## 3 Almost Anti-Periodic Oscillation of the Mechanical System
### Existence and Uniqueness of the Almost Anti-Periodic Solution
On the other hand, \(\tau\) is a \(\varepsilon\)-almost anti-period of \(f(n)\) and \(x(n)=[p(n),p(n+1)]^{T}\), we have  
\[ \begin{array}{l}\|x(n+1+\tau)+x(n+1)\|=\|q(n+1+\tau)+q(n+1)\|+\|q(n+2+\tau)+ q(n+2)\|\\ =\ \|Rx(n+\tau)+b(n+\tau)+Rx(n)+b(n)\|\\ \leq\ \|R\|\|x(n+\tau)+x(n)\|+\|b(n+\tau)+b(n)\|\\ \leq\ \|R\|\big{(}\|q(n+\tau)+q(n)\|+\|q(n+1+\tau)+q(n+1)\|\big{)}\\ +\ \|B^{-1}\|\big{(}\|f(n+\tau)+f(n)\|+\|S(q(n+\tau))+S(q(n))\|\big{)}\\ \leq\ (\|R\|+\lambda\|B^{-1}\|)\|q(n+\tau)+q(n)\|+\|R\|\|q(n+1+\tau)+q(n+1)\|+\| B^{-1}\|\varepsilon,\end{array} \]  
i.e.,  
\[ \begin{array}{l}(1-\|R\|)\|q(n+1+\tau)+q(n+1)\|+\|q(n+2+\tau)+q(n+2)\|\\ \leq(\|R\|+\lambda\|B^{-1}\|)\|q(n+\tau)+q(n)\|+\|B^{-1}\|\varepsilon.\end{array} \]  
Hence,  
\[\|q(n+1+\tau)+q(n+1)\|<\frac{\|R\|+\lambda\|B^{-1}\|}{1-\|R\|}\|q(n+\tau)+q(n)\|.\]  
The proof is completed. Now, consider the underactuated Euler-Lagrange system with \(N\) degrees of freedom and \(m\) independent controls by a discrete dynamic equation as follows:  
\[ \begin{array}{l}B\big{(}q(n)\big{)}\big{(}q(n+2)-2q(n+1)+q(n)\big{)}\\ +C\big{(}q(n),q(n+1)-q(n)\big{)}[q(n+1)-q(n)]+G(q(n))=L\omega,\end{array}  \tag{6}\]where \(B\) is a positive definite matrix denoting the inertia matrix, \(C\) is the Coriolis matrix, \(G\) is the gravity vector, \(L=[0,I_{m}]\), \(q(\cdot)=[q_{1}(\cdot),\ldots,q_{N}(\cdot)]\in\mathbb{R}^{N}\) is the generalised coordinate vector, and \(\omega\in\mathbb{R}^{m}\) is the control. Assume that \(x(n)=[q(n),q(n+1)]^{T}\), Equation (6) can be rewritten as  
\[ \begin{array}{ll}&x(n+1)=\begin{bmatrix}q(n+1)\\ q(n+2)\end{bmatrix}\\ =&\begin{bmatrix}q(n+1)\\ \gamma_{0}q(n)+\gamma_{1}q(n+1)\end{bmatrix}+\begin{bmatrix}B^{-1}\big{(}q(n) \big{)}\big{[}L\omega-G\big{(}q(n)\big{)}\big{]}\end{bmatrix}\\ =&\begin{bmatrix}0&I\\ \gamma_{0}&\gamma_{1}\end{bmatrix}x(n)+a(n)\\ =&H(n)x(n)+a(n),\end{array}  \tag{7}\]  
where  
\[ \begin{array}{ll}&\gamma_{0}=B^{-1}\big{(}q(n)\big{)}C(q(n),q(n+1)-q(n))-I,\\ &\gamma_{1}=2I-B^{-1}\big{(}q(n)\big{)}C(q(n),q(n+1)-q(n)),\end{array} \]  
and  
\[H(n)= \begin{bmatrix}0&I\\ \gamma_{0}&\gamma_{1}\end{bmatrix} ,\quad a(n)= \begin{bmatrix}0\\ B^{-1}\big{(}q(n)\big{)}\big{[}L\omega-G\big{(}q(n)\big{)}\big{]}\end{bmatrix} .\]  
**Theorem 6**. : _Let \(s<n\), the unique solution (7) with the initial value \(x(s)=x_{0}\) can be given as_  
\[x(n)=\bigg{(}\prod_{j=s}^{n-1}H(j)\bigg{)}x_{0}+\sum_{j=s}^{n-2}\bigg{[}\prod_ {i=1}^{n-(j+1)}H(n-i)\bigg{]}a(j)+a(n-1).\]  
Proof. : The proof process is similar to the proof process of Theorem 5, we will not repeat it here. **Lemma 4**. : _Let \(\tau\in\mathbb{Z}^{+}\),_  
\[ \begin{array}{c}\big{\|}B^{-1}(q)C(q,p)-I\big{\|}+\big{\|}2I-B^{-1}(q)C(q,p )\big{\|}<\frac{\eta-\lambda}{1+\eta}-\frac{1}{N}<1-\frac{1}{N},\\ \big{\|}B^{-1}(q)\big{[}L\omega-G(q)\big{]}+B^{-1}(p)\big{[}L\omega-G(p) \big{]}\leq\lambda\|p+q\|,\\ \|H(n)a(n)+H(s)a(s)\|\leq\max\{\|H(n)\|,\|H(s)\|\}\|a(n)+a(s)\|\end{array} \]  
_for any \(p,q\in\mathbb{R}^{N}\), \(n,s\in\mathbb{Z}\) and \(0<\lambda<\eta<1\). Then,_  
\[ \begin{array}{c}\|a(n+\tau)+a(n)\|\leq\lambda\|q(n+\tau)+q(n)\|,\\ \|q(n+1+\tau)+q(n+1)\|<\eta\|q(n+\tau)+q(n)\|,\\ \|H(n+\tau)a(n+\tau)+H(n)a(n)\|\leq\max\{\|H(n+\tau)\|,\|H(n)\|\}\|a(n+\tau)+a( n)\|.\end{array} \]  
Proof. : Since  
\[a(n)=\begin{bmatrix}0\\ B^{-1}\big{(}q(n)\big{)}\big{[}L\omega-G\big{(}q(n)\big{)}\big{]},\end{array}\]  
we have  
\[ \begin{array}{ll}\|a(n+\tau)+a(n)\|=&\big{\|}B^{-1}\big{(}q(n+\tau)\big{)} \big{[}L\omega-G\big{(}q(n+\tau)\big{)}\big{]}\\ &+B^{-1}\big{(}q(n)\big{)}\big{[}L\omega-G\big{(}q(n)\big{)}\big{]}\big{\|}\\ &\leq&\lambda\big{\|}q(n+\tau)+q(n)\big{\|}\end{array} \]and  
\[ \begin{array}{ll}&\|x(n+1+\tau)+x(n+1)\|\\ =&\|q(n+1+\tau)+q(n+1)\|+\|q(n+2+\tau)+q(n+2)\|\\ =&\|H(n)x(n+\tau)+a(n+\tau)+H(n)x(n)+a(n)\|\\ \leq&\|H(n)\|\|x(n+\tau)+x(n)\|+\|a(n+\tau)+a(n)\|\\ \leq&\|H(n)\|\left(\|q(n+\tau)+q(n)\|+\|q(n+1+\tau)+q(n+1)\|\right)+\lambda\| q(n+\tau)+q(n)\|\right),\end{array} \]  
i.e.,  
\[