## 3 Main Results
\begin{array}{l}\|y_{n}-y_{n-1}\|\\ \leq\sigma_{n}\|x_{n}-x_{n-1}\|+|\sigma_{n}-\sigma_{n-1}\|x_{n-1}-p_{n-1}\|+(1- \sigma_{n})\|p_{n}-p_{n-1}\|\\ \leq\sigma_{n}\|x_{n}-x_{n-1}\|+|\sigma_{n}-\sigma_{n-1}\|x_{n-1}-p_{n-1}\|+(1 -\sigma_{n})(\|u_{n}-u_{n-1}\|+M_{0}|\delta_{n}-\delta_{n-1}|)\\ \leq\sigma_{n}\|x_{n}-x_{n-1}\|+|\sigma_{n}-\sigma_{n-1}\|x_{n-1}-p_{n-1}\|+(1 -\sigma_{n})(\|x_{n}-x_{n-1}\|\\ \quad+\frac{1}{a}\|S_{n}u_{n}-S_{n-1}u_{n}\|+|\gamma_{n}-\gamma_{n-1}\|\frac{ \|x_{n-1}-S_{n-1}u_{n-1}\|}{4}+M_{0}|\delta_{n}-\delta_{n-1}|)\\ \leq\|x_{n}-x_{n-1}\|+|\sigma_{n}-\sigma_{n-1}\|x_{n-1}-p_{n-1}\|+\frac{1}{a} \|S_{n}u_{n}-S_{n-1}u_{n}\|\\ \quad+|\gamma_{n}-\gamma_{n-1}\|\frac{\|x_{n-1}-S_{n-1}u_{n-1}\|}{a}+M_{0}| \delta_{n}-\delta_{n-1}|\\ \leq\|x_{n}-x_{n-1}\|+M(|\sigma_{n}-\sigma_{n-1}|+\|S_{n}u_{n}-S_{n-1}u_{n}\|+ |\gamma_{n}-\gamma_{n-1}|+|\delta_{n}-\delta_{n-1}|),\end{array}  \tag{20}\]  
where \(\sup_{n\geq 0}\{\|x_{n}-p_{n}\|+\frac{1}{a}+\frac{\|x_{n}-S_{n}u_{n}\|}{4}+M_{0}\}\leq M\) for some \(M>0\). Furthermore, from \(q_{n}=P_{\mathrm{C}}(I-\alpha_{n}\mu A_{2})T^{n}y_{n}\) and Lemma 4, we note that  
\[ \begin{array}{l}\|q_{n}-q_{n-1}\|\\ \leq\|(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-(I-\alpha_{n-1}\mu A_{2})T^{n-1}y_{n-1}\| \\ =\|(I-\alpha_{n}\mu A_{2})T^{n}y_{n}-(I-\alpha_{n}\mu A_{2})T^{n-1}y_{n-1}-( \alpha_{n}-\alpha_{n-1})\mu A_{2}T^{n-1}y_{n-1}\|\\ \leq(1-\alpha_{n}\tau)\|T^{n}y_{n}-T^{n}y_{n-1}+T^{n}y_{n-1}-T^{n-1}y_{n-1}\|+ |\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n-1}y_{n-1}\|\\ \leq(1-\alpha_{n}\tau)\big{(}1+\theta_{n}\|y_{n}-y_{n-1}\|+\|T^{n}y_{n-1}-T^{ n-1}y_{n-1}\|\big{)}+|\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n-1}y_{n-1}\|\\ \leq(1-\alpha_{n}\tau+\theta_{n})\|y_{n}-y_{n-1}\|+\|T^{n}y_{n-1}-T^{n-1}y_{n-1 }\|+|\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n-1}y_{n-1}\|.\end{array}  \tag{21}\]  
Hence, from (16), (20) and (21), we get  
\[ \begin{array}{l}\|x_{n+1}-x_{n}\|\\ \leq\beta_{n}\|f(y_{n})-f(y_{n-1})\|+|\beta_{n}-\beta_{n-1}|\|f(y_{n-1})-q_{n- 1}\|+(1-\beta_{n})\|q_{n}-q_{n-1}\|\\ \leq\alpha_{n}\delta\|y_{n}-y_{n-1}\|+|\beta_{n}-\beta_{n-1}|\|f(y_{n-1})-q_{n -1}\|+(1-\alpha_{n}\tau+\theta_{n})\|y_{n}-y_{n-1}\|\\ \quad+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|+|\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n- 1}y_{n-1}\|\\ \leq[1-\alpha_{n}(\tau-\delta)+\frac{\alpha_{n}(\tau-\delta)}{2}]\|y_{n}-y_{n- 1}\|+|\beta_{n}-\beta_{n-1}|\|f(y_{n-1})-q_{n-1}\|\\ \quad+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|+|\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n -1}y_{n-1}\|\\ \leq(1-\frac{\alpha_{n}(\tau-\delta)}{2})[\|x_{n}-x_{n-1}\|+M(|\sigma_{n}- \sigma_{n-1}|+\|S_{n}u_{n}-S_{n-1}u_{n}\|\\ \quad+|\gamma_{n}-\gamma_{n-1}|+|\delta_{n}-\delta_{n-1}|)]+|\beta_{n}-\beta_{ n-1}|\|f(y_{n-1})-q_{n-1}\|\\ \quad+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|+|\alpha_{n}-\alpha_{n-1}|\|\mu A_{2}T^{n -1}y_{n-1}\|\\ \leq(1-\frac{\alpha_{n}(\tau-\delta)}{2})\|x_{n}-x_{n-1}\|+M_{1}(|\alpha_{n}- \alpha_{n-1}|+|\beta_{n}-\beta_{n-1}|+|\gamma_{n}-\gamma_{n-1}|\\ \quad+|\delta_{n}-\bar{\delta}_{n-1}|+|\sigma_{n}-\sigma_{n-1}|+\|S_{n}u_{n}-S_ {n-1}u_{n}\|)+\|T^{n}y_{n-1}-T^{n-1}y_{n-1}\|,\end{array}  \tag{22}\]  
where \(\sup_{n\geq 1}\{M+\|f(y_{n-1})-q_{n-1}\|+\|\mu A_{2}T^{n-1}y_{n-1}\|\}\leq M_{1}\) for some \(M_{1}>0\). From (18) and Conditions (i)-(v), we know that \(\sum_{n=0}^{\infty}\frac{\alpha_{n}(\tau-\delta)}{2}=\infty\) and  
\[ \begin{array}{l}\sum_{n=1}^{\infty}\{M_{1}(|\alpha_{n}-\alpha_{n-1}|+|\beta_{n }-\beta_{n-1}|+|\gamma_{n}-\gamma_{n-1}|+|\delta_{n}-\delta_{n-1}|\\ \quad+|\sigma_{n}-\sigma_{n-1}|+\|S_{n}u_{n}-S_{n-1}u_{n}\|)+\|T^{n}y_{n-1}-T^{ n-1}y_{n-1}\|\}<\infty.\end{array} \]  
Consequently, applying Lemma 3 to (22), we obtain that  
\[\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0. \tag{23}\]  
In terms of (18), (23) and Conditions (ii)-(iv), we deduce from (20) that  
\[\lim_{n\to\infty}\|y_{n+1}-y_{n}\|=0.\]  
**Step 3. ** We claim that \(\|x_{n}-Gx_{n}\|\to 0\) as \(n\to\infty\).