## 2 Preliminaries
\begin{array}{l}\|y_{n+1}-y_{n}\|\\ \leq\|u_{n+1}-u_{n}\|+|\alpha_{n}-\alpha_{n+1}|\|Az_{n}\|+\alpha_{n+1}\|g(x_{n +1})\|+\alpha_{n}\|g(x_{n})\|\\ \leq\|x_{n+1}-x_{n}\|+\frac{1}{a}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1} -\gamma_{n}\|\frac{\|x_{n}-S_{n}u_{n}\|}{a}+|\alpha_{n}-\alpha_{n+1}|\|Az_{n}\| \\ \quad+\alpha_{n+1}\|g(x_{n+1})\|+\alpha_{n}\|g(x_{n})\|\to 0\quad(n\to\infty), \end{array} \]  
and  
\[\|z_{n}-z_{n+1}\|=\|Gu_{n}-Gu_{n+1}\|\leq\|u_{n}-u_{n+1}\|\to 0\quad(n\to\infty).\]  
Thus,  
\[\lim_{n\to\infty}\|y_{n}-y_{n+1}\|=0,\quad\lim_{n\to\infty}\|u_{n}-u_{n+1}\|=0 \quad\text{and}\quad\lim_{n\to\infty}\|z_{n}-z_{n+1}\|=0.\]  
**Step 3. ** We prove \(\|x_{n}-Gx_{n}\|\to 0\) as \(n\to\infty\). Indeed, noticing \(w_{n}=P_{\mathbb{C}}[(I-\alpha_{n}\mu F)T^{n}y_{n}+\alpha_{n}x_{n}]\) for all \(n\geq 0\), we have  
\[\langle(I-\alpha_{n}\mu F)T^{n}y_{n}+\alpha_{n}x_{n}-P_{\mathbb{C}}[\alpha_{n }x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}],p-w_{n}\rangle\leq 0. \tag{16}\]  
From (16), we have  
\[ \begin{array}{ll}\|w_{n}-p\|^{2}&=\langle P_{\mathbb{C}}[(I-\alpha_{n}\mu F) T^{n}y_{n}+\alpha_{n}x_{n}]-p,w_{n}-p\rangle\\ &=\langle P_{\mathbb{C}}[(I-\alpha_{n}\mu F)T^{n}y_{n}+\alpha_{n}x_{n}]-\alpha _{n}x_{n}-(I-\alpha_{n}\mu F)T^{n}y_{n},w_{n}-p\rangle\\ &\quad+\langle\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}-p,w_{n}-p\rangle \\ &\leq\langle\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}-p,w_{n}-p\rangle\\ &=\langle(I-\alpha_{n}\mu F)T^{n}y_{n}-(I-\alpha_{n}\mu F)p,w_{n}-p\rangle+ \alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle\\ &\leq(1-\alpha_{n}\tau)\|T^{n}y_{n}-p\|\|w_{n}-p\|+\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle\\ &\leq\frac{1}{2}(1-\alpha_{n}\tau)^{2}\|T^{n}y_{n}-p\|^{2}+\frac{1}{2}\|w_{n}-p \|^{2}+\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle.\end{array} \]  
Hence, we have  
\[ \begin{array}{ll}\|w_{n}-p\|^{2}&\leq(1-\alpha_{n}\tau)^{2}\|T^{n}y_{n}-p\|^ {2}+2\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle\\ &\leq(1-\alpha_{n}\tau)(1+\theta_{n})^{2}\|y_{n}-p\|^{2}+2\alpha_{n}\langle x _{n}-\mu Fp,w_{n}-p\rangle\\ &=(1-\alpha_{n}\tau)(\|y_{n}-p\|^{2}+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2})+ 2\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle\\ &\leq(1-\alpha_{n}\tau)\|y_{n}-p\|^{2}+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+ 2\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle.\end{array}  \tag{17}\]  
From (9) and (17), we get \[ \begin{array}{ll}&\|x_{n+1}-p\|^{2}\\ &=\|\beta_{n}(x_{n}-p)+\alpha_{n}(f(x_{n})-f(p))+(1-\alpha_{n}-\beta_{n})(w_{n}-p )+\alpha_{n}(f(p)-p)\|^{2}\\ &\leq\|\beta_{n}(x_{n}-p)+\alpha_{n}(f(x_{n})-f(p))+(1-\alpha_{n}-\beta_{n})(w_ {n}-p)\|^{2}+2\alpha_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ &\leq\alpha_{n}\|f(x_{n})-f(p)\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n}- \beta_{n})\|w_{n}-p\|^{2}+2\alpha_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ &\leq\alpha_{n}\delta\|x_{n}-p\|^{2}+\beta_{n}\|x_{n}-p\|^{2}+(1-\alpha_{n}- \beta_{n})[(1-\alpha_{n}\tau)\|y_{n}-p\|^{2}\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle]+2\alpha_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ &\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\|+\alpha_{n}\|g(x_{n})-Ap\|)^{2}\\ &\quad+\theta_{n}(2+\theta_{n})\|y_{n}-p\|^{2}+2\alpha_{n}\langle x_{n}-\mu Fp,w_{n}-p\rangle\}+2\alpha_{n}\langle f(p)-p,x_{n+1}-p\rangle\\ &\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}\langle x_{n}-\mu Fp,w_{n}-p\rangle\}+2\alpha_{n}\langle f(p )-p,x_{n+1}-p\rangle\\ &\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\|.\end{array}