In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed point of quasi- ϕ-asymptotically nonexpansive multivalued mapping in Banach space.
Theorem 1
Let E be a uniformly smooth and uniformly convex Banach space, and Let C be a nonempty closed convex subset of E and \(\hat {C}B(C)\) be the family of nonempty, closed, and bounded subsets of C. Let \(f_{i} : C \times C \longrightarrow \mathbb {R}, i = 1,2,3,...k\) be bi functions which satisfy the conditions (A1)−(A4),A:C→E∗ be a nonlinear mapping, and \(\varphi : C \longrightarrow \mathbb {R} \cup \lbrace \infty \rbrace \) be a proper, convex, and lower semi-continuous function. Let Ti,i=1,2,3,...N be a quasi- ϕ-asymptotically nonexpansive multivalued mapping from C into \(\hat {C}B(C)\) such that F(T)∩GMEP(f,A,φ)≠∅. Let {xn} be a sequence generated by
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{0} = x\in C \\ y_{n} = J^{-1} \left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n}\right), \\ u_{n} \in C \; such\; that\; \sum_{i=1}^{K} f_{i}(u_{n}, y)+ \varphi(y) -\varphi (u_{n})+ \left\langle y - u_{n}, Au_{n} \right\rangle\\ + \frac{1}{r_{n}}\left\langle y - u_{n}, Ju_{n} - Jy_{n}\right\rangle \geq 0,\forall y\in C\\ M_{n} = \left\lbrace z\in C : \phi(z, u_{n})\leq k_{n}^{2} \phi(z, x_{n})\right\rbrace\\ W_{n}=\left\lbrace z\in C : \left\langle x_{n}- z, Jx - Jx_{n}\right\rangle \geq 0\right\rbrace\\ x_{n+1} = \Pi_{M_{n}\cap W_{n}}x, ~\forall n\geq 0, \end{array}\right. \end{array} $$
(8)
where J is the normalized duality mapping of E, {αi,n}⊂[0,1] satisfies \(\underset {n\to \infty }{\liminf }\alpha _{0,n}\alpha _{i,n} > 0, \sum _{i=0}^{N} \alpha _{i,n} = 1\) and \(w_{i,n} \in T_{i}^{n}x_{n}, \forall _{i} = 1,2,3,...N. \lbrace r_{n} \rbrace \subset [ a, \infty ]\), some a>0. Then, {xn} converges strongly to ΠF(T)∩GMEP(f,A,φ)x, where ΠF(T)∩GMEP(f,A,φ) is the generalized projection of E onto F(T)∩GMEP(f,A,φ),
Proof
Let two functions \(\tau : C\times C\longrightarrow \mathbb {R}\) and Tr:E→C be defined by
$$\begin{array}{@{}rcl@{}} \tau (x,y)= \sum_{i=1}^{k} f_{i}(x, y) + \langle Ax, y - x \rangle + \varphi (y) - \varphi(x), ~\forall x,y\in C \end{array} $$
and
$$\begin{array}{@{}rcl@{}} T_{r}(x) = \left\lbrace u\in C : \tau \left(u, y\right) + \frac{1}{r}\left\langle y - u, Ju - Jx\right\rangle \geq 0, \forall y\in C, \right\rbrace ~\forall x \in E \end{array} $$
respectively. Now, the function τ satisfies conditions (A1)−(A4) and Tr has the properties (a)−(d). Therefore, iterative sequence (8) can be rewritten as
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{0} = x\in C \\ y_{n} = J^{-1}\left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n}\right), w_{i,n}\in T_{i}^{n}x_{n}, n\geq 1\\ u_{n} \in C \; \text{such that}\; \tau\left(u_{n}, y\right) + \frac{1}{r_{n}}\left\langle y - u_{n}, Ju_{n} - Jy_{n} \right\rangle \geq 0,\forall y\in C\\ M_{n} = \left\lbrace z\in C : \phi (z, u_{n})\leq k_{n}^{2}\phi (z, x_{n})\right\rbrace\\ W_{n} =\left\lbrace z\in C : \left\langle x_{n} - z, Jx - Jx_{n} \right\rangle\geq 0\right\rbrace \\ x_{n+1} = \Pi_{M_{n} \cap W_{n}x},~ N \in \mathbb{N}.\end{array}\right. \end{array} $$
(9)
□
We first show that Mn∩Wn is closed and convex, and it is obvious that Mn is closed and convex since \(\phi (z,u_{n})\leq k_{n}^{2}\phi (z,x_{n})\Longleftrightarrow \left (1-k_{n}^{2} \right)\left \Vert z \right \Vert ^{2} -2\left (1- k_{n}^{2}\right)\left \langle z, Ju_{n}\right \rangle + 2k_{n}^{2} \left \langle z,Jx_{n}-Ju_{n}\right \rangle \leq k_{n}^{2}\left \Vert x_{n} \Vert ^{2}-\Vert u_{n}\right \Vert ^{2}. \)
Thus, Mn∩Wn is a closed and convex subset of E for all \(n\in \mathbb {N}\cup \lbrace 0\rbrace \), so that {xn} is well defined.
Let u∈F(T)∩GMEP(f,A,φ), putting \(\phantom {\dot {i}\!}u_{n} = \omega _{n}\in T_{r_{n}}y_{n}\) for all \(n\in \mathbb {N}\cup \lbrace 0\rbrace \), and since \(T_{r_{n}}\) are quasi- ϕ-asymptotically nonexpansive multivalued, we have
$$\begin{array}{@{}rcl@{}} \phi(u, u_{n}) &=&\phi(u, \omega_{n})\\ &\leq& k_{n} \phi (u, y_{n}) \\ &=& k_{n} \left[ \phi\left(u,J^{-1} \left(\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n} \right) \right)\right], \\ \\ &=& k_{n} \left[ \Vert u \Vert^{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right.\\ &\leq& k_{n} \left[ \Vert u \Vert^{2}- 2\left(\!\left\langle u, \alpha_{0,n}Jx_{n} \right\rangle \!+ \! \left\langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle \!\right) \!+ \! \alpha_{0,n} \left\| Jx_{n} \right\|^{2} \!+ \!\sum_{i=1}^{N}\alpha_{i,n} \Vert Jw_{i,n} \Vert^{2} \right]\\ &=&k_{n} \left[ \alpha_{0,n}\left(\parallel u \parallel^{2} - 2 \left\langle u, Jx_{n} \right\rangle + \Vert x_{n} \Vert^{2}\right) \!+ \! \sum_{i=1}^{N} \alpha_{i,n} \left(\Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2} \right)\right]\\ &=& k_{n}\left[ \alpha_{0,n} \phi \left(u, x_{n} \right) + \sum_{i=1}^{N} \alpha_{i,n} \phi (u,w_{i,n}) \right]\\ &\leq& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) +k_{n} \sum_{i=1}^{N}\alpha_{i,n} \phi(u, x_{n})\right]\\ &\leq& k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n}\sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n})\right] \\ &=& k_{n}\left[\left(\alpha_{0,n} + \sum_{i=1}^{N}\alpha_{i,n}\right)k_{n}\phi\left(u, x_{n}\right)\right]\\ &=&k_{n}^{2}\phi(u, x_{n}). \end{array} $$
Hence, we have u∈Mn. This implies that \(F(T) \cap GMEP(f, A,\varphi) \subset M_{n}, \forall n \in \mathbb {N}\cup \lbrace 0 \rbrace \).
Next, we show by induction that \(F(T) \cap GMEP (f,A,\varphi) \subset M_{n} \cap W_{n}, \forall n \in \mathbb {N}\cup \lbrace 0\rbrace \).
From W0=C, we have
$$\begin{array}{@{}rcl@{}} F(T) \cap GMEP(f,A,\varphi) \subset M_{0}\cap W_{0},. \end{array} $$
Suppose that F(T)∩GMEP(f,A,φ)⊂Mk∩Wk, for some \(k \in \mathbb {N}\cup \lbrace 0\rbrace.\) Then, there exists xk+1∈Mk∩Wk such that \(x_{k+1} =\Pi _{M_{k} \cap W_{k}}x \)
From the definition of xk+1, we have for all z∈Mk∩Wk,
$$\begin{array}{@{}rcl@{}} \left\langle x_{k+1} - z, Jx - Jx_{k+1} \right\rangle \geq 0. \end{array} $$
Since F(T)∩GMEP(f,A,φ)⊂Mk∩Wk, we have
$$\begin{array}{@{}rcl@{}} \left\langle x_{k+1} - z, Jx - Jx_{k+1} \right\rangle \geq 0. \end{array} $$
∀z∈F(T)∩GMEP(f,A,φ) and so z∈Wk+1. Thus, F(T)∩GMEP(f,A,φ)⊂Wk+1. Therefore, we have F(T)∩GMEP(f,A,φ)⊂Mk+1∩Wk+1. Therefore, we obtain
$$\begin{array}{@{}rcl@{}} F(T)\cap GMEP\left(f, A,\varphi\right)\subset M_{n} \cap W_{n}, ~\forall n \in \mathbb{N} \cup \lbrace 0 \rbrace. \end{array} $$
From the definition of Wn, we have \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x\); using this and Lemma 1, we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n}, x) &=& \phi \left(\Pi_{W_{n}}x, x \right) \leq \phi (u, x) - \phi \left(u, \Pi_{W_{n}}x, \right)\\ &\leq& \phi(u, x) \end{array} $$
(10)
for all u∈F(T)∩GMEP(f,A,φ)⊂Wn. Therefore, ϕ(xn,x) is bounded, and consequently {xn} and \(\left \lbrace T_{i}^{n}x_{n} \right \rbrace \) are bounded.
Since \(x_{n+1} = \Pi _{M_{n} \cap W_{n}}x \in M_{n} \cap W_{n}\subset W_{n}\) and \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x,\) we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n}, x) \leq \phi (x_{n+1}, x),~ \forall n \in \mathbb{N}\cup \lbrace 0 \rbrace. \end{array} $$
(11)
Thus, {ϕ(xn,x)} is nondecreasing. Using (10) and (11), we have the limit of {ϕ(xn,x)} exists.
From \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x\) and Lemma 1, we also have
$$\begin{array}{@{}rcl@{}} \phi (x_{n+1}, x_{n}) = \phi (x_{n+1}, \Pi_{W_{n}}x,) \leq \phi (x_{n+1}, x) - \phi (\Pi_{W_{n}}x,, x) = \phi (x_{n+1}, x) - \phi (x_{n}, x) \end{array} $$
for all \(n \in \mathbb {N} \cup \lbrace 0 \rbrace.\) This means that \(\underset {n\to \infty }{lim} \phi (x_{n+1}, x_{n}) = 0\).
From \(x_{n+1} = \Pi _{M_{n} \cap W_{n}}x \in M_{n}\) and the definition of Mn, we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n+1}, u_{n})\leq k_{n}^{2} \phi(x_{n+1}, x_{n}),\forall n \in \mathbb{N} \cup \lbrace 0 \rbrace. \end{array} $$
Therefore, we have\(\phantom {\dot {i}\!} \underset {n\to \infty }{\lim } \phi (x_{n+1}, u_{n})= 0\). As E is uniformly convex and smooth, we have from Lemma 3 that
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n+1}- x_{n} \Vert =\underset{n\to \infty}{\lim} \Vert x_{n+1} - u_{n} \Vert =0. \end{array} $$
From which, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n} - u_{n} \Vert = 0. \end{array} $$
Since J is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert Jx_{n} - Ju_{n}\Vert = 0. \end{array} $$
Let r = sup\(_{n \in \mathbb {N}} \left \lbrace \Vert x_{n} \Vert, \Vert T_{i}^{n} x_{n}\Vert \right \rbrace.\) Since E is a uniformly smooth Banach space, we know that E∗ is a uniformly convex Banach space. So, for u∈F(T)∩GMEP(f,A,φ), putting \(\phantom {\dot {i}\!}u_{n} = \omega _{n} = T_{r_{n}}y_{n}\) and using Lemma 4, we have :
$$\begin{array}{@{}rcl@{}} \phi (u, u_{n}) &=& \phi (u, \omega_{n})\\ &\leq& k_{n} \phi (u, y_{n})\\ &=& k_{n} \left[ \phi\left(u,J^{-1} \left(\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n} \right) \right) \right],\\ &=& k_{n} \left[ \Vert u \Vert_{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right] \\ &\leq& k_{n} \left[ \Vert u \Vert^{2} - 2 (\langle u, \alpha_{0,n}Jx_{n} \rangle + \langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \rangle) + \alpha_{0,n} \Vert Jx_{n} \Vert^{2} + \sum_{i=1}^{N} \alpha_{i,n} \Vert Jw_{i,n} \Vert^{2}\right.\\ & & {} - \left.\alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert){\vphantom{ \sum_{i=1}^{N}}}\right] \\ &=& k_{n} \left[ \alpha_{0,n} \left[ \Vert u \Vert^{2} - 2 \langle u, Jx_{n} \rangle + \Vert x_{n} \Vert^{2} \right] + \sum_{i=1}^{N} \alpha_{i,n} \left[ \Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2}\right]\right.\\ & & {} -\left.{\vphantom{\sum_{i=1}^{N}}} \alpha_{0,n}\alpha_{i,n} g \left(\Vert Jx_{n} - Jw_{i,n} \Vert \right)\right]\\ &=& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) + \sum_{i=1}^{N} \alpha_{i,n} \phi (u, w_{i,n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &\leq& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) + \sum_{i=1}^{N} k_{n} \alpha_{i,n} \phi (u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &\leq& k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n} \sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &=& k_{n} \left[ k_{n} \phi(u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right] \end{array} $$
$$\begin{array}{@{}rcl@{}} &=& k_{n}^{2} \phi (u, x_{n}) -k_{n} \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert). \end{array} $$
(12)
Therefore, from (12), we have\(k_{n}\alpha _{0,n}\alpha _{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\leq k_{n}^{2} \phi (u, x_{n}) - \phi (u, u_{n}),\forall n\in \mathbb {N} \cup \lbrace 0 \rbrace.\)
But
$$\begin{array}{@{}rcl@{}} \lefteqn{k_{n}^{2} \phi (u, x_{n})- \phi(u, u_{n}) = k_{n}^{2} \left[ \Vert u \Vert^{2} - 2 \langle u, Jx_{n} \rangle + \left\Vert x_{n} \right\Vert^{2} \right] - \left[ \Vert u \parallel^{2} - 2 \langle u, Ju_{n} \rangle + \parallel u_{n} \Vert^{2}\right]} \\ &=&\left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}-2\left(k_{n}^{2}-1\right) \left\langle u, Ju_{n}\right\rangle - 2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle + k_{n}^{2} \parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}\\ &=& \left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}-2\left(k_{n}^{2}-1\right) \langle u, Ju_{n}\rangle - 2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle + \left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2}\\ & &+\parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}\\ &\leq& |\left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}|+|2(k_{n}^{2}-1) \langle u, Ju_{n}\rangle| + |2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle| + |\left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2}|\\ & &+|\parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}|\\ &\leq& \left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}+2\left(k_{n}^{2}-1\right) \Vert u\Vert \Vert Ju_{n}\Vert + 2k_{n}^{2} \Vert u \Vert \Vert Jx_{n} - Ju_{n}\Vert + \left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2} \\ & &+(\parallel x_{n} - u_{n} \Vert) (\parallel x_{n}\Vert + \parallel u_{n} \Vert)\\ \end{array} $$
Hence,
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \left(k_{n}^{2} \phi(u,x_{n}) -\phi(u, u_{n})\right) = 0.\end{array} $$
(13)
Since \(\underset {n\to \infty }{\liminf } \alpha _{0,n} \alpha _{i,n} > 0,\) we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}g(\Vert Jx_{n} - Jw_{i,n}\Vert) = 0. \end{array} $$
From the property g, we have \(\underset {n\to \infty }{\lim } \Vert Jx_{n} - Jw_{i,n}\Vert = 0\).
Since J−1 is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n} - w_{i,n} \Vert = 0. \end{array} $$
Since {xn} is bounded, there exists a subsequence \(\left \lbrace x_{n_{k}}\right \rbrace \) of {xn} such that \(x_{n_{k}} \rightharpoonup \hat {x},\) for some \(\hat {x}\in E\). Since T is quasi- ϕ-asymptotically nonexpansive multivalued mapping and E is a reflexive space, then we have \(\hat {x}\in F(T_{i})\).
Next, we show that \(\hat {x}\in GMEP\left (f, A,\varphi \right).\) From \(\phantom {\dot {i}\!}u_{n} = T_{r_{n}}y_{n}\) Lemma 7 (d) and (13), we have
$$\begin{array}{@{}rcl@{}} \phi (u_{n}, y_{n})&=& \phi(T_{r_{n} }y_{n}, y_{n})\\ &\leq& \phi(u, y_{n}) - \phi(u, T_{r_{n}}y_{n})\\ &\leq& k_{n}^{2}(u, x_{n}) -\phi(u,T_{r_{n}} y_{n})\\ &=& k_{n}^{2}\phi (u, x_{n}) - \phi(u, u_{n}). \end{array} $$
Thus, \(\underset {n\to \infty }{\lim } \phi (u_{n}, y_{n}) = 0\).
Since E is uniformly convex and smooth, we have from Lemma 3 that
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}\parallel u_{n} - y_{n}\parallel = 0. \end{array} $$
(14)
From \(x_{n_{k}} \rightharpoonup \hat {x}, ~ \Vert x_{n} - u_{n} \Vert \longrightarrow 0\) and (14), we have \(y_{n_{k}} \rightharpoonup \hat {x}\) and \(u_{n_{k}} \rightharpoonup \hat {x}.\)
As J is uniformly norm-to-norm continuous on bounded sets and (14), we have\(\underset {n\to \infty }{\lim } \left \Vert Ju_{n} - Jy_{n} \right \Vert = 0\). From rn≥a, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}\left\Vert \frac{Ju_{n} - Jy_{n}}{r_{n}}\right\Vert = 0. \end{array} $$
(15)
By \(\phantom {\dot {i}\!}u_{n} = T_{r_{n}}y_{n}\), we have
$$\begin{array}{@{}rcl@{}} \tau\left(u_{n}, y\right) + \frac{1}{r_{n}}\langle y-u_{n}, Ju_{n} - Jy_{n} \rangle \geq 0, ~ \forall y\in C. \end{array} $$
Replacing n by nk, we have from (A2) that
$$\begin{array}{@{}rcl@{}} \frac{1}{r_{n_{k}}}\left\langle y - u_{n_{n}}, Ju_{n_{k}} - Jy_{n_{k}} \right\rangle \geq - \tau \left(u_{n_{k}}, y\right) \geq \tau\left(y, u_{n_{k}}\right), \forall y\in C \end{array} $$
(16)
Letting k→∞, in (16) and using (A4), we obtain
$$\begin{array}{@{}rcl@{}} \tau \left(y, \hat{x}\right)\leq 0, \forall y\in C. \end{array} $$
For t with 0<t≤1 and y∈C, let \(y_{t}= ty +(1-t)\hat {x}\). Since y∈C and \(\hat {x}\in C\), we have yt∈C and \(\tau (y_{t}, \hat {x})\leq 0, \forall y\in C\). Now, using (A1) and (A3), we have
$$\begin{array}{@{}rcl@{}} 0 &=& \tau(y_{t}, y_{t})\\ &\leq& t \tau \left(y_{t}, y\right) + (1-t) \tau\left(y_{t}, \hat{x}\right)\\ &\leq& t \tau \left(y_{t}, y\right)\!. \end{array} $$
Dividing by, t we have
$$\begin{array}{@{}rcl@{}} \tau \left(y_{t}, y \right)\geq 0, \forall y \in C. \end{array} $$
Letting t→0, and using (A3), we have
$$\begin{array}{@{}rcl@{}} \tau \left(\hat{x}, y \right)\geq 0, \forall y \in C. \end{array} $$
This shows that \(\hat {x}\in GMEP\left (f, A, \varphi \right)\).
Let ω=ΠF(T)∩GMEP(f,A,φ)x, From \(x_{n+1} = \Pi _{M_{n}\cap W_{n}}x\) and ω∈F(T)∩GMEP(f,A,φ)⊂Mn∩Wn, we have
$$\begin{array}{@{}rcl@{}} \phi(x_{n+1},x) \leq \phi(\omega,x). \end{array} $$
Since the norm is weakly lower semi-continuous and \(x_{n_{k}}\rightharpoonup \hat {x}\), we have
$$\begin{array}{@{}rcl@{}} \phi(\hat{x},x)&=& \left\Vert \hat{x} \right\Vert^{2}- 2 \left\langle \hat{x}, Jx \right\rangle + \left\Vert x \right\Vert^{2} \\&\leq& \underset{k\to \infty}{\liminf}\left(\Vert x_{n_{k}} \parallel^{2}-2\left\langle x_{n_{k}}, Jx \right\rangle + \Vert x\parallel^{2}\right)\\ &=& \underset{k\to \infty}{\liminf} \phi \left(x_{n_{k}}, x \right)\\ &\leq& \underset{k\to \infty}{\limsup}\phi \left(x_{n_{k}}, x \right)\\ &\leq& \phi(\omega, x). \end{array} $$
From the definition of ΠF(T)∩GMEP(f,A,φ), we have \(\hat {x} = \omega.\) Hence, \(\underset {k \rightarrow \infty }{lim}\phi \left (x_{n_{k}}, x \right) = \phi (\omega, x),\) Therefore,
$$\begin{array}{@{}rcl@{}} 0 &=& \underset{k\to\infty}{\lim}\left(\phi \left(x_{n_{k}}, x\right) - \phi (\omega, x)\right)\\&=& \underset{k \to \infty}{\lim}\left(\Vert x_{n_{k}} \Vert^{2} - \parallel \omega\Vert^{2} - 2 \left\langle x_{n_{k}} -\omega, Jx \right\rangle \right)\\ &=& \underset{k\to \infty}{\lim} \left(\parallel x_{n_{k}} \Vert^{2} - \parallel \omega \Vert^{2} \right). \end{array} $$
Since E has the Kadec-Klee property, we have that \(x_{n_{k}} \longrightarrow \omega = \Pi _{F(T)\cap GMEP\left (f, A, \varphi \right)}x.\)
Therefore, {xn} converges strongly to ΠF(T)∩GMEP(f,A,φ)x.