Existence Results for a System of Coupled Hybrid Fractional Differential Equations

This paper studies the existence of solutions for a system of coupled hybrid fractional differential equations with Dirichlet boundary conditions. We make use of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example.


Introduction
Fractional calculus is the study of theory and applications of integrals and derivatives of an arbitrary (noninteger) order. This branch of mathematical analysis, extensively investigated in the recent years, has emerged as an effective and powerful tool for the mathematical modeling of several engineering and scientific phenomena. One of the key factors for the popularity of the subject is the nonlocal nature of fractional-order operators. Due to this reason, fractionalorder operators are used for describing the hereditary properties of many materials and processes. It clearly reflects from the related literature that the focus of investigation has shifted from classical integer-order models to fractionalorder models. For applications in applied and biomedical sciences and engineering, we refer the reader to the books [1][2][3][4]. For some recent work on the topic, see  and the references therein. The study of coupled systems of fractionalorder differential equations is quite important as such systems appear in a variety of problems of applied nature, especially in biosciences. For details and examples, the reader is referred to the papers [26][27][28][29][30][31][32][33] and the references cited therein.
Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers (see [34][35][36][37]).
The aim of this paper is to obtain some existence results for the given problem. Our first theorem describes the uniqueness of solutions for the problem (1) by means of Banach's fixed point theorem. In the second theorem, we apply Leray-Schauder's alternative criterion to show the 2 The Scientific World Journal existence of solutions for the given problem. The paper is organized as follows. Section 2 contains some basic concepts and an auxiliary lemma, an important result for establishing our main results. In Section 3, we present the main results.

Preliminaries
In this section, some basic definitions on fractional calculus and an auxiliary lemma are presented [1,2]. Definition 1. The Riemann-Liouville fractional integral of order for a continuous function is defined as provided that the integral exists.

Definition 2.
For at least -times continuously differentiable function : [0, ∞) → R, the Caputo derivative of fractional-order is defined as where [ ] denotes the integer part of the real number .

Main Results
In view of Lemma 3, we define an operator Θ : where In the sequel, we need the following assumptions.
The Scientific World Journal 3 For brevity, let us set Now we are in a position to present our first result that deals with the existence and uniqueness of solutions for the problem (1). This result is based on Banach's contraction mapping principle.

4
The Scientific World Journal Example 5. Consider the following coupled system of hybrid fractional differential equations: sin ( ≈ 0.3762217 < 1. In our second result, we discuss the existence of solutions for the problem (1) by means of Leray-Schauder alternative.
Lemma 6 (Leray-Schauder alternative [38, page 4]). Let F : G → G be a completely continuous operator (i.e., a map that is restricted to any bounded set in G is compact). Let P(F) = { ∈ G : = F for some 0 < < 1}. Then either the set P(F) is unbounded or F has at least one fixed point.
Proof. We will show that the operator Θ : U × V → U × V satisfies all the assumptions of Lemma 6. In the first step, we prove that the operator Θ is completely continuous. Clearly, it follows by the continuity of functions 1 , 2 , ℎ 1 , and ℎ 2 that the operator Θ is continuous.
Let M ⊂ U × V be bounded. Then we can find positive constants 1 and 2 such that ℎ 1 ( , ( ) , ( )) ≤ 1 , ℎ 2 ( , ( ) , ( )) ≤ 2 , Thus for any , ∈ M, we can get Θ 1 ( , ) ( ) which yields In a similar manner, one can show that From the inequalities (27) and (28), we deduce that the operator Θ is uniformly bounded. Now we show that the operator Θ is equicontinuous. For that, we take 1 , 2 ∈ [0, 1] with 1 < 2 and obtain The Scientific World Journal 5 which tend to 0 independently of ( , ). This implies that the operator Θ( , ) is equicontinuous. Thus, by the above findings, the operator Θ( , ) is completely continuous. In the next step, it will be established that the set P = {( , ) ∈ U × V | ( , ) = Θ( , ), 0 ≤ ≤ 1} is bounded. Let ( , ) ∈ P; then we have ( , ) = Θ( , ). Thus, for any ∈ [0, 1], we can write which imply that In consequence, we have ‖ ‖ + = (] 1 0 + ] 2 0 ) which, in view of (13), can be expressed as This shows that the set P is bounded. Hence all the conditions of Lemma 6 are satisfied and consequently the operator Θ has at least one fixed point, which corresponds to a solution of the problem (1). This completes the proof.