Existence and Uniqueness of Positive Solutions for a Coupled System of Fractional Differential Equations

In this article, we investigate a boundary value problem for a coupled dierential system of fractional order that the nonlinear term depends on the unknown functions as well as their lower order fractional derivatives. Firstly, we give Green’s functions and prove their properties; secondly, the existence and uniqueness of positive solutions are obtained by using some xed point theorems. In addition, two examples are presented to demonstrate the application of our main results.


Introduction
Fractional di erential equations have received considerable attention in recent years due to their wide applications in engineering, physics, economy, and control theory (see the monographs of Das [1], Kilbas et al. [2], and Podlubny [3]).
ere are a large number of papers dealing with the solvability of nonlinear fractional di erential equations, such as [4][5][6][7][8] and references therein. e study of coupled di erential systems of fractional order is very signi cant because this kind of system can often occur in applications. Recently, a series of investigations on boundary value problems for fractional di erential equation systems with the nonlinearity depending on the fractional derivative have been presented. Most of them are devoted to the solvability of nonlinear fractional di erential equation systems by using techniques of nonlinear analysis [9][10][11][12][13][14][15]. It is worth mentioning that the nonlinear terms in these papers are independent of the fractional derivative of the unknown functions. But the opposite case is more di cult and complicated, and this work attempts to deal exactly with this case.
In [12], the author discussed the existence of solutions for a coupled di erential system of fractional order: Motivated by the above work, in this paper, we consider the system of nonlinear fractional di erential equations:

Preliminaries
For the convenience of the readers, we present here the necessary definitions from fractional calculus theory. ese definitions can be found in the recent literature [2-5, 7, 17].
Remark 1 (see [2,3]). e following properties are useful for our discussion: In the following, we present Green's function of the fractional differential equation boundary value problem.
e function u(t) is said be a solution of BVP (9) and (10) if it satisfies the factional differential equation (9) and boundary conditions (10) in the classical sense. We may apply Lemma 2 to reduce equation (9) to an equivalent integral equation for some C 1 , C 2 , C 3 ∈ R. Consequently, the general solution of equation (9) is one has erefore, the unique solution of problem (9) and (10) is For t ≤ ξ < 1, one has For t ≥ ξ, one has Discrete Dynamics in Nature and Society e proof is finished. □ Lemma 4. e function G 1 (t, s) defined by equation (11) possesses the following properties: Discrete Dynamics in Nature and Society For t ≤ s, ξ ≤ s, g 4 (t, s) > 0 holds clearly. erefore, G 1 (t, s) > 0, for t, s ∈ (0, 1).
if t > ξ, by 0 < μξ α− 2 < 1, 0 < ξ < 1, there is Discrete Dynamics in Nature and Society For t ≤ s ≤ ξ, by 0 < μξ α− 2 < 1, 0 < ξ < 1, there is On the other hand, when 0 < s ≤ t < 1, s ≤ ξ, we have When 0 < ξ ≤ s ≤ t < 1, we obtain 6 Discrete Dynamics in Nature and Society In the similar discussion, we can deduce where e proof is finished. Similarly, we can obtain G 2 (t, s) if α is replaced by β, e function G 2 (t, s) defined by (31) has the same properties as G 1 (t, s). □ Lemma 5 (see [17]). If S is a closed, bounded and convex subset of a Banach space X and T: S ⟶ S is completely continuous, then T has a fixed point in S.

Main Results and Proof
In this section, we establish the existence and uniqueness of positive solutions for (3).
By using Green's functions G i (t, s)(i � 1, 2) from Section 2, problem (3) can be written equivalently as the following nonlinear system of integral equations: We define the space X � u(t)| ∈ C 1 [0, 1] and D δ u(t) ∈ C[0, 1], 1 < δ ≤ 2} and the norm be a Cauchy sequence in the space (X, ‖ · ‖ X ); then clearly u n Note that (34) By the convergence of D δ u n ∞ n�1 , we have lim n⟶∞ I δ D δ u n (t) � I δ w(t) uniformly for t ∈ [0, 1]. On the other hand, by Remark 1 one has I δ D δ u n (t) � u n (t). Hence, v(t) � I δ w(t), Remark 1 implies that it is equivalent to the relation w � D δ v. is completes the proof.
Also we define the space and T(u, v) � (T 1 v, T 2 u), (u, v) ∈ X × Y. us, the solutions of our problem (3) are the fixed points of the operator T. Now, we give the main result of this work. For convenience, we introduce the following notations: functions and f(t, 0, 0), g(t, 0, 0) are not identically zero. Suppose that one of the following conditions is satisfied.
Proof. First, let condition (H 1 ) be valid. Define where R ≥ max (3Ac 1 ) (1/1− ρ 1 ) , (3Ac 2 ) (1/1− ρ 2 ) , (3Bd 1 ) (1/1− θ 1 ) , (3Bd 2 ) (1/1− θ 2 ) , 3k, 3l}., and Observe that S is a closed convex set of the Banach space X × Y. Now we prove that T: S ⟶ S. For any (u, v) ∈ S, from the nonnegativeness of f, g and Lemma 4, it is easy to know that Again by Remark 1 and Lemma 4, we have 8 Discrete Dynamics in Nature and Society Hence, Similarly, one has Repeating arguments similar to that above we can obtain Consequently, we have T: S ⟶ S. In view of the continuity of G 1 , G 2 , f, and g, it is easy to see that the operator T is continuous.

Conclusion
e existence and uniqueness of positive solutions for a coupled differential system of fractional order with threepoint boundary conditions are obtained. e main tools are some fixed point theorems in cones. It is worth mentioning that the nonlinear terms in the system are dependent on the fractional derivative of the unknown functions.

Data Availability
e data used to support the findings of this study are included within the article.

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