In this portion, we discuss some definitions and basic concepts which are useful for establishing the main work. We will use the notation \(\mathcal{R}\), \(\mathcal{R}^{+}\), \(\mathcal{Z}_{+}\) for the real numbers, non-native real numbers, all nonnegative integers, and the space of all n-tuples of \(\mathcal{R}\) is denoted by \(\mathcal{R}^{n}\). The set \(J=\{0,1,\ldots,k\}\) is the subset of \(\mathcal{Z}\) and \(X=\mathcal{R}^{n}\), the space of all bounded and convergent sequences from J to X is represented by \(\mathcal{CS}(J,X)\) with the norm
$$ \Vert \mathbf{G} \Vert _{cs}= \Bigl\{ \sup_{n\in J} \bigl\Vert \mathbf{G}(n) \bigr\Vert \text{ for all } \mathbf{G}\in \mathcal{CS}(J,X) \Bigr\} . $$
Besides, we define \(\mathcal{C}^{1}(J,X)=\{\mathbf{G}\in \mathcal{C}(J,X) ; \mathbf{G}'\in \mathcal{C}^{1}(J,X)\}\).
Lemma 2.1
The nonsingular delay difference system
$$ \textstyle\begin{cases} A\mathbf{G}_{n+1}=M\mathbf{G}_{n}+N\mathbf{G}_{n-k}+\mathbb{F}(n, \mathbf{G}_{n-k}), \quad n\geq 0, k\geq 0, \\ \mathbf{G}_{n}=\Phi _{n}, \quad -k\leq n\leq 0, \end{cases} $$
has the solution
$$\begin{aligned} \mathbf{G}_{n} =& M^{n}A^{-n }\Phi _{0}+M^{n-1}A^{-n}\sum_{i=0}^{k}M^{-i}A^{i} { \bigl(}N\Phi _{i-k}+\mathbb{F}(i,\Phi _{i-k}) { \bigr)} \\ &{}+ M^{n-1}A^{-n}\sum_{i=k+1}^{n}M^{-i}A^{i} { \bigl(}N\mathbf{G}_{i-k}+ \mathbb{F}(i,\mathbf{G}_{i-k}) { \bigr)}, \end{aligned}$$
where \(MN=NM\), \(NA=AN\), and \(MA=AM\).
The proof can be easily obtained by successively putting the values of \(n \in \{-k,-k+1,\ldots\}\).
Definition 2.1
The solution of system (1.1) will be exponentially stable if there exist positive real numbers \(\lambda _{1}\) and \(\lambda _{2}\) such that
$$ \Vert \mathbf{G}_{n} \Vert \leq \lambda _{1}e^{-\lambda _{2}n}, \quad \forall n \geq 0. $$
Definition 2.2
For a positive real number ϵ, the sequence \(\boldsymbol{\psi}_{n}\) is said to be an ϵ-approximate solution of (1.1) if
$$ \textstyle\begin{cases} \Vert A\boldsymbol{\psi}_{n+1}-M\boldsymbol{\psi}_{n}-N\boldsymbol{\psi}_{n-k}- \mathbb{F}(n,\boldsymbol{\psi}_{n-k}) \Vert \leq \epsilon , \quad n\geq 0, k\geq 0, \\ \Vert \boldsymbol{\psi}_{n}-\phi _{n} \Vert \leq \epsilon , \quad -k\leq n\leq 0. \end{cases} $$
(2.1)
Definition 2.3
System (1.1) will be Hyers–Ulam stable if, for every ϵ-approximate solution \(\boldsymbol{\psi}_{n}\) of system (1.1), there will be an exact solution \(\mathbf{Y}_{n}\) of (1.1) and a nonnegative real number K such that
$$ \Vert \mathbf{Y}_{n}-\boldsymbol{\psi}_{n} \Vert \leq \mathbf{K}\epsilon , \quad n\in J. $$
Definition 2.4
A function \(\Vert \cdot \Vert _{\beta }:\mathbb{V}\rightarrow [0,\infty )\) is called β-norm, with \(0<\beta \leq 1\), where \(\mathbb{V}\) is a vector space over field K, if the function satisfies the following properties:
-
(1)
\(\Vert \mathcal{H} \Vert _{\beta }=0\) if and only if \(\mathcal{G}=0\);
-
(2)
\(\Vert \kappa \mathcal{H} \Vert _{\beta }= \vert \kappa \vert ^{\beta } \Vert \mathcal{H} \Vert _{ \beta }\) for each \(\kappa \in \mathbf{K} \) and \(\mathcal{H}\in \mathbb{V}\);
-
(3)
\(\Vert \mathcal{H}+\mathcal{H}_{1} \Vert _{\beta }\leq \Vert \mathcal{H} \Vert _{\beta }+ \Vert \mathcal{H}_{1} \Vert _{\beta } \) for all \(\mathcal{H}, \mathcal{H}\in \mathbb{V}\).
And \((\mathbb{V}, \Vert \cdot \Vert _{\beta })\) is said to be a β-norm space.
Lemma 2.2
If \(z_{n}\) and \(g_{n}\) are nonnegative sequences and \(a\geq 0\), which satisfies the inequality
$$ \Vert z_{n} \Vert \leq a+\sum_{i=0}^{n} \Vert g_{i} \Vert \Vert z_{i} \Vert , \quad n\geq 0, $$
then
$$ \Vert z_{n} \Vert \leq a \exp { \Biggl(} \sum _{i=0}^{n} \Vert z_{i} \Vert { \Biggr)}. $$
Remark 2.1
It is clear from (2.1) that \(\mathbf{Y}\in \mathcal{C}^{1}(J,X)\) satisfies (2.1) if and only if there exists \(f\in \mathcal{CS}(J,X)\) satisfying
$$ \textstyle\begin{cases} \Vert f_{n} \Vert \leq \epsilon , \quad n \in J, \\ \mathbb{A}\mathcal{Y}_{n+1}=\mathbf{M}\mathcal{Y}_{n}+\mathcal{N} \mathcal{Y}_{n-k}+\mathbb{F}(n,\mathcal{Y}_{n-k})+{f}_{n}, \quad n\in \mathbb{Z}_{+} , \\ \mathcal{Y}_{n}=\phi _{n},\quad -k\leq n \leq 0. \end{cases} $$