# Systems of Variational Inequalities with Nonlinear Operators
## 4 Applications  
Now, we give an application to solve CFPPs of asymptotically nonexpansive and pseudocontractive mappings, and variational inequality problems for strict pseudocontractive mappings in Banach spaces. Let \(C\) be a nonempty, closed, and convex subset of a real Banach space \(E\). A mapping \(T:C\to C\) is said to be \(\lambda\)-strictly pseudocontractive if for every \(x,y\in C\) there exists \(j(x-y)\in J(x-y)\) such that  
\[\langle Tx-Ty,j(x-y)\rangle+\lambda\|(I-T)x-(I-Ty\|^{2}\leq\|x-y\|^{2},\quad \text{for some }\lambda\in(0,1). \tag{42}\]  
A simple computation shows that (42) is equivalent to the following inequality:  
\[\langle(I-T)x-(I-T)y,j(x-y)\rangle\geq\lambda\|(I-T)x-(I-T)y\|^{2}, \tag{43}\]  
for every \(x,y\in C\) and for some \(j(x-y)\in J(x-y)\). Therefore, \(I-T\) is \(\lambda\)-inverse-strongly accretive. By Theorem 1, we can obtain the following results easily. **Theorem 2**. : _Let \(C\) be a convex 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 sumy nonexpansive retraction from \(E\) onto \(C\). Let the mappings \(B_{1},B_{2}:C\to C\) be a-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 uniformly continuous and asymptotically nonexpansive mapping in the intermediate sense, 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 \(E\). Assume that \(\sum_{n=0}^{\infty}c_{n}<\infty\), where \(c_{n}\) is defined by (2). For arbitrarily given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence generated by_  
\[\left\{ \begin{array}{ll}&y_{n}=(1-\mu_{1})z_{n}+\mu_{1}B_{1}z_{n},\\ &z_{n}=(1-\mu_{2})u_{n}+\mu_{2}B_{2}u_{n},\\ &u_{n}=\beta_{n}(x_{n}-S_{n}u_{n})+S_{n}u_{n},\\ &x_{n+1}=\Pi_{C}[(T^{n}y_{n}-\alpha_{n}\theta FT^{n}y_{n})+\alpha_{n}f(x_{n})], \quad\forall n\geq 0,\end{array} \right. \tag{44}\]  
_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 \(\mathrm{Fix}(S)=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\). 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.