## 3 Main Results  
In this part, we analyze a relaxed Mann implicit iteration method for approximating solutions of the GSVI (3) with the hierarchical variational inequality (HVI) constraint for countable many uniformly Lipschitzian pseudocontractions and an asymptotically nonexpansive mapping, which is in the intermediate sense in a real Banach space. **Theorem 1**. : _Fix a convex nonempty and closed set \(C\) in a \(2\)-uniformly smooth and uniformly convex Banach space \(E\). The space is assumed to enjoy 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 a-inverse-strongly and \(\beta\)-inverse-strongly accretive. Let linear bounded operator \(F:E\to E\) be strongly positive with coefficient \(\widehat{\gamma}>0\). 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 pseudocontractions on \(C\) such that \(\Omega:=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\cap\operatorname{ GSVI}(C,B_{1},B_{2})\cap\operatorname{Fix}(T)\
eq\emptyset\) where \(\operatorname{GSVI}(C,B_{1},B_{2})\) 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 \(0<\mu_{2}<\frac{\beta}{\kappa^{2}}\). Suppose \(\sum_{n=0}^{\infty}c_{n}<\infty\), where \(c_{n}\) is defined by (2). For arbitrarily given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence generated by_  
\[\left\{ \begin{array}{rl}&x_{n+1}=(1-\beta_{n})\Pi_{C}(T^{n}y_{n}-\alpha_{n} FT^{n}y_{n})+\beta_{n}x_{n},\\ &z_{n}=\Pi_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &y_{n}=\Pi_{C}(z_{n}-\mu_{1}B_{1}z_{n}),\\ &u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\quad\forall n\geq 0,\end{array} \right. \tag{4}\]  
_where \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) are the real sequences in the interval \([0,1]\) satisfying:_  
_(i)_ \(\lim_{n\to\infty}\alpha_{n}=0\) _and_ \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\)_;_  
_(ii)_ \(\limsup_{n\to\infty}\beta_{n}<1\) _and_ \(\liminf_{n\to\infty}\beta_{n}>0\)_;_  
_(iii)_ \(\limsup_{n\to\infty}\gamma_{n}<1\)_,_ \(\liminf_{n\to\infty}\gamma_{n}>0\) _and_ \(\lim_{n\to\infty}|\gamma_{n}-\gamma_{n+1}|=0\)_._  
_Suppose \(\sum_{n=0}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n+1}x\|<\infty\), where \(D\) is a bounded set in \(C\), and let \(S:C\to C\) be a nonself mapping defined by \(Sx=\lim_{n\to\infty}S_{n}x\) for all \(x\in C\), and assume that \(\operatorname{Fix}(S)=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\). Therefore, \(\{x_{n}\}\) strongly converges to \(x^{*}\in\Omega\) provided \(\lim_{n\to\infty}\|T^{n}y_{n}-T^{n+1}y_{n}\|=0\). 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_ \(y^{*}=\Pi_{C}(x^{*}-\mu_{2}B_{2}x^{*})\)_._  
Proof. : From \(\lim_{n\to\infty}\alpha_{n}=0\), one may assume \(\alpha_{n}\leq\|F\|^{-1}\ \forall n\geq 0\). By Lemma 9 we get \(1-\alpha_{n}\widehat{\gamma}\geq\|I-\alpha_{n}F\|\). Since \(1>\limsup_{n\to\infty}\gamma_{n}\geq\liminf_{n\to\infty}\gamma_{n}>0\), one supposes, without loss of generality, that \(\{\gamma_{n}\}\subset[a,b]\subset(0,1)\) for some \(a,b\in(0,1)\). Also, since \(G:=\Pi_{C}(\Pi_{C}(I-\mu_{2}B_{2})-\mu_{1}B_{1}\Pi_{C}(I-\mu_{2}B_{2}))\), where \(0<\mu_{1}\kappa^{2}\leq\alpha\) and \(0<\mu_{2}\kappa^{2}\leq\beta\), according to Lemma 5\(G\) is nonexpansive. For each \(n\geq 0\), one has a unique \(u_{n}\in C\) such that  
\[u_{n}=\gamma_{n}(x_{n}-S_{n}u_{n})+S_{n}u_{n}. \tag{5}\]  
As a matter of fact, we consider \(F_{n}x:=\gamma_{n}(x_{n}-S_{n}x)+S_{n}x\ \forall x\in C\). Since mapping \(S_{n}:C\to C\) is a continuous pseudocontractive, one gets \((1-\gamma_{n})\|x-y\|^{2}\geq\langle F_{n}x-F_{n}y,j(x-y)\rangle,\ \forall x,y\in C\).