## 3 Main Results  
We suppose \(C\) is a convex nonempty closed set. Let the mappings \(A_{1},B_{i}\) be nonself monotone mappings for \(i=1,2\) from \(C\) to \(H\). We also let \(T\) be a self asymptotically nonexpansive mapping. Suppose \(\{S_{n}\}_{n=0}^{\infty}\) is a countable family of self mapping. We also assume it is \(\ell\)-uniformly Lipschitzian pseudocontractive on set \(C\). Consider the variational inequality for monotone mapping \(A_{1}\) over the common solution set \(\Omega\) of the GSVI (4) and the CFPP of \(\{S_{n}\}_{n=0}^{\infty}\) and \(T\):  
\[ \begin{array}{l}\mbox{Find }\bar{x}\in{\rm VI}(\Omega,A_{1})\\ :=\{\bar{x}\in\Omega:\langle A_{1}\bar{x},y-\bar{x}\rangle\geq 0\;\forall y\in \Omega\},\end{array} \]  
where \(\Omega:=\bigcap_{n=0}^{\infty}{\rm Fix}(S_{n})\cap{\rm GSVI}(C,B_{1},B_{2}) \cap{\rm Fix}(T)\
eq\emptyset\). This section introduces the following monotone variational inequality with the variational inequality constraint over the common solution set of the GSVI (4) and the CFPP of \(\{S_{n}\}_{n=0}^{\infty}\) and \(T\), which is called the triple hierarchical constrained variational inequality (THCVI):  
**Problem 1**: _Assume that_  
1. \(T:C\to C\) _is an asymptotically nonexpansive mapping with a sequence_ \(\{\theta_{n}\}\)_._
2. \(\{S_{n}\}_{n=0}^{\infty}\) _is a countable family of_ \(\ell\)_-uniformly Lipschitzian pseudocontractive self-nappings on_ \(C\)* \(B_{1}:C\to H\) _is an_ \(\alpha\)_-inverse-strongly monotone operator and_ \(B_{2}:C\to H\) _is a_ \(\beta\)_-inverse-strongly monotone operator._
* \(\operatorname{GSVI}(C,B_{1},B_{2}):=\operatorname{Fix}(G)\) _where_ \(G:=P_{C}(I-\mu_{1}B_{1})P_{C}(I-\mu_{2}B_{2})\) _for_ \(\mu_{1},\mu_{2}>0\)_._
* \(\Omega:=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\cap\operatorname{ GSVI}(C,B_{1},B_{2})\cap\operatorname{Fix}(T)\
eq\emptyset\)_._
* \(\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n-1}x\|<\infty\) _for any bounded subset_ \(D\) _of_ \(C\)_._
* \(S:C\to C\) _is the mapping defined by_ \(Sx=\lim_{n\to\infty}S_{n}x\ \forall x\in C\)_, such that_ \[\operatorname{Fix}(S)=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n}).\]
* \(A_{1}:C\to H\) _is an_ \(\zeta\)_-inverse-strongly monotone operator and_ \(A_{2}:C\to H\) _is a_ \(\kappa\)_-Lipschitzian and_ \(\eta\)_-strongly monotone operator._
* \(f:C\to C\) _is a contraction mapping with coefficient_ \(\delta\in[0,1)\)_._
* \(\operatorname{VI}(\Omega,A_{1})\
eq\emptyset\)_._  
_Then, the objective is to_  
\[\operatorname{find}\ x^{*} \in\operatorname{VI}(\operatorname{VI}(\Omega,A_{1}),A_{2})\] \[:=\{x^{*}\in\operatorname{VI}(\Omega,A_{1}):\langle A_{2}x^{*},v- x^{*}\rangle\geq 0\ \forall v\in\operatorname{VI}(\Omega,A_{1})\}.\]  
_Since the original problem is a variational inequality problem, we therefore call it a triple hierarchical constrained variational inequality (THCVI). We introduce the following hybrid Mann viscosity implicit iteration method to find the solution of such a problem._  
We show the main result of this paper, that is, the strong convergence analysis for Algorithm 1. ```
Step 0. Take \(\{\alpha_{n}\}_{n=0^{\prime}}^{\infty},\{\beta_{n}\}_{n=0^{\prime}}^{\infty}, \{\gamma_{n}\}_{n=0^{\prime}}^{\infty},\{\delta_{n}\}_{n=0^{\prime}}^{\infty}, \{\sigma_{n}\}_{n=0}^{\infty}\subset(0,\infty)\), and \(\mu>0\); arbitrarily choose \(x_{0}\in C\); and let \(n:=0\). Step 1. Given \(x_{n}\in C\), compute \(x_{n+1}\in C\) as \[\left\{ \begin{array}{rl}u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\\ v_{n}=P_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ z_{n}=P_{C}(v_{n}-\mu_{1}B_{1}v_{n}),\\ y_{n}=\sigma_{n}x_{n}+(1-\sigma_{n})P_{C}(I-\delta_{n}A_{1})z_{n},\\ x_{n+1}=\beta_{n}f(y_{n})+(1-\beta_{n})P_{C}(I-\alpha_{n}\mu A_{2})T^{n}y_{n}.\end{array} \right.\] (10) Update \(n:=n+1\) and go to Step 1. ```  
**Algorithm 1**Hybrid Mann viscosity-like implicit iterative algorithm. **Theorem 1**. : _Assume that \(\mu_{1}\in(0,2\alpha)\), \(\mu_{2}\in(0,2\beta)\), and \(\delta<\tau:=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\) for \(\mu\in(0,\frac{2\eta}{\kappa^{2}})\).