# Strong convergence theorems for modifying Halpern iterations for a totally quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping in reflexive Banach spaces

- Hong Bo Liu
^{1}Email author and - Yi Li
^{1}

**2013**:126

https://doi.org/10.1186/1029-242X-2013-126

© Liu and Li; licensee Springer 2013

**Received: **21 October 2012

**Accepted: **26 February 2013

**Published: **26 March 2013

## Abstract

In this paper, we discuss an iterative sequence for a totally quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping for modifying Halpern’s iterations and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang *et al.* (Appl. Math. Comput. 218:6489-6497, 2012).

**MSC:**47J05, 47H09, 49J25.

### Keywords

multi-valued mapping quasi-*ϕ*-asymptotically nonexpansive total quasi-

*ϕ*-asymptotically nonexpansive Halpern iterative sequence

## 1 Introduction

*N*and

*R*the sets of positive integers and real numbers, respectively. Let

*D*be a nonempty closed subset of a real Banach space

*X*. A mapping $T:D\to D$ is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in D$. Let $N(D)$ and $CB(D)$ denote the family of nonempty subsets and nonempty bounded closed subsets of

*D*, respectively. The Hausdorff metric on $CB(D)$ is defined by

for ${A}_{1},{A}_{2}\in CB(D)$, where $d(x,{A}_{2})=inf\{\parallel x-y\parallel ,y\in {A}_{2}\}$. The multi-valued mapping $T:D\to CB(D)$ is called nonexpansive if $H(Tx,Ty)\le \parallel x-y\parallel $ for all $x,y\in D$. An element $p\in D$ is called a fixed point of $T:D\to CB(D)$ if $p\in T(p)$. The set of fixed points of *T* is represented by $F(T)$.

*X*is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel \le 1$ for all $x,y\in S(X)$ and $x\ne y$. A Banach space is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\},\{{y}_{n}\}\subset S(X)$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0$. The norm of the Banach space

*X*is said to be Gâteaux differentiable if for each $x,y\in S(X)$, the limit

exists. In this case, *X* is said to be smooth. The norm of the Banach space *X* is said to be Fréchet differentiable if for each $x\in S(X)$, the limit (1.1) is attained uniformly for $y\in S(x)$, and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for $x,y\in S(X)$. In this case, *X* is said to be uniformly smooth.

*X*be a real Banach space with dual ${X}^{\ast}$. We denote by

*J*the normalized duality mapping from

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

where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing.

**Remark 1.1**The following basic properties for the Banach space

*X*and for the normalized duality mapping

*J*can be found in Cioranescu [1].

- (1)
*X*(${X}^{\ast}$, resp.) is uniformly convex if and only if ${X}^{\ast}$ (*X*, resp.) is uniformly smooth. - (2)
If

*X*is smooth, then*J*is single-valued and norm-to-weak^{∗}continuous. - (3)
If

*X*is reflexive, then*J*is onto. - (4)
If

*X*is strictly convex, then $Jx\cap Jy\ne \mathrm{\Phi}$ for all $x,y\in X$. - (5)
If

*X*has a Fréchet differentiable norm, then*J*is norm-to-norm continuous. - (6)
If

*X*is uniformly smooth, then*J*is uniformly norm-to-norm continuous on each bounded subset of*X*. - (7)
Each uniformly convex Banach space

*X*has the Kadec-Klee property,*i.e.*, for any sequence $\{{x}_{n}\}\subset X$, if ${x}_{n}\rightharpoonup x\in X$ and $\parallel {x}_{n}\parallel \to \parallel x\parallel $, then ${x}_{n}\to x\in X$.

*X*is a smooth, strictly convex, and reflexive Banach space and

*D*is a nonempty closed convex subset of

*X*. In the sequel, we always use $\varphi :X\times X\to {R}^{+}$ to denote the Lyapunov bifunction defined by

for all $\alpha \in [0,1]$ and $x,y,z\in X$.

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

**Remark 1.2** (see [3])

Let ${\mathrm{\Pi}}_{D}$ be the generalized projection from a smooth, reflexive and strictly convex Banach space *X* onto a nonempty closed convex subset *D* of *X*, then ${\mathrm{\Pi}}_{D}$ is a closed and quasi-*ϕ*-nonexpansive from *X* onto *D*.

*T*. However, even in a Hilbert space, the Mann iteration may fail to converge strongly [5]. Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following so-called Halpern iteration:

where $\{{\alpha}_{n}\}$ is a real sequence in $[0,1]$ and ${P}_{K}$ denotes the metric projection from a Hilbert space *H* onto a closed convex subset *K* of *H*. It should be noted here that the iteration above works only in the Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi-*ϕ*-nonexpansive mappings have been introduced by Aoyama *et al.* [8], Chang *et al.* [9, 10], Chidume *et al.* [11], Matsushita *et al.* [12–14], Qin *et al.* [15], Song *et al.* [16], Wang *et al.* [17] and others.

Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a multi-valued mapping $T:D\to CB(D)$ and prove some strong convergence theorems. The results presented in the paper improve and extend the corresponding results in [9].

## 2 Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence $\{{x}_{n}\}$ by ${x}_{n}\to x$ and ${x}_{n}\rightharpoonup x$, respectively.

**Lemma 2.1** (see [2])

*Let*

*X*

*be a smooth*,

*strictly convex and reflexive Banach space*,

*and let*

*D*

*be a nonempty closed convex subset of*

*X*.

*Then the following conclusions hold*:

- (a)$\varphi (x,y)=0$
*if and only if*$x=y$. - (b), $\mathrm{\forall}x,y\in D$.$\varphi (x,{\mathrm{\Pi}}_{D}y)+\varphi ({\mathrm{\Pi}}_{D}y,y)\le \varphi (x,y)$
- (c)
*If*$x\in X$*and*$z\in D$,*then*$z={\mathrm{\Pi}}_{D}x$*if and only if*$\u3008z-y,Jx-Jz\u3009\ge 0$, $\mathrm{\forall}y\in D$.

**Lemma 2.2** (see [9])

*Let* *X* *be a real uniformly smooth and strictly convex Banach space with the Kadec*-*Klee property*, *and let* *D* *be a nonempty closed convex subset of* *X*. *Let* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *be two sequences in* *D* *such that* ${x}_{n}\to p$ *and* $\varphi ({x}_{n},{y}_{n})\to 0$, *where* *ϕ* *is the function defined by* (1.2), *then* ${y}_{n}\to p$.

**Definition 2.1** A point $p\in D$ is said to be an asymptotic fixed point of a multi-valued mapping $T:D\to CB(D)$ if there exists a sequence $\{{x}_{n}\}\subset D$ such that ${x}_{n}\rightharpoonup x\in X$ and $d({x}_{n},T({x}_{n}))\to 0$. Denote the set of all asymptotic fixed points of *T* by $\stackrel{\u02c6}{F}(T)$.

**Definition 2.2**

- (1)
A multi-valued mapping $T:D\to CB(D)$ is said to be relatively nonexpansive if $F(T)\ne \mathrm{\Phi}$, $\stackrel{\u02c6}{F}(T)=F(T)$ and $\varphi (p,z)\le \varphi (p,x)$, $\mathrm{\forall}x\in D$, $p\in F(T)$, $z\in T(x)$.

- (2)
A multi-valued mapping $T:D\to CB(D)$ is said to be closed if for any sequence $\{{x}_{n}\}\subset D$ with ${x}_{n}\to x\in X$ and $d(y,T({x}_{n}))\to 0$, then $d(y,T(x))=0$.

**Remark 2.1** If *H* is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel}^{2}$ and ${\mathrm{\Pi}}_{D}$ is the metric projection ${P}_{D}$ of *H* onto *D*.

Next, we present an example of a relatively nonexpansive multi-valued mapping.

**Example 2.1** (see [18])

Let *X* be a smooth, strictly convex and reflexive Banach space, let *D* be a nonempty closed and convex subset of *X*, and let $f:D\times D\to R$ be a bifunction satisfying the conditions: (A1) $f(x,x)=0$, $\mathrm{\forall}x\in D$; (A2) $f(x,y)+f(y,x)\le 0$, $\mathrm{\forall}x,y\in D$; (A3) for each $x,y,z\in D$, ${lim}_{t\to 0}f(tz+(1-t)x,y)\le f(x,y)$; (A4) for each given $x\in D$, the function $y\mapsto f(x,y)$ is convex and lower semicontinuous. The so-called equilibrium problem for *f* is to find an ${x}^{\ast}\in D$ such that $f({x}^{\ast},y)\ge 0$, $\mathrm{\forall}y\in D$. The set of its solutions is denoted by $EP(f)$.

then (1) ${T}_{r}$ is single-valued, and so $\{z\}={T}_{r}(x)$; (2) ${T}_{r}$ is a relatively nonexpansive mapping, therefore, it is a closed quasi-*ϕ*-nonexpansive mapping; (3) $F({T}_{r})=EP(f)$.

**Definition 2.3**

- (1)
A multi-valued mapping $T:D\to CB(D)$ is said to be quasi-

*ϕ*-nonexpansive if $F(T)\ne \mathrm{\Phi}$ and $\varphi (p,z)\le \varphi (p,x)$, $\mathrm{\forall}x\in D$, $p\in F(T)$, $z\in Tx$. - (2)A multi-valued mapping $T:D\to CB(D)$ is said to be quasi-
*ϕ*-asymptotically nonexpansive if $F(T)\ne \mathrm{\Phi}$ and there exists a real sequence ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$, such that$\varphi (p,{z}_{n})\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in {T}^{n}x.$(2.2) - (3)A multi-valued mapping $T:D\to CB(D)$ is said to be totally quasi-
*ϕ*-asymptotically nonexpansive if $F(T)\ne \mathrm{\Phi}$ and there exist nonnegative real sequences $\{{v}_{n}\}$, $\{{\mu}_{n}\}$ with ${v}_{n},{\mu}_{n}\to 0$ (as $n\to \mathrm{\infty}$) and a strictly increasing continuous function $\zeta :{R}^{+}\to {R}^{+}$ with $\zeta (0)=0$ such that

**Remark 2.2** From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi-*ϕ*-nonexpansive multi-valued mapping, and a quasi-*ϕ*-nonexpansive multi-valued mapping is a quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping, and a quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping is a total quasi-*ϕ*-asymptotically nonexpansive multi-valued mapping, but the converse is not true.

**Lemma 2.3** *Let* *X* *and* *D* *be as in Lemma* 2.2. *Let* $T:D\to CB(D)$ *be a closed and totally quasi*-*ϕ*-*asymptotically nonexpansive multi*-*valued mapping with nonnegative real sequences* $\{{v}_{n}\}$, $\{{\mu}_{n}\}$ *and a strictly increasing continuous function* $\zeta :{R}^{+}\to {R}^{+}$ *with* $\zeta (0)=0$. *If* ${v}_{n},{\mu}_{n}\to 0$ (*as* $n\to \mathrm{\infty}$) *and* ${\mu}_{1}=0$, *then* $F(T)$ *is a closed and convex subset of* *D*.

*Proof*Let $\{{x}_{n}\}$ be a sequence in $F(T)$ such that ${x}_{n}\to {x}^{\ast}$. Since

*T*is a totally quasi-

*ϕ*-asymptotically nonexpansive multi-valued mapping, we have

By Lemma 2.1(a), we obtain $z={x}^{\ast}$. Hence, $T{x}^{\ast}=\{{x}^{\ast}\}$. So, we have ${x}^{\ast}\in F(T)$. This implies $F(T)$ is closed.

*ϕ*, letting ${z}_{n}\in {T}^{n}w$, we have

By Lemma 2.2, we have ${z}_{n}\to w$. This implies that ${z}_{n+1}\phantom{\rule{0.25em}{0ex}}(\in T{T}^{n}w)\to w$. Since *T* is closed, we have $Tw=\{w\}$, *i.e.*, $w\in F(T)$. This completes the proof of Lemma 2.3. □

**Definition 2.4** A mapping $T:D\to CB(D)$ is said to be uniformly *L*-Lipschitz continuous if there exists a constant $L>0$ such that $\parallel {x}_{n}-{y}_{n}\parallel \le L\parallel x-y\parallel $, where $x,y\in D$, ${x}_{n}\in {T}^{n}x$, ${y}_{n}\in {T}^{n}y$.

## 3 Main results

**Theorem 3.1**

*Let*

*X*

*be a real uniformly smooth and strictly convex Banach space with the Kadec*-

*Klee property*,

*let*

*D*

*be a nonempty closed convex subset of*

*X*,

*and let*$T:D\to CB(D)$

*be a closed and uniformly*

*L*-

*Lipschitz continuous totally quasi*-

*ϕ*-

*asymptotically nonexpansive multi*-

*valued mapping with nonnegative real sequences*$\{{v}_{n}\}$, $\{{\mu}_{n}\}$, ${v}_{n},{\mu}_{n}\to 0$ (

*as*$n\to \mathrm{\infty}$)

*and a strictly increasing continuous function*$\zeta :{R}^{+}\to {R}^{+}$

*with*$\zeta (0)=0$

*satisfying condition*(2.3).

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

*be a sequence in*$[0,1]$

*such that*${\alpha}_{n}\to 0$.

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

*is the sequence generated by*

*where* ${\xi}_{n}={v}_{n}{sup}_{p\in F(T)}\zeta [\varphi (p,{x}_{n})]+{\mu}_{n}$, $F(T)$ *is the fixed point set of* *T*, *and* ${\mathrm{\Pi}}_{{D}_{n+1}}$ *is the generalized projection of* *X* *onto* ${D}_{n+1}$. *If* $F(T)$ *is nonempty and* ${\mu}_{1}=0$, *then* ${lim}_{n\to \mathrm{\infty}}{x}_{n}={\mathrm{\Pi}}_{F(T)}{x}_{1}$.

*Proof* (I) First, we prove that ${D}_{n}$ is a closed and convex subset in *D*.

*ϕ*, we have

- (II)
Next, we prove that $F(T)\subset {D}_{n}$ for all $n\ge 1$.

- (III)
Now we prove that $\{{x}_{n}\}$ converges strongly to some point ${p}^{\ast}$.

*X*is reflexive, there exists a subsequence $\{{x}_{{n}_{i}}\}\subset \{{x}_{n}\}$ such that ${x}_{{n}_{i}}\rightharpoonup {p}^{\ast}$ (some point in $D={D}_{1}$). Since ${D}_{n}$ is closed and convex and ${D}_{n+1}\subset {D}_{n}$. This implies that ${D}_{n}$ is weakly closed and ${p}^{\ast}\in {D}_{n}$ for each $n\ge 1$. In view of ${x}_{{n}_{i}}={\mathrm{\Pi}}_{{D}_{{n}_{i}}}{x}_{1}$, we have

*X*, we obtain that ${x}_{{n}_{i}}\to {p}^{\ast}$. Since $\{\varphi ({x}_{n},{x}_{1})\}$ is convergent, this together with ${lim}_{{n}_{i}\to \mathrm{\infty}}\varphi ({x}_{{n}_{i}},{x}_{1})=\varphi ({p}^{\ast},{x}_{1})$ shows that ${lim}_{{n}_{i}\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi ({p}^{\ast},{x}_{1})$. If there exists some subsequence $\{{x}_{{n}_{j}}\}\subset \{{x}_{n}\}$ such that ${x}_{{n}_{j}}\to q$, then from Lemma 2.1 we have

*i.e.*, ${p}^{\ast}=q$, and hence

- (IV)
Now we prove that ${p}^{\ast}\in F(T)$.

*T*is a totally quasi-

*ϕ*-asymptotically nonexpansive multi-valued mapping, ${T}^{n}{x}_{n}$ is bounded. In view of ${\alpha}_{n}\to 0$, from (3.1), we have

*X*shows that

*T*is

*L*-Lipschitz continuous, we have

*T*, it yields that $T({p}^{\ast})=\{{p}^{\ast}\}$, which implies that ${p}^{\ast}\in F(T)$.

- (V)
Finally, we prove that ${p}^{\ast}={\mathrm{\Pi}}_{F(T)}{x}_{1}$ and so ${x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}$.

which yields that ${p}^{\ast}=w={\mathrm{\Pi}}_{F(T)}{x}_{1}$. Therefore, ${x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}$. The proof of Theorem 3.1 is completed. □

By Remark 2.2, the following corollaries are obtained.

**Corollary 3.1**

*Let*

*X*

*and*

*D*

*be as in Theorem*3.1,

*and let*$T:D\to CB(D)$

*be a closed and uniformly*

*L*-

*Lipschitz continuous relatively nonexpansive multi*-

*valued mapping*.

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

*in*$(0,1)$

*with*${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$.

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

*be the sequence generated by*

*where* $F(T)$ *is the set of fixed points of* *T*, *and* ${\mathrm{\Pi}}_{{D}_{n+1}}$ *is the generalized projection of* *X* *onto* ${D}_{n+1}$, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)}{x}_{1}$.

**Corollary 3.2** *Let* *X* *and* *D* *be as in Theorem* 3.1, *and* $T:D\to CB(D)$ *be a closed and uniformly* *L*-*Lipschitz continuous quasi*-*ϕ*-*nonexpansive multi*-*valued mapping*. *Let* $\{{\alpha}_{n}\}$ *be a sequence of real numbers such that* ${\alpha}_{n}\in (0,1)$ *for all* $n\in N$ *and satisfy* ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$. *Let* $\{{x}_{n}\}$ *be the sequence generated by* (3.14). *Then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)}{x}_{1}$.

**Corollary 3.3**

*Let*

*X*

*be a real uniformly smooth and strictly convex Banach space with the Kadec*-

*Klee property*,

*let*

*D*

*be a nonempty closed convex subset of*

*X*,

*and let*$T:D\to CB(D)$

*be a closed and uniformly*

*L*-

*Lipschitz continuous quasi*-

*ϕ*-

*asymptotically nonexpansive multi*-

*valued mapping with nonnegative real sequences*$\{{k}_{n}\}\subset [1,+\mathrm{\infty})$

*and*${k}_{n}\to 1$

*satisfying condition*(2.2).

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

*be a sequence in*$(0,1)$

*and satisfy*${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$.

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

*is the sequence generated by*

*where* ${\xi}_{n}=({k}_{n}-1){sup}_{p\in F(T)}\varphi (p,{x}_{n})$, $F(T)$ *is the fixed point set of* *T*, *and* ${\mathrm{\Pi}}_{{D}_{n+1}}$ *is the generalized projection of* *X* *onto* ${D}_{n+1}$, *if* $F(T)$ *is nonempty*, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)}{x}_{1}$.

## 4 Application

We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.

**Theorem 4.1**

*Let*

*D*,

*X*

*and*$\{{\alpha}_{n}\}$

*be the same as in Theorem*3.1.

*Let*$f:D\times D\to R$

*be a bifunction satisfying conditions*(A1)-(A4)

*as given in Example*2.1.

*Let*${T}_{r}:X\to D$

*be a mapping defined by*(2.1),

*i*.

*e*.,

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

*be the sequence generated by*

*If* $F({T}_{r})\ne \mathrm{\Phi}$, *then* $\{{x}_{n}\}$ *converges strongly to* ${\prod}_{F(T)}{x}_{1}$, *which is a common solution of the system of equilibrium problems for* *f*.

*Proof*In Example 2.1, we have pointed out that ${u}_{n}={T}_{r}({x}_{n})$, $F({T}_{r})=EP(f)$ and ${T}_{r}$ is a closed quasi-

*ϕ*-nonexpansive mapping. Hence (4.1) can be rewritten as follows:

Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.2. □

## Declarations

### Acknowledgements

The authors are very grateful to both reviewers for carefully reading this paper and for their comments.

## Authors’ Affiliations

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