## 2 Preliminaries
\(G:C\to C\) is nonexpansive if \(0\leq\mu_{1}\leq 2\alpha\) and \(0\leq\mu_{2}\leq 2\beta\)._  
**Lemma 7** ([32]). : _We suppose the Banach space \(X\) enjoys a weakly continuous duality mapping, and \(C\) is a convex closed set in \(X\). Let \(T:C\to C\) be an asymptotically nonexpansive self mapping on \(C\) with a nonempty fixed point set. Then, \(I-T\) is demiclosed at zero, i.e., if \(\{x_{n}\}\) is a sequence in \(C\) converging weakly to some \(x\in C\) and the sequence \(\{(I-T)x_{n}\}\) converges strongly to zero, then \((I-T)x=0\), where \(I\) is the identity mapping of \(X\)._  
**Lemma 8** ([33]). : _Let both \(\{x_{n}\}\) and \(\{h_{n}\}\) be a bounded sequence in a Banach space \(X\). Let \(\{\beta_{n}\}\subset(0,1)\) be a number sequence such that_  
\[0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1.\]  
_Suppose that \(x_{n+1}=\beta_{n}x_{n}+(1-\beta_{n})h_{n}\;\forall n\geq 0\) and \(\limsup_{n\to\infty}(\|h_{n+1}-h_{n}\|-\|x_{n+1}-x_{n}\|)\leq 0\). \(So\), \(\lim_{n\to\infty}\|h_{n}-x_{n}\|=0\)._  
**3. Main Results**  
Let \(C\) be a convex closed subset of a real Hilbert space \(H\). Let \(B_{1},B_{2}:C\to H\) be monotone mappings, \(A-g:C\to H\) be a monotone mapping with \(A,g:C\to H\), \(T:C\to C\) be an asymptotically nonexpansive mapping, and \(\{S_{n}\}_{n=0}^{\infty}\) be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings defined on \(C\). We suppose \(\Omega:=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\cap\mathrm{GSVI}(C,B_{1},B_{ 2})\cap\mathrm{Fix}(T)\
eq\emptyset\) and studied the variational inequality for monotone mapping \(A-g\) over the common solution set \(\Omega\) of the GSVI (3) and the CFPP of \(\{S_{n}\}_{n=0}^{\infty}\) and \(T\):  
\[\mathrm{Find}\;\bar{x}\in\mathrm{VI}(\Omega,A-g):=\{\bar{x}\in\Omega:\langle(A -g)\bar{x},y-\bar{x}\rangle\geq 0\;\forall y\in\Omega\}.\]  
This section introduces the following monotone variational inequality problem with the inequality constraint over the common solution set of the GSVI (2) and the CFPP of \(T\) and \(\{S_{n}\}_{n=0}^{\infty}\), which is named the triple hierarchical constrained variational inequality:  
Assume that  
(C1) \(T:C\to C\) is an asymptotically nonexpansive mapping with a sequence \(\{\theta_{n}\}\);  
(C2) \(\{S_{n}\}_{n=0}^{\infty}\) is a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings on \(C\);(C3) \(B_{1},B_{2}:C\to H\) are \(\alpha\)-inverse-strongly monotone and \(\beta\)-inverse-strongly monotone, respectively;  
(C4) \(\mathrm{GSVI}(C,B_{1},B_{2}):=\mathrm{Fix}(G)\) where \(G:=P_{C}(I-\mu_{1}B_{1})P_{C}(I-\mu_{2}B_{2})\) for \(\mu_{1},\mu_{2}>0\);  
(C5) \(\Omega:=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\cap\mathrm{GSVI}(C,B_{1},B _{2})\cap\mathrm{Fix}(T)\
eq\emptyset\);  
(C6) \(\sum_{n=1}^{\infty}\sup_{x\in D}\|S_{n}x-S_{n-1}x\|<\infty\) for any bounded subset \(D\) of \(C\);  
(C7) \(S:C\to C\) is the mapping defined by \(Sx=\lim_{n\to\infty}S_{n}x\ \forall x\in C\), such that \(\mathrm{Fix}(S)=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\);  
(C8) \(g:C\to H\) is \(l\)-Lipschitzian and \(A:C\to H\) is \(\zeta\)-inverse-strongly monotone such that \(A-g\) is monotone;  
(C9) \(f:C\to C\) is a contraction mapping with coefficient \(\delta\in[0,1)\) and \(F:C\to H\) is \(\kappa\)-Lipschitzian and \(\eta\)-strongly monotone;  
(C10) \(\mathrm{VI}(\Omega,A-g)\
eq\emptyset\). **Problem 1**. : _The objective is to_  
\[\mathrm{find}\ x^{*} \in\mathrm{VI}(\mathrm{VI}(\Omega,A-g),I-f)\] \[2\langle y+x,y\rangle\geq\|x+y\|^{2}-\|x\|^{2},\quad\forall x,y\in H.\]0  
Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical constrained variational inequality. Since the mapping \(f\) is a contractive, we easily get that the solution of the problem is unique. Inspired by the results announced recently, we introduce the following multiple hybrid implicit iterative algorithm to find the solution of such a problem. Step 0.