## 3 Main Results
\begin{array}{ll}\|w_{n}-p\|^{2}&\leq 2\alpha_{n}\langle F(p),j(p-w_{n}) \rangle+(1-\alpha_{n}\tilde{\gamma})^{2}\|T^{n}y_{n}-p\|^{2}\\ &\leq c_{n}(2\|y_{n}-p\|+c_{n})+(1-\alpha_{n}\tilde{\gamma})\|p-y_{n}\|^{2}+2 \alpha_{n}\langle F(p),j(p-w_{n})\rangle.\end{array}  \tag{17}\]  
From (4) and (17), we get  
\[ \begin{array}{ll}\|x_{n+1}-p\|^{2}&\leq(1-\beta_{n})(1-\alpha_{n}\tilde{ \gamma})\|y_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}\\ &\quad+c_{n}(2\|y_{n}-p\|+c_{n})+2\alpha_{n}(1-\beta_{n})\langle F(p),j(p-w_{n} )\rangle.\end{array}  \tag{18}\]We now note that \(q=\Pi_{C}(p-\mu_{2}B_{2}p)\), \(z_{n}=\Pi_{C}(u_{n}-\mu_{2}B_{2}u_{n})\) and \(y_{n}=\Pi_{C}(z_{n}-\mu_{1}B_{1}z_{n})\). Then \(y_{n}=Gu_{n}\). By using the Lemma 4,  
\[\|z_{n}-q\|^{2}\leq\|u_{n}-p-\mu_{2}(B_{2}u_{n}-B_{2}p)\|^{2}\leq\|u_{n}-p\|^{2 }-2\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}, \tag{19}\]  
and  
\[\|y_{n}-p\|^{2}\leq\|z_{n}-q-\mu_{1}(B_{1}z_{n}-B_{1}q)\|^{2}\leq\|z_{n}-q\|^{2 }-2\mu_{1}(\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2}. \tag{20}\]  
Substituting (19) for (20), we obtain from (7) that  
\[\|y_{n}-p\|^{2}+2\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}+2\mu _{1}(\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2}\leq\|x_{n}-p\|^{2}. \tag{21}\]  
Combining (18) and (21), we get  
\[ \begin{array}{l}\|p-x_{n+1}\|^{2}\\ \leq(1-\beta_{n})(1-\alpha_{n}\tilde{\gamma})[\|x_{n}-p\|^{2}-2\mu_{2}(\beta- \kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}+\beta_{n}\|p-x_{n}\|^{2}\\ \qquad+2\mu_{1}(\kappa^{2}\mu_{1}-\alpha)\|B_{1}z_{n}-B_{1}q\|^{2}]+2\alpha_{n }\|F(p)\|\|p-w_{n}\|+c_{n}(2\|y_{n}-p\|+c_{n})\\ =[1-\alpha_{n}(1-\beta_{n})\tilde{\gamma}]\|p-x_{n}\|^{2}-2(1-\beta_{n})(1- \alpha_{n}\tilde{\gamma})[\mu_{2}(\beta-\kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p \|^{2}\\ \qquad+\mu_{1}(\alpha-\kappa^{2}\mu_{1})\|B_{1}z_{n}-B_{1}q\|^{2}]+c_{n}(2\|y _{n}-p\|+c_{n})+2\alpha_{n}\|F(p)\|\|p-w_{n}\|,\end{array} \]  
which immediately yields  
\[ \begin{array}{l}2(1-\beta_{n})(\alpha_{n}\tilde{\gamma}-1)[\mu_{2}(\beta- \kappa^{2}\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}+\mu_{1}(\alpha-\kappa^{2}\mu_{1}) \|B_{1}z_{n}-B_{1}q\|^{2}]\\ \geq\|x_{n+1}-p\|^{2}-\|x_{n}-p\|^{2}-c_{n}(2\|y_{n}-p\|+c_{n})-2\alpha_{n}\|F (p)\|\|p-w_{n}\|\\ \geq-c_{n}(2\|y_{n}-p\|+c_{n})-2\alpha_{n}\|F(p)\|\|p-w_{n}\|-(p-\|x_{n}\|+\|p -x_{n+1}\|)\|x_{n}-x_{n+1}\|.\end{array} \]  
Since \(\lim\inf_{n\to\infty}(1-\beta_{n})>0\) (due to condition (ii)), \(\mu_{1}\in(0,\frac{\delta}{\kappa^{2}})\), \(\mu_{2}\in(0,\frac{\delta}{\kappa^{2}})\), \(\lim_{n\to\infty}c_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we obtain from (14) that  
\[\lim_{n\to\infty}\|B_{2}u_{n}-B_{2}p\|=\lim_{n\to\infty}\|B_{1}z_{n}-B_{1}q\|=0. \tag{22}\]  
On the other hand, from Lemma 10 and Proposition 3 we have  
\[ \begin{array}{l}2\|q-z_{n}\|^{2}&2\leq\langle j(z_{n}-q),u_{n}-\mu_{2}B_{2} u_{n}-(p-\mu_{2}B_{2}p)\rangle\\ &=2\mu_{2}\langle B_{2}p-B_{2}u_{n},j(z_{n}-q)\rangle+\langle j(z_{n}-q),u_{n} -p\rangle\\ &\leq[\|u_{n}-p\|^{2}+\|z_{n}-q\|^{2}-g_{1}(\|u_{n}-z_{n}-(p-q)\|)]+2\mu_{2} \|B_{2}p-B_{2}u_{n}\|\|z_{n}-q\|,\end{array} \]  
which implies that  
\[\|z_{n}-q\|^{2}+g_{1}(\|u_{n}-z_{n}-(p-q)\|)\leq\|u_{n}-p\|^{2}+2\mu_{2}\|B_{2} p-B_{2}u_{n}\|\|z_{n}-q\|. \tag{23}\]  
In the same manner, one reaches  
\[\|y_{n}-p\|^{2}+g_{2}(\|z_{n}-y_{n}+(p-q)\|)\leq\|z_{n}-q\|^{2}+2\mu_{1}\|B_{1} q-B_{1}z_{n}\|\|y_{n}-p\|. \tag{24}\]  
Substituting (23) for (24), we deduce from (7) that  
\[ \begin{array}{l}\|y_{n}-p\|^{2}+g_{1}(\|u_{n}-z_{n}-(p-q)\|)+g_{2}(\|z_{n}-y _{n}+(p-q)\|)\\ \leq\|x_{n}-p\|^{2}+2\mu_{1}\|B_{1}q-B_{1}z_{n}\|\|y_{n}-p\|+2\mu_{2}\|B_{2}p-B _{2}u_{n}\|\|z_{n}-q\|.\end{array}  \tag{25}\]  
From (27) and (28) it follows that \(\|x_{n}-y_{n}\|\leq\|x_{n}-u_{n}\|+\|u_{n}-Gu_{n}\|\to 0\), and  
\[\|x_{n}-Gx_{n}\|\leq\|x_{n}-y_{n}\|+\|Gu_{n}-Gx_{n}\|\leq\|x_{n}-y_{n}\|+\|u_{n}- x_{n}\|\to 0\quad(n\to\infty). \tag{29}\]  
**Step 4. ** One proves that \(\|Tx_{n}-x_{n}\|\to 0\) and \(\|S_{n}x_{n}-x_{n}\|\to 0\) as \(n\to\infty\).