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Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 59 (2014)
Abstract
In this paper, we propose and analyze an explicit type algorithm for finding a common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two countable families of total quasiφasymptotically nonexpansive mappings in a Banach space E. As an application of our result, we suggest a framework for finding a common solution of a finite family of generalized equilibrium problems and common zeros of two finite families of maximal monotone operators on E.
MSC:47H05, 47H10, 47H15, 47J25, 49M05.
1 Introduction and preliminaries
Let E be a real Banach space with the norm $\parallel \cdot \parallel $ and ${E}^{\ast}$ be its dual. Let C be a nonempty subset of E and $T:C\to C$ be a mapping. We denote by $F(T)=\{x\in C:x=Tx\}$ the set of fixed points of T. We symbolize weak convergence and strong convergence of a sequence $\{{x}_{n}\}$ in E as ${x}_{n}\rightharpoonup x$ and ${x}_{n}\to x$, respectively. Let $f:C\times C\to \mathbb{R}$ (the set of reals) be a bifunction and $A:C\to {E}^{\ast}$ be a nonlinear mapping. A generalized equilibrium problem is to find the set
where $\u3008\cdot \phantom{\rule{0.2em}{0ex}},\cdot \u3009$ stands for the duality product.
Note that:

(i)
if $A\equiv 0$, then problem (1.1) reduces to the following equilibrium problem $EP(f)$:
$$\text{find}x\in C\text{such that}f(x,y)\ge 0\text{for all}y\in C;$$ 
(ii)
if $f\equiv 0$, then problem (1.1) reduces to the classical variational inequality problem $VI(C,A)$:
$$\text{find}x\in C\text{such that}\u3008Ax,yx\u3009\ge 0\text{for all}y\in C.$$
The equilibrium problem provides a unified approach to finding a solution of a large number of problems arising in physics, optimization, economics and fixed point problems [1]. Moreover, the generalized equilibrium problem addresses monotone inclusion problems, variational inequality problems, minimization problems and vector equilibrium problem [2–4]. Since an algorithmic construction plays a key role in solving nonlinear equations in various fields of investigation, numerous implicit and explicit algorithms have been developed for the approximate solution of nonlinear equations as well as for the approximation of fixed points of various mappings [5–7].
Recall that a Banach space E is said to be:

(i)
strictly convex if for all $x,y\in {S}_{E}:=\{z\in E:\parallel z\parallel =1\}$ with $x\ne y$, we have $\parallel x+y\parallel <2$;

(ii)
uniformly convex if for any $\u03f5\in (0,2]$, there exists $\delta >0$ such that
$$x,y\in {S}_{E},\phantom{\rule{1em}{0ex}}\parallel xy\parallel \ge \u03f5\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\parallel \frac{x+y}{2}\parallel \le 1\delta .$$
A Banach space E is said to have the KadecKlee property if for any sequence $\{{x}_{n}\}$ in E with ${x}_{n}\rightharpoonup {x}_{0}\in E$ and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel =\parallel {x}_{0}\parallel $, we have ${x}_{n}\to {x}_{0}$. Note that every uniformly convex Banach space E is strictly convex and enjoys the KadecKlee property but the converse is not true.
Furthermore, define $h:{S}_{E}\times {S}_{E}\times \mathbb{R}\setminus \{0\}\to \mathbb{R}$ by
for $x,y\in {S}_{E}$ and $t\in \mathbb{R}\setminus \{0\}$. The norm of E is said to be Gâteaux differentiable if ${lim}_{t\to 0}h(x,y,t)$ exists for each $x,y\in {S}_{E}$ and in this case E is smooth.
The normalized duality mapping $J:E\to {E}^{\ast}$ is defined by
It is remarked that the setvalued mapping J is nonempty, closed and convex in a real Banach space whereas J is singlevalued in a reflexive, strictly convex and smooth Banach space. Furthermore, ${J}^{1}:{E}^{\ast}\to E$, the inverse of the normalized duality mapping J, is also a duality mapping in a uniformly convex and smooth Banach space. Both J and ${J}^{1}$ are uniformly normtonorm continuous on each bounded subset of E and ${E}^{\ast}$, respectively. If E is reflexive and strictly convex, then ${J}^{1}$ is normtoweak continuous. For more details, see [8, 9].
The Lyapunov functional $\phi :E\times E\to \mathbb{R}$ is defined by
It is obvious from the definition of φ that $\phi (x,y)\ge 0$ for all $x,y\in E$. In a real Hilbert space, $\phi (x,y)={\parallel xy\parallel}^{2}$ for all $x,y\in E$. For details, see [5, 10].
Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty, closed and convex subset of E. Then, for arbitrarily fixed $x\in E$, there exists a unique point ${y}_{x}\in C$ such that $\phi (x,{y}_{x})={min}_{y\in C}\phi (x,y)$. Following the notation of [5], we let ${\mathrm{\Pi}}_{C}(x)={y}_{x}$ and call ${\mathrm{\Pi}}_{C}$ a generalized projection onto C. Note that the generalized projection operator coincides with the metric projection in a Hilbert space.
A point $x\in C$ is said to be an asymptotic fixed point [11] of $T:C\to C$ if there exists a sequence $\{{x}_{n}\}\subset C$ such that ${x}_{n}\rightharpoonup x$ and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0$. The set of all asymptotic fixed points of T is denoted by $\stackrel{\u02c6}{F}(T)$.
Recall that a mapping $T:C\to C$ is:

(i)
nonexpansive if $\parallel TxTy\parallel \le \parallel xy\parallel $ for all $x,y\in C$;

(ii)
relatively nonexpansive if $\stackrel{\u02c6}{F}(T)=F(T)\ne \mathrm{\varnothing}$ and $\phi (p,Tx)\le \phi (p,x)$ for all $p\in F(T)$ and $x\in C$;

(iii)
quasiφnonexpansive if $F(T)\ne \mathrm{\varnothing}$ and $\phi (p,Tx)\le \phi (p,x)$ for all $p\in F(T)$ and $x\in C$;

(iv)
quasiφasymptotically nonexpansive if there exists a real sequence $\{{k}_{n}\}$ with ${k}_{n}\ge 1$; ${k}_{n}\stackrel{n\to \mathrm{\infty}}{\to}1$ and $F(T)\ne \mathrm{\varnothing}$ such that $\phi (p,{T}^{n}x)\le {k}_{n}\phi (p,x)$ for all $n\ge 1$, $p\in F(T)$ and $x\in C$;

(v)
total quasiφasymptotically nonexpansive if there exist nonnegative real sequences $\{{k}_{n}\}$ and $\{{\eta}_{n}\}$ with ${k}_{n}\stackrel{n\to \mathrm{\infty}}{\to}0$, ${\eta}_{n}\stackrel{n\to \mathrm{\infty}}{\to}0$ and $F(T)\ne \mathrm{\varnothing}$ such that
$$\phi (p,{T}^{n}x)\le \phi (p,x)+{k}_{n}\xi (\phi (p,x))+{\eta}_{n}\phantom{\rule{1em}{0ex}}\text{for all}n\ge 1,p\in F(T)\text{and}x\in C,$$
where $\xi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a strictly increasing continuous function with $\xi (0)=0$.
It is worth mentioning that the class of total quasiφasymptotically nonexpansive mappings properly contains the mappings defined in (i)(iv), but the converse is not true.
A mapping $T:C\to C$ is said to be uniformly LLipschitzian if
Recently, numerous attempts have been made to guarantee strong convergence through explicit and implicit algorithms for finding a common solution of the set of fixed points of (relatively nonexpansive, quasiφnonexpansive, quasiφasymptotically nonexpansive, total quasiφasymptotically nonexpansive) mappings and the set of solutions of equilibrium problems; see [12–21] and the references cited therein.
In 2008, Takahashi and Zembayashi [21] introduced an explicit algorithm based on the shrinking projection method for finding a common solution of the set of fixed points of a relatively nonexpansive mapping T and the set of solutions of an equilibrium problem. Their algorithm reads as follows:
where J is the duality mapping on E and ${\mathrm{\Pi}}_{C}$ is the generalized projection from E onto C. They proved that the sequence $\{{x}_{n}\}$ generated by (1.2) converges strongly to ${\mathrm{\Pi}}_{F(T)\cap EP(f)}{x}_{0}$ under some appropriate conditions.
In 2010, Chang et al. [12] proved a strong convergence theorem for finding a common element of the set of solutions for generalized equilibrium problem (1.1) and the set of common fixed points for a pair of relatively nonexpansive mappings in Banach spaces. Their algorithm reads as follows:
where J, T, S and ${\mathrm{\Pi}}_{C}$ are as in (1.2). The authors showed that the sequence $\{{x}_{n}\}$ generated by (1.3) converges strongly to ${\mathrm{\Pi}}_{F(S)\cap F(T)\cap GEP(f)}{x}_{0}$ under some appropriate conditions.
In 2009, Wattanawitton and Kumam [22] approximated a common solution for a pair of relatively quasinonexpansive mappings and an equilibrium problem. Recently, Qin et al. [18] established strong convergence results for a pair of asymptotically quasiφnonexpansive mappings in a Banach space. It is worth mentioning that the hybrid algorithms proposed in [12, 18, 22] are computationally complex. Therefore, it is natural to have improved and computationally simpler counterparts.
Quite recently, Zuo et al. [16] proposed a hybrid algorithm for total quasiφasymptotically nonexpansive mappings and established strong convergence results in a Banach space. Moreover, they characterized such strong convergence results by using the notion of Mosco convergence; see also [23]. Inspired and motivated by the work of Takahashi and Zembayashi [21], Chang et al. [12] and Zuo et al. [16], we aim to introduce and analyze a general algorithm based on the shrinking projection method for finding a common element of the set of common solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two countable families of total quasiφasymptotically nonexpansive mappings. We also characterize the set of common solutions for families of total quasiφasymptotically nonexpansive mappings and equilibrium problems in terms of Mosco convergence.
We now introduce the notion of Mosco convergence.
Let $\{{C}_{n}\}$ be a sequence of nonempty closed convex subsets of a reflexive Banach space E. We denote the set of all strong limit points of $\{{C}_{n}\}$ by $sL{i}_{n}{C}_{n}$, that is, $x\in sL{i}_{n}{C}_{n}$ if and only if there exists $\{{x}_{n}\}\subset E$ such that $\{{x}_{n}\}$ converges strongly to x and that ${x}_{n}\in {C}_{n}$ for all n. Similarly, we define the set of all weak subsequential limit points by $wL{s}_{n}{C}_{n}$; $y\in wL{s}_{n}{C}_{n}$ if and only if there exist a subsequence $\{{C}_{{n}_{i}}\}$ of $\{{C}_{n}\}$ and a sequence $\{{y}_{i}\}\subset E$ such that $\{{y}_{i}\}$ converges weakly to y and ${y}_{i}\in {C}_{{n}_{i}}$ for all i. If ${C}_{0}$ satisfies ${C}_{0}=sL{i}_{n}{C}_{n}=wL{s}_{n}{C}_{n}$, then we say that $\{{C}_{n}\}$ converges to ${C}_{0}$ in the sense of Mosco and we write ${C}_{0}=M\text{}{lim}_{n}{C}_{n}$. By definition, it always holds that $sL{i}_{n}{C}_{n}\subset wL{s}_{n}{C}_{n}$. Therefore, to prove ${C}_{0}=M\text{}{lim}_{n}{C}_{n}$, it suffices to show that $wL{s}_{n}{C}_{n}\subset {C}_{0}\subset sL{i}_{n}{C}_{n}$. One of the simplest examples of Mosco convergence is a decreasing sequence $\{{C}_{n}\}$ with respect to inclusion. The Mosco limit of such a sequence is ${\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$. For more details, we refer to [23, 24].
For a relation between a sequence of closed convex sets and the corresponding generalized projections, we state the following lemma which plays a key role in our main result.
Lemma 1.1 ([23], Theorem 2.2)
Let E be a smooth, reflexive and strictly convex Banach space having the KadecKlee property. Let $\{{C}_{n}\}$ be a sequence of nonempty closed subset of E. If ${C}_{0}=M\text{}{lim}_{n}{C}_{n}$ exists and is nonempty, then $\{{\mathrm{\Pi}}_{{C}_{n}}x\}$ converges strongly to ${\mathrm{\Pi}}_{{C}_{0}}x$ for each $x\in C$.
The following two results can be found as Remark 7.3 in [5].
Lemma 1.2 Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E, let ${x}_{0}\in E$ and let $x\in C$. Then ${\mathrm{\Pi}}_{C}{x}_{0}=x$ if and only if
Lemma 1.3 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let $x\in E$. Then
The following wellknown results are also needed in the sequel for the development of our main result.
Lemma 1.4 ([25], Proposition 2)
Let E be a uniformly convex and smooth Banach space and let $\{{x}_{n}\}$, $\{{y}_{n}\}$ be two sequences of E. If $\phi ({x}_{n},{y}_{n})\to 0$ and either $\{{x}_{n}\}$ or $\{{y}_{n}\}$ is bounded, then ${x}_{n}{y}_{n}\to 0$.
Lemma 1.5 ([6], Lemma 1.4)
Let E be a uniformly convex Banach space and let ${B}_{r}[0]$ be a closed ball in E. Then there exists a continuous strictly increasing convex function $g:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $g(0)=0$ such that
for all $x,y\in {B}_{r}[0]$ and $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$.
For solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditions (cf. [2, 26]):
(A1) $f(x,x)=0$ for all $x\in C$;
(A2) f is monotone, i.e., $f(x,y)+f(y,x)\le 0$ for all $x,y\in C$;
(A3) ${lim\hspace{0.17em}sup}_{t\downarrow 0}f(tz+(1t)x,y)\le f(x,y)$ for all $x,y,z\in C$;
(A4) $f(x,\cdot )$ is convex and lower semicontinuous for all $x\in C$.
The following result is stated as Lemma 2.7 in [21] (see also [2]).
Lemma 1.6 Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let $f:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)(A4), let $r>0$ and $x\in E$. Then there exists $z\in C$ such that
Lemma 1.7 ([21], Lemma 2.8)
Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E. Let $f:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)(A4). For $r>0$ and $x\in E$, define a mapping ${V}_{r}:E\to C$ by
for all $x\in C$. Then the following hold:

(1)
$EP(f)$ is closed and convex;

(2)
${V}_{r}$ is singlevalued;

(3)
${V}_{r}$ is firmly nonexpansivetype mapping, i.e.,
$$\u3008{V}_{r}x{V}_{r}y,J{V}_{r}xJ{V}_{r}y\u3009\le \u3008{V}_{r}x{V}_{r}y,JxJy\u3009\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in E,$$ 
(4)
$F({V}_{r})=EP(f)$.
2 Main results
Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the KadecKlee property. Let ${T}_{i},{S}_{i}:C\to C$ be two countable families of uniformly LLipschitzian and uniformly total quasiφasymptotically nonexpansive mappings with sequences $\{{k}_{n}^{T}\}$, $\{{\eta}_{n}^{T}\}$ and $\{{k}_{n}^{S}\}$, $\{{\eta}_{n}^{S}\}$, respectively. Let ${f}_{n}:C\times C\to \mathbb{R}$, $n=1,2,3,\dots ,N$, be a finite family of bifunctions such that ${f}_{n}\equiv {f}_{n}(modN)$ (here the modN function takes values in $\{1,2,3,\dots ,N\}$).
Algorithm
where ${\alpha}_{n,i}+{\beta}_{n,i}+{\gamma}_{n,i}=1$ and ${\mathrm{\Theta}}_{n}:=(1{\alpha}_{n,i})\{{k}_{n}{sup}_{u\in F}\xi (\phi (u,{x}_{n}))+{\eta}_{n}\}$.
Theorem 2.1 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the KadecKlee property and let ${T}_{i},{S}_{i}:C\to C$ be two countable families of uniformly LLipschitzian and uniformly total quasiφasymptotically nonexpansive mappings with sequences $\{{k}_{n}\}$ and $\{{\eta}_{n}\}$, where $0<L:=max\{{L}_{i}^{T},{L}_{i}^{S}\}$ for each $i\ge 1$, $\{{k}_{n}\}:={max}_{n\ge 1}\{\{{k}_{n}^{T}\},\{{k}_{n}^{S}\}\}$ and $\{{\eta}_{n}\}:={max}_{n\ge 1}\{\{{\eta}_{n}^{T}\},\{{\eta}_{n}^{S}\}\}$. Let ${f}_{n}:C\times C\to \mathbb{R}$, $n=1,2,3,\dots ,N$, be a finite family of bifunctions satisfying (A1)(A4) such that ${f}_{n}={f}_{n}(modN)$. Let $a\le {\alpha}_{n,i},{\beta}_{n,i},{\gamma}_{n,i}\le b$ for some $a,b\in (0,1)$ and $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ satisfying:
(C1) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$.
Assume that $F:=[{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})]\cap [{\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})]\cap [{\bigcap}_{n=1}^{N}GEP({f}_{n})]\ne \mathrm{\varnothing}$. Then the sequence $\{{x}_{n}\}$ generated by (2.1) converges strongly to $x={\mathrm{\Pi}}_{F}{x}_{0}$, where ${\mathrm{\Pi}}_{F}$ is the generalized projection of E onto F.
Proof For each $n=1,2,3,\dots ,N$, define ${G}_{n}:C\times C\to \mathbb{R}$ by ${G}_{n}(x,y)={f}_{n}(x,y)+\u3008{A}_{n}x,yx\u3009$. Then ${G}_{n}$ coincides with the classical equilibrium problem and satisfies (A1)(A4). Now, we show that algorithm (2.1) is well defined. By Lemma 1.7(1) we have that F is closed and convex. Next we show that ${C}_{n+1}$ is closed and convex. Clearly, ${C}_{0}=C$ is closed and convex. Suppose that ${C}_{k}$ is closed and convex for $k\in \mathbb{N}$ (set of naturals). For each $z\in {C}_{k}$, we observe that
This implies that ${C}_{k+1}$ is closed and convex. This implies, inductively, that ${C}_{n+1}$ is closed and convex for all $n\ge 0$. Now we prove that $F\subset {C}_{n+1}$ for each $n\ge 0$. Obviously, $F\subset {C}_{0}=C$. Suppose that $F\subset {C}_{k}$ for some $k\in \mathbb{N}$. It follows from Lemma 1.6 that ${u}_{k,i}={V}_{{r}_{k}}{y}_{k,i}$ and ${V}_{{r}_{k}}$ is relatively nonexpansive. Hence, for any $p\in F$, we have
That is,
where ${\mathrm{\Theta}}_{k}:=(1{\alpha}_{k,i})\{{k}_{k}{sup}_{u\in F}\xi (\phi (u,{x}_{k}))+{\eta}_{k}\}$. This shows that $p\in {C}_{k+1}$; consequently, $F\subset {C}_{k+1}$. By induction, we also get that $F\subset {C}_{n+1}$ for all $n\ge 0$ with ${\mathrm{\Theta}}_{n}:=(1{\alpha}_{n,i})\{{k}_{n}{sup}_{u\in F}\xi (\phi (u,{x}_{n}))+{\eta}_{n}\}$. Since $F\ne \mathrm{\varnothing}$ and ${C}_{n+1}$ is a nonempty closed convex subset of E, hence both ${\mathrm{\Pi}}_{{C}_{n+1}}x$ and $\{{x}_{n}\}$ are well defined.
Note that $\phi ({x}_{n},{x}^{\ast})$ is nondecreasing. In fact, from the definition of ${C}_{n+1}$, we conclude that ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}$ and ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}\in {C}_{n+1}\subset {C}_{n}$; hence
Moreover, from Lemma 1.3 we get that
for each $p\in F$.
So $\phi ({x}_{n},{x}^{\ast})$ is nondecreasing and bounded. This implies that ${lim}_{n\to \mathrm{\infty}}\phi ({x}_{n},{x}^{\ast})$ exists.
Let $m\in {\mathbb{Z}}^{+}$ (a set of positive integers). Then Lemma 1.3 implies
Letting $n\to \mathrm{\infty}$, we have $\phi ({x}_{n+m},{x}_{n})\to 0$. By Lemma 1.4, we have
Hence $\{{x}_{n}\}$ is Cauchy. Since C is a closed subset of the Banach space E, we can assume that there exists a point $x\in C$ such that
Note that $\{{C}_{n}\}$ is a decreasing sequence of closed convex subsets of E with ${C}_{0}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$ is nonempty. That is,
Hence Lemma 1.1 asserts that $\{{\mathrm{\Pi}}_{{C}_{n}}{x}_{0}\}$ converges to ${\mathrm{\Pi}}_{{C}_{0}}{x}_{0}$.
In what follows, we show that:
Step 1. $x\in [{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})]\cap [{\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})]$;
Step 2. $x\in [{\bigcap}_{n=1}^{N}EP({G}_{n})]$;
Step 3. $x={\mathrm{\Pi}}_{F}{x}_{0}$.
Proof of Step 1. As ${x}_{n+1}\in {C}_{n+1}$, so ${sup}_{i\ge 1}\phi ({x}_{n+1},{u}_{n,i})\le \phi ({x}_{n+1},{x}_{n})+{\mathrm{\Theta}}_{n}$. It follows from (2.3) and the fact that ${\mathrm{\Theta}}_{n}\stackrel{n\to \mathrm{\infty}}{\to}0$, we get ${lim}_{n\to \mathrm{\infty}}\phi ({x}_{n+1},{u}_{n,i})=0$ for all $i\ge 1$. Again by Lemma 1.4, we have
Observe that the following implication of the triangle inequality
yields that
As J is uniformly normtonorm continuous on bounded sets, so we have
Moreover, it follows from (2.4) and (2.6) that
From (2.2), we know that $\phi (p,{y}_{n,i})\le \phi (p,{x}_{n})+{\mathrm{\Theta}}_{n}$ for all $i\ge 1$ and for all $p\in F$. So, by Lemma 1.3, we have
Letting $n\to \mathrm{\infty}$ in the above estimate and using (2.7), we have ${lim}_{n\to \mathrm{\infty}}\phi ({u}_{n,i},{y}_{n,i})=0$ for all $i\ge 1$; consequently, Lemma 1.4 asserts that
From (2.6) and (2.9), we have
Hence, we conclude that
On the other hand, from Lemma 1.5, we have
Since $a\le {\alpha}_{n,i},{\beta}_{n,i}$, rearranging the terms of the above estimate and simplifying, we get
Reasoning as above and then utilizing (2.6), we get that
Note that g is a continuous function and $g(0)=0$, so we have
In view of estimate (2.4), we conclude that $J{x}_{n}\stackrel{n\to \mathrm{\infty}}{\to}Jx$. Hence, from (2.11) we obtain
Since ${J}^{1}$ is uniformly normtonorm continuous, (2.11) implies that
Observe that
So, we conclude from (2.4) and (2.13) that
Since each ${T}_{i}$ is uniformly LLipschitzian, we have
Hence, from (2.3) and (2.13), we obtain that
Moreover, it yields that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n+1}{x}_{n}x\parallel =0$, $i\ge 1$; consequently, we have ${T}_{i}({T}_{i}^{n}{x}_{n})\stackrel{n\to \mathrm{\infty}}{\to}x$, $i\ge 1$. So we infer that $x\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$.
Reasoning as above, one can also show that $x\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i})$.
Proof of Step 2. We first show that $x\in EP({G}_{1})$, where ${G}_{1}={G}_{{n}_{j}}$ for some $j\ge 1$. In view of estimate (2.9) and (C1), i.e., ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, we observe that
From ${u}_{{n}_{j},i}={V}_{{r}_{{n}_{j}}}{y}_{{n}_{j},i}$, for all $n\ge 0$ and $i\ge 1$, we have
Using (A2), the above estimate yields that
From ${u}_{{n}_{j},i}\to x$ and (A4), we obtain ${G}_{1}(y,x)\le 0$ for all $y\in C$. Let ${y}_{t}=ty+(1t){x}_{0}$ for $0<t<1$ and $y\in C$. Then ${y}_{t}\in C$ and hence ${G}_{1}({y}_{t},x)\le 0$. From (A1) and (A4), we have $0={G}_{1}({y}_{t},{y}_{t})\le t{G}_{1}({y}_{t},y)+(1t){G}_{1}({y}_{t},x)\le t{G}_{1}({y}_{t},y)$. Thus, ${G}_{1}({y}_{t},y)\ge 0$. From (A3), we have ${G}_{1}(x,y)\ge 0$ for all $y\in C$. Hence $x\in EP({G}_{1})$. In a similar fashion, we have some $k\ge 1$ such that ${G}_{2}={G}_{{n}_{k}}$ and $x\in EP({G}_{2})$. Therefore, $x\in {\bigcap}_{n=1}^{N}EP({G}_{n})$ and hence $x\in F$.
Proof of Step 3. Lemma 1.1 asserts that the sequence $\{{x}_{n}\}=\{{\mathrm{\Pi}}_{{C}_{n}}{x}_{0}\}$ converges to ${\mathrm{\Pi}}_{{C}_{0}}{x}_{0}$. Let $p={\mathrm{\Pi}}_{{C}_{0}}{x}_{0}\in F$ and F is a nonempty closed convex subset of ${C}_{0}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$. Therefore $p={\mathrm{\Pi}}_{F}{x}_{0}$. It suffices to show that $x=p$. For this, we reason as follows:
Since ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}$ and $p\in F\subset {C}_{n}$, we have
Since the norm is weakly lower semicontinuous, we have
From the definition of ${\mathrm{\Pi}}_{F}$, we have $x=p$. Hence ${lim}_{k\to \mathrm{\infty}}\phi ({x}_{{n}_{k}},{x}_{0})=\phi (p,{x}_{0})$. Therefore, we have
Since E has the KadecKlee property, we have that ${lim}_{k\to \mathrm{\infty}}{x}_{{n}_{k}}=x={\mathrm{\Pi}}_{F}{x}_{0}$.
The arbitrariness of $\{{x}_{{n}_{k}}\}$ implies that $\{{x}_{n}\}$ converges strongly to $x={\mathrm{\Pi}}_{F}{x}_{0}$. □
Remark 2.2 If ${\{{T}_{i}\}}_{i=1}^{N}$ and ${\{{S}_{i}\}}_{i=1}^{N}$ are finite families in Theorem 2.1, then its conclusion can be strengthened as follows:
The sequence $\{{x}_{n}\}$ generated by (2.1) converges strongly to some $x\in F$ if and only if ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0$, where $x={\mathrm{\Pi}}_{F}{x}_{0}$ and $d(q,F)=inf\{d(q,y):y\in F\}$.
Proof The necessity is obvious. Conversely, suppose that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0$. Since $\phi ({x}_{n},{x}^{\ast})$ is bounded and ${lim}_{n\to \mathrm{\infty}}\phi ({x}_{n},{x}^{\ast})$ exists (established above). Moreover, Lemma 1.3 and the following estimate
imply $\phi ({x}_{n+m},{x}_{n})\to 0$ when $m,n\to \mathrm{\infty}$. Hence, Lemma 1.4 asserts that $\parallel {x}_{n+m}{x}_{n}\parallel \to 0$. Thus, $\{{x}_{n}\}$ is a Cauchy sequence. Therefore, there exists a point $x\in C$ such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. Thus,
Since F is closed, this implies $x\in F$. Now, as in the proof of Step 3 in Theorem 2.1, we have $x={\mathrm{\Pi}}_{F}{x}_{0}$. □
To proceed further, we need the following concept.
Let f be a nondecreasing selfmapping on $[0,\mathrm{\infty})$ with $f(0)=0$ and $f(t)>0$ for all $t\in (0,\mathrm{\infty})$. Let $\{{T}_{i}:i\in I\}$ and $\{{S}_{i}:i\in I\}$ be two finite families of total quasiφasymptotically nonexpansive mappings on C with $F\ne \mathrm{\varnothing}$. Then the two families are said to satisfy condition (I) on C if
As an application of Theorem 2.1 to the case of finite families of mappings ${\{{T}_{i}\}}_{i=1}^{N}$ and ${\{{S}_{i}\}}_{i=1}^{N}$, we obtain the following new strong convergence result.
Theorem 2.3 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the KadecKlee property and let ${T}_{i},{S}_{i}:C\to C$ be two finite families of uniformly LLipschitzian and uniformly total quasiφasymptotically nonexpansive mappings with sequences $\{{k}_{n}\}$ and $\{{\eta}_{n}\}$, where $0<L:=max\{{L}_{i}^{T},{L}_{i}^{S}\}$ for each $i\ge 1$, $\{{k}_{n}\}:={max}_{n\ge 1}\{\{{k}_{n}^{T}\},\{{k}_{n}^{S}\}\}$ and $\{{\eta}_{n}\}:={max}_{n\ge 1}\{\{{\eta}_{n}^{T}\},\{{\eta}_{n}^{S}\}\}$. Let ${f}_{n}:C\times C\to \mathbb{R}$, $n=1,2,3,\dots ,N$, be a finite family of bifunctions satisfying (A1)(A4) such that ${f}_{n}={f}_{n}(modN)$. Let $a\le {\alpha}_{n,i},{\beta}_{n,i},{\gamma}_{n,i}\le b$ for some $a,b\in (0,1)$ and $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ satisfying
(C1) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$.
Assume that ${T}_{i}$ and ${S}_{i}$ satisfy condition (I) and $F\ne \mathrm{\varnothing}$. Then the sequence $\{{x}_{n}\}$ generated by (2.1) converges strongly to some $x\in F$, where $x={\mathrm{\Pi}}_{F}{x}_{0}$.
Proof It follows from Theorem 2.1 that
Since $\{{T}_{i}^{n}:i\in I\}$ and $\{{S}_{i}^{n}:i\in I\}$ satisfy condition (I), so we have either
or
This implies that ${lim}_{n\to \mathrm{\infty}}f(d({u}_{n,i},F))=0$. Since f is nondecreasing and $f(0)=0$, we have ${lim}_{n\to \mathrm{\infty}}d({u}_{n,i},F)=0$. The rest of the proof follows from Remark 2.2 and is, therefore, omitted. □
As another application of Theorem 2.1, we establish a result for finding a common element in the set of solutions of a finite family of generalized equilibrium problems and the set of common zeros of two finite families of maximal monotone operators on a Banach space.
First we recall some preliminary concepts as follows.
A multivalued operator $T:E\to {E}^{\ast}$ is said to be monotone if for any $x,y\in E$ and ${x}^{\ast},{y}^{\ast}\in {E}^{\ast}$ with ${x}^{\ast}\in Tx$ and ${y}^{\ast}\in Ty$ it holds that
A point $u\in E$ satisfying $0\in Tu$ is called a zero of T and the set of all such points is denoted by ${T}^{1}0$. T is said to be maximal monotone if T has no monotone extension. Equivalently, T is maximal monotone if the graph of T, i.e., $G(T)=\{(x,y):y\in Tx\}$, is not properly contained in the graph of any other monotone operator (cf. [27], p.264).
Let E be a strictly convex, reflexive and smooth Banach space and let $T:E\to {E}^{\ast}$ be a maximal monotone operator. For a positive real number r, we can define a singlevalued mapping ${J}_{r}:E\to D(T)$ by ${J}_{r}(x)={(J+rT)}^{1}Jx$ for each $x\in E$. This mapping is called resolvent of T for $r>0$. It is known that if T is maximal monotone, then ${T}^{1}0=F({J}_{r})$ for each $r>0$. Moreover, ${T}^{1}0$ is a closed convex subset of E. For each $r>0$, we can define Yosida approximation of T by ${Y}_{r}x=\frac{1}{r}(JxJ{J}_{r}x)$ for all $x\in E$. We know that ${Y}_{r}x\in T({J}_{r}x)$; for more details, see [9].
Theorem 2.4 Let C be a nonempty closed convex subset of a strictly convex, reflexive and smooth Banach space E having the KadecKlee property. Let ${T}_{i},{S}_{i}:E\to {E}^{\ast}$, $i=1,2,3,\dots ,N$, be two finite families of maximal monotone operators and let ${J}_{r,i}^{{T}_{i}}$ and ${J}_{r,i}^{{S}_{i}}$ be the corresponding finite families of resolvents of ${T}_{i}$ and ${S}_{i}$, respectively, where $r>0$. Let ${f}_{n}:C\times C\to \mathbb{R}$, $n=1,2,3,\dots ,N$, be a finite family of bifunctions satisfying (A1)(A4). Let $\{{\alpha}_{n,i}\}$, $\{{\beta}_{n,i}\}$ and $\{{\gamma}_{n,i}\}$ be three sequences in $[0,1]$ such that ${\alpha}_{n,i}+{\beta}_{n,i}+{\gamma}_{n,i}=1$ and $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ satisfying
(C1) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$.
Assume that $F:=[{\bigcap}_{i=1}^{N}{T}_{i}^{1}]\cap [{\bigcap}_{i=1}^{N}{S}_{i}^{1}]\cap [{\bigcap}_{n=1}^{N}GEP({f}_{n})]\ne \mathrm{\varnothing}$. Then the sequence $\{{x}_{n}\}$ generated by
converges strongly to ${\mathrm{\Pi}}_{F}{x}_{0}$, where ${\mathrm{\Pi}}_{F}$ is the generalized projection of E onto F and ${\mathrm{\Theta}}_{n}:=(1{\alpha}_{n,i})\{{k}_{n}{sup}_{u\in F}\xi (\phi (u,{x}_{n}))+{\eta}_{n}\}$.
Proof The first part of this proof  ${J}_{{r}_{n,i}}^{{T}_{i}}$ (resp. ${J}_{{r}_{n,i}}^{{S}_{i}}$) are relatively nonexpansive mappings for each $i\ge 1$  is essentially due to Matsushita and Takahashi (cf. [[27], p.265]) which we include for completeness. Note that $\stackrel{\u02c6}{F}({J}_{{r}_{n,i}}^{{T}_{i}})\subset {T}_{i}^{1}0$ for each $i=1,2,3,\dots ,N$. Let $p\in \stackrel{\u02c6}{F}({J}_{{r}_{n,i}}^{{T}_{i}})$. Then there exists $\{{z}_{n}\}\subset E$ such that ${z}_{n}\rightharpoonup p$ and ${lim}_{n\to \mathrm{\infty}}\parallel {z}_{n}{J}_{{r}_{n,i}}^{{T}_{i}}{z}_{n}\parallel =0$. Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, we have
It follows from $\frac{1}{{r}_{n,i}}(J{z}_{n}J{J}_{{r}_{n,i}}^{{T}_{i}}{z}_{n})\in {T}_{i}({J}_{{r}_{n,i}}^{{T}_{i}}{z}_{n})$ and the monotonicity of ${T}_{i}$ that
for all $w\in D({T}_{i})$ and ${w}^{\ast}\in {T}_{i}w$. Letting $n\to \mathrm{\infty}$, we have
for all $w\in D({T}_{i})$ and ${w}^{\ast}\in {T}_{i}w$. Therefore, from the maximality of ${T}_{i}$, we obtain that $p\in {T}_{i}^{1}0$. On the other hand, ${T}_{i}^{1}0=F({J}_{r,i}^{{T}_{i}})$ and $F({J}_{r,i}^{{T}_{i}})\subset \stackrel{\u02c6}{F}({J}_{r,i}^{{T}_{i}})$, so we have ${T}_{i}^{1}0=F({J}_{r,i}^{{T}_{i}})=\stackrel{\u02c6}{F}({J}_{r,i}^{{T}_{i}})$ for each $i=1,2,3,\dots ,N$. Similarly, we can show that $p\in {S}_{i}^{1}0$ and ${S}_{i}^{1}0=F({J}_{r,i}^{{S}_{i}})=\stackrel{\u02c6}{F}({J}_{r,i}^{{S}_{i}})$ for each $i=1,2,3,\dots ,N$. Moreover, the resolvent ${J}_{r,i}^{{T}_{i}}$ of ${T}_{i}$ (resp. ${J}_{r,i}^{{S}_{i}}$ of ${S}_{i}$) with $r>0$ satisfies $\phi (u,{J}_{r,i}^{{T}_{i}}x)\le \phi (u,x)$ for all $u\in F({J}_{r,i}^{{T}_{i}})$ and $x\in E$ (resp. $\phi (u,{J}_{r,i}^{{S}_{i}}x)\le \phi (u,x)$ for all $u\in F({J}_{r,i}^{{S}_{i}})$ and $x\in E$). Hence ${J}_{r,i}^{{T}_{i}}$ and ${J}_{r,i}^{{S}_{i}}$ are relatively nonexpansive mappings. Since the class of relatively nonexpansive mappings is properly contained in the class of total quasiφasymptotically nonexpansive mappings, for a finitely many mappings case, one can derive the desired result from Theorem 2.1. □
Remark 2.5 It is worth to mention that Theorem 2.1 and Theorem 2.4 improve and generalize various results available in the current literature. In particular, we highlight some significant features of both these theorems as follows.

(i)
Algorithm (2.1) is comparatively more general and computationally simpler than the algorithms which appeared in [12, 18] and [22], respectively, in the context of (two) countable families of total quasiφasymptotically nonexpansive mappings.

(ii)
Theorem 2.1 improves and extends [[12], Theorem 3.1], [16], [[21], Theorem 3.1] and [[22], Theorem 3.1] for two countable families of total quasiφasymptotically nonexpansive mappings and a finite family of generalized equilibrium problems.

(iii)
Theorem 2.4 improves and extends [[12], Theorem 4.1] for two finite families of maximal monotone operators. Moreover, our algorithm (2.14) is computationally simpler in comparison with algorithm (4.3) in [[12], Theorem 3.1].

(iv)
Theorem 2.4 improves and extends [[28], Theorem 4.1] for two finite families of maximal monotone operators in the more general domain of a strictly convex, reflexive and smooth Banach space E.

(v)
In the context of an approximate common solution for an equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings, algorithms (2.1) and (2.14) are more general and computationally simpler in comparison with algorithms (3.1) and (4.2), respectively, in [28].

(vi)
Algorithm (2.14) is computationally simpler in comparison with algorithms (3.6) in [29] and (3.1) in [30]. Moreover, Theorem 2.4 improves [[29], Theorem 3.2] and [[30], Theorem 3.1] by removing the ${H}_{n}$ and ${W}_{n}$ conditions in the corresponding algorithms.

(vii)
Theorem 2.4 is an analogue of [[17], Theorem 3.1] for the approximation of a common zero for a finite family of maximal monotone operators.
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Acknowledgements
We are very grateful to anonymous referees for their critical and useful comments. The author MAA Khan gratefully acknowledges the support of Higher Education Commission (HEC) of Pakistan. The authors H Fukharuddin and AR Khan are grateful to King Fahd University of Petroleum and Minerals for supporting the research project IN121037.
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Khan, M.A.A., Fukharuddin, H. & Khan, A.R. Mosco convergence results for common fixed point problems and generalized equilibrium problems in Banach spaces. Fixed Point Theory Appl 2014, 59 (2014). https://doi.org/10.1186/16871812201459
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Keywords
 total quasiφasymptotically nonexpansive mapping
 asymptotic fixed point
 equilibrium problem
 variational inequality problem
 maximal monotone operator
 Mosco convergence