A hybrid scheme for fixed points of a countable family of generalized non-expansive-type maps and generalized mixed equilibrium problems
September 27, 2025
Peter U. Nwokoro\(^{++}\), Maria A. Onyido\(^{**}\), Markjoe O. Uba\(^{**}\), and Cyril I. Udeani\(^{\&\&}\) \(^{**}\)Department of Mathematical Science,
Northern Illinois University,
DeKalb, IL 60115, USA. \(^{\&\&}\)Department of Computer Science,
University of Nevada, Las Vegas
Las Vegas, NV 89154, USA. \(^{++}\)Department of Mathematics,
University of Nigeria,
Nsukka, Nigeria.
Let \(\Omega\) be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space \(\mathcal{X}\) with dual space \(\mathcal{X}^*\). This article presents a hybrid algorithm for finding a common element of the set of solutions to a generalized mixed equilibrium problem and the set of common fixed points of a family of a general class of
nonlinear nonexpansive maps. The results obtained were employed to study optimization problem. Our results and its applications complement, generalize, and extend several results in literature.
Let \(\mathcal{X}\) be a uniformly convex and uniformly smooth real Banach space with dual space \(\mathcal{X}^*\). Let \(\Omega\) be a nonempty closed
and convex subset of \(\mathcal{X}\) such that \(J\Omega\) is closed and convex, where \(J : \mathcal{X} \rightarrow \mathcal{X}^*\) is the normalized
duality map on \(\mathcal{X}\). Let \(\langle \cdot, \cdot \rangle\) denote the duality pairing between \(\mathcal{X}\) and \(\mathcal{X}^*\). Let \(f : J\Omega\times J\Omega \rightarrow \mathbb{R}\) be a bifunction, \(\varphi : J\Omega \rightarrow \mathbb{R}\) be a real-valued
function, and \(A : \Omega \rightarrow \mathcal{X}^*\) be a nonlinear mapping. The generalized mixed equilibrium problem is to find an element \(v \in \Omega\) such that \[f(Jv,Ju) + \varphi(Ju) - \varphi(Jv) + \langle Av,u-v \rangle \ge 0 ~\forall u \in \Omega.\] In this paper, we consider equilibrium problem such that \(f\) is given by \[f(Jx,Jy) = \sum_{i=1}^k f_i(Jx,Jy), ~\forall x,y \in \Omega,\] where \(f_i : J\Omega\times J\Omega \rightarrow \mathbb{R}\) is a bifunction for each \(i\in
\{1,2,3,\cdots,k\}, \;k\geq1\). Consequently, we study the following well known generalized mixed equilibrium problem: find \(v \in \Omega\) such that \[\label{gmep} \sum_{i=1}^k f_i(Jv,Ju) + \varphi(Ju) - \varphi(Jv) + \langle Av,u-v \rangle \ge 0 ~\forall u \in \Omega.\tag{1}\] The set of solution of 1 is given by
\[\label{gmepsol} GMEP(f,A,\varphi) = \Bigg\{v\in \Omega : \sum_{i=1}^k f_i(Jv,Ju) + \varphi(Ju) - \varphi(Jv) + \langle Av,u-v \rangle \ge 0 ~\forall u \in
\Omega\Bigg\}.\tag{2}\] If \(A=0\), 1 reduces to finding \(v \in \Omega\) such that \[\label{mep} \sum_{i=1}^k f_i(Jv,Ju) + \varphi(Ju) - \varphi(Jv) \ge 0 ~\forall u \in \Omega,\tag{3}\] which is called the mixed equilibrium problem and MEP denotes the set of solutions to 3 . The
class of generalized mixed equilibrium problems is famous, well studied, and contains, as special cases, numerous important classes of nonlinear problems such as equilibrium problems, optimization problems, variational inequality problems (see e.g., [1]–[4] and the references contained in them).
For an arbitrary real normed space \(\mathcal{X}\) with dual space \(\mathcal{X}^*,\) an operator \(A:\mathcal{X}\rightrightarrows \mathcal{X}^*\) is called
monotone if \[\langle \xi-\tau,x-y \rangle \geq0\,\, \forall \,\, \xi \in Ax, \tau \in Ay.\]
It is well known that these operators appear in a wide variety of contexts since they can be found in many functional equations. Many of them also are known to appear in calculus of variations as subdifferential of convex functions. Additionally,
monotone operators are well known to have a strong connection with optimization. There are several ongoing research efforts to develop and employ fixed point techniques to approximate solution of the equation \(Au = 0\)
when \(A\) is monotone. One of such efforts lead to the introduction and study of a new notion of fixed points for maps from \(\mathcal{X}\) to \(\mathcal{X}^*\) called \(J-\)fixed points (see e.g., [5]–[8] and the references contained in them).
It is our purpose in this paper to introduce and study a new hybrid algorithm and prove a strong convergence theorems for obtaining a common element in the solutions of a generalized mixed equilibrium problem and common fixed points for a countable family
of generalized \(J_*-\)nonexpansive maps (as well as generalized \(J-\)nonexpansive maps) in a uniformly smooth and uniformly convex real Banach space. The results introduced are applicable
in classical Banach spaces such as \(L_{p},~l_{p}, or~ W^{m}_{p}(\Omega)\), \(p \in (1,\infty)\), where \(W^{m}_{p}(\Omega)\) denotes the usual Sobolev
space. Additionally, the Hilbert space case of the results obtained complement, extend and improve several results in literature.
In this section, we present definitions and lemmas which we shall use in proving our main result.
Definition 1. (Uniformly Smooth) A normed linear space \(\mathcal{X}\) of dimension \(\geq 2\) is called uniformly smooth if the modulus of
smoothness \(\rho_{\mathcal{X}}:[0,\infty)\rightarrow [0,\infty)\), defined by \[\rho_{\mathcal{X}(\tau )}:= \sup\left\{\frac{\| x+y\| +\| x-y\|}{2}-1: \| x\| =1, \| y\| =\tau,~~\tau
>0\right\}\] tends to \(0\) as \(\tau \to 0.\)
Definition 2. (Uniformly Convex) A Banach space \(\mathcal{X}\) is uniformly convex if the modulus of convexity of \(\mathcal{X}, \;
\delta_{\mathcal{X}}:(0,2]\rightarrow [0,1]\) defined by \[\delta_{\mathcal{X}(\epsilon)}:=\inf\Big\{1-\Big\|\frac{x+y}{2}\Big\|:\|x\|=\|y\|=1;\, \epsilon=\|x-y\|\Big\}\] is positive for every \(\epsilon\in (0,2].\)
Definition 3. (Normalized duality map) The map \(J:\mathcal{X}\rightarrow 2^{\mathcal{X}^*}\) defined by \[Jx:=\big \{x^*\in \mathcal{X}^*:\big <x,x^*\big
>=\|x\|.\|x^*\|,~\|x\|=\|x^*\|\big \}\] is called the normalized duality map on \(\mathcal{X}\).
It is well known that if \(\mathcal{X}\) is smooth, strictly convex and reflexive then \(J^{-1}\) exists (see e.g., [9]); \(J^{-1}:\mathcal{X}^{*}\rightarrow \mathcal{X}\) is the normalized duality mapping on \(\mathcal{X}^{*}\), and \(J^{-1}=J_{*}, ~JJ_{*}=I_{\mathcal{X}^{*}}\) and \(J_{*}J =I_{\mathcal{X}}\), where \(I_{\mathcal{X}}\) and \(I_{\mathcal{X}^{*}}\) are the identity maps on \(\mathcal{X}\) and \(\mathcal{X}^{*}\), respectively. A well known property of \(J\) is (see e.g., [9], [10]):
If \(\mathcal{X}\) is uniformly smooth, then \(J\) is uniformly continuous on bounded subsets of \(\mathcal{X}\).
Definition 4. (Lyapunov Functional) Let \(\mathcal{X}\) be a smooth real Banach space with dual \(\mathcal{X}^*\). The Lyapounov functional\(\phi:\mathcal{X}\times \mathcal{X}\to\mathbb{R}\), is defined by \[\begin{align}
\label{Lya} \phi(x,y)=\|x\|^2-2\langle x,Jy\rangle+\|y\|^2,~~\text{for}~x,y\in \mathcal{X},
\end{align}\tag{4}\] where \(J\) is the normalized duality map.
The Lyapounov functional was introduced and studied in [11]–[14]. If \(\mathcal{X}=H\), a real Hilbert space, then equation (4 ) reduces to \(\phi(x,y)=\|x-y\|^2\) for \(x,y\in H.\) It is obvious from the
definition of the function \(\phi\) that \[\begin{align}
\label{fi} (\|x\|-\|y\|)^2\leq \phi(x,y)\leq(\|x\|+\|y\|)^2~~\text{for}~x,y\in \mathcal{X}.
\end{align}\tag{5}\]
Definition 5. (Generalized nonexpansive) Let \(\Omega\) be a nonempty closed and convex subset of a real Banach space \(\mathcal{X}\) and \(T\) be a map from \(\Omega\) to \(\mathcal{X}\). The map \(T\) is called generalized nonexpansive if \(F(T):=\{x\in \Omega: Tx=x\}\neq \emptyset\) and \(\phi(Tx,p)\leq \phi(x,p)\) for all \(x\in \Omega, p\in F(T)\).
Definition 6. (Retraction) A map \(R\) from \(\mathcal{X}\) onto \(\Omega\) is said to be a retraction if \(R^{2}=R\). The map \(R\) is said to be sunny if \(R(Rx+t(x-Rx))=Rx\) for all \(x\in \mathcal{X}\) and \(t\geq 0\).
A nonempty closed subset \(\Omega\) of a smooth Banach space \(\mathcal{X}\) is said to be a sunny generalized nonexpansive retract of \(\mathcal{X}\) if there exists a sunny generalized nonexpansive retraction \(R\) from \(\mathcal{X}\) onto \(\Omega\).
Let \(\Omega\) be a closed subset of a Banach space \(\mathcal{X}\). Let \(\{T_{n}\}\) and \(\Gamma\) be two families of
generalized nonexpansive maps of \(\Omega\) into \(\mathcal{X}\) such that \(\cap_{n=1}^{\infty}F(T_{n})=F(\Gamma)\neq \emptyset,\) where \(F(T_{n})\) is the set of fixed points of \(\{T_{n}\}\) and \(F(\Gamma)\) is the set of common fixed points of \(\Gamma\).
Definition 7. The sequence \(\{T_{n}\}\) satisfies the NST-condition (see e.g., [15]) with \(\Gamma\) if for each bounded sequence \(\{x_{n}\}\subset \Omega\), \[\lim_{n\rightarrow \infty}||x_{n}-T_{n}x_{n}||=0 \Rightarrow \lim_{n\rightarrow
\infty}||x_{n}-Tx_{n}||=0, ~for ~all~T\in \Gamma.\]
Remark 1. If \(\Gamma =\{T\}\) a singleton, \(\{T_{n}\}\) satisfies the NST-condition with \(\{T\}\). If \(T_{n}=T\) for all \(n\geq 1\), then, \(\{T_{n}\}\) satisfies the NST-condition with \(\{T\}\).
Let \(\Omega\) be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space \(\mathcal{X}\) with dual space \(\mathcal{X}^*\). Let \(J\) be the normalized duality map on \(\mathcal{X}\) and \(J_{*}\) be the normalized duality map on \(\mathcal{X}^*\). Observe that under this setting, \(J^{-1}\) exists and \(J^{-1}=J_{*}\). With these notations, we have the following definitions.
Definition 8. (Closed map)[16] A map \(T:\Omega\rightarrow \mathcal{X}^*\) is called \(J_{*}-\)closed if \((J_{*}oT) : \Omega\rightarrow \mathcal{X}\) is a closed map, i.e., if \(\{x_{n}\}\) is a sequence in \(\Omega\) such that \(x_{n}\rightarrow x\) and \((J_{*}oT)x_{n}\rightarrow y\), then \((J_{*}oT)x =y\).
Definition 9. (\(J-\)fixed Point)[7] A point \(x^*\in
\Omega\) is called a \(J-\)fixed point of \(T\) if \(Tx^*=Jx^*\). The set of \(J-\)fixed points of \(T\) will be denoted by \(F_{J}(T)\).
Definition 10. (Generalized \(J_{*}-\)nonexpansive)[16] A map \(T:\Omega\rightarrow \mathcal{X}^*\) will be called generalized \(J_{*}-\)nonexpansive if \(F_{J}(T)\neq \emptyset\), and \(\phi (p, (J_{*}oT)x) \leq \phi(p,x)\) for all \(x\in \Omega\) and for all \(p\in F_{J}(T)\).
Remark 2. Exampes of generalized \(J_{*}-\)nonexpansive maps in Hilbert and more general Banach spaces were given in [6] and [16].
Let \(\Omega\) be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space \(\mathcal{X}\) such that \(J\Omega\) is closed and
convex. For solving equilibrium problem, let us assume that a bifunction \(f:J\Omega\times J\Omega\rightarrow \mathbb{R}\) satisfies the following conditions:
\(f(x^*,x^*)=0\) for all \(x^*\in J\Omega\);
\(f\) is monotone, i.e. \(f(x^*,y^*)+f(y^*,x^*)\leq0\) for all \(x^*,y^*\in J\Omega\);
for all \(x^*,y^*,z^*\in J\Omega\), \(\limsup_{t\downarrow0} f(tz^*+(1-t)x^*,y^*)\leq f(x^*,y^*)\);
for all \(x^*\in J\Omega\), \(f(x^*,\cdot)\) is convex and lower semicontinuous.
With the above definitions, we now provide the lemmas we shall use.
Lemma 1. [17]Let \(\mathcal{X}\) be a uniformly convex Banach space, \(r > 0\)
be a positive number, and \(B_r(0)\) be a closed ball of \(\mathcal{X}\). For any given points \(\{ x_1, x_2, \cdots , x_N \} \subset B_r(0)\) and any given
positive numbers \(\{ \lambda_1, \lambda_2, \cdots , \lambda_N \}\) with \(\sum_{n = 1} ^{N} \lambda_n = 1,\) there exists a continuous strictly increasing and convex function \(g : [0, 2r) \to [0, \infty)\) with \(g(0) = 0\) such that, for any \(i,j \in \{ 1,2, \cdots N \}, \; i < j,\)\[\| \sum_{n = 1}
^{N} \lambda_n x_n \|^2 \le \sum_{n = 1} ^{N} \lambda_n\|x_n\|^2 - \lambda_i \lambda_j g(\|x_i - x_j\|).\]
Lemma 2. [13]Let \(\mathcal{X}\) be a real smooth and uniformly convex Banach space, and let \(\{x_n\}\) and \(\{y_n\}\) be two sequences of \(\mathcal{X}\). If either \(\{x_n\}\) or \(\{y_n\}\) is bounded and \(\phi(x_n,y_n)\to0\) as \(n\to \infty\), then \(\|x_n-y_n\|\to0\) as \(n\to\infty\).
Lemma 3. [11]Let \(\Omega\) be a nonempty closed and convex subset of a smooth, strictly convex and reflexive
Banach space \(\mathcal{X}\). Then, the following are equivalent. \((i)\)\(\Omega\) is a sunny generalized nonexpansive retract of \(\mathcal{X}\), \((ii)\)\(\Omega\) is a generalized nonexpansive retract of \(\mathcal{X}\), \((iii)\)\(J\Omega\) is closed and convex.
Lemma 4. [11]Let \(\Omega\) be a nonempty closed and convex subset of a smooth and strictly convex Banach space
\(\mathcal{X}\) such that there exists a sunny generalized nonexpansive retraction \(R\) from \(\mathcal{X}\) onto \(\Omega\).Then, the following hold. \((i)\)\(z=Rx\) iff \(\langle x-z, Jy-Jz\rangle \leq 0\) for all \(y\in \Omega\), \((ii)\)\(\phi (x,Rx) + \phi (Rx, z) \leq \phi (x,z)\) for all \(z\in \Omega\).
Lemma 5. [18]Let \(\Omega\) be a nonempty closed sunny generalized nonexpansive retract of a smooth and
strictly convex Banach space \(\mathcal{X}\). Then the sunny generalized nonexpansive retraction from \(\mathcal{X}\) to \(\Omega\) is uniquely
determined.
Lemma 6. [19]Let \(\Omega\) be a nonempty closed subset of a smooth, strictly convex and reflexive Banach
space \(\mathcal{X}\) such that \(J\Omega\) is closed and convex, let \(f\) be a bifunction from \(J\Omega\times J\Omega\)
to \(\mathbb{R}\) satisfying \((A1)-(A4)\). For \(r>0\) and let \(x\in E\). Then there exists \(z\in \Omega\) such that \(f(Jz,Jy)+\frac{1}{r}\langle z-x, Jy-Jz\rangle\geq0,~~\forall~~y\in \Omega.\)
Lemma 7. [20]Let \(\Omega\) be a nonempty closed subset of a smooth, strictly convex and reflexive Banach
space \(\mathcal{X}\) such that \(J\Omega\) is closed and convex, let \(f\) be a bifunction from \(J\Omega\times J\Omega\)
to \(\mathbb{R}\) satisfying \((A1)-(A4)\). For \(r>0\) and let \(x\in \mathcal{X}\), define a mapping \(T_r(x):\mathcal{X}\rightarrow \Omega\) as follows: \[T_r(x)={\color{blue}\Bigg\{}z\in \Omega:\sum_{i=1}^k f_i(Jz,Jy)+\langle y-z, Az\rangle+\varphi(y)-\varphi(z)+\frac{1}{r}\langle y - z,
Jz-Jx\rangle\geq0,~~\forall~~y\in \Omega{\color{blue}\Bigg\}}.\] Then the following hold:
\(T_r\) is single valued;
for all \(x,y\in \mathcal{X}\), \(\langle T_rx-T_ry, JT_rx-JT_ry\rangle\leq\langle x-y, JT_rx-JT_ry\rangle\);
\(F(T_r)=GMEP(f,A,\varphi)\);
\(\phi (p,T_r(x)) + \phi (T_r(x), x) \leq \phi (p,x)\) for all \(p\in F(T_r)\), \(x\in \mathcal{X}\).
\(GMEP(f,A,\varphi)\) is closed and \(JGMEP(f,A,\varphi)\) is closed and convex.
Lemma 8. [16]Let \(\mathcal{X}\) be a uniformly convex and uniformly smooth real Banach space with dual space
\(\mathcal{X}^*\) and let \(\Omega\) be a closed subset of \(\mathcal{X}\) such that \(J\Omega\) is closed and convex. Let
\(T\) be a generalized \(J_{*}-\)nonexpansive map from \(\Omega\) to \(\mathcal{X}^*\) such that \(F_{J}(T) \neq \emptyset\), then \(F_{J}(T)\) and \(JF_{J}(T)\) are closed.
Lemma 9. [16] Let \(\mathcal{X}\) be a uniformly smooth and uniformly convex real Banach space and let \(\Omega\) be a closed subset of \(\mathcal{X}\) such that \(J\Omega\) is closed and convex. Let \(T\) be a generalized \(J_{*}-\)nonexpansive map from \(\Omega\) to \(\mathcal{X}^*\) such that \(F_{J}(T) \neq \emptyset\). If \(JF_{J}(T)\) is convex, then \(F_{J}(T)\) is a sunny generalized nonexpansive retract of \(\mathcal{X}\).
Example 1. Let \(\mathcal{X}\) be a uniformly smooth and uniformly convex real Banach space with dual space \(\mathcal{X}^*\) and let \(\Omega\) be a nonempty closed subset of \(\mathcal{X}\). Let \(T:\Omega\rightarrow \mathcal{X}^*\), be a generalized \(J_{*}-\)nonexpansive maps such that \(F_{J}(T)\neq \emptyset\). Let \(\alpha_n \subset (0,1)\) such that \(1 - \alpha_n \ge
\frac{1}{2}\). For all \(n \in \mathbb{N}\), define \(T_{n}:\Omega\rightarrow \mathcal{X}^*\) by \[T_{n}u = \alpha_nJu + (1 - \alpha_n)Tu, ~\forall~ u\in
\Omega.\] Then \(\{T_n\}\) is a countable family of generalized \(J_{*}-\)nonexpansive maps satisfying \(NST-\)condition with \(T\).
Proof. Clearly \(F_{J}(T_{n})=F_{J}(T)\)\(\forall~n \in \mathbb{N}\). Hence, \(\cap_{n=1}^{\infty}F_{J}(T_{n})=F_{J}(T)\). For \(u \in \Omega\), \(v \in F_{J}(T_{n})\), \[\begin{align} \phi(v, J_{*}oT_{n}u)& = &\phi(v, J_{*}(\alpha_nJu + (1 - \alpha_n)Tu)) \nonumber\\ & = &
||v||^2 -2\langle v, (\alpha_nJu + (1 - \alpha_n)J(J_{*}oT)u)\rangle + ||\alpha_nJu + (1 - \alpha_n)J(J_{*}oT)u||^2 \nonumber\\ & = & ||v||^2 -2\alpha_n\langle v, Ju\rangle -2(1 - \alpha_n)\langle v, J(J_{*}oT)u)\rangle + ||\alpha_nJu + (1 -
\alpha_n)J(J_{*}oT)u||^2 \nonumber\\ & \le & ||v||^2 -2\alpha_n\langle v, Ju\rangle -2(1 - \alpha_n)\langle v, J(J_{*}oT)u)\rangle + \alpha_n||u||^2 + (1 - \alpha_n)||J_{*}oTu||^2 \nonumber\\ & = & \alpha_n||v||^2 -2\alpha_n\langle v,
Ju\rangle + \alpha_n||u||^2 + (1 - \alpha_n)||v||^2 \nonumber\\ &&-2(1 - \alpha_n)\langle v, J(J_{*}oT)u)\rangle + (1 - \alpha_n)||J_{*}oTu||^2 \nonumber\\ & = & \alpha_n\phi(v, u) + (1 - \alpha_n)\phi(v, J_{*}oTu) \nonumber\\ & \le
& \phi(v, u).
\end{align}\] Hence, \(\{T_n\}\) is a countable family of generalized \(J_{*}-\)nonexpansive maps.
Let \(\{u_n\}\) be a bounded sequence in \(\Omega\) such that \(\lim ||Ju_n - T_nu_n|| = 0\). This implies that \(\{J_{*}oTu_n\}\) is bounded. From the definition of \(T_n\), we obtain the following inequality \[||Ju_n - Tu_n|| = \frac{1}{(1 - \alpha_n)} ||Ju_n - T_nu_n|| \le
2||Ju_n - T_nu_n||.\] This shows that \(\lim ||Ju_n - Tu_n|| = 0\). ◻
Theorem 1. Let \(\mathcal{X}\) be a uniformly smooth and uniformly convex real Banach space with dual space \(\mathcal{X}^*\) and let \(\Omega\) be a nonempty closed and convex subset of \(\mathcal{X}\) such that \(J\Omega\) is closed and convex. Let \(\varphi:J\Omega
\rightarrow \mathbb{R}\) be a lower semi-continuous and convex function. For each \(i \in \{1, 2, 3, ..., N\}\), let \(f_{i}\) be a bifunction from \(J\Omega\times J\Omega\) to \(\mathbb{R}\) satisfying \((A1)-(A4)\), \(T^i_{n}:\Omega\rightarrow \mathcal{X}^*, n=1, 2, 3, ...\)
be an infinite family of generalized \(J_{*}-\)nonexpansive maps and \(\Gamma\) be a family of closed and generalized \(J_{*}-\)nonexpansive maps from \(\Omega\) to \(\mathcal{X}^*\) such that \(\cap_{n=1}^{\infty}F_{J}(T^i_{n})=F_{J}(\Gamma) \neq \emptyset\) and \(B := F_{J}(\Gamma)\cap
GMEP(f,A, \varphi) \neq \emptyset.\) Assume that \(JF_{J}(\Gamma)\) is convex and \(\{T^i_{n}\}\) satisfies the NST-condition with \(\Gamma\). Let
\(\{x_{n}\}\) be generated by:
for all \(n\in \mathbb{N}, \; \{\alpha^i_{n}\}\in [0,1]\) such that \(\sum_{i = 0} ^{N} \alpha^i_{n} = 1\), \(\{r_n\}\subset [a,\infty)\) for some \(a>0\). Then, \(\{x_{n}\}\) converges strongly to \(R_B x\), where \(R_B\) is the sunny generalized \(J_*-\)nonexpansive retraction of \(\mathcal{X}\) onto \(B\).
We show that
\(\{x_{n}\}\) is well defined;
\(F_{J}(\Gamma)\cap GMEP(f,A, \varphi) \subset \Omega_{n} \text{ for all} ~ n \ge 1\);
\(R_B x\) exists as a point in \(\Omega_n\; \text{for all} ~ n \ge 1\);
\(x_n \to x^*\) for some \(x^* \in \Omega\);
\(x^*\in F_{J}(\Gamma)\cap GMEP(f,A, \varphi)\);
\(x^* = R_{B}x\).
Proof. The proof is given in \(6\) steps.
It is easy to see that \(J\Omega_n\) is closed and convex for each \(n\geq1\). Therefore, from Lemma 3, we have that
\(\Omega_n\) is a sunny generalized \(J_*-\)nonexpansive retract of \(\mathcal{X}\) for each \(n\geq1\). Hence, \(\{x_{n}\}\) is well defined.
Clearly, \(B \subset \Omega_1\). Suppose \(B \subset \Omega_{n}\) for some \(n\in \mathbb{N}\). Let \(u\in B\), and \(u_n=T_{r_n}y_n\) for all \(n\in \mathbb{N}\). Using the fact that \(\{T^i_{n}\}\) is an infinite family of generalized \(J_*-\)nonexpansive maps, the definition of \(y_n\), Lemmas 7, and 1, we compute as follows: \[\begin{align}
\label{222} \phi(u, u_{n}) & = & \phi(u,T_{r_n}y_n)\leq\phi(u,y_n) = \phi{\color{blue}\Big(}u, J^{-1}{\color{blue}\Big(}\alpha^0_{n}Jx_{n} + \sum_{i = 1} ^{N} \alpha^i_{n}T^i_{n}x_{n}{\color{blue}\Big)\Big)}\nonumber\\ & = & ||u||^2
-2{\color{blue}\Big\langle} u, \alpha^0_{n}Jx_{n} + \sum_{i = 1} ^{N} \alpha^i_{n}T^i_{n}x_{n}{\color{blue}\Big\rangle} + {\color{blue}\Big|\Big|}\alpha^0_{n}Jx_{n} + \sum_{i = 1} ^{N} \alpha^i_{n}T^i_{n}x_{n}{\color{blue}\Big|\Big|}^2 \nonumber\\ &
\leq & ||u||^2 -2\alpha^0_{n}\langle u, Jx_{n}\rangle + \alpha^0_{n}||x_{n}||^2 -2{\color{blue}\Big\langle} u, \sum_{i = 1} ^{N} \alpha^i_{n}T^i_{n}x_{n}{\color{blue}\Big\rangle} + \sum_{i = 1} ^{N} \alpha^i_{n}||T^i_{n}x_{n}||^2 \nonumber\\ &&
- \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||)\nonumber\\ & = & \alpha^0_{n}(||u||^2 -2\langle u, Jx_{n}\rangle + ||x_{n}||^2) + \sum_{i = 1} ^{N}\alpha^i_{n}(||u||^2 -2\langle u, J(J_{*}oT^i_{n})x_{n}\rangle + ||T^i_{n}x_{n}||^2)
\nonumber\\ && - \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||)\nonumber\\ & = & \alpha^0_{n}\phi(u, x_{n}) + \sum_{i = 1} ^{N}\alpha^i_{n}\phi(u, J_{*}oT^i_{n}x_{n}) - \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||)\nonumber\\
& \leq & \alpha^0_{n}\phi(u, x_{n}) + \sum_{i = 1} ^{N}\alpha^i_{n}\phi(u, x_{n}) - \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||).\nonumber
\end{align}\tag{7}\] Hence, we obtain the following inequality for each \(i \in \{1,2,3,\cdots, N\}\)\[\label{key}
\phi(u, u_{n}) \leq \phi(u, x_{n}) - \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||).\tag{8}\] Additionally, we have that \[\label{key2}
\phi(u, y_{n}) \leq \phi(u, x_{n}) ~\forall~ n\in \mathbb{N}.\tag{9}\] From inequality (8 ), we conclude that \(u\in \Omega_{n+1}\). Thus, \(B
\subset{\Omega_n}\) for all \(n \ge 1\).
From Lemma 7\(JGMEP(f,A, \varphi)\) is closed and convex. Also, using our assumption and Lemma 8, we have that \(J(F_{J}(\Gamma))\) is closed and convex. Since \(\mathcal{X}\) is uniformly convex, \(J\) is one-to-one. Thus,
we have that,
and so \(J(B)\) is closed and convex. Hence, using Lemma 3, we obtain that \(B\) is a sunny generalized \(J_*-\)nonexpansive retract of \(\mathcal{X}\). Thus, using Lemma 5, we have that \(R_Bx\)
exists as a point in \(\Omega_n\) for all \(n \ge 1\).
Using the fact that \(x_n=R_{\Omega_n}x\) and Lemma 4\((ii)\), we obtain \[\phi(x,x_{n})=\phi(x,R_{\Omega_{n}}x)\leq \phi(x,u),\] for all \(u\in F_{J}(\Gamma)\cap GMEP(f,A, \varphi)\subset \Omega_{n}.\) This implies that \(\{\phi(x,x_{n})\}\) is bounded. Hence, from equation 5 , \(\{x_{n}\}\) is bounded. Also, since \(x_{n+1} = R_{\Omega_{n+1}}x \in \Omega_{n+1}
\subset \Omega_n\), and \(x_n=R_{\Omega_n}x \in \Omega_n\), applying Lemma 4\((ii)\) gives
\[\phi(x,x_{n})\leq \phi(x,x_{n+1}) ~\forall~ n\in \mathbb{N}.\] So, \(\lim _{n\rightarrow \infty}\phi(x, x_{n})\) exists. Again, using Lemma 4\((ii)\) and \(x_{n}=R_{\Omega_{n}}x\), we obtain that for all \(m,n\in \mathbb{N}\) with \(m>n\), \[\begin{align}
\phi(x_{n},x_{m})& =&\phi(R_{\Omega_{n}}x,x_{m})\leq \phi(x,x_{m}) -\phi(x,R_{\Omega_{n}}x) \nonumber\\
&= & \phi(x,x_{m}) -\phi(x,x_{n}) \rightarrow 0~as~ n\rightarrow \infty.
\end{align}\] From Lemma 2, we conclude that \(||x_{n}-x_{m}||\rightarrow 0, ~as~ m, ~ n\rightarrow \infty.\) Hence, \(\{x_{n}\}\) is a Cauchy sequence in \(\Omega\), and so, there exists \(x^*\in \Omega\) such that \(x_{n}\rightarrow x^*\).
From the definitions of \(\Omega_{n+1}\) and \(x_{n+1}\), we obtain that \(\phi(x_{n+1}, u_{n})\leq \phi(x_{n+1},x_{n})\rightarrow 0\) as \(n\rightarrow \infty.\) Hence, by Lemma 2 , we have that \[\label{limun}
\underset{n \to \infty}{\text{\color{blue}lim}}||x_{n}-u_{n}||= 0.\tag{10}\]
Since \(x_n\rightarrow x^*~as~n\rightarrow \infty\), equation (10 ) implies that \(u_n\rightarrow x^*~as~n\rightarrow \infty\). Observe that since \(J\) is uniformly continuous on bounded subsets of \(\mathcal{X}\), it follows from (10 ) that \[\label{limjun}
\lim_{n\rightarrow \infty}||Ju_{n}-Jx_{n}||=0.\tag{11}\]
From inequality (8 ) and the fact that \(g\) is nonnegative, we obtain \[0 \leq \alpha^0_{n}\alpha^i_{n}g(||Jx_{n}- T^i_{n}x_{n}||) \leq \phi(u, x_{n}) - \phi(u, u_{n})
\leq 2||u||.||Jx_{n}-Ju_{n}|| + ||x_{n}-u_{n}||M,\]
for some \(M > 0.\) Let \(\liminf \alpha^0_{n}\alpha^i_{n} = a\). Since \(a > 0\), there exists \(n_0\in\mathbb{N}\): \[0<\frac{a}{2}<\alpha^0_{n}\alpha^i_{n}~\textrm{for all}~ n\ge n_0.\] Thus, \[0 \leq \frac{a}{2}g(||Jx_{n}- T^i_{n}x_{n}||) \leq
2||u||.||Jx_{n}-Ju_{n}|| + ||x_{n}-u_{n}||M ~\textrm{for all}~ n\ge n_0.\] Thus, from (10 ), (11 ), and properties of \(g\), we obtain that \(\lim_{n\rightarrow \infty}||Jx_{n}-T^i_{n}x_{n}||\;= 0\). Since \(\{T^i_{n}\}_{n=1}^{\infty}\) satisfies the NST condition with \(\Gamma\), we have that \[\lim _{n\rightarrow \infty}||Jx_{n}-T^ix_{n}||=0 ~\forall~ T^i\in \Gamma.\]
Now, since we have established that \(x_{n}\rightarrow {\color{blue}x^*\in \Omega}\). Assume that \((J_{*}oT^i)x_{n}\rightarrow y^*\). Since \(T^i\) is
closed, we have \(y^*=(J_{*}oT^i)x^*\). Furthermore, by the uniform continuity of \(J\) on bounded subsets of \(\mathcal{X}\), we have: \(Jx_{n} \rightarrow Jx^*\) and \(J(J_{*}oT^i)x_{n}\rightarrow Jy^*\) as \(n\rightarrow \infty.\) Hence, we have \[\lim_{n\rightarrow
\infty}||Jx_{n}-J(J_{*}oT^i)x_{n}|| = \lim_{n\rightarrow \infty}||Jx_{n}-T^ix_{n}||=0, ~\forall~ T^i\in \Gamma,\] which implies \(||Jx^{*}-Jy^{*}||= ||Jx^{*}-J(J_{*}oT^i)x^*||= ||Jx^*-T^ix^*||=0 .\) So, \(x^*\in F_{J}(\Gamma)\) for each \(i\).
Next, let \(u_n=T_{r_n}y_n\) for all \(n\in \mathbb{N}\). Also, from (10 ), \(u_n\rightarrow x^*~as~n\rightarrow \infty\). From
Lemma 7 and inequality (9 ), we have \[\begin{align} \phi(u_{n},y_n) &=& \phi(T_{r_n}y_n,y_n)\\ &\leq&\phi(u,y_n) -
\phi(u,T_{r_n}y_n)\nonumber\\ &\leq&\phi(u,x_n) - \phi(u,u_n)\nonumber
\end{align}\] Since \(\lim_{n\rightarrow \infty}(\phi(u,x_n) - \phi(u,u_n))=0,\) we have that \(\lim_{n\rightarrow \infty}\phi(u_n,y_n)=0.\) From Lemma 2, we have that \({\color{blue}\lim}_{n\rightarrow \infty}||y_n-u_n||=0.\) Again, since \(r_n\in [a,\infty)\) and \(J\) is uniformly continuous on bounded subsets of \(\mathcal{X}\), we have that \[\begin{align}
\label{223} {\color{blue}\lim}_{n\rightarrow \infty}\frac{||Jy_n-Ju_n||}{r_n}=0.
\end{align}\tag{12}\] Let \(k\in \mathbb{N}\) and let \(f_i:J\Omega\times J\Omega\to \mathbb{R}\) be a bifunction satisfying \((A1)\)–\((A4)\) for each \(i\in\{1,2,\ldots,k\}\). Define \(F:J\Omega\times J\Omega\to \mathbb{R}\) by \[F(u,v)=\sum_{i=1}^{k} f_i(u,v), \qquad \forall\, u,v\in J\Omega.\]
Equivalently, for all \(x,y\in\Omega\), \[F(Jx,Jy)=\sum_{i=1}^{k} f_i(Jx,Jy).\] Since \(F\) is a finite sum of bifunctions satisfying \((A1)\)–\((A4)\), it also satisfies \((A1)\)–\((A4)\).
From \(u_n=T_{r_n}y_n\), we have that \[F(Ju_n,Jy)+\frac{1}{r_n}\langle u_n-y_{n}, Jy-Ju_n\rangle\geq0,~~\forall~~y\in \Omega.\] By (A2), we have \[\begin{align}
\label{224} \frac{1}{r_n}\langle u_n-y_{n}, Jy-Ju_n\rangle\geq-F(Ju_n,Jy)\geq F(Jy,Ju_n),~~\forall~~y\in \Omega.
\end{align}\tag{13}\] Since \(F(x,\cdot)\) is convex and lower semicontinuous and \(u_n\rightarrow x^*\), it follows from equation (12 ) and inequality
(13 ) that \[F(Jy,Jx^*)\leq0,~~\forall~~y\in \Omega.\] For \(t\in(0,1]\) and \(y\in \Omega\), let \(y^*_t=tJy+(1-t)Jx^*\). Since, \(J\Omega\) is convex, we have that \(y^*_t\in J\Omega\) and hence \(F(y^*_t,Jx^*)\leq0\). From
(A1), \[0=F(y^*_t,y^*_t)\leq tF(y^*_t,Jy)+(1-t)F(y^*_t,Jx^*)\leq tF(y^*_t,Jy),~~\forall~~y\in \Omega.\] This implies that \[F(y^*_t,Jy)\geq 0,~~\forall~~y\in \Omega.\] Letting \(t\downarrow0\), from (A3), \[F(Jx^*,Jy)\geq 0,~~\forall~~y\in \Omega.\] Therefore, we have that \(Jx^*\in JGMEP(f).\) This implies that \(x^*\in GMEP(f).\)
Finally, we show that \(x^* = R_{B}x.\)
From Lemma 4\((ii)\), we obtain that \[\label{eq4463}
\phi(x,R_Bx) \leq \phi(x,x^*) - \phi(R_Bx, x^*) \leq \phi(x,x^*).\tag{14}\]
Again, using Lemma 4\((ii)\), definition of \(x_{n+1}\), and \(x^*\in B \subset
\Omega_{n},\) we compute as follows: \[\begin{align} \phi(x,x_{n+1}) &\leq &\phi(x,x_{n+1}) + \phi(x_{n+1},R_Bx)\nonumber\\ & = & \phi( x, R_{\Omega_{n+1}}x) + \phi(R_{\Omega_{n+1}}x, R_Bx) \leq \phi(x,
R_Bx).\nonumber
\end{align}\] Since \(x_{n}\rightarrow x^*\), taking limits on both sides of the last inequality, we obtain \[\label{eq4464}
\phi(x,x^*) \leq \phi(x, R_Bx).\tag{15}\] Using inequalities (14 ) and (15 ), we obtain that \(\phi(x,x^*) = \phi(x, R_Bx)\). By the uniqueness of \(R_B\) (Lemma 5), we obtain that \(x^*=R_Bx\). This completes proof of the theorem. ◻
Theorem 2. Let \(\mathcal{X}\) be a uniformly smooth and uniformly convex real Banach space with dual space \(\mathcal{X}^*\) and let \(\Omega\) be a nonempty closed and convex subset of \(\mathcal{X}\) such that \(J\Omega\) is closed and convex. Let \(\varphi:J\Omega
\rightarrow \mathbb{R}\) be a lower semi-continuous and convex function. For each \(i \in \{1, 2, 3, ..., N\}\), let \(f_{i}\) be a bifunction from \(J\Omega\times J\Omega\) to \(\mathbb{R}\) satisfying \((A1)-(A4)\), \(T^i_{n}:\Omega\rightarrow \mathcal{X}^*, n=1, 2, 3, ...\)
be an infinite family of generalized \(J-\)nonexpansive maps and \(\Gamma\) be a family of closed and generalized \(J-\)nonexpansive maps from \(\Omega\) to \(\mathcal{X}^*\) such that \(\cap_{n=1}^{\infty}F_{J}(T^i_{n})=F_{J}(\Gamma) \neq \emptyset\) and \(B := F_{J}(\Gamma)\cap
GMEP(f,A, \varphi) \neq \emptyset.\) Assume that \(\{T_{n}\}\) satisfies the NST-condition with \(\Gamma\). Let \(\{x_{n}\}\) be generated by:
for all \(n\in \mathbb{N}, \; \{\alpha^i_{n}\}\in [0,1]\) such that \(\sum_{i = 0} ^{N} \alpha^i_{n} = 1\), \(\{r_n\}\subset [a,\infty)\) for some \(a>0\). Then, \(\{x_{n}\}\) converges strongly to \(R_B x\), where \(R_B\) is the sunny generalized \(J-\)nonexpansive retraction of \(\mathcal{X}\) onto \(B\).
Proof. It is easy to see that \(\{x_{n}\}\) is well defined.
Clearly, \(B \subset \Omega_1\). Suppose \(B \subset \Omega_{n}\) for some \(n\in \mathbb{N}\). Let \(u\in B\), and \(u_n=T_{r_n}y_n\) for all \(n\in \mathbb{N}\). Using the fact that \(\{T^i_{n}\}\) is an infinite family of generalized \(J-\)nonexpansive maps, the definition of \(y_n\), Lemmas 7, and 1, we compute as follows: \[\begin{align} \phi(u_{n},u) & = & \phi(T_{r_n}y_n,u)\leq\phi(y_n,u) = \phi{\color{blue}\Big(}\alpha^0_{n}x_{n} + \sum_{i = 1} ^{N}
\alpha^i_{n}J^{-1}oT^i_{n}x_{n},u{\color{blue}\Big)}\nonumber\\ & = & {\color{blue}\Big|\Big|}\alpha^0_{n}x_{n} + \sum_{i = 1} ^{N} \alpha^i_{n}J^{-1}oT^i_{n}x_{n}{\color{blue}\Big|\Big|}^2 -2{\color{blue}\Big\langle} \alpha^0_{n}x_{n} + \sum_{i =
1} ^{N} \alpha^i_{n}J^{-1}oT^i_{n}x_{n},Ju{\color{blue}\Big\rangle} + ||u||^2\nonumber\\ & \leq & \alpha^0_{n}||x_{n}||^2 -2\alpha^0_{n}\langle x_{n}, Ju\rangle + ||u||^2 + \sum_{i = 1} ^{N} \alpha^i_{n}||J^{-1}oT^i_{n}x_{n}||^2
-2{\color{blue}\Big\langle} \sum_{i = 1} ^{N} \alpha^i_{n}J^{-1}oT^i_{n}x_{n},Ju{\color{blue}\Big\rangle}\nonumber\\ && - \alpha^0_{n}\alpha^i_{n}g(||x_{n}- J^{-1}oT^i_{n}x_{n}||)\nonumber\\ & = & \alpha^0_{n}(||x_{n}||^2 -2\langle x_{n},
Ju\rangle + ||u||^2) + \sum_{i = 1} ^{N}\alpha^i_{n}(||J^{-1}oT^i_{n}x_{n}||^2 -2\langle J^{-1}oT^i_{n}x_{n}, Ju\rangle + ||u||^2) \nonumber\\ && - \alpha^0_{n}\alpha^i_{n}g(||x_{n}- J^{-1}oT^i_{n}x_{n}||)\nonumber\\ & = &
\alpha^0_{n}\phi(x_{n},u) + \sum_{i = 1} ^{N}\alpha^i_{n}\phi(J^{-1}oT^i_{n}x_{n},u) - \alpha^0_{n}\alpha^i_{n}g(||x_{n}- J^{-1}oT^i_{n}x_{n}||)\nonumber\\ & \leq & \alpha^0_{n}\phi(x_{n},u) + \sum_{i = 1} ^{N}\alpha^i_{n}\phi(x_{n},u) -
\alpha^0_{n}\alpha^i_{n}g(||x_{n}- J^{-1}oT^i_{n}x_{n}||).\nonumber
\end{align}\] Hence, we obtain the following inequality for each \(i \in \{1,2,3,\cdots, N\}\)\[\label{now}
\phi(u_{n},u) \leq \phi(x_{n},u) - \alpha^0_{n}\alpha^i_{n}g(||x_{n}- J^{-1}oT^i_{n}x_{n}||).\tag{17}\] Additionally, we have that \[\phi(y_{n},u) \leq \phi(x_{n},u) ~\forall~ n\in \mathbb{N}.\] From
inequality (17 ), we conclude that \(u\in \Omega_{n+1}\). Thus, \(B \subset{\Omega_n}\) for all \(n \ge 1\).
The rest of the proof is similar to the proof of Theorem 1. ◻
Example 2. Let \(\mathcal{X} = l_p\), \(1< p <\infty\), \(\frac{1}{p} + \frac{1}{q} = 1\), and \(\Omega =
\overline{B_{l_p}}(0,1)\) = \(\{x \in l_p : ||x||_{l_p}\leq 1\}\). Then \(J{\color{blue}\Omega} = \overline{B_{l_q}}(0,1)\). Let \(f_i : J\Omega\times
J\Omega \longrightarrow \mathbb{R}\) defined by \(f_i(x^*, y^*) = \langle J^{-1}x^*, y^* - x^*\rangle\)\(\forall\)\(x^*,y^* \in J\Omega\) and for
each \(i \in \{1,2,3,\cdots,k\}\), \(A : \Omega \longrightarrow l_q\) defined by \({\color{blue}A}x = J(x_1, x_2, x_3, \cdots)\)\(\forall\)\(x = (x_1, x_2, x_3, \cdots) \in \Omega\), \(\varphi : J\Omega \rightarrow \mathbb{R}\) defined by \(\varphi(x^*) = ||x^*||,
\forall~ x^* \in J\Omega\), \(T : \Omega \longrightarrow l_q\) defined by \(Tx = J(0, x_1, x_2, x_3, \cdots)\)\(\forall\)\(x = (x_1, x_2, x_3, \cdots) \in \Omega\), \(\Gamma = \{T\}\), and \(T_n : \Omega \longrightarrow l_q\) defined by \(T_{n}x = \alpha_nJx
+ (1 - \alpha_n)Tx, ~\forall n \geq 1,~\forall~ x\in \Omega, \alpha_n \in (0,1) \text{ such that } 1 - \alpha_n \ge \frac{1}{2}\). Then \(\Omega\), \(J\Omega\), \(f_i\), \(A\), \(\varphi\), \(T\), and \(T_n\) satisfy the conditions of Theorems 1 and 2. Moreover, \(0 \in F_{J}(\Gamma) \cap GMEP(f,A,\varphi)\).
Let \(A = 0\) in Theorems 1 and 2. We obtain the following results.
Corollary 1. Let \(\mathcal{X}, \mathcal{X}^*, \Omega, \varphi, \{f_i\}, \{T^i_{n}\}, \{r_n\}, \{\alpha^i_{n}\}\) be as in Theorem 2.
If \(\cap_{n=1}^{\infty}F_{J}(T^i_{n})=F_{J}(\Gamma) \neq \emptyset\) and \(B := F_{J}(\Gamma)\cap MEP(f,A, \varphi) \neq \emptyset\), where \(MEP\) is the
set of solutions to mixed equilibrium problem 3 . Let \(\{x_{n}\}\) be generated by:
Theorems 1 and 2 are applicable in classical Banach spaces, such as \(L_{p},~l_{p}, or~
W^{m}_{p}(\Omega)\), \(p \in (1,\infty)\), where \(W^{m}_{p}(\Omega)\) denotes the usual Sobolev space. The analytical representations of duality maps are known in \(L_p,\)\(l_p,\) and \(W^p_m(\Omega),\)\(p \in (1,\infty)\), \(p^{-1}+q^{-1}=1\), see e.g.,
[21].
Corollary 2. Let \(\mathcal{X}=H\) be a real Hilbert space and let \(\Omega\) be a nonempty closed and convex subset of \(H\). Let \(\varphi:\Omega \rightarrow \mathbb{R}\) be a lower semi-continuous and convex function. For each \(i \in \{1, 2, 3, ..., N\}\), let \(f_{i}\) be a bifunction
from \(\Omega\times \Omega\) to \(\mathbb{R}\) satisfying \((A1)-(A4)\), \(T^i_{n}:\Omega\rightarrow H, n=1, 2, 3, ...\) be
an infinite family of nonexpansive maps and \(\Gamma\) be a family of closed and nonexpansive maps from \(\Omega\) to \(H\) such that \(\cap_{n=1}^{\infty}F(T_{n})=F(\Gamma) \neq \emptyset\) and \(B := F(\Gamma)\cap GMEP(f,A, \varphi) \neq \emptyset.\) Assume that \(\{T_{n}\}\) satisfies the
NST-condition with \(\Gamma\). Let \(\{x_{n}\}\) be generated by:
for all \(n\in \mathbb{N}, \; \{\alpha^i_{n}\}\in [0,1]\) such that \(\sum_{i = 0} ^{N} \alpha^i_{n} = 1\), \(\{r_n\}\subset [a,\infty)\) for some \(a>0\). Then, \(\{x_{n}\}\) converges strongly to \(P_B x\), where \(P_B\) is the metric projection of \(H\) onto \(B\).
Proof. In a Hilbert space, \(J\) is the identity operator and \(\phi(x,y)=||x-y||^{2} ~ \text{for all} ~ x,y\in H\). The result follows from Theorem 1 or Theorem 2. ◻
Consider the following optimization problem: \[\label{appoptm} \min_{x \in \Omega}(\psi(x) + \varphi(x))\tag{20}\] where \(\Omega\) is a
nonempty closed convex subset of a Hilbert space \(H\), and \(\psi, \varphi :\Omega\rightarrow \mathbb{R}\) are two convex and lower semi-continuous functionals. Let \(\Lambda \subset \Omega\) be the set of solutions to 20 . Clearly, \(\Lambda\) is a closed convex subset of \(\Omega\). Let \(f : \Omega \times \Omega \rightarrow \mathbb{R}\) be a bifunction defined by \(f(x, y) = \psi(y) - \psi(x)\). Consider the following mixed equilibrium problem: find \(x^* \in \Omega\) such that \[\label{appmep} f(x^*,y)+\varphi(y) - \varphi (x^*) \geq 0,~~\forall~~y\in \Omega.\tag{21}\] Then, \(f\) satisfies conditions (A1)–(A4) and \(MEP = \Lambda\), where \(MEP\) is the set of solutions to mixed equilibrium problem 21 . Let
\(\{x_n\}\) be the iterative sequence generated by:
\[\label{okdzqyfh}
\begin{cases} & x_{1} = x\in \Omega, \,\Omega_{1}=\Omega; \cr & x_{n+1} = P_{\Omega_{n+1}}x,\, \Omega_{n+1} =\{z\in \Omega_{n} : ||u_{n} - z|| \leq ||x_{n} - z||\};\cr & u_{n}\in \Omega, ~~such~~that~~ f(u_n,y)+\varphi(y) - \varphi (u_n) +
\frac{1}{r_n}\langle u_n-y_{n}, y-u_n\rangle\geq0,~~\forall~~y\in \Omega,
\end{cases}\tag{22}\] for all \(n\in \mathbb{N}\), \(\{r_n\}\subset [a,\infty)\) for some \(a>0\), where \(P_\Omega\) is the metric projection of \(H\) onto \(\Omega\). Then, \(\{x_{n}\}\) converges strongly to \(P_\Omega x\).
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