## 2 Preliminaries
 \tag{25}\]  
Combining (18) and (25), we have \[\|x_{n+1}-p\|^{2}\] \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\] \[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\] \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\] \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\] \[\quad-\|u_{n}-v_{n}-(p-q)\|^{2}-\|v_{n}-z_{n}+(p-q)\|^{2}+2\mu_{1} \|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|\] \[\quad+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|+\alpha_{n}\|g(x_{n} )-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})-Ap\|)\] \[\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}\|x_{n}- \mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\|\|x_{n+1}-p\|\] \[=[1-\alpha_{n}(1-\alpha_{n}-\beta_{n})\tau]\|x_{n}-p\|^{2}-(1- \alpha_{n}-\beta_{n})(1-\alpha_{n}\tau)[\|u_{n}-v_{n}-(p-q)\|^{2}\] \[\quad+\|v_{n}-z_{n}+(p-q)\|^{2}]+2\mu_{1}\|B_{1}q-B_{1}v_{n}\|\|z_{ n}-p\|+2\mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|\] \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]0 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]1 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]2 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]3 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]4 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]5  
which hence yields  
\[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]6 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]7 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]8 \[\leq a_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n} -\beta_{n})(1-\alpha_{n}\tau)\|z_{n}-p\|^{2}\]9 \[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]0 \[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]1 \[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]2  
Since \(\liminf_{n\to\infty}(1-\alpha_{n}-\beta_{n})>0\) (due to condition (iii)), \(\lim_{n\to\infty}\theta_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we conclude from (15) and (22) that  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]3  
It follows that  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]4  
That is,  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]5  
Additionally, according to (6), we have  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]6 \[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]7  
which implicitly yields that  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]8  
which together with (3.16), yields  
\[\quad+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+\alpha_{n}\|g(x_{n})- Ap\|)+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}\]9 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]0 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]1 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]2 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]3 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]4 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]5 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]6 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]7 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]8 \[\quad+2\alpha_{n}\|x_{n}-\mu Pp\|\|w_{n}-p\|+2\alpha_{n}\|f(p)-p\| \|x_{n+1}-p\|\]9 \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]0  
Hence, we have  
\[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]1 \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]2 \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]3 \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]4 \[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]5  
Since \(\liminf_{n\to\infty}(1-\alpha_{n}-\beta_{n})>0\), \(\lim_{n\to\infty}\theta_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we obtain from (15) that  
\[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]6  
Moreover, observe that  
\[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]7  
\[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]8  
and  
\[\leq\alpha_{n}\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{ n}-\beta_{n})(1-\alpha_{n}\tau)[\|x_{n}-p\|^{2}\]9  
Then, from (27) and (28), it follows that  
\[\quad-\|u_{n}-v_{n}-(p-q)\|^{2}-\|v_{n}-z_{n}+(p-q)\|^{2}+2\mu_{1} \|B_{1}q-B_{1}v_{n}\|\|z_{n}-p\|\]0  
**Step 4.