## 4 Applications to Finite Generalized Mixed Equilibria
\(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\)_,_ \(\lim_{n\to\infty}\|x_{n}-Gx_{n}\|=0\)_,_ \(\lim_{n\to\infty}\|x_{n}-T\Delta^{N}x_{n}\|=0\) _and_ \(\lim_{n\to\infty}\|x_{n}-Sx_{n}\|=0\)_.__(c)_ \(\{x_{n}\}_{n=0}^{\infty}\) _converges strongly to the unique element_ \(\{x^{*}\}=\mathrm{VI}(\mathrm{VI}(\Omega,\mathcal{A}_{1}),\mathcal{A}_{2})\) _(i.e., the unique solution of a THCVI), provided_ \(\|x_{n}-y_{n}\|=o(\delta_{n})\)_._  
Proof. : First, let us show that for each \(i=1,2,...,N\), the composite mapping \(T_{r_{i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})\) with \(r_{i}\in(0,2\eta_{i})\) is nonexpansive. Indeed, from Lemma 10 (iii), it is not difficult to obtain  
\[\mathrm{GMEP}(\Theta_{i},\varphi_{i},A_{i})=\mathrm{Fix}(T_{r_{i}}^{(\Theta_{ i},\varphi_{i})}(I-r_{i}A_{i})).\]  
Utilizing Lemma 5 and Lemma 10 (ii), we have  
\[ \begin{array}{l}\|T_{r_{i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})x-T_{r_{ i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})y\|^{2}\\ \leq\|(I-r_{i}A_{i})x-(I-r_{i}A_{i})y\|^{2}\\ \leq\|x-y\|^{2}+r_{i}(r_{i}-2\eta_{i})\|A_{i}-A_{i}y\|^{2}\\ \leq\|x-y\|^{2}.\end{array} \]  
Thus, each composite mapping \(T_{r_{i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})\) is nonexpansive. Moreover, we claim that \(T_{r_{i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})\) is also attracting nonexpansive for each \(i=1,2,...,N\). In fact, for all \(x\
ot\in\mathrm{GMEP}(\Theta_{i},\varphi_{i},A_{i})\) and \(p\in\mathrm{GMEP}(\Theta_{i},\varphi_{i},A_{i})\), by the firm nonexpansivity of \(T_{r_{i}}^{(\Theta_{i},\varphi_{i})}\) (due to Lemma 10 (ii)), we obtain  
\[ \begin{array}{l}\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-p\|^ {2}=\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-T_{r_{i}}^{(\[PERSON]_{ i},\varphi_{i})}(I-r_{i}A_{i})p\|^{2}\\ \leq(T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-T_{r_{i}}^{(\[PERSON]_{ i},\varphi_{i})}(I-r_{i}A_{i})p,(I-r_{i}A_{i})x-(I-r_{i}A_{i})p)\\ =\frac{1}{2}\left\{\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-T_{r_ {i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})p\|^{2}+\|(I-r_{i}A_{i})x-(I-r_{i }A_{i})p\|^{2}\right.\\ \quad-\left.\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-T_{r_{i}}^{( \[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})p-(I-r_{i}A_{i})x+(I-r_{i}A_{i})p\|^{2}\right\} \\ =\frac{1}{2}\left\{\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-p\|^ {2}+\|(I-r_{i}A_{i})x-(I-r_{i}A_{i})p\|^{2}\right.\\ \quad-\left.\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-p-(I-r_{i}A_ {i})x+(I-r_{i}A_{i})p\|^{2}\right\},\end{array} \]  
which immediately implies that  
\[ \begin{array}{l}\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-p\|^ {2}\\ \leq\|(I-r_{i}A_{i})x-(I-r_{i}A_{i})p\|^{2}-\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{ i})}(I-r_{i}A_{i})x-p-(I-r_{i}A_{i})x+(I-r_{i}A_{i})p\|^{2}.\end{array}  \tag{55}\]  
Next, we discuss two cases. Case 1. If \(A_{i}x=A_{i}p\), then from (55) we have  
\[ \begin{array}{l}\|T_{r_{i}}^{(\Theta_{i},\varphi_{i})}(I-r_{i}A_{i})x-p\|^ {2}\\ \leq\|(I-r_{i}A_{i})x-(I-r_{i}A_{i})p\|^{2}-\|T_{r_{i}}^{(\Theta_{i},\varphi_{ i})}(I-r_{i}A_{i})x-p-(I-r_{i}A_{i})x+(I-r_{i}A_{i})p\|^{2}\\ =\|x-p\|^{2}-\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-x\|^{2}\\ <\|x-p\|^{2}.\end{array} \]  
Case 2. If \(A_{i}x\
eq A_{i}p\), then from (55) we get  
\[ \begin{array}{l}\|T_{r_{i}}^{(\[PERSON]_{i},\varphi_{i})}(I-r_{i}A_{i})x-p\|^ {2}\\ \leq\|(x-r_{i}A_{i}x)-(p-r_{i}A_{i}p)\|^{2}-\|T_{r_{i}}^{(\Theta_{i},\varphi_{ i})}(I-r_{i}A_{i})x-p-(I-r_{i}A_{i})x+(I-r_{i}A_{i})p\|^{2}\\ \leq\|(x-r_{i}A_{i}x)-(p-r_{i}A_{i}p)\|^{2}\\ \leq\|x-p\|^{2}+r_{i}(r_{i}-2\eta_{i})\|A_{i}x-A_{i}p\|^{2}\\ <\|x-p\|^{2}\end{array} \](due to \(r_{i}\in(0,2\eta_{i})\)). Summing up the above two cases, we know that each composite mapping \(T_{r_{i}}^{(\Theta_{i},\varphi_{i})}\left(I-r_{i}A_{i}\right)\) is also attracting nonexpansive. Therefore, by Lemma 9, we conclude that \(\operatorname{Fix}(T\Delta^{N})=\bigcap_{i=1}^{N}\operatorname{GMEP}(\Theta_{i },\varphi_{i},A_{i})\cap\operatorname{Fix}(T)\).