## 2 Preliminaries
 \tag{18}\]  
We now note that \(q=P_{\mathbb{C}}(p-\mu_{2}B_{2}p)\), \(v_{n}=P_{\mathbb{C}}(u_{n}-\mu_{2}B_{2}u_{n})\), and \(z_{n}=P_{\mathbb{C}}(v_{n}-\mu_{1}B_{1}v_{n})\). Then, \(z_{n}=Gu_{n}\). By Lemma 5, we have  
\[ \begin{array}{ll}\|v_{n}-q\|^{2}&=\|P_{\mathbb{C}}(u_{n}-\mu_{2}B_{2}u_{n})-P _{\mathbb{C}}(p-\mu_{2}B_{2}p)\|^{2}\\ &\leq\|u_{n}-p-\mu_{2}(B_{2}u_{n}-B_{2}p)\|^{2}\\ &\leq\|u_{n}-p\|^{2}-\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}\end{array}  \tag{19}\]  
and  
\[ \begin{array}{ll}\|z_{n}-p\|^{2}&=\|P_{\mathbb{C}}(v_{n}-\mu_{1}B_{1}v_{n})-P _{\mathbb{C}}(q-\mu_{1}B_{1}q)\|^{2}\\ &\leq\|v_{n}-q-\mu_{1}(B_{1}v_{n}-B_{1}q)\|^{2}\\ &\leq\|v_{n}-q\|^{2}-\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}.\end{array}  \tag{20}\]  
Substituting (19) for (20), we obtain from (7) that  
\[ \begin{array}{ll}\|z_{n}-p\|^{2}&\leq\|u_{n}-p\|^{2}-\mu_{2}(2\beta-\mu_{2}) \|B_{2}u_{n}-B_{2}p\|^{2}-\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}\\ &\leq\|x_{n}-p\|^{2}-\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}-\mu_{1}(2 \alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}.\end{array}  \tag{21}\]  
Combining (18) and (21), we get  
\[\begin{array}{ll}&\|x_{n+1}-p\|^{2}\\ &\leq\alpha_{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 Fp\|\|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-\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}-B_{2}p\|^{2}-\mu_{1}(2\alpha-\mu_{1}) \|B_{1}v_{n}-B_{1}q\|^{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 Fp\|\|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)[\mu_{2}(2\beta-\mu_{2})\|B_{2 which immediately yields  
\[ \begin{array}{ll}&(1-\alpha_{n}-\beta_{n})(1-\alpha_{n}\tau)[\mu_{2}(2\beta-\mu_ {2})\|B_{2}u_{n}-B_{2}p\|^{2}+\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2 }]\\ &\leq\|x_{n}-p\|^{2}-\|x_{n+1}-p\|^{2}+\alpha_{n}\|g(x_{n})-Ap\|(2\|z_{n}-p\|+ \alpha_{n}\|g(x_{n})-Ap\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}\|x_{n}-\mu Fp\|\|w_ {n}-p\|+2\alpha_{n}\|f(p)-p\|\|x_{n+1}-p\|\\ &\leq(\|x_{n}-p\|+\|x_{n+1}-p\|)\|x_{n}-x_{n+1}\|+\alpha_{n}\|g(x_{n})-Ap\|(2 \|z_{n}-p\|+\alpha_{n}\|g(x_{n})-Ap\|)\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}\|x_{n}-\mu Fp\|\|w_ {n}-p\|+2\alpha_{n}\|f(p)-p\|\|x_{n+1}-p\|.\end{array} \]  
\((1-\alpha_{n}-\beta_{n})(1-\alpha_{n}\tau)[\mu_{2}(2\beta-\mu_{2})\|B_{2}u_{n}- B_{2}p\|^{2}+\mu_{1}(2\alpha-\mu_{1})\|B_{1}v_{n}-B_{1}q\|^{2}]\to 0\) as \(n\to\infty\). Since \(\liminf_{n\to\infty}(1-\alpha_{n}-\beta_{n})>0\) (due to condition (iii)), \(\mu_{1}\in(0,2\alpha)\), \(\mu_{2}\in(0,2\beta)\), \(\lim_{n\to\infty}\theta_{n}=0\) and \(\lim_{n\to\infty}\alpha_{n}=0\), we obtain from (15) that  
\[\lim_{n\to\infty}\|B_{2}u_{n}-B_{2}p\|=0\quad\text{and}\quad\lim_{n\to\infty} \|B_{1}v_{n}-B_{1}q\|=0. \tag{22}\]  
On the other hand, we have  
\[ \begin{array}{ll}\|v_{n}-q\|^{2}&=\|P_{C}(u_{n}-\mu_{2}B_{2}u_{n})-P_{C}(p- \mu_{2}B_{2}p)\|^{2}\\ &\leq\langle u_{n}-\mu_{2}B_{2}u_{n}-(p-\mu_{2}B_{2}p),v_{n}-q\rangle\\ &=\langle u_{n}-p,v_{n}-q\rangle+\mu_{2}\langle B_{2}p-B_{2}u_{n},v_{n}-q\rangle \\ &\leq\frac{1}{2}[\|u_{n}-p\|^{2}+\|v_{n}-q\|^{2}-\|u_{n}-v_{n}-(p-q)\|^{2}]+ \mu_{2}\|B_{2}p-B_{2}u_{n}\|\|v_{n}-q\|,\end{array} \]  
which implies that  
\[\|v_{n}-q\|^{2}\leq\|u_{n}-p\|^{2}-\|u_{n}-v_{n}-(p-q)\|^{2}+2\mu_{2}\|B_{2}p- B_{2}u_{n}\|\|v_{n}-q\|. \tag{23}\]  
In the same way, we derive  
\[