## 2 Almost Anti-Periodic Discrete Functions  
In this section, we will introduce the notions of the almost anti-periodic discrete functions for the \(N\times N\) matrix-valued function and the \(N\)-dimensional vector-valued function and establish some of their basic properties. Definition 1: Let \(M(\cdot)=[m_{ij}(\cdot)]_{N\times N}:\mathbb{Z}\to\mathbb{R}^{N\times N}\) be an \(N\times N\) matrix-valued discrete function and \(p(\cdot)=[p_{1}(\cdot),\ldots,p_{N}(\cdot)]^{T}:\mathbb{Z}\to\mathbb{R}^{N}\) be a \(N\)-dimensional vector-valued discrete function, we define  
\[\|M(n)\|=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{i=1}^{N}|m_{ij}(n)|,\ \ \|p(n)\|=\frac{1}{N}\sum_{j=1}^{N}|p_{j}(n)|.\]  
If for \(\forall\epsilon>0\), there exists a positive integer \(l(\epsilon)\) and \(\tau\in[m,m+l]_{\mathbb{Z}},\forall m\in\mathbb{Z}\), such that \(p(n)\) satisfies the following condition  
\[\|p(n+\tau)+p(n)\|<\epsilon,\ \ \forall n\in\mathbb{Z}.\]  
Then, \(p(n)\) is called the almost anti-periodic discrete function, \(\tau\) is called the \(\epsilon\)-almost anti-period of \(p(n)\), and \(l\) is called an inclusion length. Similarly, \(M(n)\) is an almost anti-periodic \(N\times N\) matrix-valued discrete function if  
\[\|M(n+\tau)+M(n)\|<\epsilon,\ \ \forall n\in\mathbb{Z}.\]  
The set of all almost anti-periodic \(N\times N\) matrix-valued discrete functions is denoted by \(\mathfrak{G}\). Definition 2 ([30]): Let \(\{q(n)\}\) be a discrete sequence, if for \(\forall\epsilon>0\), there exists a positive integer \(l(\epsilon)\) such that \(q(n)\) satisfies the following condition  
\[|q(n+\tau)-q(n)|<\epsilon,\ \ \forall n\in\mathbb{Z}\]  
for some \(\tau\in[m,m+l]_{\mathbb{Z}}\), where \(m\in\mathbb{Z}\). Then, \(q(n)\) is called the almost periodic discrete function and \(\tau\) is called the \(\epsilon\)-almost period of \(q(n)\). Lemma 1: _Let \(M(n)\) be an almost anti-periodic discrete function; then, it is an almost periodic discrete function._  
Proof: By Definition 1, we have  
\[\|M(n+2\tau)-M(n)\|\leq|M(n+2\tau)+M(n+\tau)\|+\|M(n+\tau)+M(n)\|<2\epsilon.\]  
The proof is completed. Through Lemma 1, the following lemma is immediate. Lemma 2 ([26]): _Let \(M(n)\) be an almost periodic discrete function (or an almost anti-periodic discrete function). Then, \(\|M(\cdot)\|:\mathbb{Z}\to\mathbb{R}\) is bounded._  
In what follows, some basic properties of the almost anti-periodic discrete functions will be established. Theorem 1: _Let \(M(\cdot)\in\mathfrak{G}\), then \(cM(\cdot)\in\mathfrak{G}\) for all \(c\in\mathbb{R}\)._Proof. : By Definition 1, one has  
\[\|M(n+\tau)+M(n)\|<\frac{\varepsilon}{|c|}\]  
for \(c\
eq 0\). Thus, we have  
\[\|cM(n+\tau)+cM(n)\|\leq|c|\|M(n+\tau)+M(n)\|<|c|\frac{\varepsilon}{|c|}=\varepsilon,\]  
which means \(cM(\cdot)\in\mathfrak{G}\). For \(c=0\), one can obtain the desired result. The proof is completed. **Theorem 2**. : _Let \(M(\cdot)\in\mathfrak{G}\) and \(Q(n)=M(n+k)\) for some \(k\in\mathbb{Z}\). Then, \(Q(\cdot)\in\mathfrak{G}\)._  
Proof. : Since \(M(\cdot)\in\mathfrak{G}\), we have  
\[\|M(n+\tau)+M(n)\|<\varepsilon\;\;\text{for all}\;\;n\in\mathbb{Z}.\]  
Hence,  
\[\|M(n+k+\tau)+M(n+k)\|<\varepsilon,\;\;n\in\mathbb{Z},\]  
i.e.,  
\[\|Q(n+\tau)+Q(n)\|<\varepsilon,\;\;n\in\mathbb{Z},\]  
which implies \(Q(\cdot)\in\mathfrak{G}\). The proof is completed. **Theorem 3**. : _Let \(B(\cdot)\in\mathfrak{G}\) and \(S(n)\) be a summable \(N\times N\) matrix sequence, i.e.,_  
\[\|p(n+\tau)+p(n)\|<\epsilon,\ \ \forall n\in\mathbb{Z}.\]0  
_If \(M(n)=\sum\limits_{m\in\mathbb{Z}}S(m)B(n-m)\) for \(n\in\mathbb{Z}\). Then, \(M(\cdot)\in\mathfrak{G}\)._  
Proof. : Since \(S(n)\) is a summable matrix sequence, for \(\forall\varepsilon>0\), there exists \(N_{0}>0\) such that  
\[\|p(n+\tau)+p(n)\|<\epsilon,\ \ \forall n\in\mathbb{Z}.\]1  
By Definition 1 and Theorem 2, one has  
\[\|p(n+\tau)+p(n)\|<\epsilon,\ \ \forall n\in\mathbb{Z}.\]2  
By Lemma 2, we have \(\|B(n)\|<\lambda\) for some \(\lambda>0\) and any \(n\in\mathbb{Z}\).