On Nonlinear Fractional Difference Equation with Delay and Impulses

: In this paper, we establish the existence results for a nonlinear fractional difference equation with delay and impulses. The Banach and Schauder’s ﬁxed point theorems are employed as tools to study the existence of its solutions. We obtain the theorems showing the conditions for existence results. Finally, we provide an example to show the applicability of our results.


Introduction
Discrete fractional calculus studies have been an interesting field of present day, because some real-world phenomena are described by using fractional difference operators (see papers [1][2][3] and the references therein). Basic knowledge of fractional difference calculus can be found in [4]. The extension of this field can be found in  and references cited therein.
For the development of the fractional difference equations theory, which is the discrete case of fractional differential equations, there are still few publications. However, there are some recent papers studying fractional difference equations with delay. In 2017, Kaewwisetkul et al. [38] studied boundary value problems for Caputo fractional functional difference equations with delay. In 2018, Wu et al. [39] proposed the finite-time stability of discrete fractional delay systems, Alzabut et al. [40] studied nonlinear delay fractional difference equations with applications on the discrete fractional Lotka-Volterra competition model, Alzabut et al.. [41] investigated the application on the uniqueness of solutions for nonlinear delay fractional difference system, and Luo et al. [42] considered the uniqueness and finite-time stability of solutions for a class of nonlinear fractional delay difference systems.
In particular, the fractional difference equations with delay and impulses have not been studied extensively. In 2018, Wu et al. [43] studied a linear fractional delay difference equations with impulse. These results are incentives for research. In this paper, we propose a nonlinear fractional difference equation with delay and impulses of the form: where N 0,T := {0, 1, . . . , T}, α, β ∈ (0, 1), ,T+α × C r × R, R , I k : C r → R and ψ is an element of the space: For r ∈ N 0,T+1 , let C r be the Banach space of all continuous functions ψ : N α−r−1,α−1 → R with the norm: If u : N α−r−1,α−1 → R, then for any t ∈ N α−1,T+α , we define the element u t of C r as, We aim to prove the existence results to the problem of Equation (1) by using the Banach and Schauder's fixed point theorems. Finally, we present an example in the last section.

Preliminaries
In this section, we recall some notations, definitions, and lemmas used in the main results. Definition 1. The generalized falling function is defined by: .
If t + 1 − α is a pole of the Gamma function and t + 1 is not a pole, then t α = 0.

Definition 3.
For α > 0, N ∈ N is satisfied with 0 ≤ N − 1 < α < N and f defined on N a , the α-order Riemann-Liouville fractional difference of f is defined by: where t ∈ N a+N−α . The α-order Caputo fractional difference of f is defined by: Lemma 1. [5] Assume that α > 0 and f defined on N a . Then, Next, we aim to find a solution of the linear variant of the mixed problem in Equation (1) as follows.

Existence and Uniqueness Result
In this section, we employ the Banach fixed point theorem to consider the existence and uniqueness result for the problem in Equation (1). Define the Banach space: with the norm defined by: where u = max In view of the definitions of u t and ψ, we have: Thus, we obtainL Next, define an operator T : X → X as: Firstly, we provide some basic knowledge that is used in this section as follows.

Definition 4.
A mapping S from a subset M of a Banach space X into X is called a contraction mapping (or simply a contraction) if there exists a positive number α < 1 such that: If one can prove that T has fixed point, we can conclude that the problem of Equation (1) has a solution.
Theorem 1. Assume the following properties: (H1) There exists a constant > 0 such that: for each u 1 , v 1 ∈ C r and u 2 , v 2 ∈ R. (H2) There exists a constant λ > 0 such that for each u t , v t ∈ C r and k = 1, 2, ..., p.
Then, the problem of Equation (1) has an unique solution.
Proof. We will show that T is a contraction. Letting, for each t ∈ N α−1,T+α , we obtain: Taking the fractional difference of order β for Equation (13) and substituting t = t − β + 1, we get: For each t ∈ N α−1,T+α , we obtain: Obviously, for each t ∈ N α−r−1,α−1 , we get (T u)(t) − (T v)(t) = 0. Therefore, we have: By (H3), it implies that T is a contraction. Therefore, by Banach fixed point theorem, T has a fixed point which is a unique solution of the problem in Equation (1).

Existence of at Least One Solution
In this section, we also present the existence of at least one solution of Equation (1) by using the Schauder's fixed point theorem. Firstly, we provide some basic knowledge that is used in this section as follows. Lemma 5. [44] A bounded set in R n is relatively compact, a closed bounded set in R n is compact. Lemma 6. [45] (Schauder's fixed point theorem) Let (D, d) be a complete metric space, U be a closed convex subset of D, and T : D → D be the map such that the set Tu : u ∈ U is relatively compact in D. Then the operator T has at least one fixed point u * ∈ U: Tu * = u * .
The following notations are defined for using in the sequel.
Then boundary value problem of Equation (1) has at least one solution.
Proof. The proof is organized into three steps as follows.
Step I. We verify that T map bounded sets into bounded sets. Let max |I k (u t k −1 )| = N for k = 1, 2, ..., p.
Suppose that (H4) holds, we choose a constant: and define the P = u ∈ X : u ≤ R, R > 0 . For any u ∈ P, we have: Hence, we have: This implies that T : P → P.
Therefore, by Theorem 1, the boundary value problem of Equation (34) has an unique solution.

Conclusions
We established the conditions for the existence and uniqueness of a solution for a nonlinear fractional difference equation with delay and impulses in Equation (1) by using the Banach fixed point theorem, and the conditions of at least one solution by using the Schauder's fixed point theorem. Our problem contained both delay and impulses, which is a new idea.