Existence and Hyers–Ulam Stability of Solutions for a Mixed Fractional-Order Nonlinear Delay Difference Equation with Parameters

This paper focuses on a kind of mixed fractional-order nonlinear delay diﬀerence equations with parameters. Under some new criteria and by applying the Brouwer theorem and the contraction mapping principle, the new existence and uniqueness results of the solutions have been established. In addition, we deduce that the solution of the addressed equation is Hyers–Ulam stable. Some results in the literature can be generalized and improved. As an application, three typical examples are delineated to demonstrate the eﬀectiveness of our theoretical results.

{ }, and f: [α − 1, b + α] N α− 1 × R ⟶ R is a continuous function with respect to the second variable, when f(·, x(·)) satisfies Lipschitz condition, that is to say, there exists a constant L > 0 such that en, the boundary value problem (2) has a unique solution if holds. Secondly, the authors researched the Hyers-Ulam stability of solutions for the following boundary value problem: and let x be a solution of boundary value problem (4).
en, the solution is the Hyers-Ulam stable provided that However, the nonlinear terms in (1), (2), and (4) are too simple to portray the development of things well. We adopt mixed fractional equation which makes the model more generalized, such as in [36][37][38]. Inspired by the abovementioned articles, in this paper, we are concerned with the existence, uniqueness, and Hyers-Ulam stability of solutions for the following discrete fractional equation: where λ, η, and ρ are positive real numbers, and t ∈ [0, m 0 denotes the fractional difference operator of order ], and Δ − β 0 is the βth fractional sum operator, ], β ∈ (1, 2] are given. Equation (6) is an important parameters system and also is a mixed fractional system, which is quite different from (1) and (2), and it can enrich the description of the mathematical model. Besides, the most interesting thing is that, in eorem 2, we find the necessary conditions are only dependent on λ (independent on η and ρ).
Compared with some new achievements in the articles, such as [16][17][18]25], the major contributions of our research contain at least the following three: (1) e Hyers-Ulam stability is introduced into the mixed fractional order nonlinear difference equation.
(2) e model we are concerned with is more generalized, and some ones in the articles are the special cases of it. Moreover, we provide more ecumenical boundary value conditions in researching Hyers-Ulam stability of solutions for the fractional difference equation. us, the comprehensive model is originally discussed in the present paper. (3) A ground-breaking approach based on contraction mapping and the Brouwer theorem is utilized to discuss the existence and uniqueness of the solutions for the mixed fractional-order difference equation. e results established are essentially new.
e following article is organized as follows. In Section 2, we will recall some known results for our consideration. Some lemmas and definitions are useful to our works. Section 3 is devoted to researching the existence and uniqueness of solutions for equation (6). In Section 4, we will investigate the Hyers-Ulam stability of this fractional order difference equation, and then we will come up with the main theorem. To explain the results clearly, we finally provide three examples in Section 5.

Preliminaries
In this section, we plan to introduce some basic definitions and lemmas which are useful throughout this paper.
Definition 1 (see [16,18]). We define as for any t and ] for which the right-hand side is defined.
Here and in what follows, Γ denotes the Gamma function.
We also appeal to the common convention that if t + 1 − ] is a pole of the gamma function and t + 1 is not a pole, then t ] � 0.
Definition 2 (see [18]). e ]th fractional sum of a function f ∈ N a ⟶ R, for ] > 0, is defined to be We also define the ]th fractional difference, where ] > 0 and 0 Lemma 1 (see [16]) en, Lemma 2 (see [18]). Let ] ∈ R and t, s Lemma 3 (see Theorem 2.40 in [15]). Assume that μ > 0 and 2 Mathematical Problems in Engineering Lemma 4 (see [24]). A function y is a solution of the boundary value problem: if and only if y has the form where and

Lemma 5. Green function G satisfies the inequalities
for any (t, s) Note that  Definition 3. We say that equation (6) has the Hyers-Ulam stability if there exists a constant K > 0 with the following property. Let ε > 0 be a given arbitrary constant. If a function for all t ∈ [0, m + 1] N 0 , then there exists a solution y: x is a solution of inequality (20) if and only if there exists a function g: subject to the fractional boundary value problem (23) if and only if y has the form where C 1 , C 2 are constants, and we have erefore, 4 Mathematical Problems in Engineering By the boundary conditions (22) and (23), we can solve C 1 , C 2 as follows: Substituting C 1 and C 2 into (26), then we can obtain (24).

Existence and Uniqueness of Solutions
In this section, we consider the following boundary value problem (BVP): where α(t) and β(t) are defined as (12) and (13) and G(t, s) is given as (14). Obviously, y is a solution of (30) if it is a fixed point of the operator T.
With all the preparatory works done, we will give the main conclusions. First, we provide the uniqueness result by contraction mapping as follows.

en, BVP (30) has a unique solution on E provided that
Proof. Let x, y ∈ E; then, for each t ∈ [] − 1, ] + m] N ]− 1 and by Definition 2 and Lemma 5, we have

| (Tx)(t) − (Ty)(t)| ≤ α(t)|u(x) − u(y)| + β(t)|g(x) − g(y)|
and by condition (34), we get that T is a contraction mapping. erefore, the Banach fixed-point theorem (see Lemma 7 in [25]) implies that the operator T has a unique fixed point which is a unique solution of (30). Now, we plan to adopt Brouwer theorem to give the existence result of solutions.
Proof. Consider the Banach space B ≔ y ∈ R: ‖y‖ ≤ M . Let T be the operator defined in (31). It is clear that T is a continuous operator.
erefore, the main objective in establishing this result is to show that T: B ⟶ B, that is, whenever ‖y‖ ≤ M, it follows that ‖Ty‖ ≤ M. Note that us, we deduce that T: B ⟶ B. Consequently, it follows from the Brouwer theorem that there exists a fixed point y 0 of the map T. is function y 0 is a solution of (30). Moreover, is completes the proof of the theorem.

Hyers-Ulam Stability
In this section, we study the Hyers-Ulam stability of the fractional-order difference system: where x ∈ E satisfies (20). According to Lemma 7, we have

Mathematical Problems in Engineering
If x ∈ E satisfies (20) and y ∈ E is a solution of (40), then the fractional difference equation (6) is Hyers-Ulam stable.
Proof. If x ∈ E satisfies (20) and due to the Remark 1, we can obtain We can solve equation (43) with the corresponding boundary value conditions (22) and (23) as follows: en, we have

Mathematical Problems in Engineering
By condition (42) in eorem 3, we have and we note that the quantity on the left-hand side of the inequality is the constant "K" in Definition 3. We can deduce that system (40) is Hyers-Ulam stable.

Examples
In this section, we will present the following three examples to illustrate our main results.

Remark 2.
Since there are few papers research solutions of the mixed fractional-order nonlinear difference equation, one can see that all the results in [16-18, 23-25, 36-38] cannot directly be applicable to (51) and (53) to obtain the existence and uniqueness of the solution. ese imply that the results in this paper are essentially new.

Conclusion
In this paper, we are concerned with the nonlinear mixed fractional order difference equations, which are quite different from the related references discussed in the literature. e fractional order difference equation studied in the present paper is more generalized and more practical. By applying the Brouwer theorem and contraction mapping principle and the definition of Hyers-Ulam stability, the easily verifiable sufficient conditions have been provided to determine the existence, uniqueness, and Hyers-Ulam stability of the solutions for the considered equation. Finally, the necessary three typical numerical examples have been presented at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion. Consequently, this paper shows theoretically and numerically that some related references known in the literature can be enriched and complemented. e proposed method by the authors could be applied to other fractional difference equation of other similar type, such as [39][40][41][42][43][44].
An interesting extension of our study would be to discuss Ulam-Hyers-Mittag-Leffler stability and finite-time stability for the mixed fractional nonlinear difference equation with time-varying delay terms or fractional stochastic system based on [45,46]. is topic will be the subject of a forthcoming paper.
Data Availability e data in this study were mainly collected via discussion during our class and obtained from the corresponding author upon request.

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