# Generalizedβ−Hyers–Ulam–Rassias Stability of Impulsive Difference Equations

Authors: Yahya Almalki; Gul Rahmat; Atta Ullah; Fatima Shehryar; Muhammad Numan; Muhammad Usman Ali
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

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.


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## Body

## 1. 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]. The theory of impulsive difference equations was studied in [2–4]. 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 H1 the group and by H2 the metric group with a metric δ and a constant ν>0. The problem is to study if there exists λ>0 satisfies for every h:H1⟶H2 such that(1)δhσν,hσhν≤λ,∀σ,ν∈H1,there exists a homomorphism f:H1⟶H2 that satisfies(2)δhσ,fσ≤ν,∀σ∈H1.The linear functional equations, of the formfx+y=fx+fy, and their solutions have been discussed in several spaces. A linear transformation is a solution of a linear functional equation. By considering the H1 and H2 as Banach spaces, Hyers [7] discussed the above problem in terms of linear functional equations. Then, 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. The 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 theβ-Hyers–Ulam–Rassias stability and generalized β-Hyers–Ulam–Rassias stability of the impulsive difference system of the form(3)Θn+1=HΘn+Bζn+fn,Θn,ζn,n≥0,Θ0,n=0,Θnk+1=Θnk−1+Ikn,Θnk−1,ζnk−1,k=1,2,3,…,m,where the constant matrix H,B∈ℝn×n, f∈ℂℤ+×X,X and Θn∈Bℤ+,X space of bounded and convergent sequences, ℤ+=0,1,2,… and X=ℝn, I=0,1,2,…,n. 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.


## 2. Preliminaries

Here, we discuss some notation and definitions, which will be needed for our main work. Then-dimensional Euclidean space will be denoted by ℝn along with the vector norm ⋅, and n×n matrices with real-valued entries will be denoted by ℝn×n. The vector infinite norm is defined as v=max1≤i≤nvi, and the matrix infinite-norm is given as A=max1≤i≤n∑j=1naij where v∈ℝn and A∈ℝn×n, also vi and aij are the elements of the vector v and the matrix A, respectively. ℂI,X will be the space of all convergent sequences from I to X with norm v=supn∈Ivn. We will use ℝ,ℤ, and ℤ+ for the set of all real, integer, and nonnegative integer numbers, respectively. The next lemma is a basic result about the solution of the difference system (1).Lemma 1.
The impulsive difference system (1) has the solution(4)Xn=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk,ζnk,n∈I.The solution can easily be obtained by consecutively placing the values ofn∈0,1,2,….Definition 1.
A function⋅β:V⟶0,∞ is called β-norm, with 0<β≤1, where V is a vector space over the field K, if the function satisfied the following properties:(1)
ℋβ=0 if and only ℋ=0(2)
κℋβ=κβℋβ,foreachkκ∈Kandℋ∈V(3)
ℋ+ℋ1β≤ℋβ+ℋ1β,forallℋ,ℋ∈V
AndV,⋅β is said to be β-norm space.Definition 2.
Letϵ>0,ψ>0 and φn∈BI,X. A sequence Θn will be an ϵ-approximate solution of (1), if(5)Θn+1−HΘn−Bζn−fn,Θn,ζn≤εφn,n≥0,Θnk+1−Θnk−1−Ikn,Θnk−1,ζnk−1≤εψ,k=1,2,3,…,m,.Definition 3.
System (1) is said to be β-Hyers–Ulam–Rassias stable if for every ϵ-approximate solution Yn of system (1), there exists an exact solution Θn of (1) and a nonnegative real number Cf,N,M,φ,Ψ such that(6)Yn−Θnβ≤Cf,N,M,φ,Ψεβφnβ+ψβ,foralln∈I.Definition 4.
System (1) is a generalized Hyers–Ulam–Rassias stable if for every ϵ-approximate solution Yn of system (1), there will be an exact solution Θn of (1) and a nonnegative real scalar JM,ηΨ,ηϕ,f such that(7)Yn−Θnβ≤JM,ηΨ,ηϕ,fεβφnβ+Ψk+1β,n∈I.Remark 1.
From (2), it is clear that Y∈ℂI,X satisfies (2) if and only if there exists h∈ℂI,X and a sequence hk,k∈M satisfying(8)hn≤εψn,n∈M,Θn+1=HΘn+Bζn+fn,Θn,ζn+hn,n∈ℤ+,Y0=Θ0+h0,Θnk+1−Θnk−1=IkΘnk−1,ζnk−1+hnk,k∈M.The solution of Remark1 is(9)Θn=HnΘ0+h0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+hi+∑nk=0nTn−nkIk(nk,Θnk,ζnk+hnk,n∈I.Lemma 2 (see [19]).
for anyn≥0 with(10)Un≤an+∑i=0nPiUi+∑0≤nk≤nγkUnk−1,n≥0,then, we have(11)Un≤an1+γkkexp∑i=0nPi,wherek∈I.Remark 2.
If we replaceγk by γkn, then(12)Yn≤c∏nk=0n1+γknexp∑i=0nPiforn≥0.


## 3. 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:G1: for f,Ik:ℤ+×X×X⟶X, k∈I, there exist constants ℒf>0 and ℒIk>0, such that(13)fn,ℑ,p−fn,ℑ′,p≤ℒfℑ−ℑ′,Ikn,w,y−Ikn,w′,y≤ℒIkw−w′.G2:(14)M=supi∈IHi,N=max0≤nk≤nTn−nk.G2∗:(15)M∑i=0n−1H−iℒf+NnℒIk<1.Theorem 1.
If assumptionsG1, G2, and G2∗ are held, then system (1) has a unique solution Θ∈ℂI,X.Proof.
DefineA:ℂI,X⟶ℂI,X by(16)AΘn=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1.
Now, forΘ,Θ′∈ℂI,X, we have(17)AΘn−AΘn′=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi′,ζi−∑nk=0nTn−nkIknk,Θnk−1′,ζnk−1.
This implies that(18)AΘn−AΘn′=Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1−Hn−1∑i=0n−1H−ifi,Θi′,ζi−∑nk=0nTn−nkIknk,Θnk−1′,ζnk−1≤+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1−Iknk,Θnk−1′,ζnk−1≤M∑i=0n−1H−iℒfΘi−Θi′+N∑nk=0nℒIkΘnk−1−Θnk−1′≤M∑i=0n−1H−iℒf+NnℒIkΘ−Θ′.
This implies thatA is a contraction map using the Banach contraction principle, we say that system (1) has a unique solution.


## 4.β-Hyers–Ulam–Rassias Stability on Bounded Discrete Interval

To determineβ-Hyers–Ulam–Rassias stability on the bounded discrete interval, we have one more assumption:G3: there exist a constant ηφ>0 and φn and a nondecreasing function φ∈BI,X such that(19)∑i=0n−1ψi≤ηϕϕn.Theorem 2.
System (1) is β-Hyers–Ulam–Rassias stable over discrete bounded interval, if G1, G2 and G3 are satisfied.Proof.
The solution of system (1) is as follows:(20)Θn=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1.
LetYn be the solution of inequality (2), we have(21)Yn−HnΘ0−Hn−1∑i=0nH−iBζi−Hn−1∑i=0nH−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1=Hnhn+Hn−1∑i=0n−1H−ihi+∑nk=0nTn−nkhnk≤MεΨ+M2ò∑i=0n−1Ψi+Nε∑nk=0mΨ=M+N∑k=0mòΨ+∑i=0n−1M2òΨi≤mMò+M2εηϕφn+φ=Mòm+Mηϕφn+φ,thus, for each n∈nk,nk+1,…, we have(22)Yn−θnβ=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Yi,ζi+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β≤Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Θi,ζiβ+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β≤Mεm+Mηϕφn+φβ+M2∑i=0n−1ℒfiYi−Θiβ+N∑k=1mℒInkYnk−ΘnkβYn−Θn≤31/β−1Mεm+Mηϕφn+φ+M2∑i=0n−1ℒfiYi−Θi+N∑k=1mℒIkYnk−Θnk,by using the relation(23)x+y+zγ≤3γ−1xγ+yγ+zγ,wherekx,y,z≥0andγ>1.
Now, using Gronwall Lemma2, we get(24)YN−Θn≤31/β−1Mεm+Mηϕφn+φ1+31/β−1NℒIkkexpM231/β−1∑i=0n−1ℒfi.
Now,(25)Yn−Θnβ≤31−βMεm+Mηϕφn+φβ1+31/β−1NℒIkkβexpM2β31/β−1∑i=0n−1ℒfi≤31−βMεm+Mηϕβφn+φβ1+31/β−1NℒIkkβexpM2β31/β−1∑i=0n−1ℒfi≤Cf,N,M,φ,Ψεβφnβ+φβ,using x+yr≤xr+yr,x,y≥0,foranyr∈0,1, where(26)Cf,N,M,φ,Ψ=31−βMεm+Mηϕβ1+31/β−1NℒIkkβexpM2β31/β−1∑i=0n−1ℒfi.
Hence, system (1) is β-Hyers–Ulam–Rassaias stable. □


## 5.β-Hyers–Ulam–Rassias Stability on Unbounded Discrete Interval

To explainβ-Hyers–Ulam–Rassias stability on an unbounded discrete interval, we must need the following assumptions:G4: the operators family Hn−i≤Meωn−i and Tn−nk≤Meωn−nk.(27)G5:∑i=0n−1ℒfi≤κfn+ϰf,G6:ℒJ=1+31/β−1MℒInkk<∞,G7:∑i=0n−1eωn−i+31/β−1MκfnΨi≤ηφφn,G8:M1=∑nk=0meωn−nk+31/β−1Mκfn.Theorem 2.
Assume thatG1 and G3−G8 are holds, then system (1) is β-Hyers–Ulam–Rassias stable over a discrete unbounded interval.Proof.
The solution of system (1) is(28)Θn=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1.
LetYn be the solution of inequality (2), we have(29)Yn−HnΘ0−Hn−1∑i=0nH−iBζi−Hn−1∑i=0nH−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1=∑i=0n−1Hn−1−ihi+∑nk=0nTn−nkhnk≤∑i=0n−1Hn−1−ihi+∑nk=0nTn−nkhnk≤M∑i=0n−1eωn−iεΨi+∑nk=0meωn−nkεΨ.
Now, for eachn∈nk,nk+1, we have(30)Yn−θnβ=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Yi,ζi+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β≤Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Θi,ζi‖β+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β≤M∑i=0n−1eωn−iεΨi+∑nk=0meωn−nkεΨβ+M∑i=0n−1eωn−iℒfiYi−Θiβ+M∑nk=0neωn−nkℒInkYnk−1−Θnk−1β,if we set Y¯n=e−ωnYn and Θ¯n=e−ωnΘn, then we have(31)Y¯n−Θ¯nβ≤M∑i=0n−1e−ωiεΨi+∑nk=0me−ωnkεΨβ+M∑i=0n−1ℒfiY¯i−Θ¯iβ+M∑nk=0nℒInkY¯nk−1−Θ¯nk−1β,with the help of relation(32)x+y+zγ≤3γ−1xγ+yγ+zγ,wherex,y,z≥0γ>1.we get(33)Y¯n−Θ¯n≤31/β−1M∑i=0n−1e−ωiεΨi+∑nk=0me−ωnkεΨ+31/β−1M∑i=0n−1ℒfiY¯i−Θ¯i+31/β−1M∑nk=0nℒInkY¯nk−1−Θ¯nk−1.
Using Lemma1, we have(34)Y¯n−Θ¯n≤31/β−1Mε∑i=0n−1e−ωiΨi+∑nk=0me−ωnkΨ1+31/β−1MℒInkkexp31/β−1M∑i=0n−1ℒfi.
Resubmitting the values, we have(35)Yn−Θn≤31/β−1Mε∑i=0n−1eωn−iΨi+∑nk=0meωn−nkΨ1+31/β−1MℒInkkexp31/β−1M∑i=0n−1ℒfi≤31/β−1MεℒJ∑i=0n−1eωn−iΨi+∑nk=0meωn−nkΨexp31/β−1Mκfn+ϰf=31/β−1MεℒJ∑i=0n−1eωn−i+31/β−1Mκfn+ϰfΨi+∑nk=0meωn−nk+31/β−1Mκfn+ϰfΨ=31/β−1MεℒJe31/β−1Mϰf∑i=0n−1eωn−i+31/β−1MκfnΨi+∑nk=0meωn−nk+31/β−1MκfnΨ≤31/β−1MεℒJe31/β−1Mϰfηφφ+M1Ψ≤31/β−1MεℒJe31/β−1MϰfM1+ηφφn+ΨYn−Θnβ≤ℋM,ℒJ,ηφ,M1εβφnβ+Ψβ,where(36)ℋM,ℒJ,ηφ,M1=31−βMℒJe31/β−1MϰfM1+ηφβ>0.
Thus, 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. The system of impulsive difference equations used in this article is analogous to the system of impulsive differential equations used in [18]. Thus, the findings of this article are the discrete version of the work of Wang et al. [18].


## 6. Generalizedβ-Hyers–Ulam–Rassias Stability

In this section, we present the generalizedβ-Hyers–Ulam–Rassias stability, for which we need the following assumptions:(37)G9:31/β−1Mε∏nk=0n1+31/β−1MℒInk∑i=0n−1eωn−i+31/β−1M∑i=0n−1ℒfiΨi≤ηφφn,G10:31/β−1Mε∏nk=0n1+31/β−1MℒInk∑nk=0meωn−nk+31/β−1M∑i=0n−1ℒfiΨ≤ηΨΨk+1.Theorem 3.
Assume thatG1 and G3−G10 are satisfied, then system (1) is generalized β-Hyers–Ulam–Rassias stable.Proof.
The solution of system (1) is as follows:(38)Θn=HnΘ0+Hn−1∑i=0n−1H−iBζi+Hn−1∑i=0n−1H−ifi,Θi,ζi+∑nk=0nTn−nkIknk,Θnk−1,ζnk−1.
LetYn be the solution of inequality (2), we have(39)Yn−HnΘ0−Hn−1∑i=0nH−iBζi−Hn−1∑i=0nH−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1=∑i=0n−1Hn−1−ihi+∑nk=0nTn−nkhnk≤∑i=0n−1Hn−1−ihi+∑nk=0nTn−nkhnk≤M∑i=0n−1eωn−iεΨi+∑nk=0meωn−nkεΨ.
Now, for eachn∈nk,nk+1, we have(40)Yn−θnβ=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β=Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Θi,ζi−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Yi,ζi+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β≤Yn−HnΘ0−Hn−1∑i=0n−1H−iBζi−Hn−1∑i=0n−1H−ifi,Yi,ζi−∑nk=0nTn−nkIknk,Ynk−1,ζnk−1β+Hn−1∑i=0n−1H−ifi,Yi,ζi−Hn−1∑i=0n−1H−ifi,Θi,ζiβ+∑nk=0nTn−nkIknk,Ynk−1,ζnk−1−∑nk=0nTn−nkIknk,Θnk−1,ζnk−1β≤M∑i=0n−1eωn−iεΨi+∑nk=0meωn−nkεΨβ+M∑i=0n−1eωn−iℒfiYi−Θiβ+M∑nk=0neωn−nkℒInkYnk−1−Θnk−1β,if we set Y¯n=e−ωnYn and Θ¯n=e−ωnΘn, then we have(41)Y¯n−Θ¯nβ≤M∑i=0n−1e−ωiεΨi+∑nk=0me−ωnkεΨβ+M∑i=0n−1ℒfiY¯i−Θ¯iβ+M∑nk=0nℒInkY¯nk−1−Θ¯nk−1β,with the help of relation(42)x+y+zγ≤3γ−1xγ+yγ+zγ,wherex,y,z≥0γ>1,we get(43)Y¯n−Θ¯n≤31/β−1M∑i=0n−1e−ωiεΨi+∑nk=0me−ωnkεΨ+31/β−1M∑i=0n−1ℒfiY¯i−Θ¯i+31/β−1M∑nk=0nℒInkY¯nk−1−Θ¯nk−1,using, we have(44)Y¯n−Θ¯n≤31/β−1Mε∏nk=0n∑i=0n−1e−ωiΨi+∑nk=0me−ωnkΨ1+31/β−1MℒInkexp31/β−1M∑i=0n−1ℒfi,resubmitting the values, we have(45)Yn−Θn≤31/β−1Mε∏nk=0n∑i=0n−1eωn−iΨi+∑nk=0meωn−nkΨ1+31/β−1MℒInkexp31/β−1M∑i=0n−1ℒfi=31/β−1Mε∏nk=0n∑i=0n−1eωn−i+31/β−1M∑i=0n−1ℒfiΨi+∑nk=0meωn−nk+31/β−1M∑i=0n−1ℒfiΨ1+31/β−1MℒInk,
At last, we obtain(46)Yn−Θn≤31/β−1Mεηφφn+ηΨΨk+1≤31/β−1Mεηφ+ηΨφn+Ψk+1Yn−Θnβ=31−βMβεβηφ+ηΨβφn+Ψk+1β=JM,ηΨ,ηϕ,fεβφnβ+Ψk+1β,where(47)JM,ηΨ,ηϕ,f=31−βMβηφ+ηΨβ>0.
Hence, system (1) is generalized β-Hyers–Ulam–Rassias stable.


## 7. Example

The impulsive difference system is as follows:(48)Θn+1=HΘn+Bζn+fn,Θn,ζn,n∈0,1,2,3,Θ0,n=0,ΔΘnk=Ikn,Θnk−1,ζnk−1=13r2Θnk−1,k=1,2,3,…,m.AssumptionG1 and G2 are holds if N=1 and M=1. Clearly, assumptions G5 and G6 are hold if κf=0 and(49)ℒJ=sup∏r=1∞1+31/βMℒInk=sup∏r=1∞1+1r2≤e∑l=1∞1/r2≤eπ2/6.Also,ùf=0. Now, set φn=en and ψ=1, then G7 holds if ηϕ=1/2. G8 holds if e2/e2−1. Thus, system (3) is 1/2− Hyers–Ulam–Rassias stable with respect to en,1 on Z+ and ℋM,ℒJ,ηφ,M1=3eπ2/r21/2+e2/e2−11/2.


## 8. Conclusion

Nowadays, studies on the qualitative behavior of impulsive difference equations have a significant contribution to the literature. In particular, the discussion regarding theβ-Hyers–Ulam–Rassias stability of difference equations has been considered as one of the important topics of the literature, in which different types of conditions have been used in the form of inequalities, and most results have been obtained through discrete Gronwall inequality. In this paper, we have investigated the existence and uniqueness of the solution through the Banach contraction principle and β-Hyers–Ulam–Rassias stability of the impulsive difference system with the help of Gronwall inequality.



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*Source: 9462424-2022-09-15.xml*