## 3 Main Results
\begin{array}{ll}\|p-z_{n}\|^{2}&\leq\|p-u_{n}\|^{2}-\|u_{n}-v_{n}-(p-q)\|^{2}- \|v_{n}-z_{n}+(p-q)\|^{2}\\ &\quad+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|+2\mu_{1}\|B_{1}q-B_{1}v_{n}\| \|z_{n}-p\|\\ &\leq\|p-x_{n}\|^{2}-\|u_{n}-v_{n}-(p-q)\|^{2}-\|v_{n}-z_{n}+(p-q)\|^{2}\\ &\quad+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|+2\mu_{2}\|B_{2}p-B_{2}u_{n}\| \|v_{n}-q\|.\end{array}  \tag{32}\]  
Combining (25) and (32), we have  
\[ \begin{array}{ll}&\|x_{n+1}-p\|^{2}\\ &\leq[1-\alpha_{n}(\tau-\delta)][\sigma_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})\|z_{ n}-p\|^{2}]+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|+\alpha_{n}\|A_{1}p\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|A_{2}p\|\|q_{n}- p\|+\|f(p)-p\|\|x_{n+1}-p\|)\\ &\leq[1-\alpha_{n}(\tau-\delta)]\{\sigma_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})[\| x_{n}-p\|^{2}-\|u_{n}-v_{n}-(p-q)\|^{2}\\ &\quad-\|v_{n}-z_{n}+(p-q)\|^{2}+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|+2\mu_ {2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|]\}\\ &\quad+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|+\alpha_{n}\|A_{1}p\|)+\theta_{n}(2+ \theta_{n})\|y_{n}-p\|^{2}\\ &\quad+2\alpha_{n}(\|\mu A_{2}p\|\|q_{n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|)\\ &\leq[1-\alpha_{n}(\tau-\delta)]\|x_{n}-p\|^{2}-[1-\alpha_{n}(\tau-\delta)](1 -\sigma_{n})[\|u_{n}-v_{n}-(p-q)\|^{2}\\ &\quad+\|v_{n}-z_{n}+(p-q)\|^{2}]+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|+2 \mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|\\ &\quad+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|+\alpha_{n}\|A_{1}p\|)+\theta_{n}(2+ \theta_{n})\|y_{n}-p\|^{2}\\ &\quad+2\alpha_{n}(\|\mu A_{2}p\|\|q_{n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|)\\ &\leq\|x_{n}-p\|^{2}-[1-\alpha_{n}(\tau-\delta)](1-\sigma_{n})[\|u_{n}-v_{n}- (p-q)\|^{2}+\|v_{n}-z_{n}+(p-q)\|^{2}]\\ &\quad+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|+2\mu_{2}\|B_{2}p-B_{2}u_{n}\| \|v_{n}-q\|+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|\\ &\quad+\alpha_{n}\|A_{1}p\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{ n}(\|\mu A_{2}p\|\|q_{n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|),\end{array} \]  
which hence yields  
\[ \begin{array}{ll}&[1-\alpha_{n}(\tau-\delta)](1-\sigma_{n})[\|u_{n}-v_{n}-(p -q)\|^{2}+\|v_{n}-z_{n}+(p-q)\|^{2}]\\ &\leq\|x_{n}-p\|^{2}-\|x_{n+1}-p\|^{2}+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\| \\ &\quad+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|+\alpha_{n}\|A_{1}p\|(2\|z_{n}- p\|+\alpha_{n}\|A_{1}p\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|\mu A_{2}p\|\|q_{n }-p\|+\|f(p)-p\|\|x_{n+1}-p\|)\\ &\leq(\|x_{n}-p\|+\|x_{n+1}-p\|)\|x_{n}-x_{n+1}\|+2\mu_{1}\|B_{1}q-B_{1}v_{n}\| \|z_{n}-p\|\\ &\quad+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|+\alpha_{n}\|A_{1}p\|(2\|z_{n}- p\|+\alpha_{n}\|A_{1}p\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|\mu A_{2}p\|\|q_{n }-p\|+\|f(p)-p\|\|x_{n+1}-p\|).\end{array} \]  
Since \(\liminf_{n\to\infty}(1-\sigma_{n})>0\) (due to Condition (iii)), \(\lim_{n\to\infty}\theta_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we conclude from (23) and (29) that  
\[\lim_{n\to\infty}\|u_{n}-v_{n}-(p-q)\|=0\quad\text{and}\quad\lim_{n\to\infty} \|v_{n}-z_{n}+(p-q)\|=0. \tag{33}\]  
It follows that  
\[\|u_{n}-z_{n}\|\leq\|u_{n}-v_{n}-(p-q)\|+\|v_{n}-z_{n}+(p-q)\|\to 0\quad(n\to\infty).\]  
That is,  
\[\lim_{n\to\infty}\|u_{n}-Gu_{n}\|=\lim_{n\to\infty}\|u_{n}-z_{n}\|=0. \tag{34}\]  
In addition, according to (12), we have  
\[ \begin{array}{ll}\|p-u_{n}\|^{2}&=\gamma_{n}\langle p-x_{n},p-u_{n}\rangle+(1 -\gamma_{n})\langle S_{n}u_{n}-p,u_{n}-p\rangle\\ &\leq\gamma_{n}\langle x_{n}-p,u_{n}-p\rangle+(1-\gamma_{n})\|u_{n}-p\|^{2}, \end{array} \]which, together with (9), yields  
\[ \begin{array}{ll}\|p-u_{n}\|^{2}&\leq\langle x_{n}-p,u_{n}-p\rangle\\ &=\frac{1}{2}[\|x_{n}-p\|^{2}+\|u_{n}-p\|^{2}-\|x_{n}-u_{n}\|^{2}].\end{array} \]  
This immediately implies that  
\[\|p-u_{n}\|^{2}\leq\|p-x_{n}\|^{2}-\|u_{n}-x_{n}\|^{2},\]  
which together with (14) and (26), yields  
\[