CUBO, A Mathematical Journal Vol.22, N◦03, (361–377). December 2020 http://dx.doi.org/10.4067/S0719-06462020000300361 Received: 29 May, 2020 | Accepted: 10 November, 2020 Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions Abdeldjalil Aouane1,3, Smäıl Djebali2,3 and Mohamed Aziz Taoudi4 1Département de Sciences Exactes et Informatique, École Normale Supérieure, Constantine, Algeria. 2Department of Mathematics, Faculty of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), PB 90950. Riyadh 11623, Saudi Arabia. 3Laboratoire Théorie du Point Fixe et Applications ENS, BP 92 Kouba, Algiers, 16006. Algeria. 4 Cadi Ayyad University, National School of Applied Science Marrakesh, Morocco. abdeldjalilens@hotmail.com, djebali@hotmail.com, a.taoudi@uca.ma ABSTRACT In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order β ∈ (1,2] subjected to noncompact initial nonlo- cal conditions. We assume that the linear part generates an arbitrarily strongly continu- ous β-order fractional cosine family, while the nonlinear forcing term is of Carathéodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncom- pactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000300361 362 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) RESUMEN En este art́ıculo, probamos la existencia de soluciones leves de una clase de ecuaciones integro-diferenciales fraccionales semilineales de orden β ∈ (1,2] con condiciones no- compactas iniciales no-locales. Asumimos que la parte lineal genera una familia coseno de orden fraccional β arbitrariamente fuertemente continua , mientras que el término no-lineal de forzamiento es de tipo Carathéodory y satisface algunas condiciones de crecimiento bastante generales. Nuestro enfoque combina el teorema de punto fijo de Monch con algunos resultados recientes sobre la medida de no-compacidad de oper- adores integrales. Nuestras conclusiones mejoran y generalizan muchos trabajos ante- riores relacionados . Se provee un ejemplo para ilustrar los resultados principales. Keywords and Phrases: Cosine operator, fractional integro-differential operator, abstract dif- ferential equation, noncompact nonlocal condition. 2020 AMS Mathematics Subject Classification: 34A08, 34G20, 35F25, 47D09, 47D60, 47H08, 47H10, 47G20. 1 Introduction In recent years, the investigation of fractional differential equations in Banach spaces has attracted many research works due to its applications in various areas of engineering, physics, bio-engineering, and other applied sciences. Notable contributions have been made to both theory and applications of fractional differential equations; we refer, e.g., to [1, 6, 13, 14, 15, 16, 18, 19, 25] and the refer- ences therein. Actually, it has been found that differential equations involving fractional derivatives in time are more realistic to describe many phenomena in practical situations than those of integer order. The most significant advantage of fractional derivatives compared with integer derivatives is that it can be used to describe the property of memory and heredity of various materials and processes [5, 8, 22]. For more details about fractional calculus and fractional differential equations, we refer the reader to [2, 4, 10]. In this paper, we are concerned with the existence of mild solutions of the following class of fractional semilinear integro-differential equations:    cD β t u(t) = Au(t) + f(t,u(t),Gu(t),Su(t)), t ∈ [0,a], u(0) = u0 + q(u), u′(0) = v0 + p(u), (1.1) where β ∈ (1,2] and cD β t is the standard Caputo fractional derivative of order β. The operator A is the infinitesimal generator of a strongly continuous β-order fractional cosine family {Cβ(t) : t ≥ 0} CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 363 in a Banach space E, f, q,p are suitably defined functions satisfying certain conditions to be specified later, x0, y0 are given elements of E and G,S are two linear operators defined by Gu(t) = ∫ t 0 K(t,s)u(s)ds and Su(t) = ∫ a 0 H(t,s)u(s)ds, t ∈ [0,a], (1.2) where H ∈ C [[0,a] × [0,a],R+] , K ∈ C [U,R+] , and U = { (t,s) ∈ R2 : 0 ≤ s ≤ t ≤ a } . Here R+ refers to the set of nonnegative real numbers. The problem of the existence of mild solutions to (1.1) has been addressed by many investigators in the case where β ∈ (0,1]. We quote for instance the contributions by Shu and Wang [21], Qin et al. [20], and the pioneering works of Travis and Webb [23, 24]. However, only a few papers have been up to now devoted to the case β ∈ (1,2]. We quote the paper [25], where the authors proved the existence of mild solutions to (1.1) with β ∈ (1,2] when p and q are compact. In many applications, nonlocal conditions are not compact. Specifically, periodic p(u) = u(a), anti-periodic p(u) = −u(a), or multipoint discrete nonlocal conditions p(u) = ∑m i=1 ciu(ti), 0 < t1 < · · · < tm are not compact. As a matter of fact, the first and major aim of this paper is to address the problem of existence of mild solutions to (1.1) in the case where p and q are not necessarily compact. Moreover, we merely assume that the operator A generates an arbitrarily strongly continuous β-order fractional cosine family, which is an extra interesting feature. Our approach combines the Monch fixed point theorem with some recent results concerning the measure of noncompactness of integral operators. The outline of the paper is as follows: In Section 2, we present the main technical tools which will be used in this work. In Section 3, we investigate the existence of mild solution to problem (1.1) by means of a fixed point method. Finally, in Section 4, we include an example to illustrate our results. 2 Preliminaries and auxiliary results In this section, we recall some background and collect several useful results which are crucial for our further work. To do this, let (E,‖ · ‖) be a Banach space and C([0,a],E) be the space of all continuous functions defined on [0,a] with values in E, equipped with the standard sup-norm. Let L(E) denote the space of all bounded linear operators on E endowed with the classical operator norm. We first list some basic definitions and properties of the fractional calculus theory which are used further in this paper. Definition 2.1. [4] For 0 < γ < 1, consider the function of Wright type defined by Φγ(z) = ∞ ∑ n=0 (−z)n n!Γ(−γn + 1 − γ) = 1 2πi ∫ Γ µγ−1 exp (µ − zµγ) dµ, (2.1) 364 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) where Γ is a contour which starts and ends at −∞ and encircles the origin once counterclockwise. Φγ(t) is a probability density function: Φγ(t) ≥ 0 for t > 0 and ∫ ∞ 0 Φγ(t)dt = 1. (2.2) Definition 2.2. [4] The Riemann-Liouville fractional integral of order β > 0 of a function f ∈ L1([0,a];E) is defined by I β t f(t) = 1 Γ(β) ∫ t 0 (t − s)β−1f(s)ds, t > 0, (2.3) where Γ(·) stands for the Gamma function. Definition 2.3. [4] The Riemann-Liouville fractional derivative of order 1 < β ≤ 2 is defined by D β t f(t) = d2 dt2 I 2−β t f(t), (2.4) where f ∈ L1([0,a];E) and D β t f ∈ L 1([0,a];E). Definition 2.4. [4] The Caputo fractional derivative of order β ∈ (1,2] is defined by cD β t f(t) = D β t (f(t) − f(0) − f ′(0)t) , (2.5) where f ∈ L1([0,a];E) ∩ C1([0,a];E) and D β t f ∈ L 1([0,a];E). Consider the following problem cD β t x(t) = Ax(t), x(0) = η, x ′(0) = 0, (2.6) where β ∈ (1,2], A : D(A) ⊂ E → E is a closed densely defined linear operator in Banach space E. Definition 2.5. [4] Let β ∈ (1,2]. A family {Cβ}β≥0 ⊂ L(E) is called a solution operator (or a strongly continuous β-order fractional cosine family) for the problem (2.6) if the following conditions are satisfied: (a) Cβ(t) is strongly continuous for t ≥ 0 and Cβ(0) = I, (b) Cβ(t)D(A) ⊂ D(A) and ACβ(t)η = Cβ(t)Aη for all η ∈ D(A), t ≥ 0, (c) Cβ(t)η is a solution of x(t) = η + ∫ t 0 (t−s)β−1 Γ(β) Ax(s)ds for all η ∈ D(A), t ≥ 0. In this case, A is called the infinitesimal generator of Cβ(t). Definition 2.6. [15] The fractional sine family Sβ : R + → L(E) associated with Cβ is defined by Sβ(t) = ∫ t 0 Cβ(s)ds, t ≥ 0. (2.7) CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 365 Definition 2.7. [15] The fractional Riemann-Liouville family Pβ : R + → L(E) associated with Cβ is defined by Pβ(t) = I β−1 t Cβ(t) = 1 Γ(β − 1) ∫ t 0 (t − s)β−2Cβ(s)ds, t ≥ 0. (2.8) Definition 2.8. [4] The strongly continuous β-order fractional cosine family Cβ(t) is called expo- nentially bounded if there are constants M ≥ 1 and ω ≥ 0 such that ‖Cβ(t)‖ ≤ Me ωt, t ≥ 0. (2.9) An operator A is said to belong to Cβ(M,ω), if the problem (2.6) has a strongly continuous β-order fractional cosine family Cβ(t) satisfying (2.9). Denote C β(ω) = ⋃ {Cβ(M,ω);M ≥ 1}. Theorem 2.1. [4, Theorem 3.1] Let 0 < β′ < β ≤ 2, γ = β ′ β , ω ≥ 0. If A ∈ Cβ(ω) then A ∈ Cβ ′ (ω 1 γ ) and the following representation holds Cβ′(t) = ∫ ∞ 0 ϕt,γ(s)Cβ(s)ds, t > 0, (2.10) where ϕt,γ(s) := t −γΦγ (st −γ) and Φγ(z) is defined by (2.1). For more details regarding β-order fractional cosine families, we refer the reader to [4]. Definition 2.9. A function ψ defined on the set of all bounded subsets of a Banach space E with values in R+ is called a measure of noncompactness (MNC in short) on E if for any bounded subset M of E we have ψ(coM) = ψ(M), where coM stands for the closed convex hull of M. An MNC is said to be (i) Full: ψ(M) = 0 if and only if M is a relatively compact set. (ii) Monotone: for all bounded subsets M1 and M2 of E, we have M1 ⊂ M2 =⇒ ψ(M1) ≤ ψ(M2). (iii) Nonsingular: ψ(M ∪ {x}) = ψ(M), for every bounded subset M of E and for all x ∈ E. One of most important measures of noncompactness is the Hausdorff measure of noncompact- ness defined by χ(M) = inf{r > 0;M can be covered by finitely many balls with radii ≤ r}, for each bounded subset M of E. The Hausdorff measure of noncompactness is full, monotone and nonsingular. Moreover, it enjoys the following additional properties. 366 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) Lemma 2.1. [3] (i) χ(M1 + M2) ≤ χ(M1) + χ(M2). (ii) χ(λM) = |λ|χ(M), for all λ ∈ R. (iii) χ(co(M)) = χ(M). (iv) χ(A + x) = χ(A), ∀x ∈ E. (v) if B : E −→ E is a Lipschitz continuous map with constant k, then χ(B(M)) ≤ kχ(M) for all bounded subset M of E. Lemma 2.2. [17, 9] If {un}n∈N ⊂ L 1([0,a];E) is uniformly integrable, then the function t 7→ χ({un(t)}n∈N) for t ∈ [0,a] is measurable and χ ({ ∫ t 0 un(s)ds }∞ n=1 ) ≤ ∫ t 0 χ ( {un(s)} ∞ n=1 ) ds. In the sequel, we use a measure of noncompactness in the space C(I;E) which was investigated in [11, 12]. In order to define this measure, let us fix a nonempty bounded subset Ω of the space C(I;E). Let modC(Ω) = sup {modC(Ω(t)) : t ∈ I} , where modC(Ω(t)) = lim δ→0 sup x∈Ω {sup {|x(t2) − x(t1)| : t1, t2 ∈ (t − δ,t + δ)}} , and χ∞(Ω) = sup {χ(Ω(t)) : t ∈ I} , where χ denotes the Hausdorff measure of noncompactness in E. It is worth noticing that χ∞ and modC are monotone nonsingular MNCs on C(I;E) (see [3, 12]). From an application view point, one of the main disadvantages of these MNCs is the lack of fullness. To overcome this problem, we can define the function ψC on the family of bounded subsets in C(I;E) by taking ψC(Ω) = χ∞(Ω) + modC(Ω) Lemma 2.3. [11, Lemma 3.1] ψC is a full monotone and nonsingular MNC on the space C(I;E). Finally, we will make use of Monch’s fixed point theorem. Theorem 2.2. [17] Let C be a closed, convex subset of a Banach space E with x0 ∈ C. Suppose there is a continuous map T : C → C with the following property: { D ⊆ C countable and D ⊆ co ({x0} ∪ T(D)) imply that D is relatively compact. Then, T has at least one fixed point in C. CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 367 Let F be a function from [0,+∞) into L(E). Suppose that F is continuous for the strong operator topology, namely The mapping [0,+∞) ∋ t → F(t)x ∈ E is continuous for every x ∈ E. (2.11) Notice that from the uniform boundedness principle, we know that F is uniformly bounded on any interval [0,a], i.e., Ma := supt∈[0,a] ‖F(t)‖L(E) < +∞. For later use, let us define the quantity ω(F(t)) = lim δ→0 sup ‖x‖≤1 {‖F (t2) x − F (t1)x‖E : t1, t2 ∈ (t − δ,t + δ)} . Recall that a family (F(t))t≥0 is said to be equicontinuous if {F(·)x : x ∈ Ω} is equicontinuous at any t > 0 for any bounded subset Ω ⊂ X. It is easily seen that a family (F(t))t≥0 is equicontinuous if and only if ω(F(t)) = 0 for any t > 0. Theorem 2.3. [7] Let F be a function from [0,+∞) into L(E). Suppose that F is continuous for the strong operator topology. Then, for any bounded set Ω ⊂ E and for any t ≥ 0, we have modC(F(t)Ω) ≤ ω(F(t))χ(Ω). In particular, for any t ∈ [0,a] we have modC(F(t)Ω) ≤ 2Maχ(Ω). Now, we present two crucial results concerning the integral operator: (S0f) (t) = ∫ t 0 F(t − s)f(s)ds for t ∈ [0,a] where f ∈ L1([0,a];E) and F : [0,+∞) → L(E) verifies (2.11). Theorem 2.4. [7] Let {fn} ∞ n=1 ⊂ L 1([0,a];E) be integrably bounded, that is, ‖fn(t)‖ ≤ ν(t) for all n = 1,2, · · · and a.e. t ∈ [0,a], (2.12) where ν ∈ L1([0,a]). Assume that χ({fn(t)} ∞ n=1) ≤ q(t) (2.13) for a.e. t ∈ [0,a] where q ∈ L1([0,a]). Then, for every t ∈ [0,a] we have: mod C ({S0fn(t)} ∞ n=1) ≤ 4Ma ∫ t 0 q(s)ds. (2.14) Theorem 2.5. [7] Let {fn} ∞ n=1 ⊂ L 1([0,a];E) be as in (2.12) Assume that (2.13) holds. Then χ({S0fn(t)} ∞ n=1) ≤ 2Ma ∫ t 0 q(s)ds, for all t ∈ [0,a] 368 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) 3 Existence results In this section, we discuss the existence of mild solutions to the semilinear fractional integro- differential equation (1.1). Before doing so, it is appropriate to clarify the definition of solution we will consider. Definition 3.1. Assume A ∈ Cβ(M,ω) and Cβ(t) is the solution operator. We say that u ∈ C[I,E] is a mild solution of (1.1) if u satisfies u(t) = Cβ(t) (u0 + q(u)) + Sβ(t) (v0 + p(u)) + ∫ t 0 Pβ(t − s)f(s,u(s),Gu(s),Su(s))ds, t ∈ I. (3.1) To allow the abstract formulation of our problem, we define the operator T : C([0,a];E) → C([0,a];E) by Tu(t) = Cβ(t) (u0 + q(u)) + Sβ(t) (v0 + p(u)) + ∫ t 0 Pβ(t − s)f(s,u(s),Gu(s),Su(s))ds, t ∈ [0,a] (3.2) for all t ∈ [0,a]. It is clear that u is a mild solution of (1.1) if and only if it is a fixed point of T . Our problem will be investigated under the following assumptions: (C1) p,q : C([0,a];E) → E are continuous functions and there exist nonnegative constants kp and kq, such that for all bounded subset D ⊂ C([0,a];E), we have Maχ(q(D)) + aMaχ(p(D)) ≤ (Makq + aMakp)χ∞(D), where Ma = supt∈[0,a] ‖Cβ(t)‖L(E). (C2) There exist nondecreasing continuous functions σ1,σ2 : R + → R+ such that { ‖q(u)‖E ≤ σ1 (‖u‖∞) , for all u ∈ C([0,a];E), ‖p(u)‖E ≤ σ2 (‖u‖∞) , for all u ∈ C([0,a];E). (C3)                f : [0,a] × E × E × E → E is a Carathéodory function, i.e., (i) the map t 7→ f(t,u1,u2,u3) is measurable for all (u1,u2,u3) ∈ E × E × E, (ii) the functions u1 7→ f(t,u1,u2,u3), u2 7→ f(t,u1,u2,u3) and u3 7→ f(t,u1,u2,u3) are continuous for almost t ∈ [0,a], (C4) There exist functions ρ1,ρ2,ρ3 ∈ L 1((0,a); R+) and nondecreasing continuous functions Ω1,Ω2,Ω3 : R + → R+ such that ‖f(t,u1,u2,u3)‖E ≤ 3 ∑ i=1 ρi(t)Ωi(‖ui‖E), for all t ∈ [0,a] and ui ∈ E. CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 369 (C5) There exist functions m1,m2,m3 ∈ L 1([0,a]; R+) such that for all bounded subset D1,D2,D3 ⊂ E χ(f(t,D1,D2,D3)) ≤ 3 ∑ i=1 mi(t)χ(Di), for almost every t ∈ [0,a]. (C6) Makq + aMakp + 2 Maa β−1 Γ(β) ‖m‖1 < 1, where m(s) = m1(s) + ak0m2(s) + ah0m3(s), k0 = sup{K(t,s); (t,s) ∈ U}, h0 = sup{H(t,s); (t,s) ∈ U}, and U = { (t,s) ∈ R2 : 0 ≤ s ≤ t ≤ a } . Remark 3.1. It is easy to prove that for every t ≥ 0, we have sup t∈[0,a] ‖Sβ(t)‖L(E) ≤ aMa and sup t∈[0,a] ‖Pβ(t)‖L(E) ≤ Maa β−1 Γ(β) . (3.3) In light of this, we shall show that operator T fulfills all conditions of Theorem 2.2. This will be done in a series of lemmas. Lemma 3.1. T : C([0,a];E) → C([0,a];E) is continuous. Proof. Let (un) ⊂ C([0,a];E) be a sequence which converges to u ∈ C([0,a];E). Then ‖Tun − Tu‖∞ ≤ Ma‖q(un) − q(u)‖E + aMa‖p(un) − p(u)‖E +Maa β−1 Γ(β) ∫ a 0 ‖f(s,un(s),Gun(s),Sun(s)) −f(s,u(s),Gu(s),Su(s))‖Eds. With assumptions (C1) and (C3) in mind, the continuity of G and S entails lim n→∞ f(s,un(s),Gun(s),Sun(s)) = f(s,u(s),Gu(s),Su(s)). Since (un) is convergent then there exists r > 0 such that ‖un‖∞ ≤ r, for all n ∈ N and ‖u‖∞ ≤ r. So by (C4) we have ‖f(s,un(s),Gun(s),Sun(s)) − f(s,u(s),Gu(s),Su(s))‖∞ ≤ 2 (ρ1(s)Ω1(r) + ρ2(s)Ω2(ak0r) + ρ3(s)Ω3(ah0r)) . Using the dominated convergence theorem, we deduce that T is continuous. Lemma 3.2. Assume that Ma lim inf r→∞ ( σ(r) r + aβ−1 Γ(β) Ω(r) r ) < 1, (3.4) where σ(r) = σ1(r) + aσ2(r) and Ω(r) = Ω1(r)‖ρ1‖L1 + Ω2(ak0r)‖ρ2‖L1 + Ω3(ah0r)‖ρ3‖L1. Then, there is a r0 > 0 such that T selfmaps the closed ball Br0 = {u ∈ C([0,a];E) : ‖u‖∞ ≤ r0} . 370 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) Proof. For u ∈ Br and t ∈ [0,a], we have ‖(Tu)(t)‖E ≤ ‖Cβ(t) (u0 + q(u))‖E + ‖Sβ(t) (v0 + p(u))‖E + ∥ ∥ ∥ ∫ t 0 Pβ(t − s)f(s,u(s),Gu(s),Su(s))ds ∥ ∥ ∥ E ≤ Ma (‖u0‖E + σ1(r)) + aMa (‖v0‖E + σ2(r)) +Maa β−1 Γ(β) ∫ a 0 Ω1(r)ρ1(s) +Ω2(ak0r)ρ2(s) + Ω3(ah0r)ρ3(s)ds. We claim that there exists r0 > 0 such that Tu ∈ Br0 whenever u ∈ Br0. If is not the case, then for each r > 0 there exists u ∈ Br such that Tu /∈ Br, that is r < ‖Tu‖∞ ≤ Ma (‖u0‖E + σ1(r)) + aMa (‖v0‖E + σ2(r)) + Maa β−1 Γ(β) Ω(r), which implies when dividing by r that 1 < Ma ‖u0‖E + aMa‖v0‖E r + Ma σ(r) r + Maa β−1 Γ(β) Ω(r) r . Taking the lim inf as r → ∞, we obtain 1 ≤ Ma lim inf r→∞ ( σ(r) r + aβ−1 Γ(β) Ω(r) r ) , which contradicts the assumption (3.4) Therefore, there exists r0 > 0 such that ‖Tu‖∞ ≤ r0, for all ‖u‖ ≤ r0. Thus, Tu ∈ Br0 for all u ∈ Br0. Lemma 3.3. Let r0 be as in Lemma 3.2 and let x0 ∈ Br0. Let D be a countable subset of Br0. Then D ⊆ co ({x0} ∪ T(D)) implies that D is relatively compact. Proof. Let D = {un} ∞ n=1 be any countable subset of Br0 such that D ⊆ co ({x0} ∪ T(D)) . (3.5) We show that D is relatively compact. Notice first that for each t ∈ [0,a], we have χ(T(D)(t)) ≤ χ(Cβ(t)(u0 + q(D))) + χ(Sβ(t)(v0 + p(D))) + χ ( { ∫ t 0 Pβ(t − s)f(s,un(s),Gun(s),Sun(s))ds }∞ n=1 ) . Since ‖f(s,un(s),Gun(s),Sun(s))‖E ≤ Ω1(r0)ρ1(s) + Ω2(ak0r0)ρ2(s) + Ω3(ah0r0)ρ3(s) CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 371 and ρ1,ρ2,ρ3 ∈ L 1([0,a];R+), then, in view of Theorem 2.5 and Lemma 2.2, we obtain the following estimates: χ(T(D)(t)) ≤ Maχ(q(D)) + aMaχ(p(D)) +2Maa β−1 Γ(β) ∫ t 0 m1(s)χ(D(s)) + m2(s)χ(G(D(s))) + m3(s)χ(S(D(s)))ds ≤ (Makq + aMakp)χ∞(D) +2Maa β−1 Γ(β) ∫ t 0 m1(s)χ(D(s)) + ak0m2(s)χ(D(s)) + ah0m3(s)χ(D(s))ds ≤ ( Makq + aMakp + 2 Maa β−1 Γ(β) ‖m‖1 ) χ∞(D). Thus, χ∞(T(D)) ≤ [ Makq + aMakp + 2 Maa β−1 Γ(β) ‖m‖1 ] χ∞(D). (3.6) On the other hand, referring to Theorem 2.3, Theorem 2.4, and Lemma 2.2, we can see that modC(T(D)(t)) ≤ modC(Cβ(t)q(D)) + modC(Sβ(t)p(D)) +4Maa β−1 Γ(β) ∫ t 0 m(s)χ(D(s))ds ≤ 2(Makq + aMakp)χ∞(D) + 4 Maa β−1 Γ(β) ‖m‖1χ∞(D). Thus, mod C(T(D)) ≤ [ 2(Makq + aMakp) + 4 Maa β−1 Γ(β) ‖m‖1 ] χ∞(D). (3.7) Combining (3.5) and (3.6), we arrive at χ∞(TD) = χ∞(D) = 0. By (3.7) we get mod C(T(D)) = 0 and therefore T(D) is equicontinuous. Going back to (3.5) we deduce that D is equicontinuous and so relatively compact in C([0,a];E). This achieves the proof. Theorem 3.1. Assume that (C1) − −(C6) hold. Then, the nonlocal problem (1.1) has at least one mild solution in C([0,a];E), provided that (3.4) holds. Proof. Invoking Theorem 2.2 together with Lemmas 3.1, 3.2, and 3.3, we infer that T has at least one fixed point in Br0 which is, in turn, a mild solution of (1.1). 372 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) 4 Application To illustrate the application of the theoretical results of this work, we consider the following integro-differential equation:                                  cD β t w(t,x) = ∂2w(t,x) ∂2x + ρ1(t)f1(w(t,x)) + ρ2(t)f2 ( ∫ t 0 ts 2 w(s,x)ds ) +ρ3(t)f3 ( ∫ 1 0 t 2 s 2 2 w(s,x)ds ) , t ∈ I = [0,1], x ∈ [0,π], w(t,0) = w(t,π) = 0, t ∈ I, w(0,x) = w0(x) + m ∑ i=1 ciw(si,x), x ∈ [0,π], s1 < s2 < ... < sm, ti ∈ I, ci ∈ R, ∂w(t,x) ∂t ∣ ∣ t=0 = y0(x) + n ∑ i=1 diw(ti,x), x ∈ [0,π], t1 < t2 < ... < tn, ti ∈ I, di ∈ R, (4.1) where β ∈ (1,2], the functions ρi : I → R and fi : E → E for i ∈ {1,2,3} satisfy appropriate conditions which are specified later. To allow the abstract formulation of (4.1), let E = L2([0,π]; R) be the Banach space of square integrable functions from [0,π] into R. Define the operator A : D(A) ⊂ E → E by Aw = w′′ with domain D(A) = {w ∈ E : w,w′ are absolutely continuous ,w′′ ∈ E,w(0) = w(π) = 0}. It is well known that A is the generator of strongly continuous cosine functions {C(t) : t ∈ R} on E. Moreover A has a discrete spectrum whose eigenvalues are −n2, n ∈ N with corresponding normalized eigenvectors zn(τ) = √ 2 π sin(nτ), and the following properties hold: (a) {zn : n ∈ N} is an orthonormal basis of E. (b) If z ∈ E, then Az = − ∑∞ n=1 n 2 < z, zn > zn. (c) For z ∈ E, C(t)z = ∑∞ n=1 cos(nt) < z, zn > zn, and the associated sine family is S(t)z = ∑∞ n=1 sin(nt) n < z,zn > zn. S(t) is compact for every t ∈ I and ‖C(t)‖L(E) = ‖S(t)‖L(E) ≤ 1, for every t ∈ R. For β ∈ (1,2], since A is the infinitesimal generator of a strongly continuous cosine family C(t), from the subordinate principle (Theorem 2.1), it follows that A is the infinitesimal generator of a strongly continuous exponentially bounded fractional cosine family Cβ(t). CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 373 With u(t) = w(t, ·), Equation (4.1) may be written in the abstract form:    cD β t u(t) = Au(t) + f(t,u(t),Gu(t),Su(t)), t ∈ I, u(0) = u0 + q(u), u′(0) = v0 + p(u), (4.2) where the function f : I × E × E × E → E is given by f(t,x,y,z) = ρ1(t)f1(x) + ρ2(t)f2(y) + ρ3(t)f3(z). Here ρi : I → R is integrable on I, fi : E → E is a Lipschitz continuous function with a Lipschitz constant Li, the functions p,q : C(I,E) → E are given by q(u) = m ∑ i=1 ciu(si), 0 < s1 < s2 < · · · < sm ≤ 1, and p(u) = n ∑ i=1 diu(ti), 0 < t1 < t2 < · · · < tn ≤ 1, and the functions G,S : C(I,E) → C(I,E) are defined by Gu(t) = ∫ t 0 ts 2 u(s)ds, Su(t) = ∫ 1 0 t2s2 2 u(s)ds, where h0 = k0 = 1 2 . In order to obtain a mild solution, our strategy is to apply Theorem 3.1. First, by (c) we have ‖C(t)‖L(E) ≤ 1, for every t ∈ R +. In view of Theorem 2.1 and (2.2) we see that there exists a real number Ma = 1 > 0 such that ‖Cβ(t)‖L(E) ≤ Ma for t ≥ 0. Observe further that the function f : I × E × E × E → E is given by f(t,x,y,z) = ρ1(t)f1(x) + ρ2(t)f2(y) + ρ3(t)f3(z), where ρi : I → R is integrable on I and fi : E → E is a Lipschitz continuous function with a Lipschitz constant Li (i = 1,2,3). This shows that (C3) is satisfied. On one hand, ‖q(u)‖E ≤ ( m ∑ i=1 |ci| ) ‖u‖∞ = σ1(‖u‖∞) (4.3) and ‖p(u)‖E ≤ ( n ∑ i=1 |di| ) ‖u‖∞ = σ2(‖u‖∞), (4.4) where σ1(r) = ( ∑m i=1 |ci|)r and σ2(r) = ( ∑n i=1 |di|) r. In addition, it is easily seen that for any bounded subset D of C([0,1],E) we have χ(q(D)) ≤ m ∑ i=1 |ci|χ(D (si)) ≤ ( m ∑ i=1 |ci| ) χ∞(D) = kqχ∞(D) (4.5) 374 Abdeldjalil Aouane, Smäıl Djebali, Mohamed Aziz Taoudi CUBO 22, 3 (2020) and χ(p(D)) ≤ n ∑ i=1 |di|χ(D (ti)) ≤ ( n ∑ i=1 |di| ) χ∞(D) = kpχ∞(D). (4.6) Thus Maχ(q(D)) + aMaχ(p(D)) ≤ (Makq + aMakp)χ∞(D), (4.7) for any bounded subset D of C([0,1];E). This shows that (C1) and (C2) are satisfied. Moreover the function f satisfies ‖f(t,u1,u2,u3)‖E ≤ |ρ1(t)|‖f1(u1)‖E + |ρ2(t)|‖f2(u2)‖E + |ρ3(t)|‖f3(u3)‖E ≤ |ρ1(t)|(‖f1(0)‖E + L1‖u1‖E) + |ρ2(t)|(‖f2(0)‖E + L2‖u2‖E) + |ρ3(t)|(‖f3(0)‖E + L3‖u3‖E) ≤ |ρ1(t)|Ω1(‖u1‖E) + |ρ2(t)|Ω2(‖u2‖E) + |ρ3(t)|Ω3(‖u3‖E) ≤ 3 ∑ i=1 |ρi(t)|Ωi(‖ui‖E), where Ωi(‖ui‖E) = ‖fi(0)‖E + Li‖ui‖E. By virtue of Lemma 2.1, (v) we have χ(f(t,D1,D2,D3)) ≤ |ρ1(t)|χ(f1(D1)) + |ρ2(t)|χ(f2(D2)) + |ρ3(t)|χ(f3(D3)) ≤ |ρ1(t)|L1χ(D1) + |ρ2(t)|L2χ(D2) + |ρ3(t)|L3χ(D3) ≤ 3 ∑ i=1 mi(t)χ(Di), for any t ∈ [0,a] and for any bounded subsets D1,D2,D3 of E. Thus, (C4) and (C5) are satisfied. Now the condition (C6) is given by taking 2 aβ−1Ma Γ(β) ( L1‖ρ1‖L1 + 1 2 L2‖ρ2‖L1 + 1 2 L3‖ρ3‖L1 ) + (Makq + aMakp) < 1, because, we have m(s) = m1(s) + ak0m2(s) + ah0m3(s) = L1|ρ1(s)| + 1 2 L2|ρ2(s)| + 1 2 L3|ρ3(s)|. Then ‖m‖1 = L1‖ρ1‖L1 + 1 2 L2‖ρ2‖L1 + 1 2 L3‖ρ3‖L1. Finally, for Ω(r) = Ω1(r)‖ρ1‖L1 + Ω2(ak0r)‖ρ2‖L1 + Ω3(ah0r)‖ρ3‖L1 = Ω1(r)‖ρ1‖L1 + Ω2( 1 2 r)‖ρ2‖L1 + Ω3( 1 2 r)‖ρ3‖L1, we have lim r→∞ Ω(r) r = L1‖ρ1‖L1 + 1 2 L2‖ρ2‖L1 + 1 2 L3‖ρ3‖L1, CUBO 22, 3 (2020) Mild solutions of a class of semilinear fractional . . . 375 and for σ(r) = σ1(r) + aσ2(r) = (kq + akp)r, notice that lim r→∞ σ(r) r = kq + akp. 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