A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications

Markjoe O. Uba University of Nigeria
Department of Mathematics
Nsukka - Onitsha Rd, 410001 Nsukka, Nigeria [email protected] , Maria A. Onyido Northern Illinois University
Department of Mathematical Sciences,
DeKalb, IL 60115, United States [email protected] , Cyril I. Udeani Comenius University in Bratislava
Faculty of Mathematics, Physics and Informatics,
Mlynská dolina F1, 842 48 Bratislava, Slovak Republic [email protected] and Peter U. Nwokoro University of Nigeria
Department of Mathematics
Nsukka - Onitsha Rd, 410001 Nsukka, Nigeria [email protected]

Key words and phrases:

equilibrium problem,

J_{*}-

nonexpansive, fixed points, variational inequality, strong convergence.

2010 Mathematics Subject Classification:

47H09, 47H05, 47J25, 47J05.

1. Introduction

Let $E$ be a real Banach space with topological dual $E^{*}$ . Let $C\subset{E}$ be closed and convex with $JC$ also closed and convex, where $J$ is the normalized duality map (see definition 2.1). The variational inequality problem, which has its origin in the 1964 result of Stampacchia [20], has engaged the interest of researchers in the recent past (see, e.g., [24, 25] and many others). This is concerned with the following: For a monotone operator $A:C\to E$ , find a point $x^{*}\in C$ such that

(1.1)

\langle y-x^{*},Ax^{*}\rangle\geq 0~{}~{}for~{}~{}all~{}~{}y\in C.

The set of solutions of (1.1) is denoted by $VI(C,A)$ . This problem, which plays a crucial role in nonlinear analysis, is also related to fixed point problems, zeros of nonlinear operators, complementarity problems, and convex minimization problems (see, for example, [28, 29]).

A related problem is the equilibrium problem, which has been studied by several researchers and is mostly applied in solving optimization problems (see [3]). For a map $f:C\to E$ , the equilibrium problem is concerned with finding a point $x^{*}\in C$ such that

(1.2)

f(x^{*},y)\geq 0~{}~{}for~{}~{}all~{}~{}y\in C.

The set of solutions of (1.2) is denoted by $EP(f)$ . The variational inequality and equilibrium problems are special cases of the so-called generalized mixed equilibrium problem (see [15]). Another related problem is the fixed point problem. For a map $T:D(T)\subset E\to E$ , the fixed points of $T$ are the points $x^{*}\in D(T)~{}~{}such~{}~{}that~{}~{}Tx^{*}=x^{*}$ . Recently, owing to the need to develop methods for solving fixed points of problems for functions from a space to its dual, a new concept of fixed points for maps from a real normed space $E$ to its dual space $E^{*}$ , called $J-$ fixed point has been introduced and studied (see [5, 12, 23]).

With this evolving fixed point theory, we study the $J-$ fixed points of certain maps and the following equilibrium problem. Let $f:JC\times JC\rightarrow\mathbb{R}$ be a bifunction. The equilibrium problem for $f$ is finding

(1.3)

\displaystyle x^{*}\in C~{}such~{}that~{}f(Jx^{*},Jy)\geq 0,\forall~{}y\in C.

We denote the solution set of (1.3) by $EP(f)$ . Several problems in physics, optimization and economics reduce to finding a solution of (1.3) (see, e.g., [7, 24] and the references in them). Most of the equilibrium problems studied in the past two decades centered on their existence and applications (see, e.g., [3, 7] ). However, recently, several researchers have started working on finding approximate solutions of equilibrium problems and their generalizations (see, e.g., [11, 25]). Not long ago, some researchers investigated the problem of establishing a common element in the solution set of an equilibrium problem, fixed point of a family of nonexpansive maps and solution set of a variational inequality problem for different classes of maps (see [26] and references therein).

In this paper, inspired by the above results especially the works in [4, 22, 26], we present an algorithm for finding a common element of the fixed point of an infinite family of generalized $J_{*}-$ nonexpansive maps, the solution set of the variational inequality problem of a finite family of continuous monotone maps and the solution set of the equilibrium point of a finite family of bifunctions satisfying some given conditions. Our results complement, generalize and extend results in [13, 18, 19, 26] (see the section on conclusion) and other recent results in this direction. It is worth noting that very recently, the authors in [4] introduced a new class of maps which they called relatively weak $J-$ nonexpasive and developed an algorithm for approximating a common element of the $J-$ fixed point of a countable family of such maps and zeros of some other class of maps in certain Banach spaces. Previously, maps with similar requirements as these relatively weak $J-$ nonexpasive maps have also been studied in [6] where they were called quasi $-\phi-J-$ nonexpansive. We observe that these two sets of maps (relatively weak $J-$ nonexpasive and quasi $-\phi-J-$ nonexpansive) coincide in definition with the $J_{*}-$ nonexpansive maps in our results.

2. Preliminaries

In this section, we present definitions and lemmas used in proving our main results.

Definition 2.1.

(Normalized duality map) The map $J:E\rightarrow 2^{E^{*}}$ defined by

Jx:=\big{\{}x^{*}\in E^{*}:\big{<}x,x^{*}\big{>}=\|x\|.\|x^{*}\|,~{}\|x\|=\|x^{*}\|\big{\}}

is called the normalized duality map on $E$ .

It is well known that if $E$ is smooth, strictly convex and reflexive then $J^{-1}$ exists (see e.g., [10]); $J^{-1}:E^{*}\rightarrow E$ is the normalized duality mapping on $E^{*}$ , and $J^{-1}=J_{*},~{}JJ_{*}=I_{E^{*}}$ and $J_{*}J=I_{E}$ , where $I_{E}$ and $I_{E^{*}}$ are the identity maps on $E$ and $E^{*}$ , respectively. A well known property of $J$ is, see e.g., [8, 10], if $E$ is uniformly smooth, then $J$ is uniformly continuous on bounded subsets of $E$ .

Definition 2.2.

(Lyapunov Functional) [1, 11] Let $E$ be a smooth real Banach space with dual $E^{*}$ . The Lyapounov functional $\phi:E\times E\to\mathbb{R}$ , is defined by

(2.4)

\displaystyle\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2},~{}~{}\text{for}~{}x,y\in E,

where $J$ is the normalized duality map. If $E=H$ , a real Hilbert space, then equation (2.4) reduces to $\phi(x,y)=\|x-y\|^{2}$ for $x,y\in H.$ Additionally,

(2.5)

\displaystyle(\|x\|-\|y\|)^{2}\leq\phi(x,y)\leq(\|x\|+\|y\|)^{2}~{}~{}\text{for}~{}x,y\in E.

Definition 2.3.

(Generalized nonexpansive) [16, 17] Let $C$ be a nonempty closed and convex subset of a real Banach space $E$ and $T$ be a map from $C$ to $E$ . The map $T$ is called generalized nonexpansive if $F(T):=\{x\in C:Tx=x\}\neq\emptyset$ and $\phi(Tx,p)\leq\phi(x,p)$ for all $x\in C,p\in F(T)$ .

Definition 2.4.

(Retraction) [16, 17] A map $R$ from $E$ onto $C$ 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 $x\in E$ and $t\leq 0$ .

A nonempty closed subset $C$ of a smooth Banach space $E$ is said to be a sunny generalized nonexpansive retract of $E$ if there exists a sunny generalized nonexpansive retraction $R$ from $E$ onto $C$ .

NST-condition. Let $C$ be a closed subset of a Banach space $E$ . Let $\{T_{n}\}$ and $\Gamma$ be two families of generalized nonexpansive maps of $C$ into $E$ 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 2.5.

[16] The sequence $\{T_{n}\}$ satisfies the NST-condition (see e.g., [14]) with $\Gamma$ if for each bounded sequence $\{x_{n}\}\subset C$ ,

\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 2.1.

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 $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$ . Let $J$ be the normalized duality map on $E$ and $J_{*}$ be the normalized duality map on $E^{*}$ . Observe that under this setting, $J^{-1}$ exists and $J^{-1}=J_{*}$ . With these notations, we have the following definitions.

Definition 2.6.

(Closed map) [22] A map $T:C\rightarrow E^{*}$ is called $J_{*}-$ closed if $(J_{*}\circ T):C\rightarrow E$ is a closed map, i.e., if $\{x_{n}\}$ is a sequence in $C$ such that $x_{n}\rightarrow x$ and $(J_{*}\circ T)x_{n}\rightarrow y$ , then $(J_{*}\circ T)x=y$ .

Definition 2.7.

( $J-$ fixed Point) [5] A point $x^{*}\in C$ 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 2.8.

(Generalized $J_{*}-$ nonexpansive) [22] A map $T:C\rightarrow E^{*}$ will be called generalized $J_{*}-$ nonexpansive if $F_{J}(T)\neq\emptyset$ , and $\phi(p,(J_{*}\circ T)x)\leq\phi(p,x)$ for all $x\in C$ and for all $p\in F_{J}(T)$ .

Remark 2.2.

Exampes of generalized $J_{*}-$ nonexpansive maps in Hilbert and more general Banach spaces were given in [4, 22].

Let $C$ be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space $E$ such that $JC$ is closed and convex. For solving the equilibrium problem, let us assume that a bifunction $f:JC\times JC\rightarrow\mathbb{R}$ satisfies the following conditions:

(A1)

$f(x^{*},x^{*})=0$ for all $x^{*}\in JC$ ;
(A2)

$f$ is monotone, i.e. $f(x^{*},y^{*})+f(y^{*},x^{*})\leq 0$ for all $x^{*},y^{*}\in JC$ ;
(A3)

for all $x^{*},y^{*},z^{*}\in JC$ , $\limsup_{t\downarrow 0}f(tz^{*}+(1-t)x^{*},y^{*})\leq f(x^{*},y^{*})$ ;
(A4)

for all $x^{*}\in JC$ , $f(x^{*},\cdot)$ is convex and lower semicontinuous.

With the above definitions, we now provide the lemmas we shall use.

Lemma 2.1.

[27] Let $E$ be a uniformly convex Banach space, $r>0$ be a positive number, and $B_{r}(0)$ be a closed ball of $E$ . 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 $i,j\in\{1,2,\cdots N\},\;i<j,$

(2.6)

\|\sum_{n=1}^{N}\lambda_{n}x_{n}\|^{2}\leq\sum_{n=1}^{N}\lambda_{n}\|x_{n}\|^{2}-\lambda_{i}\lambda_{j}g(\|x_{i}-x_{j}\|).

Lemma 2.2.

[11] Let $X$ be a real smooth and uniformly convex Banach space, and let $\{x_{n}\}$ and $\{y_{n}\}$ be two sequences of $X$ . If either $\{x_{n}\}$ or $\{y_{n}\}$ is bounded and $\phi(x_{n},y_{n})\to 0$ as $n\to\infty$ , then $\|x_{n}-y_{n}\|\to 0$ as $n\to\infty$ .

Lemma 2.3.

[1] Let $C$ be a nonempty closed and convex subset of a smooth, strictly convex and reflexive Banach space $E$ . Then, the following are equivalent.
$(i)$ $C$ is a sunny generalized nonexpansive retract of $E$ ,
$(ii)$ $C$ is a generalized nonexpansive retract of $E$ ,
$(iii)$ $JC$ is closed and convex.

Lemma 2.4.

[1] Let $C$ be a nonempty closed and convex subset of a smooth and strictly convex Banach space $E$ such that there exists a sunny generalized nonexpansive retraction $R$ from $E$ onto $C$ . Then, the following hold.
$(i)$ $z=Rx$ iff $\langle x-z,Jy-Jz\rangle\leq 0$ for all $y\in C$ ,
$(ii)$ $\phi(x,Rx)+\phi(Rx,z)\leq\phi(x,z)$ .

Lemma 2.5.

[9] Let $C$ be a nonempty closed sunny generalized nonexpansive retract of a smooth and strictly convex Banach space $E$ . Then the sunny generalized nonexpansive retraction from $E$ to $C$ is uniquely determined.

Lemma 2.6.

[3] Let $C$ be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space $E$ such that $JC$ is closed and convex, let $f$ be a bifunction from $JC\times JC$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ . For $r>0$ and let $x\in E$ . Then there exists $z\in C$ such that $f(Jz,Jy)+\frac{1}{r}\langle z-x,Jy-Jz\rangle\geq 0,~{}~{}\forall~{}~{}y\in C.$

Lemma 2.7.

[21] Let $C$ be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space $E$ such that $JC$ is closed and convex, let $f$ be a bifunction from $JC\times JC$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ . For $r>0$ and let $x\in E$ , define a mapping $T_{r}(x):E\rightarrow C$ as follows:

T_{r}(x)=\{z\in C:f(Jz,Jy)+\frac{1}{r}\langle y-z,Jz-Jx\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\}.

Then the following hold:

(i)

$T_{r}$ is single valued;
(ii)

for all $x,y\in E$ , $\langle T_{r}x-T_{r}y,JT_{r}x-JT_{r}y\rangle\leq\langle x-y,JT_{r}x-JT_{r}y\rangle$ ;
(iii)

$F(T_{r})=EP(f)$ ;
(iv)

$\phi(p,T_{r}(x))+\phi(T_{r}(x),x)\leq\phi(p,x)$ for all $p\in F(T_{r})$ .
(v)

$JEP(f)$ is closed and convex.

Lemma 2.8.

[22] Let $C$ be a nonempty closed subset of a smooth, strictly convex and reflexive Banach space $E$ . Let $A:C\rightarrow E^{*}$ be a continuous monotone mapping. For $r>0$ and let $x\in E$ , define a mapping $F_{r}(x):E\rightarrow C$ as follows:

F_{r}(x)=\{z\in C:\langle y-z,Az\rangle+\frac{1}{r}\langle y-z,Jz-Jx\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\}.

Then the following hold:

(i)

$F_{r}$ is single valued;
(ii)

for all $x,y\in E$ , $\langle F_{r}x-T_{r}y,JF_{r}x-JF_{r}y\rangle\leq\langle x-y,JF_{r}x-JF_{r}y\rangle$ ;
(iii)

$F(F_{r})=VI(C,A)$ ;
(iv)

$\phi(p,F_{r}(x))+\phi(F_{r}(x),x)\leq\phi(p,x)$ for all $p\in F(F_{r})$ .
(v)

$JVI(C,A)$ is closed and convex.

Lemma 2.9.

[22] Let $E$ be a uniformly convex and uniformly smooth real Banach space with dual space $E^{*}$ and let $C$ be a closed subset of $E$ such that $JC$ is closed and convex. Let $T$ be a generalized $J_{*}-$ nonexpansive map from $C$ to $E^{*}$ such that $F_{J}(T)\neq\emptyset$ , then $F_{J}(T)$ and $JF_{J}(T)$ are closed. Additionally, if $JF_{J}(T)$ is convex, then $F_{J}(T)$ is a sunny generalized nonexpansive retract of $E$ .

3. Main Results

(3.7)

\begin{cases}&x_{1}=x\in C;C_{1}=C,\cr&z_{n}:=\{z\in C:f_{n}(Jz,Jy)+\frac{1}{r_{n}}\langle y-z,Jz-Jx_{n}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\},\cr&u_{n}:=\{z\in C:\langle y-z,A_{n}z\rangle+\frac{1}{r_{n}}\langle y-z,Jz-Jx_{n}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\},\cr&y_{n}=J^{-1}(\alpha_{1}Jx_{n}+\alpha_{2}Jz_{n}+\alpha_{3}T_{n}u_{n}),\cr&C_{n+1}=\{z\in C_{n}:\phi(z,y_{n})\leq\phi(z,x_{n})\},\cr&x_{n+1}=R_{C_{n+1}}x,\end{cases}

for all $n\in\mathbb{N},$ with $\alpha_{1},\alpha_{2},\alpha_{3}\in(0,1)$ satisfying $\alpha_{1}+\alpha_{2}+\alpha_{3}=1$ , $\{r_{n}\}\subset[a,\infty)$ for some $a>0$ , $A_{n}=A_{n(mod~{}N)}$ and $f_{n}(\cdot,\cdot)=f_{n(mod~{}L)}(\cdot,\cdot)$ .

Lemma 3.10.

The sequence $\{x_{n}\}$ generated by (3.7) is well defined.

Proof.

Observe that $JC_{1}$ is closed and convex. Moreover, it is easy to see that $\phi(z,y_{n})\leq\phi(z,x_{n})$ is equivalent to

0\leq||x_{n}||^{2}-||y_{n}||^{2}-2\langle z,Jx_{n}-Jy_{n}\rangle,

which is affine in $z$ . Hence, by induction $JC_{n}$ is closed and convex for each $n\geq 1$ . Therefore, from Lemma 2.3, we have that $C_{n}$ is a sunny generalized retract of $E$ for each $n\geq 1$ . This shows that $\{x_{n}\}$ is well defined. ∎

Theorem 3.1.

Let $E$ be a uniformly smooth and uniformly convex real Banach space with dual space $E^{*}$ and let $C$ be a nonempty closed and convex subset of $E$ such that $JC$ is closed and convex. Let $f_{l},l=1,2,3,...,L$ be a family of bifunctions from $JC\times JC$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ , $T_{n}:C\rightarrow E^{*},n=1,2,3,...$ be an infinite family of generalized $J_{*}-$ nonexpansive maps, $A_{k}:C\rightarrow E^{*},k=1,2,3,...,N$ be a finite family of continuous monotone mappings and $\Gamma$ be a family of $J_{*}-$ closed and generalized $J_{*}-$ nonexpansive maps from $C$ to $E^{*}$ such that $\cap_{n=1}^{\infty}F_{J}(T_{n})=F_{J}(\Gamma)\neq\emptyset$ and $B:=F_{J}(\Gamma)\cap\Big{[}\cap_{l=1}^{L}EP(f_{l})\Big{]}\cap\Big{[}\cap_{k=1}^{N}VI(C,A_{k})\Big{]}\neq\emptyset.$ Assume that $JF_{J}(\Gamma)$ is convex and $\{T_{n}\}$ satisfies the NST-condition with $\Gamma$ . Then, $\{x_{n}\}$ generated by (3.7) converges strongly to $R_{B}x$ , where $R_{B}$ is the sunny generalized nonexpansive retraction of $E$ onto $B$ .

Proof.

The proof is given in $6$ steps.

Step 1: We show that the expected limit $R_{B}x$ exists as a point in $C_{n}\;\text{for all}~{}n\geq 1$ .

First, we show that $B\subset C_{n}\;\text{for all}\;n\geq 1$ and $B$ is a sunny generalized retract of $E$ .
Since $C_{1}=C$ , we have $B\subset C_{1}$ . Suppose $B\subset C_{n}$ for some $n\in\mathbb{N}$ . Let $u\in B$ . We observe from algorithm (3.7) that $u_{n}=F_{r_{n}}x_{n}$ and $z_{n}=T_{r_{n}}x_{n}$ for all $n\in\mathbb{N}$ , using this and the fact that $\{T_{n}\}$ is an infinite family of generalized $J_{*}-$ nonexpansive maps, the definition of $y_{n}$ , Lemmas 2.7, 2.8, and 2.1, we compute as follows:

$\displaystyle\phi(u,y_{n})$	$\displaystyle=$	$\displaystyle\phi(u,J^{-1}(\alpha_{1}Jx_{n}+\alpha_{2}Jz_{n}+\alpha_{3}T_{n}u_{n})$
	$\displaystyle\leq$	$\displaystyle\alpha_{1}\big{[}\|\|u\|\|^{2}-2\langle u,Jx_{n}\rangle+\|\|x_{n}\|\|^{2}\big{]}+\alpha_{2}\big{[}\|\|u\|\|^{2}-2\langle u,Jz_{n}\rangle+\|\|z_{n}\|\|^{2}\big{]}$
		$\displaystyle+\alpha_{3}\big{[}\|\|u\|\|^{2}-2\langle u,J(J_{*}\circ T_{n})u_{n}\rangle+\|\|T_{n}u_{n}\|\|^{2}\big{]}$
		$\displaystyle-\alpha_{1}\alpha_{3}g(\|\|Jx_{n}-J(J_{*}\circ T_{n})u_{n}\|\|)$
	$\displaystyle=$	$\displaystyle\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,z_{n})+\alpha_{3}\phi(u,(J_{*}\circ T_{n})u_{n})-\alpha_{1}\alpha_{3}g(\|\|Jx_{n}-T_{n}u_{n}\|\|)$
	$\displaystyle\leq$	$\displaystyle\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,z_{n})+\alpha_{3}\phi(u,u_{n})-\alpha_{1}\alpha_{3}g(\|\|Jx_{n}-T_{n}u_{n}\|\|)$
	$\displaystyle=$	$\displaystyle\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,T_{r_{n}}x_{n})+\alpha_{3}\phi(u,u_{n})-\alpha_{1}\alpha_{3}g(\|\|Jx_{n}-T_{n}u_{n}\|\|)$
	$\displaystyle\leq$	$\displaystyle\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,x_{n})+\alpha_{3}\phi(u,u_{n})-\alpha_{1}\alpha_{3}g(\|\|Jx_{n}-T_{n}u_{n}\|\|),$

which yields

(3.9)

\phi(u,y_{n})\leq\phi(u,x_{n})-\alpha_{1}\alpha_{3}g(||Jx_{n}-T_{n}u_{n}||).

Hence, $\phi(u,y_{n})\leq\phi(u,x_{n})$ and we have that $u\in C_{n+1}$ , which implies that $B\subset{C_{n}}$ for all $n\geq 1$ . Moreover, From Lemma 2.7 and 2.8 both $JVI(C,A_{k})$ and $JEP(f_{l})$ are closed and convex for each $l$ and for each $k$ . Also, using our assumption and lemma 2.9, we have that $J(F_{J}(\Gamma)$ is closed and convex. Since $E$ is uniformly convex, $J$ is one-to-one. Thus, we have that,

$J\Big{(}F_{J}(\Gamma)\cap\Big{[}\cap_{l=1}^{L}EP(f_{l})\Big{]}\cap\Big{[}\cap_{k=1}^{N}VI(C,A_{k})\Big{]}\Big{)}=JF_{J}(\Gamma)\cap J\Big{[}\cap_{l=1}^{L}EP(f_{l})\Big{]}\cap J\Big{[}\cap_{k=1}^{N}VI(C,A_{k})\Big{]}$

so $J(B)$ is closed and convex. Using Lemma 2.3, we obtain that $B$ is a sunny generalized retract of $E$ . Therefore, from Lemma 2.5 , we have that $R_{B}x$ exists as a point in $C_{n}$ for all $n\geq 1$ . This completes step 1.

Step 2: We show that the sequence $\{x_{n}\}$ defined by (3.7) converges to some $x^{*}\in C.$

Using the fact that $x_{n}=R_{C_{n}}x$ and Lemma 2.4 $(ii)$ , we obtain

\phi(x,x_{n})=\phi(x,R_{C_{n}}x)\leq\phi(x,u)-\phi(R_{C_{n}}x,u)\leq\phi(x,u),

for all $u\in F_{J}(\Gamma)\cap EP(f_{l})\cap VI(C,A_{k})\subset C_{n};(l=1,2,\dots,L;\;k=1,2,\dots,K).$ This implies that $\{\phi(x,x_{n})\}$ is bounded. Hence, from equation (2.5), $\{x_{n}\}$ is bounded. Also, since $x_{n+1}=R_{C_{n+1}}x\in C_{n+1}\subset C_{n}$ , and $x_{n}=R_{C_{n}}x\in C_{n}$ , applying Lemma 2.4 $(ii)$ 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 2.4 $(ii)$ and $x_{n}=R_{C_{n}}x$ , we obtain that for all $m,n\in\mathbb{N}$ with $m>n$ ,

(3.10)		$\displaystyle\phi(x_{n},x_{m})$	$\displaystyle=$	$\displaystyle\phi(R_{C_{n}}x,x_{m})\leq\phi(x,x_{m})-\phi(x,R_{C_{n}}x)$
(3.10)			$\displaystyle=$	$\displaystyle\phi(x,x_{m})-\phi(x,x_{n})\rightarrow 0~{}as~{}n\rightarrow\infty.$

From Lemma 2.2, we conclude that $||x_{n}-x_{m}||\rightarrow 0,~{}as~{}m,~{}n\rightarrow\infty.$ Hence, $\{x_{n}\}$ is a Cauchy sequence in $C$ , and so, there exists $x^{*}\in C$ such that $x_{n}\rightarrow x^{*}$ completing step 2.

Step 3: We prove $x^{*}\in\cap_{k=1}^{N}VI(C,A_{k})$ .
From the definitions of $C_{n+1}$ and $x_{n+1}$ , we obtain that $\phi(x_{n+1},y_{n})\leq\phi(x_{n+1},x_{n})\rightarrow 0$ as $n\rightarrow\infty.$ Hence, by Lemma 2.2 , we have that

(3.11)

\underset{n\to\infty}{lim}||x_{n}-y_{n}||=0.

Since from step 2 $x_{n}\rightarrow x^{*}~{}as~{}n\rightarrow\infty$ , equation (3.11) implies that $y_{n}\rightarrow x^{*}~{}as~{}n\rightarrow\infty$ . Using the fact that $u_{n}=F_{r_{n}}x_{n}$ for all $n\in\mathbb{N}$ and Lemma 2.2, we get for $u\in B,$

$\displaystyle\phi(u_{n},x_{n})$	$\displaystyle=$	$\displaystyle\phi(F_{r_{n}}x_{n},x_{n})$
	$\displaystyle\leq$	$\displaystyle\phi(u,x_{n})-\phi(u,F_{r_{n}}x_{n})$
	$\displaystyle=$	$\displaystyle\phi(u,x_{n})-\phi(u,u_{n}).$

From equations (3) and (3.9) we have

(3.13)

\phi(u,y_{n})\leq\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,x_{n})+\alpha_{3}\phi(u,u_{n})\leq\phi(u,x_{n}).

Since $x_{n},y_{n}\to x^{*}$ as $n\to\infty$ , equation (3.13) implies that $\phi(u,u_{n})\to\phi(u,x^{*})$ as $n\to\infty$ . Therefore, from (3), we have $\phi(u,x_{n})-\phi(u,u_{n})\to 0$ as $n\to\infty$ which implies that $\lim_{n\rightarrow\infty}\phi(u_{n},x_{n})=0.$ Hence, from Lemma 2.2, we have

(3.14)

\lim_{n\rightarrow\infty}||u_{n}-x_{n}||=0.

Observe that since $J$ is uniformly continuous on bounded subsets of $E$ , it follows from (3.14) that $||Ju_{n}-Jx_{n}||\rightarrow 0.$

Again, since $r_{n}\in[a,\infty),$ we have that

(3.15)

\displaystyle\lim_{n\rightarrow\infty}\frac{||Ju_{n}-Jx_{n}||}{r_{n}}=0.

From $u_{n}=F_{r_{n}}x_{n}$ , we have

(3.16)

\displaystyle\langle y-u_{n},A_{n}u_{n}\rangle+\frac{1}{r_{n}}\langle y-u_{n},Ju_{n}-Jx_{n}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C.

Let $\{n_{l}\}_{l=1}^{\infty}\subset{\mathbb{N}}$ be such that $A_{n_{l}}=A_{1}\;\forall\;l\geq 1.$ Then, from (3.16), we obtain

(3.17)

\displaystyle\langle y-u_{n_{l}},A_{1}u_{n_{l}}\rangle+\frac{1}{r_{n_{l}}}\langle y-u_{n_{l}},Ju_{n_{l}}-Jx_{n_{l}}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C.

If we set $v_{t}=ty+(1-t)x^{*}$ for all $t\in(0,1]$ and $y\in C$ , then we get that $v_{t}\in C.$ Hence, it follows from (3.17) that

(3.18)

\displaystyle\langle v_{t}-u_{n_{l}},A_{1}u_{n_{l}}\rangle+\langle y-u_{n_{l}},\frac{Ju_{n_{l}}-Jx_{n_{l}}}{r_{n_{l}}}\rangle\geq 0.

This implies that

	$\displaystyle\langle v_{t}-u_{n_{l}},A_{1}v_{t}\rangle$	$\displaystyle\geq$	$\displaystyle\langle v_{t}-u_{n_{l}},A_{1}v_{t}\rangle-\langle v_{t}-u_{n_{l}},A_{1}u_{n_{l}}\rangle-\langle y-u_{n_{l}},\frac{Ju_{n_{l}}-Jx_{n_{l}}}{r_{n_{l}}}\rangle$
		$\displaystyle=$	$\displaystyle\langle v_{t}-u_{n_{l}},A_{1}v_{t}-A_{1}u_{n_{l}}\rangle-\langle y-u_{n_{l}},\frac{Ju_{n_{l}}-Jx_{n_{l}}}{r_{n_{l}}}\rangle.$

Since $A_{1}$ is monotone, $\langle v_{t}-u_{n_{l}},A_{1}v_{t}-Au_{n_{l}}\rangle\geq 0$ . Thus, using (3.15), we have that

\displaystyle 0\leq\lim_{l\rightarrow\infty}\langle v_{t}-u_{n_{l}},A_{1}v_{t}\rangle=\langle v_{t}-x^{*},A_{1}v_{t}\rangle,

therefore,

\displaystyle\langle y-x^{*},A_{1}v_{t}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C.

Letting $t\rightarrow 0$ and using continuity of $A_{1}$ , we have that

\displaystyle\langle y-x^{*},A_{1}x^{*}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C.

This implies that $x^{*}\in VI(C,A_{1})$ . Similarly, if $\{n_{i}\}_{i=1}^{\infty}\subset{\mathbb{N}}$ is such that $A_{n_{i}}=A_{2}\;~{}\text{for all}~{}\;i\geq 1$ , then we have again that $x^{*}\in VI(C,A_{2})$ . If we continue in similar manner, we obtain that $x^{*}\in\cap_{k=1}^{N}VI(C,A_{k}).$

Step 4: We prove that $x^{*}\in F_{J}(\Gamma)$ .
First, we show that $\lim_{n\rightarrow\infty}||Jx_{n}-Tu_{n}||=0~{}\forall~{}T\in\Gamma$ .
From inequality (3.9) and the fact that $g$ is nonnegative, we obtain

0\leq\alpha_{1}\alpha_{3}g(||Jx_{n}-T_{n}u_{n}||)\leq\phi(u,x_{n})-\phi(u,y_{n})\leq 2||u||.||Jx_{n}-Jy_{n}||+||x_{n}-y_{n}||M,

for some $M>0.$ Thus, using (3.11) and properties of $g$ , we obtain that
$\lim_{n\rightarrow\infty}||Jx_{n}-T_{n}u_{n}||\ =0$ . Using the above and triangle inequality gives $\|Ju_{n}-T_{n}u_{n}|\ \to 0~{}~{}as~{}~{}n\to\infty.$ Since $\{T_{n}\}_{n=1}^{\infty}$ satisfies the NST condition with $\Gamma$ , we have that

(3.19)

\lim_{n\rightarrow\infty}||Ju_{n}-Tu_{n}||=0~{}\forall~{}T\in\Gamma.

Now, from equation (3.14), we have $u_{n}\rightarrow x^{*}\in C$ . Assume that $(J_{*}\circ T)u_{n}\rightarrow y^{*}$ . Since $T$ is $J_{*}-$ closed, we have $y^{*}=(J_{*}\circ T)x^{*}$ . Furthermore, by the uniform continuity of $J$ on bounded subsets of $E$ , we have: $Ju_{n}\rightarrow Jx^{*}$ and $J(J_{*}\circ T)u_{n}\rightarrow Jy^{*}$ as $n\rightarrow\infty.$ Hence, we have

\lim_{n\rightarrow\infty}||Ju_{n}-J(J_{*}\circ T)u_{n}||=\lim_{n\rightarrow\infty}||Ju_{n}-Tu_{n}||=0,~{}\forall~{}T\in\Gamma,

which implies $||Jx^{*}-Jy^{*}||=||Jx^{*}-J(J_{*}\circ T)x^{*}||=||Jx^{*}-Tx^{*}||=0.$ So, $x^{*}\in F_{J}(\Gamma)$ .

Step 5: We prove that $x^{*}\in\cap_{l=1}^{L}EP(f_{l})$ .
This follows by similar argument as in step 3 but for the sake of completeness we provide the details. Using the fact that $z_{n}=T_{r_{n}}x_{n}$ and Lemma 2.7, we obtain that for $u\in F_{J}(\Gamma)\cap EP(f_{l})\cap VI(C,A_{k})\;\text{for all}~{}i,k,$

$\displaystyle\phi(z_{n},x_{n})$	$\displaystyle=$	$\displaystyle\phi(T_{r_{n}}x_{n},x_{n})$
	$\displaystyle\leq$	$\displaystyle\phi(u,x_{n})-\phi(u,T_{r_{n}}x_{n})$
	$\displaystyle=$	$\displaystyle\phi(u,x_{n})-\phi(u,z_{n}).$

From equations (3) and (3.9), we have

(3.21)

\phi(u,y_{n})\leq\alpha_{1}\phi(u,x_{n})+\alpha_{2}\phi(u,z_{n})+\alpha_{3}\phi(u,x_{n})\leq\phi(u,x_{n}).

Since $x_{n},\;y_{n},\;u_{n}\to x^{*}$ as $n\to\infty$ , from equation (3.21) we have $\phi(u,z_{n})\to\phi(u,x^{*})$ as $n\to\infty$ . Therefore, from (3), we have $\phi(u,x_{n})-\phi(u,u_{n})\to 0$ as $n\to\infty$ . Hence $\lim_{n\rightarrow\infty}\phi(z_{n},x_{n})=0.$ From Lemma 2.2, we have

(3.22)

\lim_{n\rightarrow\infty}||z_{n}-x_{n}||=0,

which implies that $z_{n}\to x^{*}~{}~{}as~{}~{}n\to\infty.$ Again, since $J$ is uniformly continuous on bounded subsets of $E$ , (3.22) implies $\|Jz_{n}-Jx_{n}\|\to 0$ . Since $r_{n}\in[a,\infty),$ we have that

(3.23)

\displaystyle\lim_{n\rightarrow\infty}\frac{||Jz_{n}-Jx_{n}||}{r_{n}}=0.

Since $z_{n}=T_{r_{n}}x_{n}$ , we have that

\frac{1}{r_{n}}\langle y-z_{n},Jz_{n}-Jx_{n}\rangle\geq-f_{n}(Jz_{n},Jy),~{}~{}\forall~{}~{}y\in C.

Let $\{n_{l}\}_{l=1}^{\infty}\subset{\mathbb{N}}$ be such that $f_{n_{l}}=f_{1}\;\forall\;l\geq 1.$ Then, using (A2), we have

(3.24)

\displaystyle\langle y-z_{n},\frac{Jz_{n}-Jx_{n}}{r_{n}}\rangle\geq-f_{1}(Jz_{n},Jy)\geq f_{1}(Jy,Jz_{n}),~{}~{}\forall~{}~{}y\in C.

Since $f_{1}(x,\cdot)$ is convex and lower-semicontinuous and $z_{n}\rightarrow x^{*}$ , it follows from equation (3.23) and inequality (3.24) that

f_{1}(Jy,Jx^{*})\leq 0,~{}~{}\forall~{}~{}y\in C.

For $t\in(0,1]$ and $y\in C$ , let $y^{*}_{t}=tJy+(1-t)Jx^{*}$ . Since $JC$ is convex, we have that $y^{*}_{t}\in JC$ and hence $f_{1}(y^{*}_{t},Jx^{*})\leq 0$ . Applying (A1) gives,

0=f_{1}(y^{*}_{t},y^{*}_{t})\leq tf_{1}(y^{*}_{t},Jy)+(1-t)f_{1}(y^{*}_{t},Jx^{*})\leq tf_{1}(y^{*}_{t},Jy),~{}~{}\forall~{}~{}y\in C.

This implies that

f_{1}(y^{*}_{t},Jy)\geq 0,~{}~{}\forall~{}~{}y\in C.

Letting $t\downarrow 0$ and using (A3), we get

f_{1}(Jx^{*},Jy)\geq 0,~{}~{}\forall~{}~{}y\in C.

Therefore, we have that $Jx^{*}\in JEP(f_{1}).$ This implies that $x^{*}\in EP(f_{1}).$ Applying similar argument, we can show that $x^{*}\in EP(f_{l})$ for $l=2,3,\dots,L.$ Hence, $x^{*}\in\cap_{l=1}^{L}EP(f_{l}).$

Step 6: Finally, we show that $x^{*}=R_{B}x.$
From Lemma 2.4 $(ii)$ , we obtain that

(3.25)

\phi(x,R_{B}x)\leq\phi(x,x^{*})-\phi(R_{B}x,x^{*})\leq\phi(x,x^{*}).

Again, using Lemma 2.4 $(ii)$ , definition of $x_{n+1}$ , and $x^{*}\in B\subset C_{n},$ we compute as follows:

	$\displaystyle\phi(x,x_{n+1})$	$\displaystyle\leq$	$\displaystyle\phi(x,x_{n+1})+\phi(x_{n+1},R_{B}x)$
		$\displaystyle=$	$\displaystyle\phi(x,R_{C_{n+1}}x)+\phi(R_{C_{n+1}}x,R_{B}x)\leq\phi(x,R_{B}x).$

Since $x_{n}\rightarrow x^{*}$ , taking limits on both sides of the last inequality, we obtain

(3.26)

\phi(x,x^{*})\leq\phi(x,R_{B}x).

Using inequalities (3.25) and (3.26), we obtain that $\phi(x,x^{*})=\phi(x,R_{B}x)$ . By the uniqueness of $R_{B}$ (Lemma 2.5), we obtain that $x^{*}=R_{B}x$ . This completes proof of the theorem. ∎

4. Applications

Corollary 4.1.

Let $E$ be a uniformly smooth and uniformly convex real Banach space with dual space $E^{*}$ and let $C$ be a nonempty closed and convex subset of $E$ such that $JC$ is closed and convex. Let $f$ be a bifunction from $JC\times JC$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ , $A:C\rightarrow E^{*},$ be a continuous monotone mapping, $T:C\rightarrow E^{*},$ be a generalized $J_{*}-$ nonexpansive and $J_{*}-$ closed map such that $B:=F_{J}(T)\cap EP(f)\cap VI(C,A)\neq\emptyset.$ Assume that $JF_{J}(T)$ is convex. Then, $\{x_{n}\}$ generated by (3.7) converges strongly to $R_{B}x$ , where $R_{B}$ is the sunny generalized nonexpansive retraction of $E$ onto $B$ .

Proof.

Set $T_{n}:=T$ for all $n\in\mathbb{N}$ , $A:=A_{i}$ for any $i=1,2,\cdots,N$ , and $f:=f_{l}$ for any $l=1,2,\cdots,L$ . Then, from remark 2.1, $\{T_{n}\}$ satisfies the NST-condition with $\{T\}$ . The conclusion follows from Theorem 3.1. ∎

Corollary 4.2.

Let $E$ be a uniformly smooth and uniformly convex real Banach space with dual space $E^{*}$ and let $C$ be a nonempty closed and convex subset of $E$ such that $JC$ is closed and convex. Let $f_{l},l=1,2,3,...,L$ be a family of bifunctions from $JC\times JC$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ , $T_{n}:C\rightarrow E^{*},n=1,2,3,...$ be an infinite family of generalized $J_{*}-$ nonexpansive maps and $\Gamma$ be a family of $J_{*}-$ closed and generalized $J_{*}-$ nonexpansive maps from $C$ to $E^{*}$ such that $\cap_{n=1}^{\infty}F_{J}(T_{n})=F_{J}(\Gamma)\neq\emptyset$ and $B:=F_{J}(\Gamma)\cap\Big{[}\cap_{l=1}^{L}EP(f_{l})\Big{]}\neq\emptyset.$ Assume that $JF_{J}(\Gamma)$ is convex and $\{T_{n}\}$ satisfies the NST-condition with $\Gamma$ . Then, $\{x_{n}\}$ generated by (3.7) converges strongly to $R_{B}x$ , where $R_{B}$ is the sunny generalized nonexpansive retraction of $E$ onto $B$ .

Proof.

Setting $A_{k}=0$ for any $k=1,2,3,...,N$ , then result follows from Theorem 3.1. ∎

Remark 4.3.

We note here that the theorem and corollaries presented above are applicable in classical Banach spaces, such as $L_{p},~{}l_{p},or~{}W^{m}_{p}(\Omega),1<p<\infty$ , where $W^{m}_{p}(\Omega)$ denotes the usual Sobolev space.

Remark 4.4.

([2]; p. 36) The analytical representations of duality maps are known in a number of Banach spaces, for example, in the spaces $L_{p},$ $l_{p},$ and $W^{p}_{m}(\Omega),$ $p\in(1,\infty)$ , $p^{-1}+q^{-1}=1$ .

Corollary 4.3.

Let $E=H$ , a real Hilbert space and let $C$ be a nonempty closed and convex subset of $H$ . Let $f_{l},l=1,2,3,...,L$ be a family of bifunctions from $C\times C$ to $\mathbb{R}$ satisfying $(A1)-(A4)$ , $T_{n}:C\rightarrow H,n=1,2,3,...$ be an infinite family of nonexpansive maps, $A_{k}:C\rightarrow H,k=1,2,3,...,N$ be a finite family of continuous monotone mappings and $\Gamma$ be a family of closed and generalized nonexpansive maps from $C$ to $H$ such that $\cap_{n=1}^{\infty}F(T_{n})=F(\Gamma)\neq\emptyset$ and $B:=F(\Gamma)\cap\Big{[}\cap_{l=1}^{L}EP(f_{l})\Big{]}\cap\Big{[}\cap_{k=1}^{N}VI(C,A_{k})\Big{]}\neq\emptyset.$ Assume that $\{T_{n}\}$ satisfies the NST-condition with $\Gamma$ . Let $\{x_{n}\}$ be generated by:

(4.27)

\begin{cases}&x_{1}=x\in C;C_{1}=C,\cr&z_{n}:=\{z\in C:f_{n}(z,y)+\frac{1}{r_{n}}\langle y-z,z-x_{n}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\},\cr&u_{n}:=\{z\in C:\langle y-z,A_{n}z\rangle+\frac{1}{r_{n}}\langle y-z,z-x_{n}\rangle\geq 0,~{}~{}\forall~{}~{}y\in C\},\cr&y_{n}=\alpha_{1}Jx_{n}+\alpha_{2}z_{n}+\alpha_{3}T_{n}u_{n},\cr&C_{n+1}=\{z\in C_{n}:||z-y_{n}||\leq||z-x_{n}||\},\cr&x_{n+1}=P_{C_{n+1}}x,\end{cases}

for all $n\in\mathbb{N},\;\alpha_{1},\alpha_{2},\alpha_{3}\in(0,1)$ such that $\alpha_{1}+\alpha_{2}+\alpha_{3}=1$ , $\{r_{n}\}\subset[a,\infty)$ for some $a>0$ , $A_{n}=A_{n(mod~{}N)}$ and $f_{n}(\cdot,\cdot)=f_{n(mod~{}L)}(\cdot,\cdot)$ . 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 3.1. ∎

Example 4.1.

Let $E=l_{p}$ , $1<p<\infty$ , $\frac{1}{p}+\frac{1}{q}=1$ , and $C=\overline{B_{l_{p}}}(0,1)$ = $\{x\in l_{p}:||x||_{l_{p}}\leq 1\}$ . Then $JC=\overline{B_{l_{q}}}(0,1)$ . Let $f:JC\times JC\longrightarrow\mathbb{R}$ defined by $f(x^{*},y^{*})=\langle J^{-1}x^{*},x^{*}-y^{*}\rangle$ $\forall$ $x^{*}\in JC$ , $A:C\longrightarrow l_{q}$ defined by $Ax=J(x_{1},x_{2},x_{3},\cdots)$ $\forall$ $x=(x_{1},x_{2},x_{3},\cdots)\in C$ , $T:C\longrightarrow l_{q}$ defined by $Tx=J(0,x_{1},x_{2},x_{3},\cdots)$ $\forall$ $x=(x_{1},x_{2},x_{3},\cdots)\in C$ , and $T_{n}:C\longrightarrow l_{q}$ defined by $T_{n}x=\alpha_{n}Jx+(1-\alpha_{n})Tx,~{}\forall n\geq 1,~{}\forall~{}x\in C,\alpha_{n}\in(0,1)\text{ such that }1-\alpha_{n}\geq\frac{1}{2}$ . Then $C$ , $JC$ , $f$ , $A$ , $T$ , and $T_{n}$ satisfy the conditions of Corollary 4.1. Moreover, $0\in F_{J}(\Gamma)\cap EP(f)\cap VI(C,A)$ .

5. Conclusion

Our theorem and its applications complement, generalize, and extend results of Uba et al. [22], Zegeye and Shahzad [26], Kumam [13], Qin and Su [18], and Nakajo and Takahashi [19]. Theorem 3.1 is a complementary analogue and extension of Theorem 3.2 of [26] in the following sense: while Theorem 3.2 of [26] is proved for a finite family of self-maps in uniformly smooth and strictly convex real Banach space which has the Kadec–Klee property, Theorem 3.1 is proved for countable family of non-self maps in uniformly smooth and uniformly convex real Banach space; in Hilbert spaces, Corollary 4.3 is an extension of Corollary 3.5 of [26] from finite family of nonexpansive self-maps to countable family of nonexpansive non-self maps. Additionally, Theorem 3.1 extends and generalizes Theorem 3.7 of [22] in the following sense: while Theorem 3.7 of [22] studied equilibrium problem and countable family of generalized $J_{*}-$ nonexpansive non-self maps, Theorem 3.1 studied finite family of equilibrium and variational inequality problems and countable family of generalizes $J_{*}-$ nonexpansive non-self maps; corollary 4.2 generalized Theorem 3.7 of [22] to a finite family of equilibrium problems and countable family of generalized $J_{*}-$ nonexpansive non-self maps. Furthermore, Corollary 4.1 extends Theorem 3.1 of [13] from Hilbert spaces to a more general uniformly smooth and uniformly convex Banach spaces and to a more general class of continuous monotone mappings. Finally, Corollary 4.1 improves and extends the results in [18, 19] from a nonexpansive self-map to a generalized $J_{*}-$ nonexpansive non-self map.

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