Existence of solutions for a coupled system of fractional differential equations by means of topological degree theory

This paper investigates the existence of solutions for a coupled system of fractional differential equations. The existence is proved by using the topological degree theory, and an example is given to show the applicability of our main result.


Introduction
In this manuscript, the following coupled system of fractional differential equations is discussed: Fractional differential equations are widely used in many fields such as chemistry, physics, biology, and optimization theory [1][2][3][4]. In addition, coupled systems of fractional differential equations have attracted particular concern from scholars considering their appearance in the mathematical modeling of physical phenomena like chaos synchronization [5], anomalous diffusion [6], disease models [7], and so on. The existence theory to fractional differential equations with integral boundary conditions has widespread applications in optimization theory, many researchers have studied [8][9][10][11][12][13], and the existence of solutions is the basis of studying the stability and numerical solutions of differential equations [14]. For the existence of solutions of fractional differential equations, the authors use diverse methods, such as fixed point theory [15][16][17][18][19], upper and lower solutions method [20], monotone iterative technique and Mawhin's continuation theorem [21], and topological degree theory [22]. When studying the existing literature, we find that fractional differential equations with integral boundary conditions are not properly tested via topological degree theory. Thus we investigate the existence result to a coupled system of fractional differential equations(1.1) through applying topological degree theory.
Bashiri et al. [23]investigated the existence of solutions for fractional differential equations by means of the coupled fixed point theorem of Krasnoselskii type where D θ denotes the Riemann-Liouville fractional derivative, θ ∈ (0, 1), α > 0.
Ahmad et al. [24]established existence results as well as studied qualitative aspects of the proposed coupled system of fractional hybrid delay differential equations where A = [0, τ ], C D +0 , τ > 0 is Caputo's derivative, and r 0 , h 0 are real numbers, while the delay parameter is denoted by ν ∈ (0, 1). Muthaiah et al. [25]considered the existence and Hyers-Ulam type stability results for the nonlinear coupled system of Caputo-Hadamard type fractional differential equations where C D (·) denotes the Caputo-Hadamard fractional derivative, H I (·) denotes the Hadamard fractional integrals, 2 < , ς ≤ 3, 0 < 1 , ς 1 < 1, α 1 , α 2 , β 1 , β 2 are real constants and ζ j , υ j , j = 1, 2, . . . , k -2, are positive real constants. The consequence of existence is obtained by employing the alternative of Leray-Schauder and Krasnoselskii's, whereas the uniqueness result is based on the principle of Banach contraction mapping. Motivated especially by the aforementioned work, we consider the existence of solutions to a coupled system of fractional differential equations (1.1). According to our literature review, no scholars have studied equation (1.1), the results are entirely new. The remainder of this paper is as follows. In the second part, we display some definitions, facts, and results. We confirm the existence of solutions for system (1.1) in the third part. Finally, we provide an example to prove our results.

Preliminaries
In this part, we recollect a number of facts, definitions, and conclusions. Let the map y → h(t, x, y) is continuous for each y ∈ R. Let X be a Banach space and B ⊂ P(X), where P(X) stands for the family of all bounded subsets of X. Next, we introduce some concepts.  (2) strict α-contraction if there exists 0 ≤ k < 1 such that α(F (S)) ≤ kα(S) for all bounded subsets S ⊆ ; (3) α-condensing if α(F (S)) < α(S) for all bounded subsets S ⊆ with α(S) > 0. In other words, α(F (S)) ≥ α(S) implies α(S) = 0.
All classes of strict α-contraction F : → X and all classes of α-condensing maps F : → X are represented by C α ( ) and C α ( ), respectively. Then C α ( ) ⊂ C α ( ) and Further, F will be a strict contraction if k < 1.
If is a bounded set in X, so we have r > 0 such that ⊂ B r (0), then Consequently, F has at least one fixed point, and the set of the fixed points of F lies in B r (0).

Definition 2.7 ([28]) The fractional integral of order
provided that the right-hand side is pointwise defined on (0, ∞).

Main results
In this part, we discuss the existence result for (1.1).
The space X = C([0, 1], R) of all continuous functions is a Banach space under the topological norm x = sup{|x(t)| : t ∈ [0, 1]} and the product space X × X is a Banach space under the norm (x, y) = x + y or (x, y) = max{ x , y }.
In order to get the result of our result, we need the following hypotheses.
. . , n -1, then the consequence of fractional differential equations (1.1) is a conclusion of the following system of integral equations: Proof Applying the fractional integrable operator I θ on the equation of system (1.1) and through applying Lemma 2.9, we get By applying the initial conditions x (0) = 0 and ∂ i f (t,x(t)) ∂t i | t=0 = 0, we obtain C 1 = 0 and Now, applying the boundary conditions By rearranging, we obtain Thus equation (3.2) becomes Analogously, following the same steps in the process for the second equation of system (1.1), we get Define the operator F, H, T : X × X → X × X by

Theorem 3.2 The operator F is Lipschitz with constant k. Therefore F is α-Lipschitz with the equal constant k and meets the following growth condition:
Proof Now, we shall display that the operator F is Lipschitz with constant k. Let x 1 , x 2 ∈ X, then we get By using conditions (H 1 ) and (H 5 ), we can write where k 2 = λ 2 + (|b 2 |+|c 2 |)λ 2 |a 2 -(b 2 +c 2 )| + b ψ |a 2 -(b 2 +c 2 )| (θ+1) . Thus ). Then F satisfies the Lipschitz condition, thus F is Lipschitz with constant k. According to Proposition 2.4, F is α-Lipschitz with constant k.
Moreover, we get
Proof Consider a bounded subset of X × X as Let {(x n , y n )} be a sequence in B r such that (x n , y n ) → (x, y) as n → ∞. To show that H is continuous, we consider The same as the other terms approach 0 as n → ∞, thus Then H 1 is continuous. By the same steps as above, one lightly gets that H 2 (y n ) -H 2 (y) → 0 as n → ∞. which means that H is continuous. Moreover, by (H 2 ), we have Similarly, Hence H satisfies the growth condition.

Theorem 3.4 The operator H
Proof Let be a bounded subset of B r ⊆ X × X and {(x n , y n )} be a sequence in , through applying the growth condition of H, it is obvious that H( ) is uniformly bounded in X × X. Now, we need to reveal that H is equicontinuous. Let 0 ≤ t ≤ τ ≤ 1, then we obtain Similarly, Taking limit as t → τ , we get , ∈ [0, 1), where q 1 , q 2 ∈ [0, 1). Thus R is bounded in X × X. According to Theorem 2.6, there exists r > 0 such that R ⊂ B r (0), then Therefore, T has at least one fixed point, then coupled system (1.1) has at least one solution.