## 2 Preliminaries
Take \(\{\alpha_{n}\}_{n=0}^{\infty},\{\beta_{n}\}_{n=0}^{\infty},\{\gamma_{n}\}_{ n=0}^{\infty}\subset(0,\infty)\), and \(\mu>0\), choose \(x_{0}\in C\) arbitrarily, and let  
\(n:=0\). Step 1. Given \(x_{n}\in C\), compute \(x_{n+1}\in C\) as  
\[\left\{ \begin{array}{l}u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\\ v_{n}=P_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ z_{n}=P_{C}(v_{n}-\mu_{1}B_{1}v_{n}),\\ y_{n}=P_{C}[\alpha_{n}g(x_{n})+(I-\alpha_{n}A)z_{n}],\\ w_{n}=P_{C}[\alpha_{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}],\\ x_{n+1}=\alpha_{n}f(x_{n})+\beta_{n}x_{n}+(1-\alpha_{n}-\beta_{n})w_{n}.\end{array} \right. \tag{4}\]  
Update \(n:=n+1\) and go to Step 1. **Algorithm 1**Multiple hybrid implicit iterative algorithm. We remark here that our algorithm is quite general. It includes mean-valued techniques, gradient techniques, and implicit iteration techniques. Our algorithm can also generate a strong convergence without any compact assumptions in infinite dimensional spaces. We now state and prove the main result of this paper, that is, the following convergence analysis is presented for our Algorithm 1. **Theorem 1**. : _We suppose \(\mu_{1}\in(0,2\alpha),\ \mu_{2}\in(0,2\beta)\), and \(l+\delta<\tau:=1-\sqrt{1-\mu(2\eta-\mu x^{2})}\in(0,1]\) for \(\mu\in(0,\frac{2\eta}{\kappa^{2}})\). Let number sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) lie in \((0,1]\) such that (i) \(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\); (ii) \(\lim_{n\to\infty}\frac{\beta_{n}}{\alpha_{n}}=0\); (iii) \(0<\lim\inf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\) and \(\alpha_{n}+\beta_{n}\leq 1\ \forall n\geq 0\); (iv) \(0<\lim\inf_{n\to\infty}\gamma_{n}\leq\limsup_{n\to\infty}\gamma_{n}<1\) and \(\lim_{n\to\infty}|\gamma_{n+1}-\gamma_{n}|=0\); (v) \(\lim_{n\to\infty}\|T^{n+1}y_{n}-T^{n}y_{n}\|=0\).Then, we have the following conclusions:__(a) \(\{x_{n}\}_{n=0}^{\infty}\) is bounded;_  
_(b) \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) and \(\lim_{n\to\infty}\|x_{n}-w_{n}\|=0\);_  
_(c) \(\lim_{n\to\infty}\|x_{n}-Gx_{n}\|=0\), \(\lim_{n\to\infty}\|x_{n}-Tx_{n}\|=0\) and \(\lim_{n\to\infty}\|x_{n}-Sx_{n}\|=0\);_  
_(d) If \(\left(\|x_{n}-y_{n}\|+\|x_{n}-w_{n}\|\right)=o(\alpha_{n})\), then \(\{x_{n}\}_{n=0}^{\infty}\) converges strongly to the unique solution of the Problem 1._  
Proof. : Observe that the metric projection \(P_{\mathrm{VI}(\Omega,A-g)}\) is nonexpansive. Indeed, it is firmly nonexpansive. The mapping \(f\) is contractive. Thus, the composition mapping \(P_{\mathrm{VI}(\Omega,A-g)}f\) is a contraction mapping and hence \(P_{\mathrm{VI}(\Omega,A-g)}f\) has a unique fixed point. Say \(x^{*}\in C\), that is, \(x^{*}=P_{\mathrm{VI}(\Omega,A-g)}f(x^{*})\). By Lemma 5,  
\[\{x^{*}\}=\mathrm{Fix}(P_{\mathrm{VI}(\Omega,A-g)}f)=\mathrm{VI}(\mathrm{VI} (\Omega,A-g),I-f).\]  
Therefore, Problem 1 has a unique solution. Without loss of the generality, we can assume that \(\{\alpha_{n}\}\subset(0,2\zeta]\) and \(\{\gamma_{n}\}\subset[a,b]\subset(0,1)\) for some \(a,b\in(0,1)\). By Lemma 6, we know that \(G\) is nonexpansive. It is easy to see that for each \(n\geq 0\) there exists a unique element \(u_{n}\in C\) such that  
\[u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n}. \tag{5}\]  
Therefore, it can be seen that the multiple hybrid implicit iterative scheme (4) can be rewritten as  
\[\left\{ \begin{array}{rl}&u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})S_{n}u_{n},\\ &z_{n}=Gu_{n},\\ &y_{n}=P_{C}[\alpha_{n}g(x_{n})+(I-\alpha_{n}A)z_{n}],\\ &x_{n+1}=\alpha_{n}f(x_{n})+\beta_{n}x_{n}+(1-\alpha_{n}-\beta_{n})P_{C}[\alpha _{n}x_{n}+(I-\alpha_{n}\mu F)T^{n}y_{n}],\quad\forall n\geq 0.\end{array} \right. \tag{6}\]  
Next, we divide the rest of the proof into several steps. **Step 1. ** We prove \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{z_{n}\}\), \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{T^{n}y_{n}\}\), and \(\{F(T^{n}y_{n})\}\) are bounded. Indeed, We can take an element \(p\in\Omega=\bigcap_{n=0}^{\infty}\mathrm{Fix}(S_{n})\cap\mathrm{GSVI}(C,B_{1}, B_{2})\cap\mathrm{Fix}(T)\) arbitrarily.