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

- Chunyan Huang
^{1}and - Xiaoyan Ma
^{2}Email author

**2014**:202

https://doi.org/10.1186/1029-242X-2014-202

© Huang and Ma; licensee Springer. 2014

**Received: **29 January 2014

**Accepted: **2 May 2014

**Published: **22 May 2014

## 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.

### Keywords

asymptotically quasi-*ϕ*-nonexpansive mapping asymptotically quasi-

*ϕ*-nonexpansive mapping in the intermediate sense generalized projection equilibrium problem fixed point

## 1 Introduction-preliminaries

*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 $\{{x}_{n}\}$ and $\{{y}_{n}\}$ 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 ${U}_{E}=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of

*E*. Then the Banach space

*E*is said to be smooth if

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 $\{{x}_{n}\}\subset E$, and $x\in E$ with ${x}_{n}\rightharpoonup x$, 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.

*J*from

*E*to ${2}^{{E}^{\ast}}$ is defined by

where $\u3008\cdot ,\cdot \u3009$ 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.

*E*is a smooth Banach space. Consider the functional defined by

*H*, the equality is reduced to $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$. 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 $\varphi (x,y)$, that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

*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

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

*C*be a nonempty subset of

*E*and let $T:C\to C$ be a mapping. In this paper, we use $F(T)$ 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*,

*T* is said to be closed if for any sequence $\{{x}_{n}\}\subset C$ 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 $\{{x}_{n}\}$ 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 $\tilde{F}(T)$.

*T*is said to be relatively nonexpansive iff

*T*is said to be relatively asymptotically nonexpansive iff

where $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ 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.

*T*is said to be quasi-

*ϕ*-nonexpansive iff

*T*is said to be asymptotically quasi-

*ϕ*-nonexpansive iff there exists a sequence $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that

**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 $F(T)=\tilde{F}(T)$.

**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.

*T*is said to be asymptotically quasi-

*ϕ*-nonexpansive in the intermediate sense iff $F(T)\ne \mathrm{\varnothing}$ and the following inequality holds:

**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.

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

*p*is a solution of the following variational inequality. Find

*p*such that

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

(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 $x,y\in C$;

(A4) for each $x\in C$, $y\mapsto F(x,y)$ 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, [12–33] 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*

**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*

**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*$C\times C$

*to*ℝ

*satisfying*(A1)-(A4).

*Let*$r>0$

*and*$x\in E$.

*Then*

- (a)[34]
*There exists*$z\in C$*such that*$f(z,y)+\frac{1}{r}\u3008y-z,Jz-Jx\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C.$ - (b)

*Then the following conclusions hold*:

- (1)
${S}_{r}$

*is single*-*valued*; - (2)${S}_{r}$
*is a firmly nonexpansive*-*type mapping*,*i*.*e*.,*for all*$x,y\in E$,$\u3008{S}_{r}x-{S}_{r}y,J{S}_{r}x-J{S}_{r}y\u3009\le \u3008{S}_{r}x-{S}_{r}y,Jx-Jy\u3009$ - (3)
$F({S}_{r})=EP(f)$;

- (4)
${S}_{r}$

*is quasi*-*ϕ*-*nonexpansive*; - (5)
$\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x)$, $\mathrm{\forall}q\in F({S}_{r})$;

- (6)
$EP(f)$

*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*$g:[0,2r]\to R$

*such that*$g(0)=0$

*and*

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

## 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*$C\times C$

*to*ℝ

*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*${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* ${\xi}_{n}=max\{0,{sup}_{p\in F({T}_{i}),x\in C}(\varphi (p,{T}_{i}^{n}x)-\varphi (p,x))\}$, $\{{\alpha}_{n,i}\}$ *is a real number sequence in* $(0,1)$ *for every* $1\le i\le N$, $\{{r}_{n}\}$ *is a real number sequence in* $[k,\mathrm{\infty})$, *where* *k* *is some positive real number*. *Assume that* ${\sum}_{i=0}^{N}{\alpha}_{n,i}=1$ *and* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0$ *for every* $1\le i\le N$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}{x}_{1}$, *where* ${\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}$ *is the generalized projection from* *E* *onto* ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)$.

*Proof*First, we show that ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)$ is closed and convex. From [9], we find that ${\bigcap}_{i=1}^{N}F({T}_{i})$ is closed and convex, which combines with Lemma 1.10 shows that ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)$ 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 $\varphi (z,{u}_{h})\le \varphi (z,{x}_{h})+{\xi}_{h}$ is equivalent to

*ϕ*-nonexpansive. Now, we are in a position to prove that ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{n}$. Indeed, ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{1}=C$ is obvious. Assume that ${\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{h}$ for some positive integer

*h*. Then, for $\mathrm{\forall}w\in {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{h}$, we have

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

*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 $\varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1})$. This shows that $\{\varphi ({x}_{n},{x}_{1})\}$ is nondecreasing. We find from its boundedness that ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})$ exists. It follows that

*E*and ${E}^{\ast}$ are reflexive. We may assume, without loss of generality, that $J{u}_{n}\rightharpoonup {u}^{\ast}\in {E}^{\ast}$. In view of the reflexivity of

*E*, we see that $J(E)={E}^{\ast}$. This shows that there exists an element $u\in E$ such that $Ju={u}^{\ast}$. It follows that

*E*enjoys the Kadec-Klee property, we obtain ${u}_{n}\to \overline{x}$, as $n\to \mathrm{\infty}$. Note that

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

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

*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

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 $\overline{x}\in {\bigcap}_{i=1}^{N}F({T}_{i})$.

*E*enjoys the Kadec-Klee property, we obtain ${y}_{n}\to \overline{x}$ as $n\to \mathrm{\infty}$. Note that

*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

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

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

In view of Lemma 1.9, we find that $\overline{x}={\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}{x}_{1}$. 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*$C\times C$

*to*ℝ

*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*$F(T)\cap EF(f)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* ${\xi}_{n}=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,{T}^{n}x)-\varphi (p,x))\}$, $\{{r}_{n}\}$ *is a real number sequence in* $[k,\mathrm{\infty})$, *where* *k* *is some positive real number*, $\{{\alpha}_{n,0}\}$ *and* $\{{\alpha}_{n}n,1\}$ *are two real number sequence in* $(0,1)$. *Assume that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,1}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)\cap EF(f)}{x}_{1}$, *where* ${\mathrm{\Pi}}_{F(T)\cap EF(f)}$ *is the generalized projection from* *E* *onto* $F(T)\cap EF(f)$.

*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 $F(T)\cap EF(f)$ 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*$C\times C$

*to*ℝ

*satisfying*(A1)-(A4).

*Assume that*$EF(f)$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

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

## Declarations

### Acknowledgements

The authors thank the reviewers for useful suggestions which improved the contents of this paper.

## Authors’ Affiliations

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