Generalized β−Hyers–Ulam–Rassias Stability of Impulsive Difference Equations

This paper describes the existence and uniqueness of the solution, β-Hyers–Ulam–Rassias stability and generalized β-Hyers–Ulam–Rassias stability of an impulsive difference system on bounded and unbounded discrete intervals. At the end, an example is given to illustrate the theoretical result.


Introduction
Many physical problems can be expressed in mathematical models using differential equations. Differential equations enable us to study the rapid changes in physical problems, for example, blood flows, river flows, biological systems, control theory, and mechanical systems with impact. A system of differential equations with impulses can be used to model several above-listed problems. A few existing results for a general class of impulsive systems were discussed by Ahmad [1]. e theory of impulsive difference equations was studied in [2][3][4]. In [5], the existence of solutions for semilinear abstract differential equations without instantaneous impulses was discussed.
At the University of Wisconsin, Ulam [6] proposed the stability problem, stated as follows. Let us denote by H 1 the group and by H 2 the metric group with a metric δ and a constant ] > 0. e problem is to study if there exists λ > 0 satisfies for every h: H 1 ⟶ H 2 such that there exists a homomorphism f: H 1 ⟶ H 2 that satisfies δ(h(σ), f(σ)) ≤ ], ∀σ ∈ H 1 .
e linear functional equations, of the form f(x + y) � f(x) + f(y), and their solutions have been discussed in several spaces. A linear transformation is a solution of a linear functional equation. By considering the H 1 and H 2 as Banach spaces, Hyers [7] discussed the above problem in terms of linear functional equations. en, Aoki [8] and Rassias [9] extended the concept of Hyers and Ulam. In the last decade, we have seen some worthwhile generalizations in the direction of Ulam stability.
In 2012, Wang et al. [10] studied the Ulam-type stability of first-order nonlinear impulsive differential equations by utilizing the bounded interval with finite impulses. In 2014, Wang et al. [11] studied the Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability for impulsive evolution equations on a closed and bounded interval. In 2015, Zada et al. [12] studied the Hyers-Ulam stability of differential systems in terms of a dichotomy. e existence and Hyers-Ulam stability of the periodic fractional stochastic and Riemann-Liouville fractional neutral functional stochastic impulsive differential equations were given [13,14]. Recently, Rahmat et al. [15] studied the Hyers-Ulam stability of delay differential equations. In 2019, Hu and Zhu [16] presented the stability criteria for an impulsive stochastic functional differential system with distributed delay-dependent impulsive effects. Furthermore, Hu et al. [17] provided the improved Razumikhin stability criteria for an impulsive stochastic delay differential system, and for a detail study, we refer to the readers to [17] and the references therein.
In this paper, we will explain the β-Hyers-Ulam-Rassias stability and generalized β-Hyers-Ulam-Rassias stability of the impulsive difference system of the form Θ n+1 � HΘ n + Bζ n + f n, Θ n , ζ n , n ≥ 0, where the constant matrix H, B ∈ R n×n , f ∈ C(Z +×X,X ) and Θ n ∈ B(Z + , X) space of bounded and convergent sequences, In fact, we are presenting a discrete version of the work given in [18], in which β-Hyers-Ulam-Rassias stability was discussed for differential equations. With the help of [15,18], we find out β-Hyers-Ulam-Rassias stability of the difference equation.

Preliminaries
Here, we discuss some notation and definitions, which will be needed for our main work. e n-dimensional Euclidean space will be denoted by R n along with the vector norm ‖ · ‖, and n × n matrices with real-valued entries will be denoted by R n×n . e vector infinite norm is defined as ‖v‖ � max 1≤i≤n |v i |, and the matrix infinite-norm is given as ‖A‖ � max 1≤i≤n n j�1 |a ij | where v ∈ R n and A ∈ R n×n , also v i and a ij are the elements of the vector v and the matrix A, respectively. C(I, X) will be the space of all convergent sequences from I to X with norm ‖v‖ � sup n∈I ‖v n ‖. We will use R, Z, and Z + for the set of all real, integer, and nonnegative integer numbers, respectively. e next lemma is a basic result about the solution of the difference system (1).

Lemma 1. e impulsive difference system (1) has the solution
T n − n k I k n k , Θ n k , ζ n k , n ∈ I. (4) e solution can easily be obtained by consecutively placing the values of n ∈ 0, 1, 2, . . . { }.
where V is a vector space over the field K, if the function satisfied the following properties: (1) ‖H‖ β � 0 if and only H � 0 (2) ‖κH‖ β � |κ| β ‖H‖ β , foreachk κ ∈ K and H ∈ V Definition 2. Let ϵ > 0, ψ > 0 and φ n ∈ B(I, X). A sequence Θ n will be an ϵ-approximate solution of (1), if (1), there exists an exact solution Θ n of (1) and a nonnegative real is a generalized Hyers-Ulam-Rassias stable if for every ϵ-approximate solution Y n of system (1), there will be an exact solution Θ n of (1) and a nonnegative real scalar J M,η Ψ ,η ϕ ,f such that Lemma 2 (see [19]). for any n ≥ 0 with then, we have Remark 2. If we replace c k by c k n , then 2 Computational Intelligence and Neuroscience

Uniqueness and Existence of Solution of an Impulsive Difference System
To describe the uniqueness and existence of the solution of system (1), we will use the following assumptions: Theorem 1. If assumptions G 1 , G 2 , and G 2 * are held, then system (1) has a unique solution Θ ∈ C(I, X).
is implies that Computational Intelligence and Neuroscience 3 is implies that A is a contraction map using the Banach contraction principle, we say that system (1) has a unique solution.

β-Hyers-Ulam-Rassias Stability on Bounded Discrete Interval
To determine β-Hyers-Ulam-Rassias stability on the bounded discrete interval, we have one more assumption: G 3 : there exist a constant η φ > 0 and φ n and a nondecreasing function φ ∈ B(I, X) such that Proof. e solution of system (1) is as follows: Let Y n be the solution of inequality (2), we have thus, for each n ∈ n k , n k+1 , . . . , we have Computational Intelligence and Neuroscience by using the relation where k x, y, z ≥ 0 and c > 1.
Proof. e solution of system (1) is T n − n k I k n k , Θ n k −1 , ζ n k −1 .

(28)
Let Y n be the solution of inequality (2), we have Now, for each n ∈ n k , n k+1 , we have Computational Intelligence and Neuroscience if we set Y n � e − ωn Y n and Θ n � e − ωn Θ n , then we have with the help of relation we get Using Lemma 1, we have Computational Intelligence and Neuroscience Resubmitting the values, we have where us, system (1) is β-Hyers-Ulam-Rassias stable. □ Remark 3. Wang et al. [18] studied the β-Hyers-Ulam stability and β-Hyers-Ulam-Rassias stability for a system of impulsive differential equations as we know that difference equations relate to differential equations as discrete mathematics relate to continuous mathematics. e system of impulsive difference equations used in this article is analogous to the system of impulsive differential equations used in [18]. us, the findings of this article are the discrete version of the work of Wang et al. [18].
Proof. e solution of system (1) is as follows: T n − n k I k n k , Θ n k −1 , ζ n k −1 . (38)