# Systems of Variational Inequalities with Nonlinear Operators
## 4 Applications
: Since the mappings \(B_{1},B_{2}:C\to C\) are \(\alpha\)-strictly pseudocontractive and \(\beta\)-strictly pseudocontractive, respectively, it can be seen readily that \(I-B_{1},I-B_{2}:C\to E\) are \(\alpha\)-inverse-strongly accretive and \(\beta\)-inverse-strongly accretive, respectively. Please note that \(0<\mu_{1}\kappa^{2}<\min\{1,\alpha\}\) and \(0<\mu_{2}\kappa^{2}<\min\{1,\beta\}\) for \(\kappa\) the \(2\)-uniformly smooth constant of \(E\). Then, \(\mathrm{GSVI}(C,I-B_{1},I-B_{2})\) is the fixed-point set of the following mapping  
\[Gx: =\Pi_{C}[(1-\mu_{1})I+\mu_{1}B_{1}]\Pi_{C}[(1-\mu_{2})x+\mu_{2}B_{ 2}x]\] \[=\Pi_{C}\{(1-\mu_{1})[(1-\mu_{2})x+\mu_{2}B_{2}x]+\mu_{1}B_{1}[(1- \mu_{2})x+\mu_{2}B_{2}x]\}\] \[=[(1-\mu_{1})I+\mu_{1}B_{1}][(1-\mu_{2})I+\mu_{2}B_{2}]x,\quad \forall x\in C.\]  
In this case, it is easy to see that the iterative scheme (7) reduces to (44). Therefore, by Theorem 1 we obtain the desired result. **Theorem 3**. : _Let \(C\) be a bounded, convex and closed set in a \(2\)-uniformly smooth and uniformly convex Banach space \(E\) which admits a weakly sequentially continuous duality mapping. Let \(\Pi_{C}\) be a sunny nonexpansive retraction from \(E\) onto \(C\). Let the mappings \(B_{1},B_{2}:C\to C\) be \(\alpha\)-strictly pseudocontractive and \(\beta\)-strictly pseudocontractive, respectively. Let \(f:C\to C\) be a contraction mapping with coefficient \(\gamma\in[0,1)\) and \(F:E\to E\) be a strongly positive linear bounded operator with the coefficient \(\bar{\gamma}\) such that \(0<\gamma<\bar{\gamma}\theta\) and \(0<\theta\leq\|F\|^{-1}\). Let \(T:C\to C\) be an asymptotically nonexpansive mapping with a sequence \(\{\theta_{n}\}\subset[0,\infty)\) satisfying \(\sum_{n=0}^{\infty}\theta_{n}<\infty\), and \(\{S_{n}\}_{n=0}^{\infty}\) be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings on \(C\) such that \(\Omega:=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\cap\mathrm{GSVI}(C,I-B_{1}, I-B_{2})\cap\mathrm{Fix}(T)\
eq\emptyset\) where \(\mathrm{GSVI}(C,I-B_{1},I-B_{2})\) is the fixed-point set of the mapping \(G:=[(1-\mu_{1})I+\mu_{1}B_{1}][(1-\mu_{2})I+\mu_{2}B_{2}]\) with \(0<\mu_{1}\kappa^{2}<\min\{1,\alpha\}\)and \(0<\mu_{2}\kappa^{2}<\min\{1,\beta\}\) for \(\kappa\) the \(2\)-uniformly smooth constant of \(E\). For arbitrarily given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence generated by_  
\[\left\{ \begin{aligned} z_{n}&=(1-\mu_{2})u_{n}+\mu_{2} B_{2}u_{n},\\ u_{n}&=\beta_{n}(x_{n}-S_{n}u_{n})+S_{n}u_{n},\\ y_{n}&=(1-\mu_{1})z_{n}+\mu_{1}B_{1}z_{n},\\ x_{n+1}&=\Pi_{C}[\alpha_{n}f(x_{n})+(I-\alpha_{n} \theta F)T^{n}y_{n}],\quad\forall n\geq 0,\end{aligned} \right. \tag{45}\]  
_where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are the sequences in \([0,1]\) satisfying the following conditions:_  
1. \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\)_,_ \(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty\) _and_ \(\lim_{n\to\infty}\alpha_{n}=0\)_;_
2. \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\) _and_ \(\sum_{n=1}^{\infty}|\beta_{n}-\beta_{n-1}|<\infty\)_._  
_Assume that \(\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n-1}x\|<\infty\) for any bounded subset \(D\) of \(C\), and let \(S\) be a mapping of \(C\) into itself defined by \(Sx=\lim_{n\to\infty}S_{n}x\) for all \(x\in C\), and suppose that \(\operatorname{Fix}(S)=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\). If \(\sum_{n=0}^{\infty}\|T^{n+1}y_{n}-T^{n}y_{n}\|<\infty\), then \(\{x_{n}\}\) converges strongly to \(x^{*}\in\Omega\). In this case,_  
1. \(x^{*}\in\Omega\) _solves the VI:_ \(\langle f(x^{*})-\theta F(x^{*}),j(x-x^{*})\rangle\leq 0,\ \forall x\in\Omega\)_;_
2. \((x^{*},y^{*})\) _is a solution of GSVI (_1.3_) for two inverse-strongly accretive mappings_ \(I-B_{i},\ i=1,2\)_, where_ \(y^{*}=(1-\mu_{2})x^{*}+\mu_{2}B_{2}x^{*}\)_._  
Proof. : Since set \(C\) is a bounded set, we know that \(\operatorname{diam}(C)=\sup_{x,y\in C}\|x-y\|<\infty\).