On the existence of mild solutions to some semilinear fractional integro-differential equations

This paper deals with the existence of a mild solution for some fractional semilinear differential equations with non local conditions. Using a more appropriate definition of a mild solution than the one given in [12], we prove the existence and uniqueness of such solutions, assuming that the linear part is the infinitesimal generator of an analytic semigroup that is compact for t > 0 and the nonlinear part is a Lipschitz continuous function with respect to the norm of a certain interpolation space. An example is provided.


Introduction
Let X be a Banach space and let T > 0. This paper is aimed at discussing about the existence and the uniqueness of a mild solution for the fractional semilinear integro-differential equation with nonlocal conditions in the form: x(0) + g(x) = x 0 , where the fractional derivative D β (0 < β < 1) is understood in the Caputo sense, the linear operator −A is the infinitesimal generator of an analytic semigroup (R(t)) t≥0 that is uniformly bounded on X and compact for t > 0, the function a(•) is real-valued such that (2) a T = T 0 a(s) ds < ∞, the functions f, g and h are continuous, and the non local condition with c k , k = 1, 2, ...p, are given constants and 0 < t 1 < t 2 < ... < t p ≤ T .Let us recall that those nonlocal conditions were first utilized by K. Deng [4].In his paper, K. Deng indicated that using the nonlocal condition x(0) + g(x) = x 0 to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy Problem x(0) = x 0 .Let us observe also that since Deng's paper, such problem has attracted several authors including A. Aizicovici, L. Byszewski, K. Ezzinbi, Z. Fan, J. Liu, J. Liang, Y. Lin, T.-J.Xiao, H. Lee, etc. (see for instance [1,2,3,4,9,8,7,14,11,13] and the references therein).
This problem has been studied in Mophou and N'Guérékata [12].In this paper, we revisit that work and use a more appropriate definition for mild solutions.Namely, we investigate the existence and the uniqueness of a mild solution for the fractional semilinear differential equation (1), assuming that f is defined on [0, T ] × X α × X α where X α = D(A α ) (0 < α < 1), the domain of the fractional powers of A.
The rest of this paper is organized as follows.In Section 2 we give some known preliminary results on the fractional powers of the generator of an analytic compact semigroup.In Section 3, we study the existence and the uniqueness of a mild solution for the fractional semilinear differential equation (1).We give an example to illustrate our abstract results.

Preliminaries
Let I = [0, T ] for T > 0 and let X be a Banach space with norm • .Let B(X), • B(X) be the Banach space of all linear bounded operators on X and A : D(A) → X be a linear operator such that −A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators (R(t)) t≥0 , which is compact for t > 0. In particular, this means that there exists M > 1 such that Moreover, we assume without loss of generality that 0 ∈ ρ(A).This allows us to define the fractional power A α for 0 < α < 1, as a closed linear operator on its domain D(A α ) with inverse A −α (see [8]).We have the following basic properties for fractional powers A α of A: Theorem 2.1.( [15], pp.69 -75).Under previous assumptions, then: (iv) For every t > 0, A α R(t) is bounded on X and there exist M α > 0 and δ > 0 such that Remark 2.2.Observe as in [9] that by Theorem 2.1 (ii) and (iii), the restriction and as t decreases to 0 for all x ∈ X α , it follows that (R(t)) t≥0 is a family of strongly continuous semigroup on X α and R α (t) B(X) ≤ R(t) B(X) for all t ≥ 0.
Lemma 2.3.[9] The restriction R α (t) of R(t) to X α is an immediately compact semigroup in X α , and hence it is immediately norm-continuous.Now, let Φ β be the Mainardi function: For more details we refer to [10].We set Then we have the following results EJQTDE, 2010 No. 58, p. 3 Lemma 2.4.
[16] Let S β and P β be the operators defined respectively by ( 6) and (7).Then x for all x ∈ X and t ≥ 0.
In the sequel, we set ( 9) We set α ∈ (0, 1) and we will denote by C α , the Banach space C([0, T ], X α ) endowed with the supnorm given by

Main Results
In addition to the previous assumptions, we assume that the following hold.
(H 1 ) The function f : I × X α → X is continuous and satisfies the following condition: there exists a function µ The function h : I × X α → X is continuous and satisfies the following condition: there exists a function Theorem 3.1.Suppose assumptions (H 1 )-(H 3 ) hold and that Ω α,β,T < 1 where where K is given by ( 9).In view of Lemma 2.4-(ii), the integral operator F is well defined.Now take t ∈ I and x, y ∈ C α .We have which according to Lemma 2.5 and (H 3 ) gives Since (H 2 ) and ( 2) hold, we can write Thus, using (H 1 ) we obtain So we get Since Ω α,β,T < 1, the contraction mapping principle enables us to say that, F has a unique fixed point in C α , which is the mild solution of (1).Now we assume that (H 4 ) The function f : I × X α → X is continuous and satisfies the following condition: there exists a positive function The function h : I × X α → X is continuous and satisfies the following condition: there exists a positive function Proof.Define the integral operator F : and choose r such that We proceed in three main steps.EJQTDE, 2010 No. 58, p. 7 Step 1.We show that F (B r ) ⊂ B r .For that, let x ∈ B r .Then for t ∈ I, we have which according to (H 4 )-(H 6 ) and Lemma 2.5 gives Consequently, using the inequality M λ < 1 2 , which yields M λ x ∞ < r 2 and the choice of r above, we get In view of (10) and the choice of r, we obtain (F x) ∞ ≤ r.
Step 2. We prove that F is continuous.For that, let (x n ) be a sequence of B r such that x n → x in B r .Then as both f and h are jointly continuous on I × X α .Now, for all t ∈ I, we have which in view of Lemma 2.5 gives for all t ∈ I. Therefore, on the one hand using ( 2), (H 4 ) and (H 5 ), we get for each and on the other hand using the fact that the functions s → 2µ 1 (s)(t − s) β(1−α)−1 and s → (t − s) β(1−α)−1 are integrable on I, by means of the Lebesgue Dominated Convergence Theorem yields Hence, since g(x n ) → g(x) as n → ∞ because g is completely continuous on C α , it can easily be shown that lim In other words, F is continuous.
Step 3. We show that F is compact.To this end, we use the Ascoli-Arzela's theorem.For that, we first prove that EJQTDE, 2010 No. 58, p. 9 Let t ∈ (0, T ].For each h ∈ (0, t), ǫ > 0 and x ∈ B r , we define the operator F h,ǫ by Then the sets {(F h,ǫ x)(t) : x ∈ B r } are relatively compact in X α since by Lemma 2.3, the operators R α (t), t > 0 are compact on X α .Moreover, using (H 1 ) and ( 4) we have Then using ( 4) and (H 4 ), we obtain Since by (H 5 ) and ( 2), No. 58, p. 10 using (5c), we deduce for all ǫ > 0 that In other words Therefore, the set {(F x)(t) : x ∈ B r } is relatively compact in X α for all t ∈ (0, T ] and since it is compact at t = 0 we have the relatively compactness in X α for all t ∈ I. Now, let us prove that F (B r ) is equicontinuous.By the compactness of the set g(B r ), we can prove that the functions F x, x ∈ B r are equicontinuous a t = 0.For 0 < t 2 < t 1 ≤ T , we have where Actually, I 1 , I 2 , I 3 and I 4 tend to 0 independently of x ∈ B r when t 2 → t 1 .Indeed, let x ∈ B r and G = sup x∈Cα g(x) α .In view of Lemma 2.5, we have B(X) from which we deduce that lim t2→t1 I 1 = 0 since by Lemma 2.3 the function t → R α (t) α is continuous for t ≥ 0 Therefore using the continuity of P β (t) (Lemma 2.4) and the fact that both f and K are bounded we conclude that lim t2→t1 I 2 = 0 we deduce that Hence lim EJQTDE, 2010 No. 58, p. 12 Since β(1 − α) > 0, we deduce that lim t2→t1 I 4 = 0.
In short, we have shown that F (B r ) is relatively compact, for t ∈ I, {F x : x ∈ B r } is a family of equicontinuous functions.Hence by the Arzela-Ascoli Theorem, F is compact.By Schauder fixed point theorem F has a fixed point x ∈ B r , which obviously is a mild solution to (1).

Example
Let X = L 2 [0, π] equipped with its natural norm and inner product defined respectively for all u, v ∈ L 2 [0, π] by Consider the following integro-partial differential equation First of all, note that f, h, a are given by Let A be the operator given by Au = −u ′′ with domain It is well known that A has a discrete spectrum with eigenvalues of the form n 2 , n ∈ N, and corresponding normalized eigenfunctions given by z n (ξ) := 2 π sin(nξ).
In addition to the above, the following properties hold: and µ = 0. Therefore, the condition M λ < 1 2 holds under assumption (11).Using Theorem 3.2 and inequality Eq. ( 11) it follows that the system (E) at least one mild solution.