## 3 Main Results
Also, from \(\{\gamma_{n}\}\subset[a,b]\subset(0,1)\) we get \(0<1-\gamma_{n}<1\) for all \(n\geq 0\). Thus, \(F_{n}\) is a continuous and strongpseudocontraction of \(C\) into itself. By Proposition 2, we know that for each \(n\geq 0\) there exists a unique element \(u_{n}\in C\), satisfying (5). Therefore, it can be readily seen that the relaxed Mann implicit scheme (4) can be rewritten as  
\[ \begin{cases}&u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\\ &y_{n}=Gu_{n},\\ &x_{n+1}=(1-\beta_{n})\Pi_{\text{C}}(T^{n}y_{n}-\alpha_{n}FT^{n}y_{n})+\beta_{n }x_{n},\quad\forall n\geq 0,\end{cases}  \tag{6}\]  
Next, we divide the rest of the proof into several steps. **Step 1. ** One proves that \(\{x_{n}\},\{y_{n}\},\{z_{n}\},\{u_{n}\},\{T^{n}y_{n}\}\) and \(\{F(T^{n}y_{n})\}\) are bounded. Indeed, take an arbitrary \(p\in\Omega=\bigcap_{n=0}^{\infty}\text{Fix}(S_{n})\cap\text{GSVI}(C,B_{1},B_{2 })\cap\text{Fix}(T)\). So \(S_{n}p=p\), \(Gp=p\) and \(Tp=p\). Since each \(S_{n}\) is pseudocontractive on \(C\), it follows that  
\[\|u_{n}-p\|^{2}\leq\gamma_{n}\|x_{n}-p\|\|p-u_{n}\|+(1-\gamma_{n})\|u_{n}-p\|^ {2},\]  
which hence yields  
\[\|u_{n}-p\|\leq\|x_{n}-p\|,\quad\forall n\geq 0. \tag{7}\]  
Then we observe  
it follows from (10) that  
\[ \begin{array}{ll}\|u_{n+1}-u_{n}\|&\leq\|x_{n}-x_{n+1}\|+|\gamma_{n}-\gamma_{n+ 1}|\frac{\|x_{n}-S_{n}u_{n}\|}{\gamma_{n+1}}+\frac{1-\gamma_{n+1}}{\gamma_{n+1} }\|S_{n}u_{n+1}-S_{n+1}u_{n+1}\|\\ &\leq\frac{1}{a}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1}-\gamma_{n}\| \frac{\|x_{n}-S_{n}u_{n}\|}{a}+\|x_{n}-x_{n+1}\|.\end{array}  \tag{11}\]  
Also, since \(D:=\{u_{n}:n\geq 0\}\subset C\) is bounded, one gets \(\sum_{n=0}^{\infty}\sup_{x\in D}\|S_{n+1}x-S_{n}x\|<\infty\). Hence we have  
\[\sum_{n=0}^{\infty}\|S_{n}u_{n+1}-S_{n+1}u_{n+1}\|<\infty. \tag{12}\]  
Therefore, from (9) and (11) we deduce that  
\[ \begin{array}{ll}a\|w_{n}-w_{n+1}\|&\leq a\|x_{n}-x_{n+1}\|+\|S_{n+1}u_{n+1} -S_{n}u_{n+1}\|+a|\gamma_{n+1}-\gamma_{n}\|\frac{\|x_{n}-S_{n}u_{n}\|}{a}\\ &\quad+ac_{n+1}+\|T^{n+1}y_{n}-T^{n}y_{n}\|+a\pi_{n+1}\|F(T^{n+1}y_{n+1})\|+a \pi_{n}\|F(T^{n}y_{n})\|,\end{array} \]  
which immediately attains  
\[ \begin{array}{ll}\|w_{n}-w_{n+1}\|-\|x_{n}-x_{n+1}\|\\ \leq\frac{1}{a}\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+|\gamma_{n+1}-\gamma_{n}\| \frac{\|x_{n}-S_{n}u_{n}\|}{a}+c_{n+1}\\ \quad+\|T^{n+1}y_{n}-T^{n}y_{n}\|+a\pi_{n}\|F(T^{n}y_{n})\|+a_{n+1}\|F(T^{n+1} y_{n+1})\|.\end{array}  \tag{13}\]  
Since \(\lim_{n\to\infty}c_{n}=0\) (due to (2)) and \(\lim_{n\to\infty}\|T^{n}y_{n}-T^{n+1}y_{n}\|=0\) (due to the assumption), from (12) and conditions (i), (iii) it follows that \(\limsup_{n\to\infty}(\|w_{n+1}-w_{n}\|-\|x_{n+1}-x_{n}\|)\leq 0\). Hence, by condition (ii) and Lemma 8, we get \(\lim_{n\to\infty}\|w_{n}-x_{n}\|=0\). Consequently,  
\[\lim_{n\to\infty}\|x_{n}-x_{n+1}\|=\lim_{n\to\infty}(1-\beta_{n})\|x_{n}-w_{n} \|=0. \tag{14}\]  
Again from (6) and (11) we conclude that  
\[ \begin{array}{ll}a\|y_{n+1}-y_{n}\|=a\|Gu_{n+1}-Gu_{n}\|\leq a\|u_{n}-u_{n+1 }\|\\ \leq\|S_{n+1}u_{n+1}-S_{n}u_{n+1}\|+a\|x_{n+1}-x_{n}\|+a|\gamma_{n+1}-\gamma_{ n}\|x_{n}-S_{n}u_{n}\|\to 0\quad(n\to\infty).\end{array}  \tag{15}\]  
**Step 3. ** One proves \(\lim_{n\to\infty}\|x_{n}-Gx_{n}\|=0\). Indeed, since \(w_{n}=\Pi_{C}(I-\alpha_{n}F)T^{n}y_{n}\), according to Proposition 3, we have that for each \(p\in\Omega\),  
\[\langle\Pi_{C}(I-\alpha_{n}F)T^{n}y_{n}-(I-\alpha_{n}F)T^{n}y_{n},j(p-w_{n}) \rangle\geq 0. \tag{16}\]  
From (16), we have  
\[ \begin{array}{ll}2\|w_{n}-p\|^{2}&=2\langle\Pi_{C}(I-\alpha_{n}F)T^{n}y_{n} -(I-\alpha_{n}F)T^{n}y_{n},j(w_{n}-p)\rangle\\ &\quad+2\langle(I-\alpha_{n}F)T^{n}y_{n}-p,j(w_{n}-p)\rangle\\ &\leq 2\langle(I-\alpha_{n}F)(T^{n}y_{n}-p),j(w_{n}-p)\rangle+2\alpha_{n} \langle F(p),j(p-w_{n})\rangle\\ &\leq(1-\alpha_{n}\tilde{\gamma})^{2}\|T^{n}y_{n}-p\|^{2}+\|p-w_{n}\|^{2}+2 \alpha_{n}\langle F(p),j(p-w_{n})\rangle.\end{array} \]  
It follows from (2) that  
\[