Existence of solutions for nonlinear fractional integro-differential equations

In this paper, by means of the Krasnoselskii fixed point theorem, the existence of solutions for a boundary value problem of nonlinear sequential fractional integro-differential equations are investigated. Two examples are given to illustrate our results.


Introduction
Fractional differential equations have attracted much attention and have been the focus of many studies due mainly to their varied applications in many fields of science and engineering. In other words, fractional differential equations are widely used to describe many important phenomena in various fields such as physics, biophysics, chemistry, biology, control theory, economy and so on; see [14,19,23,29,33]. For an extensive literature in the study of fractional differential equations, we refer the reader to [2,11,15,16,18,20,21,24,26,30,32]. However, it should be noted that in recent years, there have been many works related to fractional integro-differential equations, see [1,3,4,6,8,12,17,22,28,29] and the references therein. For some interesting and considerable applied works, we refer to [5,7,9,10].
In [13], Baleanu et al. studied the existence and uniqueness of solutions for the multiterm nonlinear fractional integro-differential equation In [31], Wang et al. proved the existence and uniqueness of positive solutions for the following fractional integro-differential equation: Motivated by the previous results, we discuss in this paper the existence of solutions for the following nonlinear sequential fractional boundary value problem:

Preliminaries
For convenience, in this section we recall some basic definitions and properties of the fractional calculus theory and auxiliary lemmas which will be used throughout this paper, see [23,25,27].

Definition 2.1
The Caputo fractional derivative of order α > 0 of a continuous function u : (0, ∞) → R is defined by provided the right-hand side is pointwise defined on (0, ∞).

Definition 2.2
The Riemann-Liouville fractional integral of order α > 0 of a continuous function u : (0, ∞) → R is defined by provided the right-hand side is pointwise defined on (0, ∞).

Lemma 2.1
If α > 0, then the differential equation c D α u(t) = 0 has a unique solution given by where c i ∈ R, i = 0, 1, . . . , n -1 (n is the smallest integer such that n ≥ α).
has the unique solution given by Proof In view of Lemma 2.2, FBVP (2.1) is equivalent to the following integral equation: Differentiating both sides of (2.3), we get Using the boundary conditions u(1) = u(0) = u (1) = 0, we obtain Substituting the values of c 0 , c 1 , c 2 in (2.3) we obtain (2.2). This completes the proof. Let X = C(I) be the space of all continuous real-valued functions on I = [0, 1] endowed with the norm u = max t∈I |u(t)|.

Theorem 3.2
Assume that α + β -2 ≥ 0 and there exists a nonnegative function θ (t) ∈ L 1 (0, 1) such that for all t ∈ [0, 1] and t, x, y, z, t , y , z ∈ R. Then problem (1.1) has at least one solution on X whenever and set = max{f (t, 0, 0, 0) : t ∈ I}. Consider the set B R = {u ∈ X : u ≤ R}, then B R is a closed, bounded, and convex set of X. We define the operators A and B on X as

s, u(s), ϕu(s), ψu(s) ds.
For any u ∈ B R and t ∈ I, we get with the help of inequality (3.1) Hence, we get Similarly, we estimate Bv . Let v ∈ B R and t ∈ I, then Hence, we get Taking estimates (3.3) and (3.4) into account, we get for any u, v ∈ B R and t ∈ I, Now, we prove that B is a contraction. Let v, u ∈ B R and t ∈ I. Then, thanks to (3.1), it yields so by (3.2) we conclude that B is a contraction. Let us prove that A is compact and continuous. The continuity of f implies that A is continuous. Also A is uniformly bounded on B R , indeed, from (3.3) we have We have

s, u(s), ϕu(s), ψu(s) ds
Hence, if t 2 → t 1 , then |Au(t 2 ) -Au(t 1 )| → 0. Then A is equicontinuous and so, by Arzela-Ascoli theorem, we deduce that A is compact on B R . So the operator A is completely continuous. Thus, by Theorem 3.1, problem (1.1) has at least one solution in X. The proof is complete.
Conclusion. In the present work, we have studied the existence of solutions for a fractional sequential boundary value problem. To demonstrate the existence results, we transformed the posed problem into a sum of a contraction and a compact operator, then we applied the Krasnoselskii's fixed point theorem. We ended the article with some numerical examples illustrating the obtain results.