## 2 Preliminaries
Additionally, from Lemma 1 and \(p\in\Omega\subset\mathrm{GSVI}(C,B_{1},B_{2})\), it follows that \((p,q)\) is a solution of the GSVI (3), where \(q=P_{C}(I-\mu_{2}B_{2})p\). Noting that \(v_{n}=P_{C}(I-\mu_{2}B_{2})u_{n}\) for all \(n\geq 0\), by Lemma 5, we have  
\[ \begin{array}{ll}\|v_{n}\|&\leq\|P_{C}(I-\mu_{2}B_{2})u_{n}-q\|+\|q\|\\ &=\|P_{C}(I-\mu_{2}B_{2})u_{n}-P_{C}(I-\mu_{2}B_{2})q\|+\|q\|\\ &\leq\|u_{n}-q\|+\|q\|,\end{array} \]  
which shows that \(\{v_{n}\}\) also is bounded. **Step 2. ** We prove that \(\|x_{n+1}-x_{n}\|\to 0\) and \(\|y_{n+1}-y_{n}\|\to 0\) as \(n\to\infty\). Indeed, we set  
\[x_{n+1}=\beta_{n}x_{n}+(1-\beta_{n})h_{n}\]  
and notice  
\[w_{n}=P_{\mathbb{C}}[(I-\alpha_{n}\mu F)T^{n}y_{n}+\alpha_{n}x_{n}].\]Then,  
\[h_{n}=\frac{\alpha_{n}}{1-\beta_{n}}f(x_{n})+(1-\frac{\alpha_{n}}{1-\beta_{n}})P_{ C}[\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}].\]  
Simple calculations show that  
\[ \begin{array}{l}h_{n+1}-h_{n}\\ =\frac{\alpha_{n+1}}{1-\beta_{n+1}}(f(x_{n+1})-f(x_{n}))+(1-\frac{\alpha_{n+1} }{1-\beta_{n+1}})\{P_{C}[\alpha_{n+1}x_{n+1}+(I-\alpha_{n+1}\mu F)T^{n+1}y_{n+1}] \\ \quad-P_{C}[\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n})\}+(\frac{\alpha_{n+1} }{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}})(f(x_{n})-w_{n}).\end{array} \]  
It follows from (6) that  
\[ \begin{array}{l}\|h_{n+1}-h_{n}\|\\ \leq\frac{\alpha_{n+1}}{1-\beta_{n+1}}\|f(x_{n})-f(x_{n+1})\|+(1-\frac{\alpha_{ n+1}}{1-\beta_{n+1}})\|P_{C}[\alpha_{n+1}x_{n+1}+(I-\alpha_{n+1}\mu F)T^{n+1}y_{n+1}] \\ \quad-P_{C}[\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}]\|+|\frac{\alpha_{n +1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|\|w_{n}-f(x_{n})\|\\ \leq\frac{\alpha_{n+1}}{1-\beta_{n+1}}\|x_{n}-x_{n+1}\|+(1-\frac{\alpha_{n+1} }{1-\beta_{n+1}})\|T^{n+1}y_{n+1}-T^{n+1}y_{n}+T^{n+1}y_{n}-T^{n}y_{n}\\ \quad-\alpha_{n+1}(x_{n+1}-\mu F(T^{n+1}y_{n+1}))-\alpha_{n}(x_{n}-\mu F(T^{n} y_{n}))\|\\ \quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|\|f(x _{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|\\ \leq\frac{\alpha_{n+1}}{1-\beta_{n+1}}\|x_{n+1}-x_{n}\|+(1-\frac{\alpha_{n+1} }{1-\beta_{n+1}})(1+\theta_{n+1})\|y_{n+1}-y_{n}\|+\|T^{n}y_{n}-T^{n+1}y_{n}\| \\ \quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|f(x _{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|\\ \quad\leq\|y_{n+1}-y_{n}\|+\theta_{n+1}\|y_{n}-y_{n+1}\|+\frac{\alpha_{n+1}}{1- \beta_{n+1}}\|x_{n+1}-x_{n}\|+\|T^{n+1}y_{n}-T^{n}y_{n}\|\\ \quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|f(x _{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|\\ \quad\leq\|y_{n+1}-y_{n}\|+\theta_{n+1}\|y_{n}-y_{n+1}\|+\frac{\alpha_{n+1}}{1- \beta_{n+1}}\|x_{n+1}-x_{n}\|+\|T^{n+1}y_{n}-T^{n}y_{n}\|\\ \quad+|\frac{\alpha_{n+1}}{1-\beta_{n+1}}-\frac{\alpha_{n}}{1-\beta_{n}}\|f(x _{n})-T^{n}y_{n}-\alpha_{n}(x_{n}-\mu F(T^{n}y_{n}))\|.\end{array}  \tag{10}\]  
Since \(\{\alpha_{n}\}\subset(0,2\zeta]\) and \(A\) is \(\zeta\)-inverse-strongly monotone, by Lemma 5, we obtain  
\[ \begin{array}{l}\|y_{n+1}-y_{n}\|\\ =\|P_{C}[\alpha_{n}g(x_{n})+(I-\alpha_{n}A)z_{n}]-P_{C}[\alpha_{n+1}g(x_{n+1}) +(I-\alpha_{n+1}A)z_{n+1}]\|\\ \leq\|(I-\alpha_{n+1}A)z_{n+1}-(I-\alpha_{n}A)z_{n}+\alpha_{n+1}g(x_{n+1})- \alpha_{n}g(x_{n})\|\\ =\|(I-\alpha_{n+1}A)z_{n+1}-(I-\alpha_{n+1}A)z_{n}+(\alpha_{n}-\alpha_{n+1})Az_ {n}+\alpha_{n+1}g(x_{n+1})-\alpha_{n}g(x_{n})\|\\ \leq\|(I-\alpha_{n+1}A)z_{n+1}-(I-\alpha_{n+1}A)z_{n}\|+\|\alpha_{n}-\alpha_{n+1 }\|Az_{n}\|+\|\alpha_{n+1}g(x_{n+1})-\alpha_{n}g(x_{n})\|\\ \leq\|z_{n+1}-z_{n}\|+|\alpha_{n}-\alpha_{n+1}\|Az_{n}\|+\alpha_{n+1}\|g(x_{n+1 })\|+\alpha_{n}\|g(x_{n})\|\\ \leq\|u_{n+1}-u_{n}\|+\|Az_{n}\|\|\alpha_{n}-\alpha_{n+1}\|+a_{n+1}\|g(x_{n+1 })\|+\alpha_{n}\|g(x_{n})\|.\end{array}