## 3 Main Results
For all \(x,y\in C\), we get by Lemma 9  
\[\|\Pi_{\Omega}(I-\rho F)(x)-\Pi_{\Omega}(I-\rho F)(y)\|\leq\|(I-\rho F)(x)-(I -\rho F)(y)\|\leq(1-\bar{\gamma}\rho)\|x-y\|.\]  
Since \(0<\rho<\|F\|^{-1}\leq\bar{\gamma}^{-1}\), we know that \(\Pi_{\Omega}(I-\rho F)\) is a contractive operator with the contractive coefficient \(1-\bar{\gamma}\rho\in(0,1)\). [PERSON]'s mapping principle tell us that \(\Pi_{\Omega}(I-\rho F)\) has a unique fixed point, say \(x^{*}\in C\), i.e., \(x^{*}=\Pi_{\Omega}(I-\rho F)(x^{*})\). Since \(\{w_{n}\}\subset C\) is bounded, one supposes that \(w_{n_{i}}\to z\in C\). Since \(\lim_{n\to\infty}\|x_{n}-T^{n}y_{n}\|=0\) (due to (32)) and \(\lim_{n\to\infty}\alpha_{n}=0\), it follows that  
\[ \begin{array}{ll}\|x_{n}-w_{n}\|&\leq\|x_{n}-T^{n}y_{n}\|+\|T^{n}y_{n}-\Pi_ {C}(I-\alpha_{n}F)T^{n}y_{n}\|\\ &\leq\|x_{n}-T^{n}y_{n}\|+\alpha_{n}\|F(T^{n}y_{n})\|\to 0\quad(n\to\infty). \end{array}  \tag{39}\]  
Hence, from \(w_{n_{i}}\rightharpoonup z\in C\) it follows that \(x_{n_{i}}\rightharpoonup z\in C\). Note the situation both \(G\) and \(\overline{S}\) are nonexpansive operators and that \(T\) is an asymptotically nonexpansive operator in the intermediate sense. Since \((I-G)x_{n}\to 0\) and \((I-\overline{S})x_{n}\to 0\) (due to (29) and (37)), by Lemma 11 we have that \(z\in\operatorname{Fix}(G)=\operatorname{GSVI}(C,B_{1},B_{2})\) and \(z\in\operatorname{Fix}(\overline{S})=\operatorname{Fix}(S)=\bigcap_{n=0}^{ \infty}\operatorname{Fix}(S_{n})\). From (34), we have that \(\lim_{i\to\infty}\|x_{n_{i}}-Tx_{n_{i}}\|=0\) for the subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\). It follows from Lemma 6 that \(z\in\operatorname{Fix}(T)\). Then, \(z\in\Omega=\bigcap_{n=0}^{\infty}\operatorname{Fix}(S_{n})\cap\operatorname{ GSVI}(C,B_{1},B_{2})\cap\operatorname{Fix}(T)\). Since duality mapping \(j(\cdot)\) is weakly sequentially continuous and \(w_{n_{i}}\rightharpoonup z\), we obtain  
\[\limsup_{n\to\infty}\langle F(x^{*}),j(x^{*}-w_{n})\rangle=\lim_{i\to\infty} \langle F(x^{*}),j(x^{*}-w_{n_{i}})\rangle=\langle F(x^{*}),j(x^{*}-z)\rangle \leq 0,\]  
which implies that (38) holds. **Step 7. ** One proves that \(x_{n}\to x^{*}\) as \(n\to\infty\). Indeed, putting \(p=x^{*}\) in (18) and using (8), we have  
\[ \begin{array}{ll}\|x^{*}-x_{n+1}\|^{2}&\leq\beta_{n}\|x^{*}-x_{n}\|^{2}+(1- \beta_{n})(1-\alpha_{n}\bar{\gamma})\|x^{*}-x_{n}\|^{2}\\ &\quad+c_{n}(2\|x_{n}-x^{*}\|+c_{n})+2\alpha_{n}(1-\beta_{n})\langle F(x^{*}),j (x^{*}-w_{n})\rangle\\ &=[1-\alpha_{n}(1-\beta_{n})\bar{\gamma}]\|x_{n}-x^{*}\|^{2}+\alpha_{n}(1-\beta _{n})\bar{\gamma}\cdot\frac{2}{\bar{\gamma}}\langle F(x^{*}),j(x^{*}-w_{n}) \rangle\\ &\quad+c_{n}(2\|x_{n}-x^{*}\|+c_{n}).\end{array}  \tag{40}\]  
Since \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\), \(\liminf_{n\to\infty}(1-\beta_{n})>0\) (due to conditions (i), (ii)), and \(\sum_{n=0}^{\infty}c_{n}<\infty\) (due to the assumption), we deduce from (38) that \(\sum_{n=0}^{\infty}\alpha_{n}(1-\beta_{n})\bar{\gamma}=\infty\), \(\sum_{n=0}^{\infty}c_{n}(2\|x_{n}-x^{*}\|+c_{n})<\infty\) and \(\limsup_{n\to\infty}\frac{2}{\bar{\gamma}}\langle F(x^{*}),j(x^{*}-w_{n})\rangle\leq 0\). We can employ Lemma 2.12 to the relation (40) and get that \(x_{n}\to x^{*}\) as \(n\to\infty\). This completes the proof. From [37], we give an example to illustrate the partial condition of Theorem 1 to be satisfied. **Example 1**. : _Let \(H=\mathbf{R}\) with the inner product \(\langle a,b\rangle=ab\) and induced norm \(\|\cdot\|=|\cdot|\). Let \(C=H\), and \(T\) be defined as \(Tx:=\frac{3}{4}\sin x\ \forall x\in C\). It is not hard to see that for all \(x,y\in C\),_  
\[\|T^{n}x-T^{n}y\| =\frac{3}{4}\|\sin T^{n-1}x-\sin T^{n-1}y\|\] \[\leq\frac{3}{4}\|T^{n-1}x-T^{n-1}y\|\] \[\leq\cdots\] \[\leq(\frac{3}{4})^{n}\|x-y\|\] \[=(1+\theta_{n})\|x-y\|,\]  
_with \(\theta_{n}=(\frac{3}{4})^{n}\). Then \(T:C\to C\) is an asymptotically nonexpansive mapping with \(\{\theta_{n}\}\).