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# On generalized quasi-*ϕ*-nonexpansive mappings and their projection algorithms

- Yan Hao
^{1}Email author

**2013**:204

https://doi.org/10.1186/1687-1812-2013-204

© Hao; licensee Springer 2013

**Received:**29 April 2013**Accepted:**11 July 2013**Published:**29 July 2013

## Abstract

A fixed point problem of a generalized asymptotically quasi-*ϕ*-nonexpansive mapping and an equilibrium problem are investigated. A strong convergence theorem for solutions of the fixed point problem and the equilibrium problem is established in a Banach space.

## Keywords

- asymptotically quasi-
*ϕ*-nonexpansive mapping - equilibrium problem
- fixed point
- generalized asymptotically quasi-
*ϕ*-nonexpansive mapping - generalized projection

## 1 Introduction and preliminaries

*E*be a real Banach space, and let ${E}^{\ast}$ be the dual space of

*E*. We denote by

*J*the normalized duality mapping from

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

*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 provided

exists for each $x,y\in {U}_{E}$. It is also said to be uniformly smooth if the above limit is attained uniformly for $x,y\in {U}_{E}$. 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 *E* is uniformly smooth if and only if ${E}^{\ast}$ is uniformly convex.

Recall that a Banach space *E* enjoys the 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 on 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.

*C*be a nonempty subset of

*E*. Let

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

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

*p*such that

*f*satisfies the following conditions:

- (A1)
$f(x,x)=0$, $\mathrm{\forall}x\in C$;

- (A2)
*f*is monotone,*i.e.*, $f(x,y)+f(y,x)\le 0$, $\mathrm{\forall}x,y\in C$; - (A3)$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}f(tz+(1-t)x,y)\le f(x,y),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y,z\in C;$
- (A4)
for each $x\in C$, $y\mapsto f(x,y)$ is convex and weakly lower semi-continuous.

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 analogue of the metric projection ${P}_{C}$ in Hilbert spaces.

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

*H*, the equality is reduced to $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$. 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] and [2]. In Hilbert spaces, ${\mathrm{\Pi}}_{C}={P}_{C}$. It is obvious from the definition of 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] and [3].

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*, ${lim}_{n\to \mathrm{\infty}}{sup}_{x\in K}\parallel {T}^{n+1}x-{T}^{n}x\parallel =0$. *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 ${lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}$, $T{x}_{0}={y}_{0}$. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.

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 [4, 5] iff $\tilde{F}(T)=F(T)\ne \mathrm{\varnothing}$ and $\varphi (p,Tx)\le \varphi (p,x)$ for all $x\in C$ and $p\in F(T)$. *T* is said to be relatively asymptotically nonexpansive [6, 7] iff $\tilde{F}(T)=F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{\mu}_{n}\}\subset [1,\mathrm{\infty})$ with ${\mu}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that $\varphi (p,Tx)\le {\mu}_{n}\varphi (p,x)$ for all $x\in C$, $p\in F(T)$ and $n\ge 1$. *T* is said to be quasi-*ϕ*-nonexpansive [8, 9] iff $F(T)\ne \mathrm{\varnothing}$ and $\varphi (p,Tx)\le \varphi (p,x)$ for all $x\in C$ and $p\in F(T)$. *T* is said to be asymptotically quasi-*ϕ*-nonexpansive [10–12] iff $F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{\mu}_{n}\}\subset [1,\mathrm{\infty})$ with ${\mu}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that $\varphi (p,Tx)\le {\mu}_{n}\varphi (p,x)$ for all $x\in C$, $p\in F(T)$ and $n\ge 1$.

**Remark 1.2** The class of asymptotically quasi-*ϕ*-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings which requires the restriction $F(T)=\tilde{F}(T)$.

**Remark 1.3** The classes of asymptotically quasi-*ϕ*-nonexpansive mappings and quasi-*ϕ*-nonexpansive mappings are the generalizations of asymptotically quasi-nonexpansive mappings and quasi-nonexpansive mappings in Hilbert spaces.

Recently, Qin *et al.* [13] introduced a class of generalized asymptotically quasi-*ϕ*-nonexpansive mappings. Recall that a mapping *T* is said to be generalized asymptotically quasi-*ϕ*-nonexpansive iff $F(T)\ne \mathrm{\varnothing}$ and there exist a sequence $\{{\mu}_{n}\}\subset [1,\mathrm{\infty})$ with ${\mu}_{n}\to 1$ as $n\to \mathrm{\infty}$ and a sequence $\{{\nu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that $\varphi (p,Tx)\le {\mu}_{n}\varphi (p,x)+{\nu}_{n}$ for all $x\in C$, $p\in F(T)$ and $n\ge 1$.

**Remark 1.4** The class of generalized asymptotically quasi-*ϕ*-nonexpansive mappings is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings which was studied in [14].

Recently, fixed point and equilibrium problems (1.1) have been intensively investigated based on iterative methods; see [15–28]. The projection method which grants strong convergence of the iterative sequences is one of efficient methods for the problems. In this paper, we investigate the equilibrium problem (1.1) and a fixed point problem of the generalized quasi-*ϕ*-nonexpansive mapping based on a projection method. A strong convergence theorem for solutions of the equilibrium and the fixed point problem is established in a Banach space.

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

**Lemma 1.5** [2]

*Let*

*E*

*be a reflexive*,

*strictly convex*,

*and smooth Banach space*,

*let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of*

*E*,

*and*$x\in E$.

*Then*

**Lemma 1.6** [2]

*Let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of a smooth Banach space*

*E*,

*and*$x\in E$.

*Then*${x}_{0}={\mathrm{\Pi}}_{C}x$

*if and only if*

**Lemma 1.7** [11]

*Let* *E* *be a reflexive*, *strictly convex*, *and smooth Banach space such that both* *E* *and* ${E}^{\ast}$ *have the Kadec*-*Klee property*. *Let* *C* *be a nonempty closed and convex subset of* *E*. *Let* $T:C\to C$ *be a closed asymptotically quasi*-*ϕ*-*nonexpansive mapping*. *Then* $F(T)$ *is a closed convex subset of* *C*.

*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 there exists*$z\in C$

*such that*$f(z,y)+\frac{1}{r}\u3008y-z,Jz-Jx\u3009\ge 0$, $\mathrm{\forall}y\in C$.

*Define a mapping*${S}_{r}:E\to C$

*by*${S}_{r}x=\{z\in C:f(z,y)+\frac{1}{r}\u3008y-z,Jz-Jx\u3009\ge 0,\mathrm{\forall}y\in C\}$.

*Then the following conclusions hold*:

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

*is closed and convex*; - (3)
${S}_{r}$

*is quasi*-*ϕ*-*nonexpansive*; - (4)
$\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x)$, $\mathrm{\forall}q\in F({S}_{r})$.

**Lemma 1.9** [31]

*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*.

*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*$T:C\to C$

*be a closed generalized asymptotically quasi*-

*ϕ*-

*nonexpansive mapping*.

*Assume that*

*T*

*is asymptotically regular on*

*C*

*and that*$\mathcal{F}=F(T)\cap EP(f)$

*is nonempty and bounded*.

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

*be a sequence generated in the following manner*:

*where* ${M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathcal{F}\}$, $\{{\alpha}_{n}\}$ *is a real sequence in* $[0,1]$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* $\{{r}_{n}\}$ *is a real sequence in* $[a,\mathrm{\infty})$, *where* *a* *is some positive real number*. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

*Proof*In view of Lemma 1.7 and Lemma 1.8, we find that ℱ is closed and convex, so that ${\mathrm{\Pi}}_{\mathcal{F}}x$ is well defined for any $x\in C$. Next, we show that ${C}_{n}$ is closed and convex. It is obvious that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{m}$ is closed and convex for some $m\in \mathbb{N}$. We now show that ${C}_{m+1}$ is also closed and convex. For ${z}_{1},{z}_{2}\in {C}_{m+1}$, we see that ${z}_{1},{z}_{2}\in {C}_{m}$. It follows that $z=t{z}_{1}+(1-t){z}_{2}\in {C}_{m}$, where $t\in (0,1)$. Notice that

*t*and $(1-t)$ on both sides of (2.1) and (2.2), respectively, yields that

where $z\in {C}_{m}$. This gives that ${C}_{m+1}$ is closed and convex. Then ${C}_{n}$ is closed and convex. This shows that ${\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}$ is well defined.

which implies that ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi (\overline{x},{x}_{1})$. 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 $n\to \mathrm{\infty}$.

*E*and ${E}^{\ast}$ are reflexive. We may assume 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 that ${u}_{n}\to \overline{x}$ 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

*E*is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. In view of Lemma 1.9, we see that

*E*has the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{n}{x}_{n}-\overline{x}\parallel =0$. Since

we find from the asymptotic regularity of *T* that ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{n+1}{x}_{n}-\overline{x}\parallel =0$, that is, $T{T}^{n}{x}_{n}-\overline{x}\to 0$ as $n\to \mathrm{\infty}$. It follows from the closedness of *T* that $T\overline{x}=\overline{x}$.

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

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

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

In view of Lemma 1.6, we find that $\overline{x}={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$. This completes the proof. □

If *T* is asymptotically quasi-*ϕ*-nonexpansive, then Theorem 2.1 is reduced to the following.

**Corollary 2.2**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property*.

*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*$T:C\to C$

*be a closed asymptotically quasi*-

*ϕ*-

*nonexpansive mapping*.

*Assume that*

*T*

*is asymptotically regular on*

*C*

*and that*$\mathcal{F}=F(T)\cap EP(f)$

*is nonempty and bounded*.

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

*be a sequence generated in the following manner*:

*where* ${M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathcal{F}\}$, $\{{\alpha}_{n}\}$ *is a real sequence in* $[0,1]$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* $\{{r}_{n}\}$ *is a real sequence in* $[a,\mathrm{\infty})$, *where* *a* *is some positive real number*. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

If *T* is quasi-*ϕ*-nonexpansive, then Theorem 2.1 is reduced to the following.

**Corollary 2.3**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property*.

*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*$T:C\to C$

*be a closed quasi*-

*ϕ*-

*nonexpansive mapping*.

*Assume that*$\mathcal{F}=F(T)\cap EP(f)$

*is nonempty*.

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

*be a sequence generated in the following manner*:

*where* $\{{\alpha}_{n}\}$ *is a real sequence in* $[0,1]$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* $\{{r}_{n}\}$ *is a real sequence in* $[a,\mathrm{\infty})$, *where* *a* *is some positive real number*. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

If *T* is the identity, then Theorem 2.1 is reduced to the following.

**Corollary 2.4**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property*.

*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*$EP(f)$

*is nonempty*.

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

*be a sequence generated in the following manner*:

*where* ${M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathcal{F}\}$, $\{{\alpha}_{n}\}$ *is a real sequence in* $[0,1]$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* $\{{r}_{n}\}$ *is a real sequence in* $[a,\mathrm{\infty})$, *where* *a* *is some positive real number*. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{EP(f)}{x}_{1}$.

## Declarations

### Acknowledgements

The study was supported by the Natural Science Foundation of Zhejiang Province (Y6110270).

## Authors’ Affiliations

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