## 2 Preliminaries  
Let \(\{T_{n}\}_{n=0}^{\infty}\) be a sequence of continuous pseudocontractive self-mappings on \(C\). Then, \(\{T_{n}\}_{n=0}^{\infty}\) is said to be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings on \(C\) if there exists a constant \(\ell>0\) such that each \(T_{n}\) is \(\ell\)-Lipschitz continuous. We fix an element \(x\) in \(H\) to see that there exists a unique nearest point in \(C\), denoted by \(P_{C}x\), such that  
\[\|x-P_{C}x\|\leq\|x-y\|,\quad\forall y\in C.\]  
\(P_{C}\) is called a metric projection of \(H\) onto \(C\). It may be a set-valued operator. Further, \(C\) is assumed to be convex and closed, and \(X\) is assumed to be Hilbert, \(P_{C}\) is, in such a situation, a single-valued operator. We need the following propositions and lemmas to prove our main results. **Proposition 1** ([27]). : _We suppose \(C\) is a convex closed subset of a Banach space \(X\). Let \(S_{0},S_{1},\ldots\) be a self-mapping sequence on \(C\). Let \(\sum_{n=1}^{\infty}\sup\{\|S_{n}x-S_{n-1}x\|:x\in C\}<\infty\). We conclude \(\{S_{n}y\}\), where \(y\in C\), converges strongly to some point in \(C\). Moreover, we assume \(S\) is a self mapping on \(C\) generated by \(Sy=\lim_{n\to\infty}S_{n}y\) for all \(y\in C\). Therefore, \(\lim_{n\to\infty}\sup\{\|Sx-S_{n}x\|:x\in C\}=0\)._  
**Proposition 2** ([28]). : _We suppose \(C\) is a convex closed subset of a Banach space \(X\) and \(T\) is a continuous strong pseudocontraction self-mapping. Therefore, \(T\) enjoys fixed points. Indeed, it has a unique fixed point._  
The following lemma is trivial. **Lemma 2**. : _In a real Hilbert space \(H\), there holds the inequality_  
\[2\langle y+x,y\rangle\geq\|x+y\|^{2}-\|x\|^{2},\quad\forall x,y\in H.\]  
**Lemma 3** ([29]). : _We suppose that \(\{a_{n}\}\) is a nonnegative number sequence satisfying the restrictions_  
\[a_{n+1}\leq a_{n}+\lambda_{n}\gamma_{n}-\lambda_{n}a_{n},\quad\forall n\geq 0,\]  
_where \(\{\lambda_{n}\}\) and \(\{\gamma_{n}\}\) are sequences of real sequences such that (i) \(\limsup_{n\to\infty}\gamma_{n}\leq 0\) or \(\sum_{n=0}^{\infty}|\lambda_{n}\gamma_{n}|<\infty\); (ii) \(\{\lambda_{n}\}\subset[0,1]\) and \(\sum_{n=0}^{\infty}\lambda_{n}=\infty\), or equivalently,_  
\[\lim_{n\to\infty}\prod_{k=0}^{n}(1-\lambda_{k})=0.\]  
_Hence, \(a_{n}\to 0\) as \(n\to\infty\)._  
The following lemma is a direct consequence of Yamada [30]. **Lemma 4**. : _Let \(F:H\to H\) be a \(\kappa\)-Lipschitzian and \(\eta\)-strongly monotone. We suppose \(\lambda\) is a positive real number in \((0,1]\) and \(T:C\to H\) is a nonexpansive nonself mapping, and we define the mapping \(T^{\lambda}:C\to H\) by_  
\[T^{\lambda}x:=Tx-\lambda\mu F(Tx),\quad\forall x\in C.\]_If_ \(0<\mu<\frac{2\eta}{\kappa^{2}}\)_, then_ \(T^{\lambda}\) _is a contraction operator, that is,_  
\[\|T^{\lambda}x-T^{\lambda}y\|\leq(1-\lambda\tau)\|x-y\|,\quad\forall x,y\in C,\]  
_where_ \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\)_._  
**Lemma 5** ([31]). : _We suppose that the nonself mapping \(A:C\to H\) is \(\alpha\)-inverse-strongly monotone. Then, for a given \(\lambda\geq 0\),_  
\[\|(I-\lambda A)x-(I-\lambda A)y\|^{2}\leq\|x-y\|^{2}+\lambda(\lambda-2\alpha)\| Ax-Ay\|^{2}.\]  
_In particular, if \(0\leq\lambda\leq 2\alpha\), then \(I-\lambda A\) is nonexpansive. Further, we suppose \(A:C\to H\) is a monotone and hemicontinuous mapping. Then, the following hold:_  
\((i)\)__\(\mathrm{VI}(C,A)=\{x^{*}\in C:(Ay,y-x^{*})\geq 0,\;\forall y\in C\}\)_;_  
\((ii)\)__\(\mathrm{VI}(C,A)=\mathrm{Fix}(P_{C}(I-\lambda A))\) _for all_ \(\lambda>0\)_;_  
\((iii)\)__\(\mathrm{VI}(C,A)\) _consists of one point if_ \(A\) _is strongly monotone and Lipschitz continuous._  
**Lemma 6** ([8]). : _We suppose the nonself operators \(B_{1},B_{2}:C\to H\) are \(\alpha\)-inverse-strongly monotone and \(\beta\)-inverse-strongly monotone, respectively. Let the self operator \(G:C\to C\) be defined in \(G:=P_{C}(I-\mu_{1}B_{1})P_{C}(I-\mu_{2}B_{2})\).