# Strong convergence theorems of the Halpern-Mann’s mixed iteration for a totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping in Banach spaces

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

**2014**:225

https://doi.org/10.1186/1029-242X-2014-225

© Bo and Yi; licensee Springer. 2014

**Received: **27 November 2013

**Accepted: **26 May 2014

**Published: **3 June 2014

## Abstract

In this paper, we introduce a class of totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping to modify the Halpern-Mann-type iteration algorithm for a totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation problem of solution to a system of equilibrium problems. Also, the results presented in the paper improve and extend the corresponding results of Chang *et al.* (Appl. Math. Comput. 218:7864-7870, 2012) and others.

## Keywords

*ϕ*-asymptotically nonexpansive nonself multi-valued mappingiterative sequenceHalpern and Mann-type iteration algorithmnonexpansive retractiongeneralized projection

## 1 Introduction and preliminaries

A Banach space *X* is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel \le 1$ for all $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$ 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 $\{{x}_{n}\},\{{y}_{n}\}\subset X$ with $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0$.

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

exists, where $S(x)=\{x:\parallel x\parallel =1,x\in X\}$. In this case, *X* is said to be smooth. The norm of 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.

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 $x,y\in D$. An element $p\in D$ is called a fixed point of a nonself multi-valued mapping $T:D\to X$ if $p\in Tp$. The set of fixed points of *T* is represented by $F(T)$.

A subset *D* of *X* is said to be retract of *X*, if there exists a continuous mapping $P:X\to D$ such that $Px=x$, for all $x\in X$. It is well known that every nonempty, closed, convex subset of a uniformly convex Banach space *X* is a retract of *X*. A mapping $P:X\to D$ is said to be a retraction, if ${P}^{2}=P$. It follows that if a mapping *P* is a retraction, then $Py=y$ for all *y* in the range of *P*. A mapping $P:X\to D$ is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from *X* to *D*.

*X*is a real Banach space with the dual ${X}^{\ast}$,

*D*is a nonempty, closed, convex subset of

*X*. We also 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.

*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 functional defined by

*ϕ*that

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

**Lemma 1.1** (see [3])

*Let* *X* *be a uniformly convex and smooth Banach space and let* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *be two sequences of* *X* *such that* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *is bounded*, *if* $\varphi ({x}_{n},{y}_{n})\to 0$, *then* $\parallel {x}_{n}-{y}_{n}\parallel \to 0$.

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

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 1.2** (see [2])

*Let*

*X*

*be a smooth*,

*strictly convex*,

*and reflexive Banach space and*

*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)
$\varphi (x,{\mathrm{\Pi}}_{D}y)+\varphi ({\mathrm{\Pi}}_{D}y,y)\le \varphi (x,y)$, $\mathrm{\forall}x,y\in D$;

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

**Remark 1.1** (see [4])

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*.

**Remark 1.2** (see [4])

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

**Definition 1.1**Let $P:X\to D$ be the nonexpansive retraction.

- (1)A nonself multi-valued mapping $T:D\to X$ is said to be quasi-
*ϕ*-nonexpansive, if $F(T)\ne \mathrm{\Phi}$, and$\varphi (p,{z}_{n})\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n-1}x,\mathrm{\forall}n\ge 1;$(1.7) - (2)A nonself multi-valued mapping $T:D\to X$ is said to be quasi-
*ϕ*-asymptotically nonexpansive, if $F(T)\ne \mathrm{\Phi}$ and there exists a real sequence ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$ (as $n\to \mathrm{\infty}$) such that$\varphi (p,{z}_{n})\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n-1}x,\mathrm{\forall}n\ge 1;$(1.8) - (3)A nonself multi-valued mapping $T:D\to X$ 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$\begin{array}{r}\varphi (p,{z}_{n})\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\\ \phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n-1}x,\mathrm{\forall}n\ge 1.\end{array}$(1.9)

**Remark 1.3** From the definitions, it is obvious that a quasi-*ϕ*-nonexpansive nonself multi-valued mapping is a quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping, and a quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping is a totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping, but the converse is not true.

Now, we give an example of totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping.

**Example 1.1** (see [4])

*D*be a unit ball in a real Hilbert space ${l}^{2}$ and let $T:D\to {l}^{2}$ be a nonself multi-valued mapping defined by

where $\{{a}_{i}\}$ is a sequence in $(0,1)$ such that ${\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}$.

- (i)
$\parallel Tx-Ty\parallel \le 2\parallel x-y\parallel $, $\mathrm{\forall}x,y\in D$;

- (ii)
$\parallel {T}^{n}x-{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}$, $\mathrm{\forall}x,y\in D$, $n\ge 2$.

*D*is a unit ball in a real Hilbert space ${l}^{2}$, it follows from Remark 1.2 that $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $\mathrm{\forall}x,y\in D$. The inequality above can be written as

where *P* is the nonexpansive retraction. This shows that the mapping *T* defined above is a totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping.

**Lemma 1.3** *Let* *X* *be a smooth*, *strictly convex and reflexive Banach space and* *D* *be a nonempty*, *closed*, *convex subset of* *X*. *Let* $T:D\to X$ *be a totally quasi*-*ϕ*-*asymptotically nonexpansive nonself multi*-*valued mapping with* ${\mu}_{1}=0$. *Then* $F(T)$ *is a closed and convex subset of* *D*.

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

*T*is a totally quasi-

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

By Lemma 1.2, we obtain $p=z\in Tp$. So we have $p\in F(T)$. This implies that $F(T)$ is closed.

*ϕ*, let $\{{u}_{n}\}$ be a sequence generated by ${u}_{1}\in Tw,{u}_{2}\in T(PT)w,{u}_{3}\in T{(PT)}^{2}w,\dots ,{u}_{n}\in T{(PT)}^{n-1}w\subset TP{u}_{n-1}$, we have

Hence, ${u}_{n}\to w$ holds, which yields ${u}_{n+1}\to w$. Since *TP* is closed and ${u}_{n+1}\in T{(PT)}^{n}w\subset TP{u}_{n}$, we have $w\in TPw$. It follows from $w\in D$ that $w\in Tw$, *i.e.*, $w\in F(T)$. This implies that $F(T)$ is convex. This completes the proof of Lemma 1.3. □

**Definition 1.2** (see [1])

*L*-Lipschitz continuous, if there exists a constant $L>0$, such that

**Definition 1.3**A nonself multi-valued mapping $T:D\to X$ is said to be uniformly

*L*-Lipschitz continuous, if there exists a constant $L>0$, such that

where $d(\cdot ,\cdot )$ is Hausdorff metric.

Strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-*ϕ*-nonexpansive and quasi-*ϕ*-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [1–4, 6–24]). In recent years, by hybrid projection methods, strong and weak convergence problems for totally quasi-*ϕ* quasi-*ϕ*-asymptotically nonexpansive nonself and multi-valued mapping, respectively, was also studied by Kim *et al.* (see [6, 7]), Li *et al.* (see [8]), Chang *et al.* (see [9]) and Yang *et al.* (see [10]).

Inspired by specialists above, the purpose of this paper is to modify the Halpern-Mann’s mixed type iteration algorithm for a totally quasi-*ϕ*-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang *et al.* [1, 11–13], Hao *et al.* [14], Guo *et al.* [15], Yildirim *et al.* [16], Thianwan [17], Nilsrakoo *et al.* [18], Pathak *et al.* [19], Qin *et al.* [20], Su *et al.* [21], Wang [22, 23], Yang *et al.* [24] and others.

## 2 Main results

**Theorem 2.1**

*Let*

*X*

*be a real uniformly smooth and uniformly convex Banach space*,

*D*

*be a nonempty*,

*closed*,

*convex subset of*

*X*.

*Let*$P:X\to D$

*be the nonexpansive retraction*.

*Let*$T:D\to X$

*be a totally quasi*-

*ϕ*-

*asymptotically nonexpansive nonself multi*-

*valued mapping with sequence*$\{{v}_{n}\}$, $\{{\mu}_{n}\}$ (${\mu}_{1}=0$),

*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*

*T*

*is uniformly*

*L*-

*Lipschitz continuous*.

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

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

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

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

*satisfying the following conditions*:

- (i)
${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$;

- (ii)
$0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1$.

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

*be a sequence generated by*

*where* ${\xi}_{n}={v}_{n}{sup}_{p\in F(T)}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}$, ${\mathrm{\Pi}}_{{D}_{n+1}}$ *is the generalized projection of* *X* *onto* ${D}_{n+1}$. *If* $F(T)\ne \mathrm{\varnothing}$, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)}{x}_{1}$.

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

*D*. By the assumption, ${D}_{1}=D$ is closed and convex. Suppose that ${D}_{n}$ is closed and convex for some $n\ge 1$. In view of the definition of

*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*${p}^{\ast}$.

Therefore, $\{\varphi ({x}_{n},{x}_{1})\}$ is bounded, and so is $\{{x}_{n}\}$. Since ${x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}$ and ${x}_{n+1}={\mathrm{\Pi}}_{{D}_{n+1}}{x}_{1}\in {D}_{n+1}\subset {D}_{n}$, we have $\varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1})$. This implies that $\{\varphi ({x}_{n},{x}_{1})\}$ is nondecreasing. Hence ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})$ exists.

It follows from Lemma 1.1 that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{m}-{x}_{n}\parallel =0$. Hence $\{{x}_{n}\}$ is a Cauchy sequence in *D*. Since *D* is complete, without loss of generality, we can assume that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}$ (some point in *D*).

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

*T*is a totally quasi-

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

This implies that $\{{z}_{n}\}$ is also bounded.

this implies that $\{{w}_{n}\}$ is also bounded.

*J*is uniformly continuous on each bounded subset of

*X*, we have

*J*is uniformly continuous, this shows that

*L*-Lipschitz continuous, thus we have

We get ${lim}_{n\to \mathrm{\infty}}d(T{(PT)}^{n}{x}_{n},T{(PT)}^{n-1}{x}_{n})=0$, since ${lim}_{n\to \mathrm{\infty}}{z}_{n}={p}^{\ast}$ and ${lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}$.

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

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

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

By Remark 1.3, the following corollary is obtained.

**Corollary 2.1** *Let* *X*, *D*, $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be the same as in Theorem * 2.1. *Let* $T:D\to X$ *be a quasi*-*ϕ*-*asymptotically nonexpansive nonself multi*-*valued mapping with sequence* ${k}_{n}\subset [1,+\mathrm{\infty})$, ${k}_{n}\to 1$, $T:D\to X$ *be uniformly* *L*-*Lipschitz continuous*.

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

*be a sequence generated by*

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

**Corollary 2.2** *Let* *X*, *D*, $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be the same as in Theorem * 2.1. *Let* $T:D\to X$ *be a quasi*-*ϕ*-*nonexpansive nonself multi*-*valued mapping*, $T:D\to X$ *be uniformly* *L*-*Lipschitz continuous*.

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

*is a sequence generated by*

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

## 3 Application

First, we present an example of a quasi-*ϕ*-nonexpansive nonself multi-valued mapping.

**Example 3.1** (see [4])

Let *H* be a real Hilbert space, *D* be a nonempty closed and convex subset of *H* and $f:D\times D\to R$ 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 a ${x}^{\ast}\in D$ 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, so $z={T}_{r}(x)$; (2) ${T}_{r}$ is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-*ϕ*-nonexpansive nonself mapping; (3) $F({T}_{r})=EP(f)$ and $F({T}_{r})$ is a nonempty and closed convex subset of *D*; (4) ${T}_{r}:D\to D$ is a nonexpansive. Since $F({T}_{r})$ nonempty, it is a quasi-*ϕ*-nonexpansive nonself mapping from *D* to *H*, where $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$.

In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.

**Theorem 3.1**

*Let*

*H*

*be a real Hilbert space*,

*D*

*be a nonempty closed and convex subset of*

*H*, $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$

*be the same as in Theorem*2.1.

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

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

*as given in Example*3.1.

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

*be mapping defined by*(3.1),

*i*.

*e*.,

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

*be the sequence generated by*

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

*Proof*In Example 3.1, we have pointed out that ${u}_{n}={T}_{r}({x}_{n})$, $F({T}_{r})=EP(f)$ is nonempty and convex, ${T}_{r}$ is a quasi-

*ϕ*-nonexpansive nonself mapping. Since $F({T}_{r})$ is nonempty, and so ${T}_{r}$ is a quasi-

*ϕ*-nonexpansive mapping and ${T}_{r}$ is uniformly 1-Lipschitzian mapping. Hence (3.1) can be rewritten as follows:

Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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