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# Some results on asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and equilibrium problems

## Abstract

In this paper, we investigate a common fixed point problem of a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and an equilibrium problem. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

MSC:47H09, 47J25, 90C33.

## 1 Introduction-preliminaries

Let E be a real Banach space. Recall that E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. It is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ in E such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1$. Let ${U}_{E}=\left\{x\in E:\parallel x\parallel =1\right\}$ be the unit sphere of E. Then the Banach space E is said to be smooth if

$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$

exists for each $x,y\in {U}_{E}$. It is said to be uniformly smooth if the above limit is attained uniformly for $x,y\in {U}_{E}$.

Recall that E has Kadec-Klee property if for any sequence $\left\{{x}_{n}\right\}\subset E$, and $x\in E$ with ${x}_{n}⇀x$, and $\parallel {x}_{n}\parallel \to \parallel x\parallel$, then $\parallel {x}_{n}-x\parallel \to 0$ as $n\to \mathrm{\infty }$. For more details of the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Recall that the normalized duality mapping J from E to ${2}^{{E}^{\ast }}$ is defined by

$Jx=\left\{{f}^{\ast }\in {E}^{\ast }:〈x,{f}^{\ast }〉={\parallel x\parallel }^{2}={\parallel {f}^{\ast }\parallel }^{2}\right\},$

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that if E is uniformly smooth if and only if ${E}^{\ast }$ is uniformly convex.

Next, we assume that E is a smooth Banach space. Consider the functional defined by

$\varphi \left(x,y\right)={\parallel x\parallel }^{2}-2〈x,Jy〉+{\parallel y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E.$

Observe that, in a Hilbert space H, the equality is reduced to $\varphi \left(x,y\right)={\parallel x-y\parallel }^{2}$, $x,y\in H$. As we all know if C is a nonempty closed convex subset of a Hilbert space H and ${P}_{C}:H\to C$ is the metric projection of H onto C, then ${P}_{C}$ is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [2] recently introduced a generalized projection operator ${\mathrm{\Pi }}_{C}$ in a Banach space E which is an analog of the metric projection ${P}_{C}$ in Hilbert spaces. Recall that the generalized projection ${\mathrm{\Pi }}_{C}:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi \left(x,y\right)$, that is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

$\varphi \left(\overline{x},x\right)=\underset{y\in C}{min}\varphi \left(y,x\right).$

The existence and uniqueness of the operator ${\mathrm{\Pi }}_{C}$ follow from the properties of the functional $\varphi \left(x,y\right)$ and strict monotonicity of the mapping J; see, for example, [1, 2]. In Hilbert spaces, ${\mathrm{\Pi }}_{C}={P}_{C}$. It is obvious from the definition of the function ϕ that

${\left(\parallel x\parallel -\parallel y\parallel \right)}^{2}\le \varphi \left(x,y\right)\le {\left(\parallel y\parallel +\parallel x\parallel \right)}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E,$
(1.1)

and

$\varphi \left(x,y\right)=\varphi \left(x,z\right)+\varphi \left(z,y\right)+2〈x-z,Jz-Jy〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y,z\in E.$
(1.2)

Remark 1.1 If E is a reflexive, strictly convex, and smooth Banach space, then $\varphi \left(x,y\right)=0$ if and only if $x=y$; for more details, see [1, 2] and the references therein.

Let C be a nonempty subset of E and let $T:C\to C$ be a mapping. In this paper, we use $F\left(T\right)$ to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C,

$\underset{n\to \mathrm{\infty }}{lim sup}\left\{\parallel {T}^{n+1}x-{T}^{n}x\parallel :x\in K\right\}=0.$

T is said to be closed if for any sequence $\left\{{x}_{n}\right\}\subset C$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}={x}_{0}$ and ${lim}_{n\to \mathrm{\infty }}T{x}_{n}={y}_{0}$, then $T{x}_{0}={y}_{0}$. In this paper, we use → and to denote the strong convergence and weak convergence, respectively.

Recall that a point p in C is said to be an asymptotic fixed point of T [3] iff C contains a sequence $\left\{{x}_{n}\right\}$ which converges weakly to p such that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The set of asymptotic fixed points of T will be denoted by $\stackrel{˜}{F}\left(T\right)$.

A mapping T is said to be relatively nonexpansive iff

$\stackrel{˜}{F}\left(T\right)=F\left(T\right)\ne \mathrm{\varnothing },\phantom{\rule{2em}{0ex}}\varphi \left(p,Tx\right)\le \varphi \left(p,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,\mathrm{\forall }p\in F\left(T\right).$

A mapping T is said to be relatively asymptotically nonexpansive iff

$\stackrel{˜}{F}\left(T\right)=F\left(T\right)\ne \mathrm{\varnothing },\phantom{\rule{2em}{0ex}}\varphi \left(p,{T}^{n}x\right)\le \left(1+{\mu }_{n}\right)\varphi \left(p,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,\mathrm{\forall }p\in F\left(T\right),\mathrm{\forall }n\ge 1,$

where $\left\{{\mu }_{n}\right\}\subset \left[0,\mathrm{\infty }\right)$ is a sequence such that ${\mu }_{n}\to 0$ as $n\to \mathrm{\infty }$.

Remark 1.2 The class of relatively asymptotically nonexpansive mappings were first considered in [4]; see also, [5] and the references therein.

Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff

$F\left(T\right)\ne \mathrm{\varnothing },\phantom{\rule{2em}{0ex}}\varphi \left(p,Tx\right)\le \varphi \left(p,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,\mathrm{\forall }p\in F\left(T\right).$

Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence $\left\{{\mu }_{n}\right\}\subset \left[0,\mathrm{\infty }\right)$ with ${\mu }_{n}\to 0$ as $n\to \mathrm{\infty }$ such that

$F\left(T\right)\ne \mathrm{\varnothing },\phantom{\rule{2em}{0ex}}\varphi \left(p,{T}^{n}x\right)\le \left(1+{\mu }_{n}\right)\varphi \left(p,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,\mathrm{\forall }p\in F\left(T\right),\mathrm{\forall }n\ge 1.$

Remark 1.3 The class of quasi-ϕ-nonexpansive mappings was considered in [6]. The class of asymptotically quasi-ϕ-nonexpansive mappings which was investigated in [7] and [8] includes the class of quasi-ϕ-nonexpansive mappings as a special case.

Remark 1.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive do not require the restriction $F\left(T\right)=\stackrel{˜}{F}\left(T\right)$.

Remark 1.5 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

Recall that T is said to be asymptotically quasi-ϕ-nonexpansive in the intermediate sense iff $F\left(T\right)\ne \mathrm{\varnothing }$ and the following inequality holds:

$\underset{n\to \mathrm{\infty }}{lim sup}\underset{p\in F\left(T\right),x\in C}{sup}\left(\varphi \left(p,{T}^{n}x\right)-\varphi \left(p,x\right)\right)\le 0.$
(1.3)

Putting

${\xi }_{n}=max\left\{0,\underset{p\in F\left(T\right),x\in C}{sup}\left(\varphi \left(p,{T}^{n}x\right)-\varphi \left(p,x\right)\right)\right\},$

it follows that ${\xi }_{n}\to 0$ as $n\to \mathrm{\infty }$. Then (1.3) is reduced to the following:

$\varphi \left(p,{T}^{n}x\right)\le \varphi \left(p,x\right)+{\xi }_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in F\left(T\right),\mathrm{\forall }x\in C.$
(1.4)

Remark 1.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [9].

Remark 1.7 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [10], in the framework of Banach spaces.

Let f be a bifunction from $C×C$ to , where denotes the set of real numbers. Recall the following equilibrium problem. Find $p\in C$ such that

$f\left(p,y\right)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(1.5)

We use $EP\left(f\right)$ to denote the solution set of the equilibrium problem (1.5). That is,

$EP\left(f\right)=\left\{p\in C:f\left(p,y\right)\ge 0,\mathrm{\forall }y\in C\right\}.$

We remark here that the equilibrium problem was first introduced by Fan [11]. Given a mapping $Q:C\to {E}^{\ast }$, let

$f\left(x,y\right)=〈Qx,y-x〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Then $p\in EP\left(f\right)$ if and only if p is a solution of the following variational inequality. Find p such that

$〈Qp,y-p〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(1.6)

To study the equilibrium problems (1.5), we may assume that F satisfies the following conditions:

(A1) $F\left(x,x\right)=0$ for all $x\in C$;

(A2) F is monotone, i.e., $F\left(x,y\right)+F\left(y,x\right)\le 0$ for all $x,y\in C$;

(A3) for each $x,y,z\in C$,

$\underset{t↓0}{lim sup}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right);$

(A4) for each $x\in C$, $y↦F\left(x,y\right)$ is convex and weakly lower semi-continuous.

Numerous problems in physics, optimization, and economics reduce to find a solution of (1.5). Recently, many authors have investigated common solutions of fixed point and equilibrium problems in Banach spaces; see, for example, [1233] and the references therein.

In this paper, we consider a projection algorithm for treating the equilibrium problem and fixed point problems of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.

In order to prove our main results, we need the following lemmas.

Lemma 1.8 [2]

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

$\varphi \left(y,{\mathrm{\Pi }}_{C}x\right)+\varphi \left({\mathrm{\Pi }}_{C}x,x\right)\le \varphi \left(y,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

Lemma 1.9 [2]

Let C be a nonempty closed convex subset of a smooth Banach space E and let $x\in E$. Then ${x}_{0}={\mathrm{\Pi }}_{C}x$ if and only if

$〈{x}_{0}-y,Jx-J{x}_{0}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

Lemma 1.10 Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4). Let $r>0$ and $x\in E$. Then

1. (a)

[34]There exists $z\in C$ such that

$f\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
2. (b)

[6, 24]Define a mapping ${T}_{r}:E\to C$ by

${S}_{r}x=\left\{z\in C:f\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉,\mathrm{\forall }y\in C\right\}.$

Then the following conclusions hold:

1. (1)

${S}_{r}$ is single-valued;

2. (2)

${S}_{r}$ is a firmly nonexpansive-type mapping, i.e., for all $x,y\in E$,

$〈{S}_{r}x-{S}_{r}y,J{S}_{r}x-J{S}_{r}y〉\le 〈{S}_{r}x-{S}_{r}y,Jx-Jy〉$
3. (3)

$F\left({S}_{r}\right)=EP\left(f\right)$;

4. (4)

${S}_{r}$ is quasi-ϕ-nonexpansive;

5. (5)

$\varphi \left(q,{S}_{r}x\right)+\varphi \left({S}_{r}x,x\right)\le \varphi \left(q,x\right)$, $\mathrm{\forall }q\in F\left({S}_{r}\right)$;

6. (6)

$EP\left(f\right)$ is closed and convex.

Lemma 1.11 [35]

Let E be a smooth and uniformly convex Banach space and let $r>0$. Then there exists a strictly increasing, continuous and convex function $g:\left[0,2r\right]\to R$ such that $g\left(0\right)=0$ and

${\parallel tx+\left(1-t\right)y\parallel }^{2}\le t{\parallel x\parallel }^{2}+\left(1-t\right){\parallel y\parallel }^{2}-t\left(1-t\right)g\left(\parallel x-y\parallel \right)$

for all $x,y\in {B}_{r}=\left\{x\in E:\parallel x\parallel \le r\right\}$ and $t\in \left[0,1\right]$.

## 2 Main results

Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4) and let N be some positive integer. Let ${T}_{i}:C\to C$ an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense for every $1\le i\le N$. Assume that ${T}_{i}$ is closed asymptotically regular on C and ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)$ is nonempty and bounded. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:

where ${\xi }_{n}=max\left\{0,{sup}_{p\in F\left({T}_{i}\right),x\in C}\left(\varphi \left(p,{T}_{i}^{n}x\right)-\varphi \left(p,x\right)\right)\right\}$, $\left\{{\alpha }_{n,i}\right\}$ is a real number sequence in $\left(0,1\right)$ for every $1\le i\le N$, $\left\{{r}_{n}\right\}$ is a real number sequence in $\left[k,\mathrm{\infty }\right)$, where k is some positive real number. Assume that ${\sum }_{i=0}^{N}{\alpha }_{n,i}=1$ and ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,i}>0$ for every $1\le i\le N$. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1}$, where ${\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}$ is the generalized projection from E onto ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)$.

Proof First, we show that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)$ is closed and convex. From [9], we find that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ is closed and convex, which combines with Lemma 1.10 shows that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)$ is closed and convex. Next, we show that ${C}_{n}$ is closed and convex. It is obvious that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{h}$ is closed and convex for some positive integer h. For $z\in {C}_{h}$, we see that $\varphi \left(z,{u}_{h}\right)\le \varphi \left(z,{x}_{h}\right)+{\xi }_{h}$ is equivalent to

$2〈z,J{x}_{h}-J{u}_{h}〉\le {\parallel {x}_{k}\parallel }^{2}-{\parallel {u}_{k}\parallel }^{2}+{\xi }_{h}.$

It is to see that ${C}_{h+1}$ is closed and convex. This proves that ${C}_{n}$ is closed and convex. This in turn shows that ${\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}$ is well defined. Putting ${u}_{n}={S}_{{r}_{n}}{y}_{n}$, we from Lemma 1.10 see that ${S}_{{r}_{n}}$ is quasi-ϕ-nonexpansive. Now, we are in a position to prove that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{n}$. Indeed, ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{1}=C$ is obvious. Assume that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{h}$ for some positive integer h. Then, for $\mathrm{\forall }w\in {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{h}$, we have

$\begin{array}{rl}\varphi \left(w,{u}_{h}\right)& =\varphi \left(w,{S}_{{r}_{h}}{y}_{h}\right)\\ \le \varphi \left(w,{y}_{h}\right)\\ =\varphi \left(w,{J}^{-1}\left({\alpha }_{h,0}J{x}_{h}+\sum _{i=1}^{N}{\alpha }_{h,i}J{T}_{i}^{h}{x}_{h}\right)\right)\\ ={\parallel w\parallel }^{2}-2〈w,{\alpha }_{h,0}J{x}_{h}+\sum _{i=1}^{N}{\alpha }_{h,i}J{T}_{i}^{h}{x}_{h}〉+{\parallel {\alpha }_{h,0}J{x}_{h}+\sum _{i=1}^{N}{\alpha }_{h,i}J{T}_{i}^{h}{x}_{h}\parallel }^{2}\\ \le {\parallel w\parallel }^{2}-2{\alpha }_{h,0}〈w,J{x}_{h}〉-2\sum _{i=1}^{N}{\alpha }_{h,i}〈w,J{T}_{i}^{h}{x}_{h}〉+{\alpha }_{h,0}{\parallel {x}_{h}\parallel }^{2}+\sum _{i=1}^{N}{\alpha }_{h,i}{\parallel {T}_{i}^{h}{x}_{h}\parallel }^{2}\\ ={\alpha }_{h,0}\varphi \left(w,{x}_{h}\right)+\sum _{i=1}^{N}{\alpha }_{h,i}\varphi \left(w,{T}_{i}^{h}{x}_{h}\right)\\ \le {\alpha }_{h,0}\varphi \left(w,{x}_{h}\right)+\sum _{i=1}^{N}{\alpha }_{h,i}\varphi \left(w,{x}_{h}\right)+\sum _{i=1}^{N}{\alpha }_{h,i}{\xi }_{h}\\ =\varphi \left(w,{x}_{h}\right)+\sum _{i=1}^{N}{\alpha }_{h,i}{\xi }_{h}\\ \le \varphi \left(w,{x}_{h}\right)+\sum _{i=1}^{N}{\xi }_{h},\end{array}$
(2.1)

which shows that $w\in {C}_{h+1}$. This implies that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{n}$.

Next, we prove that the sequence $\left\{{x}_{n}\right\}$ is bounded. Notice that ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$. We find from Lemma 1.9 that $〈{x}_{n}-z,J{x}_{1}-J{x}_{n}〉\ge 0$, for any $z\in {C}_{n}$. Since ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)\subset {C}_{n}$, we find that

$〈{x}_{n}-w,J{x}_{1}-J{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }w\in \bigcap _{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right).$
(2.2)

It follows from Lemma 1.8 that

$\begin{array}{rl}\varphi \left({x}_{n},{x}_{1}\right)& \le \varphi \left({\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1},{x}_{1}\right)-\varphi \left({\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1},{x}_{n}\right)\\ \le \varphi \left({\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1},{x}_{1}\right).\end{array}$

This implies that the sequence $\left\{\varphi \left({x}_{n},{x}_{1}\right)\right\}$ is bounded. It follows from (1.1) that the sequence $\left\{{x}_{n}\right\}$ is also bounded. Since the space is reflexive, we may assume, without loss of generality, that ${x}_{n}⇀\overline{x}$. Next, we prove that $\overline{x}\in {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)$. Since ${C}_{n}$ is closed and convex, we find that $\overline{x}\in {C}_{n}$. This implies from ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$ that $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left(\overline{x},{x}_{1}\right)$. On the other hand, we see from the weakly lower semicontinuity of $\parallel \cdot \parallel$ that

$\begin{array}{rl}\varphi \left(\overline{x},{x}_{1}\right)& ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},J{x}_{1}〉+{\parallel {x}_{1}\parallel }^{2}\\ \le \underset{n\to \mathrm{\infty }}{lim inf}\left({\parallel {x}_{n}\parallel }^{2}-2〈{x}_{n},J{x}_{1}〉+{\parallel {x}_{1}\parallel }^{2}\right)\\ =\underset{n\to \mathrm{\infty }}{lim inf}\varphi \left({x}_{n},{x}_{1}\right)\\ \le \underset{n\to \mathrm{\infty }}{lim sup}\varphi \left({x}_{n},{x}_{1}\right)\\ \le \varphi \left(\overline{x},{x}_{1}\right),\end{array}$

which implies that ${lim}_{n\to \mathrm{\infty }}\varphi \left({x}_{n},{x}_{1}\right)=\varphi \left(\overline{x},{x}_{1}\right)$. Hence, we have ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}\parallel =\parallel \overline{x}\parallel$. In view of the Kadec-Klee property of E, we find that ${x}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$. Since ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$, and ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$, we find that $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left({x}_{n+1},{x}_{1}\right)$. This shows that $\left\{\varphi \left({x}_{n},{x}_{1}\right)\right\}$ is nondecreasing. We find from its boundedness that ${lim}_{n\to \mathrm{\infty }}\varphi \left({x}_{n},{x}_{1}\right)$ exists. It follows that

$\begin{array}{rl}\varphi \left({x}_{n+1},{x}_{n}\right)& =\varphi \left({x}_{n+1},{\mathrm{\Pi }}_{{C}_{n}}{x}_{1}\right)\\ \le \varphi \left({x}_{n+1},{x}_{1}\right)-\varphi \left({\mathrm{\Pi }}_{{C}_{n}}{x}_{1},{x}_{1}\right)\\ =\varphi \left({x}_{n+1},{x}_{1}\right)-\varphi \left({x}_{n},{x}_{1}\right).\end{array}$

This implies that

$\underset{n\to \mathrm{\infty }}{lim}\varphi \left({x}_{n+1},{x}_{n}\right)=0.$
(2.3)

In the light of ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}$, we find that

$\varphi \left({x}_{n+1},{u}_{n}\right)\le \varphi \left({x}_{n+1},{x}_{n}\right)+{\xi }_{n}.$

It follows from (2.3) that

$\underset{n\to \mathrm{\infty }}{lim}\varphi \left({x}_{n+1},{u}_{n}\right)=0.$
(2.4)

In view of (1.1), we see that ${lim}_{n\to \mathrm{\infty }}\left(\parallel {x}_{n+1}\parallel -\parallel {u}_{n}\parallel \right)=0$. This implies that ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}\parallel =\parallel \overline{x}\parallel$. That is,

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {u}_{n}\parallel =\parallel J\overline{x}\parallel .$
(2.5)

This implies that $\left\{J{u}_{n}\right\}$ is bounded. Note that both E and ${E}^{\ast }$ are reflexive. We may assume, without loss of generality, that $J{u}_{n}⇀{u}^{\ast }\in {E}^{\ast }$. In view of the reflexivity of E, we see that $J\left(E\right)={E}^{\ast }$. This shows that there exists an element $u\in E$ such that $Ju={u}^{\ast }$. It follows that

$\begin{array}{rl}\varphi \left({x}_{n+1},{u}_{n}\right)& ={\parallel {x}_{n+1}\parallel }^{2}-2〈{x}_{n+1},J{u}_{n}〉+{\parallel {u}_{n}\parallel }^{2}\\ ={\parallel {x}_{n+1}\parallel }^{2}-2〈{x}_{n+1},J{u}_{n}〉+{\parallel J{u}_{n}\parallel }^{2}.\end{array}$

Taking ${lim inf}_{n\to \mathrm{\infty }}$ on both sides of the equality aboven yields

$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^{2}-2〈\overline{x},{u}^{\ast }〉+{\parallel {u}^{\ast }\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Ju〉+{\parallel Ju\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Ju〉+{\parallel u\parallel }^{2}\\ =\varphi \left(\overline{x},u\right).\end{array}$

That is, $\overline{x}=u$, which in turn implies that ${u}^{\ast }=J\overline{x}$. It follows that $J{u}_{n}⇀J\overline{x}\in {E}^{\ast }$. Since ${E}^{\ast }$ enjoys the Kadec-Klee property, we obtain from (2.5) that ${lim}_{n\to \mathrm{\infty }}J{u}_{n}=J\overline{x}$. Since ${J}^{-1}:{E}^{\ast }\to E$ is demicontinuous and E enjoys the Kadec-Klee property, we obtain ${u}_{n}\to \overline{x}$, as $n\to \mathrm{\infty }$. Note that

$\parallel {x}_{n}-{u}_{n}\parallel \le \parallel {x}_{n}-\overline{x}\parallel +\parallel \overline{x}-{u}_{n}\parallel .$

It follows that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{u}_{n}\parallel =0.$
(2.6)

Since J is uniformly norm-to-norm continuous on any bounded sets, we have

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n}-J{u}_{n}\parallel =0.$
(2.7)

On the other hand, we have

$\begin{array}{rl}\varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)& ={\parallel {x}_{n}\parallel }^{2}-{\parallel {u}_{n}\parallel }^{2}-2〈w,J{x}_{n}-J{u}_{n}〉\\ \le \parallel {x}_{n}-{u}_{n}\parallel \left(\parallel {x}_{n}\parallel +\parallel {u}_{n}\parallel \right)+2\parallel w\parallel \parallel J{x}_{n}-J{u}_{n}\parallel .\end{array}$

We, therefore, find that

$\underset{n\to \mathrm{\infty }}{lim}\left(\varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)\right)=0.$
(2.8)

Since E is uniformly smooth, we know that ${E}^{\ast }$ is uniformly convex. In view of Lemma 1.11, we find that

$\begin{array}{rl}\varphi \left(w,{u}_{n}\right)& =\varphi \left(w,{S}_{{r}_{n}}{y}_{n}\right)\\ \le \varphi \left(w,{y}_{n}\right)\\ =\varphi \left(w,{J}^{-1}\left({\alpha }_{n,0}J{x}_{n}+\sum _{i=1}^{N}{\alpha }_{n,i}J{T}_{i}^{n}{x}_{n}\right)\right)\\ ={\parallel w\parallel }^{2}-2〈w,{\alpha }_{n,0}J{x}_{n}+\sum _{i=1}^{N}{\alpha }_{n,i}J{T}_{i}^{n}{x}_{n}〉+{\parallel {\alpha }_{n,0}J{x}_{n}+\sum _{i=1}^{N}{\alpha }_{n,i}J{T}_{i}^{n}{x}_{n}\parallel }^{2}\\ \le {\parallel w\parallel }^{2}-2{\alpha }_{n,0}〈w,J{x}_{n}〉-2\sum _{i=1}^{N}{\alpha }_{n,i}〈w,J{T}_{i}^{n}{x}_{n}〉+{\alpha }_{n,0}{\parallel {x}_{n}\parallel }^{2}+\sum _{i=1}^{N}{\alpha }_{n,i}{\parallel {T}_{i}^{n}{x}_{n}\parallel }^{2}\\ -{\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)\\ ={\alpha }_{n,0}\varphi \left(w,{x}_{n}\right)+\sum _{i=1}^{N}{\alpha }_{n,i}\varphi \left(w,{T}_{i}^{n}{x}_{n}\right)-{\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)\\ \le {\alpha }_{n,0}\varphi \left(w,{x}_{n}\right)+\sum _{i=1}^{N}{\alpha }_{n,i}\varphi \left(w,{x}_{n}\right)+\sum _{i=1}^{N}{\alpha }_{n,i}{\xi }_{h}\\ -{\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)\\ =\varphi \left(w,{x}_{n}\right)+\sum _{i=1}^{N}{\alpha }_{n,i}{\xi }_{n}-{\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)\\ \le \varphi \left(w,{x}_{n}\right)+{\xi }_{n}-{\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right).\end{array}$

It follows that

${\alpha }_{n,0}{\alpha }_{n,1}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)\le \varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)+{\xi }_{n}.$

In view of the restriction on the sequences, we find from (2.8) that ${lim}_{n\to \mathrm{\infty }}g\left(\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel \right)=0$. It follows that

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n}-J{T}_{1}^{n}{x}_{n}\parallel =0.$

In the same way, we obtain

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n}-J{T}_{i}^{n}{x}_{n}\parallel =0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }1\le i\le N.$

Notice that $\parallel J{T}_{i}^{n}{x}_{n}-J\overline{x}\parallel \le \parallel J{T}_{i}^{n}{x}_{n}-J{x}_{n}\parallel +\parallel J{x}_{n}-J\overline{x}\parallel$. It follows that

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{T}_{i}^{n}{x}_{n}-J\overline{x}\parallel =0.$
(2.9)

The demicontinuity of ${J}^{-1}:{E}^{\ast }\to E$ implies that ${T}_{i}^{n}{x}_{n}⇀\overline{x}$. Note that

$|\parallel {T}_{i}^{n}{x}_{n}\parallel -\parallel \overline{x}\parallel |=|\parallel J{T}_{i}^{n}{x}_{n}\parallel -\parallel J\overline{x}\parallel |\le \parallel J{T}_{i}^{n}{x}_{n}-J\overline{x}\parallel .$

This implies from (2.9) that ${lim}_{n\to \mathrm{\infty }}\parallel {T}_{i}^{n}{x}_{n}\parallel =\parallel \overline{x}\parallel$. Since E has the Kadec-Klee property, we obtain ${lim}_{n\to \mathrm{\infty }}\parallel {T}_{i}^{n}{x}_{n}-\overline{x}\parallel =0$. On the other hand, we have

$\parallel {T}_{i}^{n+1}{x}_{n}-\overline{x}\parallel \le \parallel {T}_{i}^{n+1}{x}_{n}-{T}_{i}^{n}{x}_{n}\parallel +\parallel {T}_{i}^{n}{x}_{n}-\overline{x}\parallel .$

It follows from the uniformly asymptotic regularity of ${T}_{i}$ that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {T}_{i}^{n+1}{x}_{n}-\overline{x}\parallel =0.$

That is, ${T}_{i}{T}_{i}^{n}{x}_{n}\to \overline{x}$. From the closedness of ${T}_{i}$, we find $\overline{x}={T}_{i}\overline{x}$ for every $1\le i\le N$. This proves $\overline{x}\in {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$.

Next, we show that $\overline{x}\in EF\left(f\right)$. In view of Lemma 1.8, we find that

$\begin{array}{rl}\varphi \left({u}_{n},{y}_{n}\right)& \le \varphi \left(w,{y}_{n}\right)-\varphi \left(w,{u}_{n}\right)\\ \le \varphi \left(w,{x}_{n}\right)+{\mu }_{n}-\varphi \left(w,{u}_{n}\right).\end{array}$

It follows from (2.8) that ${lim}_{n\to \mathrm{\infty }}\varphi \left({u}_{n},{y}_{n}\right)=0$. This implies that ${lim}_{n\to \mathrm{\infty }}\left(\parallel {u}_{n}\parallel -\parallel {y}_{n}\parallel \right)=0$. It follows from (2.6) that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}\parallel =\parallel \overline{x}\parallel .$

It follows that

$\underset{n\to \mathrm{\infty }}{lim}\parallel J{y}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}\parallel =\parallel \overline{x}\parallel =\parallel J\overline{x}\parallel .$

This shows that $\left\{J{y}_{n}\right\}$ is bounded. Since ${E}^{\ast }$ is reflexive, we may assume that $J{y}_{n}⇀{v}^{\ast }\in {E}^{\ast }$. In view of $J\left(E\right)={E}^{\ast }$, we see that there exists $v\in E$ such that $Jv={v}^{\ast }$. It follows that

$\begin{array}{rl}\varphi \left({u}_{n},{y}_{n}\right)& ={\parallel {u}_{n}\parallel }^{2}-2〈{u}_{n},J{y}_{n}〉+{\parallel {y}_{n}\parallel }^{2}\\ ={\parallel {u}_{n}\parallel }^{2}-2〈{u}_{n},J{y}_{n}〉+{\parallel J{y}_{n}\parallel }^{2}.\end{array}$

Taking ${lim inf}_{n\to \mathrm{\infty }}$ the both sides of equality above yields that

$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^{2}-2〈\overline{x},{v}^{\ast }〉+{\parallel {v}^{\ast }\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Jv〉+{\parallel Jv\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Jv〉+{\parallel v\parallel }^{2}\\ =\varphi \left(\overline{x},v\right).\end{array}$

That is, $\overline{x}=v$, which in turn implies that ${v}^{\ast }=J\overline{x}$. It follows that $J{y}_{n}⇀J\overline{x}\in {E}^{\ast }$. Since ${E}^{\ast }$ enjoys the Kadec-Klee property, we obtain $J{y}_{n}-J\overline{x}\to 0$ as $n\to \mathrm{\infty }$. Note that ${J}^{-1}:{E}^{\ast }\to E$ is demicontinuous. It follows that ${y}_{n}⇀\overline{x}$. Since E enjoys the Kadec-Klee property, we obtain ${y}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$. Note that

$\parallel {u}_{n}-{y}_{n}\parallel \le \parallel {u}_{n}-\overline{x}\parallel +\parallel \overline{x}-{y}_{n}\parallel .$

This implies that ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}-{y}_{n}\parallel =0$. Since J is uniformly norm-to-norm continuous on any bounded sets, we have ${lim}_{n\to \mathrm{\infty }}\parallel J{u}_{n}-J{y}_{n}\parallel =0$. From the assumption ${r}_{n}\ge k$, we see that

$\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel J{u}_{n}-J{y}_{n}\parallel }{{r}_{n}}=0.$
(2.10)

Since ${u}_{n}={S}_{{r}_{n}}{y}_{n}$, we find that

$f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

It follows from (A2) that

$\parallel y-{u}_{n}\parallel \frac{\parallel J{u}_{n}-J{y}_{n}\parallel }{{r}_{n}}\ge \frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge f\left(y,{u}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

In view of (A4), we find from (2.10) that

$f\left(y,\overline{x}\right)\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

For $0 and $y\in C$, define ${y}_{t}=ty+\left(1-t\right)\overline{x}$. It follows that ${y}_{t}\in C$, which yields $f\left({y}_{t},\overline{x}\right)\le 0$. It follows from (A1) and (A4) that

$0=f\left({y}_{t},{y}_{t}\right)\le tf\left({y}_{t},y\right)+\left(1-t\right)f\left({y}_{t},\overline{x}\right)\le tf\left({y}_{t},y\right).$

That is,

$f\left({y}_{t},y\right)\ge 0.$

Letting $t↓0$, we obtain from (A3) that $f\left(\overline{x},y\right)\ge 0$, $\mathrm{\forall }y\in C$. This implies that $\overline{x}\in EP\left(f\right)$.

Finally, we turn our attention to proving that $\overline{x}={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1}$.

Letting $n\to \mathrm{\infty }$ in (2.2), we obtain

$〈\overline{x}-w,J{x}_{1}-J\overline{x}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }w\in \bigcap _{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right).$

In view of Lemma 1.9, we find that $\overline{x}={\mathrm{\Pi }}_{{\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap EF\left(f\right)}{x}_{1}$. This completes the proof. □

From the definition of quasi-ϕ-nonexpansive mappings, we see that every quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive in the intermediate sense. We also know that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property (note that every uniformly convex Banach space enjoys the Kadec-Klee property).

Remark 2.2 Theorem 2.1 can be viewed an extension of the corresponding results in Qin et al. [6], Kim [12], Qin et al. [22], Takahashi and Zembayashi [24], respectively. The space ${L}^{p}$, where $p>1$, satisfies the restriction in Theorem 2.1.

## 3 Applications

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4). Let $T:C\to C$ an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and $F\left(T\right)\cap EF\left(f\right)$ is nonempty and bounded. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:

where ${\xi }_{n}=max\left\{0,{sup}_{p\in F\left(T\right),x\in C}\left(\varphi \left(p,{T}^{n}x\right)-\varphi \left(p,x\right)\right)\right\}$, $\left\{{r}_{n}\right\}$ is a real number sequence in $\left[k,\mathrm{\infty }\right)$, where k is some positive real number, $\left\{{\alpha }_{n,0}\right\}$ and $\left\{{\alpha }_{n}n,1\right\}$ are two real number sequence in $\left(0,1\right)$. Assume that ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n,0}{\alpha }_{n,1}>0$. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{F\left(T\right)\cap EF\left(f\right)}{x}_{1}$, where ${\mathrm{\Pi }}_{F\left(T\right)\cap EF\left(f\right)}$ is the generalized projection from E onto $F\left(T\right)\cap EF\left(f\right)$.

Proof Putting $N=1$, we draw from Theorem 2.1 the desired conclusion immediately. □

Remark 3.2 If the mapping T in Theorem 3.1 is quasi-ϕ-nonexpansive, then the restrictions that T is closed asymptotically regular on C and $F\left(T\right)\cap EF\left(f\right)$ is bounded will not be required anymore.

If ${T}_{i}=I$, where I is the identity for every $1\le i\le N$, then we find from Theorem 2.1 the following.

Theorem 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4). Assume that $EF\left(f\right)$ is nonempty. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:

where $\left\{{r}_{n}\right\}$ is a real number sequence in $\left[k,\mathrm{\infty }\right)$, where k is some positive real number. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{EF\left(f\right)}{x}_{1}$, where ${\mathrm{\Pi }}_{EF\left(f\right)}$ is the generalized projection from E onto $EF\left(f\right)$.

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Huang, C., Ma, X. Some results on asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and equilibrium problems. J Inequal Appl 2014, 202 (2014). https://doi.org/10.1186/1029-242X-2014-202

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