# Systems of Variational Inequalities with Nonlinear Operators
## 3 Main Results
** We claim that \(\|x_{n}-\overline{S}x_{n}\|\to 0\) as \(n\to\infty\) where \(\overline{S}:=(2I-S)^{-1}\). Indeed, first, let us show that \(S:C\to C\) is pseudocontractive and \(\ell\)-Lipschitzian such that \(\lim_{n\to\infty}\|Sx_{n}-x_{n}\|=0\) where \(Sx=\lim_{n\to\infty}S_{n}x\forall x\in C\). Observe that for all \(x,y\in C,\lim_{n\to\infty}\|S_{n}x-Sx\|=0\) and \(\lim_{n\to\infty}\|S_{n}y-Sy\|=0\). Since \(S_{n}\) is pseudocontractive operator, we get  
\[\langle Sx-Sy,j(x-y)\rangle\leq\|x-y\|^{2}.\]  
This means that \(S\) is pseudocontractive. Noting that \(\{S_{n}\}_{n=0}^{\infty}\) is \(\ell\)-uniformly Lipschitzian on \(C\), we have  
\[\|Sx-Sy\|=\lim_{n\to\infty}\|S_{n}x-S_{n}y\|\leq\ell\|x-y\|,\quad\forall x,y \in C.\]This means that \(S\) is \(\ell\)-Lipschitzian. Taking into account the boundedness of \(\{x_{n}\}\) and putting \(D=\overline{\operatorname{conv}}\{x_{n}:n\geq 0\}\) (the closed convex hull of the set \(\{x_{n}:n\geq 0\}\)), by the assumption we have \(\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n-1}x-S_{n}x\|<\infty\). Hence, by Proposition 1 we get  
\[\lim_{n\to\infty}\sup_{x\in D}\|S_{n}x-Sx\|=0,\]  
which immediately yields  
\[\lim_{n\to\infty}\|S_{n}x_{n}-Sx_{n}\|=0. \tag{36}\]  
Thus, combining (32) with (36) we have  
\[\|x_{n}-Sx_{n}\|\leq\|x_{n}-S_{n}x_{n}\|+\|S_{n}x_{n}-Sx_{n}\|\to 0\quad(n\to\infty).\]  
That is,  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]0  
Now, let us show that if we define \(\overline{S}:=(2I-S)^{-1}\), then \(\overline{S}:C\to C\) is nonexpansive, \(\operatorname{Fix}(\overline{S})=\operatorname{Fix}(S)=\bigcap_{n=0}^{ \infty}\operatorname{Fix}(S_{n})\) and \(\lim_{n\to\infty}\|x_{n}-\overline{Sx}_{n}\|=0\). Indeed, put \(\overline{S}:=(2I-S)^{-1}\), where \(I\) is the identity mapping on \(E\). Then it is known that \(\overline{S}\) is nonexpansive and the fixed-point set \(\operatorname{Fix}(S)=\operatorname{Fix}(\overline{S})=\bigcap_{n=0}^{ \infty}\operatorname{Fix}(S_{n})\). From (37) it follows that  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]1 \[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]2 \[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]3  
That is,  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]4  
**Step 6. ** We claim that  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]5  
where \(x^{*}=\Pi_{\Omega}(f+I-\theta F)(x^{*})\). Indeed, there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]6  
Now we show that \(\Pi_{\Omega}(f+I-\theta F)\) is a contraction mapping. Since \(F\) is bounded linear strongly positive, for all \(x,y\in C\), we have  
\[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]7 \[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]8 \[\|y_{n}-Ty_{n}\|\leq\|y_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\|+\|T^{n+1} y_{n}-Ty_{n}\|.\]9  
which implies that \(\Pi_{\Omega}(f+I-\theta F)\) is a contraction mapping. [PERSON]'s contraction mapping principle guarantees that \(\Pi_{\Omega}(f+I-\theta F)\) has a unique fixed point. Say \(x^{*}\in C\), that is, \(x^{*}=\Pi_{\Omega}(f+I-\theta F)(x^{*})\). Since \(\{x_{n}\}\) is a bounded sequence in \(C\), we may assume that \(x_{n_{i}}\rightharpoonup\bar{x}\in C\). Please note that \(G\) and \(\overline{S}\) are nonexpansive and that \(T\) is asymptotically nonexpansive in the intermediate sense.