Existence Results for a System of Coupled Hybrid Differential Equations with Fractional Order

This paper studies the existence of solutions for a system of coupled hybrid fractional differential equations. 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.
is 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, fractional order 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 fractional order 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 [5][6][7][8][9][10][11][12] and the references therein. e study of coupled systems of fractional order 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 [13,14] and the references cited therein.
Hybrid fractional differential equations have also been studied by several researchers.
is 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 [15,16].
Motivated by some recent studies on hybrid fractional differential equations, we consider the following value problem of coupled hybrid fractional differential equations: where C D α and C D β denote the Caputo fractional derivative of . e aim of this paper is to obtain some existence results for the given problem. Our first theorem describes the uniqueness of solutions for problem (1) by means of Banach's fixed point theorem. In the second theorem, we apply Leray-Schauder's alternative criterion to show the existence of solutions for the given problem. e 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.

Coupled System of Hybrid Differential Equations with Fractional Order
In this section, some basic definitions on fractional calculus and an auxiliary lemma are presented [1,2].
Definition 1 (see [6]). e fractional integral of the function h ∈ L 1 ([a, b], R + ) of order α ∈ R + is defined by where Γ is the gamma function.
Definition 2 (see [6]). For a function h given on the interval [a, b], the Caputo fractional-order derivative of h is defined by where n � [α] + 1 and [α] denote the integer part of α.

Lemma 1 (Auxiliary Lemma). Given h ∈ C([0, T], R) and
a, b, and c are real constants with a + b ≠ 0, the integral solution of the problem Proof. Applying the Caputo integral operator of the order α, we obtain the first equation in (4).
Again, substituting we get International Journal of Differential Equations us, implies that Consequently, is completes the proof.

Main Result
We define an operator F: where In the sequel, we need the following assumptions: (A 1 ) e functions f i (i � 1, 2) are continuous and bounded; that is, there exist positive numbers μ f i such that (A 2 ) ere exist real constants ρ 0 , σ 0 > 0 and ρ i , σ i ≥ 0(i � 1, 2) such that For brevity, let us set , , 3.1. First Result. Now, we are in a position to present our first result that deals with the existence and uniqueness of solutions for problem (1). is result is based on Banach's contraction mapping principle.
So Banach's fixed point theorem applies and hence the operator F has a unique fixed point. is, in turn, implies that problem (1) has a unique solution on [0, T].
is completes the proof.  International Journal of Differential Equations Lemma 2 (Leray-Schauder alternative [17]). 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) � x ∈ G: { x � λFx for some 0 < λ < 1}. en, either the set P(F) is unbounded or F has at least one fixed point.
Proof. We will show that the operator F: U × V ⟶ U × V satisfies all the assumptions of Lemma 2.
In the first step, we prove that the operator F is completely continuous. Clearly, it follows the continuity of functions f 1 , f 2 , h 1 , and h 2 that the operator F is continuous.
Let M ⊂ U × V be bounded. en, we can find positive constants N 1 and N 2 such that us, for any x, y ∈ M, we can get which yields In a similar manner, one can show that From inequalities (29) and (30), we deduce that the operator F is uniformly bounded. Now, we show that the operator F is equicontinuous. For that, we take τ 1 , τ 2 ∈ [0, T] with τ 1 < τ 2 and obtain Similarly, one can get which tends to 0 independent of (x, y).
International Journal of Differential Equations is implies that the operator F(x, y) is equicontinuous.
us, by the above findings, the operator F(x, y) is completely continuous.

Data Availability
ere is no data used in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.