## 3 Main Results
Let \(A_{1}\) be a \(\zeta\)-inverse-strongly monotone nonself mapping on \(C\) and \(A_{2}\) be a strongly positive bounded linear self operator one \(H\) with the parameter \(\gamma>0\), where \(\delta<\tau:=1-\sqrt{1-\mu(2\gamma-\mu\|A_{2}\|^{2})}\in(0,1]\), \(0<\mu<\frac{2\gamma}{\|A_{2}\|^{2}}\). Let the mappings \(B_{1},B_{2}:C\to H\) be \(\mathfrak{a}\)-inverse-strongly monotone and \(\beta\)-inverse-strongly monotone, respectively. Let \(T\) be an asymptotically nonexpansive self mapping on set \(C\) with a sequence \(\{\theta_{n}\}\). Let \(\{S_{n}\}_{n=0}^{\infty}\) be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings on \(C\) satisfying the assumptions in Problem 1. For any given \(x_{0}\in C\), we suppose \(\{x_{n}\}\) is a vector sequence through_  
\[ \begin{cases}&u_{n}=\frac{x_{n}+S_{n}u_{n}}{2},\\ &v_{n}=P_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &z_{n}=P_{C}(v_{n}-\mu_{1}B_{1}v_{n}),\\ &y_{n}=\sigma_{n}x_{n}+(1-\sigma_{n})P_{C}(I-\delta_{n}A_{1})z_{n},\\ &x_{n+1}=\beta_{n}f(y_{n})+(1-\beta_{n})P_{C}(I-\alpha_{n}\mu A_{2})T^{n}y_{n},\quad\forall n\geq 0,\end{cases}  \tag{50}\]  
_where \(\mu_{1}\in(0,2\alpha)\) and \(\mu_{2}\in(0,2\beta)\). Suppose that \(\{\alpha_{n}\},\ \{\beta_{n}\},\ \{\sigma_{n}\}\subset(0,1]\) and \(\{\delta_{n}\}\subset(0,2\zeta]\) are the sequences as in Theorem 1. Then, the sequence \(\{x_{n}\}_{n=0}^{\infty}\) generated by (50) satisfies the following properties:_  
1. \(\{x_{n}\}_{n=0}^{\infty}\) _is bounded._
2. \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\)_,_ \(\lim_{n\to\infty}\|x_{n}-Gx_{n}\|=0\)_,_ \(\lim_{n\to\infty}\|x_{n}-Tx_{n}\|=0\) _and_ \(\lim_{n\to\infty}\|x_{n}-Sx_{n}\|=0\)_._
3. \(\{x_{n}\}_{n=0}^{\infty}\) _reaches to the unique solution of Problem 1 if_ \(\frac{\|x_{n}-y_{n}\|}{\delta_{n}}\to 0\) _as_ \(n\to\infty\)_._  
Proof. : Since the linear bounded operator \(A_{2}:H\to H\) is positive and strong with the parameter \(\gamma>0\), we know that \(A_{2}\) is \(\kappa\)-Lipschitzian and \(\eta\)-strongly monotone where \(\kappa=\|A_{2}\|\) and \(\eta=\gamma\). In this case, we obtain that \(0<\mu<\frac{2\eta}{\kappa^{2}}=\frac{2\gamma}{\|A_{2}\|^{2}}\), and  
\[\delta<\tau:=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}=1-\sqrt{1-\mu(2\gamma-\mu\| A_{2}\|^{2})}\in(0,1].\]  
Therefore, utilizing Theorem 1, we derive the desired result. **Corollary 2**. : _We suppose \(C\) is a convex nonempty closed set of a real Hilbert space \(H\). Let \(f:C\to C\) be a contraction with the parameter \(\delta\in[0,1)\). Let \(A_{1}:C\to H\) be a \(\zeta\)-inverse-strongly monotone mapping and \(A_{2}:C\to H\) be \(\kappa\)-Lipschitzian and \(\eta\)-strongly monotone with the parameters \(\kappa,\eta>0\), where \(\delta<\tau:=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\), \(0<\mu<\frac{2\eta}{\kappa^{2}}\). We suppose the nonself mappings \(B_{1},B_{2}:C\to H\) are \(\kappa\)-inverse-strongly monotone and \(\beta\)-inverse-strongly monotone, respectively. Let \(T:C\to C\) be a nonexpansive mapping and \(\{S_{n}\}_{n=0}^{\infty}\) be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings on \(C\) satisfying the assumptions in Problem 1. For any given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence generated by_  
\[ \begin{cases}&u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\\ &v_{n}=P_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &z_{n}=P_{C}(v_{n}-\mu_{1}B_{1}v_{n}),\\ &y_{n}=\sigma_{n}x_{n}+(1-\sigma_{n})P_{C}(I-\delta_{n}A_{1})z_{n},\\ &x_{n+1}=\beta_{n}f(y_{n})+(1-\beta_{n})P_{C}(I-\alpha_{n}\mu A_{2})Ty_{n}, \quad\forall n\geq 0,\end{cases}  \tag{51}\]  
_where \(\mu_{1}\in(0,2\alpha)\) and \(\mu_{2}\in(0,2\beta)\). Suppose that \(\{\alpha_{n}\},\ \{\beta_{n}\},\ \{\gamma_{n}\}\subset(0,1]\) and \(\{\delta_{n}\}\subset(0,2\zeta]\) are the sequences as in Theorem 1. Then, the sequence \(\{x_{n}\}_{n=0}^{\infty}\) generated by (51) satisfies the following properties:_  
1. \(\{x_{n}\}_{n=0}^{\infty}\) _is bounded._
2.