# Systems of Variational Inequalities with Nonlinear Operators
## 3 Main Results  
In this section, we suggest and analyze a hybrid viscosity implicit iteration method for solving the GSVI (3) with the hierarchical variational inequality (HVI) constraint for countably many uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in the intermediate sense in a \(2\)-uniformly smooth and uniformly convex Banach space. Theorem 1: _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 sunny nonexpansive retraction from \(E\) onto \(C\). Let the mappings \(B_{1},B_{2}:C\to E\) be \(\alpha\)-inverse-strongly accretive and \(\beta\)-inverse-strongly accretive, 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 \(\gamma\) such that \(0<\gamma<\gamma\theta\) 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 \(\Omega:=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\cap\mathrm{GSVI}(C,B_{1},B_ {2})\cap\mathrm{Fix}(T)\
eq\emptyset\) where \(\mathrm{GSVI}(C,B_{1},B_{2})\) is the fixed-point set of the mapping \(G:=\Pi_{C}(\Pi_{C}(I-\mu_{2}B_{2})-\mu_{1}B_{1}\Pi_{C}(I-\mu_{2}B_{2}))\) with \(0<\mu_{1}\kappa^{2}<\alpha\) and \(0<\mu_{2}\kappa^{2}<\beta\) 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_  
\[\left\{ \begin{array}{rl}&u_{n}=\beta_{n}(x_{n}-S_{n}u_{n})+S_{n}u_{n},\\ &z_{n}=\Pi_{\rm C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &y_{n}=\Pi_{\rm C}(z_{n}-\mu_{1}B_{1}z_{n}),\\ &x_{n+1}=\Pi_{\rm C}[(T^{n}y_{n}-\alpha_{n}\theta FT^{n}y_{n})+\alpha_{n}f(x_{n })],\quad\forall n\geq 0,\end{array} \right. \tag{7}\]  
_where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are the sequences in \([0,1]\) satisfying the following conditions:_  
1. \(\lim_{n\to\infty}\alpha_{n}=0\)_,_ \(\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty\) _and_ \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\)_,_
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 \(S_{X}=\lim_{n\to\infty}S_{n}x\) 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) with_ \(y^{*}=\Pi_{\rm C}(I-\mu_{2}B_{2})x^{*}\)_._  
Proof. : First of all, from \(\alpha_{n}\to 0\ (n\to\infty)\), we may assume, without loss of generality, that \(\alpha_{n}\leq\frac{\|F\|^{-1}}{\theta}\) for all \(n\geq 0\). Since \(F\) is strongly positive linear bounded, it follows that \(1-\alpha_{n}\tilde{\gamma}\theta\geq\|I-\alpha_{n}\theta F\|\). Taking into account that \(\{\beta_{n}\}\) is bounded away from \(0\) and \(1\), we may assume that \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\) for some \(a,b\in(0,1)\). Please note that the mapping \(G:C\to C\) is defined as \(G:=\Pi_{\rm C}(\Pi_{\rm C}(I-\mu_{2}B_{2})-\mu_{1}B_{1}\Pi_{\rm C}(I-\mu_{2}B_ {2}))\), where \(0<\mu_{1}x^{2}\leq\alpha\) and \(0<\mu_{2}x^{2}\leq\beta\) for \(\kappa\) the \(2\)-uniformly smooth constant of \(E\). Therefore, by Lemma 3, we obtain that \(G\) is nonexpansive.