## 3 Main Results
We take \(S_{n}:=T^{n}\) and obtain_  
\[\sup_{x\in D}\|S_{n}x-S_{n-1}x\| =\sup_{x\in D}\|T^{n}x-T^{n-1}x\|\] \[\leq(\frac{3}{4})^{n-1}\cdot\sup_{x\in D}\|Tx-x\|\quad\forall n \geq 1,\]  
_for any bounded subset \(D\) of \(C\). Therefore, it follows that_  
\[\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n-1}x\|\leq\sum_{n=1}^{\infty}( \frac{3}{4})^{n-1}\cdot\sup_{x\in D}\|Tx-x\|<\infty.\]  
_In addition, for any sequence \(\{y_{n}\}\subset C\), we get_  
\[\|T^{n+1}y_{n}-T^{n}y_{n}\| \leq(\frac{3}{4})^{n-1}\|T^{2}y_{n}-Ty_{n}\|\] \[=(\frac{3}{4})^{n}\|\sin Ty_{n}-\sin y_{n}\|\] \[\leq\frac{3}{4}\|T^{n-1}x-T^{n-1}y\|\]0  
_It is easy to see that \(Fix(T)=\{0\}\). We define \(Sx:=0\ \forall x\in C\). It is clear that \(Sx=\lim_{n\to\infty}S_{n}x\ \forall x\in C\) and \(Fix(S)=\bigcap_{n=0}^{\infty}Fix(S_{n})\)._  
The following results can be easily obtained by virtue of the argument techniques in Theorem 1, and hence we omit the details. **Theorem 2**. : _Fix a bounded, convex, nonempty, and closed set in a \(2\)-uniformly smooth and uniformly convex Banach space \(E\). We suppose \(E\) has a weakly sequentially continuous duality mapping. Let \(\Pi_{C}:E\to C\) be a nonexpansive sunny retraction. Let the accretive mappings \(B_{1},B_{2}:C\to E\) be \(\alpha\)-inverse-strongly and \(\beta\)-inverse-strongly. Let linear bounded operator \(F\) be a strongly positive on \(E\) with coefficient \(\gamma>0\). Let \(T\) be an asymptotically nonexpansive self-mapping on \(C\) with a sequence \(\{\theta_{n}\}\subset[0,\infty)\) satisfying \(\sum_{n=0}^{\infty}\theta_{n}<\infty\), and \(\{S_{n}\}_{n=0}^{\infty}:C\to C\) be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontructractive mappings 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 \(G:=\Pi_{C}(I-\mu_{1}B_{1})\Pi_{C}(I-\mu_{2}B_{2})\) with \(0<\mu_{1}<\frac{\alpha}{\kappa^{2}}\) and \(0<\mu_{2}<\frac{\beta}{\kappa^{2}}\). For arbitrarily given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence given by_  
\[\left\{ \begin{array}{rl}&u_{n}=\gamma_{n}(x_{n}-S_{n}u_{n})+S_{n}u_{n},\\ &z_{n}=\Pi_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &y_{n}=\Pi_{C}(z_{n}-\mu_{1}B_{1}z_{n}),\\ &x_{n+1}=(1-\beta_{n})\Pi_{C}(T^{n}y_{n}-\alpha_{n}FT^{n}y_{n})+\beta_{n}x_{n},\quad\forall n\geq 0,\end{array} \right.\]_where \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) are the sequences in \([0,1]\) satisfying the following conditions:_  
_(i)_ \(\lim_{n\to\infty}\alpha_{n}=0\) _and_ \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\)_;_  
_(ii)_ \(1\geq\limsup_{n\to\infty}\beta_{n}\geq\liminf_{n\to\infty}\beta_{n}\geq 0\)_;_  
_(iii)_ \(1\geq\limsup_{n\to\infty}\gamma_{n}\geq\liminf_{n\to\infty}\gamma_{n}\geq 0\) _and_ \(\lim_{n\to\infty}|\gamma_{n+1}-\gamma_{n}|=0\)_._  
_Assume that_ \(\sum_{n=0}^{\infty}\sup_{x\in\mathbb{C}}\|S_{n+1}x-S_{n}x\|<\infty\)_, 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_ \(\operatorname{Fix}(S)=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\)_. Then_ \(\{x_{n}\}\) _converges strongly to_ \(x^{*}\in\Omega\) _provided_ \(T^{n+1}y_{n}-T^{n}y_{n}\to 0\) _as_ \(n\to\infty\)_. In this case,_  
_(a)_ \(x^{*}\in\Omega\) _solves the VI:_ \(\langle F(x^{*}),j(x^{*}-x)\rangle\leq 0\)_,_ \(\forall x\in\Omega\)_;_  
_(b)_ \((x^{*},y^{*})\) _solves the GSVI (3) with_ \(y^{*}=\Pi_{C}(I-\mu_{2}B_{2})x^{*}\)_._  
Proof. : Since the set \(C\) is bounded, one gets \(\operatorname{diam}(C)=\sup_{x,y\in C}\|x-y\|<\infty\). Please note that \(T\) is an asymptotically nonexpansive mapping equipped \(\{\theta_{n}\}\subset[0,\infty)\) satisfying \(\sum_{n=0}^{\infty}\theta_{n}<\infty\).