## Preliminaries
Then \(\Pi(C,B)=\operatorname{Fix}(\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I- \Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C}(I-\Pi_{C(-\Pi_ {C(-\Pi_{C(((((((((((((((((((((((((((((((((( )))))))))))))))}}}}\\\)\)\\\\\\\ \\\\\\\\\ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \\ \ \\ \\ \\ \\(\forall\lambda>0\), where \(\mathrm{VI}(C,B)\) denotes the set of solutions to the VI of approximating \(x^{*}\in C\) such that \(\langle Bx^{*},j(x-x^{*})\rangle\geq 0\)\(\forall x\in C\)._  
Using Propositions 3 and 5, respectively, we immediately obtain the lemmas. **Lemma 3**. : _Fix a convex nonempty and closed set \(C\) in a smooth Banach space \(E\). Let \(B_{1},B_{2}:C\to E\) be two nonlinear mappings. Let \(\Pi_{C}:E\to C\) be a nonexpansive sunny retraction. For given \(x^{*},y^{*}\in C\), \((x^{*},y^{*})\) solves the GSVI (3) iff \(x^{*}\in\mathrm{GSVI}(C,B_{1},B_{2})\) where \(\mathrm{GSVI}(C,B_{1},B_{2})\) stands for fixed-point set of \(G:=\Pi_{C}(I-\mu_{1}B_{1})\Pi_{C}(I-\mu_{2}B_{2})\) and \(y^{*}=\Pi_{C}(I-\mu_{2}B_{2})x^{*}\)._  
**Lemma 4**. : _Fix a convex nonempty and closed set \(C\) in a \(2\)-uniformly smooth Banach space \(E\). Let the accretive mapping \(A:C\to E\) be \(\alpha\)-inverse-strongly. Then, for any given \(\lambda\geq 0\), \(\|x-y\|^{2}+2\lambda(x^{2}\lambda-\alpha)\|Ay-Ax\|^{2}\geq\|(I-\lambda A)y-(I- \lambda A)x\|^{2}\). In particular, if \(0\leq\lambda\leq\frac{\alpha}{x^{2}}\), then \(I-\lambda A\) is nonexpansive._  
Using Lemma 4, we immediately obtain the following lemma. **Lemma 5**. : _Fix a convex nonempty and closed set \(C\) in a \(2\)-uniformly smooth Banach space \(E\). Let \(\Pi_{C}:E\to C\) be a nonexpansive sunny retraction. Let the accretive mappings \(B_{1},B_{2}:C\to E\) be \(\alpha\)-inverse-strongly and \(\beta\)-inverse-strongly accretive. Let the mapping \(G:C\to C\) be defined as \(G:=\Pi_{C}(\Pi_{C}(I-\mu_{2}B_{2})-\mu_{1}B_{1}\Pi_{C}(I-\mu_{2}B_{2}))\). If \(\alpha\geq\kappa^{2}\mu_{1}\geq 0\) and \(\beta\geq\kappa^{2}\mu_{2}\geq 0\), then \(G:C\to C\) is a nonexpansive operator._  
**Proposition 6** ([32]). : _Fix a convex nonempty and closed set \(C\) in a uniformly convex Banach space \(E\) and set \(T\) an asymptotically nonexpansive self-mapping in the intermediate sense on \(C\). Given any bounded subset \(\mathcal{K}\subset C\). For every \(\varepsilon>0\) and every \(n\geq 2\) it is an integer \(N_{\epsilon}\geq 1\) and \(\delta_{\varepsilon}>0\), where both \(N_{\epsilon}\) and \(\delta_{\varepsilon}\) are independent of \(n\), such that if \(z_{1},z_{2},...,z_{n}\in\mathcal{K}\), \(k\geq N_{\epsilon}\), and if \(\|z_{i}-z_{j}\|-\|T^{k}z_{i}-T^{k}z_{j}\|\leq\delta_{\varepsilon}\) for \(1\leq i,j\leq n\), then_  
\[\|T^{k}(\sum_{i=1}^{n}\lambda_{i}z_{i})-\sum_{i=1}^{n}\lambda_{i}T^{k}z_{i}\|<\varepsilon\]  
_for all \(\lambda:=(\lambda_{1},\lambda_{2},...,\lambda_{n})\) such that \(\lambda_{i}\geq 0\) for \(i=1,2,...,n\) and \(\sum_{i=1}^{n}\lambda_{i}=1\)._  
One can make use of Proposition 6 to prove the lemmas below. We omit the proof. **Lemma 6**. : _Fix a convex nonempty and closed set \(C\) in a uniformly convex Banach space and set \(T\) a uniformly continuous self-mapping on \(C\), which is asymptotically nonexpansive in the intermediate sense. If \(\{x_{m}\}_{m=0}^{\infty}\) converges in weak topology to \(x\) and if \(\|x_{m}-Tx_{m}\|\to 0\) as \(m\to\infty\), then \(Tx=x\)._  
**Lemma 7**. : _Fix a convex nonempty and closed set \(C\) in a smooth Banach space \(E\). If the accretive operator \(A:C\to E\) is weakly continuous along segments (i.e., \(A(ty+x)\rightharpoonup Ax\) as \(t\to 0\)), then the VI of approximating \(x^{*}\in C\) such that \(\langle Ax^{*},j(x-x^{*})\rangle\geq 0\)\(\forall x\in C\). This is equivalent to the dual VI of approximating \(x^{*}\in C\) such that \(\langle Ax,j(x-x^{*})\rangle\geq 0\)\(\forall x\in C\)._  
**Lemma 8** ([33]).