# Systems of Variational Inequalities with Nonlinear Operators
## 3 Main Results
\begin{split}&\|x_{n+1}-x_{n}\|\\ &\leq\|\alpha_{n}f(x_{n})+(I-\alpha_{n}\theta F)T^{n}y_{n}-\alpha_ {n-1}f(x_{n-1})-(I-\alpha_{n-1}\theta F)T^{n-1}y_{n-1}\|\\ &\leq\alpha_{n}\gamma\|x_{n}-x_{n-1}\|+|\alpha_{n}-\alpha_{n-1} |\|f(x_{n-1})-\theta FT^{n}y_{n-1}\|+(1-\alpha_{n}\tilde{\tau}\theta)(\|y_{n}- y_{n-1}\|+c_{n})\\ &\quad+(1-\alpha_{n-1}\tilde{\tau}\theta)\|T^{n}y_{n-1}-T^{n-1}y _{n-1}\|\\ &\leq\alpha_{n}\gamma\|x_{n}-x_{n-1}\|+|\alpha_{n}-\alpha_{n-1} |M_{1}+(1-\alpha_{n}\tilde{\tau}\theta)\|u_{n}-u_{n-1}\|\\ &\quad+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|+c_{n},\end{split}  \tag{12}\]  
where  
\[M_{1}=\sup_{n\geq 1}\|\theta FT^{n}y_{n-1}-f(x_{n-1})\|.\]  
Also, simple calculations show that  
\[ \begin{split}&\|u_{n}-u_{n-1}\|^{2}\\ &\leq\beta_{n}\|x_{n}-x_{n-1}\|\|u_{n}-u_{n-1}\|+(1-\beta_{n})(\|S _{n}u_{n}-S_{n-1}u_{n}\|\|u_{n}-u_{n-1}\|\\ &\quad+\|u_{n}-u_{n-1}\|^{2})+|\beta_{n}-\beta_{n-1}|\|x_{n-1}-S _{n-1}u_{n-1}\|\|u_{n}-u_{n-1}\|.\end{split}  \tag{13}\]  
So, it follows from (13) that  
\[ \begin{split}\|u_{n}-u_{n-1}\|&\leq\beta_{n}\|x_{n}-x_{n-1} \|+(1-\beta_{n})(\|S_{n}u_{n}-S_{n-1}u_{n}\|\\ &\quad+\|u_{n}-u_{n-1}\|]+|\beta_{n}-\beta_{n-1}|\|x_{n-1}-S_{n-1 }u_{n-1}\|,\end{split} \]  
which immediately leads to  
\[ \begin{split}\|u_{n}-u_{n-1}\|-\|x_{n}-x_{n-1}\|& \leq\frac{1-\beta_{n}}{\beta_{n}}\|S_{n}u_{n}-S_{n-1}u_{n}\|+|\beta_{n}-\beta_ {n-1}\|\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{\beta_{n}}\\ &\leq\frac{1}{a}\|S_{n}u_{n}-S_{n-1}u_{n}\|+|\beta_{n}-\beta_{n-1 }|\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{a}.\end{split}  \tag{14}\]  
Putting \(D=\{u_{n}:n\geq 0\}\), we know that \(D\) is a bounded subset of \(C\). Then by the assumption we get \(\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n-1}x\|<\infty\). Noticing that  
\[\sup_{x\in D}\|S_{n}x-S_{n-1}x\|\geq\|S_{n}u_{n}-S_{n-1}u_{n}\|,\quad\forall n \geq 1,\]  
we have  
\[\sum_{n=1}^{\infty}\|S_{n}u_{n}-S_{n-1}u_{n}\|<\infty. \tag{15}\]  
Therefore, from (12) and (14) we deduce that  
\[ \begin{split}\|x_{n+1}-x_{n}\|&\leq\alpha_{n}\gamma\| x_{n}-x_{n-1}\|+|\alpha_{n}-\alpha_{n-1}|M_{1}+(1-\alpha_{n}\tilde{\tau}\theta) \{\|x_{n}-x_{n-1}\|\\ &\quad+\frac{1}{a}\|S_{n}u_{n}-S_{n-1}u_{n}\|+|\beta_{n}-\beta_{n -1}\|\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{a}\}\\ &\quad+c_{n}+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|\\ &\leq|\alpha_{n}-\alpha_{n-1}|M_{1}+\frac{1}{a}\|S_{n}u_{n}-S_{n-1 }u_{n}\|+[1-\alpha_{n}(\tilde{\tau}\theta-\gamma)]\|x_{n}-x_{n-1}\|\\ &\quad+|\beta_{n}-\beta_{n-1}|\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{a} +c_{n}+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|\\ &\leq M_{2}(|\alpha_{n}-\alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|+[1- \alpha_{n}(\tilde{\tau}\theta-\gamma)]\|x_{n}-x_{n-1}\|\\ &\quad+\|S_{n}u_{n}-S_{n-1}u_{n}\|+c_{n}+\|T^{n}y_{n-1}-T^{n-1}y_ {n-1}\|),\end{split}  \tag{16}\]where  
\[M_{2}=\sup_{n\geq 1}\{\frac{1}{a}+\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{a}+M_{1}+1\}.\]  
From (15), the assumption \(\sum_{n=0}^{\infty}c_{n}<\infty\) and conditions (i) and (ii), it can be readily seen that \(\sum_{n=0}^{\infty}\alpha_{n}(\Upsilon\theta-\gamma)=\infty\) and  
\[\sum_{n=1}^{\infty}M_{2}(\|S_{n}u_{n}-S_{n-1}u_{n}\|+c_{n}+|\alpha_{n}-\alpha_ {n-1}|+|\beta_{n}-\beta_{n-1}|+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|)<\infty.\]  
So, it follows from Lemma 7 and (16) that  
\[\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0. \tag{17}\]  
Again from (9) and (14) we conclude that  
\[ \begin{array}{l}a\|y_{n}-y_{n-1}\|=a\|Gu_{n}-Gu_{n-1}\|\leq a\|u_{n}-u_{n-1}\| \\ \leq a\|x_{n}-x_{n-1}\|+\|S_{n}u_{n}-S_{n-1}u_{n}\|+|\beta_{n}-\beta_{n-1}\|x_ {n-1}-S_{n-1}u_{n-1}\|\to 0\quad(n\to\infty).\end{array} \]  
That is,  
\[\lim_{n\to\infty}\|y_{n+1}-y_{n}\|=0. \tag{18}\]  
**Step 3. ** We claim that \(\|x_{n}-Gx_{n}\|\to 0\) as \(n\to\infty\). Indeed, note that \(q=\Pi_{\text{C}}(p-\mu_{2}B_{2}p)\), \(z_{n}=\Pi_{\text{C}}(u_{n}-\mu_{2}B_{2}u_{n})\) and \(y_{n}=\Pi_{\text{C}}(z_{n}-\mu_{1}B_{1}z_{n})\). Then \(y_{n}=Gu_{n}\). By Lemma 4 we have  
\[ \begin{array}{ll}\|z_{n}-q\|^{2}&\leq\|u_{n}-p-\mu_{2}(B_{2}u_{n}-B_{2}p)\|^ {2}\\ &\leq 2\mu_{2}(\kappa^{2}\mu_{2}-\beta)\|B_{2}u_{n}-B_{2}p\|^{2}+\|u_{n}-p\| ^{2}\end{array}