L-Solutions of Boundary Value Problems for Implicit Fractional Order Differential Equations with Integral Conditions

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications of differential equations of fractional order in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [4, 14, 17, 18, 20]). There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Abbas et al. [3], Kilbas et al. [15], Lakshmikantham et al. [16], and the papers by Agarwal et al. [1, 2], Benchohra et al. [5], and the references therein. To our knowledge, the literature on integral solutions for fractional differential equations is very limited. El-Sayed and Hashem [13] studies the existence of integral and continuous solutions for quadratic integral equations. El-Sayed and Abd El Salam[12] considered L-solutions for a weighted Cauchy problem for differential equations involving the Riemann-Liouville fractional derivative. Motivated by the above papers, in this paper we deal with the existence of solutions for boundary value problem (BVP for short), for fractional order implicit differential equation


Introduction
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications of differential equations of fractional order in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [4,14,17,18,20]). There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Abbas et al. [3], Kilbas et al. [15], Lakshmikantham et al. [16], and the papers by Agarwal et al. [1,2], Benchohra et al. [5], and the references therein.
To our knowledge, the literature on integral solutions for fractional differential equations is very limited. El-Sayed and Hashem [13] studies the existence of integral and continuous solutions for quadratic integral equations. El-Sayed and Abd El Salam [12] considered L p -solutions for a weighted Cauchy problem for differential equations involving the Riemann-Liouville fractional derivative. Motivated by the above papers, in this paper we deal with the existence of solutions for boundary value problem (BVP for short), for fractional order implicit differential equation where f, g 1 , g 2 : J × IR × IR → IR is a given functions, and c D α is the Caputo fractional derivative. This paper is organized as follows. In Section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following section. In Section 3, we give two results, the first one is based on Schauder's fixed point theorem (Theorem 9) and the second one on the Banach contraction principle (Theorem 10). An example is given in Section 4 to demonstrate the application of our main results. These results can be considered as a contribution to this emerging field.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Let L 1 (J) denotes the class of Lebesgue integrable functions on the interval J = [0, T ], with the norm ∥u∥ L 1 = ∫ J |u(t)|dt. Definition 1. ( [15,19]) The fractional (arbitrary) order integral of the function h ∈ L 1 ([a, b], R + ) of order α ∈ R + is defined by where Γ(.) is the gamma function. When a = 0, we write Here n = [α] + 1 and [α] denotes the integer part of α. If α ∈ (0, 1], then The following properties are some of the main ones of the fractional derivatives and integrals.
The following theorems will be needed.
Theorem Schauder fixed point theorem [11]. Let E a Banach space and Q be a convex subset of E and T : Q −→ Q is compact, and continuous map. Then T has at least one fixed point in Q.

Existence of solutions
Let us start by defining what we mean by an integrable solution of the problem (1) − (3). (1) and (2) and (3).
For the existence of solutions for the problem (1) − (3), we need the following auxiliary lemma.
is equivalent to the integral equation where x is the solution of the functional integral equation

|G(t, s)|ds, t ∈ J,
and let us introduce the following assumptions: Let us introduce the following assumptions: Our first result is based on Schauder fixed point theorem.
Proof. Transform the problem (1) − (2) into a fixed point problem. Consider the operator Let

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Consider the set Clearly B r is nonempty, bounded, convex and closed. Now, we will show that HB r ⊂ B r , indeed, for each x ∈ B r , from assumption (H2) and (13) we get Then HB r ⊂ B r . Assumption (H1) implies that H is continuous. Now, we will show that H is compact, this is HB r is relatively compact. Clearly HB r is bounded in L 1 (J, IR), i.e condition (i) of Kolmogorov compactness criterion is satisfied. It remains to show (Hx Let x ∈ B r , then we have dt . Since x ∈ B r ⊂ L 1 (J, IR) and assumption (H2) that implies f ∈ L 1 (J, IR), then we have Then by Kolmogorov compactness criterion, HB r is relatively compact. As a consequence of Schauder's fixed point theorem the BVP (1) − (2) has at least one solution in B r .
The following result is based on the Banach contraction principle.
Theorem 10. Assume that (H1) and the following condition hold.
Proof. We shall use the Banach contraction principle to prove that H defined by (9) has a fixed point. Let x, y ∈ L 1 (J, IR), and t ∈ J. Then we have, Consequently by (15) H is a contraction. As a consequence of the Banach contraction principle, we deduce that H has a fixed point which is a solution of the problem (1) − (2).