Existence results for coupled nonlinear fractional differential equations of different orders with nonlocal coupled boundary conditions

This paper is concerned with the solvability of coupled nonlinear fractional differential equations of different orders supplemented with nonlocal coupled boundary conditions on an arbitrary domain. The tools of the fixed point theory are applied to obtain the criteria ensuring the existence and uniqueness of solutions of the problem at hand. Examples illustrating the main results are presented.

Recently, in [32], the authors studied a new class of coupled systems of mixed-order fractional differential equations equipped with nonlocal multi-point coupled boundary conditions of the form: In [33], the existence and uniqueness of solutions for the following system were investigated by using the Leray-Schauder alternative and the contraction mapping principle: In the present research, inspired by the published articles [32] and [33], we consider a coupled system (1.1) consisting of fractional differential equations of two different fractional-orders: (1,2] and (2,3] on an arbitrary domain supplemented with a new set of coupled nonlocal multi-point boundary conditions. We emphasize that the present study is novel and more general, and contributes significantly to the existing literature on the topic. Moreover, several new results follow as special cases of the results presented in this work (see Sect. 5).
The rest of the paper is organized as follows: In Sect. 2 we recall some definitions and prove a basic lemma helping us to transform system (1.1) into equivalent integral equations. The main results are established in Sect. 3. An existence result is proved via the Leray-Schauder alternative, and the existence of a unique solution is established by using Banach's contraction mapping principle. Examples illustrating the obtained results are also constructed in Sect. 4.

Preliminaries
Let us begin this section with some definitions related to our study [34].

Definition 2.1
The Riemann-Liouville fractional integral of order ω ∈ R (ω > 0) for a locally integrable real-valued function h defined on -∞ ≤ a < t < b ≤ +∞, denoted by I ω a + h, is defined by where denotes the Euler gamma function.
is given by a pair of integral equations Proof The solution of system (2.1) can be written as

Main results
Let X = C([a, b], R) be a Banach space endowed with the norm In view of Lemma 2.4, we define an operator T : Here (X × X, (x, y) ) is a Banach space equipped with the norm (x, y) = x + y , x, y ∈ X.
In our first result, we establish the existence of a solution for system (1.1) by applying the Leray-Schauder alternative [35]. For computational convenience, we set where (2.4)). In addition, we assume that:

Theorem 3.2 Let = 0 ( is defined by
(H 1 ) ϕ, ψ : [a, b] × R × R → R are continuous functions and there exist real constants k i , γ i ≥ 0 (i = 1, 2) and k 0 > 0, γ 0 > 0 such that, for all t ∈ [a, b] and x, y ∈ R, Proof Observe that the continuity of the operator T : X × X → X × X follows that of the functions ϕ and ψ. Next, let ⊂ X × X be bounded such that ϕ t, x(t), y(t) ≤ K 1 , ψ t, x(t), y(t) ≤ K 2 , ∀(x, y) ∈ , for positive constants K 1 and K 2 . Then, for any (x, y) ∈ , we have

x(s), y(s) ds
which implies that In a similar way, in view of notation (3.1), we have

x(s), y(s) ds
which yields From the above argument, we deduce that the operator T is uniformly bounded, as Next, we show that T is equicontinuous. Let t 1 , t 2 ∈ [a, b] with t 1 < t 2 . Then we have Analogously, we can obtain Clearly the right-hand sides of inequalities (3.3) and (3.4) tend to zero independently of x and y as t 1 → t 2 . This shows that the operator T(x, y) is equicontinuous. In consequence, we deduce that the operator T(x, y) is completely continuous. Finally, we consider the set P = {(x, y) ∈ X × X : (x, y) = νT(x, y), 0 ≤ ν ≤ 1} and show that it is bounded.
Let (x, y) ∈ P with (x, y) = νT(x, y). For any t ∈ [a, b], we have x(t) = νT 1 (x, y)(t), y(t) = νT 2 (x, y)(t). Then, by (H 1 ), we have and In consequence of the above arguments, we deduce that In the next theorem we prove the existence of a unique solution of system (1.1) by using the contraction mapping principle due to Banach. (2.4)). In addition, we assume that:

Theorem 3.3 Let = 0 ( is defined by
(H 2 ) ϕ, ψ : [a, b] × R × R → R are continuous functions and there exist positive constants l 1 and l 2 such that, for all t ∈ [a, b] and x i , y i ∈ R, i = 1, 2, we have In the first step, we show that Similarly, we get ψ t, x(t), y(t) ≤ l 2 x + y + N 2 ≤ l 2 r + N 2 .
Now, for (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × X and for any t ∈ [a, b], we get which implies that Similarly, we find that T 2 (x 2 , y 2 ) -T 2 (x 1 , y 1 ) ≤ (L 2 l 1 + M 2 l 2 ) x 2x 1 + y 2y 1 . (3.10) It follows from (3.9) and (3.10) that From the above inequality and (3.8), we deduce that T is a contraction. Hence it follows by Banach's fixed point theorem that there exists a unique fixed point for the operator T, which corresponds to a unique solution of problem (1.1) on [a, b]. This completes the proof.

Conclusions
In this paper, we have studied the existence of solution for a boundary value problem consisting of a coupled system of nonlinear fractional differential equations of different orders and five-point nonlocal coupled boundary conditions on an arbitrary domain. The given problem is transformed into an equivalent fixed point problem, which is solved by applying the standard tools of the modern functional analysis to obtain the existence and uniqueness results for the original problem. Our results are not only new in the given setting, but also reduce to some new results as special cases by fixing the parameters involved in the boundary conditions. For example, if we take p 1 = 0 = p 2 in the obtained results, we get the ones associated with four-point nonlocal boundary conditions: x(a) = 0, x(b) = 0, y(θ 1 ) = 0, y(θ 2 ) = 0, y(b) = 0.