## 3 Main Results
\begin{array}{ll}&\|p-x_{n+1}\|^{2}\\ &\leq[1-\alpha_{n}(\tau-\delta)][\sigma_{n}\|p-x_{n}\|^{2}+(1-\sigma_{n})\|p- u_{n}\|^{2}]+\alpha_{n}\|A_{1}p\|(2\|p-z_{n}\|+\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[1-\alpha_{n}(\tau-\delta)]\{\sigma_{n}\|x_{n}-p\|^{2}+(1-\sigma_{n})\| x_{n}-p\|^{2}-\|x_{n}-u_{n}\|^{2}]\}\\ &\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})\|x_{n}-u_{n}\|^{2}\\ &\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})\|x_{n}-u_{n}\| ^{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}(\|\mu A_{2}p\|\|q_ {n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|).\end{array} \]  
Hence, we have  
\[ \begin{array}{ll}&[1-\alpha_{n}(\tau-\delta)](1-\sigma_{n})\|x_{n}-u_{n}\|^{ 2}\\ &\leq\|x_{n}-p\|^{2}-\|x_{n+1}-p\|^{2}+\alpha_{n}\|A_{1}p\|(2\|z_{n}-p\|+ \alpha_{n}\|A_{1}p\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|\mu A_{2}p\|\|q_ {n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|)\\ &\leq(\|x_{n}-p\|+\|x_{n+1}-p\|)\|x_{n}-x_{n+1}\|+\alpha_{n}\|A_{1}p\|(2\|z_{n }-p\|+\alpha_{n}\|A_{1}p\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}(\|\mu A_{2}p\|\|q_ {n}-p\|+\|f(p)-p\|\|x_{n+1}-p\|).\end{array} \]  
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 obtain from (23) that  
\[\lim_{n\to\infty}\|x_{n}-u_{n}\|=0. \tag{35}\]  
In addition, observe that  
\[\|x_{n}-z_{n}\|\leq\|x_{n}-u_{n}\|+\|u_{n}-Gu_{n}\|,\]  
\[\|x_{n}-Gx_{n}\|\leq\|x_{n}-z_{n}\|+\|Gu_{n}-Gx_{n}\|\leq\|x_{n}-z_{n}\|+\|u_ {n}-x_{n}\|,\]  
and  
\[\|x_{n}-y_{n}\|\leq(1-\sigma_{n})\|x_{n}-(1-\delta_{n}A_{1})z_{n}\|\leq\|x_{n} -z_{n}\|+\alpha_{n}\|A_{1}z_{n}\|.\]  
Then, from (34) and (35), it follows that  
\[\lim_{n\to\infty}\|x_{n}-z_{n}\|=0,\quad\lim_{n\to\infty}\|x_{n}-Gx_{n}\|=0 \quad\text{and}\quad\lim_{n\to\infty}\|x_{n}-y_{n}\|=0. \tag{36}\]  
**Step 4. ** We claim that \(\|x_{n}-S_{n}x_{n}\|\to 0,\ \|x_{n}-q_{n}\|\to 0\) and \(\|x_{n}-Tx_{n}\|\to 0\) as \(n\to\infty\). Indeed, combining (11) and (25), we obtain that  
\[\|S_{n}u_{n}-u_{n}\|=\frac{\gamma_{n}}{1-\gamma_{n}}\|x_{n}-u_{n}\|\leq\frac{b }{1-b}\|x_{n}-u_{n}\|\to 0\quad(n\to\infty).\]That is,  
\[\lim_{n\to\infty}\|S_{n}u_{n}-u_{n}\|=0. \tag{37}\]  
Since \(\{S_{n}\}_{n=0}^{\infty}\) is \(\ell\)-uniformly Lipschitzian on \(C\), we deduce from (35) and (37) that  
\[ \begin{array}{ll}\|S_{n}x_{n}-x_{n}\|&\leq\|S_{n}x_{n}-S_{n}u_{n}\|+\|S_{n}u_ {n}-u_{n}\|+\|u_{n}-x_{n}\|\\ &\leq\ell\|x_{n}-u_{n}\|+\|S_{n}u_{n}-u_{n}\|+\|u_{n}-x_{n}\|\\ &=(\ell+1)\|x_{n}-u_{n}\|+\|S_{n}u_{n}-u_{n}\|\to 0\quad(n\to\infty).\end{array} \]  
That is,  
\[\lim_{n\to\infty}\|x_{n}-S_{n}x_{n}\|=0. \tag{38}\]  
In addition, we observe that  
\[ \begin{array}{ll}\|x_{n}-T^{n}y_{n}\|\leq\|x_{n+1}-x_{n}\|+\|x_{n+1}-T^{n}y_ {n}\|\\ \leq\|x_{n}-x_{n+1}\|+\beta_{n}\|f(y_{n})-T^{n}y_{n}\|+(1-\beta_{n})\|(I- \alpha_{n}A_{2})T^{n}y_{n}-T^{n}y_{n}\|\\ \leq\|x_{n}-x_{n+1}\|+\alpha_{n}(\|f(y_{n})-T^{n}y_{n}\|+\|uA_{2}(T^{n}y_{n})\|).\end{array} \]  
Hence, we get  
\[ \begin{array}{ll}\|y_{n}-T^{n}y_{n}\|&\leq\|y_{n}-x_{n}\|+\|x_{n}-T^{n}y_{n} \|\\ &\leq\|y_{n}-x_{n}\|+\|x_{n}-x_{n+1}\|+\alpha_{n}(\|f(y_{n})-T^{n}y_{n}\|+\|uA _{2}(T^{n}y_{n})\|).\end{array} \]  
Consequently, from (23), (36) and \(\lim_{n\to\infty}\alpha_{n}=0\), we obtain that  
\[\lim_{n\to\infty}\|x_{n}-T^{n}y_{n}\|=0\quad\text{and}\quad\lim_{n\to\infty}\| y_{n}-T^{n}y_{n}\|=0. \tag{39}\]  
Thus, it follows that  
\[