# Systems of Variational Inequalities with Nonlinear Operators
## 3 Main Results
 \tag{19}\]  
and  
\[ \begin{array}{ll}\|y_{n}-p\|^{2}&\leq\|z_{n}-q-\mu_{1}(B_{1}z_{n}-B_{1}q)\| ^{2}\\ &\leq 2\mu_{1}(\kappa^{2}\mu_{1}-\alpha)\|B_{1}z_{n}-B_{1}q\|^{2}+\|z_{n}-q \|^{2}.\end{array}  \tag{20}\]  
Substituting (19) for (20), we obtain  
\[\|y_{n}-p\|^{2}+2\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}+2 \mu_{1}(\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2}\leq\|u_{n}-p\|^{2}. \tag{21}\]  
Let \(v_{n}:=(T^{n}y_{n}-\alpha_{n}\theta FT^{n}y_{n})+\alpha_{n}f(x_{n})\). Then, from (7) and Lemma 2 we obtain  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}&\leq\|v_{n}-p\|^{2}\\ &\leq 2\alpha_{n}\langle f(x_{n})-\theta FT^{n}y_{n},j(v_{n}-p)\rangle+\| T^{n}y_{n}-p\|^{2}\\ &\leq 2\alpha_{n}\|f(x_{n})-\theta FT^{n}y_{n}\|\|v_{n}-p\|+(\|y_{n}-p\|+c_{n}) ^{2}\\ &\leq\|y_{n}-p\|^{2}+c_{n}M_{2}+\alpha_{n}M_{3},\end{array}  \tag{22}\]  
where \(M_{2}=\sup_{n\geq 0}\{2\|y_{n}-p\|+c_{n}\}\) and  
\[M_{3}=\sup_{n\geq 0}\{2\|f(x_{n})-\theta FT^{n}y_{n}\|\|v_{n}-p\|\}.\]  
Substituting (21) to (22), we deduce from (10) that  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}+2\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u _{n}-B_{2}p\|^{2}+2\mu_{1}(\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2} \\ \leq\|x_{n}-p\|^{2}+c_{n}M_{2}+\alpha_{n}M_{3},\end{array} \]which immediately yields  
\[2\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}+2\mu_{1} (\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2}\] \[\leq\alpha_{n}M_{3}-\|x_{n+1}-p\|^{2}+c_{n}M_{2}+\|x_{n}-p\|^{2}\] \[\leq c_{n}M_{2}+\alpha_{n}M_{3}+(\|x_{n}-p\|+\|x_{n+1}-p\|)\|x_{n}- x_{n+1}\|.\]  
Since \(\mu_{1}\kappa^{2}\in(0,\alpha),\ \mu_{2}\kappa^{2}\in(0,\beta),\ \lim_{n\to\infty}c_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we obtain from (17) that  
\[\lim_{n\to\infty}\|B_{2}u_{n}-B_{2}p\|=0\quad\text{and}\quad\lim_{n\to\infty} \|B_{1}z_{n}-B_{1}q\|=0. \tag{23}\]  
On the other hand, from Proposition 3 and Lemma 5 we have  
\[ \begin{array}{ll}2\|z_{n}-q\|^{2}&\leq 2\langle u_{n}-\mu_{2}B_{2}u_{n}-(p- \mu_{2}B_{2}p),j(z_{n}-q)\rangle\\ &\leq[\|u_{n}-p\|^{2}+\|z_{n}-q\|^{2}-g_{1}(\|u_{n}-z_{n}-(p-q)\|)]+2\mu_{2}\|B _{2}p-B_{2}u_{n}\|\|z_{n}-q\|,\end{array} \]  
which implies that  
\[\|z_{n}-q\|^{2}+g_{1}(\|u_{n}-z_{n}-(p-q)\|)\leq\|u_{n}-p\|^{2}+2\mu_{2}\|B_{2 }p-B_{2}u_{n}\|\|z_{n}-q\|. \tag{24}\]  
In the same way, we derive  
\[ \begin{array}{ll}2\|y_{n}-p\|^{2}&\leq 2\langle z_{n}-\mu_{1}B_{1}z_{n}-(q- \mu_{1}B_{1}q),j(y_{n}-p)\rangle\\ &\leq[\|z_{n}-q\|^{2}+\|y_{n}-p\|^{2}-g_{2}(\|z_{n}-y_{n}+(p-q)\|)]+2\mu_{1}\| B_{1}q-B_{1}z_{n}\|\|y_{n}-p\|,\end{array} \]  
which implies that  
\[\|y_{n}-p\|^{2}+g_{2}(\|z_{n}-y_{n}+(p-q)\|)\leq\|z_{n}-q\|^{2}+2\mu_{1}\|B_{1 }q-B_{1}z_{n}\|\|y_{n}-p\|. \tag{25}\]  
Substituting (24) for (25), we deduce from (10) that  
\[ \begin{array}{ll}\|y_{n}-p\|^{2}&\leq\|u_{n}-p\|^{2}-g_{1}(\|u_{n}-z_{n}-(p -q)\|)-g_{2}(\|z_{n}-y_{n}+(p-q)\|)\\ &\quad+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|z_{n}-q\|+2\mu_{1}\|B_{1}q-B_{1}z_{n}\| \|y_{n}-p\|\\ &\leq\|x_{n}-p\|^{2}-g_{1}(\|u_{n}-z_{n}-(p-q)\|)-g_{2}(\|z_{n}-y_{n}+(p-q)\|) \\ &\quad+2\mu_{1}\|B_{1}q-B_{1}z_{n}\|\|y_{n}-p\|+2\mu_{2}\|B_{2}p-B_{2}u_{n}\| \|z_{n}-q\|.\end{array}  \tag{26}\]  
Substituting (26) for (22), we have  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}+g_{1}(\|u_{n}-z_{n}-(p-q)\|)+g_{2}(\|z_{n }-y_{n}+(p-q)\|)\\ \leq\|x_{n}-p\|^{2}+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|z_{n}-q\|+2\mu_{1}\|B_{1}q- B_{1}z_{n}\|\|y_{n}-p\|+c_{n}M_{2}+\alpha_{n}M_{3},\end{array} \]  
which hence yields  
\[ \begin{array}{ll}&g_{1}(\|u_{n}-z_{n}-(p-q)\|)+g_{2}(\|z_{n}-y_{n}+(p-q)\|) \\ &\leq(\|x_{n}-p\|+\|x_{n+1}-p\|)\|x_{n}-x_{n+1}\|+2\mu_{2}\|B_{2}p-B_{2}u_{n} \|\|z_{n}-q\|\\ &\quad+2\mu_{1}\|B_{1}q-B_{1}z_{n}\|\|y_{n}-p\|+\alpha_{n}M_{3}+c_{n}M_{2}. \end{array}