized forward sensitivityindex. in section 5, the model is modified by adding time-dependent and solved with pontryagin’smaximum principle. section 6 tackles computational investigations of the optimal control modelbased on the three control strategies, and the results are illustrated. then, finally, we provideconclusions and discussions of the work in section 7. 2. mathematical model in the current section, a deterministic model for meningitis disease that partitions the totalpopulation into shh, susceptible, ehh, exposed, ahh, asymptomatic, ihh, symptomatic, and rhh,recovered is formulated. the population n is given as n = shh + ehh + ahh + ihh + rhh. themodel assumes that people are recruited into the population by birth at the rate λ. the susceptiblebecome exposed through contact with the symptomatic at rate η1. the exposed leaves at a rate τ1and enters the symptomatic while a fraction k1 enters the asymptomatic. the asymptomatic andsymptomatic die at rates τ3 and ψ2, respectively. the asymptomatic can leave to recovery classdue to natural immunity at rate τ2. the symptomatic enters the recovered compartments at a rate ψ1. the recovered individuals could return to the susceptible class due to loss of immunity at rate ω. with all the compartments, natural death occurs at a rate µ. figure 1. schematic of the meningitis model https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 5 d dt shh = λ− µshh − η1ihhshh + ωrhh, d dt ehh = η1ihhshh − k1τ1ehh − (1− k1)τ1ehh − µehh, d dt ahh = (1− k1)τ1ehh − (τ2 + τ3 + µ)ahh, (1) d dt ihh = k1τ1ehh − (ψ1 + ψ2 + µ)ihh, d dt rhh = τ2ahh + ψ1ehh − (ω + µ)rhh, with: shh0 ≥ 0, ehh0 ≥ 0, ahh0 ≥ 0, ihh0 ≥ 0 and rhh0 ≥ 0. (2) 3. qualitative properties 3.1. positivity and boundedness. theorem 3.1. the set {shh, ehh, ahh, ihh, rhh} being the solution of the state system (1) with parameters which are non-negatives is positive with the initial condition given by; {shh0 ≥ 0, ehh0 ≥ 0, ahh0 ≥ 0, ihh0 ≥ 0, rhh0 ≥ 0} . proof. by inspection, the third equation of model (1) can be structured into a first-order differentialequation standard form as: d dt ahh + (τ2 + τ3 + µ)ahh = (1− k1)τ1ehh. (3) when equation (3) is solved with the integrating factor method, we get ahh(t) = e−(τ2+τ3+µ)t [ ahh(0) + (1− k1)τ1 ∫ t 0 e(s)e−(τ2+τ3+µ)sds ] . the same method, when applied to the fourth equation, gives ihh(t) = e−(ψ1+ψ2+µ)t [ ihh(0) + k1τ1 ∫ t 0 e(s)e−(ψ1+ψ2+µ)sds ] . hence, we observe that d dt ahh ≥ 0 at t0, d dt ihh ≥ 0 at t0. thus, we can generalise that the otherstate variables remain positive at t = 0. hence, the state model system 1 is positively invariant in r5+. � theorem 3.2. the model equation (1) is bounded within the invariant region, ϑ ∈ r5+ given as; ϑ = { (shh, ehh, ahh, ihh, rhh) ∈ r5+, shh + ehh + ahh + ihh + rhh ≤ λ− µn } . https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 6 proof. we add the respective compartments to prove the boundedness of model system (1). thus,we get n(t) = λ− µshh − τ3ahh − µahh − µehh − ψ2ihh − µihh − µrhh, dn(t) dt = λ− τ3ahh − ψ2ihh − µn. (4) it follows that from equation (4), setting h to be a solution of (4), we have a unique initial valueproblem,  d dt h1(t) = λ− µh1(t) t ≥ 0 h1(0) = n(0). (5) the solution of (5) gives; h1(t) = n(0)e−µt + λ µ (1− e(−µt)). (6) what happens next is that, from the comparison theorem in [1], we notice that, n(t) = n(0)e−µt + λ µ (1− e(−µt)). (7) therefore, from equation (7) the state variables (shh, ehh, ahh, ihh, rhh) has the possible solutionset which bounded and the model equation (1) is invariant ϑ ∈ r5+. as a result, model (1) ismathematically well-posed and epidemiologically feasible. � 3.2. existence of disease-free equilibrium (dfe) point. model system (1) has a trivial point (0, 0, 0, 0, 0), which is usually ignored in the model’s analysis. the right-hand side of (1) is set tozero and solved, the disease-free equilibrium becomes e0 = ( λ µ , 0, 0, 0, 0 ) . (8) 3.3. basic reproduction number. the basic reproduction number, r0, is one of the things thatmodellers look for when it comes to infectious disease modelling. the basic reproduction numberis sufficient for determining the condition of the disease. in a completely naive population, thebasic reproduction number is defined as the number of persons one infected person may infect. itis denoted by r0, and when r0 > 1, it means the disease will spread unless preventive strategiesare cautiously enacted. however, when r0 < 1, the infection dies without strenuous effort. thederivation of r0 is important in modelling and can be derived by the method of [19]. the (9) is theformulae guaranteeing r0 derivation. r0 = ρ(fv−1). (9) the ρ is considered as the largest entry in the derivation of the next generation matrix of r0 = ρ(fv−1), where f is the coming infection into compartment i and v . thus, the transfer of indi-viduals out of compartment i by death. technically, the r0 becomes the largest eigenvalue of the https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 7matrix resulting from the partial derivative of (9). what happens next is the infected compartmentsof model (1) are given by d dt ehh = η1ihhshh − k1τ1ehh − (1− k1)τ1ehh − µehh, d dt ahh = (1− k1)τ1ehh − (τ2 + τ3 + µ)ahh, d dt ihh = k1τ1ehh − (ψ1 + ψ2 + µ)ihh. we notice from the diseased compartment that, f = η1ihhshh0 0  , and v =  k1τ1ehh + (1− k1)τ1ehh + µehh −(1− k1)τ1ehh + (τ2 + τ3 + µ)ahh −k1τ1ehh + (ψ1 + ψ2 + µ)ihh  . (10) when f is evaluated at e0, then fe0 becomes; fe0 =  0 0 η1λ µ 0 0 0 0 0 0  . (11) evaluate v at e0, which gives; ve0 =  ( k1τ1 + µ+ (1− k1)τ1 ) 0 0 −(1− k1)τ1 (τ2 + τ3 + µ) 0 −k1τ1 0 (ψ1 + ψ2 + µ)  . (12) the basic reproduction number of model system (1) is determined by using the method of [19],which gives; r0 = η1λk1τ1 µ(k1τ1 + µ+ (1− k1)τ1)(ψ1 + ψ2 + µ) . (13) 3.4. existence of an endemic equilibrium point (eep). endemic equilibrium exists when there isa presence of infection. the model (1) has a unique endemic equilibrium given by; e∗ = (s∗hh, e ∗ hh, a ∗ hh, i ∗ hh, r ∗ hh), (14) https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 8where s∗hh = λ + ωr∗hh µ+ η1i ∗ hh , e∗hh = η1i ∗ hhs ∗ hh (µ+ τ1) , a∗hh = (1− k1)τ1e∗hh (τ2 + τ3 + µ) , i∗hh = k1τ1e ∗ hh (ψ1 + ψ2 + µ) , r∗hh = τ2a ∗ hh + ψ1e ∗ hh (ω + µ) . 3.5. stability of the disease-free equilibrium point. here, the global and local stability analysesof the meningitis model (1) at the disease-free equilibrium are studied. the geometrical approachof lyapunov function theory by [20] would be used to prove that model (1) is globally asymptoticallystable at the disease-free equilibrium. the results are provided as follows; j =  −µ− η1ihh 0 0 −η1shh ω η1ihh − ( k1τ1 + µ+ (1− k1)τ1 ) 0 η1shh 0 0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0 0 k1τ1 0 −(ψ1 + ψ2 + µ) 0 0 0 τ2 ψ1 −(ω + µ)  . (15)evaluating the jacobian in (15) at the e0 gives; j =  −µ 0 0 −η1 λ µ ω 0 − ( k1τ1 + µ+ (1− k1)τ1 ) 0 η1 λ µ 0 0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0 0 k1τ1 0 −(ψ1 + ψ2 + µ) 0 0 0 τ2 ψ1 −(ω + µ)  . clearly, λ1 = −µ, λ2 = −(ω + µ), λ3 = −(τ2 + τ3 + µ). the remaining matrix becomes; ĵ = −(k1τ1 + µ+ (1− k1)τ1 ) η1 λ µ k1τ1 −(ψ1 + ψ2 + µ)  . the characteristic equation is given by λ2 + b1λ+ b2 = 0 (16) https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 9where b1 = (k1τ1 + µ+ (1− k1)τ1 + (ψ1 + ψ2 + µ), b2 = (k1τ1 + µ+ (1− k1)τ1(ψ1 + ψ2 + µ)− λ µ η1k1τ1, then λ3,4 = −(b1)± √ t1 2 , where t1 = b21−4b2. if λ3 ≤ 0 and λ4 ≤ 0, then the disease free equilibrium is stable. otherwise,it is unstable. theorem 3.3. when r0 < 1, the disease-free equilibrium e0 for the meningitis model (1) is globally asymptotically stable in r5+. proof. we construct a lyapunov function l = k1 ( τ1 d1d2 ) ehh + 1 d2 ihh, where d1 = (k1τ1 + µ + (1 − k1)τ1) and d2 = (ψ1 + ψ2 + µ). taking the derivative of l withrespect to ehh and ihh gives; dl dt = k1 ( τ1 d1d2 ) d dt ehh + 1 d2 d dt ihh, dl dt = k1 ( τ1 d1d2 ) (η1ihhshh − k1τ1ehh − (1− k1)τ1ehh − µehh) + 1 d2 (k1τ1ehh − (ψ1 + ψ2 + µ)ihh) , dl dt = k1 ( τ1 d1d2 ) (η1ihhshh − d1ehh) + 1 d2 (kτ1ehh − d2ihh) . it follows that shh = λ µ at t0. hence dl dt = k1 ( τ1λ d1d2µ ) η1ihh − k1τ1 d2 ehh + k1τ1 d2 ehh − ihh, = (r0 − 1)ihh. from the model equation (1), the system variables and parameters are all non-negative, implyingthat dl dt < 0 when r0 < 1, with dl dt = 0 in the disease-free equilibrium. hence, l is a lyapunovfunction in ψ. hence, from [20] principle, (ehh(t), ihh(t))→ (0, 0) as t →∞. � 3.6. stability of the endemic equilibrium point. here, we study the global and local stabilityof the meningitis model (1) at the endemic equilibrium. the lyapunov function method by [21]is employed to prove the globally asymptotic stability of model (1) at endemic equilibrium. the https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 10underlying steps are, therefore, followed. the jacobian evaluated at e∗ gives; j =  −µ− η1i∗hh 0 0 −η1s∗hh ω η1i ∗ hh − ( k1τ1 + µ+ (1− k1)τ1 ) 0 η1s ∗ hh 0 0 (1− k1)τ1 −(τ2 + τ3 + µ) 0 0 0 k1τ1 0 −(ψ1 + ψ2 + µ) 0 0 0 τ2 ψ1 −(ω + µ)  (17)we denote a = −µ − η1i∗hh, b = −η1s∗hh, c = ω, d = η1i ∗ hh, e = − ( k1τ1 + µ + (1 − k1)τ1 ), f = η1s ∗ hh, g = (1 − k1)τ1, h = −(τ2 + τ3 + µ), i = k1τ1, j = −(ψ1 + ψ2 + µ), k = τ2, l = ψ1,and m = −(ω + µ). then, the characteristics equation of model (1) is given by y 5 + a0y 4 + a1y 3 + a2y 2 + a3y + a4 = 0 (18) with a0 = (a + e + h + j +m), a1 = ae + ah + aj + eh + am + ej +−f i + em + hj + hm + jm, a2 = aeh + bdi + aej − af i + aem + ahj + ahm + ehj − f hi + ajm + ehm + ejm − f im + hjm, a3 = ehjm − f him + bdhi + aehj − af hi − cdgk + aehm + bdim − cdi l + aejm − af im + ahjm, a4 = −cdgjk + bdhim − cdhi l + aehjm − af him. based on the routh-hurwitz stability by [22], the condition for the characteristics equation (18) isgiven by yi =  y1 y3 y5 y0 y2 y4 0 y1 y3 0 y0 y2 0 0 y1 0 0 y0 0 0 0  > 0. https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 11the condition requires all the coefficients of the characteristics equation (18) to be positive, implyingthat all eigenvalues have negative real parts. if the condition yi is satisfied, we conclude that themeningitis model at the endemic equilibrium is stable and otherwise unstable. theorem 3.4. when r0 ≥ 1, the endemic equilibrium e∗ of model 1 is stable when shh = s∗hh, ehh = e∗hh, ahh = a∗hh, ihh = i∗hh, and rhh = r∗hh, otherwise unstable. proof. we construct a lyapunov function lp = ( shh − s∗hh − s∗hh ln ( shh s∗hh )) + ( ehh − e∗hh − e∗hh ln ( ehh e∗hh )) + ( ahh − a∗hh − a∗hh ln ( ahh a∗hh )) + ( ihh − i∗hh − i∗hh ln ( ihh i∗hh )) + ( rhh − r∗hh − r∗hh ln ( rhh r∗hh )) . the derivative of lp with respect to t gives; dlp dt = ( shh − s∗hh shh ) dshh dt + ( ehh − e∗hh ehh ) dehh dt + ( ahh − a∗hh ahh ) dahh dt + ( ihh − i∗hh ihh ) dihh dt + ( rhh − r∗hh rhh ) drhh dt . (19) hence, substituting dshh dt , dehhdt , dahhdt , dihhdt and drhh dt into equation (19) gives; dlp dt = ( shh − s∗hh shh ) (λ− µshh − η1ihhshh + ωrhh) + ( ehh − e∗hh ehh ) (η1ihhshh − kτ1ehh − (1− k1)τ1ehh − µehh) + ( ahh − a∗hh ahh ) ((1− k1)τ1ehh − (τ2 + τ3 + µ)ahh) + ( ihh − i∗hh ihh ) (kτ1ehh − (ψ1 + ψ2 + µ)ihh) + ( rhh − r∗hh rhh ) (τ2ahh + ψ1ehh − (ω + µ)rhh) . hence, for shh = s∗hh, ehh = e∗hh, ahh = a∗hh, ihh = i∗hh, and rhh = r∗hh. we have that, dlp dt = λ− λ ( s∗hh shh ) − µ ( (shh − s∗hh)2 shh ) − η1(ihh − i∗hh) ( (shh − s∗hh)2 shh ) + ω(rhh − r∗hh) ( shh − s∗hh shh ) + η1 ( ehh − e∗hh ehh ) (ihh − i∗hh)(shh − s∗hh) − kτ1 ( (ehh − e∗hh)2 ehh ) − (1− k1)τ1 ( (ehh − e∗hh)2 ehh ) − µ ( (ehh − e∗hh)2 ehh ) https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 12 + (1− k1)τ1(ehh − e∗hh) ( ahh − a∗hh ahh ) − (τ2 + τ3 + µ) ( (ahh − a∗hh)2 ahh ) + kτ1(ehh − e∗hh) ( ihh − i∗hh ihh ) − (ψ1 + ψ2 + µ) ( (ihh − i∗hh)2 ihh ) + τ2(ahh − a∗hh) ( rhh − r∗hh rhh ) + ψ1(ehh − e∗hh) ( rhh − r∗hh rhh ) − (ω + µ) ( (rhh − r∗hh)2 rhh ) . we generate the below equation after thorough algebraic manipulations; dlp dt = g1 − g2, (20) where g1 = λ + ω(rhh − r∗hh) ( shh − s∗hh shh ) + η1 ( ehh − e∗hh ehh ) (ihh − i∗hh)(shh − s∗hh) + (1− k1)τ1(ehh − e∗hh) ( ahh − a∗hh ahh ) + kτ1(ehh − e∗hh) ( ihh − i∗hh ihh ) + τ2(ahh − a∗hh) ( rhh − r∗hh rhh ) , and g2 = λ ( s∗hh shh ) + µ ( (shh − s∗hh)2 shh ) + η1(ihh − i∗hh) ( (shh − s∗hh)2 shh ) + kτ1 ( (ehh − e∗hh)2 ehh ) µ ( (ehh − e∗hh)2 ehh ) + (τ2 + τ3 + µ) ( (ahh − a∗hh)2 ahh ) + (ψ1 + ψ2 + µ) ( (ihh − i∗hh)2 ihh ) + (ω + µ) ( (rhh − r∗hh)2 rhh ) . hence, dlpdt = 0 when shh = s∗hh, ehh = e∗hh, ahh = a∗hh, ihh = i∗hh, and rhh = r∗hh. it canbe shown that the inequality g1 ≤ g2. evidently, it can be verified that dlp dt ≤ 0 when g1 ≤ g2.hence dlp dt = 0, when shh = s∗hh, ehh = e∗hh, ahh = a∗hh,ihh = i∗hh and rhh = r∗hh. thisindicates that the largest compact invariant set is a singleton. hence, from [20], e∗ is globallystable. � 4. sensitivity analysis of r0 getting the correct estimation of the r0 in infection disease modelling is crucial because ithelps us in the decisions concerning the management of the infection. however, the possibilityof the parameters linked to the r0 to change makes sensitivity analysis an important subject inepidemiology. https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 13 definition 4.1. the normalized forward sensitivity index of r0 computed using the formula used by [23] for a given parameter α1 is ϑr0α1 = ∂r0 ∂α1 α1 r0 . (21) the parameters with positive indices contribute to the epidemic spreading since they enhancethe r0. the parameters with a negative index, on the other hand, aid in disease control by lowering r0. from table 1, λ, τ1, ψ1, η1, k1, µ, and ψ2 are the parameters which are most sensitive on r0. table 1. model parameter sensitivity indices for the reproduction number parameter sensitivity index λ 1.000 τ1 −2.333 ψ1 −0.526 ψ2 −0.473 η1 1.000 µ −1.000 k1 1.000 this is because, any increment in the parameter values of λ, η1, and k1 will lead to a 100% increasein r0. also, an increase in µ, τ1, ψ1, and ψ2, will decrease r0 by 100%, 233.3%, 52.6% and 47.3%respectively. therefore, effective measures must be put in place to decrease λ, η1, and k1 and toincrease µ, τ1ψ1, and ψ2. although intervention measures are geared towards increasing and/orincreasing the most significant parameters, it is paramount that the control of the other parametersnot be completely ignored. 5. optimal control analysis in this section, model system (1) is modified by putting in three time-dependent controls, viz. per-sonal protection, vaccination and treatment controls, to examine the impact of the control schemeson the meningitis disease. in model system (1), the associated infection force is lowered by afactor of (1 − u1), where u1 is the personal protection control that ensures the attempt to reduceroom heat and avoid close contact with the infected. the rate of vaccinating susceptible individualsagainst meningitis is represented by the control function u2. as a result, the model assumes thatvaccinated individuals shift from the susceptible compartment to the removed compartment at anytime. furthermore, we assume that the control function u3 reflects the rate at which sick patients https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 14are treated. hence, the modified nonlinear control system becomes; dshh dt = λ− µshh − (1− u1)η1ihhshh + ωrhh − u2shh, dehh dt = (1− u1)η1ihhshh − kτ1ehh − (1− k1)τ1ehh − µehh, dahh dt = (1− k1)τ1ehh − (τ2 + τ3 + µ)ahh, (22) dihh dt = kτ1ehh − (ψ1 + ψ2 + u3 + µ)ihh, drhh dt = τ2ahh + ψ1ehh + u2shh + u3ihh − (ω + µ)rhh. to examine the efforts needed to control the disease, we define an optimal functional j thatminimizes the exposed, asymptomatic and symptomatic individuals and maximizes the recoverythrough personal protection, vaccination and treatment controls of u1, u2 and u3. hence, theobjective functional j is given by; j (u1, u2, u3) = ∫ tf 0 [ b1ehh + b2ahh + b3ihh + 1 2 (u21d1 + u22d2 + u23d3) ] dt. (23) referring to (23), the quantities b1, b2, and b3 are the weight coefficients of the exposed, asymp-tomatic and symptomatic individuals. in addition, the terms u21d1 2 , u22d22 and u23d3 2 represents the costrelated to minimizing the exposed, asymptomatic and symptomatic individual. the control modelconsiders a quadratic cost on the controls as in other works. we target optimal control u∗1, u∗2, u∗3such that j (u∗1, u ∗ 2, u ∗ 3) = min{j (u1, u2, u3) : (u1, u2, u3) ∈ u}, (24) where u = {(u1, u2, u3)|0 ≤ ui ≤ 1, i = 1, 2, 3 lebesgue measurable} (25) with the method of pontryagin’s maximum principle [24], system (22) and (23) are transformed intoa problem of hamiltonian minimization h with respect to the controls u1, u2 and u3 where; h = [ b1ehh + b2ahh + b3ihh + 1 2 (u21d1 + u22d2 + u23d3) ] , + λ1 {λ− µshh − (1− u1)η1ihhshh + ωrhh − u2shh} , + λ2 {(1− u1)η1ihhshh − kτ1ehh − (1− k1)τ1ehh − µehh} , (26) + λ3 {(1− k1)τ1ehh − (τ2 + τ3 + µ)ahh} , + λ4 {kτ1ehh − (ψ1 + ψ2 + u3 + µ)ihh} , + λ5 {τ2ahh + ψ1ehh + u2shh + u3ihh − (ω + µ)rhh} . https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 15 theorem 5.1. there exists an optimal control u∗ = (u∗1, u ∗ 2, u ∗ 3) ∈ u such that j (u∗1, u ∗ 2, u ∗ 3) = min u∈u j (u1, u2, u3), (27) subject to the control system 22 with the initial conditions. proof. the work of [25] would be considered the grounds for proving the existence of optimal control.in minimizing the control problem, the necessary and convexity of the objective functional in u1, u2and u3 are satisfied. the control space u is also convex and closed by definition.the optimal control system is bounded, which verifies the compactness necessary for the optimalcontrol. also, the integrand in the functional (23) is convex on u . therefore, we notice that thereexist a constant q > 1, positive numbers u1, u2 and u3 such that, j(u1, u2, u3) ≥ u1 ( |u1|2 + |u2|2 + |u3|2 ) q 2 − u2 . � hence, there exists an optimal control. it follows that determining the optimal solution, thepontryagins’s maximum principle by [26] is applied to the hamiltonian (26) such that given (y , u)is an optimal solution of the optimal control problem, then there exist a non-trivial vector function λ = (λ1, · · · , λ5) satisfying the below equation; dy dt = − ∂h(t, y , u, λ) ∂λ , 0 = ∂h(t, y , u, λ) ∂u , (28) dλ dt = ∂h(t, y , u, λ) ∂y . hence, the necessary condition related to the hamiltonian (26) is applied. theorem 5.2. given that s∗hh, e∗hh, ahh, i∗hh and r∗hh are optimal state solutions with associated optimal control variables (u∗1, u ∗ 2, u ∗ 3) for the optimal control problem (22) and (23), then there exist adjoint variables λi for i = 1, . . . , 5, satisfying; dλ1 dt = (λ1 − λ2)(1− u1)η1ihh + (λ2 − λ5)u2 + µλ1, dλ2 dt = −b1 + (λ2 − λ3)(1− k1)τ1 + (λ2 − λ4)kτ1 + µλ2, dλ3 dt = −b2 + (τ3 + µ)λ3 + (λ3 − λ5)τ2, (29) dλ4 dt = −b3 + (λ1 − λ2)(1− u1)η1shh + (λ4 − λ5)ψ1 + (ψ2 + µ)λ4 + (λ4 − λ5)u3, dλ5 dt = (λ5 − λ1)ω + µλ5, with boundary condition; λi(tf ) = 0, i = 1, 2, . . . , 5, (30) https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 16 and the optimal control u∗1 , u∗2 and u∗3 are given by u∗1 = min { 1,max { 0, ( (λ2 − λ1) η1ihhshh d1 )}} u∗2 = min { 1,max { 0, (λ1 − λ5) shh d2 }} u∗3 = min { 1,max { 0, (λ4 − λ5) ihh d3 }} (31) proof. the adjoint and transversality conditions are derived using the hamiltonian (26). thus weequate shh = s∗hh, ehh = e∗hh, ahh = a∗hh, ihh = i∗hh and rhh = r∗hh and differentiating thehamiltonian with respect to shh, ehh, ahh, ihh and rhh to obtain (29). further, the equations ∂h ∂u1 = 0, ∂h ∂u2 = 0, ∂h ∂u3 = 0. (32) are determined on the interior of the control set, and using the optimality conditions and theproperty of the control space u1 and u2, we can determine (22). from (22), we can characterize thecontrol found by solving the optimality system. in solving the optimality system, the transversalityand the characterization of the optimal control (u1, u2, u3) are used. the controls u∗1, u∗2 and u∗3when substituted into the control system (22) gives; dshh dt = λ− µshh − ( 1−min { 1,max { 0, ( (λ2 − λ1) η1ihhshh d1 )}}) η1ihhshh +ωrhh − (λ1 − λ5) shh d2 shh, dehh dt = ( 1−min { 1,max { 0, ( (λ2 − λ1) η1ihhshh d1 )}}) η1ihhshh −kτ1ehh − (1− k1)τ1ehh − µehh, dahh dt = (1− k1)τ1ehh − (τ2 + τ3 + µ)ahh, d dt ihh = kτ1ehh − (ψ1 + ψ2 + min { 1,max { 0, (λ1 − λ5) shh d2 }} + µ)ihh, d dt rhh = τ2ahh + ψ1ehh + min { 1,max { 0, (λ1 − λ5) shh d2 }} shh + min { 1,max { 0, (λ4 − λ5) ihh d3 }} ihh − (ω + µ)rhh. (33) � 6. numerical simulation and discussion the present section focuses on obtaining a numerical solution for the model. besides the qual-itative analysis that has been carried out, it becomes imperative to find a numerical solution forthe model. hence, our task here is to derive a numerical solution that solves the without and withcontrol models and evaluates the effectiveness of the considered control strategies. a numericalalgorithm uses a 4th-order runge-kutta method and matlab to solve the optimality system. thus,a numerical solution of the control optimality system involves running the adjoint system backwards https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 17and the state forward in time, with the associated boundary conditions and the controls. the pro-cess involves continuously upgrading the controls and the characterization value until the earlierresults become close to the currently obtained value; then, the algorithm stops, and a solution isobtained. the matlab simulation was done with values taken from the published work. beloware the parameters considered for the simulation. table 2. meningitis model parameters parameter description range estimated value reference λ recruitment rate (100− 100000) 1000 [3] τ1 modification parameter (0.001− 0.8) 0.3 [3] τ2 rate at individuals leaves theasymptomatic class (0.002− 0.3) 0.03 [3] τ3 disease induced death rate (0.002− 0.2) 0.2 [3] ψ1 rate at which individuals leavethe infected class to the recov-ered class (0.001− 0.1) 0.02 [27] ψ2 disease induced death rate (0.002− 0.1) 0.018 [3] ω loss of immunity (0.01− 0.1) 0.084 [48] η1 contact rate (0.1− 0.9) 0.5 [28] µ rate at which individuals nat-urally leaves the compartment (0.00001− 0.2) 0.0000391 [29] k1 rate at which individualsleaves the exposed class (0.01− 0.5) 0.3 [30] 6.1. strategy a : u3 = 0. strategy a uses the controls u1 and u2, with u3 set to zero. the graphsof 2a, 2b, 2c and 2d indicate the exposed, asymptomatic, and symptomatic people, as well as thecontrols. the without-control graph of 2a showed a swift increase of an estimated 5000 in the first 20 days. moreover, it remained at this level throughout the remaining simulated time. with theasymptomatic non-control graph of 2b, we notice a gentle rise of the graph in the first 20 days to 4300 of the asymptomatic population. the asymptomatic control graph progressed steadily to a new 5000 in 140 days and retained it till the end of the simulation. the symptomatic non-control graphof 2c increased smoothly and moved to the maximum height of 430 of the symptomatic populationin 130 days, which remained until the end of the simulation. control figures of the exposed,asymptomatic and symptomatic produced results with substantially minimized graphs. from theexposed graph of 2a, the graph increases similarly but could not rise to the level of the without https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 18control plot. we see that the control graph has been dramatically minimized. the control figures2b of the asymptomatic lie far below the non-control figure. the symptomatic control figure of2c lay slightly above zero and maintained the level throughout the simulated time. figure 2d isthe control profile plot of strategy a. the plot shows that the personal protection procedure u1remained at the upper bound throughout the simulation, while the vaccine control u2 remained atthe upper bound until 178 days before dropping to the lower bound. 6.2. strategy b : u2 = 0. strategy b sets u2 = 0 and generates the exposed, asymptomatic, andsymptomatic graphs. from the exposed graph of a, we observed that the without control curve swiftlyraised to 5000 when t = 20. it moved gently to about 5100 for the next 20 days and remainedat that level for the remaining time. the asymptomatic without control curve steadily increasedin the early days of 20 to 4500, increased gently for the next 120 days to 5000, and maintainedthe level for the remaining time. the symptomatic without control graph increased smoothly andprogressed to a height of 430 at the final time. the exposed, asymptomatic and symptomatic controlgraphs produced graphs with desired results. thus, we noticed a completely minimized exposure,asymptomatic and symptomatic, with the control simulations. figure 3d is the control profile graph.the graph shows that the personal protection and treatment controls remained at the upper bounduntil 100 and 98 when they dropped to the lower bound. the simulated plots of the exposed,asymptomatic and symptomatic confirmed that strategy b is effective. 6.3. strategy c : u1 = 0. strategy c uses the control u1 = 0 and the remaining controls forthe simulated. the graphs of 4a, 4b, 4c, and 4d are the exposed, asymptomatic, symptomatic andcontrol profile plots of strategy c. in the early days, the exposure surged swiftly for the withoutcontrol graphs but maintained a stable level after 40 days. the asymptomatic graph moved steeplyin the early days of 20 to 4500, increased further to 5000 for the next 120, and retained the levelfor the remaining days. the symptomatic graph gently increased throughout the simulation andmaintained a steady progression. the exposed control plot showed a swift increase in the graphin the early days and progressed gently for the entire simulated time. the asymptomatic controlgraph smoothly increased in the early days and progressed with the same momentum for the rest ofthe simulation. the symptomatic control curve was noticed to be minimized throughout the entiresimulation. the plot of figure 4d is the optimal control profile of strategy c. the vaccine (u2) andtreatment (u3) controls remained at the upper bound throughout the simulation until 180 days,when they decreased to the lower bound. 6.4. strategy d : u1 6= 0, u2 6= 0 and u3 6= 0. strategy d considered the three controls in itssimulation and generated the exposed, asymptomatic and symptomatic plots. without control, theexposed graph sparked to 2000 at t = 0. the graphs increased steadily to 5000 in 20 days andthen retained it for the rest of the time. the asymptomatic graph also increased early in the first https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 19 0 20 40 60 80 100 120 140 160 180 time (days) 0 1000 2000 3000 4000 5000 6000 e xp os ed u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 =0 (a) 0 20 40 60 80 100 120 140 160 180 time (days) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 a sy m pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 =0 (b) 0 20 40 60 80 100 120 140 160 180 time (days) 0 50 100 150 200 250 300 350 400 450 s ym pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 =0 (c) 0 20 40 60 80 100 120 140 160 180 time (days) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 c on tr ol p ro fil e u 1 u 2 (d) figure 2. plot of phase portraits with u3 = 0 https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 20 0 20 40 60 80 100 120 140 160 180 time (days) 0 1000 2000 3000 4000 5000 6000 e xp os ed u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 =0, u 3 0 (a) 0 20 40 60 80 100 120 140 160 180 time (days) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 a sy m pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 =0, u 3 0 (b) 0 20 40 60 80 100 120 140 160 180 time (days) 0 50 100 150 200 250 300 350 400 450 s ym pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 =0, u 3 0 (c) 0 20 40 60 80 100 120 140 160 180 time (days) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 c on tr ol p ro fil e u 1 u 3 (d) figure 3. plot of phase portraits with u2 = 0 https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 21 0 20 40 60 80 100 120 140 160 180 time (days) 0 1000 2000 3000 4000 5000 6000 e xp os ed u 1 =0, u 2 =0, u 3 =0 u 1 =0, u 2 0, u 3 0 (a) 0 20 40 60 80 100 120 140 160 180 time (days) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 a sy m pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 =0, u 2 0, u 3 0 (b) 0 20 40 60 80 100 120 140 160 180 time (days) 0 50 100 150 200 250 300 350 400 450 s ym pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 =0, u 2 0, u 3 0 (c) 0 20 40 60 80 100 120 140 160 180 time (days) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 c on tr ol p ro fil e u 2 u 3 (d) figure 4. plot of phase portraits with u1 = 0 https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 22 20 days to 4500 and then smoothly progressed to 5000 for the next 100 days and maintained thelevel. the symptomatic level gradually rose from 10 in a day (1) to 430 in 180 days. with thecontrol application, witnessed the exposure raised 1000 to 1900 in 180 days. the asymptomaticslightly raised slightly below 500 for the 1180 days. the symptoms could barely move above 10.plot 4d is the control profile plot of strategy d, we notice from the control plot that the personalprotection, vaccination and treatment controls remained at the upper bound until 180, 178, and177 respectively when they dropped to the lower bound. 7. discussion and conclusion the study considered a non-linear compartmental model of meningitis disease to explain thetransmission dynamics. the compartmental meningitis model was presented, with compartmentsfor susceptible(s), exposed(e), asymptomatic(a), symptomatic(i), and recovered(r). the model’sequilibria and the basic reproduction number were determined. the model’s local stability at thetwo equilibrium points, the disease-free and endemic, was determined by using the linearisationapproach. the global stabilities of the equilibria were investigated by employing the geometricmethod of the lyapunov function. in addition, a sensitivity analysis was carried out on the r0to determine the parameters that significantly affect the r0. it was seen that the most sensitiveparameters on r0 are λ, τ1, ψ1, η1, k1, µ, and ψ2.an optimal control model was formulated by adding time-dependent optimal controls. by defin-ing time-dependent policies that might help decrease or eradicate the disease, the model waschanged into an optimal control problem. to discover the optimality conditions of the systems, thecontrol model was solved using pontryagin’s maximum principle. several works have been done onmeningitis transmission, but few have considered optimal control. as a result, we formulated themeningitis model that was modified to optimal control problems to characterise a range of possiblestrategies to help control the disease. therefore, we considered possible pairing of the controls toexamine their combined effect on the disease.with strategy a, the controls of personal protection and vaccination were considered. thegraphs of 2a, 2b, 2c, and 2d denote the exposed, asymptomatic and symptomatic individuals andthe control profile. without control, the graph of 2a showed a swift increase of an estimated 5000in the first 20 days. moreover, it remained at this level throughout the remaining simulated time.with the asymptomatic non-control graph of 2b, we notice a gentle rise of the graph in the first 20 days to 4300 of the asymptomatic population. the asymptomatic control graph progressedsteadily to a new level of 5000 in 140 days and retained at that level till the end of the simulation.the symptomatic non-control graph of2c increased smoothly and moved to the maximum height of 430 of the symptomatic population in 130 days, which remained until the end of the simulation.control figures of the exposed, asymptomatic and symptomatic produced results with substantiallyminimized graphs. from the exposed graph of 2c, the graph increases similarly but could not rise https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 23 0 20 40 60 80 100 120 140 160 180 time (days) 0 1000 2000 3000 4000 5000 6000 e xp os ed u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 0 (a) 0 20 40 60 80 100 120 140 160 180 time (days) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 a sy m pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 0 (b) 0 20 40 60 80 100 120 140 160 180 time (days) 0 50 100 150 200 250 300 350 400 450 s ym pt om at ic u 1 =0, u 2 =0, u 3 =0 u 1 0, u 2 0, u 3 0 (c) 0 20 40 60 80 100 120 140 160 180 time (days) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 c on tr ol p ro fil e u 1 u 2 u 3 (d) figure 5. plot of phase portraits with u1 6= 0,u2 6= 0 and u3 6= 0 https://doi.org/10.28924/ada/ma.4.21 eur. j. math. anal. 10.28924/ada/ma.4.21 24to the level of the without control plot. we see that the control graph has been dramaticallyminimized. the control figures 2b of the asymptomatic lies far below the non-control figure. thesymptomatic control figure of 2d lay slightly above zero and maintained the level throughout thesimulated time.strategy b considered personal protection and treatment control. from the exposed graph of2a, we observed that the without control curve swiftly raised to 5000 when t = 20. it movedgently to about 5100 for the next 20 days and remained at that level for the remaining time. theasymptomatic without control curve steadily increased in the early days of 20 to 4500, increasedgently for the next 120 days to 5000, and maintained the level for the remaining time. thesymptomatic without control graph increased smoothly and progressed to a height of 430 at the finaltime. the exposed, asymptomatic and symptomatic control graphs produced graphs with desiredresults. thus, we noticed a completely minimized exposure, asymptomatic and symptomatic, withthe control simulations.strategy c paired the vaccination and treatment controls. we noticed that the exposure surgedswiftly for the graphs without control in the early days but maintained a stable level after 40 days.the asymptomatic graph moved steeply in the early days of 20 to 4500, increased further to 5000for the next 120, and retained it for the remaining days. the symptomatic graph gently increasedthroughout the simulation and maintained a steady progression. the exposed control plot showeda swift increase in the graph in the early days and progressed gently for the entire simulated time.the asymptomatic control graph climbed gradually in the early days and continued throughout thesimulation. throughout the simulation, the symptomatic control curve was found to be minimized.strategy d paired all three controls. without control, the exposed graph rose to 2000 at t = 0.the graphs increased steadily to 5000 in day 20 and then maintained that level for the rest of thetime. the asymptomatic graph also increased early in the first 20 days to 4500 and then smoothlyprogressed to 5000 for the next 100 days and retained the level. the symptomatic level graduallyrose from 10 in a day (1) to 430 in 180 days. with the control application, we witnessed theexposed raised 1000 to 1900 in 180 days. the asymptomatic slightly raised slightly below 500for the 1180 days. the symptomatic could not move above 10. the simulated results showed thatthe strategies significantly minimise the disease. it can be 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stability and simulations, alex. eng. j. 60 (2021),647–658. https://doi.org/10.28924/ada/ma.4.21 https://doi.org/10.1017/s0950268818002625 eur. j. math. anal. 10.28924/ada/ma.4.21 27 [47] x. yan, y. zou, optimal and sub-optimal quarantine and isolation control in sars epidemics, math. comp. model.47 (2008), 235–245.[48] a. karachaliou, a. j. k. conlan, m. p. preziosi, c. l. trotter, modeling long-term vaccination strategies withmenafrivac in the african meningitis belt, clin. infect. dis. 61 (2015), s594–s600. https://doi.org/10.28924/ada/ma.4.21 1. introduction 2. mathematical model 3. qualitative properties 3.1. positivity and boundedness 3.2. existence of disease-free equilibrium (dfe) point 3.3. basic reproduction number 3.4. existence of an endemic equilibrium point (eep) 3.5. stability of the disease-free equilibrium point 3.6. stability of the endemic equilibrium point 4. sensitivity analysis of 5. optimal control analysis 6. numerical simulation and discussion 6.1. strategy a : 6.2. strategy b : 6.3. strategy c : 6.4. strategy d : 7. discussion and conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 18doi: 10.28924/ada/ma.5.18 on local and semi-local convergence analysis of a high-order iterative method for solving nonlinear systems without high derivatives ioannis k. argyros1,∗, stepan shakhno2, yurii shunkin2, samundra regmi3,christopher i. argyros4 1department of computing and mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2department of theory of optimal processes, ivan franko national university of lviv, universytetska str. 1, 79000 lviv, ukraine stepan.shakhno@lnu.edu.ua, yuriy.shunkin@lnu.edu.ua 3department of mathematics, university of houston, houston, tx 77014, usa sregmi5@uh.edu 4georgia institute of technology, 225 north avenue nw, atlanta, ga 30313, usa cargyros3@gatech.edu ∗correspondence: iargyros@cameron.edu abstract. in this paper, we study a general high-order iterative method for solving nonlinear sys-tems in banach spaces without requiring higher-order derivatives. the proposed method constructseach iteration by combining evaluations of the operator and its derivative, together with an adaptedcorrection scheme. a detailed local convergence analysis under majorant conditions is provided, es-tablishing the convergence to the solution. we also show a semi-local convergence by introducingnew majorizing sequences. the theoretical results are illustrated with examples, and confirm thetheoretical predictions. 1. introduction numerous problems in applied mathematics, scientific computing, and engineering are modeledby nonlinear systems of the form g : d ⊂ b0 → b, where d is an open and convex subset of the banach space b0, and b is another banach space.the goal is to find x∗ ∈ d satisfying g(x) = 0. (1) received: 1 jun 2025. key words and phrases. high-order methods; nonlinear systems; semi-local convergence; iterative schemes.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 2finding exact analytical solutions to such nonlinear systems is typically difficult or impossible.consequently, iterative methods are commonly employed. newton’s method, defined by xn+1 = xn − g′(xn)−1g(xn), is among the most classical and efficient iterative schemes, offering quadratic convergence undersuitable conditions. however, many real-world problems demand faster convergence with lowercomputational cost, motivating the development of high-order methods.following this line, we study a general high-order iterative method, which extends classicalschemes by incorporating additional correction steps while using only evaluations of g and g′(without requiring higher derivatives). inspired by the high-order framework of behl et al. [9], whichattains order 3(k−1) for systems in rm by reusing a frozen inverse jacobian and relying solely onfirst-order information, the present study generalizes that scheme to banach spaces, discards theseventh-derivative assumptions underpinning their local taylor analysis, and furnishes a unifiedlocal and semi-local convergence theory with computable error bounds, larger attraction regions,and sharpened uniqueness criteria—thereby achieving broader applicability.let k ≥ 3 be a natural number and x0 ∈ d an initial point. then, the method is defined for each n = 0, 1, 2, . . . by y (1) n = xn − g′(xn)−1g(xn), y (2) n = xn − 2t−1g(xn), y (3) n = y (2) n −mg(y (2)) n , · · · xn+1 = y (k) n = y (k−1) n −mg(y (k−1) n ), (2) where the operators t , l, and m are given by t = tn = g′(xn) + g′(y (1)) n , l = ln = 3f ′(y (2)) n − g′(xn), m = mn = l−1tg′(xn)−1. the method (2) is shown in [9] to possess convergence order 3(j − 1) for j = 3, . . . , k usingtaylor expansions when b0 = b = rm (m natural number), assuming the existence of at least theseventh derivative g(7), although the derivatives g′′, g(3), ..., g(7) are not explicitly required in theiteration steps.unlike many classical methods that impose strong smoothness assumptions, this method is de-signed to work under weaker differentiability conditions. it builds upon ideas from previous studies,including parhi and gupta [14], wang et al. [18], and cordero et al. [10], while aiming for a betterbalance between convergence speed, computational cost, and robustness. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 3to illustrate the need for such an approach, consider the scalar function g : d → r defined by g(t) = b1t7 log t + b1t 8 + b2t 9, t 6= 0, 0, t = 0, where d = [−2, 2), and b1, b2 are real constants with b1 6= 0 and b1 + b2 = 0. although g hasa zero at t∗ = 1, the seventh derivative g(7)(t) does not exist at t = 0, showing that classicalconvergence assumptions based on higher derivatives are not satisfied.this observation motivates the use of generalized majorant conditions rather than strict smooth-ness hypotheses. moreover, method not only achieves high-order local convergence but also pro-vides a semi-local convergence analysis by constructing suitable majorizing sequences.the main contributions of this paper are as follows: • a local convergence analysis is established that depends only on g, g′, and suitablyconstructed auxiliary operators, thereby eliminating any need for higher-order derivatives. • semi-local convergence results are provided through the use of majorizing sequences, guar-anteeing convergence even when the initial guess is relatively far from the solution. • computable radii of convergence and explicit error bounds are derived, permitting a prioriestimates of the number of iterations required to achieve a prescribed accuracy. • conditions are specified that ensure the uniqueness of the solution within a neighborhoodof the limit point.the remainder of the paper is organized as follows. in section 2, we introduce the assumptionsand establish the local convergence theorems, including uniqueness and error estimates. sec-tion 3 presents the semi-local convergence analysis via majorizing sequences. section 4 discussesexamples and practical aspects. finally, conclusions are drawn in section 5. 2. convergence analysis 2.1. local. some real functions which are defined on the interval a = [0,+∞) play a crucial rolein the local convergence analysis of the method (2).suppose (h1) there exists a nondecreasing and continuous function φ0 : a → a such that the function 1 − φ0(t) has a smallest positive zero in a, which is denoted by s0. define the interval a0 = [0, s0). (h2) there exists a nondecreasing and continuous function φ : a0 → a such that for h1 : a0 → adefined by h1(t) = 1∫ 0 φ((t − η)t)dη (3) https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 4the function 1− h1(t) has a smallest positive zero in the interval a0, which is denoted by r1. (h3) for p : a0 → a defined by p(t) = 1 2 (φ0(t) + φ0(h1(t)t)) (4) the function 1 − p(t) has a smallest positive zero in the interval a0, which is denoted by s1. define the interval a1 = [0, s1). (h4) for φ : a1 → a, h2 : a1 → a, p : a1 → a, defined by φ(t) =  φ((1 + h1(t))t) or φ0(t) + φ0(h1(t)t), p(t) = 1 2 (φ0(t) + φ0(h1(t)t)) , and h2(t) = 1∫ 0 φ0((1− η)t)dη 1− φ0(t) + φ(t) ( 1 + 1∫ 0 φ0(ηt)dη ) 2(1− φ0(t))(1− p(t)) , the function 1− h2(t) has a smallest positive zero in the interval a1, which is denoted by r2. (h5) for q : a1 → a defined by q(t) = 1 2 (φ0(t) + 3φ0(h1(t)t)) the function 1 − q(t) has a smallest positive zero, which is denoted by s2. define theinterval a2 = [0, s2). (h6) for j = 3, . . . , k , aj−1 = [0, sj−1), hj : aj−1 → athe functions 1− φ0(hj−1(t)t) and 1− hj(t) have smallest positive zeros in the interval aj−1, which are denoted by sj−1 and rj , respectively, where hj(t) =  1∫ 0 φ0((1− η)hj−1(t)t)dη 1− φ0(hj−1(t)t) + (1 + φ0(h1(t)t)) ( 1 + 1∫ 0 φ0(ηhj−1(t)t)dη ) 2(1− φ0(t))(1− q(t))  hj−1(t) define r∗ = min{rj}, m = 1, 2, . . . , k and a∗ = [0, r∗) (5) it follows by these definitions and conditions (h1)− (h6) that for each t ∈ a∗ https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 5 0 ≤ φ0(t) < 1, (6) 0 ≤ p(t) < 1, (7) 0 ≤ q(t) < 1, (8) 0 ≤ φ0(hj−1(t)t) < 1, (9)and (10) 0 ≤ hj(t) < 1. (11) notice also that the parameter r is shown to be a radius of convergence for the method(2) (see theorem 1).next, we relate functions φ0 and φ to the operators in the method (2). (h7) there exists a solution x∗ ∈ d and a linear operator e ∈ l(b0, b) which is invertible suchthat for each z ∈ d ‖e−1(g(z)− e)‖ ≤ φ0(‖z − x∗‖). define the region d0 = d∩u(x∗, s0), where u(x, s) stands for an open ball in b0 centeredat x and of some radius s > 0. the set u[x, s] denotes the closure of u(x, s), which is aclosed set. (h8) ‖e−1(g′(z2)− g′(z1))‖ ≤ φ(‖z2 − z1‖) for each z1, z2 ∈ d0. (h9) u[x∗, r∗] ⊂ d. remark 1. some possible selections for the linear operator e can be e = i , the identity operator,or e = g′(z̄) for some z̄ ∈ d with z̄ 6= x∗ or e = g′(x∗). the last choice of e implies x∗ is asimple solution of the equation g(x) = 0. it is worth noting, though, that such an assumption isnot made or implied by the conditions (h1)–(h9). the local convergence analysis of the method (2) is provided in the next result. let u0 = u(x∗, r∗)− {x∗}. theorem 1. suppose that the conditions (h1)–(h9) hold. then, the sequence {xn} generated for the starting point x0 ∈ u0 is convergent to the solution x∗ of the equation g(x) = 0. proof. the following assertions shall be established using induction on n = 0, 1, 2, . . . ‖y (1)n − x∗‖ ≤ g1(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖ < r∗, (12) ‖y (2)n − x∗‖ ≤ g2(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖, (13) ‖y (j)n − x∗‖ ≤ gj(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖, (14) · · · https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 6 ‖xn+1 − x∗‖ = ‖y (k)n − x∗‖ ≤ gk(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖. (15) pick u ∈ u0. it follows by conditions (h1), (h7), and (5), (6) ‖e−1(g′(u)− e)‖ ≤ φ0(‖u − x∗‖) ≤ φ0(r∗) < 1. (16) so, the linear operator g′(u) is invertible by the lemma due to banach [13] and ‖g′(u)−1e‖ ≤ 1 1− φ0(‖u − x∗‖) . (17) in particular, if u = x0, the iterate y (1)0 exists by the first substep of the method (2) for n = 0,and we can write y (1) 0 − x ∗ = x0 − x∗ − g′(x0)−1g(x0) = [ g′(x0) −1e ] 1∫ 0 e−1(g′(x0 + η(x∗ − x0))− g′(x0))dη(x0 − x∗)  (18) using the condition (h8), (5), (11) (for i = 1), (17), and (18), we get from (18) ‖y (1)0 −x ∗‖ ≤ 1∫ 0 φ((1− η)‖x0 − x∗‖)dη‖x0 − x∗‖ 1− φ0(‖x0 − x∗‖) ≤ q1(‖x0−x∗‖)‖x0−x∗‖ ≤ ‖x0−x∗‖ < r∗. (19) thus, the assertion (12) holds if n = 0 and the iterate y (1)0 ∈ u0. next, we show t0 is alsoinvertible. in view of the conditions (h7), (5), (7), and (19), we can have ‖(2e)−1(t0 − 2e)‖ ≤ 1 2 ( φ0(‖x0 − x∗‖) + φ0(‖y (1)0 − x ∗‖) ) ≤ 1 2 ( φ0(‖x0 − x∗‖) + φ0 ( g1(‖x0 − x∗‖)‖x0 − x∗‖ )) = p0 < 1. thus, the linear operator t0 is invertible and ‖t−10 e‖ ≤ 1 2(1− p0) . (20) moreover, the iterate y (2)0 is well defined by the second substep of method (2), from which wecan also write y (2) 0 − x ∗ = x0 − x∗ − g′(x0)−1g(x0) + (g′(x0) −1 − 2t−10 )g(x0) = x0 − x∗ − g′(x0)−1g(x0)− (2t−10 − g ′(x0) −1)g(x0) = x0 − x∗ − g′(x0)−1g(x0)− t−10 (g′(x0)− g′(y (1)0 )g′(x0))−1g(x0) = x0 − x∗ − g′(x0)−1g(x0) + [t−10 e][e−1(g′(y (1) 0 )− g′(x0))]g′(x0) −1g(x0) (21) https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 7by the conditions (h1), (h7), (h8), (17) (for u = x0), (19), (11) (for i = 2), (20) and (21), weobtain ‖y (2)0 − x ∗‖ ≤  1∫ 0 φ((1− η)‖x0 − x∗‖)dη 1− φ0(‖x0 − x∗‖) + φ0(1 + 1∫ 0 φ0(η‖x0 − x∗‖)dη) 2(1− φ0(‖x0 − x∗‖))(1− p0)  ‖x0 − x∗‖ ≤ g2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖. (22) hence, the assertion (13) holds if n = 0 and the iterate y (2)0 ∈ u0. similarly, from (8) and (h7)we can have ‖(2e−1)(l− 2e)‖ = 1 2 ‖e−1 ( 3(g′(y (1)) 0 − e) + (g′(x0)− e) ) ‖ ≤ 1 2 ( 3φ0(‖y (1)0 − x ∗‖) + φ0(‖x0 − x∗‖) ) ≤ q0 < 1,so ‖l−1e‖ ≤ 1 2(1− q0) . (23) thus, the iterates y (3)0 , . . . , y (k) 0 = x1 exist, since l is invertible and we can write for j = 3, 4, . . . , k y (j) 0 − x ∗ = y (j−1) 0 − x∗ − g′(y (j−1)0 )−1g(y (j−1) 0 ) + (i −m0)g(y (j−1) 0 ) = y (j−1) 0 − x∗ − g′(y (j−1)0 )−1g(y (j−1) 0 )− l−1g′(y (j−1)0 )g′(x0) −1g(y (2) 0 ) (24) which can be implied by (5), (11), (22), and (23) ‖y (j)0 − x ∗‖ ≤  1∫ 0 φ0((1− η)‖y (j−1)0 − x∗‖)dη 1− φ0(‖y (j−1)0 − x∗‖) + (1 + φ0(‖y (j−1) 0 − x∗‖))(1 + 1∫ 0 φ0(η‖y (j−1)0 − x∗‖)dη) 2(1− φ0(‖x0 − x∗‖))(1− q0)  ‖y (j−1)0 − x∗‖ ≤ gj(‖y (j−1)0 − x∗‖)‖y (j−1)0 − x∗‖ ≤ ‖x0 − x∗‖ (25) where we have also used the estimates ‖e−1(g′(y (j)0 )− g′(x0))‖ ≤ φ(‖y (1)0 − x0‖) ≤ φ(‖y (1)0 − x ∗‖+ ‖x0 − x∗‖) ≤ φ0or ‖e−1(g′(y (1)0 )− g′(x0))‖ ≤ ‖e−1(g′(y (1)0 )− g′(x∗))‖+ ‖e−1(g′(x0)− g′(x∗))‖ ≤ φ0(‖y (1)0 − x ∗‖) + φ0(‖x0 − x∗‖) ≤ φ0, https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 8 ‖e−1g(y (1) 0 )‖ = ‖e−1(g(y (1) 0 )− e + e)‖ ≤ 1 + ‖e−1(g′(y (1)0 )− e)‖ ≤ 1 + φ0(‖y (1)0 − x ∗‖), and ‖e−1g(y (1) 0 )‖ = ∥∥∥∥∥∥ 1∫ 0 e−1(g′(x∗ + η(y (1) 0 − x ∗)))dη(y (1) 0 − x ∗) ∥∥∥∥∥∥ = ∥∥∥∥∥∥ 1∫ 0 e−1(g′(x∗ + η(y (1) 0 − x ∗))− e + e)dη(y (1) 0 − x ∗) ∥∥∥∥∥∥ ≤ ( 1 + ∫ 1 0 φ0(η‖y (1)0 − x ∗‖)dη ) ‖y (1)0 − x ∗‖. therefore, the assertions (14) and (15) hold if n = 0 and the iterate y (j,)0 x1 ∈ u0.the induction for assertions (12)–(15) is completed if x0, y (1,) 0 y (2,) 0 y (j) 0 are replaced by xj , y (1,) j y (2,) j y (j) j respectively.furthermore, by estimate (15) and dk = gk(‖x0 − x∗‖) ∈ [0, 1), we can get ‖xn+1 − x∗‖ = ‖y (k)n − x∗‖ ≤ dk‖y (k−1)n − x∗‖ ≤ dkdk−1‖y (k−2)n − x∗‖ ≤ · · · ≤ dkdk−1 · · · d3‖y (2)n − x∗‖ ≤ dkdk−1 · · · d2‖xn − x∗‖. (26) it follows by the definition of dk that there exists d ∈ [0, 1) such that d2, d3, . . . , dk ≤ d (27) so, by (26) and (27), we get ‖xn+1 − x∗‖ ≤ dk−1‖xn − x∗‖ ≤ d (k−1)(n+1)‖x0 − x∗‖ < r∗. (28) finally, if we let n → +∞ in (28), we conclude that lim n→+∞ xn = x∗, and all the iterates {xn} ⊆ u0. � the uniqueness of the solution x∗ is established in a neighborhood of it next. proposition 1. suppose that the condition (h7) holds in the ball u(x∗, r1), for some r1 > 0 and there exists r2 ≥ r1 such that 1∫ 0 φ0(ηr2)dη < 1. (29) define the region d1 = d ∩ u[x∗, r2]. then, x∗ is the only solution of the equation g(x) = 0 in the region d1. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 9 proof. suppose that there exists a solution z∗ ∈ d1 of the equation g(x) = 0 such that z∗ 6= x∗.then, define the linear operator e1 = 1∫ 0 g′(x∗ + η(z∗ − x∗))dη. by this definition, the condition (h7) and (29) we get in turn ‖e−11 (e1 − e)‖ ≤ 1∫ 0 φ0(η‖z∗ − x∗‖)dη ≤ 1∫ 0 φ0(ηr2)dη < 1. thus, the linear operator e1 is invertible. it follows by the identity z∗ − x∗ = e−11 (g(z∗)− g(x∗)) = e−11 (0) = 0, and we conclude z∗ = x∗. � remark 2. under all the conditions (h4)–(h9), one can set r1 = r∗ in proposition 1. 2.2. semi-local. the calculations and formulae are as in section 2.1, but x∗, φ0, φ are exchangedby x0, ψ0, and ψ, respectively.suppose (c1) there exists a nondecreasing and continuous function ψ0 : a → a such that the function 1−ψ0(t) has a smallest positive solution in the interval a, which is denoted by t0. definethe interval s = [0, t0). (c2) there exists a nondecreasing and continuous function ψ : s → a.define the sequences {αin} for α00 = 0, some α10 ≥ 0, i = 0, . . . , k , and each n = 0, 1, 2, . . . by ψn =  ψ(α1n − α0n),or ψ0(α 0 n) + ψ0(α 1 n), pn = 1 2 ( ψ0(α 0 n) + ψ0(α 1 n) ) , α2n = α1n + ψn(α1n − α0n) 2(1− pn) , (30) λj−1n = 1∫ 0 ψ((1− η)(αj−1n − α0n))dη · (αj−1n − α0n) + (1 + ψ0(α 0 n))(αj−1n − α1n), qn = 1 2 ( 3ψ0(α 1 n) + ψ0(α 0 n) ) , https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 10 αjn = αj−1n + (ψ0(α 0 n) + ψ0(α 1 n) + 2)λj−1n 2(1− ψ0(α0n))(1− qn) , µn+1 = 1∫ 0 ψ((1− η)(α0n+1 − α0n))dη(α0n+1 − α0n) + (1 + ψ0(α 0 n))(α0n − α0n), and α1n+1 = α0n+1 + µn+1 1− ψ0(α0n+1) , where again α0n+1 = αkn .the sequence {αin}n is shown to be majorizing for {y (i)n }n in theorem 2.but let us first provide a convergence condition for it. (c3) there exists t ∈ [0, t0) such that for each i = 0, 1, 2, . . . , k and each n = 0, 1, 2, . . . ψ0(α 0 n) < 1, pn < 1, qn < 1, and αin ≤ t. it follows by this condition and (30) that the sequence {αin} is nondecreasing andbounded from above by t and as such it converges to some α∗ ∈ [0, t]. the limit point α∗is the unique least upper bound of the sequence {αin}.as in the local analysis, the operators on the method (2) connect to the functions ψ0 and ψ. (c4) there exist x0 ∈ d and a linear operator e such that for each u ∈ d ‖e−1(g′(u)− e)‖ ≤ ψ0(‖u − x0‖). it follows by the conditions (c1), (c4), and (30) that if u = x0, we get ‖e−1(g′(x0)− e)‖ ≤ ψ0(0) < 1. so, the linear operator g′(x0) is invertible, in which case we can take α10 ≥ ‖g′(x0)−1g(x0)‖. define the region d2 = u[x0, α ∗] ∩d. (c5) ‖e−1(g′(u2)− g′(u1))‖ ≤ ψ(‖u2 − u1‖), for each u2, u1 ∈ d2. (c6) u[x0, α ∗] ⊂ d. remark 3. as in the local analysis, possible selections for e can be e = i or e = g′(z̄) for someauxiliary point x̄ ∈ d such that x̄ 6= x0, or e = g′(x0), or some other selection. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 11the semi-local analysis of the method (2) follows in the next result. theorem 2. suppose the conditions (c1) − (c6) hold. then, the sequence {xn} generated by the method (2) is well-defined in u(x0, α ∗), remains in u(x0, α ∗), and is convergent to a solution x∗ ∈ u[x0, α ∗] of the equation g(x) = 0 such that for each n = 0, 1, 2, . . . ‖x∗ − xn‖ ≤ α∗αn. proof. as in the local analysis, induction is used to first establish the assertions ‖y (1)n − xn‖ ≤ α1n − α0n, (31) ‖y (2)n − y (1)n ‖ ≤ α2n − α1n, (32) ‖y (j)n − y (j−1)n ‖ ≤ αjn − αj−1n . (33) by switching the conditions (h1)− (h9) by (c1)− (c5) but using the same formulas, we get inturn y (2) n − y (1)n = t ( g′(y (1) n )− g′(xn) ) g′(xn)−1g(xn) = − [ te−1 ] [ e−1(g′(y (1) n )− g′(xn)) ] (y (1) n − xn), ‖y (2)n − y (1)n ‖ ≤ ψn(α1n − α0n) 2(1− pn) ≤ α2n − α1n, ‖y (2)n − x0‖ ≤ ‖y (2)n − y (1)n ‖+ ‖y (1)n − x0‖ ≤ α2n − α1n + α1n − α0n = α2n < α∗. so, the estimate (32) holds and the iterate y (2)n ∈ u[x0, α ∗].then, by the identity g(y (j−1) n ) = g(y (j−1) n )− g(xn)− g′(xn)(y (1) n − xn), = g(y (j−1) n )− g(xn)− g′(xn)(y (j−1) n − xn) + g′(xn)(y (j−1) n − y (1)n ), which can imply ‖e−1g(y (j−1) n )‖ ≤ 1∫ 0 ψ((1−η)(αj−1n −α0n))dη(αj−1n −α0n)+(1+ψ0(α 0 n))(αj−1n −α1n) = λj−1n (34) ‖y (j)n − y (j−1)n ‖ ≤ (ψ0(α 0 n) + ψ0(α 1 n) + 2)λj−1n 2(1− ψ0(α0n))(1− qn) = αjn − αj−1n . thus, ‖y (j)n − x0‖ ≤ ‖y (j)n − y (j−1)n ‖+ ‖y (j−1)n − x0‖ ≤ αjn − αj−1n + αj−1n − α00 = αjn < α∗. thus, the assertions (33) hold and all the iterates {y (j)n } ⊂ u(x0, α ∗).it is left to show that assertion (31) holds if n + 1 replaces n. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 12but we can write in turn g(xn+1) = g(xn+1)− g(xn)− g′(xn)(y1 − xn) = g(xn+1)− g(xn)− g′(xn)(xn+1 − xn) + g′(xn)(xn+1 − y (1)n ),which can give, as in (34), ‖e−1g(xn+1)‖ ≤ 1∫ 0 ψ((1−η)(α0n+1−α0n))dη(α0n+1−α0n)+(1+ψ0(α 0 n))(α0n+1−α1n) = µn+1. (35) consequently, we obtain ‖y (1)n+1 − xn+1‖ ≤ ‖g ′(xn+1) −1e‖‖e−1g(xn+1)‖, ≤ µn+1 1− ψ0(α0n+1) = α1n+1 − α0n+1 and ‖y (1)n+1 − x0‖ ≤ ‖y (1) n+1 − xn+1‖+ ‖xn+1 − x0‖ ≤ (α1n+1 − α0n+1) + (α0n+1 − α00) = α1n+1 < α∗. thus, the induction for assertions (31)–(33) is completed, and all the iterates {y (i)n } ∈ u(x0, α ∗).it also follows that the sequence {x jn} is complete in banach space b0, since {αin} is alsocomplete as convergent by the condition (c4). therefore, there exists x∗ ∈ u[x0, α ∗] such that lim n→+∞ y (k) n = x∗ or lim n→+∞ xn = x∗. moreover, by letting n → +∞ in (35), we obtain g(x∗) = 0, where the continuity of the operator g has also been used. furthermore, by noticing that αkn = αn+1 and αkn = α0n+1, estimate (33)can be rewritten for j = k as ‖xn+1 − xn‖ ≤ αn+1 − αn,so ‖xn+h − xn‖ ≤ αn+h − αn, h = 0, 1, 2, . . . (36)finally, by letting h → +∞ in (36), we show the assertion (2). � next, we study the uniqueness of a solution in a certain region. proposition 2. suppose there exists a solution y∗ ∈ u(x0, r3) of the equation g(x) = 0 for some r3 > 0; the condition (c4) holds in the ball u(x0, r3), and there exists r4 ≥ r3 such that 1∫ 0 ψ0((1− η)r3 + ηr4)dη < 1. (37) define the region d3 = d ∩ u[x0, r4]. then, y∗ is the only solution of the equation g(x) = 0 in the region d3. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 13 proof. suppose there exists y∗∗ ∈ d3 solving the equation g(x) = 0 such that y∗∗ 6= y∗.then define the linear operator e2 = 1∫ 0 g′(y∗ + η(y∗∗ − y∗))dη. by applying the condition (c4) and (37), we obtain in turn ‖e−1(e2 − e)‖ ≤ 1∫ 0 ψ0((1− η)‖y∗∗ − x0‖+ η‖y∗ − x0‖)dη ≤ 1∫ 0 ψ0((1− η)r3 + ηr4)dη < 1. it follows that the linear operator e2 is invertible. hence, again we conclude y∗∗ = y∗. � remark 4. • the limit point a∗ given in the condition (c1) can be replaced by t0 in (c6). • if all conditions (c1)− (c6) hold, then we can set r3 = α∗ and y∗ = x∗ in proposition 2. 3. numerical results to comprehensively evaluate the performance and robustness of the proposed high-order iterativemethods, we present five numerical examples of different complexity and dimensionality. thesetest problems have been selected from the literature and include systems with diverse nonlinearcharacteristics, such as trigonometric, exponential, and polynomial structures. the examples aredesigned to assess the methods’ accuracy, convergence speed, and stability.in all numerical experiments, the stopping criterion was based on achieving a residual norm belowcertain ε, ensuring a high level of numerical precision. a maximum of 50 iterations was imposedto prevent excessive computational effort. this limit is justified by empirical evidence indicatingthat well-designed, high-order methods typically achieve convergence within this range. to ensurereliable performance metrics, cpu execution times were averaged over 50 independent runs, therebymitigating the influence of background system noise and transient operational conditionsall simulations were conducted within a google colaboratory runtime environment. this envi-ronment was configured with an intel xeon cpu operating at 2.20 ghz, 13 gb of system ram,and an nvidia tesla k80 gpu equipped with 12 gb of vram. numerical computations wereperformed using the python library mpmath, with the arithmetic precision set to 100 decimal dig-its. this standardized setup was maintained across all test cases to ensure fair and reproduciblecomparisons.we compare method (2) with several established iterative methods, specifically the sixth-ordermethod (29) of wang et al. [18], the method (14) by hueso et al. [12], the scheme (6) of cordero etal. [10] and the method (14) proposed by abbasbandy et al. [1]. these benchmark techniques arewell known in the literature for their high-order convergence properties and are frequently used fortesting nonlinear solvers. our method is evaluated against these in terms of number of iterations, https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 14residual norm ‖g(x)‖, step difference ‖xn+1 − xn‖, and cpu time. the results are demonstratedin tables 1–5. example 1. consider the nonlinear system defined as: gi(x) = arctan(xi) + 1− 2 20∑ j=1 j 6=i x2j = 0, i = 1, 2, . . . , 20. the methods converge to the zero x∗ = (0.1757683, 0.1757683, . . . , 0.1757683)t , starting fromthe initial approximation x0 = (0.15, 0.15, . . . , 0.15)t . table 1. results for example 1 method iterations ‖g(x)‖ ‖xn+1 − xn‖ cpu time (s) method (2) 2 6.7121× 10−31 8.1782× 10−30 0.342917wang et al. 3 1.8677× 10−52 1.4708× 10−26 0.656792hueso et al. 4 6.4643× 10−44 8.6687× 10−46 1.112088cordero et al. 2 1.8677× 10−52 1.0138× 10−10 0.333056abbasbandy et al. 3 1.2046× 10−23 2.0006× 10−23 0.307392 example 2. consider the nonlinear system: gi(x) = xi − cos 2πxi − 50∑ j=1 xj  = 0, i = 1, 2, . . . , 50. the solution is x∗ = (0.5018261, 0.5018261, . . . , 0.5018261)t , with the initial guess x0 = (0.51, 0.51, . . . , 0.51)t . table 2. results for example 2 method iterations ‖g(x)‖ ‖xn+1 − xn‖ cpu time (s) method (2) 2 2.221× 10−25 4.9027× 10−12 4.127071wang et al. 7 6.9532× 10−22 3.6049× 10−20 17.881168hueso et al. 8 5.4021× 10−24 2.8007× 10−22 29.956985cordero et al. 7 5.5536× 10−23 2.8792× 10−21 15.002068abbasbandy et al. 9 6.4707× 10−16 7.4808× 10−16 12.348433 https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 15 example 3. let us consider the following system of 99 nonlinear equations: gi(x) = xixi+1 − 1 = 0, 1 ≤ i ≤ 98, x99x1 − 1 = 0, i = 99. the exact solution is x∗ = (1, 1, . . . , 1)t , with the starting vector x0 = (2, 2, . . . , 2)t . table 3. results for example 3 method iterations ‖g(x)‖ ‖xn+1 − xn‖ cpu time (s) method (2) 2 0.0 9.3704× 10−5 21.838247wang et al. 4 0.0 4.0358× 10−25 56.250490hueso et al. 5 0.0 8.9985× 10−39 99.436670cordero et al. 3 0.0 7.0416× 10−15 33.705407abbasbandy et al. 4 0.0 1.3554× 10−17 24.382990 example 4. consider the following system of nonlinear equations g(x) =  x1 + x2 − 1 = 0, 2x1 + x2 + 2x3 − 2 = 0, x1 + x2 + x3 − x4 = 0, x22 x3 x21 x4 − (0.647)2 = 0. the solution vector is x∗ = (0.422499, 0.577501, 0.288751, 1.288751)t , obtained from theinitial estimate x0 = (0.8, 0.2, 0.9, 1.8)t . table 4. results for example 4 method iterations ‖g(x)‖ ‖xn+1 − xn‖ cpu time (s) method (2) 4 1.4315× 10−51 4.334× 10−16 0.04373wang et al. n/a n/a n/a n/ahueso et al. n/a n/a n/a n/acordero et al. 4 2.3346× 10−53 1.243× 10−21 0.01554abbasbandy et al. n/a n/a n/a n/a n/a indicates that the method did not converge to the required solution within theprescribed iteration or tolerance limits. https://doi.org/10.28924/ada/ma.5.18 eur. j. math. anal. 10.28924/ada/ma.5.18 16 example 5. consider the nonlinear system of equations of size 200 gi(x) = e−xi − 200∑ j=1 j 6=i xj = 0, i = 1, 2, . . . , 200. the initial approximation is set as x0 = ( 3 2 , 3 2 , . . . , 3 2 )t , with parameters a = −2.0 and b = 2.0,leading to the solution: x∗ = (0.0050, 0.0050, . . . , 0.0050)t . table 5. results for example 5 method iterations ‖g(x)‖ ‖xn+1 − xn‖ cpu time (s) method (2) 2 3.4388× 10−51 4.7064× 10−37 50.414524wang et al. 3 6.8725× 10−52 7.9328× 10−26 86.904696hueso et al. 3 1.1438× 10−51 1.4069× 10−18 129.807516cordero et al. 3 2.5137× 10−51 3.0846× 10−48 71.584528abbasbandy et al. 3 6.5891× 10−52 1.7579× 10−20 98.961347 4. conclusions this paper presented a general high-order iterative method for solving nonlinear systems withoutrequiring higher-order derivatives. we established both local and semi-local convergence resultsusing majorant conditions and majorizing sequences, providing rigorous guarantees even from dis-tant initial guesses. numerical experiments on benchmark problems confirmed the method’s ac-curacy, fast convergence, and low residual errors compared to existing high-order methods. theresults validate the theoretical findings and highlight the method’s applicability to a wide range ofnonlinear problems, suggesting potential for further extensions and refinements. author contributions. authors contributed equally. all authors have read and agreed to thepublished version of the manuscript. references [1] s. abbasbandy, p. bakhtiari, a. cordero, j.r. torregrosa, t. lotfi, new efficient methods for solving nonlinear systemsof equations with arbitrary even order, appl. math. comput. 287-288 (2016), 94–103. https://doi.org/10.1016/ j.amc.2016.04.038.[2] c. amorós, i.k. argyros, r. gonzález, á.a. magreñán, l. orcos, í. sarría, study of a high-order family: localconvergence and dynamics, mathematics 7 (2019), 225. https://doi.org/10.3390/math7030225.[3] i.k. argyros, convergence and application of newton-type 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convergence, appl. math. lett. 22 (2009), 1798–1802. https://doi.org/10.1016/j.aml.2009.06.022.[19] x.y. xiao, h.m. yin, increasing the order of convergence for iterative methods to solve nonlinear systems, calcolo53 (2016), 285–300. https://doi.org/10.1007/s10092-015-0149-9. https://doi.org/10.28924/ada/ma.5.18 https://doi.org/10.1007/s40819-020-0784-y https://doi.org/10.3390/math7010099 https://doi.org/10.3390/sym11020128 https://doi.org/10.3390/sym11020128 https://doi.org/10.1016/j.cam.2020.113249 https://doi.org/10.1007/s11075-009-9359-z https://doi.org/10.1016/j.cam.2018.06.006 https://doi.org/10.1016/j.cam.2014.06.010 https://doi.org/10.1016/j.cam.2014.06.010 https://doi.org/10.1016/j.amc.2008.03.037 https://doi.org/10.1016/j.cam.2016.12.019 https://doi.org/10.1016/j.aml.2009.06.022 https://doi.org/10.1007/s10092-015-0149-9 1. introduction 2. convergence analysis 2.1. local 2.2. semi-local 3. numerical results example 1 example 2 example 3 example 4 example 5 4. conclusions references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 14doi: 10.28924/ada/ma.5.14 modulation instability, dark and singular soliton for weakly nonlocal schrodinger equation ali danladi1,∗, alhaji tahir2, hadi rezazadeh3 1department of mathematics, federal university, dutse, nigeria alidanladi14@gmail.com, ali.danladi@fud.edu.ng 2department of mathematics, faculty of physical sciences, modibbo adama university, yola, nigeria atahir@mau.edu.ng 3faculty of engineering technology, amol university of special modern technologies, amol, iran h.rezazadeh@ausmt.ac.ir ∗correspondence: alidanladi14@gmail.com, ali.danladi@fud.edu.ng abstract. the optical soliton solution of nonlinear complex models holds significant importance innonlinear optics and communication systems. considering nonlinear complex models, often describedby equations like the nonlinear schrodinger equation (nlse), plays a crucial role in defining thebalance between dispersive and nonlinear effects, enabling the formation and maintenance of solitonsover long distances. this stability is crucial for signal integrity in optical communication systems.the investigation of optical soliton solutions from nonlinear complex models is sometimes compli-cated. with this in mind, we employed an effective method of extended tanh function with a riccattidifferential equation to retrieve the dark, singular and periodic wave solutions for the weakly nonlocalnonlinear schrodinger equation with parabolic law. the obtained solutions were verified by back-substitution in the original equations, with the aid of a mathematica to affirm the robustness of thechosen approach. respective 2d and 3d graphs for some of the obtained results was portrayed bychoosing suitable values of the parameters that were involved. an analysis of instability that resultsin the modulation of the steady-state as a result of co-action between the nonlinear and dispersiveeffects was performed on the proposed model where the condition for stable wave under small per-turbation was obtained and presented. the gain spectrum plot for the modulation instability wasportrayed. 1. introductionpartial differential equations (pdes) serve as powerful mathematical tools for modeling and under-standing diverse phenomena in science and engineering. these equations involve multiple variablesand their partial derivatives, making them well-suited for describing complex physical processessuch as heat conduction, fluid dynamics, and electromagnetic fields. the investigation of pdesencompasses various aspects, including analytical and numerical methods for solving these equa-tions. analytically, researchers explore techniques like separation of variables, integral transforms, received: 3 mar 2025. key words and phrases. nonlocal schrodinger equation with pl; extended tanh approach; modulation instability.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 2and series solutions to obtain exact solutions or gain insights into the behavior of solutions. onthe other hand, numerical methods, such as finite difference, finite element, and spectral methods,provide computational tools to approximate solutions for pdes in cases where analytical solu-tions are challenging or impossible to obtain. additionally, the study of stability, existence, anduniqueness of solutions, as well as the development of new solution methods, continues to be avibrant area of research within the broader field of partial differential equations. understandingand harnessing the mathematical intricacies of pdes play a crucial role in advancing our compre-hension of the natural world and technological applications. numerous scholars have focused theirefforts on determining exact solutions for npdes, with these equations finding applications acrossdiverse scientific and technological domains, including but not limited to mathematical physics,fluid dynamics, optical fibers, and economics [10, 14, 27].in recent years, various methodologies have been developed for determining precise solutions tonpdes. these approaches include the lie symmetry method [16, 18, 20, 26, 30], the kudryashovmethod [21, 22, 33], the sine-gordon expansion method [6], the invariant subspace method [13, 28],the sardar subequation method [3, 11], and others [1, 5, 8, 9, 29]. these techniques contributeto the exploration of exact solutions for a wide range of pdes. the nlse is a complex pdethat plays a crucial role in various fields of physics and engineering, describing the evolution ofcomplex wave packets. it arises in different contexts, including nonlinear optics, plasma physics,fluid dynamics, and condensed matter physics. the nlse and its variants have significant impor-tance in understanding and predicting the behavior of wave-like phenomena in diverse real-worldapplications.in nonlinear optics, the nlse governs the propagation of intense laser beams through nonlinearmedia. this equation describes the interactions among optical waves, considering effects suchas self-focusing, self-phase modulation, and optical solitons. optical solitons, which are stable,localized wave packets that can maintain their shape during propagation, and applications in long-distance communication systems. the nlse helps optimize and control these phenomena in thedesign of optical communication systems and laser technologies.the nlse also appears in the study of ultra-cold atomic gases, particularly in the context ofbose-einstein condensates. in this scenario, the nlse describes the dynamics of the macroscopicwave function of the condensate. understanding the nlse for becs is crucial for investigatingphenomena such as matter-wave solitons and vortices, which have applications in precision mea-surements and quantum information processing.the nlse arises in the study of langmuir waves in plasmas, where it describes the evolution of theelectron plasma wave. nonlinear effects become significant in high-intensity laser-plasma inter-actions and can lead to the generation of harmonics and other phenomena. this has applicationsin areas such as controlled nuclear fusion research and the development of high-power particle https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 3accelerators.nlse is fundamental in understanding the behavior of optical pulses in fiber optic communica-tion systems. fiber optic channels exhibit nonlinear effects such as self-phase modulation andcross-phase modulation, which can distort transmitted signals. the nlse is essential for modelingand mitigating these effects, ensuring the reliability and efficiency of long-distance communicationnetworks.variants of the nlse are used to model the propagation of water waves in oceans and otherbodies of water. nonlinear effects, such as wave steepening and wave breaking, can be describedusing these equations. understanding these phenomena is crucial for predicting and mitigating theimpact of tsunamis, storm surges, and other oceanic events.nlse variants are employed in the study of biological systems, such as modeling the propagationof nerve impulses. the nlse can describe the nonlinear dynamics of excitable media, providinginsights into the behavior of electrical signals in biological tissues.moreover, the nlse [4, 12, 23], which is an essential fully-integrated nonlinear dispersive par-tial differential equation (pde), has found extensive application in elucidating diverse phenomenalike deep water waves, rogue waves, plasmas, and nonlinear optics, including atomic physics. thenlse serves as a comprehensive description of nonlinear dispersive processes. zhou et al. [34]considered the weakly nonlocal nlse having pl nonlinearity with external potential as iφt + λ1φxx + ( λ2|φ|2xx + λ3|φ|2 + λ4|φ|4 ) φ = 0. (1) the study of schrodinger equations with nonlinearity is an important area of research in math-ematical physics. in recent years, the weakly nonlocal schrodinger equation with parabolic lawnonlinearity has gained significant attention due to its numerous applications in various fields suchas quantum mechanics, nonlinear optics, and fluid dynamics.in this paper, we present modulation instability analysis and the novel exact solutions for theweakly nonlocal schrodinger equation with parabolic law nonlinearity, derived using the methodextended tanh function. the solutions we obtain, which include dark soliton, bright soliton, andtraveling wave solutions, have not been previously reported in the literature and demonstrate thecomplex dynamics of the considered equation. dark optical soliton describes the solitary waveswith lower intensity than the background, bright optical soliton describes the solitary waves whosepeak intensity is larger than the background and the singular optical soliton is a solitary wavewith discontinuous derivatives; examples of such solitary waves include compactions, which havefinite (compact) support, and peakons, whose peaks have a discontinuous first derivative [35, 36].dark soliton propagates without changing its shape, but it is not made by a normal pulse; rather,it is a lack of energy in a continuous-time beam. the intensity is constant, but for a short timeduring which it jumps to zero and back again, thus generating a "dark pulse"’. those solitons can https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 4actually be generated introducing short dark pulses in much longer standard pulses. dark soli-tons are more difficult to handle than standard solitons, but they have shown to be more stableand robust to losses. bright optical soliton causes a temporary increase in an associated waveamplitude [52]. the combined dark-bright optical soliton carries the combine features of the darkand bright optical solitons. moreover, the modulation instability analysis for the weakly nonlocalnlse having pl nonlinearity with external potential will also be presented.our findings have significant implications for the study of nonlinear phenomena in physical sys-tems. the ability to obtain exact solutions to such complex equations is crucial in understandingthe underlying physics and designing new experiments. additionally, with the aid of mathemat-ica, our results are verified by back-substitution in the original equations, affrming the robustnessof our approach. the suggested is not only direct and simple but also suitable for constructingnew results, paving the way for future applications on nlses with dual power law and perturbednlses with kerr law. this is particularly useful for identifying solitons in photorefractive andpolymer materials. the proposed technique holds potential for further applications in natural sci-ence models, aiding in the investigation of other mathematical challenges and characterizing thebehavior of nonlinear models.the remaining sections of this manuscript are organized as follows: in section 2, we give thedescription of the approach.in section 3, we apply the method the governing equation. insection4, we give the graphical representation of some of the obtained results. in section 5, we presentthe analysis of modulation instability for the model and its physical description. in section 6, weprovide the conclusion of the study. 2. description of the approachthe method of extended tanh is often employed in obtaining soliton solutions due to is effectivenessin capturing the inherent characteristics of solitons such as their amplitude, width, and velocity.the following are some of the rationale for using the method: flexibility, asymptotic behavior, easeof manipulation accuracy, existence of exact solutions.in order to present the modified extended tanh approach, we need to consider the nonlinear partialdifferential equation of the form: ρ (u,ux , uxx , utux , ...) = 0; (2) in transforming (2); we use the wave transformations: u(x, t) = u(ξ); ξ = (x − µt) (3) where µ is a nonzero constant. substitution of the transformation (3) into eq. (2), it reduces to anordinary differential equation of the polynomial form: ξ ( u(ξ), u ′(ξ), u ′′(ξ), u ′′′(ξ), ... ) = 0. (4) https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 5furthermore, the solution is considered to be a finite series of the form: u(ξ) = a0 + n=n∑ n=1 anψ n(ξ) + n=n∑ n=1 bn ψn (ξ) (5) where, a0, an, bn, n = 1, 2, 3, ...n are constants which to be computed; and n is a positive integerwhich is to be determined by balancing the highest order derivative and with the highest nonlinearterms in the equation. also, ψ(ξ) satisfies the riccati’s differential equation: ψ′(ξ) = ψ2(ξ) + b (6) where b is a constant. furthermore, the riccati differential equation in (6) has solutions of theform: • for hyperbolic solution, if b < 0, then φ(ξ) = − √ −b tanh √ −b ξ, φ(ξ) = − √ −b coth √ −b ξ • for rational solution, if b = 0, then 1 ξ • for periodic solution, if b > 0, then φ(ξ) = √ b tan √ b ξ, φ(ξ) = √ b cot − √ −b ξnow, after we substitute eq. (5) and its derivatives together with the riccati eq. (6), into eq.(4), it gives a polynomial in φ(ξ) . collecting the coefficients of the same power of φ(ξ) in thepolynomial and setting each of them to zero, we shall get a set of algebraic equations. solvingthese system of algebraic equation with the aid of symbolic computation using mathematica to getthe values of a0, an, bn, (n = 1, 2, 3, ...) and b. finally, substituting these values into eq. (5) fromwhich we obtain the solution of eq. (4). 3. applicationin this section, the application of the modified extended tanh expansion method with riccati differ-ential equation [49] to our equation (1) shall be presented.consider the transformation: φ(x, t) = φ(ξ)e i$; ξ = x − νt and $ = ωx − r t + δ (7) substituting equation (7) into equation (1), gives the following nonlinear ode: 2λ2φ2φ′′ + λ1φ′′ + λ4φ5 + λ3φ3 + φ ( 2λ2φ′2 − λ1ω 2 + r ) = 0. (8) from the real part and (v − 2λ1ω) φ′ = 0. (9) from the imaginary part. thus, we have the constraint condition as v = 2λ1ω. (10) https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 6balancing between φ2φ′′ and φ5 , that is: 2n + n + 2 = 5n =⇒ n = 1. (11) with equation (11), our equation (5) takes the form: φ(ω) = a1φ(ω) + a0 + b1 φ(ω) (12) substituting eq. (12), its first and second derivative along with equation (6) into equation (8), weget a polynomial in powers of φ(ω). summing the coefficients of φ(ω) with the same power andequating each summation to zero, gives a set of algebraic equations. solving these set of algebraicequations, yields the following cases of solutions: case onewhen a0 = 0; a1 = − i √ 6 √ λ2√ λ4 ; b1 = 0; r → 1 16 ( λ1 ( 2λ3 λ2 + 16ω2 ) − λ4λ 2 1 λ2 2 + 3λ2 3 λ4 ) ; w = λ1λ4−3λ2λ3 24λ2 2 , we obtain the following solutions to equation (3):if w < 0, then we have φ1(ξ) = ± i √ 3λ2λ3 − λ1λ4 tanh (√ 3λ2λ3−λ1λ4ω 2 √ 6λ2 ) 2 √ λ2 √ λ4 e i(ωx−r t+δ). (13) φ2(ξ) = ± i √ 3λ2λ3 − λ1λ4 coth (√ 3λ2λ3−λ1λ4ω 2 √ 6λ2 ) 2 √ λ2 √ λ4 e i(ωx−r t+δ). (14)if w > 0, then we have φ3(ξ) = ∓− i √ λ1λ4 − 3λ2λ3 tan (√ λ1λ4−3λ2λ3ω 2 √ 6λ2 ) 2 √ λ2 √ λ4 e i(ωx−r t+δ). (15) φ4(ξ) = ± i √ λ1λ4 − 3λ2λ3 cot (√ λ1λ4−3λ2λ3ω 2 √ 6λ2 ) 2 √ λ2 √ λ4 e i(ωx−r t+δ). (16) case twowhen a0 = 0; a1 = i √ 6 √ λ2√ λ4 ; b1 = i(3λ2λ3−λ1λ4) 8 √ 6λ 3/2 2 √ λ4 ; r = λ1(3λ2(4λ2ω 2+λ3)−λ1λ4) 12λ2 2 ; w = 3λ2λ3−λ1λ4 48λ2 2 ; we obtain the following solutions to equation (1):if w < 0, then we have φ5(ξ) = ± i √ λ1λ4 − 3λ2λ3csch(√ λ1λ4 3 −λ2λ3ω 2λ2 ) √ 2 √ λ2 √ λ4 e i(ωx−r t+δ). (17) if w > 0, then we have φ6(ξ) = ∓ i √ 3λ2λ3 − λ1λ4 csc (√ λ2λ3− λ1λ4 3 ω 2λ2 ) √ 2 √ λ2 √ λ4 e i(ωx−r t+δ). (18) https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 7 4. graphical representations of resultsin this section, the 2d and 3d graphical representation for some of the obtained results for theweakly nonlocal nlse having pl nonlinearity, which is given by equations (1) shall be presentedby choosing different values of parameters that are involved. figure 1. 3d and 2d abs plot for φ1(ξ). in figure (1) above, we have the surface profile of 2d and 3d absolute plot represen-tation of equation (13) ( that is, φ1(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of xfor which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t at 0, 2, 4, . figure 2. 3d and 2d abs plot for φ3(ξ). in figure (2) below, we have the surface profile of 2d and 3d absolute plot represen-tation of equation (15) ( that is, φ3(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of xfor which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t at https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 8 figure 3. 3d and 2d im plot for φ3(ξ). the origin. in figure (3) above, we have the surface profile of 2d and 3d imaginary plot represen-tation of equation (15) ( that is, φ3(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of xfor which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t atthe origin. figure 4. 3d and 2d abs plot for φ5(ξ). in figure (4) above, we have the surface profile of 2d and 3d absolute plot represen-tation of equation (17) ( that is, φ5(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of xfor which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t atthe origin.in figure (5) above, we have the surface profile of 2d and 3d real plot representa-tion of equation (17) ( that is, φ5(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of x https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 9 figure 5. 3d and 2d real plot for φ5(ξ). for which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t atthe origin. figure 6. 3d and 2d real plot for φ6(ξ). in figure (6) above, we have the surface profile of 2d and 3d real plot representa-tion of equation (18) ( that is, φ6(ξ)) by taking the following parameter values (δ = −1.5; ) (λ4 = 1.5; ) (λ1 = 0.5; ) (λ3 = 1.5; ) (λ2 = 1.15; ) (ω = 1.5; ). the range of values of xfor which the graphs were plotted is [-10,10]. the 2d graphs was plotted by taking values of t atthe origin. 5. modulation instability (mi) analysisvarious nonlinear phenomena display an instability that results in the modulation of the steady-state as a result of co-action between the nonlinear and dispersive effects [37]. in this section, wederive modulation instability for the weakly nonlocal schrodinger equation with parabolic law. theorem 1: suppose that equation (1) characterizes a wave system featuring a non-trivial disper-sion relation, alongside nonlinear components. given that φ(x, t) represents a solution to equation,it is possible, under appropriate circumstances, for a range of parameters to exist where mi takesplace. over time, the amplitude and configuration of the wave undergo significant alterations dueto instability, resulting in a rapid growth of small perturbations. https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 10 proof : to explore the mi of equation (1), let take the initial assumption that equation (1) issubjected to a small perturbation in the following manner: φ(x, t) = (√ m + φ(x, t) ) e imx ; (19) where m is the normalized optical power,φ(x, t) shows real valued amplitude of perturbation withrelatively dispersion. substituting equation (19) into equation (1) and linearizing, gives iφt + (λ1 + λ2m) φxx + iλ1m (φx + φ∗x) + ( (λ4 − λ1)m2 + λ3m ) (φ + φ∗) = 0, (20) where φ∗ denotes complex conjugate.assume the solutions of equation (20) to be of the form φ(x, t) = a1e (i(x$−ξt)) + a2e (−i(x$−ξt)) (21) φ(x, t)∗ = a1e (−i(x$−ξt)) + a2e (i(x$−ξt)) (22)where $ and ξ represent normalize wave number and frequency of perturbation respectively.substituting (21) and (22) into (20) and collect the coefficients of e(−i(x$−ξt)), and e(i(x$−ξt)), and solve the determinant of the resulting matrix of coefficients we obtain the followingdispersion relation: −λ2 1$ 4 − 2λ1λ2m 3$2 + 2λ2λ4m 3$2 − λ2 2m 2$4 − 2λ2 1m 2$2 + 2λ2λ3m 2$2 2λ1λ4m 2$2 − 2λ1mξ$ − 2λ1λ2m$ 4 + 2λ1λ3m$ 2 + ξ2 = 0 (23) solving the dispersion relation(23) for ξ, we obtain: ξ = λ1m$ ∓ ( λ2 1$ 4 + 2λ1λ2m 3$2 + λ2 2m 2$4 + 3λ2 1m 2$2 + 2λ1λ2m$ 4 −2λ2λ4m 3$2 − 2λ2λ3m 2$2 − 2λ1λ4m 2$2 − 2λ1λ3m$ 2 ) 1 2 . (24) in a situation whereby( λ2 1$ 4 + 2λ1λ2m 3$2 + λ2 2m 2$4 + 3λ2 1m 2$2 + 2λ1λ2m$ 4 −2λ2λ4m 3$2 − 2λ2λ3m 2$2 − 2λ1λ4m 2$2 − 2λ1λ3m$ 2 ) > 0 ,the wave number is real for any real value of m and the steady-state is stable against small perturbations. however, in contraryto the above condition, the steady-state solution turns to be unstable, that is, the wave number becomes imaginary, when ( λ2 1$ 4 + 2λ1λ2m 3$2 + λ2 2m 2$4 + 3λ2 1m 2$2 + 2λ1λ2m$ 4 −2λ2λ4m 3$2 − 2λ2λ3m 2$2 − 2λ1λ4m 2$2 − 2λ1λ3m$ 2 ) < 0 and the perturbation grows exponentially. under this condition, the growth rate of modulationstability gain spectrum g(m) may be given as g(r) = 2im(ξ)= 2im ( λ1m$ ∓ ( λ2 1$ 4 + 2λ1λ2m 3$2 + λ2 2m 2$4 + 3λ2 1m 2$2 + 2λ1λ2m$ 4 −2λ2λ4m 3$2 − 2λ2λ3m 2$2 − 2λ1λ4m 2$2 − 2λ1λ3m$ 2 )) . (25) the figure below shows the gained dispersion relation to investigate the steady-state stabilityby taking the following parameter values λ1 → 2, λ2 → 2.4, λ3 → 1.5, λ4 → 2.3 under distinct https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 11 figure 7. gain spectrum of mi under different values of m values of m. 6. conclusionsvariants of the nlse are used to model the propagation of water waves in oceans and otherbodies of water. nonlinear effects, such as wave steepening and wave breaking, can be describedusing these equations. understanding these phenomena is crucial for predicting and mitigatingthe impact of tsunamis, storm surges, and other oceanic events. furthermore, nlse variant areemployed in the study of biological systems, such as modeling the propagation of nerve impulses.the nlse can describe the nonlinear dynamics of excitable media, providing insights into the be-havior of electrical signals in biological tissues. also, nlse which are essential in fully-integratednonlinear dispersive partial differential equation (pde), have found extensive application in elu-cidating diverse phenomena like deep water waves, rogue waves, plasmas, and nonlinear optics,including atomic physics. the nlse serves as a comprehensive description of nonlinear dispersiveprocesses. with all of these in mind, we found the courage and motivation in this study to providethe distinct types of exact soliton solutions for the weakly nonlocal schrodinger equation. we ob-tain dark and periodic singular soliton solutions via the reliable approach of the modified extendedtanh function method. the obtained solutions were verified by back-substitution in the originalequations, with the aid of a mathematica to affirm the robustness of the chosen approach. the ob-tained results were portrayed using two-dimensional and three-dimensional graphs. additionally, https://doi.org/10.28924/ada/ma.5.14 eur. j. math. anal. 10.28924/ada/ma.5.14 12modulation instability was performed to study the stationary state of the governing model. resultsare helpful in the progress of the concerned system. the gained results will be of high importancein the interaction of quantum-mechanical fluctuations, granular matters, and other fields of weaklynonlocal schrodinger with parabolic law applications. the achieved results are also useful in var-ious naturally occurring phenomena, industries, geophysics, civil engineering, pharmaceutical, andmany others. it is suggested that the method used is also useful for the other nonlinear models ofdifferent fields of science and engineering. references [1] m.j. ablowitz, nonlinear dispersive waves: asymptotic analysis and solitons, cambridge univ. press, 47 (2011).[2] a.r.z.u. akbulut, m. mirzazadeh, m.s. hashemi, k. hosseini, s. salahshour, c. park, triki?biswas model: itssymmetry reduction, nucci’s reduction and conservation laws, int. j. mod. phys. b 37 (2023), 2350063.[3] l. akinyemi, u. akpan, p. veeresha, h. rezazadeh, m. inc, computational techniques to study the dynamics ofgeneralized unstable nonlinear schrodinger equation, j. ocean eng. sci. 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space. the estimation ofcommutators plays an important role in studying the regularity of solutions of elliptic, parabolicand ultraparabolic partial differential equations of second order, and their boundedness can beused to characterize certain function spaces (see, for instance [3, 18,20]).we denote by l0(rd) the complex vector space of equivalent classes (modulo equality lebesguealmost everywhere) of lebesgue measurable complex-valued functions on rd . for f ∈ l0(rd) and x ∈ rd , the hardy-littlewood maximal function is defined by the formula mf (x) = sup r>0 |b(x, r)|−1 ∫ b(x,r) |f (y)| dy, (1.1) where |b(x, r)| is the lebesgue measure of the ball b(x, r) = {y ∈ rd : |x − y | < r}. the maximal commutatormb generated by the maximal operatorm and a locally integrable function b is defined by mb(f )(x) = sup r>0 |b(x, r)|−1 ∫ b(x,r) |b(x)− b(y)||f (y)|dy. furthermore the commutator generated by the operator m and a suitable function b is defined by [b,m]f (x) = b(x)m(f )(x)−m(bf )(x). recall that the operators mb and [b, m] essentially differ from each other since mb is positiveand sublinear and [b,m] is neither positive nor sublinear. the operators m , [b,m] and mb play received: 28 jul 2024. key words and phrases. total fofana spaces; maximal operator; commutator; sublinear operators; bmo spaces.1 https://adac.ee https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 2an important role in real and harmonic analysis and applications (see for example [1, 2, 10, 15, 16]and the references therein). the boundedness of these operators on lebesgue spaces has beenextended to several other spaces. for example, on morrey spaces ( [4], [11]), modified morreyspaces ( [14]), total morrey spaces ( [13]), fofana spaces ( [6]) to name but a few.recently, p. nagacy and b. a. kpata [17] introduced the total fofana spaces and established inthese spaces, under certain conditions, the boundedness of the hardy-littlewood maximal operators,the riesz potential and fractional maximal operators.the main purpose of the present paper is to establish the boundedness of the commutators of thehardy-littlewood maximal operator and some sublinear operators in total fofana spaces. beforestating our main results, let us start with some notations and basic definitions. let 1 ≤ q, p ≤ ∞and r > 0. for f ∈ l0(rd) we define r ‖f ‖q,p := ∥∥∥∥∥ [∫ rd |f χb(y,r)|q(x)dx ] 1 q ∥∥∥∥∥ pwith the lp(rd)-norm taken with respect to the variable y . we adopt the usual convention 1 ∞ = 0.in 1988, fofana [7] introduced the functions spaces (lq, lp)α (rd), 1 ≤ q ≤ α ≤ p ≤ ∞, whichconsists of the set of all functions f ∈ l0(rd) satisfying ‖f ‖q,p,α <∞, where ‖f ‖q,p,α = sup r>0 rd( 1 α − 1 q − 1 p ) r ‖f ‖q,p . it is proved in [7] the following properties: • for 1 ≤ q < α fixed and p going from α to ∞, the spaces (lq, lp)α (rd) form a chain of distinctbanach spaces beginning with the lebesgue space lα(rd) and ending by the classical morreyspace lq,d(1− q α )(rd) = (lq, l∞)α (rd); • there exists a constant c > 0 such that ‖f ‖q,p,α ≤ c ‖f ‖α . (1.2) for an in-depth study of fofana spaces, please consult the following references (see [6, 8, 9]).let 1 ≤ q ≤ α, λ ≤ p ≤ ∞ and [r ]1 = min{1, r}, r > 0. the total fofana spaces (lq, lp)α,λ(rd)are defined by (lq, lp)α,λ(rd) = { f ∈ l0(rd) : ‖f ‖(lq ,lp)α,λ(rd ) <∞ } where ‖f ‖(lq ,lp)α,λ(rd ) = sup r>0 [r ] d( 1 α − 1 q − 1 p ) 1 [1/r ] d(− 1 λ + 1 q + 1 p ) 1 r ‖f ‖q,p .note that(1) the space (lq, lp)α,λ(rd) is a complex vector subspace of l0(rd).(2) from the definition of (lq, lp)α,λ(rd) spaces we deduce that the map l0(rd) 3 f 7→ ‖f ‖(lq ,lp)α,λ(rd ) defines a norm on (lq, lp)α,λ(rd). https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 3 (3) for 1 ≤ q ≤ α, λ <∞, the space (lq, l∞)α,λ(rd) is the total morrey space lq,d(1− q α ),d(1− q λ )(rd)defined in [13] and lq,d(1− q α ),d(1− q λ )(rd) = lq,d(1− q α )(rd) ∩ lq,d(1− q λ )(rd) with λ ≤ α.we define the space bmo(rd) as the set of all locally integrable functions b with finite norm ‖b‖bmo(rd ) = sup r>0,x∈rd |b(x, r)|−1 ∫ b(x,r) |b(y)− bb(x,r)|dy where bb(x,r) = |b(x, r)|−1 ∫ b(x,r) b(y)dy.for a function b defined on rd , we denote by b−(x) := { 0 if b(x) ≥ 0 |b(x)| if b(x) < 0 and b+(x) = |b(x)| − b−(x). obviously b+(x)− b−(x) = b(x).our first result reads as follows theorem 1.1. let 1 < q ≤ λ ≤ α < p <∞ such that 1q + 2 p < 1 α + 1 λ . then the operator [b,m] is bounded on (lq, lp)α,λ(rd) if and only if b ∈ bmo(rd) such that b− ∈ l∞(rd). the last two theorems concern the sublinear operators t satisfying the condition |t (x)| ≤ c ∫ rd |f (y)| |x − y |d dy x /∈ supp f , (1.3) for any f ∈ l1(rd) with compact support. we point out that the condition (1.3) was first introducedby soria and weiss [19]. this condition is satisfied by many operators such as the hardy-littewoodmaximal operator, calderón-zygmund singular integral operators, bochner-riesz operators at thecritical index, c. fefferman’s singular multiplier. it is proved in [5] that t is bounded on morreyspaces. it is also bounded on classical fofana spaces (see [6]). in the setting of total fofana spaces,we have the following result holds true. theorem 1.2. let 1 < q ≤ λ,α < p < ∞ such that 1q + 2 p < 1 α + 1 λ . if t is sublinear operator with is bounded on lq and satisfies the condition (1.3) then t is also bounded on (lq, lp)α,λ(rd). if t is a linear operator and b ∈ bmo(rd), we define the linear commutator [b, t ] by [b, t ]f (x) = t (bf )(x)− b(x)t (f )(x) x ∈ rd , with locally integrable functions f on rd . it is also proved in [6] that [b, t ] is bounded on classicalfofana spaces and bounded on morrey spaces in [5]. the next result shows the boundedness ontotal fofana spaces of [b, t ]. https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 4 theorem 1.3. let 1 < q ≤ λ,α < p <∞ such that 1q + 2 p < 1 α + 1 λ and b ∈ bmo(rd). if a linear operator t satisfies (1.3) and [b, t ] is bounded on lq , then t is also bounded on (lq, lp)α,λ(rd). the remainder of this note is organized as follows: in section 2 we recall some properties oftotal fofana spaces. section 3 is devoted to the proofs of theorem 1.1 and section 4 deals withthe proofs of theorem 1.2 and theorem 1.3. the letter c will be used for positive constants not depending on the relevant variables, andtheses constants may change from one occurrence to another. we propose the following abbreviation a <∼ b for the inequalities a ≤ cb. if a <∼ b and b <∼ a, then we write a ≈ b. 2. some properties of total fofana spaces the results of this section are proved in [17].the following result examines the relationship between total fofana spaces and fofana spaces. proposition 2.1. let 1 ≤ q ≤ α, λ ≤ p ≤ ∞. then (lq, lp)α(rd) ∩ (lq, lp)λ(rd) ↪→ (lq, lp)α,λ(rd) and for f ∈ (lq, lp)α(rd) ∩ (lq, lp)λ(rd) ‖f ‖(lq ,lp)α,λ(rd ) ≤ max{‖f ‖q,p,α , ‖f ‖q,p,λ}. proposition 2.2. let 1 ≤ q ≤ λ ≤ α ≤ p ≤ ∞. then (lq, lp)α,λ(rd) = (lq, lp)α(rd) ∩ (lq, lp)λ(rd) and for f ∈ (lq, lp)α,λ(rd) ‖f ‖(lq ,lp)α,λ(rd ) = max{‖f ‖q,p,α , ‖f ‖q,p,λ}. total fofana spaces are generalizations of classical fofana spaces since proposition 2.2 assertsthat (lq, lp)α,α(rd) = (lq, lp)α(rd).the family of spaces (lq, lp)α,λ(rd) is increasing with respect to the p power. more precisely,we have the following. proposition 2.3. let 1 ≤ q ≤ α, λ ≤ p1 ≤ p2 ≤ ∞. then: ‖f ‖(lq ,lp2)α,λ(rd ) <∼ ‖f ‖(lq ,lp1)α,λ(rd ) , f ∈ l0(rd) and consequently, (lq, lp1)α,λ(rd) ⊂ (lq, lp2)α,λ(rd). the following result states the boundedness property of m (the hardy-littlewood maximal op-erator) on total fofana spaces. https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 5 theorem 2.4. (1) let 1 < q ≤ α, λ < p <∞ such that 1q + 2 p < 1 α + 1 λ . then ‖mf ‖(lq ,lp)α,λ(rd ) <∼ ‖f ‖(lq ,lp)α,λ(rd ) , f ∈ (lq, lp)α,λ(rd). (2) let q = 1 < α,λ < p <∞. then ‖mf ‖(l1,∞,lp)α,λ(rd ) <∼ ‖f ‖(l1,lp)α,λ(rd ) , f ∈ (l1, lp)α,λ(rd), where ‖f ‖(l1,∞,lp)α,λ(rd ) := sup r>0 [r ] d( 1 α −1− 1 p ) 1 [1/r ] d(− 1 λ +1+ 1 p ) 1 [∫ rd (∥∥f χb(y,r)∥∥∗1,∞)p dy] 1p with ∥∥f χb(y,r)∥∥∗1,∞ = sup r>0 r |{x ∈ b(y , r) : |f (x)| > r}| . 3. proof of theorem 1.1 for the proof of this theorem, we need some results.the following result (see [1, corollary 1.11]) will be useful in the proof of theorem 1.1. lemma 3.1. if b ∈ bmo(rd), then there exists a positive constant c such that mbf (x) ≤ c ‖b‖bmo(rd )m(mf )(x) for almost every x ∈ rd and any locally integrable functions f on rd . proposition 3.2. let 1 < q ≤ α, λ < p <∞ such that 1q + 2 p < 1 α + 1 λ and b ∈ bmo(rd). then mb is bounded on (lq, lp)α,λ(rd). proof. let 1 < q ≤ α, λ < p < ∞ such that 1q + 2 p < 1 α + 1 λ , b ∈ bmo(rd) and f ∈ (lq, lp)α,λ(rd).by taking the (lq, lp)α,λ(rd)-norm of both sides of the estimate in lemma 3.1, we obtain ‖mbf ‖(lq ,lp)α,λ(rd ) <∼ ‖b‖bmo(rd ) ‖m(mf )‖(lq ,lp)α,λ(rd ) .according to the first point of theorem 2.4, we have ‖m(mf )‖(lq ,lp)α,λ(rd ) <∼ ‖mf ‖(lq ,lp)α,λ(rd ) <∼ ‖f ‖(lq ,lp)α,λ(rd ) .we deduce that ‖mbf ‖(lq ,lp)α,λ(rd ) <∼ ‖b‖bmo(rd ) ‖f ‖(lq ,lp)α,λ(rd ) . (3.1) � lemma 3.3. let 1 ≤ q ≤ λ ≤ α <∞ and r > 0. then r− d q [r ] d(− 1 α + 1 q ) 1 [1/r ] d( 1 λ − 1 q ) 1 max{r d α , r d λ } ≤ 2. https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 6 proof. let 1 ≤ q ≤ λ ≤ α <∞ and r > 0.put c(r) = r −dq [r ]d(− 1α+ 1q )1 [1/r ] d( 1 λ − 1 q ) 1 max{r d α , r d λ }. c(r) ≤ { r− d α (r d α + r d λ ), 0 < r ≤ 1 r− d λ (r d α + r d λ ), r > 1 ≤ { 1 + rd(− 1 α + 1 λ ), 0 < r ≤ 1 1 + rd( 1 α − 1 λ ), r > 1. thus, c(r) ≤ 2 for all r > 0. � proof of theorem 1.1. let 1 < q ≤ λ ≤ α < p <∞ such that 1 q + 2 p < 1 α + 1 λ . (1) assume that b ∈ bmo(rd) such that b− ∈ l∞(rd) and f ∈ (lq, lp)α,λ(µ).proceeding as in the proof of theorem 4 in [13], we have ‖[b,m]f ‖(lq ,lp)α,λ(rd ) ≤ ∥∥mbf + 2b −mf ∥∥ (lq ,lp)α,λ(rd ) ≤ ‖mbf ‖(lq ,lp)α,λ(rd ) + 2 ∥∥b−∥∥∞ ‖mf ‖(lq ,lp)α,λ(rd ) . from (3.1) and the first point of theorem 2.4, we deduce that ‖[b,m]f ‖(lq ,lp)α,λ(rd ) <∼ ( ‖b‖bmo(rd ) + ∥∥b−∥∥∞) ‖f ‖(lq ,lp)α,λ(rd ) . (2) conversely, assume that [b,m] is bounded on (lq, lp)α,λ(rd).let t > 0 and x ∈ rd . put b = b(x, t). denote by mbf the local maximal function of f definedby: mbf (x) = sup b ′3x :b′⊂b |b′ |−1 ∫ b ′ |f (y)| dy. since χb ∈ lα(rd) ∩ lλ(rd), it follows from (1.2) that χb ∈ (lq, lp)α (rd)∩(lq, lp)λ (rd). from proposition 2.2, we deduce that χb ∈ (lq, lp)α,λ(rd).therefore, there exists a constant c > 0 such that ‖[b,m]χb‖(lq ,lp)α,λ(rd ) ≤ c ‖χb‖(lq ,lp)α,λ(rd ) . we also have |mb(b)− bχb| = |m(bχb)χb − bm(χb)χb| ≤ |m(bχb)− bm(χb)| = |[b,m]χb|. https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 7applying hölder’s inequality, proposition 2.3 and proposition 2.2, we get |b|−1 ∫ b |b(z)−mb(b)(z)|dz ≤ ( |b|−1 ∫ b |b(z)−mb(b)(z)|qdz ) 1 q ≤ |b|− 1 q (∫ b |[b,m]χb(z)|qdz ) 1 q <∼ t− d q [t] d(− 1 α + 1 q ) 1 [1/t] d( 1 λ − 1 q ) 1 ‖[b,m]χb‖(lq ,l∞)α,λ(rd ) <∼ t− d q [t] d(− 1 α + 1 q ) 1 [1/t] d( 1 λ − 1 q ) 1 ‖[b,m]χb‖(lq ,lp)α,λ(rd ) <∼ t− d q [t] d(− 1 α + 1 q ) 1 [1/t] d( 1 λ − 1 q ) 1 ‖χb‖(lq ,lp)α,λ(rd ) <∼ t− d q [t] d(− 1 α + 1 q ) 1 [1/t] d( 1 λ − 1 q ) 1 max{‖χb‖q,p,α , ‖χb‖q,p,λ}. it follows from (1.2) that |b|−1 ∫ b |b(z)−mb(b)(z)|dz <∼ t − d q [t] d(− 1 α + 1 q ) 1 [1/t] d( 1 λ − 1 q ) 1 max{t d α , t d λ }. so, by lemma 3.3, we obtain |b|−1 ∫ b |b(z)−mb(b)(z)|dz <∼ 2.denote by e := {y ∈ b : b(y) ≤ bb}, f := {y ∈ b : b(y) > bb}.since ∫ e |b(z)− bb|dz = ∫ f |b(z)− bb|dz,in view of the inequality b(x) ≤ bb ≤ mb(b), for x ∈ e, we get |b|−1 ∫ b |b(z)− bb|dz = 2|b|−1 ∫ e |b(z)− bb|dz ≤ 2|b|−1 ∫ e |b(z)−mb(b)(z)|dz ≤ 2|b|−1 ∫ b |b(z)−mb(b)(z)|dz <∼ 4. by taking in the left hand side the supremum over all t > 0 and x ∈ rd , we obtain ‖b‖bmo(rd ) <∞.in order to show that b− ∈ l∞(rd), note that mb(b) ≥ |b|. hence 0 ≤ b− = |b| − b+ ≤ mb(b)− b+ ≤ mb(b)− b+ + b− = mb(b)− b. thus (b−)b <∼ 2, https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 8and by the lebesgue differentiation theorem we get b−(x) <∼ 2for almost every x ∈ rd . � 4. proof of theorem 1.2 and theorem 1.3 we recall that the proofs of theorem 1.2 and theorem 1.3 are simply an adaptation of thosegiven in [5] (see also [6]). proof of theorem 1.2. let 1 < q ≤ λ,α < p <∞ such that 1q+ 2p < 1 α+ 1 λ and f ∈ (lq, lp)α,λ(rd).fix y ∈ rd and r > 0 we have f = f χb(y,2r) + ∞∑ i=1 f χb(y,2i+1r)\b(y,2i r). by the sublinearity of t and the condition (1.3) we obtain |t f | <∼ |t (f χb(y,2r))|+ ∞∑ i=1 |b(y , 2i+1r)|−1 ∫ b(y,2i+1r) |f (x)|dx and therefore, an application of hölder inequality leads to |t f | <∼ |t (f χb(y,2r))|+ ∞∑ i=1 |b(y , 2i+1r)|− 1 q ‖f χb(y,2i+1r)‖q. taking the lq-norm of both sides on the ball b(y , r) and using the boundedness of t on lq , weget ‖(t f )χb(y,r)‖q <∼ ‖f χb(y,2r)‖q + ∞∑ i=1 (2i)− d q ‖f χb(y,2i+1r)‖q. taking the lp-norm of both sides with respect to y , it comes that r ‖t f ‖q,p <∼ 2r ‖f ‖q,p + ∞∑ i=1 (2i)− d q 2i+1r ‖f ‖q,p . on the one hand, we have, 2r ‖f ‖q,p = [2r ] d( 1 α − 1 q − 1 p ) 1 [1/2r ] d(− 1 λ + 1 q + 1 p ) 1 [2r ] d( 1 α − 1 q − 1 p ) 1 [1/2r ] d(− 1 λ + 1 q + 1 p ) 1 2r ‖f ‖q,p ≤ [2r ] d(− 1 α + 1 q + 1 p ) 1 [1/2r ] d( 1 λ − 1 q − 1 p ) 1 ‖f ‖(lq ,lp)α,λ(rd ) ≤ (2[r ]1) d(− 1 α + 1 q + 1 p ) ( 1 2 [1/r ]1 )d( 1 λ − 1 q − 1 p ) ‖f ‖(lq ,lp)α,λ(rd ) . hence, 2r ‖f ‖q,p <∼ [r ] d(− 1 α + 1 q + 1 p ) 1 [1/r ] d( 1 λ − 1 q − 1 p ) 1 ‖f ‖(lq ,lp)α,λ(rd ) . (4.1) https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 9on the other hand, ∞∑ i=1 (2i)− d q 2i+1r ‖f ‖q,p = ∞∑ i=1 (2i)− d q [2i+1r ] d( 1 α − 1 q − 1 p ) 1 [1/2i+1r ] d(− 1 λ + 1 q + 1 p ) 1 [2i+1r ] d( 1 α − 1 q − 1 p ) 1 [1/2i+1r ] d(− 1 λ + 1 q + 1 p ) 1 2i+1r ‖f ‖q,p ≤ ‖f ‖(lq ,lp)α,λ(rd ) ∞∑ i=1 (2i)− d q [2i+1r ] d(− 1 α + 1 q + 1 p ) 1 [1/2i+1r ] d( 1 λ − 1 q − 1 p ) 1 <∼ [r ] d(− 1 α + 1 q + 1 p ) 1 [1/r ] d( 1 λ − 1 q − 1 p ) 1 ‖f ‖(lq ,lp)α,λ(rd ) ∞∑ i=1 (2i)d(− 1 α − 1 λ + 1 q + 2 p ). since − 1α − 1λ + 1q + 2p < 0, ∑∞i=1(2i)d(− 1α− 1λ+ 1q+ 2p ) <∞. therefore, ∞∑ i=1 (2i)− d q 2i+1r ‖f ‖q,p (4.2) <∼ [r ] d(− 1 α + 1 q + 1 p ) 1 [1/r ] d( 1 λ − 1 q − 1 p ) 1 ‖f ‖(lq ,lp)α,λ(rd ) . from (4.1) and (4.2), we deduce that r ‖t f ‖q,p <∼ [r ] d(− 1 α + 1 q + 1 p ) 1 [1/r ] d( 1 λ − 1 q − 1 p ) 1 ‖f ‖(lq ,lp)α,λ(rd ) . it follows that [r ] d( 1 α − 1 q − 1 p ) 1 [1/r ] d(− 1 λ + 1 q + 1 p ) 1 r ‖t f ‖q,p <∼ ‖f ‖(lq ,lp)α,λ(rd ) . (4.3) we obtain the desired result by taking the supremum over all r > 0 in the left hand side of (4.3). � proof of theorem 1.3. let 1 < q ≤ λ,α < p <∞ such that 1q + 2p < 1 α + 1 λ and b ∈ bmo(rd).let f be any element of f ∈ (lq, lp)α,λ(rd). we recall that (lq, lp)α,λ(rd) is a subspace of themorrey space lq,d(1− q α )(rd). proceeding as in the proof of [5, theorem 2.2 ], we have that for all y ∈ rd and r > 0, ‖[b, t ]f χb(y,r)‖q <∼ ‖f χb(y,2r)‖q + ∞∑ i=1 (2i r)−d [∫ b(y,r) (∫ b(y,2i+1r) |b(x)− b(z)||f (x)|dx )q dz ] 1 p . therefore, using the john-nirenberg theorem on bmo-functions (see [12, corollary 7.1.8]), weobtain ‖[b, t ]f χb(y,r)‖q <∼ ‖f χb(y,2r)‖q + ‖b‖bmo(rd ) ∞∑ i=1 (2i)− d q ‖f χb(y,2i+1r)‖q. using the same argument as in the proof of theorem 1.2, we end the proof. � https://doi.org/10.28924/ada/ma.4.22 eur. j. math. anal. 10.28924/ada/ma.4.22 10references [1] m. agcayazi, a. gogatishvili, k. koca, r. ch. mustafayev, a note on maximal commutators and commutators ofmaximal functions, j. math. soc. japan, 67 (2015), 581-593.[2] j. bastero, m. milman, f. j. ruiz, commutators for the maximal and sharp functions, proc. amer. math. soc., 128(2000), 3329-3334.[3] s. chanillo, a note on commutators, indiana univ. math. j., 31 (1982), 7-16.[4] f. chiarenza, m. frasca, morrey spaces and hardy-littlewood maximal function, rend. math. 7 (1987), 273-279.[5] d. fan, s. lu and d. yang, regularity in morrey spaces of strong solutions to nondivergence elliptic equations withvmo coefficients, georgian math. j. 5 (1998), 425-440.[6] j. feuto, norm inequalities in some subspaces of morrey space, ann. math. blaise pascal 21 (2014), 21-37.[7] i. fofana, etude d’une classe d’espaces de fonctions contenant les espaces de lorentz, afr. mat. 2 (1988), 29-50.[8] i. fofana, continuité de l’intégrale fractionnaire et espace (lq, lp)α, c. r. acad. sci. paris 308 (1989), 525-527.[9] i. fofana, espace (lq, lp)α et continuité de l’opérateur maximal fractionnaire de hardy-littlewood, afr. mat. 3(2001), 23-37.[10] j. garciá-cuerva, e. harboure, c. segovia, j. l. torrea, weighted norm inequalities for commutators of stronglysingular integrals, indiana univ. math. j., 40 (1991), 1397-1420.[11] a. gogatishvili, r.ch. mustafayev, m. agcayazi, weak-type estimates in morrey spaces for maximal commuatatorand commutator of maximal function, tokyo j. math. 41 (2018), 193-218.[12] l. grafakos, modern fourier analysis, 2nd ed., graduate texts in mathematics vol. 250, springer, new york, 2009.[13] v.s. guliyev, maximal commutator and commutator of maximal function on total morrey spaces, j. math. inequal.16 (2022), 1509-1524[14] v. s. guliyev, j. j. hasanov , y. zeren, necessary and sufficient conditions for the boundedness of the riesz potentialin modified morrey spaces, j. math. inequal. 5 (2011), 491-506.[15] g. hu, d. yang, maximal commutators of bmo functions and singular integral operators with non-smooth kernelson spaces of homogeneous type, j. math. anal. appl. 354 (2009), 249-262.[16] s. janson, mean oscillation and commutators of singular integral operators, ark. mat., 16 (1978), 263-270.[17] p. nagacy, b. a. kpata, norm inequalities for fractional integral and fractional maximal operators in the total fofanaspaces, preprint.[18] y. sawano, g. di fazio, d. i. hakim, morrey spaces-introduction and applications to integral operators and pde’s,vol. i, monographs and research notes in mathematics, crc press, boca raton, fl, 409 pp. (2020).[19] f. soria, g. weiss and d. i. hakim, a remark on singular integrals and power weights, indiana univ. math. j. 43(1994), 187-204.[20] e.m. stein, harmonic analysis: real variable methods, orthogonality, and oscillatory integrals princeton mathe-matical series, vol. 43, princeton university press, princeton, new jersey (1993). https://doi.org/10.28924/ada/ma.4.22 1. introduction and main results 2. some properties of total fofana spaces 3. proof of theorem 1.1 4. proof of theorem 1.2 and theorem 1.3 references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 22doi: 10.28924/ada/ma.5.22 estimates of variable kernel parameterized littlewood–paley operators on variable herz spaces afif abdalmonem1,∗ , omer khalil2, omer abdalrhman3 1faculty of science, department of mathematics, university of dalanj, dalanj, sudan afeefy86@gmail.com 2 college of mathematics and statistics, northwest normal university, china us.omer2008@yahoo.com 3college of education, shendi university, sudan humoora@gmail.com ∗correspondence: afeefy86@gmail.com abstract. in this article, we prove some boundedness results for variable kernel parameterizedlittlewood−paley operators on the homogeneous herz spaces k̇α(·),q(·) p(·) (rn). several known resultsare extended. 1. introduction suppose that ψ(x1, z1) ∈ l∞(rn)× lb(sn−1) (where b ≥ 1) satisfies:(1) ψ(x1, αz1) = ψ(x1, z1) and ∫ sn−1 ψ(x1, z ′ 1)dσ(z ′1) = 0, for all z1, x1 ∈ rn, α > 0; (2) ‖ψ‖l∞(rn)×lb(sn−1) := sup x1∈rn (∫ sn−1 |ψ(x1, z ′ 1)|bdz ′1 ) 1 b <∞, where sn−1 (for n ≥ 2) is the unit sphere in rn equipped with lebesgue measure dz ′1. theparameterized littlewood−paley operators, denoted by µσψ,s and µ∗,σψ,λ, are closely associated withthe lusin area integral and littlewood−paley g∗λ function. these operators are defined as follows µσψ,s(f )(x) = (∫ ∫ σ(x) ∣∣∣∣ 1 tσ ∫ |y1−z1|≤t ψ(y1, y1 − z1) |y1 − z1|n−σ f (z1)dz1 ∣∣∣∣2 dy1dt tn+1 ) 1 2 and µ∗,σψ,λ(f )(x) = (∫ ∫ rn+1 + ( t t + |x1 − y1| )λn ∣∣∣∣ 1 tσ ∫ |y1−z1|≤t ψ(y1, y1 − z1) |y1 − z1|n−σ f (z1)dz1 ∣∣∣∣2 dy1dt tn+1 ) 1 2 , where σ(x) = {(y1, t) ∈ rn+1 + : |x1 − y1| < t and λ > 1}. received: 17 jul 2025. key words and phrases. littlewood−paley operator; variable kernel; herz space; variable exponent..1 https://adac.ee https://doi.org/10.28924/ada/ma.5.22 https://orcid.org/0000-0002-6391-4243 https://orcid.org/0000-0003-0663-068x eur. j. math. anal. 10.28924/ada/ma.5.22 2the parameterized littlewood−paley operators µσψ,s and µ∗,σψ,λ were initially investigated bysakamoto and yabuta in [1]. they proved that if ψ ∈ libβ(sn−1) and 1 < p <∞, then µ∗,σψ,λ and µσψ,s operators are bounded on lp(rn) space. xue and ding [2] established sharp lp(w) boundedfor these operators (µ∗,σψ,λ , µσψ,s) in terms of the aq characteristic of w , under the condition ψ ∈ lb(sn−1). deringoz, guliyev and ragusa [3] obtained the boundedness of intrinsic squarefunctions and their commutators in the framework of morrey-orlicz spaces. the boundedness ofparametric littlewood−paley operators on musielak−orlicz hardy spaces was further studiedin [4].as is well known, over the past thirty years, variable kernel integral operators have become anincreasingly active area of research. for example, tao et al. [5] obtained the lp(rn) boundednessof variable kernel fractional integral operators tψ,α, chen and ding [6] proved the lp(rn) bound-edness of variable kernel littlewood−paley operators, shao [7] investigated the weighted estimatesfor variable kernel fractional integrals and their commutators on generalized morrey spaces, in [8]the author proved the boundedness properties of marcinkiewicz integral operator µψ with variablekernel on the hardy space hp(rn). recently, abdalmonem et. al. [9] obtained the boundedness oflittlewood−paley operators with variable kernel on the weighted variable herz-morrey spaces.moreover, variable exponents herz spaces have been extensively studied by many authors us-ing different methods( [14–20, 24, 25]). izuki [23] defined the variable exponent homogeneous herzspace k̇α,q p(·)(rn) and investigated the boundedness of some integral operators on these spaces.wang [20] considered the boundedness results for certain rough kernel littlewood−paley opera-tors in homogeneous and homogeneous herz spaces k̇α,q(·) p(·) (rn). in [21] the authors studied theboundedness of the vector-valued inequality for the intrinsic square function in variable exponentshomogeneous herz spaces k̇α(·),q p(·) (rn). izuki and noi [12] considered the generalized herz spaces k̇ α(·) q(·),p(·)(rn) and obtained some boundedness results for integral operators and their commutatorson those spaces. in [13] the author established the boundedness properties of the rough kernelfractional integral operators in k̇α(·) q(·),p(·)(rn) spaces.motivated by the work of [9, 13, 19],this paper discusses the boundedness of variable kernelparameterized littlewood-paley operators on homogeneous herz spaces k̇α(·) q(·),p(·)(rn) with threevariable exponents. the results are also new for the case when α(·) is constant. 2. mathematical background consider a lebesgue measurable set e ⊂ rn with positive measure |e| > 0. denote by χe thecharacteristic function of e. in this paper, c represents a positive constant that may vary betweenoccurrences. we write g . f means g ≤ cf , for some constant c > 0. definition 2.1 ( [22]). (variable lebesgue space ) suppose that p(·) : γ→ [1,∞) is a measurablefunction. the lp(·)(γ) space is defined by https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 3 lp(·)(γ) = { g is measurable : ∫ γ ( |g(x)| β )p(x) dx <∞ for some constant β > 0 } . the local lp(·) loc (γ) space is defined as l p(·) loc (γ) = {g is measurable : g ∈ lp(·)(k) for any compact set k ⊂ γ}. with the given norm, the lebesgue space lp(·) loc (γ) is a banach space ‖f ‖lp(·)(γ) = inf { η > 0 : ∫ e ( |g(x)| β )p(x) dx ≤ 1 } . let p− = ess inf{p(x) : x ∈ γ}, p+ = ess sup{p(x) : x ∈ γ} denote the essential infimum andsupremum of p(γ), respectively. p(γ) represents the collection of all measurable functions p(·)with p− > 1. p+ < +∞. p0(γ) consists of all measurable functions p(·) such that p− > 0and p+ < +∞. furthermore, b(rn) is defined as the subset of p(·) ∈ p(rn) for which thehardy−littlewood maximal operator m∗ is bounded in variable lp(·) space.we know that, if p(·) ∈ p(rn), then the operator m∗, m∗g(x) = sup b⊆rn,b3x 1 |b| ∫ b |g(y)|dy, is bounded in variable lp(·) space [24], where m∗ denotes the hardy−littlewood maximal operator.let us now recall the definition of herz space k̇α(·),q(·) p(·) (rn). let bk = {y ∈ rn : |y | ≤ 2k}, k ∈ z, ck = bk\bk−1, χck = χk . definition 2.2 ( [12]). let α(·) : rn −→ r, −∞ < α− ≤ α+ < ∞ and q(·), p(·) ∈ p(rn). thehomogeneous variable exponents herz k̇α(·),q(·) p(·) (rn) space is defined by k̇ α(·),q(·) p(·) (rn) = {f ∈ lp(·) loc (rn\{0}) : ‖f ‖ k̇ α(·),q(·) p(·) (rn) <∞}, where ‖f ‖ k̇ α(·),q(·) p(·) (rn) := ∥∥∥{2kα(·)|f χk |}∞k=−∞ ∥∥∥ lq(·)(lp(·)) = inf { β > 0 : ∞∑ k=−∞ ∥∥∥∥∥ ( 2kα(·)|f χk | β )q(·)∥∥∥∥∥ l p(·) q(·) ≤ 1 } . the nonhomogeneous variable exponents herz k̇α(·),q(·) p(·) (rn) space is defined by k α(·),q(·) p(·) (rn) = {f ∈ lp(·) loc (rn\{0}) : ‖f ‖ k α(·),q(·) p(·) (rn) <∞}, where ‖f ‖ k α(·),q(·) p(·) (rn) := ∥∥∥{2kα(·)|f χk |}∞k=0 ∥∥∥ lq(·)(lp(·)) = inf β > 0 : ∞∑ k=0 ∥∥∥∥∥∥ ( 2kα(·)|f χk | β )q(·) ∥∥∥∥∥∥ l p(·) q(·) ≤ 1  . https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 4 remark. (1) if k̇ α(·),q(·) p(·) (rn) = k̇ α(·),q p(·) (rn), then, q(·) is a constant.(2) if k̇ α(·),q(·) p(·) (rn) = k̇α,q p(·)(rn), then, both α(·), q(·) are constants.(3) if k̇ α(·),q(·) p(·) (rn) = k̇α,qp (rn), then, α(·), p(·), q(·) are all constants.(4) moreover, if p(·) = q(·) and α(·) = 0 , then k̇ α(·),q(·) p(·) (rn) = lp(·)(rn). next, we present some key lemmas needed to prove our main theorems. lemma 2.3 ( [22]). (generalized hölder’s inequality) let f ∈ lp1(·)(rn), g ∈ lp ′ 1(·)(rn), and p(·) ∈ p(rn). then, the following inequality is satisfied:∫ rn |g(x)f (x)|dx ≤ c‖g‖ lp ′ 1(·)(rn) ‖f ‖lp1(·)(rn), here c = 1− 1 p+ + 1 p− . lemma 2.4 ( [23]). suppose p(·) ∈ b(rn). for a given c > 0, the following inequality is satisfied: c ≥ 1 |b|‖χb‖lp1(·)(rn)‖χb‖lp′1(·)(rn) , here b ⊂ rn. lemma 2.5 ( [23]). suppose p(·) ∈ b(rn). for n = 1, 2, there are constants δn1, δn2 > 0 for which the following inequalities hold: ‖χb‖lp(·)(rn) ‖χs‖lp(·)(rn) . |b| |s| , ‖χs‖lp′1(·)(rn) ‖χb‖lp′1(·)(rn) . ( |s| |b| )δn1 , ‖χs‖lp1(·)(rn) ‖χb‖lp1(·)(rn) . ( |s| |b| )δn2 , here b ⊂ rn, s ⊂ b. lemma 2.6 ( [20]). let p1(·), q1(·) ∈ p0(rn), g ∈ lp1(·)q1(·)(rn), and 0 < q− ≤ p1(·) ≤ q+. then, we have min(‖g‖q+ lp1(·)q1(·) , ‖g‖ q− lp1(·)q1(·) ) ≤ ‖|g|q1(·)‖lp1(·) ≤ max(‖g‖q+ lp1(·)q1(·) , ‖g‖ q− lp1(·)q1(·) ). https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 5 lemma 2.7 ( [10]). suppose that α(·) ∈ l∞(rn) and r0 > 0. if α(·) be a function that is loghölder continuous both both at the origin and at infinity, then for any x ∈ b(0, r0) \ b(0, r0/2), x ′ ∈ b(0, r1) \ b(0, r1/2), we have r α(x) 0 . rα(x ′) 1 ×  [ r0r1 ]α+ , 0 < r1 ≤ r0/2, 1, r0/2 < r1 ≤ 2r0, [ r0r1 ]α− , r1 > 2r0. 3. boundedness of the parameterized littlewood-paley operators in this section, we discuss the boundedness of variable kernel parameterized littlewood-paleyoperators on homogeneous herz spaces k̇α(·) q(·),p(·)(rn). the results are also new for the case when α(·) is constant.let 1 < q <∞, q′ = q q−1 and w be a weight. for every cube q ⊆ rn, we say w ∈ aq if thereexists c > 0, the following inequality is satisfied:( 1 |q| ∫ q w(x)dx )( 1 |q| ∫ q w(x)1−q′dx )q−1 ≤ c <∞. xue et al. [2] proved the following lp−boundedness of µσψ,s and µ∗,σψ,λ. lemma 3.1 ( [2]). let 1 < p < ∞ and ψ ∈ l∞(rn) × l2(sn−1) satisfies (1) and (2). then, we have ‖µσψ,s f ‖lp(w) . ‖f ‖lp(w) and ‖µ∗,σψ,λf ‖lp(w) . ‖f ‖lp(w). lemma 3.2 ( [21]). given a family of functions f , if for some p1, 1 < p1 < ∞, p1 ≤ p− and (p(·) p1 )′ ∈ b(e) and every w1 ∈ ap1 ,∫ rn f1(x)p1w1(x)dx . ∫ rn g1(x)p1w1(x)dx, (f , g) ∈ f . if p(·) ∈ p(e) and f1 ∈ lp(·)(e), then for all (f1, g1) ∈ f , ‖f1‖lp(·)(e) . ‖g1‖lp(·)(e). since aq/s ′ ⊂ a∞, using lemma 3.1 and lemma 3.2, its simple to obtain the lp(·)-boundednessof µσψ,s and µ∗,σψ,λ. theorem 3.3. assume that p1(·) ∈ b(rn), q1(·), q2(·) ∈ p(rn), λ > 2, 2σ − n > 0, and ψ ∈ l∞(rn) × l2(sn−1) satisfies (1) and (2). let α(·) ∈ l∞(rn) be a function that is log-hölder continuous both at the origin and at infinity, such that −nδ11 < α− ≤ α+ < nδ12, https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 6 where δn1, δn2 (n = 1, 2) are the same as in lemma 2.4. then, µσψ,s operator is bounded from k̇ α(·) p1(·),q1(·)(rn) to k̇α(·) p1(·),q2(·)(rn) for all f ∈ k̇α(·) p1(·),q1(·)(rn). theorem 3.4. assume that p1(·) ∈ b(rn), q1(·), q2(·) ∈ p(rn), λ > 2, 2σ − n > 0, and ψ ∈ l∞(rn) × l2(sn−1) satisfies (1) and (2). let α(·) ∈ l∞(rn) be a function that is log-hölder continuous both at the origin and at infinity, such that −nδ11 < α− ≤ α+ < nδ12, where δn1, δn2 (n = 1, 2) are the same as in lemma 2.4. then, µ∗,σψ,λ operator is bounded from k̇ α(·) p1(·),q1(·)(rn) to k̇α(·) p1(·),q2(·)(rn) for all f ∈ k̇α(·) p1(·),q1(·)(rn). before proving theorems, we first establish a necessary inequality. remark. let 1 ≤ pm <∞, am ≥ 0, m ∈ n. we have ∞∑ m=0 apmm ≤ ( ∞∑ m=0 am )p• , here p• =  min m∈n pm if ∞∑ m=0 am ≤ 1, max m∈n pm if ∞∑ m=0 am > 1. remark. from ( [11], p.89]), we recall the estimate µσψ,s f (x) ≤ 2nλµ∗,σσ,λf (x). therefore, we presentonly the proof of theorem 3.4. proof. we present the proof of k̇α(·),q(·) p(·) (rn) (homogeneous case). the same argument holds truefor kα(·),q(·) p(·) (rn) (nonhomogeneous case).let f ∈ k̇α(·),q1(·) p1(·) (rn). decomposet f as: f (x) = ∞∑ j=−∞ f (x)χj(x) = ∞∑ j=−∞ fj(x). from homogeneous k̇α(·),q(·) p(·) (rn) (definition 2.2), we have ‖µ∗,σψ,λ(f )‖ k̇ α(·),q2(·) p1(·) (rn) = inf η > 0 : ∞∑ k=−∞ ∥∥∥∥∥∥ ( 2kα(·)|µ∗,σψ,λ(f )χk | β )q2(·) ∥∥∥∥∥∥ l p1(·) q2(·) ≤ 1  . we have ∥∥∥∥∥ ( 2kα(·)|µ∗,σψ,λ(f )χk | β )q2(·) ∥∥∥∥∥ l p1(·) q2(·) ≤ ∥∥∥∥∥∥∥ 2kα(·)| ∞∑ j=−∞ µ∗,σψ,λ(fj )χk | β01+β02+β03 q2(·)∥∥∥∥∥∥∥ l p1(·) q2(·) https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 7 ≤ c ∥∥∥∥∥∥∥∥ 2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj )χk | β01  q2(·) ∥∥∥∥∥∥∥∥ l p1(·) q2(·) + c ∥∥∥∥∥∥∥∥ 2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj )χk | β02  q2(·) ∥∥∥∥∥∥∥∥ l p1(·) q2(·) +c ∥∥∥∥∥∥∥ 2kα(·)| ∞∑ j=k+2 µ∗,σψ,λ(fj )χk | β03 q2(·)∥∥∥∥∥∥∥ l p1(·) q2(·) , where β01 = ∥∥∥∥∥ { 2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj)χk | }∞ k=−∞ ∥∥∥∥∥ lq2(·)(lp1(·)) , β02 = ∥∥∥∥∥ { 2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj)χk | }∞ k=−∞ ∥∥∥∥∥ lq2(·)(lp1(·)) , β03 = ∥∥∥∥∥ { 2kα(·)| ∞∑ j=k+2 µ∗,σψ,λ(fj)χk | }∞ k=−∞ ∥∥∥∥∥ lq2(·)(lp1(·)) . if β0 = β01 + β02 + β03 thus ∞∑ k=−∞ ∥∥∥∥∥∥ ( 2kα(·)|µ∗,σψ,λ(fj)χk | β0 )q2(·) ∥∥∥∥∥∥ l p1(·) q2(·) . 1. then ‖µ∗,σψ,λ(f )χk‖k̇α(·),q1(·) p1(·) (rn) . β0 . [β01 + β02 + β03].therefore, if we can conclude that β01 ≤ c‖f ‖k̇α(·),q1(·) p1(·) (rn) , β02 ≤ c‖f ‖k̇α(·),q1(·) p1(·) (rn) , β03 ≤ c‖f ‖k̇α(·),q1(·) p1(·) (rn) , we are finished. let us set β0 = ‖f ‖ k̇ α(·),q1(·) p1(·) (rn) .first, we estimate β02. from lemma 2.6 and lemma 2.7, we have ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥∥  2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥ 2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj)χk | β0 ∥∥∥∥∥∥∥∥∥ (q0 2 )k lp1(·) . ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥ | k+1∑ j=k−1 µ∗,σψ,λ(2αj fj)χk | β0 ∥∥∥∥∥∥∥∥∥ (q0 2 )k lp1(·) https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 8 . ∞∑ k=−∞  k+1∑ j=k−1 ∥∥∥∥∥ |µ∗,σψ,λ(2αj fj)χk | β0 ∥∥∥∥∥ lp1(·) (q0 2 )k , where (q0 2)k =  (q2)+ ∥∥∥∥∥∥∥∥ 2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj )χk | β0  q2(·) ∥∥∥∥∥∥∥∥ l p1(·) q2(·) ≥ 1, (q2)− otherwise. by the boundedness of µ∗,σψ,λ on lp(·), we have ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥∥  2kα(·)| k+1∑ j=k−1 µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞  k+1∑ j=k−1 ∥∥∥∥∥ |(2jα(·)fj)| β0 ∥∥∥∥∥ lp1(·) (q0 2 )k , . ∞∑ k=−∞ ∥∥∥∥∥∥ ( 2kα(·)|fk | β0 )q1(·) ∥∥∥∥∥∥ (q1 2 )k (q1)+ lp1(·) lq1(·) .  ∞∑ k=−∞ ∥∥∥∥∥∥ ( 2kα(·)|fk | β0 )q1(·) ∥∥∥∥∥∥ lp1(·) lq1(·)  q• . 1, here q• = min k∈n (q0 2 )k (q1)+ ≥ 1.the previous calculations imply that β02 . β0 . ‖f ‖k̇α(·),q1(·) p1(·) (rn) . we must examine µ∗,σψ,λfj . by applying the minkowski inequality, we have |µ∗,σψ,λ(fj)(x)| = (∫ ∞ 0 ∫ rn ( (t)(t + |x1 − y1|)−1 )λn ∣∣∣∣ 1 tσ ∫ |y1−z1|≤t ψ(y1, y1 − z1) |y1 − z1|n−σ fj(z1)dz1 ∣∣∣∣2 dy1dt tn+1 ) 1 2 ≤ ∫ rn fj(z1) (∫ ∞ 0 ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ+n+1 ) 1 2 dz1 ≤ ∫ rn fj(z1) (∫ |x1−z1| 0 ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ+n+1 ) 1 2 dz1 https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 9 + ∫ rn fj(z1) (∫ ∞ |x1−z1| ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ+n+1 ) 1 2 dz1. let 2ρ− n > 0 and ψ ∈ l∞(rn)× l2(sn−1). then, the following inequality is satisfied∫ |y1−z1|≤t |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1 ≤ ∫ sn−1 ∫ t 0 |ψ(sy ′1 + z1, y ′ 1)|2 s2n−2σ sn−1dsdσ(y ′1) . ‖ψ‖2 l∞(rn)×l2(sn−1)t 2σ−n. because |x1− z1| ≤ |y1− z1|+ |x1− y1| ≤ |x1− y1|+ t , for λ > 2 and 0 < ε < (λ− 2)n, we obtain∫ |x1−z1| 0 ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ+n+1 . ∫ |x1−z1| 0 ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn−2n−ε 1 |x1 − z1|2n+ε |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ−n−ε+1 . 1 |x1 − z1|2n+ε ∫ |x1−z1| 0 ∫ |y1−z1|≤t |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ−n−ε+1 . ‖ψ‖2 l∞(rn)×l2(sn−1) |x1 − z1|2n+ε ∫ |x1−z1| 0 tε−1dt . |x1 − z1|−2n. let λ0n − 2n < 0, λ0n − n > 0 and 1 < λ0 < 2. then, we get∫ ∞ |x1−z1| ∫ |y1−z1|≤t ( (t)(t + |x1 − y1|)−1 )λn |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ+n+1 . ∫ ∞ |x1−z1| ∫ |y1−z1|≤t |x1 − z1|−λ0n |ψ(y1, y1 − z1)|2 |y1 − z1|2n−2σ dy1dt t2σ−λ0n+n+1 . ∫ ∞ |x1−z1| |x1 − z1|−λ0n ∫ |y1−z1|≤t |ψ(y1, y1 − z1)|2 |y1 − z1|2n−λ0n dy1dt tn+1 . ∫ ∞ |x1−z1| |x1 − z1|−λ0n ∫ sn−1 ∫ t 0 |ψ(y ′1, (y1 − z1)′)|2 s2n−λ0n sn−1dsdσ(y ′1) dt tn+1 . ‖ψ‖2 l∞(rn)×l2(sn−1)|x1 − z1|−λ0n ∫ ∞ |x1−z1| tλ0n−2n−1dt . |x1 − z1|−2n. by combining these estimates, we find µ∗,σψ,λ(f )(x) . ∫ rn |fj(z1)| |x1 − z1|n dz1. (∗) next, we consider β01. since j ≤ k − 2, by applying hölder’s inequality (lemma 2.3), we obtain µ∗,σψ,λ(f )(x) . ∫ rn |fj(z1)| |x1 − z1|n dz . 2−kn‖χj‖lp′1(·)(rn) ‖fj‖lp1(·)(rn). https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 10then, by lemmas 2.5 2.7, we deduce that ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥∥  2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥ 2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj)χk | β0 ∥∥∥∥∥∥∥∥∥ (q01 2 )k lp1(·) . ∞∑ k=−∞ 2kα(·) k−2∑ j=−∞ ∥∥∥∥ fjβ0 ∥∥∥∥ lp1(·)(rn) ‖χbj‖lp′1(·)‖χbk‖lp1(·) 2−kn (q01 2 )k . ∞∑ k=−∞ 2kα(·) k−2∑ j=−∞ ∥∥∥∥ fjβ0 ∥∥∥∥ lp1(·)(rn) ‖χbj‖lp′1(·) ‖χbk‖lp′1(·) (q01 2 )k . ∞∑ k=−∞ 2kα(·) k−2∑ j=−∞ 2(j−k)nδ11 ∥∥∥∥ fjβ0 ∥∥∥∥ lp1(·)(rn) (q01 2 )k . ∞∑ k=−∞  k−2∑ j=−∞ 2(k−j)(−nδ11+α+) ∥∥∥∥∥∥ ( |2jα(·)f χj | β0 )q1(·) ∥∥∥∥∥∥ 1 (q1)+ lp1(·)q1(·)(rn)  (q01 2 )k , where (q01 2 )k =  (q2)− ∥∥∥∥∥∥∥∥ 2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj )χk | β0  q2(·) ∥∥∥∥∥∥∥∥ l p1(·) q2(·) ≥ 1, (q2)+ otherwise. if (q1)+ 6 1, then by lemma 2.6, we have ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥∥  2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞  k−2∑ j=−∞ 2(k−j)(q1)+(α+−nδ11) ∥∥∥∥∥∥ ( |2jα(·)f χj | β0 )q1(·) ∥∥∥∥∥∥ lp1(·)q1(·)(rn)  (q01 2 )k (q1)+ https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 11 .  ∞∑ j=−∞ ∥∥∥∥∥∥ ( |2jα(·)f χj | β0 )q1(·) ∥∥∥∥∥∥ lp1(·)q1(·)(rn) ∞∑ k=j+2 2(k−j)(q1)+(α+−nδ11)  q• . 1, here q• = min k∈n (q01 2 )k (q1)+ .if (q1)+ > 1, using hölder’s inequality, we deduce that ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥∥  2kα(·)| k−2∑ j=−∞ µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞  k−2∑ j=−∞ 2(k−j)(α+−nδ11) (q1)+ 2 ∥∥∥∥∥∥ ( |2jα(·)f χj | β0 )q1(·) ∥∥∥∥∥∥ lp1(·)q1(·)(rn)  (q01 2 )k (q1)+ ×  k−2∑ j=−∞ 2(k−j)(α+−nδ11) ((q1)+)′ 2  (q01 2 )+ ((q1)+)′ .  ∞∑ j=−∞ ∥∥∥∥∥∥ ( |2jα(·)f χj | β0 )q1(·) ∥∥∥∥∥∥ lp1(·)q1(·)(rn) ∞∑ k=j+2 2(k−j)(α+−nδ11) (q1 )+ 2  q• . 1. therefore, we conclude the estimate β01 . β0 . ‖f ‖k̇α,q1(·) p1(·) (rn) . finally, we estimate β03. since j ≥ k + 2, by (∗) and applying hölder’s inequality (lemma 2.3), we have µ∗,σψ,λ(f )(x) . ∫ rn |fj(z1)| |x1 − z1|n dz1 . 2−jn‖χj‖lp′1(·)(rn) ‖fj‖lp1(·)(rn). then, by lemma2.5, lemma2.6 and lemma2.7, we get ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥ 2kα(·)| ∞∑ k+2 µ∗,σψ,λ(fj)χk | β0  q2(·)∥∥∥∥∥∥∥∥∥ l p1(·) q2(·) . ∞∑ k=−∞ ∥∥∥∥∥∥∥∥∥ 2kα(·)| ∞∑ j=k+2 µ∗,σψ,λ(fj)χk | β0 ∥∥∥∥∥∥∥∥∥ (q02 2 )k lp1(·) https://doi.org/10.28924/ada/ma.5.22 eur. j. math. anal. 10.28924/ada/ma.5.22 12 . ∞∑ k=−∞ 2kα(·) ∞∑ j=k+2 2−jn‖χbj‖lp′1(·)‖χbk‖lp1(·) ∥∥∥∥ fjβ0 ∥∥∥∥ lp1(·)(rn) (q02 2 )k . ∞∑ k=−∞ 2kα(·) ∞∑ j=k+2 2−jn ‖χbk‖lp1(·) ‖χbj‖lp1(·) |bj | ∥∥∥∥ fjβ0 ∥∥∥∥ lp1(·)(rn) (q02 2 )k . ∞∑ k=−∞  ∞∑ j=k+2 2(k−j)(nδ12+α−) ∥∥∥∥∥∥ ( 2jα(·)f χj β0 )q1(·) ∥∥∥∥∥∥ 1 (q1)+ lp1(·)q1(·)  (q02 2 )k , where (q02 2 )k =  (q2)− ∥∥∥∥∥∥∥ 2kα(·)| ∞∑ j=k+2 µ∗,σψ,λ(fj )χk | β0 q2(·)∥∥∥∥∥∥∥ l p1(·) q2(·) ≥ 1, (q2)+, otherwise.notice that α− > −nδ12, and 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(2019), 19–29. https://doi.org/10.28924/ada/ma.5.22 https://doi.org/10.1007/s11401-021-0268-3 https://doi.org/10.1007/s11401-021-0268-3 https://doi.org/10.1007/s10114-016-4617-1 https://www.utgjiu.ro/math/sma/v15/v15.html https://www.utgjiu.ro/math/sma/v15/v15.html https://doi.org/10.1016/j.jmaa.2012.04.043 https://doi.org/10.1112/blms/5.1.121 https://ocu-omu.repo.nii.ac.jp/records/201676 https://doi.org/10.1016/j.na.2009.02.075 https://doi.org/10.1016/j.na.2009.02.075 https://doi.org/10.14232/ejqtde.2018.1.82 https://doi.org/10.1186/s13661-018-1116-6 https://doi.org/10.1016/j.aml.2011.11.022 https://doi.org/10.1016/j.aml.2011.11.022 https://doi.org/10.4236/am.2016.710104 https://doi.org/10.4236/am.2016.710104 https://doi.org/10.1002/mma.7487 https://doi.org/10.3906/mat-1412-52 https://doi.org/10.3390/math10071168 https://doi.org/10.1007/978-3-0348-0548-3 https://doi.org/10.1007/s12215-010-0034-y http://eudml.org/doc/126704 1. introduction 2. mathematical background 3. boundedness of the parameterized littlewood-paley operators author contributions: conflicts of interest: references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 12doi: 10.28924/ada/ma.5.12 some new series expansions of a special type of functions involving the logarithmic function christophe chesneau department of mathematics, lmno, university of caen-normandie, 14032 caen, france christophe.chesneau@gmail.com abstract. in this article, we present new series expansions for a certain family of functions thatdepend on the logarithmic function. a general result is demonstrated by considering a tunable inter-mediate function. this result has the interest of unifying several important results in the literature,including a well-known series expansion established by srinivasa ramanujan. several precise ex-amples are given and discussed in detail. in addition, we recover the so-called seidel formula andderive new product expansions, with an emphasis on the so-called einstein function. some inequali-ties involving logarithmic functions are also applications of our series expansion approach. selectedresults are supported by graphical work. 1. introduction the logarithmic function, denoted log(x), plays a crucial role in several mathematical contexts,including calculus, number theory, and computer science. see [13], and the references therein.understanding its properties, especially its various series expansions, is fundamental to manymathematical analyses. the classical (taylor) expansion of log(x) is given by log(x) = +∞∑ k=1 (−1)k−1 k (x − 1)k . it is valid for x ∈ (0, 2] only (see [8], among others). from this expansion, we immediately get x − 1− log(x) = +∞∑ k=2 1 k (1− x)k . (1) a consequence of this result is the following inequality: log(x) ≤ x − 1 for x ∈ (0, 2]. however, asdiscussed in [4], we know that it actually holds for x > 0. thus, the use of the classical logarithmicexpansion is somehow inadequate for a full understanding of this inequality. there is a kind received: 26 sep 2024. key words and phrases. series expansions; logarithmic function; product expansions; inequalities.1 https://adac.ee https://doi.org/10.28924/ada/ma.5.12 https://orcid.org/0000-0002-1522-9292 eur. j. math. anal. 10.28924/ada/ma.5.12 2of "gap in understanding". a solution is given in [4]. using a telescoping technique and carefuldevelopment, the following series expansion is demonstrated: x − 1− log(x) = +∞∑ k=1 2k−1(x2 −k − 1)2, and it is valid for x > 0. from this result, we immediately observe that log(x) ≤ x−1 for x > 0; theconstraint x ∈ (0, 2] is relaxed. this key inequality is now fully understandable using the seriesexpansion tool. in addition, the underlying telescoping technique provides an original alternativeproof, making it very interesting from a mathematical point of view. it can also be used for otherpurposes; the proof of natural logarithmic inequalities is just one example.on the other hand, from a completely different perspective, a famous result of srinivasa ramanu-jan ensures that, for x > 0 with x 6= 1, we have 1 log(x) + 1 1− x = +∞∑ k=1 1 2k(1 + x2 −k ) . see [12, page 364]. the proof is based on an iteration technique, noting that 1/(1 − x) = (1/2)[1/(1 + √ x) + 1/(1 − √ x)]. it received special attention in [2, chapter 31, entry 29, page399] and was the object of an in-depth study in [5]. as a new visual note, this extension can bereformulated as 1 x − 1 − 1 log(x) = − +∞∑ k=1 1 2k(1 + x2 −k ) . based on this form, doing a parallel with the formula in equation (1), a functional pattern seemsto be present. in fact, both expansions can be expressed as φ(x − 1)− φ[log(x)] = +∞∑ k=1 ak(x), (2) where φ(t) = t and φ(t) = 1/t , respectively, and ak(x) = 2k−1(x2−k−1)2 and ak(x) = −1/2k(1+ x2 −k ), respectively. given this, a unified approach seems possible.in this article, we formalize such an approach. it aims to generate a wide range of new seriesexpansions of certain functions that depend on the logarithmic function, i.e., functions of the form φ(x − 1) − φ[log(x)]. the proof is based on telescoping techniques inspired by [4] and precisefactorization developments. the results established have interesting consequences, including thederivation of old and new product expansions. among other things, a new product expansion ofthe einstein function, i.e., e2(x) = x/(ex − 1) (see [1]), is established. in addition, inequalitiesinvolving logarithmic functions are obtained almost immediately, in the spirit of [3, 6, 7, 9–11, 16].some graphics illustrate the results.the following sections structure the article: section 2 is devoted to the general result on theseries expansion of φ(x − 1) − φ[log(x)] and emphasizes several examples. section 3 deals with https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 3some of its implications, including product expansions and inequalities. a conclusion is given insection 4. 2. results our general and specific findings are presented in this section. 2.1. a general result. the theorem below suggests a series expansion for the difference function φ(x − 1)− φ[log(x)], defined with a certain function φ. theorem 2.1. let x > 0 and φ be a continuous function such that |φ(x−1)| < +∞ and |φ[log(x)]| < +∞. then we have φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] . proof. introducing an integer n ≥ 1 and using the telescoping technique, we get φ(x − 1)− φ [ 2n(x2 −n − 1) ] = φ [ 20(x2 −0 − 1) ] − φ [ 2n(x2 −n − 1) ] = n∑ k=1 { φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ]} = n∑ k=1 αk(φ)(x). since limn→+∞ 2n(x2−n−1) = limn→+∞ 2n(e2−n log(x)−1) = limn→+∞ 2n {[1 + 2−n log(x)]− 1} = log(x), thanks to the continuity of φ, we obtain φ(x − 1)− φ[log(x)] = φ(x − 1)− φ [ lim n→+∞ 2n(x2 −n − 1) ] = lim n→+∞ { φ(x − 1)− φ [ 2n(x2 −n − 1) ]} = lim n→+∞ n∑ k=1 αk(φ)(x) = +∞∑ k=1 αk(φ)(x). this ends the proof of the theorem. � from this theorem, if φ is bijective, then the following series expansion of the logarithmic functionholds: log(x) = φ−1 { φ(x − 1)− +∞∑ k=1 αk(φ)(x) } . it can be useful in several mathematical contexts. https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 4in fact, this theorem has the advantage of being general and tunable thanks to the function φ.it also unifies several results in the literature, as will be developed in the next part. 2.2. specific results. the proposition below shows some consequences of theorem 2.1, includingsome new results. proposition 2.2. the series expansions below are valid.(1) for x > 0, we have x − 1− log(x) = +∞∑ k=1 2k−1(x2 −k − 1)2. as mentioned in the introduction, this result is not new; it was established in [4].(2) for x > 0, we have 1 x − 1 − 1 log(x) = − +∞∑ k=1 1 2k(1 + x2 −k ) . as mentioned in the introduction, this result is not new; it is a famous result proved by srinivasa ramanujan, as highlighted in [2, chapter 31, page 399] and [5]. to the best of our knowledge, the eight results below are new.(3) for x > 0, we have (x − 1)2 − [log(x)]2 = +∞∑ k=1 22(k−1)(x2 −k − 1)3(3 + x2−k ). (4) for x ≥ 1, we have √ x − 1− √ log(x) = +∞∑ k=1 2(k−1)/2(x2 −k − 1)3/2√ 1 + x2 −k + √ 2 . (5) for x > 1, we have 1√ x − 1 − 1√ log(x) = − +∞∑ k=1 1 2k/2 [√ 1 + x2 −k + √ 2 ]√x2 −k − 1 1 + x2 −k . (6) for x > 1, we have log(x − 1)− log [log(x)] = +∞∑ k=1 log ( 1 + x2 −k 2 ) . (7) for x > 0, we have sin(x − 1)− sin[log(x)] = 2 +∞∑ k=1 sin [ 2k−2(x2 −k − 1)2 ] cos [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . (8) for x > 0, we have cos(x − 1)− cos[log(x)] = −2 +∞∑ k=1 sin [ 2k−2(x2 −k − 1)2 ] sin [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 5(9) for x > 0, we have sinh(x − 1)− sinh[log(x)] = 2 +∞∑ k=1 sinh [ 2k−2(x2 −k − 1)2 ] cosh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . (10) for x > 0, we have cosh(x − 1)− cosh[log(x)] = 2 +∞∑ k=1 sinh [ 2k−2(x2 −k − 1)2 ] sinh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . proof. let us prove each result, one by one.(1) for x > 0, by applying theorem 2.1 with φ(t) = t , we obtain x − 1− log(x) = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = 2k−1(x2 −(k−1) − 1)− 2k(x2−k − 1) = 2k−1(x2 −k − 1)(1 + x2−k )− 2k(x2−k − 1) = 2k−1(x2 −k − 1) [ (1 + x2 −k )− 2 ] = 2k−1(x2 −k − 1)2. the used factorization arguments are the same as those in [4]. hence, we have x − 1− log(x) = +∞∑ k=1 2k−1(x2 −k − 1)2. (2) for x > 0, applying theorem 2.1 to φ(t) = 1/t , we establish that 1 x − 1 − 1 log(x) = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = 1 2k−1(x2−(k−1) − 1) − 1 2k(x2 −k − 1) = − 2k−1(x2 −(k−1) − 1)− 2k(x2−k − 1) 22k−1(x2−(k−1) − 1)(x2−k − 1) = − 2k−1(x2 −k − 1)2 22k−1(x2−(k−1) − 1)(x2−k − 1) = − (x2 −k − 1)2 2k(1 + x2 −k )(x2 −k − 1)2 = − 1 2k(1 + x2 −k ) . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 6hence, we have 1 x − 1 − 1 log(x) = − +∞∑ k=1 1 2k(1 + x2 −k ) . (3) for x > 0, applying theorem 2.1 to φ(t) = t2, we get (x − 1)2 − [log(x)]2 = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = [ 2k−1(x2 −(k−1) − 1) ]2 − [ 2k(x2 −k − 1) ]2 = [ 2k−1(x2 −(k−1) − 1)− 2k(x2−k − 1) ] [ 2k−1(x2 −(k−1) − 1) + 2k(x2−k − 1) ] = 2k−1(x2 −k − 1)22k−1(x2−k − 1)(3 + x2−k ) = 22(k−1)(x2 −k − 1)3(3 + x2−k ). hence, we have (x − 1)2 − [log(x)]2 = +∞∑ k=1 22(k−1)(x2 −k − 1)3(3 + x2−k ). (4) for x ≥ 1, using theorem 2.1 with φ(t) = √t , we obtain √ x − 1− √ log(x) = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = √ 2k−1(x2−(k−1) − 1)− √ 2k(x2 −k − 1) = 2k−1(x2 −(k−1) − 1)− 2k(x2−k − 1)√ 2k−1(x2−(k−1) − 1) + √ 2k(x2 −k − 1) = 2k−1(x2 −k − 1)2√ 2k−1(x2−k − 1)(1 + x2−k ) + √ 2[2k−1(x2−k − 1)] = 2(k−1)/2(x2 −k − 1)3/2√ 1 + x2 −k + √ 2 . hence, we have √ x − 1− √ log(x) = +∞∑ k=1 2(k−1)/2(x2 −k − 1)3/2√ 1 + x2 −k + √ 2 . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 7(5) for x > 1, applying theorem 2.1 to φ(t) = 1/√t , we have 1√ x − 1 − 1√ log(x) = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = 1√ 2k−1(x2−(k−1) − 1) − 1√ 2k(x2 −k − 1) = − √ 2k−1(x2−(k−1) − 1)− √ 2k(x2 −k − 1)√ 22k−1(x2−(k−1) − 1)(x2−k − 1) = − 2k−1(x2 −(k−1) − 1)− 2k(x2−k − 1)[√ 2k−1(x2−(k−1) − 1) + √ 2k(x2 −k − 1) ]√ 22k−1(x2−(k−1) − 1)(x2−k − 1) = − 2k−1(x2 −k − 1)2[√ 2k−1(x2−k − 1)(1 + x2−k ) + √ 2[2k−1(x2−k − 1)] ]√ 22k−1(x2−k − 1)2(1 + x2−k ) = − 1 2k/2 [√ 1 + x2 −k + √ 2 ]√x2 −k − 1 1 + x2 −k . hence, we have 1√ x − 1 − 1√ log(x) = − +∞∑ k=1 1 2k/2 [√ 1 + x2 −k + √ 2 ]√x2 −k − 1 1 + x2 −k . (6) for x > 1, it follows from theorem 2.1 with φ(t) = log(t) that log(x − 1)− log[log(x)] = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = log [ 2k−1(x2 −(k−1) − 1) ] − log [ 2k(x2 −k − 1) ] = log [ 2k−1(x2 −(k−1) − 1) 2k(x2 −k − 1) ] = log [ (x2 −k − 1)(1 + x2−k ) 2(x2 −k − 1) ] = log ( 1 + x2 −k 2 ) . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 8hence, we have log(x − 1)− log [log(x)] = +∞∑ k=1 log ( 1 + x2 −k 2 ) . (7) for x > 0, applying theorem 2.1 to φ(t) = sin(t), we obtain sin(x − 1)− sin[log(x)] = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = sin [ 2k−1(x2 −(k−1) − 1) ] − sin [ 2k(x2 −k − 1) ] . using the standard trigonometric formula sin(t)− sin(u) = 2 sin[(t − u)/2] cos[(t + u)/2],we get αk(φ)(x) = 2 sin [ 2k−2(x2 −(k−1) − 1)− 2k−1(x2−k − 1) ] cos [ 2k−2(x2 −(k−1) − 1) + 2k−1(x2−k − 1) ] = 2 sin [ 2k−2(x2 −k − 1)2 ] cos [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . hence, we have sin(x − 1)− sin[log(x)] = 2 +∞∑ k=1 sin [ 2k−2(x2 −k − 1)2 ] cos [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . (8) for x > 0, it follows from theorem 2.1 with φ(t) = cos(t) that cos(x − 1)− cos[log(x)] = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = cos [ 2k−1(x2 −(k−1) − 1) ] − cos [ 2k(x2 −k − 1) ] . using the standard trigonometric formula cos(t)−cos(u) = −2 sin[(t−u)/2] sin[(t+u)/2],we obtain αk(φ)(x) = −2 sin [ 2k−2(x2 −(k−1) − 1)− 2k−1(x2−k − 1) ] sin [ 2k−2(x2 −(k−1) − 1) + 2k−1(x2−k − 1) ] = −2 sin [ 2k−2(x2 −k − 1)2 ] sin [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . hence, we have cos(x − 1)− cos[log(x)] = −2 +∞∑ k=1 sin [ 2k−2(x2 −k − 1)2 ] sin [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 9(9) for x > 0, applying theorem 2.1 to φ(t) = sinh(t), we establish that sinh(x − 1)− sinh[log(x)] = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = sinh [ 2k−1(x2 −(k−1) − 1) ] − sinh [ 2k(x2 −k − 1) ] . using the standard hyperbolic formula sinh(t)− sinh(u) = 2 sinh[(t−u)/2] cosh[(t+u)/2],we get αk(φ)(x) = 2 sinh [ 2k−2(x2 −(k−1) − 1)− 2k−1(x2−k − 1) ] cosh [ 2k−2(x2 −(k−1) − 1) + 2k−1(x2−k − 1) ] = 2 sinh [ 2k−2(x2 −k − 1)2 ] cosh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . hence, we have sinh(x − 1)− sinh[log(x)] = 2 +∞∑ k=1 sinh [ 2k−2(x2 −k − 1)2 ] cosh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . (10) for x > 0, using theorem 2.1 with φ(t) = cosh(t), we find that cosh(x − 1)− cosh[log(x)] = φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x), where αk(φ)(x) = φ [ 2k−1(x2 −(k−1) − 1) ] − φ [ 2k(x2 −k − 1) ] = cosh [ 2k−1(x2 −(k−1) − 1) ] − cosh [ 2k(x2 −k − 1) ] . using the standard trigonometric formula cosh(t) − cosh(u) = 2 sinh[(t − u)/2] sinh[(t + u)/2], we obtain αk(φ)(x) = 2 sinh [ 2k−2(x2 −(k−1) − 1)− 2k−1(x2−k − 1) ] sinh [ 2k−2(x2 −(k−1) − 1) + 2k−1(x2−k − 1) ] = 2 sinh [ 2k−2(x2 −k − 1)2 ] sinh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . hence, we have cosh(x − 1)− cosh[log(x)] = 2 +∞∑ k=1 sinh [ 2k−2(x2 −k − 1)2 ] sinh [ 2k−2(x2 −k − 1)(3 + x2−k ) ] . all the claimed expansions are established, ending the proof. � https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 10thus, proposition 2.2 is derived from theorem 2.1 by using various functions φ, namely φ(t) = t , φ(t) = 1/t , φ(t) = t2, φ(t) = √t , φ(t) = 1/√t , φ(t) = log(t), φ(t) = sin(t), φ(t) = cos(t), φ(t) = sinh(t) and φ(t) = cosh(t), one for each sub-result, in order. in addition, some extendedresults can be proved. for example, for x > 0, based on the proof theorem 2.1 with the function φ(t) = √ |t|, we can extend the item numbered 4 as√ |x − 1| − √ | log(x)| = sign(x − 1) +∞∑ k=1 2(k−1)/2|x2−k − 1|3/2√ 1 + x2 −k + √ 2 , where sign(x − 1) =  1 if x > 1, 0 if x = 1, −1 if x < 1. analogous extension of the item numbered 5 is possible. with a little mathematical effort, we can get similar series expansions by considering the trans-lated version of φ, i.e., φ(t; a) = φ(t + a) for some a ∈ r. of course, other interesting functionscan also be examined for φ, such as φ(t) = arctanh(t), which benefits from an interesting additionformula, among other things.it is important to note that the convergence of the series expansions in proposition 2.2 hasbeen checked on the basis of theoretical and practical work. let us illustrate graphically theconvergence of the series expansion in the item numbered 3. to do this, we consider the followingtruncated-series function: ϕ(x ;m) = (x − 1)2 − [log(x)]2 − m∑ k=1 22(k−1)(x2 −k − 1)3(3 + x2−k ), where m denotes an integer such that m ≥ 1. figure 1 displays the plots of ϕ(x ;m) for m = 1, 2, . . . , 15 and four arbitrary values of x . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 11 0 5 10 15 − 0 .1 4 − 0 .1 0 − 0 .0 6 − 0 .0 2 m 0 5 10 15 0 .0 0 0 .0 1 0 .0 2 0 .0 3 m (a) (b) 0 5 10 15 0 .0 0 .5 1 .0 1 .5 2 .0 m 0 5 10 15 0 5 1 0 1 5 2 0 2 5 3 0 3 5 m (c) (d) figure 1. plots of ϕ(x ;m) for m = 1, 2, . . . , 15 and (a) x = 0.5, (b) x = 1.5, (c) x = 4, and (d) x = 18. from this figure we see that, for all the values of x considered, ϕ(x ;m) converges very quicklyto 0; it begins to be very close to the y = 0 axis from m = 7.let us mention that the proof of the item numbered 2 has a different construction than the onein [2, chapter 31, page 399], even though it is based on the same functional basis.also, from proposition 2.2, several expansions of the logarithmic function can be deduced. forinstance, based on the item numbered 4, for x ≥ 1, we have log(x) = [ √ x − 1− +∞∑ k=1 2(k−1)/2(x2 −k − 1)3/2√ 1 + x2 −k + √ 2 ]2 . such expansions are innovative, to the best of our knowledge.concerning the item numbered 9 in proposition 2.2, one can remark that sinh(x − 1)− sinh[log(x)] = 1 2x − x 2 − sinh(1− x) https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 12and, by using the series expansion of the hyperbolic sine function, the following alternative expres-sion is obtained: sinh(x − 1)− sinh[log(x)] = 1 2x − x 2 − +∞∑ k=0 (1− x)2k+1 (2k + 1)! . clearly, it can be more manageable to use in comparison to the one in the item numbered 9,depending on the context. the same remark holds for the item numbered 10; we have cosh(x − 1)− cosh[log(x)] = − 1 2x − x 2 + +∞∑ k=0 (1− x)2k (2k)! . thus, the items numbered 9 and 10 are mainly interesting because of their originality, i.e., thetrigonometric and hyperbolic functions involved, respectively. 3. applications some consequences of our results are described in this part. 3.1. product expansions. proposition 2.2 can be used for many purposes, including product ex-pansions. the result below illustrates this claim with an example. proposition 3.1. the infinite product expansions below are valid.(1) for x > 1, we have x − 1 log(x) = +∞∏ k=1 1 + x2 −k 2 . this expansion is, in fact, valid for x ∈ (0,+∞) \ {1}, as discussed later. it "almost" corresponds to the so-called seidel formula (see [15]).(2) for x > 1, we have log(x) x − 1 = +∞∏ k=1 2 1 + x2 −k . this expansion is, in fact, valid for x ∈ (0,+∞) \ {1}, as discussed later.(3) for x > 0, we have ex − 1 x = +∞∏ k=1 1 + e2 −kx 2 . this expansion is, in fact, valid for x ∈ r \ {0}, as discussed later.(4) for x > 0, we have the following product expansion of the einstein function: e2(x) = x ex − 1 = +∞∏ k=1 2 1 + e2 −kx . this expansion is, in fact, valid for x ∈ r \ {0}, as discussed later. https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 13 proof. let us prove each result, one by one.(1) we propose two different proofs. proof 1: use of proposition 2.2.: for x > 1, it follows from the item numbered 6 inproposition 2.2 that log(x − 1)− log [log(x)] = +∞∑ k=1 log ( 1 + x2 −k 2 ) . by the continuity of the logarithmic function over its domain, it can be rewritten as log [ x − 1 log(x) ] = log [ +∞∏ k=1 1 + x2 −k 2 ] . composing with the exponential function, we get the desired result, i.e., x − 1 log(x) = +∞∏ k=1 1 + x2 −k 2 . proof 2: iterative scheme.: to provide an alternative proof, we now revisit the originalproof of the seidel formula in [15]. we can write x − 1 = (1 + √ x)( √ x − 1) = (1 + x2 −1 )(x2 −1 − 1) and, with the same principle, we can write the last term as x2 −1 − 1 = (1 + x2−2)(x2−2 − 1), and the same for x2−2 − 1, etc. so, for any integer n ≥ 1, we have x − 1 = (1 + x2−1)(x2−1 − 1) = (1 + x2−1)(1 + x2−2)(x2−2 − 1) = . . . = [ n∏ k=1 (1 + x2 −k ) ] (x2 −n − 1) = [ n∏ k=1 1 + x2 −k 2 ] 2n(x2 −n − 1). therefore, by considering the limit when n → +∞, we obtain x − 1 = [ lim n→+∞ n∏ k=1 1 + x2 −k 2 ] [ lim n→+∞ 2n(x2 −n − 1) ] = [ +∞∏ k=1 1 + x2 −k 2 ] log(x), which implies that x − 1 log(x) = +∞∏ k=1 1 + x2 −k 2 . thus, proof 1 offers an alternative to these known developments by using proposition2.2. proof 2, however, has the advantage of being valid for x ∈ (0,+∞) \ {1}, not just x > 1. we will show later why this is actually not a problem.(2) for x > 1, by using the previous result, we get log(x) x − 1 = 1 (x − 1)/ log(x) = 1∏+∞ k=1 [ (1 + x2 −k )/2 ] = +∞∏ k=1 1 (1 + x2 −k )/2 = +∞∏ k=1 2 1 + x2 −k . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 14(3) for y > 0, by applying the result in the item numbered 1 with x = ey > 1, we establishthat ey − 1 y = x − 1 log(x) = +∞∏ k=1 1 + x2 −k 2 = +∞∏ k=1 1 + e2 −ky 2 . (4) for x > 0, by using the result in the previous item, we have e2(x) = x ex − 1 = 1 [(ex − 1)/x ] = 1∏+∞ k=1[(1 + e 2−kx)/2] = +∞∏ k=1 1 (1 + e2 −kx)/2 = +∞∏ k=1 2 1 + e2 −kx . this ends the proof. � from the item numbered 2, for x ≥ 1, we get the following product expansion of the logarithmicfunction: log(x) = (x − 1) +∞∏ k=1 2 1 + x2 −k , (3) which corresponds to the seidel formula restricted to (1,+∞). we can complete it for x ∈ (0, 1)based on the case x ≥ 1. indeed, for x ∈ (0, 1), since 1/x > 1 and ∑+∞k=1 2−k = 1, we get log(x) = − log ( 1 x ) = − ( 1 x − 1 ) +∞∏ k=1 2 1 + x−2−k = (x − 1) 1 x [ +∞∏ k=1 2 1 + x2 −k ][ +∞∏ k=1 x2 −k ] = (x − 1) [ +∞∏ k=1 2x2 −k 1 + x2 −k ] [ x−1+ ∑+∞ k=1 2 −k ] = (x − 1) +∞∏ k=1 2 1 + x2 −k . we thus find the seidel formula in its entirety, as mentioned in the second proof of the itemnumbered 1 in proposition 3.1. the advantages of this decomposition are that it has no constraintson the natural domain of definition, i.e., x > 0, to satisfy log(x) = − log(1/x), which is not the casefor most series expansions of log(x), and also that log(x) and x − 1 have the same sign accordingto x ∈ (0, 1) and x > 1. it is also underexploited in the literature to determine sharp logarithmicinequalities. we will emphasize this aspect in the next section. thus, in a sense, the results intheorem 2.1 unify three known results, one by srinivasa ramanujan in [12], one by ludwig seidelin [15], and a more recent one by david m. bradley in [4].as an additional numerical contribution, let us illustrate this expansion by considering thefollowing function: ζ(x ;m) = log(x)− (x − 1) m∏ k=1 2 1 + x2 −k , https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 15where m denotes an integer such that m ≥ 1. figure 2 displays the plots of ζ(x ;m) for m = 1, 2, . . . , 15 and four arbitrary values of x , including x = 0.5 ∈ (0, 1) to check the previousstatement. 0 5 10 15 − 0 .1 0 − 0 .0 6 − 0 .0 2 m 0 5 10 15 − 0 .0 4 − 0 .0 3 − 0 .0 2 − 0 .0 1 0 .0 0 m (a) (b) 0 5 10 15 − 0 .6 − 0 .4 − 0 .2 0 .0 m 0 5 10 15 − 3 .5 − 2 .5 − 1 .5 − 0 .5 m (c) (d) figure 2. plots of ζ(x ;m) for m = 1, 2, . . . , 15 and (a) x = 0.5, (b) x = 1.5, (c) x = 4, and (d) x = 18. this figure shows that ζ(x ;m) converges very quickly to 0 for all the considered values of x ; itstarts very close to the axis y = 0 from m = 7.sophisticated infinite product formulas can be derived from the natural properties of the loga-rithmic function and equation (3). in particular, the formulas below are true. • for x > 0, we have x = e log(x), which is equivalent to x = e (x−1) ∏+∞ k=1 2 1+x2 −k . • for x > 0 and y ∈ r, we have log(xy ) = y log(x), which yields (xy − 1) +∞∏ k=1 2 1 + x2 −ky = y(x − 1) +∞∏ k=1 2 1 + x2 −k . https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 16similarly, for x > 0 and y ∈ r \ {0}, we have log(x) = (1/y) log(xy ), giving the followingexpansions: log(x) = 1 y (xy − 1) +∞∏ k=1 2 1 + x2 −ky . in particular, for y = 2m with an arbitrary m > 0, we have log(x) = 1 2m (x2 m − 1) +∞∏ k=1 2 1 + x2 m−k . • for x > 0 and y > 0, we have log(xy) = log(x) + log(y), which is equivalent to (xy − 1) +∞∏ k=1 2 1 + (xy)2 −k = (x − 1) +∞∏ k=1 2 1 + x2 −k + (y − 1) +∞∏ k=1 2 1 + y2 −k . in addition, it is known that the einstein function can be expressed as a series expansion involvingbernoulli numbers as e2(x) = +∞∑ k=0 b−k k! xk , where b−k = k∑ `=0 ∑̀ v=0 (−1)v ( ` v ) v k `+ 1 and (`v) = `!/[v !(`− v)!].in some sense, item numbered 4 completes this result by investigating a simple product expansionfor x > 0, given as e2(x) = +∞∏ k=1 2 1 + e2 −kx . (4) it is interesting to note that this formula is also valid for x < 0. indeed, in this case, we can remarkthat e2(x) = e −xe2(−x) = e−x +∞∏ k=1 2 1 + e−2−kx = e−x [ +∞∏ k=1 2 1 + e2 −kx ][ +∞∏ k=1 e2 −kx ] = [ +∞∏ k=1 2 1 + e2 −kx ] [ e−x+x ∑+∞ k=1 2 −k ] = +∞∏ k=1 2 1 + e2 −kx . thus, for x ∈ r\{0}, the formula in equation (4) is true. to the best of our knowledge, this specialrepresentation of the einstein function is a new result.the next part is about some inequalities derived from our findings. https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 173.2. inequalities. inequalities of various kinds can be derived from our previous results. the propo-sition below proposes an original one. proposition 3.2. for x ≥ 1, we have √ x − 1− √ log(x) ≤ 1 4 +∞∑ k=1 2k/2(x2 −k − 1)3/2. this last series expansion converges. proof. thanks to the item numbered 4 in proposition 2.2, for x ≥ 1, we have √ x − 1− √ log(x) = +∞∑ k=1 2(k−1)/2(x2 −k − 1)3/2√ 1 + x2 −k + √ 2 . since x ≥ 1, for any integer k ≥ 1, we have x2−k ≥ 1, so √1 + x2−k ≥ √2, which implies that√ 1 + x2 −k + √ 2 ≥ 2 √ 2. furthermore, it is clear that 2(k−1)/2(x2−k − 1)3/2 ≥ 0. therefore, weobtain √ x − 1− √ log(x) ≤ 1 2 √ 2 +∞∑ k=1 2(k−1)/2(x2 −k − 1)3/2 = 1 4 +∞∑ k=1 2k/2(x2 −k − 1)3/2. the convergence of this series expansion can be shown by the equivalence technique. moreprecisely, when k → +∞, we have 2k/2(x2 −k − 1)3/2 = 2−k [2k(x2−k − 1)]3/2 ∼ 2−k [log(x)]3/2, and 2−k is the term of a convergent geometric series. the desired result is demonstrated. � this inequality may be more interesting for the lower bound of the series term than for the upperbound of √x − 1−√log(x). in fact, it is difficult to capture the analytic function associated withthe series term.the proposition below is a general inequality setting based on theorem 2.1. proposition 3.3. in the framework of theorem 2.1, the inequalities below are true.(1) if φ is non-decreasing, then, for any sets of integers m ⊆ {1, 2, . . .}, we have φ(x − 1)− φ[log(x)] ≥ ∑ k∈m αk(φ)(x). (2) if φ is non-increasing, then, for any sets of integers m ⊆ {1, 2, . . .}, we have φ(x − 1)− φ[log(x)] ≤ ∑ k∈m αk(φ)(x). proof. for x > 0, let us consider the following function: ψ(y) = y(x1/y − 1), y ∈ (0,+∞). https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 18then we have ψ′(y) = x1/y − 1 y x1/y log(x)− 1 = x1/y − x1/y log(x1/y )− 1. using the well-known logarithmic identity log(t) ≥ t−1(t − 1) for t > 0, with t = x1/y , we obtain ψ′(y) ≤ x1/y − x1/yx−1/y (x1/y − 1)− 1 = 0. as a result, ψ is a non-increasing function. then, for any integer k ≥ 1, since 2k−1 ≤ 2k , we have ψ(2k) ≤ ψ(2k−1). let us now distinguish two cases: • if φ is non-decreasing, then we have φ [ψ(2k)] ≤ φ [ψ(2k−1)], implying that αk(φ)(x) = φ [ ψ(2k−1) ] − φ [ ψ(2k) ] ≥ 0. it follows from theorem 2.1 that, for any sets of integers m ⊆ {1, 2, . . .}, we have φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x) = ∑ k∈m αk(φ)(x) + ∑ k 6∈m αk(φ)(x) ≥ ∑ k∈m αk(φ)(x). • with similar arguments, if φ is non-increasing, then we have φ [ψ(2k)] ≥ φ [ ψ(2k−1) ],implying that αk(φ)(x) = φ [ψ(2k−1)]− φ [ψ(2k)] ≤ 0. it follows from theorem 2.1 that,for any sets of integers m ⊆ {1, 2, . . .}, φ(x − 1)− φ[log(x)] = +∞∑ k=1 αk(φ)(x) = ∑ k∈m αk(φ)(x) + ∑ k 6∈m αk(φ)(x) ≤ ∑ k∈m αk(φ)(x). the desired inequalities are demonstrated. � let us exemplify this general result with an immediate application. taking m = {m, . . . , n},where m and n are integers such that n ≥ m ≥ 1, with regard to the item numbered 3 inproposition 2.2 using φ(t) = t2, the following inequalities hold: • for x > 1, we have (x − 1)2 − [log(x)]2 ≥ n∑ k=m 22(k−1)(x2 −k − 1)3(3 + x2−k ). • for x ∈ (0, 1), we have (x − 1)2 − [log(x)]2 ≤ n∑ k=m 22(k−1)(x2 −k − 1)3(3 + x2−k ). we can also remark that, for any integer k ≥ 1, if x > 1, then we have (x2−k − 1)3 > 0, and if x ∈ (0, 1), then we have (x2−k − 1)3 < 0, and the above inequalities follow.another simple application is the inequality formulated in the lemma below. lemma 3.4. for x > 0, we have log(x) ≤ 2( √ x − 1). proof. we propose three different proofs. https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 19 proof 1: judicious approach.: the following inequality is well known: log(y) ≤ y − 1 for y > 0. applying it to the judicious choice y = √x , we get log(x) = 2 log(√x) ≤ 2(√x−1). proof 2: use of our series expansion.: we can apply the first items in propositions 2.2 and3.3 with m = {1}. indeed, since φ(t) is non-decreasing (or the coefficients of the relatedseries expansion are clearly non-negative), we have x − 1− log(x) ≥ ∑ k∈m 2k−1(x2 −k − 1)2 = ( √ x − 1)2, implying that log(x) ≤ x − 1− ( √ x − 1)2 = 2( √ x − 1). proof 3: use of differentiation.: in the proof of proposition 3.3, we showed that, for x > 0,the function ψ(y) = y(x1/y − 1) is non-increasing. this implies that, for any θ > 0, wehave limy→0+ ψ(y) ≤ ψ(θ), i.e., log(x) ≤ θ(x1/θ − 1). the desired result is just a special case; it is enough to take θ = 2.this completes the proof. � this lemma is not new; it has been demonstrated with other differentiation techniques in [17],and its sharpness has also been illustrated.another logarithmic inequality is highlighted in the lemma below. lemma 3.5. for x > 0 and any integers m and n such that n ≥ m ≥ 1, we have log(x) ≤ (x − 1) n∏ k=m 2 1 + x2 −k . proof. the proof is a consequence of the infinite product expansion in equation (3). let us distin-guish the cases x ≥ 1 and x ∈ (0, 1). • for x ≥ 1 and any integer k ≥ 1, we have x2−k ≥ 1, implying that 2/(1 + x2−k ) ≤ 1. sowe have log(x) = (x − 1) [ m−1∏ k=1 2 1 + x2 −k ][ n∏ k=m 2 1 + x2 −k ][ +∞∏ k=n+1 2 1 + x2 −k ] ≤ (x − 1) n∏ k=m 2 1 + x2 −k , with the convention ∏0k=1[2/(1 + x2−k )] = 1. • for x ∈ (0, 1) and any integer k ≥ 1, we have x2−k ≤ 1, implying that 2/(1 + x2−k ) ≥ 1and x − 1 ≤ 0. the exact same inequality as above is obtained.the proof is therefore finished. � https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 20this lemma generalizes the famous inequality log(x) ≤ x − 1 for x > 0. it also shows how theseidel formula, i.e., equation (3), can be applied to refine it.a similar inequality involving the einstein function is examined below. lemma 3.6. for x > 0 and any integers m and n such that n ≥ m ≥ 1, we have e2(x) ≤ n∏ k=m 2 1 + e2 −kx . for x < 0, the reversed inequality holds. proof. the proof follows from the infinite product expansion in equation (4). indeed, for x > 0 andany integer k ≥ 1, we have e2−kx > 1, implying that 2/(1 + e2−kx) < 1. as a result, we have e2(x) = [ m−1∏ k=1 2 1 + e2 −kx ][ n∏ k=m 2 1 + e2 −kx ][ +∞∏ k=n+1 2 1 + e2 −kx ] ≤ n∏ k=m 2 1 + e2 −kx , with the convention ∏0k=1[2/(1 + e2−kx)] = 1. for x < 0, for x > 0 and any integer k ≥ 1, wehave e2−kx < 1, implying that 2/(1 + e2−kx) > 1. using this, the reversed inequality above isimmediately established. the desired results are obtained. � the inequalities above are just a sample of what can be derived from our results; other explo-rations are left for future studies. 4. conclusion in conclusion, the logarithmic function log(x) is central to several mathematical disciplines. inparticular, its series expansions play a crucial role in mathematical analysis. while the classicalseries expansion initially presented limitations for the natural domain, i.e., x ∈ (0,+∞), a re-fined telescoping technique elaborated in [4] allows to relax the constraint. this allows a broaderunderstanding of the logarithmic function. in our first investigations, based on this result and awell-known series expansion established by srinivasa ramanujan (see [2] and [5]), we discovereda unified functional pattern. this connection can be expressed in the form "φ(x − 1) − φ[log(x)]".in light of this, using telescoping techniques and thorough factorization developments, we generatenew series extensions for such functions. several examples are given and discussed. as illustrated,these results lead to new product expansions, including one for the einstein function, and to in-equalities involving the logarithmic function. this article thus contributes to a better understandingof the logarithmic function and series expansions in general, and lays some foundations for futurework. https://doi.org/10.28924/ada/ma.5.12 eur. j. math. anal. 10.28924/ada/ma.5.12 21references [1] m. abramowitz, i.a. stegun, "debye functions." 27.1 in handbook of mathematical functions with formulas, graphs,and mathematical tables, 9th printing. new york: dover, 999-1000, 1972.[2] b.c. berndt, ramanujan’s notebooks, part iv, springer, new york, 1994. https://doi.org/10.1007/ 978-1-4612-0879-2.[3] l. bougoffa, p. krasopoulos, new optimal bounds for logarithmic and exponential functions, journal of inequalitiesand special functions, 12, 24-32, 2021.[4] d.m. bradley, an infinite series that displays the concavity of the natural logarithm, math. mag. 90 (2017), 353-354. https://doi.org/10.4169/math.mag.90.5.353.[5] d.m. bradley, concerning an infinite series of ramanujan related to the natural logarithm, ramanujan j. 47 (2018),253-265. https://doi.org/10.1007/s11139-017-9961-y.[6] f. burk, the geometric, logarithmic, and arithmetic mean inequality, am. math. mon. 94 (1987), 527-528. https: //doi.org/10.1080/00029890.1987.12000678.[7] c. chesneau, y.j. bagul, new sharp bounds for the logarithmic function, elec. j. math. anal. appl. 8 (2020), 140-145. https://doi.org/10.21608/ejmaa.2020.312813.[8] i.s. gradshteyn, i.m. ryzhik, table of integrals, series, and products; translated from the russian, eighth edition,revised from the seventh edition; zwillinger, d., moll, v., translators; elsevier/academic press: amsterdam, thenetherlands, 2015.[9] g. jameson, p.r. mercer, the logarithmic mean revisited, am. math. mon. 126 (2019), 641-645. https://doi.org/ 10.1080/00029890.2019.1605799.[10] m. kostić, new inequalities for the function y = t ln t , electronic journal of mathematical analysis and applications,8 (2020), 291-296. https://doi.org/10.21608/ejmaa.2020.312859.[11] e.r. love, some logarithm inequalities, math. gaz. 64 (1980), 55-57. https://doi.org/10.2307/3615890.[12] s. ramanujan, notebooks (2 volumes), tata institute of fundamental research, bombay, 1957.[13] h.j. ricardo, the equivalence of definitions of the natural logarithm function, coll. math. j. 53 (2022), 1-7. https: //doi.org/10.1080/07468342.2022.2039553.[14] w. rudin, real and complex analysis, mcgraw-hill, new york, 1986.[15] l. seidel, ueber eine darstellung des kreisbogens, des logarithmus und des elliptischen integrales erster art durchunendliche producte, j. reine angew. math. 73 (1871), 273-277.[16] n. thomas, a. chandran, k. namboothiri, on some logarithmic inequalities, adv. math.: sci. j. 10 (2021), 2483-2489. https://doi.org/10.37418/amsj.10.5.14.[17] a. vallejo, finding and improving bounds of real functions by thermodynamic arguments, 2023. preprint arxiv, 1-7. https://doi.org/10.48550/arxiv.2309.02479. https://doi.org/10.28924/ada/ma.5.12 https://doi.org/10.1007/978-1-4612-0879-2 https://doi.org/10.1007/978-1-4612-0879-2 https://doi.org/10.4169/math.mag.90.5.353 https://doi.org/10.1007/s11139-017-9961-y https://doi.org/10.1080/00029890.1987.12000678 https://doi.org/10.1080/00029890.1987.12000678 https://doi.org/10.21608/ejmaa.2020.312813 https://doi.org/10.1080/00029890.2019.1605799 https://doi.org/10.1080/00029890.2019.1605799 https://doi.org/10.21608/ejmaa.2020.312859 https://doi.org/10.2307/3615890 https://doi.org/10.1080/07468342.2022.2039553 https://doi.org/10.1080/07468342.2022.2039553 https://doi.org/10.37418/amsj.10.5.14 https://doi.org/10.48550/arxiv.2309.02479 1. introduction 2. results 2.1. a general result 2.2. specific results 3. applications 3.1. product expansions 3.2. inequalities 4. conclusion references ©2025 ada academica https://adac.eeeur. j. math. anal. 5 (2025) 7doi: 10.28924/ada/ma.5.7 a proposal of new extended symmetric cosine distribution christophe chesneau department of mathematics, lmno, university of caen-normandie, 14032 caen, france christophe.chesneau@gmail.com abstract. this article presents an extended symmetric version of the cosine distribution. the cor-responding probability density function is constructed by a special linear combination of cosine andsine functions. these trigonometric functions are activated by two adjustable parameters with the aimof generating modulable oscillatory shapes. this gives the new distribution greater flexibility andapplicability than the cosine distribution. its main characteristics are then examined, focusing on itsfunctional properties, the key moment measures and the generation of distributions with different sup-port. a new skewed version of the standard normal distribution is also derived. potential applicationsin various fields are discussed. two simulated data examples are presented and analyzed, showingthe superior performance of the new distribution compared to another two-parameter extended versionof the cosine distribution. 1. introduction symmetric distributions model variables whose values are equally likely to deviate from a centralvalue in both directions. well-known examples include the normal (gaussian), logistic, student(often symbolized by t), cauchy and laplace distributions, all of which are defined over the entirereal line, i.e., r. however, many practical situations require distributions defined over a boundedinterval. for example, physical measurements, such as lengths, weights and concentrations cannotbe negative and often have upper limits. in addition to measurements with positive values, otherexamples include proportions, cosine or sine values of an angle, correlations, normalized test scoresor normalized risk metrics, which are naturally bounded over an interval. in such cases, dependingon the exact context, it may be a good idea to retain the symmetry property while ensuring thatthe support of the distribution is bounded.several symmetric distributions over bounded intervals have been extensively studied in theliterature. these include the uniform distribution with support of the form [−υ, υ] with υ > 0,which assigns equal probability to all points within [−υ, υ]; it is the simplest form of boundedsymmetric distribution. we can also mention the truncated normal distribution, which attempts to received: 12 aug 2024. key words and phrases. mathematical analysis; symmetric distribution; cosine distribution; moment measures; skewednormal distribution; statistical modeling. 1 https://adac.ee https://doi.org/10.28924/ada/ma.5.7 https://orcid.org/0000-0002-1522-9292 eur. j. math. anal. 10.28924/ada/ma.5.7 2apply the properties of the normal distribution, including its symmetry, to a given interval. this isdone by truncating the tails beyond certain limits and renormalizing the probability density function(pdf) accordingly. see [8], [10] and [4]. another notable example is the cosine (c) distribution, whosepdf has a symmetric bell-shaped curve similar to that of the normal distribution, but is defined overa finite interval. let us develop it for the purposes of this article. the (standard) c distributionwith support [−1, 1] is defined by the following pdf: f◦(x) = 1 2 [1 + cos(πx)], x ∈ [−1, 1], (1) and f◦(x) = 0 for x 6∈ [−1, 1]. it is clear that f◦(−x) = f◦(x) for any x ∈ [−1, 1], justifyingthe symmetry of this pdf around x = 0. furthermore, since cos(−π) = −1, the curve of this pdfstarts at 0, increases smoothly for x ∈ [−1, 0), reaches its maximum value 1 at x = 0 and thendecreases at the same rate thanks to the symmetry. it thus resembles a "single cosine wave",which is very close to the symmetric bell-shaped curve of the pdf of the normal distribution. the cdistribution is the best known trigonometric distribution with bounded support. we also note thata scaled version of this distribution exists in the literature, with support [−π, π]. for more details,see [18], [12], [13], [16], and [20].the c distribution has been the subject of recent developments in distribution theory and practice.in particular, various characterisations of the c distribution were examined in [1], an asymmetricsystem based on it to produce a new skewed standard normal distribution was considered in [19], anatural two-parameter symmetric version of the c distribution was proposed in [2], original distri-butions based on the deformation of the cumulative distribution function (cdf) of the c distributionwere constructed in [7], and two different two-parameter asymmetric versions of the c distributionwere proposed in [5] and [6].despite the variety of existing symmetric distributions over bounded intervals, there is a contin-uous need for new candidates that offer greater flexibility and innovative modeling capabilities. inthis article, we propose a new extended symmetric c (esc) distribution. it is designed to retain thesymmetric properties of the c distribution, while introducing two additional parameters to enhanceits adaptability. in particular, one of the parameters activates an additional trigonometric termcapable of introducing oscillatory shapes in the curves of the corresponding pdf. from a statisti-cal point of view, this feature allows the pdf to accommodate nuanced patterns for a normalizedhistogram of the data, and thus may be preferable in some situations. we illustrate this claimusing the maximum likelihood (ml) estimation for the two parameters and the means of two sim-ulated data sets. we also show that it can be more accurate in the fitting exercise than anothertwo-parameter modified c distribution introduced in [5]. complementing this practical aspect, weexamine some understandable properties of the esc distribution that can be used beyond thepurposes of the article. these include its functional properties, the key moment measures, and https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 3the generation of distributions with different supports. a new skewed normal distribution is alsoderived and discussed.the rest of this article is structured as follows: section 2 presents a general distribution resultfrom which we derive the mathematical formulation of the esc distribution. section 3 derives itsmain properties and additional results. section 4 discusses possible applications and illustratesthe flexibility of the esc distribution with examples. finally, section 5 concludes the article andsuggests directions for future research. 2. the esc distribution 2.1. a general distribution result. the result below is theoretical; it shows how to choose someparameters of a special linear combination of trigonometric functions in order to satisfy the condi-tions of a valid pdf. the function considered is inspired by the pdf of the c distribution as definedin equation (1), with the aim of making it more flexible in a functional sense. theorem 2.1. for any a ∈ r, b ∈ r, and c ∈ r, let us set f (x ; a, b, c) = c [ 1 + a cos(πx){1− b[sin(πx)]2} ] , x ∈ [−1, 1], and f (x ; a, b, c) = 0 for any x 6∈ [−1, 1]. if the following conditions on a, b, and c are satisfied, then f (x ; a, b, c) is a valid pdf for a random variable with support [−1, 1]: c = 1 2 , a ∈ [−1, 1], b ∈ [0, 1]. proof. to define a valid pdf, we need to check that the following three conditions are met: • (condi): f (x ; a, b, c) is continuous on r, except possibly, for a finite number of values for x . • (condii): f (x ; a, b, c) ≥ 0 for any x ∈ r. • (condiii): ∫ +∞−∞ f (x ; a, b, c)dx = 1.the condition (condi) is immediate; f (x ; a, b, c) is a linear combination of continuous trigonometricfunctions, it is continuous on r, except possibly, at the extremes, i.e., x = −1 and x = 1.let us now investigate the condition (condii). this condition is obvious for any x 6∈ [−1, 1], since f (x ; a, b, c) = 0. let us concentrate on the case x ∈ [−1, 1]. using c > 0, and the standardtriangle inequality, i.e., |u − v | ≥ |u| − |v | for any u ∈ r and v ∈ r, we have f (x ; a, b, c) ≥ c { 1− |a|| cos(πx)||1− b[sin(πx)]2| } . since b ∈ [0, 1] and sin(πx) ∈ [0, 1], we have |1 − b[sin(πx)]2| = 1 − b[sin(πx)]2 ∈ [0, 1]. thiscombined with | cos(πx)| ≤ 1 and a ∈ [−1, 1] gives f (x ; a, b, c) ≥ c [1− |a|| cos(πx)|] ≥ c(1− |a|) ≥ 0. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 4the condition (condiii) is now examined. since c = 1/2, using standard integral rules and sin(−π) = sin(π) = 0, we have∫ +∞ −∞ f (x ; a, b, c)dx = ∫ 1 −1 f (x ; a, b, c)dx = c ∫ 1 −1 [ 1 + a cos(πx){1− b[sin(πx)]2} ] dx = c ∫ 1 −1 { 1 + a cos(πx)− ab cos(πx)[sin(πx)]2 } dx = c { x + a π sin(πx)− ab 3π [sin(πx)]3 }∣∣∣∣x=+1 x=−1 = c { 1 + a π sin(π)− ab 3π [sin(π)]3 − (−1)− a π sin(−π) + ab 3π [sin(−π)]3 } = c { 1 + a π × 0− ab 2π × 0 + 1− a π × 0 + ab 2π × 0 } = 2c = 1. this ends the proof. � this theorem is the key theoretical result for deriving the esc distribution. it is described inthe next subsection. 2.2. definition of the esc distribution. based on theorem 2.1, we define the esc distribution bythe following pdf: f∗(x ; a, b) = 1 2 [ 1 + a cos(πx){1− b[sin(πx)]2} ] , x ∈ [−1, 1], and f∗(x ; a, b) = 0 for any x 6∈ [−1, 1], where a ∈ [−1, 1] and b ∈ [0, 1]. we say "symmetric"because, for any x ∈ r, f∗(x ; a, b) satisfies f∗(−x ; a, b) = f∗(x ; a, b); it is immediate for any x 6∈ [−1, 1] since f∗(x ; a, b) = 0, and, for any x ∈ [−1, 1], this follows from the facts that cos(−πx) = cos(πx) and [sin(−πx)]2 = [− sin(πx)]2 = [sin(πx)]2.obviously, for a = 1 and b = 0, we have f∗(x ; a, b) = f◦(x) as given in equation (1); the escdistribution is thus reduced to the c distribution. more generally, note that, for any x ∈ [−1, 1],we can write f∗(x ; a, b) = f◦(x) + g(x ; a, b), where g(x ; a, b) = 1 2 cos(πx) { a − 1− ab[sin(πx)]2 } , which can be thought of as a two-parameter perturbative trigonometric function of f◦(x) in thiscontext.figure 1 shows the curves of the pdf of the esc distribution under different but complementaryconfigurations: when a is fixed at a positive value and b varies, when a is fixed at a negative valueand b varies, when b is fixed and a varies, and a summary of the different plots. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 5 −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x p d f a = 1 b = 0.001 a = 1 b = 0.2 a = 1 b = 0.4 a = 1 b = 0.7 a = 1 b = 1 −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x p d f a = − 1 b = 0.001 a = − 1 b = 0.2 a = − 1 b = 0.4 a = − 1 b = 0.7 a = − 1 b = 1 (i) (ii) −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .5 1 .0 1 .5 x p d f a = 1 b = 0.2 a = 0.5 b = 0.2 a = 0.1 b = 0.2 a = − 0.5 b = 0.2 a = − 1 b = 0.2 −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .5 1 .0 1 .5 x p d f a = 1 b = 0.001 a = 0 b = 0.6 a = − 1 b = 0.001 a = 0.6 b = 0.2 a = − 1 b = 1 (iii) (iv) figure 1. curves of the pdf of the esc distribution (i) when a = 1 and b varies,(ii) when a = −1 and b varies, (iii) when b = 0.2 and a varies, and (iv) a summaryof the different plots https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 6in this figure, we can see that all the curves are symmetrical with respect to the axis x = 0, witha general bell or inverted bell shape. we distinguish typical smooth curves and curves with smalloscillatory shapes, mainly caused by the additional sine term in the pdf. compared to the curves ofthe pdf of the c distribution, these nuanced shapes can be a plus for the fitting perspective whenthe histogram shows some complex symmetric features. this claim will be made more concrete inthe application section, i.e., section 5. 3. functions and properties the main functions of the esc distribution, moment measures and some transformed esc dis-tributions are determined in this section. 3.1. the cdf. an important function, the cdf of the esc distribution, is shown in the result below. proposition 3.1. the cdf of the esc distribution is given as f∗(x ; a, b) = 1 2 [ 1 + x + a π sin(πx) { 1− b 3 [sin(πx)]2 }] , x ∈ [−1, 1], f∗(x ; a, b) = 0 for any x < −1 and f∗(x ; a, b) = 1 for any x > 1. proof. since the support of the esc distribution is [−1, 1], we immediately know that f∗(x ; a, b) = 0 for any x < −1 and f∗(x ; a, b) = 1 for any x > 1. for any x ∈ [−1, 1], we have f∗(x ; a, b) = ∫ x −∞ f∗(t; a, b)dt = ∫ x −1 f∗(t; a, b)dt = 1 2 ∫ x −1 [ 1 + a cos(πt){1− b[sin(πt)]2} ] dt = 1 2 ∫ x −1 { 1 + a cos(πt)− ab cos(πt)[sin(πt)]2 } dt = 1 2 { t + a π sin(πt)− ab 3π [sin(πt)]3 }∣∣∣∣t=x t=−1 = 1 2 { x + a π sin(πx)− ab 3π [sin(πx)]3 − (−1)− a π sin(−π) + ab 3π [sin(−π)]3 } = 1 2 { 1 + x + a π sin(πx)− ab 3π [sin(πx)]3 } = 1 2 [ 1 + x + a π sin(πx) { 1− b 3 [sin(πx)]2 }] . the specified expression is found. � clearly, we have f∗(0; a, b) = 1 2 [ 1 + 0 + a π sin(π × 0) { 1− b 3 [sin(π × 0)]2 }] = 1 2 , which means that the median of the esc distribution is 0 as expected, since the symmetry pointof the corresponding pdf is x = 0. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 7figure 2 represents the curves of the obtained cdf for different values of a and b (those consideredin the subfigure (iv) of figure 1). −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 x c d f a = 1 b = 0.001 a = 0 b = 0.6 a = − 1 b = 0.001 a = 0.6 b = 0.2 a = − 1 b = 1 figure 2. curves of the cdf of the esc distribution for different values of a and b this figure confirms the flexibility of the esc distribution, as various concave and convex in-creasing curves are observed. however, the "degree of distortion" is limited; in a sense, some areasof the rectangle [−1, 1]× [0, 1] cannot be reached. 3.2. quantile function. the quantile function (qf) of the esc distribution is defined as the inversefunction of f∗(x ; a, b), i.e., f−1∗ (x ; a, b). let us denote it by q∗(p; a, b), for p ∈ [0, 1]. due to thetrigonometric complexity of f∗(x ; a, b), it has no closed form expression. however, we can determineit numerically, for fixed values of a and b. table 1 illustrates this claim, with also different valuesof p. table 1. some values of q∗(p; a, b) for different values of a, b, and p. p → 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 a = 1, b = 0.001 -1.00 -0.48 -0.33 -0.21 -0.10 0.00 0.10 0.21 0.33 0.48 1.00 a = 0, b = 0.6 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 a = −1, b = 0.001 -1.00 -0.90 -0.79 -0.67 -0.52 0.00 0.52 0.67 0.79 0.90 1.00 a = 0.6, b = 0.2 -1.00 -0.64 -0.43 -0.26 -0.13 0.00 0.13 0.26 0.43 0.64 1.00 a = −1, b = 1 -1.00 -0.90 -0.78 -0.61 -0.41 0.00 0.41 0.61 0.78 0.90 1.00 https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 8for example, for a = 0.6, b = 0.2 and p = 0.6 we have q∗(p; a, b) = 0.13. for p = 0.5,we refind that the median of the esc distribution is 0. this table shows that a computationalquantile analysis of the esc distribution is possible, including the determination of various quantilemeasures (see [11] and [14]). 3.3. hazard rate function. complementary to the cdf, we can determine the survival function of theesc distribution. it is obtained as s∗(x ; a, b) = 1− f∗(x ; a, b), so that s∗(x ; a, b) = 1 2 [ 1− x − a π sin(πx) { 1− b 3 [sin(πx)]2 }] , x ∈ [−1, 1], s∗(x ; a, b) = 1 for any x < −1 and s∗(x ; a, b) = 0 for any x > 1.the hazard rate function (hrf) of the esc distribution is obtained by the following ratio formula: h∗(x ; a, b) = f∗(x ; a, b)/s∗(x ; a, b), which can be expressed as h∗(x ; a, b) = 1 + a cos(πx){1− b[sin(πx)]2} 1− x − (a/π) sin(πx) {1− (b/3)[sin(πx)]2} , x ∈ [−1, 1], and h∗(x ; a, b) = 0 for any x 6∈ [−1, 1]. the shape behavior of this function is informative aboutthe flexibility of the esc distribution. with this in mind, figure 3 shows the curves of this hrf fordifferent values of a and b. −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 x c d f a = 1 b = 0.001 a = 0 b = 0.6 a = − 1 b = 0.001 a = 0.6 b = 0.2 a = − 1 b = 1 figure 3. curves of the hrf of the esc distribution for different values of a and b in this figure, we distinguish smooth increasing curves as well as oscillating curves, which arecharacteristics of the trigonometric type distributions. this shows that the esc distribution is ableto model both simple and complex statistical situations involving data in [−1, 1]. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 93.4. moment analysis. moment analysis is particularly interesting for distributions with boundedsupports, as most of the derived moment measures and functions are automatically well defined. 3.4.1. main moment results. we start this analysis with the mean and variance associated withthe esc distribution in the result below. proposition 3.2. let x be a random variable with the esc distribution. then we have e(x) = 0 and v(x) = 1 3 + 2a π2 ( 2b 9 − 1 ) , where e and v denote the mean and variance operators, respectively. proof. the symmetry of the esc distribution (around 0) implies that x and −x follow the escdistribution (with the same parameters). we thus have e(x) = e(−x) = −e(x), so that e(x) = 0. on the other hand, we have v(x) = e(x2)− [e(x)]2 = e(x2), where e(x2) = ∫ +∞ −∞ x2f∗(x ; a, b)dx = ∫ 1 −1 x2f∗(x ; a, b)dx = 1 2 ∫ 1 −1 x2 [ 1 + a cos(πx){1− b[sin(πx)]2} ] dx = 1 2 ∫ 1 −1 x2dx + a 2 ∫ 1 −1 x2 cos(πx){1− b[sin(πx)]2}dx = 1 3 + a 2 ∫ 1 −1 x2 cos(πx){1− b[sin(πx)]2}dx. for the remaining integral term, using two integrations by parts in a row and [sin(πx)]2 = 1 − [cos(πx)]2, we get∫ 1 −1 x2 cos(πx){1− b[sin(πx)]2}dx = [ x2 1 π sin(πx) { 1− b 3 [sin(πx)]2 }]∣∣∣∣x=1 x=−1 − 2 π ∫ 1 −1 x sin(πx) { 1− b 3 [sin(πx)]2 } dx = 0− 2 π ∫ 1 −1 x sin(πx) { 1− b 3 + b 3 [cos(πx)]2 } dx = − 2 π { [ −x 1 π cos(πx) { 1− b 3 + b 9 [cos(πx)]2 }]∣∣∣∣x=1 x=−1 + 1 π ∫ 1 −1 cos(πx) { 1− b 3 + b 9 [cos(πx)]2 } dx } = − 2 π { 2 π ( 1− 2b 9 ) + ( 1− b 3 ) 1 π ∫ 1 −1 cos(πx)dx + b 9π ∫ 1 −1 [cos(πx)]3dx } = − 2 π { 2 π ( 1− 2b 9 ) + 0 + b 9π ∫ 1 −1 cos(πx){1− [sin(πx)]2}dx } https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 10 = − 2 π { 2 π ( 1− 2b 9 ) + b 9π [ 1 π sin(πx) { 1− 1 3 [sin(πx)]2 }]∣∣∣∣x=1 x=−1 } = − 2 π { 2 π ( 1− 2b 9 ) + 0 } = 8b 9π2 − 4 π2 = 4 π2 ( 2b 9 − 1 ) . hence, we have v(x) = 1 3 + a 2 × 4 π2 ( 2b 9 − 1 ) = 1 3 + 2a π2 ( 2b 9 − 1 ) . this concludes the proof. � now let us complete this result. let x be a random variable with the esc distribution. then,for any positive integer n, since the esc distribution is symmetric (around 0), we have e(x2n+1) = 0. more generally, for any odd function p(x), we have e[p(x)] = 0. in particular, we rediscover e(x) = 0, and we can mention that e(x3) = 0. on the other hand, using several integraltechniques and trigonometric formulas, we find that e(x4) = 1 5 + 4a π4 [ 2(3π2 − 20)b 27 − π2 + 6 ] . since e(x) = 0, the skewness coefficient of x is equal to γ(x) = 1 [e(x2)]3/2 e(x3) = 0. the esc distribution therefore has a neutral skewness. furthermore, the kurtosis coefficient of xcan be calculated as β(x) = 1 [e(x2)]2 e(x4) = 81π4 + 60a [ (6π2 − 40)b − 27(π2 − 6) ] 5 [2a(2b − 9) + 3π2]2 . since the above moment measures are not easy to handle, especially for β(x), we propose anumerical work. table 2 gives some values of the moment measures for different values of a and b. table 2. some values of moment measures of a random variable x with the escdistribution for different values of a and b. e(x) e(x2) e(x3) e(x4) γ(x) β(x) a = 1, b = 0.001 0.000 0.131 0.000 0.041 0.000 2.406 a = 0, b = 0.6 0.000 0.333 0.000 0.200 0.000 1.800 a = −1, b = 0.001 0.000 0.536 0.000 0.359 0.000 1.249 a = 0.6, b = 0.2 0.000 0.217 0.000 0.108 0.000 2.294 a = −1, b = 1 0.000 0.491 0.000 0.330 0.000 1.368 https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 11for the values considered, we have β(x) < 3, indicating that the esc distribution is mainlyplatykurtic. thus, for the same variance, the distribution is relatively "flattened".we now determine the incomplete mean associated with the esc distribution. proposition 3.3. let x be a random variable with the esc distribution. then, for any x ∈ [−1, 1], we have e(x1{x≤x}) = 1 4 (x2 − 1) + a 2π2 [ πx sin(πx) { 1− b 3 [sin(πx)]2 } + cos(πx) { 1− b 3 + b 9 [cos(πx)]2 } + 1− 2b 9 ] , where 1{.} denotes the indicator operator. proof. for any x ∈ [−1, 1], we have e(x1{x≤x}) = ∫ x −∞ tf∗(t; a, b)dt = ∫ x −1 tf∗(t; a, b)dt = 1 2 ∫ x −1 t [ 1 + a cos(πt){1− b[sin(πt)]2} ] dt = 1 2 ∫ x −1 tdt + a 2 ∫ x −1 t cos(πt){1− b[sin(πt)]2}dt = 1 4 (x2 − 1) + a 2 ∫ x −1 t cos(πt){1− b[sin(πt)]2}dt. for the remaining integral term, using an integration by parts and [sin(πx)]2 = 1− [cos(πx)]2, weget ∫ x −1 t cos(πt){1− b[sin(πt)]2}dt = [ t 1 π sin(πt) { 1− b 3 [sin(πt)]2 }]∣∣∣∣t=x t=−1 − 1 π ∫ x −1 sin(πt) { 1− b 3 [sin(πt)]2 } dt = 1 π x sin(πx) { 1− b 3 [sin(πx)]2 } − 1 π ∫ x −1 sin(πt) { 1− b 3 + b 3 [cos(πt)]2 } dt = 1 π x sin(πx) { 1− b 3 [sin(πx)]2 } − 1 π [ − 1 π cos(πt) { 1− b 3 + b 9 [cos(πt)]2 }]∣∣∣∣t=x t=−1 = 1 π x sin(πx) { 1− b 3 [sin(πx)]2 } + 1 π2 cos(πx) { 1− b 3 + b 9 [cos(πx)]2 } + 1 π2 ( 1− 2b 9 ) . hence, we have e(x1{x≤x}) = 1 4 (x2 − 1) + a 2π2 [ πx sin(πx) { 1− b 3 [sin(πx)]2 } + cos(πx) { 1− b 3 + b 9 [cos(πx)]2 } + 1− 2b 9 ] . https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 12this ends the proof. � clearly, if we take x = 1, we refind that e(x1{x≤1}) = e(x) = 0. more generally, theexpression of this incomplete mean can be useful in assessing the expected value of x within acertain range, which is crucial in areas such as reliability. it also helps in the calculation andinterpretation of other related functions, such as the mean residual life function. the incompletemean also plays a theoretical role in some characterization results (see [17]).we now determine the mean deviation associated with the esc distribution. proposition 3.4. let x be a random variable with the esc distribution. then we have e(|x|) = 1 2 + 2a π2 ( 2b 9 − 1 ) . proof. since the pdf f∗(x ; a, b) is symmetric (around 0), we have e(|x|) = ∫ +∞ −∞ |x |f∗(x ; a, b)dx = ∫ 1 −1 |x |f∗(x ; a, b)dx = 2 ∫ 1 0 xf∗(x ; a, b)dx = ∫ 1 0 x [ 1 + a cos(πx){1− b[sin(πx)]2} ] dx = ∫ 1 0 xdx + a ∫ 1 0 x cos(πx){1− b[sin(πx)]2}dx = 1 2 + a ∫ 1 0 x cos(πx){1− b[sin(πx)]2}dx. for the remaining integral term, using an integration by parts and [sin(πx)]2 = 1− [cos(πx)]2, weget ∫ 1 0 x cos(πx){1− b[sin(πx)]2}dx = [ x 1 π sin(πx) { 1− b 3 [sin(πx)]2 }]∣∣∣∣x=1 x=0 − 1 π ∫ 1 0 sin(πx) { 1− b 3 [sin(πx)]2 } dx = 0− 1 π ∫ 1 0 sin(πx) { 1− b 3 + b 3 [cos(πx)]2 } dx = − 1 π [ − 1 π cos(πx) { 1− b 3 + b 9 [cos(πx)]2 }]∣∣∣∣x=1 x=0 = 1 π2 ( −1 + 2b 9 − 1 + 2b 9 ) = 2 π2 ( 2b 9 − 1 ) . we therefore have e(|x|) = 1 2 + 2a π2 ( 2b 9 − 1 ) . this concludes the proof. � https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 13as the esc distribution is symmetric, the associated mean deviation of x can be considered asa measure of dispersion of the same unit as the modeled variable. it is comparable to the standarddeviation of x , which is defined as the square of the corresponding variance. we notice also thefollowing formula: v(x)− e(|x|) = 1 3 − 1 2 = − 1 6 , which has the property of being independent of a and b.the moment generating function of the esc distribution can be determined with mathematicalefforts. this is developed in the result below. proposition 3.5. let x be a random variable with the esc distribution. then the moment generating function of x is given as ϕ(t; a, b) = e [ etx ] , and can be expressed as ϕ(t; a, b) = sinh(t) { 1 t − at[π2(9− 2b) + t2] (t2 + π2)(t2 + 9π2) } , t ∈ r, where sinh(t) = (et − e−t)/2. proof. we have ϕ(t; a, b) = e [ etx ] = ∫ +∞ −∞ etx f∗(x ; a, b)dx = ∫ 1 −1 etx f∗(x ; a, b)dx = 1 2 ∫ 1 −1 etx [ 1 + a cos(πx){1− b[sin(πx)]2} ] dx = 1 2 ∫ 1 −1 etxdx + a 2 ∫ 1 −1 cos(πx)etxdx − ab 2 ∫ 1 −1 cos(πx)[sin(πx)]2etxdx. the last two integral terms require special treatment. it follows from [9, formula number 2.663.3]that ∫ 1 −1 cos(πx)etxdx = − 2t sinh(t) t2 + π2 . on the other hand, after a work on the formula in [9, formula number 2.664.2], we obtain∫ 1 −1 cos(πx)[sin(πx)]2etxdx = − 4π2t sinh(t) (t2 + 9π2)(t2 + π2) . we therefore have ϕ(t; a, b) = 1 2 [ 1 t etx ]∣∣∣∣x=1 x=−1 − at sinh(t) t2 + π2 + 2abπ2t sinh(t) (t2 + 9π2)(t2 + π2) = sinh(t) t − at sinh(t) t2 + π2 + 2abπ2t sinh(t) (t2 + 9π2)(t2 + π2) = sinh(t) { 1 t − at[π2(9− 2b) + t2] (t2 + π2)(t2 + 9π2) } . the stated formula is established, concluding the proof. � https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 14we rediscover the moment generating function associated with the c distribution by taking a = 1and b = 0, i.e, ϕ◦(t) = π2 sinh(t) t(t2 + π2) , t ∈ r. see [1] (with the following configuration: µ = 0 and σ = 1). the characteristic function associatedwith the esc distribution follows from the same mathematical development. by introducing i , theimaginary unit solution of the equation x2 + 1 = 0, since sinh(i t) = i sin(t), we have φ(t; a, b) = ”ϕ(i t; a, b)” = sinh(i t) { 1 i t − ait[π2(9− 2b) + (i t)2] ((i t)2 + π2)((i t)2 + 9π2) } = sin(t) { 1 t − at(2π2b + t2 − 9π2) (t2 − 9π2)(t2 − π2) } , t ∈ r. this function also fully defines the esc distribution. in fact, we have the following formulalinking the pdf and the characteristic function: f∗(x ; a, b) = 1 2π ∫ +∞ −∞ φ(t; a, b)e−itxdt, x ∈ [−1, 1], which can be written as, for any x ∈ [−1, 1], 1 2 [ 1 + a cos(πx){1− b[sin(πx)]2} ] = 1 2π ∫ +∞ −∞ sin(t) { 1 t − at(2π2b + t2 − 9π2) (t2 − 9π2)(t2 − π2) } e−itxdt or, equivalently, 1 + a cos(πx){1− b[sin(πx)]2} = 1 π ∫ +∞ −∞ sin(t) { 1 t − at(2π2b + t2 − 9π2) (t2 − 9π2)(t2 − π2) } e−itxdt. this two-parameter formula can be of some mathematical interest, especially in harmonic analysisdealing with the fourier transform of various trigonometric functions.we can also note that, since x is symmetric (around 0), we have ϕ(t; a, b) = ϕ(−t; a, b), aproperty we can also check using the expression we found. as a direct consequence, for any t ∈ r,we have e[cosh(tx)] = 1 2 [ϕ(t; a, b) + ϕ(−t; a, b)] = ϕ(t; a, b) = sinh(t) { 1 t − at[π2(9− 2b) + t2] (t2 + π2)(t2 + 9π2) } , recalling that cosh(x) = (ex + e−x)/2.we end this part by investigating the moments of the power of the sine transformed random vari-able with the esc distribution. these moments have the property of being very simple, dependenton an adjustable positive integer, and independent of a and b. proposition 3.6. let n be a positive integer, and x be a random variable with the esc distribution. then we have e { [sin(πx)]2n } = (2n)! 22n(n!)2 https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 15 and e { [sin(πx)]2n+1 } = 0. proof. we have e { [sin(πx)]2n } = ∫ +∞ −∞ [sin(πx)]2nf∗(x ; a, b)dx = ∫ 1 −1 [sin(πx)]2nf∗(x ; a, b)dx = 1 2 ∫ 1 −1 [sin(πx)]2n [ 1 + a cos(πx){1− b[sin(πx)]2} ] dx = 1 2 ∫ 1 −1 [sin(πx)]2ndx + a 2 ∫ 1 −1 cos(πx)[sin(πx)]2n{1− b[sin(πx)]2}dx. after a work on the formula in [9, formula number 2.513.1], we find that∫ 1 −1 [sin(πx)]2ndx = (2n)! 22n−1(n!)2 . (2) we therefore have e { [sin(πx)]2n } = (2n)! 22n(n!)2 + a 2 [ 1 π [sin(πx)]2n+1 { 1 2n + 1 − b 2n + 3 [sin(πx)]2 }]∣∣∣∣x=1 x=−1 = (2n)! 22n(n!)2 + 0 = (2n)! 22n(n!)2 . on the other hand, since p(x) = [sin(πx)]2n+1 is an odd function, we have already discussed that e { [sin(πx)]2n+1 } = e [p(x)] = 0. the desired results are obtained. � 3.4.2. secondary moment results. during our investigations of the esc distribution, we found othermoment results of potential interest. we present them below, considering a random variable x withthe esc distribution. the mathematical details are omitted for reasons of space. • a special moment formula is as follows: e [ 1√ x + 1 ] = 1 12 √ 2 [ 3a(b − 4)c(2)− ab √ 3c[2 √ 3] + 24 ] , where c(x) denotes the fresnel c integral defined by c(x) = ∫ x0 cos(πt2/2)dt (see [9,section 8.25]). this integral is implemented in most mathematical software, such as r withthe package entitled pracma. • another special moment formula is as follows: e [ e−|x| ] = e[aπ2(9− 2b) + 1 + a + 10π2 + 9π4]− 1 + a + π2(9a − 10− 2ab)− 9π4 e(1 + π2)(1 + 9π2) . more succinct moment formulas are given below, some of which may be useful for varioustheoretical or practical applications involving the esc distribution. • we have e { 1 1 + [cos(πx)]2 } = 1√ 2 . https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 16 • we have e { 1 1 + [sin(πx)]2 } = 1√ 2 . the two results for the moments above are therefore the same. • we have e { 1 1 + [tan(πx)]2 } = 1 2 . • we have e {cos[π|x|]} = a 8 (4− b). • we have e {sin[π|x|]} = 2 π . • we have e [| cos(πx)|] = 2 π . • we have e [| sin(πx)|] = 2 π .the three results for the moments above are therefore the same. • we have e [√ 1 + cos(πx) ] = − 2 √ 2 105π [a(8b − 35)− 105]. • we have e { cos(πx) 1 + [sin(πx)]2 } = a {√ 2− 1 + b [√ 2− 3 2 ]} . 3.5. some derived distributions. 3.5.1. transformed esc distributions. since the esc distribution is new in the literature, it maybe interesting to derive transformed esc distributions with different properties, such as differentsupports. for this purpose, we consider a random variable x with the esc distribution andinvestigate the following transformed random variable: y = m(x), where m(x) is a well-defined function on [−1, 1] or (−1, 1). • for m(x) = πx , we get y = πx , which is of support [−π, π]. based on proposition 3.1, yhas the following cdf: f∧(x ; a, b) = f∗ ( x π ; a, b ) = 1 2π [ π + x + a sin(x) { 1− b 3 [sin(x)]2 }] , x ∈ [−π, π], f∧(x ; a, b) = 0 for any x < −π and f∧(x ; a, b) = 1 for any x > π (still with a ∈ [−1, 1]and b ∈ [0, 1]). https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 17 • for m(x) = |x |, we get y = |x|, which is of support [0, 1]. based on proposition 3.1, yhas the following cdf: f�(x ; a, b) = 2f∗(x ; a, b)− 1 = x + a π sin(πx) { 1− b 3 [sin(πx)]2 } , x ∈ [0, 1], f�(x ; a, b) = 0 for any x < 0 and f�(x ; a, b) = 1 for any x > 1. • for m(x) = (x + 1)/2, we get y = (x + 1)/2, which is of support [0, 1]. based onproposition 3.1, y has the following cdf: f?(x ; a, b) = f∗(2x − 1; a, b) = x − a 2π sin(πx) { 1− b 3 [sin(πx)]2 } , x ∈ [0, 1], f?(x ; a, b) = 0 for any x < 0 and f?(x ; a, b) = 1 for any x > 1. this and the previous cdfcan be seen as modified versions of the cdf of the uniform distribution over [0, 1]. • for m(x) = tan[(π/2)x ], we get y = tan[(π/2)x], which is of support r. based onproposition 3.1, y has the following cdf: f�(x ; a, b) = f∗ [ 2 π arctan(x); a, b ] = 1 2 { 1 + 2 π arctan(x) + a π sin[2 arctan(x)] [ 1− b 3 {sin[2 arctan(x)]}2 ]} = 1 2 { 1 + 2 π arctan(x) + 2ax π(1 + x2) [ 1− 4bx2 3(1 + x2)2 ]} , x ∈ r. this cdf can be seen as a modified cdf of the standard cauchy distribution. • for m(x) = sin[(π/2)x ], we get y = sin[(π/2)x], which is of support [−1, 1]. based onproposition 3.1, y has the following cdf: f∨(x ; a, b) = f∗ [ 2 π arcsin(x); a, b ] = 1 2 { 1 + 2 π arcsin(x) + a π sin[2 arcsin(x)] [ 1− b 3 {sin[2 arcsin(x)]}2 ]} = 1 2 { 1 + 2 π arcsin(x) + 2a π x √ 1− x2 [ 1− 4b 3 x2(1− x2) ]} , x ∈ [−1, 1], f∨(x ; a, b) = 0 for any x < −1 and f∨(x ; a, b) = 1 for any x > 1. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 18 • for m(x) = (1 − x)/(1 + x), we get y = (1 − x)/(1 + x), which is of support (0,+∞).based on proposition 3.1, y has the following cdf: f/(x ; a, b) = 1− f∗ ( 1− x 1 + x ; a, b ) = 1 2 { 1− 1− x 1 + x − a π sin [ π(1− x) 1 + x ][ 1− b 3 { sin [ π(1− x) 1 + x ]}2]} = x x + 1 − a 2π sin ( 2πx 1 + x ){ 1− b 3 [ sin ( 2πx 1 + x )]2} , x > 0 and f/(x ; a, b) = 0 for any x ≤ 0. this cdf can be seen as a modified cdf of the standardlomax distribution, i.e., defined by q(x) = x/(x + 1) = 1− (x + 1)−1. these are only examples of new distributions derived from the esc distribution. others may beconsidered depending on the objectives. 3.5.2. a new skewed normal distribution. in [19], the c distribution is used to create a trigonometricskewed version of the standard normal distribution by applying the azzalini scheme introduced in [3].we can follow the same idea but using the esc distribution, also for more flexibility thanks to thetwo additional parameters. to be precise, let us consider the pdf of the standard normal distributiondefined by g(x) = 1√ 2π e−x 2/2, x ∈ r. we then construct a new trigonometric skewed version of this pdf by considering the followingfunction: f4(x ; a, b, λ) = 2g(x)f∗(λx ; a, b), x ∈ r, where λ > 0 in an additional skew parameter, and f∗(x ; a, b) is still the cdf of the esc distributiongiven in proposition 3.1. thus defined, f4(x ; a, b, λ) is a valid pdf. it can be expressed as f4(x ; a, b, λ) = 1√ 2π e−x 2/2 [ 1 + λx + a π sin(λπx) { 1− b 3 [sin(λπx)]2 }] , x ∈ ( − 1 λ , 1 λ ) , completed by f4(x ; a, b, λ) = √ 2 π e−x 2/2, x ≥ 1 λ and f4(x ; a, b, λ) = 0 for any x ≤ −1/λ.figure 4 shows the curves of this pdf for different values of a, b and λ. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 19 −2 −1 0 1 2 3 0 .0 0 .2 0 .4 0 .6 0 .8 x p d f a = 1 b = 0.001 λ = 1 a = − 1 b = 1 λ = 1.5 a = − 0.5 b = 0.4 λ = 0.5 a = 0.4 b = 0.7 λ = 12 a = − 1 b = 0.1 λ = 1 figure 4. curves of the pdf of the derived skewed normal distribution for differentvalues of a, b and λ this figure illustrates the complexity of the deformations applied to the pdf of the standardnormal distribution based on our trigonometric skew scheme. in particular, we see some abruptchanges and pronounced peaks in the mode, which are quite rare properties in the family of skewednormal distributions.the derived skewed normal distribution may be of interest for the analysis of versatile skeweddata. it is more adaptable than the one developed in [19], mainly due to the presence of theadditional parameter b. we leave this practical aspect to future work.of course, the same methodology can be applied to other basic symmetric distributions around 0 than the standard normal. for example, we can consider the pdf of the logistic, student, cauchyand laplace distributions for g(x), which opens up new research directions. 4. applications this section looks at some statistical aspects of the esc distribution, mainly when it is goingto be used to analyze data. 4.1. estimation method. in practice, a variable with values in [−1, 1] can potentially be modeledby a random variable x with the esc distribution. to get a precise idea of this probabilisticassumption, we assume that the parameters a and b as unknown, and try to estimate them efficientlybased on the available information of x that make up the data. a wide range of estimation methods https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 20can be used to do this. one of the most common and efficient is the ml estimation method. let usformalize this in the context of the esc distribution below.suppose we have n data, observations of x , classically denoted as x1, . . . , xn. then the mlestimates of a and b are given as (â, b̂) = argmax(a,b)∈[−1,1]×[0,1] `(a, b), where `(a, b) = n∑ i=1 ln [f∗(xi ; a, b)] = −n ln(2) + n∑ i=1 ln [ 1 + a cos(πxi){1− b[sin(πxi)]2} ] . this procedure of maximization can be carried out with any mathematical software, such as r withthe function nlminb. once â and b̂ are obtained, the corresponding estimated pdf is given by f̂∗(x) = f∗(x ; â, b̂) = 1 2 [ 1 + â cos(πx){1− b̂[sin(πx)]2} ] , x ∈ [−1, 1]. (3) using this function, we can visually assess the fit of the esc distribution to the data. morespecifically, we have an acceptable result if the corresponding normalized histogram of the datahas a shape that is well fitted by the curve of f̂∗(x) for x ∈ [−1, 1].the goodness of fit of different distributions can be compared using well-established mathemat-ical tools, such as the akaike information criterion (aic) and the bayesian information criterion(bic). in the context of the esc distribution, these criteria are defined as aic = 2[k − `(â, b̂)], bic = k ln(n)− 2`(â, b̂), respectively, where k is the number of unknown parameters, i.e., k = 2.as the main competitor of the esc distribution, we consider another variant of the c distribution:the ac distribution introduced in [5]. it is defined by the following two-parameter pdf: f†(x ;α, β) = β(π2 + β2) 2[π2 + (1− α)β2] sinh(β) [1 + α cos(πx)]e βx , x ∈ [−1, 1], (4) and f†(x ;α, β) = 0 for all x 6∈ [−1, 1], where α ∈ [−1, 1] and β ∈ r. unlike the esc distribution,the ac distribution is not designed to be exclusively symmetric. this makes it a challenger ofthe esc distribution when dealing with "possibly symmetric" data, especially when the data arenot "perfectly symmetric". for this distribution, we can also use the ml estimation method toestimate the parameters α and β, say α̂ and β̂, respectively, determine the estimated pdf as f̂†(x) = f†(x ; α̂, β̂), and compute the aic and bic for comparison purposes.for more general information on the ml estimation method, see [4].in the remaining parts of this section, for illustrative purposes, we test the ml estimation methodon the esc distribution using two different examples of simulated data, i.e., data that can be https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 21obtained in credible scenarios (not real data). in particular, we compare the fit obtained with thatof the ac distribution. 4.2. simulated example 1. in this simulated example, we look at the normalized daily air qualityindex (aqi) in a particular urban area over a period of 57 days. each measurement of the aqiis contained in the range [−1, 1], where a value close to 0 indicates moderate air quality, whilevalues close to −1 represent poor air quality and values close to 1 represent good air quality.conceptually, the data capture the natural variability of air quality due to various environmentalfactors, such as traffic patterns, weather conditions and industrial activities.the data are as follows: 0.877, -0.803, 0.623, -0.132, 0.063, 0.282, -0.725, 0.452, -0.290, 0.114,-0.029, -0.520, 0.711, -0.775, 0.906, 0.106, -0.506, 0.194, -0.744, 0.549, -0.339, 0.186, -0.263,0.109, -0.888, 0.739, -0.591, 0.731, -0.253, 0.533, -0.037, 0.307, -0.205, 0.122, -0.106, 0.858,-0.760, 0.586, -0.723, 0.280, -0.496, 0.037, -0.236, 0.159, -0.253, 0.105, -0.901, 0.709, -0.590,0.725, -0.298, 0.432, -0.052, 0.347, -0.194, 0.297, -0.108.to summarize these data, the minimum is −0.926, the first quartile is −0.194, the median is 0.002, the mean is −0.005122, the third quartile is 0.199, and the maximum is 0.920.this information indicates some symmetry in the distribution of the data. the main quantilefeatures of the data are illustrated by a boxplot in figure 5. −0.5 0.0 0.5 d a ta figure 5. boxplot of the data of the simulated example 1 this boxplot confirms a certain symmetry of the data, centered around 0, with no outliers. theesc distribution may therefore be appropriate for their analysis.using the ml estimation method, we obtain â = 0.5688681 and b̂ = 1. from these estimates,we derive the estimated pdf f̂∗(x). note that the high value of b̂ makes the estimated pdf of the https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 22esc distribution really different from that of the c distribution. we also find aic = 77.49385 and bic = 81.57996.the ac distribution and the ml estimation method are used for comparison. the estimates ofthe parameters involved in the ac distribution are α̂ = 0.36297403 and β̂ = 0.02174637, andwe obtain aic = 79.11784 and bic = 83.20395. since the aic and bic associated with theesc distribution are lower than those associated with the ac distribution, the esc distributionprovides the best fit to the data. as a visual check, in figure 6, we plot the histogram of the dataand overlay the estimated pdfs of the esc and ac distributions. x −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 estimated pdf of the esc distribution estimated pdf of the ac distribution figure 6. histogram of the data of the simulated example 1 and estimated pdfs ofthe esc and ac distributions the normalized histogram is well fitted by the two estimated pdfs, but that of the esc distributionseems to be closer to the overall shape thanks to oscillatory features. this follows the resultsinterpreted from the values of the aic and bic. 4.3. simulated example 2. this example simulates a different scenario. we consider a series ofmeasurements taken over time from a sensor array monitoring water quality parameters in a river.there are 49 values, ranging from −1 to 1, reflecting the variation in a particular water qualityparameter. most values are centered around 0, due to the overall stability of the condition of theriver, but with a complex symmetry feature, including a slight asymmetry caused by occasionalfluctuations due to external factors such as run-off, pollution or weather changes. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 23the data are as follows: 0.096, -0.590, 0.811, -0.194, -0.309, 0.593, -0.926, 0.378, -0.635, 0.285,0.116, -0.382, 0.439, 0.020, -0.682, 0.724, -0.771, 0.920, -0.048, -0.121, 0.199, -0.251, 0.445, -0.599, 0.536, -0.807, 0.899, -0.797, -0.044, 0.030, 0.055, -0.035, -0.001, -0.064, 0.072, -0.073,0.131, 0.002, 0.150, -0.140, 0.217, -0.128, 0.192, -0.167, 0.186, -0.136, 0.264, -0.274, 0.163.to summarize these data, the minimum is −0.901, the first quartile is −0.298, the median is 0.037, the mean is 0.005649, the third quartile is 0.347, and the maximum is 0.906. this informationindicates some symmetry in the distribution of the data. as for the previous example, the mainquantile features of the data are illustrated by a boxplot in figure 7. −0.5 0.0 0.5 d a ta figure 7. boxplot of the data of the simulated example 2 there are some outliers in this boxplot, but also some symmetry. again, the esc distributionmay be able to analyze these data.using the ml estimation method for the esc distribution, we obtain â = 0.7993845 and b̂ = 1.from these estimates, we derive the estimated pdf f̂∗(x). note that the high value of b̂ makes theestimated pdf of the esc distribution really different from that of the c distribution. we also find aic = 57.03759 and bic = 60.82123.for comparison, we look at the ac distribution. we also use the ml estimation method. thecorresponding estimates of the parameters are α̂ = 0.65030333 and β̂ = −0.02541488, andwe get aic = 58.46521 and bic = 62.24885. since the aic and bic associated with the escdistribution are lower than those associated with the ac distribution, the esc distribution providesthe best fit for the data. to support this claim, in figure 8, we show the histogram of the data andoverlay the estimated pdfs of the esc and ac distributions. https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 24 x −1.0 −0.5 0.0 0.5 1.0 0 .0 0 .5 1 .0 1 .5 estimated pdf of the esc distribution estimated pdf of the ac distribution figure 8. histogram of the data of the simulated example 2 and estimated pdfs ofthe esc and ac distributions we see that both estimated pdf fit the general shape of the normalized histogram. the advantageof the pdf of the esc distribution is that it better captures the bars around x = 0, and also theoriginal variability of the data represented by the other bars.it is worth noting that during our statistical investigations, other simulated data examples wereproduced and it results that the esc distribution was not uniformly the best compared to the acdistribution. this is mainly due to the fact that the esc distribution is only designed to analyzedata with a symmetric distribution, whereas the ac distribution offers more possibilities to capturedifferent skewnesses and is clearly more appropriate in some scenarios. if the normalized histogramof the data is symmetric in shape, the esc distribution is more recommended. if there is skewness,the ac distribution is preferable. 5. conclusion in this article, we have developed a new symmetric extension of the famous c distribution,called the esc distribution. in addition to being symmetric, it has the property of having thesupport [−1, 1], while maintaining a high degree of flexibility thanks to two adjustable parameters.in particular, one of them activates an original sine term in the main functions. this results insome oscillations in the pdf and hrf, which can be useful in different data fitting scenarios. wehave investigated some of the main properties of the esc distributions. in particular, a momentanalysis expresses several key measures in closed form, including the mean, variance, skewness, https://doi.org/10.28924/ada/ma.5.7 eur. j. math. anal. 10.28924/ada/ma.5.7 25kurtosis, incomplete mean and mean deviation. a number of transformed esc distributions withdifferent supports were also defined, together with a new skewed version of the standard normaldistribution. the statistical aspects of the esc distribution were then examined using the mlestimation of the two parameters and two examples of simulated data. for these examples, thefit of the new distribution is satisfactory and outperforms that of another two-parameter extended(mainly asymmetric) version of the c distribution.thus, this article provides a valuable extension of the c distribution and opens some perspectivesfor new statistical models with trigonometric properties. the esc 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properties 3.1. the cdf 3.2. quantile function 3.3. hazard rate function 3.4. moment analysis 3.5. some derived distributions 4. applications 4.1. estimation method 4.2. simulated example 1 4.3. simulated example 2 5. conclusion references 