## 3 Almost Anti-Periodic Oscillation of the Mechanical System
### Existence and Uniqueness of the Almost Anti-Periodic Solution  
Consider the general \(N\)-dimensional mechanical system as follows:  
\[B[(q(n+2)-2q(n+1)+q(n)]+C[q(n+1)-q(n)]+S\big{(}q(n)\big{)}=f(n), \tag{4}\]where \(q(\cdot)=[q_{1}(\cdot),\ldots,q_{N}(\cdot)]\in\mathbb{R}^{N}\), \(B,C\in\mathbb{R}^{N\times N}\) is symmetric and \(B\) is positive definite, the vector \(S(q)\) collects all position dependent forces, such a linear and nonlinear stiffness forces or non-potential forces, \(f(n)\) is an almost anti-periodic discrete function. Moreover, Equation (4) can be rewritten as  
\[Bq(n+2)+(C-2B)q(n+1)+(B-C)q(n)+S\big{(}q(n)\big{)}=f(n).\]  
Assume that \(x(n)=[q(n),q(n+1)]^{T}\), i.e.,  
\[x(n)=[q_{1}(n),\ldots,q_{N}(n),q_{1}(n+1),\ldots,q_{N}(n+1)]^{T},\]  
Equation (4) can be rewritten as  
\[ \begin{array}{ll}x(n+1)&=\begin{bmatrix}q(n+1)\\ q(n+2)\end{bmatrix}=\begin{bmatrix}q(n+1)\\ \xi\end{bmatrix}+\begin{bmatrix}0\\ f(n)-S\big{(}q(n)\big{)}\end{bmatrix}\\ &=\begin{bmatrix}B^{-1}(C-B)&B^{-1}(2B-C)\end{bmatrix}\begin{bmatrix}q(n)\\ q(n+1)\end{bmatrix}\\ &+\begin{bmatrix}B^{-1}f(n)-B^{-1}S\big{(}q(n)\big{)}\end{bmatrix}\\ &=Rx(n)+b(n),\end{array}  \tag{5}\]  
where \(\xi=B^{-1}(C-B)q(n)+B^{-1}(2B-C)q(n+1)\) and  
\[R= \begin{bmatrix}0&I\\ B^{-1}(C-B)&B^{-1}(2B-C)\end{bmatrix} ,\ b(n)= \begin{bmatrix}0\\ B^{-1}f(n)-B^{-1}S\big{(}q(n)\big{)}\end{bmatrix} .\]  
In the sequel, we will establish some sufficient conditions of the existence and uniqueness of the almost anti-periodic solution of Equation (5). **Theorem 5**. : _Let \(s<n\), the unique solution of Equation (5) with the initial value condition \(x(s)=x_{0}\) can be given as:_  
\[x(n)=[q(n),q(n+1)]^{T}=R^{n-s}x(s)+\sum\limits_{j=n-1-s}^{0}R^{j}b(n-1-j),\]  
_i.e.,_  
\[ \begin{array}{ll}x(n)=[q(n),q(n+1)]^{T}=\begin{bmatrix}0&I\\ B^{-1}(C-B)&B^{-1}(2B-C)\end{bmatrix}^{n-s}x(s)\\ &\qquad\qquad+\sum\limits_{j=n-1-s}^{0}\begin{bmatrix}0&I\\ B^{-1}(C-B)&B^{-1}(2B-C)\end{bmatrix}^{j}\begin{bmatrix}0\\ \eta\end{bmatrix},\end{array} \]  
_where \(R^{0}=I\), \(\eta=B^{-1}f(n-1-j)-B^{-1}S\big{(}q(n-1-j)\big{)}\)._  
Proof. : Step 1, we will prove the existence of the solution of Equation (4). For \(n>s\), we have  
\[ \begin{array}{ll}x(n+1)&=R^{n+1-s}x(s)+\sum\limits_{j=n-s}^{0}R^{j}b(n-j)\\ &=R^{n+1-s}x(s)+\sum\limits_{j=n-s}^{1}R^{j}b(n-j)+b(n)\\ &=R\bigg{[}R^{n-s}x(s)+\sum\limits_{j=n-s}^{1}R^{j-1}b(n-j)\bigg{]}+b(n)=Rx(n)+ b(n)\end{array} \]  
and \(x(s)=x_{0}\), which means that \(x(n)\) is a solution of Equation (4). Step 2, we will prove the uniqueness of the solution of Equation (5). Let \(x(n)\) and \(y(n)\) be two solutions of Equation (5) with the initial value condition \(x(s)=y(s)=x_{0}\), then we have  
\[x(n)=Rx(n-1)+b(n-1)\ \ \text{and}\ \ \ y(n)=Ry(n-1)+b(n-1).\]  
Assume that \(V(n)=x(n)-y(n)\), then \(V(n)=RV(n-1)\). Hence,  
\[V(n)=R^{n-s}V(s).\]  
On the other hand, \(x(s)=y(s)=x_{0}\), i.e., \(V(s)=0\), which implies \(V(n)=0\), i.e., \(x(n)=y(n)\). The proof is completed. **Lemma 3**. : _Let \(\tau\) be a \(\varepsilon\)-almost anti-period of \(f(n)\), if \(\|R\|+\lambda\|B^{-1}\|<1-\lambda\), \(\|R\|\leq\lambda\), \(\|S(q)+S(p)\|\leq\frac{\lambda}{\|B^{-1}\|}\|p+q\|\) for \(p,q\in\mathbb{R}^{N}\) and \(0<\lambda<1\). Then,_  
\[ \begin{array}{l}\|b(n+\tau)+b(n)\|\leq v\|q(n+\tau)+q(n)\|\\ \|q(n+1+\tau)+q(n+1)\|<\zeta\|q(n+\tau)+q(n)\|\end{array} \]  
_where \(v\in(\lambda,1)\) and \(0<\zeta=\frac{\|R\|+\lambda\|B^{-1}\|}{1-\|R\|}<1\)._  
Proof. : Since \(b(n)=\big{[}0,B^{-1}f(n)-B^{-1}S\big{(}q(n)\big{)}\big{]}^{T}\), we have  
\[ \begin{array}{l}\|b(n+\tau)+b(n)\|\\ \leq\Big{\|}B^{-1}f(n+\tau)+B^{-1}f(n)\Big{\|}+\big{\|}B^{-1}S\big{(}q(n+\tau) \big{)}+B^{-1}S\big{(}q(n)\big{)}\big{\|}\\ \leq\|B^{-1}\|\varepsilon+\|B^{-1}\|\frac{\lambda}{\|B^{-1}\|}\big{\|}q(n+\tau )+q(n)\big{\|}\\ =\|B^{-1}\|\varepsilon+\lambda\big{\|}q(n+\tau)+q(n)\big{\|}.\end{array} \]  
Hence, there exists \(v\in(\lambda,1)\) such that \(\|b(n+\tau)+b(n)\|\leq v\|q(n+\tau)+q(n)\|\).