## 2 Preliminaries
 \tag{11}\]  
Furthermore, simple calculations show that  
\[u_{n+1}-u_{n}=\gamma_{n+1}(x_{n+1}-x_{n})+(1-\gamma_{n+1})(S_{n+1}u_{n+1}-S_{n}u _{n})+(\gamma_{n+1}-\gamma_{n})(x_{n}-S_{n}u_{n}),\]  
which hence yields  
\[\begin{array}{l}\|u_{n+1}-u_{n}\|^{2}\\ =\gamma_{n+1}\langle x_{n+1}-x_{n},u_{n+1}-u_{n}\rangle+(1-\gamma_{n+1}) \langle S_{n+1}u_{n+1}-S_{n}u_{n},u_{n+1}-u_{n}\rangle\\ \quad+(\gamma_{n+1}-\gamma_{n})\langle x_{n}-S_{n}u_{n},u_{n+1}-u_{n}\rangle\\ =\gamma_{n+1}\langle x_{n+1}-x_{n},u_{n+1}-u_{n}\rangle+(1-\gamma_{n+1})[ \langle S_{n+1}u_{n+1}-Swhich immediately leads to  
\[ \begin{array}{ll}\|u_{n+1}-u_{n}\|&\leq\|x_{n+1}-x_{n}\|+\frac{1-\gamma_{n+1}}{ \gamma_{n+1}}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1}-\gamma_{n}\|\frac{ \|x_{n}-S_{n}u_{n}\|}{\gamma_{n+1}}\\ &\leq\|x_{n+1}-x_{n}\|+\frac{1}{a}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1 }-\gamma_{n}\|\frac{\|x_{n}-S_{n}u_{n}\|}{a}.\end{array}  \tag{12}\]  
Put \(D=\{u_{n}:n\geq 0\}\). Since \(\{u_{n}\}\) is a bounded sequence, we know that \(D\) is a bounded set. Then, by the assumption of this theorem, we get  
\[\sum_{n=0}^{\infty}\sup_{x\in D}\|S_{n+1}x-S_{n}x\|<\infty.\]  
Noticing that  
\[\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|\leq\sup_{x\in D}\|S_{n+1}x-S_{n}x\|,\quad\forall n \geq 0,\]  
we have  
\[\sum_{n=0}^{\infty}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|<\infty. \tag{13}\]  
Therefore, from (10)-(12) we deduce that  
\[ \begin{array}{ll}&\|h_{n+1}-h_{n}\|\\ &\leq\theta_{n+1}\|y_{n+1}-y_{n}\|+\frac{\alpha_{n+1}\delta}{1-\beta_{n+1}}\|x _{n+1}-x_{n}\|+\|T^{n+1}y_{n}-T^{n}y_{n}\|+\|y_{n+1}-y_{n}\|\\ &\quad+\alpha_{n+1}\|x_{n+1}-\mu F(T^{n+1}y_{n+1})\|+\alpha_{n}\|x_{n}-\mu F(T ^{n}y_{n})\|\\ &\quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|f( x_{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|\\ &\leq\|u_{n+1}-u_{n}\|+|\alpha_{n}-\alpha_{n+1}|\|Az_{n}\|+\alpha_{n+1}\|g(x_ {n+1})\|+\alpha_{n}\|g(x_{n})\|\\ &\quad+\theta_{n+1}\|y_{n+1}-y_{n}\|+\frac{\alpha_{n+1}}{1-\beta_{n+1}}\|x _{n+1}-x_{n}\|+\|T^{n+1}y_{n}-T^{n}y_{n}\|\\ &\quad+\alpha_{n+1}\|x_{n+1}-\mu F(T^{n+1}y_{n+1})\|+\alpha_{n}\|x_{n}-\mu F(T ^{n}y_{n})\|\\ &\quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|f( x_{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|\\ &\leq\|x_{n+1}-x_{n}\|+\frac{1}{a}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1 }-\gamma_{n}\|\frac{\|x_{n}-S_{n}u_{n}\|}{a}\\ &\quad+|\alpha_{n}-\alpha_{n+1}|\|Az_{n}\|+\alpha_{n+1}\|g(x_{n+1})\|+\alpha_ {n}\|g(x_{n})\|+\theta_{n+1}\|y_{n+1}-y_{n}\|\\ &\quad+\frac{\alpha_{n+1}\delta}{1-\beta_{n+1}}\|x_{n+1}-x_{n}\|+\|T^{n+1}y_{n }-T^{n}y_{n}\|+\alpha_{n+1}\|x_{n+1}-\mu F(T^{n+1}y_{n+1})\|\\ &\quad+\alpha_{n}\|x_{n}-\mu F(T^{n}y_{n})\|+|\frac{\alpha_{n+1}}{1-\beta_{n+1 }}-\frac{\alpha_{n}}{1-\beta_{n}}\|f(x_{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F( T^{n}y_{n}))\|,\end{array} \]  
which immediately attains  
\[ \begin{array}{ll}&\|h_{n+1}-h_{n}\|-\|x_{n+1}-x_{n}\|\leq\frac{1}{a}\|S_{n+1 }u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1}-\gamma_{n}\|\frac{\|x_{n}-S_{n}u_{n}\|}{a} \\ &\quad+|\alpha_{n}-\alpha_{n+1}|\|Az_{n}\|+\alpha_{n+1}\|g(x_{n+1})\|+\alpha_ {n}\|g(x_{n})\|+\theta_{n+1}\|y_{n+1}-y_{n}\|\\ &\quad+\frac{\alpha_{n+1}\delta}{1-\beta_{n+1}}\|x_{n+1}-x_{n}\|+\|T^{n+1}y_{n }-T^{n}y_{n}\|+\alpha_{n+1}\|x_{n+1}-\mu F(T^{n+1}y_{n+1})\|\\ &\quad+\alpha_{n}\|x_{n}-\mu F(T^{n}y_{n})\|+|\frac{\alpha_{n+1}}{1-\beta_{n+1 }}-\frac{\alpha_{n}}{1-\beta_{n}}\|\|f(x_{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F (T^{n}y_{n}))\|.\end{array}  \tag{14}\]  
Since \(\lim_{n\to\infty}\theta_{n}=0\) and \(\lim_{n\to\infty}\|T^{n+1}y_{n}-T^{n}y_{n}\|=0\) (due to condition (v)), from (13) and conditions (i), (iii), (iv), it follows that  
\[\limsup_{n\to\infty}(\|h_{n+1}-h_{n}\|-\|x_{n+1}-x_{n}\|)\leq 0.\]Hence, by condition (iii) and Lemma 8, we get \(\lim_{n\to\infty}\|h_{n}-x_{n}\|=0\). Consequently,  
\[\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=\lim_{n\to\infty}(1-\beta_{n})\|h_{n}-x_{n}\|=0. \tag{15}\]  
Again from (11) and (12), we conclude that  
\[