# Strong convergence of a CQ method for *k*-strictly asymptotically pseudocontractive mappings

- Hossein Dehghan
^{1}and - Naseer Shahzad
^{2}Email author

**2012**:208

https://doi.org/10.1186/1687-1812-2012-208

© Dehghan and Shahzad; licensee Springer 2012

**Received: **6 February 2012

**Accepted: **26 October 2012

**Published: **22 November 2012

## Abstract

Let *E* be a real *q*-uniformly smooth Banach space, which is also uniformly convex (for example, ${L}_{p}$ or ${\ell}_{p}$ spaces, $1<p<\mathrm{\infty}$), and *C* be a nonempty bounded closed convex subset of *E*. Let $T:C\to C$ be a *k*-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the proposed algorithm is established.

## Keywords

*k*-strictly asymptotically pseudocontractive mapping

## 1 Introduction

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

*E*. We denote the value of ${x}^{\ast}\in {E}^{\ast}$ at $x\in E$ by $\u3008x,{x}^{\ast}\u3009$. The normalized duality mapping

*J*from

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

for all $x\in E$. It is known that a Banach space *E* is smooth if and only if the normalized duality mapping *J* is single valued. Some properties of the duality mapping have been given in [1, 2].

*C*be a nonempty subset of

*E*. The mapping $T:C\to C$ is called

*nonexpansive*if

*T*is called

*uniformly*

*L*-

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

*k*-

*strictly asymptotically pseudocontractive*if there exist a sequence $\{{k}_{n}\}$ in $[1,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ and a constant $k\in [0,1)$, and for any $x,y\in C$, there exists $j(x-y)\in J(x-y)$ such that

*I*denotes the identity operator, then (1.1) can be written in the form

*k*-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces,

*j*is the identity and it is shown [4] that (1.1) (and hence (1.2)) is equivalent to the inequality

which is the inequality considered by Qihou [3]. In the same paper, the author proved strong convergence of the modified Mann iteration processes for *k*-strictly asymptotically pseudocontractive mappings in Hilbert spaces. The modified Mann iteration scheme was introduced by Schu [5, 6] and has been used by several authors (see, for example, [7–12]). In [13] Osilike extended Qihou’s result from Hilbert spaces to much more general real *q*-uniformly smooth Banach spaces, $1<q<\mathrm{\infty}$.

The classes of nonexpansive and asymptotically nonexpansive mappings are important classes of mappings because they have applications to solutions of differential equations which have been studied by several authors (see, *e.g.*, [14–16] and references contained therein). It would be of interest to study the class of *k*-strictly asymptotically pseudocontractive mappings in view of the fact that it is closely related to the above two classes.

where $\overline{co}D$ denotes the convex closure of the set *D*, *J* is the normalized duality mapping, $\{{t}_{n}\}$ is a sequence in $(0,1)$ with ${t}_{n}\to 0$, and ${P}_{{C}_{n}\cap {D}_{n}}$ is the metric projection from *E* onto ${C}_{n}\cap {D}_{n}$. Then, they proved that $\{{x}_{n}\}$ generated by (1.3) converges strongly to a fixed point of the mapping *T*.

*k*-strictly asymptotically pseudocontractive mapping

*T*in a uniformly convex and

*q*-uniformly smooth Banach space: ${x}_{1}=x\in C$, ${C}_{0}={D}_{0}=C$ and

where $\overline{co}D$ denotes the convex closure of the set *D*, *J* is the normalized duality mapping, $\{{t}_{n}\}$ is a sequence in $(0,1)$ with ${t}_{n}\to 0$, and ${P}_{{C}_{n}\cap {D}_{n}}$ is the metric projection from *E* onto ${C}_{n}\cap {D}_{n}$.

The purpose of this paper is to establish a strong convergence theorem of the iterative algorithm (1.4) for *k*-strictly asymptotically pseudocontractive mappings in a uniformly convex and *q*-uniformly smooth Banach space.

## 2 Preliminaries

*modulus of smoothness*of a Banach space

*E*is the function ${\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by

*E* is *uniformly smooth* if and only if ${lim}_{t\to {0}^{+}}{\rho}_{E}(t)/t=0$. Let $q>1$. The Banach space *E* is said to be *q*-*uniformly smooth* if there exists a constant $c>0$ such that ${\rho}_{E}(t)\le c{t}^{q}$. Hilbert spaces, ${L}_{p}$ (or ${\ell}_{p}$) spaces, $1<p<\mathrm{\infty}$, and the Sobolev spaces, ${W}_{m}^{p}$, $1<p<\mathrm{\infty}$, are *q*-uniformly smooth.

When $\{{x}_{n}\}$ is a sequence in *E*, we denote strong convergence of $\{{x}_{n}\}$ to $x\in E$ by ${x}_{n}\to x$ and weak convergence by ${x}_{n}\rightharpoonup x$. The Banach space *E* is said to have the Kadec-Klee property if for every sequence $\{{x}_{n}\}$ in *E*, ${x}_{n}\rightharpoonup x$ and $\parallel {x}_{n}\parallel \to \parallel x\parallel $ imply that ${x}_{n}\to x$. Every uniformly convex Banach space has the Kadec-Klee property [1].

*C*be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space

*E*. Then for any $x\in E$, there exists a unique point ${x}_{0}\in C$ such that

*metric projection*from

*E*onto

*C*. Let $x\in E$ and $u\in C$. Then it is known that $u={P}_{C}x$ if and only if

for all $y\in C$ ( see [1, 18]).

In the sequel, we need the following results.

**Proposition 2.1** (See [19])

*Let*

*C*

*be a bounded closed convex subset of a uniformly convex Banach space*

*E*.

*Then there exists a strictly increasing convex continuous function*$\gamma :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

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

*depending only on the diameter of*

*C*

*such that*

*holds for any nonexpansive mapping* $T:C\to E$, *any elements* ${x}_{1},\dots ,{x}_{n}$ *in* *C*, *and any numbers* ${\lambda}_{1},\dots ,{\lambda}_{n}\ge 0$ *with* ${\lambda}_{1}+\cdots +{\lambda}_{n}=1$.

**Corollary 2.2** [[20], Corollary 1.2]

*Under the same suppositions as in Proposition*2.1,

*there exists a strictly increasing convex continuous function*$\gamma :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

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

*depending only on the diameter of*

*C*

*such that*

*holds for any nonexpansive mapping* $T:C\to E$, *any elements* ${x}_{1},\dots ,{x}_{n}$ *in* *C*, *and any numbers* ${\lambda}_{1},\dots ,{\lambda}_{n}\ge 0$ *with* ${\lambda}_{1}+\cdots +{\lambda}_{n}=1$. (*Note that* *γ* *does not depend on* *T*.)

In order to utilize Corollary 2.2 for *k*-strictly asymptotically pseudocontractive mappings, we need the following lemmas.

**Lemma 2.3** [4]

*Let* *E* *be a real Banach space*, *C* *be a nonempty subset of* *E*, *and* $T:C\to C$ *be a* *k*-*strictly asymptotically pseudocontractive mapping*. *Then* *T* *is uniformly* *L*-*Lipschitzian*.

**Lemma 2.4** [[21], Lemma 3.1]

*Let*

*E*

*be a real*

*q*-

*uniformly smooth Banach space and*

*C*

*be a nonempty convex subset of*

*E*.

*Let*$T:C\to C$

*be a*

*k*-

*strictly asymptotically pseudocontractive map*,

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

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

*Define*${S}_{n}:C\to C$

*by*${S}_{n}x:=(1-{\alpha}_{n})x+{\alpha}_{n}{T}^{n}x$

*for all*$x\in C$.

*Then for all*$x,y\in C$,

*we have*

*where* *L* *is the uniformly Lipschitzian constant of* *T* *and* ${c}_{q}>0$ *is the constant which appeared in* [[21], *Theorem * 2.1].

for all $x,y\in C$ and each $n\ge 1$.

**Theorem 2.5** [[21], Theorem 3.1]

*Let* *E* *be a real* *q*-*uniformly smooth Banach space which is also uniformly convex*. *Let* *C* *be a nonempty closed convex subset of* *E* *and* $T:C\to C$ *be a k*-*strictly asymptotically pseudocontractive mapping with a nonempty fixed point set*. *Then* $(I-T)$ *is demiclosed at zero*, *i*.*e*., *if* ${x}_{n}\rightharpoonup x$ *and* ${x}_{n}-T{x}_{n}\to 0$, *then* $x\in F(T)$, *where* $F(T)$ *is the set of all fixed points of* *T*.

## 3 Strong convergence theorem

In this section, we study the iterative algorithm (1.4) for finding fixed points of *k*-strictly asymptotically pseudocontractive mappings in a uniformly convex and *q*-uniformly smooth Banach space. We first prove that the sequence $\{{x}_{n}\}$ generated by (1.4) is well defined. Then, we prove that $\{{x}_{n}\}$ converges strongly to ${P}_{F(T)}x$, where ${P}_{F(T)}$ is the metric projection from *E* onto $F(T)$.

**Lemma 3.1** *Let* *C* *be a nonempty closed convex subset of a reflexive*, *strictly convex*, *and smooth Banach space* *E*, *and let* $T:C\to C$ *be a mapping*. *If* $F(T)\ne \mathrm{\varnothing}$, *then the sequence* $\{{x}_{n}\}$ *generated by* (1.4) *is well defined*.

*Proof*It is easy to check that ${C}_{n}\cap {D}_{n}$ is closed and convex and $F(T)\subset {C}_{n}$ for each $n\in \mathbb{N}$. Moreover, ${D}_{1}=C$ and so $F(T)\subset {C}_{1}\cap {D}_{1}$. Suppose $F(T)\subset {C}_{k}\cap {D}_{k}$ for $k\in \mathbb{N}$. Then there exists a unique element ${x}_{k+1}\in {C}_{k}\cap {D}_{k}$ such that ${x}_{k+1}={P}_{{C}_{k}\cap {D}_{k}}x$. If $u\in F(T)$, then it follows from (2.1) that

which implies $u\in {D}_{k+1}$. Therefore, $F(T)\subset {C}_{k+1}\cap {D}_{k+1}$. By the mathematical induction, we obtain that $F(T)\subset {C}_{n}\cap {D}_{n}$ for all $n\in \mathbb{N}$. Therefore, $\{{x}_{n}\}$ is well defined. □

In order to prove our main result, the following lemma is needed.

**Lemma 3.2**

*Let*

*C*

*be a nonempty bounded closed convex subset of a real*

*q*-

*uniformly smooth and uniformly convex Banach space*

*E*.

*Let*$T:C\to C$

*be a*

*k*-

*strictly asymptotically pseudocontractive mapping with*$\{{k}_{n}\}$

*such that*$F(T)\ne \mathrm{\varnothing}$.

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

*be the sequence generated by*(1.4),

*then for any*$j\in \mathbb{N}$,

*Proof*Fix $j\in \mathbb{N}$ and put $m=n-j$. Since ${x}_{n}={P}_{{C}_{n-1}\cap {D}_{n-1}}x$, we have ${x}_{n}\in {C}_{n-1}\subseteq \cdots \subseteq {C}_{m}$. Since ${t}_{m}>0$, there exist ${y}_{1},\dots ,{y}_{N}\in C$ and ${\lambda}_{1},\dots ,{\lambda}_{N}\ge 0$ with ${\lambda}_{1}+\cdots +{\lambda}_{N}=1$ such that

*T*is uniformly

*L*-Lipschitzian. Put $M={sup}_{x\in C}\parallel x\parallel $, $u={P}_{F(T)}x$ and ${r}_{0}={sup}_{n\ge 1}(1+L)\parallel {x}_{n}-u\parallel $. Thus,

Since and , it follows from the last inequality that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{H}_{m}{x}_{n}\parallel =0$. Thus, ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{S}_{m}{x}_{n}\parallel =0$ and so ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}^{m}{x}_{n}\parallel =0$. This completes the proof. □

**Theorem 3.3** *Let* *C* *be a nonempty bounded closed convex subset of a real* *q*-*uniformly smooth and uniformly convex Banach space* *E*. *Let* $T:C\to C$ *be a k*-*strictly asymptotically pseudocontractive mapping with* $\{{k}_{n}\}$ *such that* $F(T)\ne \mathrm{\varnothing}$. *Let* $\{{x}_{n}\}$ *be the sequence generated by* (1.4). *Then* $\{{x}_{n}\}$ *converges strongly to the element* ${P}_{F(T)}x$ *of* $F(T)$, *where* ${P}_{F(T)}$ *is the metric projection from* *E* *onto* $F(T)$.

*Proof*Put $u={P}_{F(T)}x$. Since $F(T)\subset {C}_{n}\cap {D}_{n}$ and ${x}_{n+1}={P}_{{C}_{n}\cap {D}_{n}}x$, we have that

*T*) that $v\in F(T)$. From the weakly lower semicontinuity of norm and (3.4), we obtain

Since *E* is uniformly convex, using the Kadec-Klee property, we have that $x-{x}_{n}\to x-u$. It follows that ${x}_{n}\to u$. This completes the proof. □

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for their useful comments.

## Authors’ Affiliations

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