## 4 Applications to Finite Generalized Mixed Equilibria  
We suppose set \(C\) is convex nonempty closed and a mapping \(T\) with fixed points is named as a attracting nonexpansive mapping if it is nonexpansive and satisfies:  
\[\|Tx-p\|<\|x-p\|\quad\text{for all }x\
ot\in\operatorname{Fix}(T)\text{ and }p\in \operatorname{Fix}(T).\]  
**Lemma 9** ([27]). : _Let \(X\) be a strictly convex space, \(T_{1}\) be an attracting nonexpansive mapping and \(T_{2}\) be a nonexpansive mapping. We suppose they have common fixed points. Then, \(\operatorname{Fix}(T_{1}T_{2})=\operatorname{Fix}(T_{2}T_{1})=\operatorname {Fix}(T_{1})\cap\operatorname{Fix}(T_{2})\)._  
_Let \(A:C\to H\) be nonself mapping, \(\varphi:C\to\mathbf{R}\) be a single-valued real function, and \(\Theta:C\times C\to\mathbf{R}\) be a bifunction to \(R\). The mixed generalized equilibrium problem (MGEP) is to find \(x\in C\) such that_  
\[\Theta(x,y)+\langle Ax,y-x\rangle\geq 0+\varphi(y)-\varphi(x),\quad\forall y \in C. \tag{52}\]  
_We borrow the collection of solutions of MGEP (52) by MGEP\((\Theta,\varphi,A)\). The GMEP (52) is quite useful in the sense that it includes many problems, namely, vector optimization problems, minimax problems, classical variational inequalities, Nash equilibrium problems in noncooperative games and others. For different aspects and solution methods, we refer to [28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38] and the references therein._  
_In particular, if \(\varphi=0\), then MGEP (52) become the generalized equilibrium problem (GEP) of finding \(x\in C\) such that_  
\[\Theta(x,y)+\langle Ax,y-x\rangle\geq 0,\quad\forall y\in C. \tag{53}\]  
_The collection of solutions of GEP is used by GEP\((\Theta,A)\)._  
_If \(A=0\), then MGEP (52) become the mixed equilibrium problem (MEP). which is to find \(x\in C\) such that_  
\[\Theta(x,y)+\varphi(y)-\varphi(x)\geq 0,\quad\forall y\in C.\]  
_The collection of solutions of MEP is used by MEP\((\Theta,\varphi)\)._  
_If \(\varphi=0\) and \(A=0\), then MGEP (52) become to the equilibrium problem (EP) (see [PERSON] and [PERSON] [30]), which will approximate \(x\in C\) with_  
\[\Theta(x,y)\geq 0,\quad\forall y\in C.\]  
_The collection of solutions of EP is used by EP\((\Theta)\)._  
_Here, we list some elementary conclusions for the MEP._  
_It is first used in [38] that \(\Theta:C\times C\to\mathbf{R}\) is a bifunction and \(\varphi:C\to\mathbf{R}\) is a convex lower semicontinuous function restricted to the following items_  
1. \(\forall x\in C\)_,_ \(\Theta(x,x)\equiv 0\)_._
2. \(\Theta\) _has the monotonicity, i.e.,_ \(\forall x,y\in C\)_,_ \(\Theta(x,y)+\Theta(y,x)\leq 0\)_._
3. \[\limsup_{t\to 0^{+}}\Theta(tz+(1-t)x,y)\leq\Theta(x,y).\]
4. \(\forall x\in C\)_,_ \(\Theta(x,\cdot)\) _is lower semicontinuous convex._
5. \(\forall x\in H\) _and_ \(\forall r>0\)_, we fix a set_ \(D_{x}\subset C\) _and_ \(y_{x}\in C\) _with_ \[\Theta(z,y_{x})+\varphi(y_{x})-\varphi(z)+\frac{1}{r}\langle y_{x}-z,z-x\rangle<0\] \(\forall z\in C\setminus D_{x}\)_._
6. \(C\) _acts as a bounded set._  
**Lemma 10** ([38]). : _We suppose that \(\Theta:C\times C\to\mathbf{R}\) has conditions (A1)-(A4) and \(\varphi:C\to\mathbf{R}\) has the properties proper lower semicontinuous and convex, if either condition (B1) or condition (B2) is true. For \(r>0\) and \(x\in H\), generate an operator \(T_{r}^{(\Theta,\varphi)}:H\to C\) through_  
\[T_{r}^{(\Theta,\varphi)}(x):=\{z\in C:\varphi(y)+\Theta(z,y)-\varphi(z)+\frac{ 1}{r}\langle y-z,z-x\rangle\geq 0,\forall y\in C\}\]  
_for all \(x\in H\). Then,_  
1. _Set_ \(T_{r}^{(\Theta,\varphi)}(x)\) _is a singleton set._
2. \(\forall x,y\in H\)_,_ \[\|T_{r}^{(\Theta,\varphi)}x-T_{r}^{(\Theta,\varphi)}y\|^{2}\leq\langle T_{r}^ {(\Theta,\varphi)}x-T_{r}^{(\Theta,\varphi)}y,x-y\rangle.\]
3. \(\mathrm{MEP}(\Theta,\varphi)=\mathrm{Fix}(T_{r}^{(\Theta,\varphi)})\)_._
4. \(\mathrm{MEP}(\[PERSON],\varphi)\) _is convex closed._
5.