A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line

In this article, we propose a coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line and study its existence result by using Schauder’s fixed point theorem. An example is provided to illustrate the results.


Introduction
Recently, many mathematicians and researchers have extensively studied fractional difference calculus since this subject can be used for describing many problems of real-world phenomena such as mechanical, control systems, flow in porous media, and electrical networks (see [1,2] and the references therein).The basic definitions and properties of fractional difference calculus are given in the book [3].The applications and developments of the theory can be found in  and the references cited therein.For example, Ferreira [20] studied the fractional difference equation of order less than one.Goodrich [22] presented the fractional difference equation of order 1 < α ≤ 2 with a constant boundary condition.Chen et al. [28] proposed the initial value problem of order less than one.Chen and Zhou [29] studied the antiperiodic boundary value problem of order 1 < α ≤ 2. Sitthiwirattham et al. [38] initiated the study of the fractional sum boundary value problem of order 1 < α ≤ 2. Sitthiwirattham [40] proposed the sequential fractional difference equation with the fractional sum boundary condition.We observe that these research works are fractional problems containing only one equation.
The study of coupled systems of fractional differential equations is an important topic in this area (see [48][49][50][51][52][53] and the references cited therein), and a recent example of the application of systems of fractional difference equations is [54].
For the boundary value problems for systems of discrete fractional equations, there are some studies in this area (see [55][56][57][58][59][60] and the references cited therein).
In this paper, we considered the coupled system of fractional difference equations: for t ∈ N 0 , subject to the nonlocal fractional sum boundary conditions on the discrete half-line N 0 : R + are given functions; φ i (u 1 , u 2 ) are given functionals; and ∆ −θ i are fractional sums of order θ i .
The goal of this study is to show the existence of solutions of the governing problems ( 5) and ( 6).The paper is structured as follows.Some definitions and basic lemmas are recalled in Section 2. In Section 3, we prove the existence of solutions of the boundary value problem (5) by employing Schauder's fixed point theorem.Finally, we present an example to illustrate our result in the last section.

Preliminaries
In what follows, the notation, definitions, and lemmas used in the main results are given.

Definition 1. The generalized falling function is defined by t
, for any t and α for which the right-hand side is defined.If t + 1 − α is a pole of the Gamma function and t + 1 is not a pole, then t α = 0.
Definition 2. For α > 0 and f defined on N a , the α-order fractional sum of f is defined by: where t ∈ N a+α and σ(s) = s + 1. Definition 3.For α > 0 and f defined on N a , the α-order Riemann-Liouville fractional difference of f is defined by: The following lemma deals with the linear variant of the boundary value problems ( 5) and ( 6) and gives a representation of the solution.
The following lemma deals with the solutions u i (t i ), i = 1, 2 of the problems ( 7)-( 9), and The solution u i (t i ) of the problems ( 7)-( 9) and ∆ β i u i (t i − β i + 1) are uniformly bounded on N α i −2 , if and only if u i (t i ) and ∆ β i u i (t i − β i + 1) satisfy the following properties: Proof.Firstly, taking the fractional difference of order (10) and (11), we obtain: and: ≤ max lim Furthermore, considering u 1 (t i ) and ∆ β 1 u i (t i ), we obtain: and: where: Consequently, the conditions (A1) and (A2) hold.We next show that the condition (A3) holds.By using the conditions (A1) and (A2), we obtain: and: where: with Therefore, the condition (A3) holds.Finally, if the conditions (A1)-(A3) hold, it is clear that u i (t i ) and ∆ β i u i (t i − β i + 1) are uniformly bounded on N α i −2 .Our proof is complete.
We next provide the following theorems used for proving the existence result for the problems ( 5) and ( 6).Theorem 2. [61] If a set is closed and relatively compact, then it is compact.Theorem 3. (Schauder's fixed point theorem [61]) If S is a convex compact subset of a normed space, every continuous mapping of S into itself has a fixed point.

Main Result
In this section, we aim to establish the existence result for the problems (5) and (6).To accomplish this, we let C i = C(N α i −2 , R) be a Banach space of all functions on N α i −2 , for each i, j ∈ {1, 2} and i = j.Obviously, the product spaces: is also the Banach space endowed with the norm defined by: where: Let U = U 1 ∩ U 2 ; clearly, the space U , (u 1 , u 2 ) U is the Banach space with the norm: Next, we define the operator F : U → U by: and: where Λ is defined as (12), and: We next make the following assumptions: (H 1 ) There exist positive numbers ip ρ 2 ∈ (−1, ρ 2 ) and M ip , m ip > 0 (i = 1, 2 and p = 1, 2, 3) such that, for each t i ∈ N α i −2 and v i ∈ R, (H 2 ) There exist positive numbers ip ρi ∈ (−1, ρ i ) and N ip , n ip > 0 (i = 1, 2 and p = 1, 2, 3) such that, for v i ∈ C i , Lemma 5. Suppose that (H 1 )-(H 3 ) hold.Then, the fixed point of F coincides with the solution of the problems (5) and (6), and F : U → U is completely continuous.
Proof.Let (u 1 , u 2 ) ∈ U , for each i, j ∈ {1, 2} and i = j.By the above assumptions (H 1 ) and (H 2 ), it follows that: and The rest of the proof follows from Lemmas 3 and 4.This implies that the fixed point of F coincides with the solution of the problems (5) and (6).
To show that F is completely continuous, we organize the proof as the following four steps.
Step I. F is well defined and maps bounded sets into bounded sets.
Hence, we obtain: and: Similarly, we have: Therefore, F i (u 1 , u 2 ) ∈ U .This implies that F : U −→ U is well defined.
Furthermore, we obtain: Hence, Thus, F maps bounded sets into bounded sets.
Step II.F is continuous.
Let > 0 be given.Since F i and φ i are continuous, then F i and φ i are uniformly continuous.Therefore, there exists δ = min δ i , δi > 0 such that, for each For each Similar to Step I, we obtain: Thus, we have: This means that each F i , i = 1, 2 is continuous.This shows F is continuous.
In order to prove that F maps bounded sets of U ⊂ C 1 × C 2 to relatively compact sets of U ⊂ C 1 × C 2 , it suffices to show that both F 1 and F 2 map bounded sets to relatively compact sets.Let Θ i ⊂ C i , i = 1, 2 be bounded sets and Θ 1 × Θ 2 ⊂ U .Recall that Θ i are relatively compact if: both χΘ i are equicontinuous on any closed subintervals of N α i −2 , • both χΘ i are equiconvergent as t i → ∞.
It has been shown from in Step I that both F i are uniformly bounded.Now, we show that F i maps bounded sets into equicontinuous sets of U .
Step III.Both F i : For any > 0, there exists δ > 0 such that, for each t i1 , where: Hence, for each t i1 , t i2 ∈ N α i −2 ∩ [a i , b i ], and u i ∈ Θ i , we have: and: Similarly, for each i, j ∈ {1, 2} and j = i, we obtain: Hence: This implies that both F 1 and F 2 are equicontinuous on D, which shows that F is equicontinuous on D. Therefore, by the Arzelá-Ascoli theorem and Theorem 2, we can conclude that F is completely continuous.
Finally, we present the main result of the article.For the sake of convenience, we set: where Ω1p , Ω2p , p = 1, 2, 3, 4 are defined as ( 48) and (49).

Example
In order to illustrate our result, we consider the following fractional sum boundary value problem:

Theorem 1 .
(Arzelá-Ascoli theorem [61]) A set of functions in C[a, b] with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on [a, b].
e e