## 4 Applications to Finite Generalized Mixed Equilibria
\(\|T_{s}^{(\[PERSON],\varphi)}x-T_{t}^{(\[PERSON],\varphi)}x\|^{2}\leq\frac{s-t}{s} \langle T_{s}^{(\[PERSON],\varphi)}x-T_{t}^{(\[PERSON],\varphi)}x,T_{s}^{(\[PERSON], \varphi)}x-x\rangle\)_,_ \(\forall s,t>0\) _and_ \(\forall x\in H\)_._  
Next, under some mild control conditions, we establish the strong convergence of the proposed algorithm to the unique element \(\{x^{*}\}=\mathrm{VI}(\mathrm{VI}(\Omega,\mathcal{A}_{1}),\mathcal{A}_{2})\) (i.e., the unique solution of a THCVI), where \(\Omega:=\bigcap_{i=1}^{N}\mathrm{GMEP}(\Theta_{i},\varphi_{i},A_{i})\cap \mathrm{GSVI}(C,B_{1},B_{2})\cap\mathrm{Fix}(S)\cap\mathrm{Fix}(T)\). **Theorem 2**. : _We suppose \(C\) is a convex nonempty closed set. Assume that, \(\forall i=1,2,...,N\), \(\Theta_{i}:C\times C\to\mathbf{R}\) a bifunction has Conditions (A1)-(A4), \(\varphi_{i}:C\to\mathbf{R}\cup\{+\infty\}\) is a lower semicontinuous, convex proper function with Condition (B1) or Condition (B2), and \(A_{i}:C\to H\) is an \(\eta_{i}\)-inverse-strongly monotone nonself mapping. Let \(f:C\to C\) be a contraction with the parameter \(\delta\in[0,1)\), \(\mathcal{A}_{1}:C\to H\) be a \(\zeta\)-inverse-strongly monotone nonself mapping and \(\mathcal{A}_{2}:C\to H\) be \(\kappa\)-Lipschitzian and \(\eta\)-strongly monotone with parameters \(\kappa,\eta>0\), where \(\delta<\tau:=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\), \(0<\mu<\frac{2\eta}{\kappa^{2}}\). Let the nonself mappings \(B_{1},B_{2}:C\to H\) be \(\alpha\)-inverse-strongly monotone and \(\beta\)-inverse-strongly monotone, respectively. Let self mapping \(T\), defined on \(C\), be a nonexpansive mapping and self mapping \(S\), also defined on \(C\), be an \(\ell\)-Lipschitzian pseudocontractive mapping such that \(\Omega:=\bigcap_{i=1}^{N}\mathrm{GMEP}(\Theta_{i},\varphi_{i},A_{i})\cap \mathrm{GSVI}(C,B_{1},B_{2})\cap\mathrm{Fix}(S)\cap\mathrm{Fix}(T)\
eq\emptyset\) and \(\mathrm{VI}(\Omega,\mathcal{A}_{1})\
eq\emptyset\), where \(\mathrm{GSVI}(C,B_{1},B_{2})\) is the fixed point set of the mapping \(G:=P_{C}(I-\mu_{1}B_{1})P_{C}(I-\mu_{2}B_{2})\) with \(\mu_{1}\in(0,2\alpha)\) and \(\mu_{2}\in(0,2\beta)\). For any given \(x_{0}\in C\), let \(\{x_{n}\}\) be the sequence generated by_  
\[\left\{ \begin{array}{ll}&u_{n}=\gamma_{n}x_{n}+(1-\gamma_{n})Su_{n},\\ &v_{n}=P_{C}(u_{n}-\mu_{2}B_{2}u_{n}),\\ &z_{n}=P_{C}(v_{n}-\mu_{1}B_{1}v_{n}),\\ &y_{n}=\sigma_{n}x_{n}+(1-\sigma_{n})P_{C}(I-\delta_{n}\mathcal{A}_{1})z_{n}, \\ &x_{n+1}=\beta_{n}f(y_{n})+(1-\beta_{n})P_{C}(I-\alpha_{n}\mu_{2})T\Lambda^{N}y _{n},\quad\forall n\geq 0,\end{array} \right. \tag{54}\]  
_where \(\Delta^{N}=T_{r_{1}}^{(\Theta_{1},\varphi_{1})}(I-r_{1}A_{1})\cdots T_{r_{N}} ^{(\Theta_{N},\varphi_{N})}(I-r_{N}A_{N})\) with \(r_{i}\in(0,2\eta_{i})\) for each \(i=1,2,...,N\). Suppose that \(\{\alpha_{n}\},\{\beta_{n}\},\{\gamma_{n}\},\{\sigma_{n}\}\subset(0,1]\) and \(\{\delta_{n}\}\subset(0,2\zeta]\) are the sequences such that_  
1. \(\alpha_{n}\to 0\) _as_ \(n\to\infty\)_,_ \(\sum_{n=0}^{\infty}\alpha_{n}=\infty\) _and_ \(\sum_{n=0}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\)_._
2. \(\frac{\beta_{n}}{\alpha_{n}}\to 0\) _as_ \(n\to\infty\) _and_ \(\sum_{n=0}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty\)_._
3. \(0<\liminf_{n\to\infty}\sigma_{n}\leq\limsup_{n\to\infty}\sigma_{n}<1\) _and_ \(\sum_{n=0}^{\infty}|\sigma_{n+1}-\sigma_{n}|<\infty\)_._
4. \(0<\liminf_{n\to\infty}\gamma_{n}\leq\limsup_{n\to\infty}\gamma_{n}<1\) _and_ \(\sum_{n=0}^{\infty}|\gamma_{n+1}-\gamma_{n}|<\infty\)_._
5. \(\delta_{n}\leq\alpha_{n}\ \forall n\geq 0\) _and_ \(\sum_{n=0}^{\infty}|\delta_{n+1}-\delta_{n}|<\infty\)_._  
_Then, the sequence \(\{x_{n}\}_{n=0}^{\infty}\) generated by (54) satisfies the following properties:_  
1. \(\{x_{n}\}_{n=0}^{\infty}\) _is bounded._
2.