/compile/output.dvi A Note on Positivity of One-Dimensional Elliptic Differential Operators Allaberen Ashyralyev1 and Sema Akturk2,∗ 1 Department of Elementary Mathematics Education, Fatih University, 34500 Buyukcekmece, Istan- bul, Turkey, Department of Mathematics, ITTU, 74400 Gerogly Street,Ashgabat, Turkmenistan 2 Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey Abstract. We consider the structure of fractional spaces Eα(C � R+ � ,A) generated by the positive dif- ferential operator A defined by the formula Au(t) = −ut t(t) + u(t) with domain D(A) = {u : ut t ,u ∈ C � R+ � ,u(0) = 0,u(∞) = 0}, where R+ = [0,∞). It is established that for any 0 < α < 1/2, the norms in the spaces Eα(C � R+ � ,A) and C2α � R+ � are equivalent. The positivity of the differential operator A in C2α � R+ � is established. 2010 Mathematics Subject Classifications: 35J08, 35J58, 47B65 Key Words and Phrases: Positive operator, Fractional spaces, Green’s function, Hölder spaces 1. Introduction It is well-known that various local and nonlocal boundary value problems for partial differ- ential equation can be considered as an abstract boundary value problem for ordinary differ- ential equation in a Banach space E with a densely defined unbounded operator A. Therefore, the study of various properties of partial differential equations is based on the positivity prop- erty of the differential operator in a Banach space [6–8]. Many researcher have studied the positivity of wider class of differential operators (see [12] through [23]). An differential operator A densely defined in a Banach space E with domain D(A) is called positive in E, if its spectrum σA lies in the interior of the sector of angle ϕ, 0 < ϕ < π, symmetric with respect to the real axis, and moreover on the edges of this sector S1 � ϕ � ={ρeiϕ : 0≤ ρ ≤∞} ∗Corresponding author. Email addresses: aashyr@fatih.edu.tr (A. Ashyralyev), semathakturk@gmail.com (S. Akturk) http://www.ejpam.com 165 c© 2016 EJPAM All rights reserved. EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 2, 2016, 165-174 ISSN 1307-5543 – www.ejpam.com A. Ashyralyev, S. Akturk / Eur. J. Pure Appl. Math, 9 (2016), 165-174 166 S2 � ϕ � ={ρe−iϕ : 0≤ ρ ≤∞}, and outside of the sector the resolvent (A−λ)−1 is a subject to the bound (see, [6]) (A−λ)−1 E→E ≤ M 1+ |λ| . The infimum of all such angles ϕ is called the spectral angle of the positive operator A and is denoted by ϕ(A) = ϕ(E,A). The operator A is said to be strongly positive in a Banach space E, if ϕ(E,A)< π 2 . Throughout the paper, M will denote positive constants which can be different from time to time and we are not interested to precise. To stress the fact that the constant depends only on α,β , . . . , we will write M(α,β , . . .). For a positive operator A in the Banach space E, let us define the fractional spaces Eα = Eα(E,A)(0< α < 1) consisting of those v ∈ E for which the norm ‖v‖Eα = sup λ>0 λα‖A(λ+ A)−1v‖E + ‖v‖E is finite. It is well-known that from the positivity of operator A in the Banach space E it follows the positivity of this operator in fractional spaces Eα = Eα(E,A)(0< α < 1). In this study, we consider the second order differential operator Au(t) = −ut t(t) + u(t) (1) with domain D(A) = {u : ut t ,u ∈ C � R+ � ,u(0) = 0,u(∞) = 0}, where R+ = [0,∞). The Green’s function of A is constructed. The positivity of the operator A in the Banach space E = C � R+ � with norm ϕ C(R+) = sup t≥0 |ϕ(t)| is proved. Moreover, the structure of the fractional spaces Eα(E,A),α ∈ (0,1/2) are established and the positivity of A in the Hölder spaces C2α � R+ � , α ∈ (0,1/2) is established. 2. Green’s Function of A and Positivity of A in C � R+ � To find the Green’s function of operator A we need to solve the resolvent equation Au(t) +λu(t) = ϕ(t), 0< t <∞ or ¨ −ut t(t) + (1+λ)u(t) = ϕ(t), 0< t <∞, u(0) = 0, u(∞) = 0 . (2) Let us give a lemma that will be needed below. A. Ashyralyev, S. Akturk / Eur. J. Pure Appl. Math, 9 (2016), 165-174 167 Lemma 1. For λ ≥ 0, equation (2) is uniquely solvable and the following formula holds: u(t) = (A+λ)−1ϕ(t) = ∫ ∞ 0 G(t, s)ϕ(s)ds (3) where G(t, s) = e− p 1+λ|t−s| − e− p 1+λ(t+s) 2 p 1+λ , t, s ≥ 0. Now, we will prove the positivity of A in the Banach space C � R+ � . Theorem 1. For λ in the sector Σϕ0 = {λ = ̺eiθ ; |θ | ≤ ϕ0 < π/2}, the following estimate holds: (A+λ)−1 C(R+)→C(R+) ≤ M(ϕ0) 1+ |λ| (4) where the resolvent (A+λ)−1 defined by formula (3). Proof. For λ = |λ| eiϕ ∈ Σϕ0 , we have 1 + λ = |1+λ| eiψ, ψ ≤ ϕ < ϕ0 < π 2 . Then,p 1+λ = |1+λ|1/2 ei ψ 2 with ψ< π 4 . Clearly, we have � � � p 1+λ � � �= 4 q 1+ 2 |λ| cosϕ + |λ|2 ≥ M(ϕ0) Æ 1+ |λ|. (5) Using formula (3), estimate (5) and the triangle inequality, we get � �(A+λ)−1 f (t) � �≤ f C(R+) M(ϕ0) p 1+ |λ| ∞ ∫ 0 e−M(ϕ0) p 1+|λ||t−s|ds ≤ f C(R+) M(ϕ0) p 1+ |λ|   t ∫ 0 e−M(ϕ0) p 1+|λ|(t−s)ds+ ∞ ∫ t e−M(ϕ0) p 1+|λ|(s−t)ds   ≤M(ϕ0) 1+ |λ| . This finishes the proof of Theorem 1. Now, we will introduce the Banach space C2α � R+ � (0< α < 1) of all continuous functions ϕ(x) defined on R+ and satisfying a Hölder condition for which the following norm is finite: ‖ϕ‖C2α(R+) = ‖ϕ‖C(R+) + sup t1 6=t2 t1,t2∈R+ |ϕ(t1)−ϕ(t2)| |t1 − t2|2α . A. Ashyralyev, S. Akturk / Eur. J. Pure Appl. Math, 9 (2016), 165-174 168 3. The Structure of Fractional Spaces Eα(C � R+ � , A) Theorem 2. For α ∈ (0,1/2), the Banach spaces Eα(C � R+ � ,A) and C2α � R+ � are equivalent. Proof. Let λ > 0 and t ≥ 0. From formula (3) it follows that A(A+λ)−1 f (t) =λ � 1 λ f (t)− (A+λ)−1 f (t) � = 1 λ+ 1 f (t) +λ � 1 λ+ 1 f (t)− (A+λ)−1 f (t) � = 1 λ+ 1 f (t) +λ 1 2 p 1+λ ∞ ∫ 0 � e− p 1+λ|t−s| − e− p 1+λ(t+s) � � f (t)− f (s) � ds. Then, by this formula, the triangle inequality, and the definition of C2α � R+ �−norm, we have � �λαA(A+λ)−1 f (t) � �≤M f C2α   λα 1+λ + λα+1 2 p 1+λ ∞ ∫ 0 � � �e− p 1+λ|t−s| − e− p 1+λ(t+s) � � � |t − s|2α ds   ≤M f C2α   λα 1+λ +λα+1 1p 1+λ ∞ ∫ 0 e− p 1+λ|t−s| |t − s|2α ds   . (6) The substitution p 1+λ|t − s|= p yields that ∞ ∫ 0 e− p 1+λ|t−s| |t − s|2α ds = t ∫ 0 e− p 1+λ|t−s| |t − s|2α ds+ ∞ ∫ t e− p 1+λ(s−t) 2 p 1+λ |s− t|2α ds =− 0 ∫ p 1+λt e−p p2α (1+λ)α+ 1 2 dp+ ∞ ∫ 0 e−p p2α (1+λ)α+ 1 2 dp ≤ 2 (1+λ)α+ 1 2 Γ(2α+ 1) (7) where Γ(·) is the gamma function. Thus, from estimate (7) it follows that estimate (6) becomes � �λαA(A+λ)−1 f (t) � �≤ M f C2α � λα 1+λ +λα+1 1 2 (1+λ)1+α M(α) � ≤ M(α) f C2α . Hence, we get sup λ>0 sup t∈[0,∞) � �λαA(A+λ)−1 f (t) � �≤ M(α) f C2α A. Ashyralyev, S. Akturk / Eur. J. Pure Appl. Math, 9 (2016), 165-174 169 or f Eα(A,C) ≤ M(α) f C2α . Therefore, we prove C2α � R+ � ⊂ Eα(C � R+ � ,A). Next, let us prove that Eα(C � R+ � ,A) ⊂ C2α � R+ � . Clearly, for a positive operator A in a Banach space E, we have v = ∞ ∫ 0 A(λ+ A)−2vdλ. By this fact, for t +τ > t ≥ 0, we have f (t) = ∞ ∫ 0 A(λ+ A)−2 f (t)dλ= ∞ ∫ 0 (λ+ A)−1A(λ+ A)−1 f (t)dλ = ∞ ∫ 0 ∞ ∫ 0 1 2 p 1+λ � e− p 1+λ|t−s| − e− p 1+λ(t+s) � A(λ+ A)−1 f (s)dsdλ, (8) and f (t +τ) = ∞ ∫ 0 ∞ ∫ 0 1 2 p 1+λ � e− p 1+λ|t+τ−s| − e− p 1+λ(t+τ+s) � A(λ+ A)−1 f (s)dsdλ. (9) Clearly, ‖ f ‖C(R+) ≤ M(α). (10) From equations (8) and (9) it follows that f (t +τ)− f (t) τ2α = ∞ ∫ 0 λ−α 2 p 1+λ 1 τ2α t ∫ 0 � e− p 1+λ|t+τ−s| − e− p 1+λ(t+τ+s) − e− p 1+λ|t−s| − e− p 1+λ(t+s) � ×λαA(λ+ A)−1 f (s)dsdλ + ∞ ∫ 0 λ−α 2 p 1+λ 1 τ2α t+τ ∫ t � e− p 1+λ|t+τ−s| − e− p 1+λ(t+τ+s) − e− p 1+λ(s−t) − e− p 1+λ(s+t) � ×λαA(λ+ A)−1 f (s)dsdλ + ∞ ∫ 0 λ−α 2 p 1+λ 1 τ2α ∞ ∫ t+τ � e− p 1+λ|t+τ−s| − e− p 1+λ(t+τ+s) − e− p 1+λ(s−t) − e− p 1+λ(s+t) � A. Ashyralyev, S. Akturk / Eur. J. Pure Appl. Math, 9 (2016), 165-174 170 ×λαA(λ+ A)−1 f (s)dsdλ =J1 + J2 + J3. Clearly, we have 1− e− p 1+λτ ≤ (1+λ)ατ2α. (11) Using estimate (11), the triangle inequality, and the definition of Eα−norm, we obtain |J1| ≤‖ f ‖C(Eα) ∞ ∫ 0 λ−α 2 p 1+λ × 1 τ2α t ∫ 0 � � �e− p 1+λ|t+τ−s| − e− p 1+λ(t+τ+s) − e− p 1+λ|t−s| − e− p 1+λ(t+s) � � � dsdλ ≤‖ f ‖C(Eα) ∞ ∫ 0 λ−α 2 (1+λ) 1 τ2α � 1− e− p 1+λτ � dλ =‖ f ‖C(Eα)   1 ∫ 0 λ−α 2 (1+λ) 1 τ2α � 1− e− p 1+λτ � dλ+ ∞ ∫ 1 λ−α 2 (1+λ) 1 τ2α � 1− e− p 1+λτ � dλ   ≤M(α)‖ f ‖C(Eα). (12) In the same manner, we get |J2| ≤M(α)‖ f ‖C(Eα), (13) |J3| ≤M(α)‖ f ‖C(Eα). (14) Estimates (12)-(14) yield that sup 0≤t0 λα A1/2 � λ+ A1/2 �−1 v E + ||v||E is finite. Theorem 4 ([5]). The spaces Eα(E,A) and E′2α(A 1/2, E) coincide for any 0 < α < 1 2 , and their norms are equivalent. Theorem 5 ([7]). Let A be positive operator in a Banach space E and f ∈ C([0, T] , E′α) (0 < α < 1). Then, for the solution of boundary value problem ¨ −u′′(t) + Au(t) = f (t), 0< t < T, u(0) = ϕ, u(T ) =ψ (17) in a Banach space E with positive operator A the coercive inequality ‖u′′‖C([0,T],E′α) + ‖Au‖C([0,T],E′α) ≤ M � ‖Aϕ‖E′α + ‖Aψ‖E′α + M α (1−α)‖ f ‖C([0,T],E′α) � holds. Second, we will consider the nonlocal-boundary value problem for the elliptic equation      − ∂ 2u(t,x) ∂ t2 − ∂ 2u(t,x) ∂ x2 +δu(t, x) = f (t, x), 0< t < T, x ∈ R+, u(0, x) = u(T, x), ut(0, x) = ut(T, x), x ∈ R+, u(t, 0) = 0, 0≤ t ≤ T . 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