# Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

- Rabian Wangkeeree
^{1}Email author and - Uthai Kamraksa
^{1}

**2010**:579725

https://doi.org/10.1155/2010/579725

© R.Wangkeeree and U. Kamraksa. 2010

**Received: **23 June 2010

**Accepted: **13 August 2010

**Published: **17 August 2010

## Abstract

For a countable family
of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of
in either a *p*-uniformly convex Banach space which admits a weakly continuous duality mapping or a *p*-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.

## 1. Introduction

*family of uniformly*-

*strict pseudo-contractions with respect to*, if there exists a constant such that

We denote by the set of fixed points of , that is, .

for all where is a nonempty closed convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and is a sequence in . They proved that defined by (1.5) converges strongly to a common fixed point of Very recently, Song and Zheng [4] also studied the strong convergence theorem of Halpern iteration (1.5) for a countable family of nonexpansive mappings satisfying some conditions in either a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm or a reflexive Banach space with a weakly continuous duality mapping. Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in [3, 5–10] and many results not cited here.

On the other hand, in the last twenty years or so, there are many papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudo-contractive mappings by using the Mann and Ishikawa iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and a more general class of mappings (see, e.g., [1, 11–13] and the references therein).

In 2007, Marino and Xu [12] proved that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions in Hilbert spaces, which extend Reich's theorem [14, Theorem ] from nonexpansive mappings to -strict pseudo-contractions in Hilbert spaces.

Recently, Zhou [13] obtained some weak and strong convergence theorems for -strict pseudo-contractions in Hilbert spaces by using Mann iteration and modified Ishikawa iteration which extend Marino and Xu's convergence theorems [12].

More recently, Hu and Wang [11] obtained that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions with respect to in -uniformly convex Banach spaces. To be more precise, they obtained the following theorem.

Theorem HW

Let be a real -uniformly convex Banach space which satisfies one of the following:

(i) has a Fréchet differentiable norm;

(ii) satisfies Opial's property.

converges weakly to a fixed point of .

Very recently, Hu [15] obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for -strict pseudo-contractions in -uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. To be more precise, Hu [15] obtained the following theorem.

Theorem H.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Then the sequence converges strongly to a fixed point of .

In this paper, motivated by Hu and Wang [11], Hu [15], Aoyama et al. [3] and Song and Zheng [4], we introduce a viscosity iterative approximation method for finding a common fixed point of a countable family of strictly pseudo-contractions which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in either -uniformly convex Banach space which admits a weakly continuous duality mapping or -uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The results presented in this paper improve and extend the corresponding results announced by Hu and Wang [11], Hu [15], Aoyama et al. [3] Song and Zheng [4], and many others.

## 2. Preliminaries

Throughout this paper, let be a real Banach space and its dual space. We write (resp., ) to indicate that the sequence weakly (resp., weak*) converges to ; as usual will symbolize strong convergence. Let denote the unit sphere of a Banach space . A Banach space is said to have

(ii)*a uniformly Gâteaux differentiable norm*, if for each
in
, the limit (2.1) is uniformly attained for
,

(iii)*a Fréchet differentiable norm*, if for each
, the limit (2.1) is attained uniformly for
,

(iv)*a uniformly Fréchet differentiable norm* (we also say that
is uniformly smooth), if the limit (2.1) is attained uniformly for
.

is uniformly convex if and only if, for all such that . is said to be -uniformly convex, if there exists a constant such that .

The following facts are well known which can be found in [16, 17]:

(i)the normalized duality mapping in a Banach space with a uniformly Gâteaux differentiable norm is single-valued and strong-weak* uniformly continuous on any bounded subset of ;

(ii)each uniformly convex Banach space is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings;

(iii)every uniformly smooth Banach space is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1 (see [11]).

where is a constant in [18, Theorem ]. In addition, if , , and , then , for all , .

Let be a nonempty closed convex subset of a Banach space which has uniformly Gâteaux differentiable norm, a nonexpansive mapping with and a -contraction. Assume that every nonempty closed convex bounded subset of has the fixed points property for nonexpansive mappings. Then there exists a continuous path: , satisfying , which converges to a fixed point of as .

Lemma 2.3 (see [21]).

Definition 2.4 (see [3]).

Remark 2.5.

The example of the sequence of mappings satisfying AKTT-condition is supported by Lemma 4.1.

Lemma 2.6 (see [3, Lemma ]).

Then for each bounded subset of ,

Lemma 2.7 (see [22]).

where is a sequence in and is a sequence such that

where denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function at is the set .

The following lemma is an immediate consequence of the subdifferential inequality. The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [24].

Lemma 2.8 (see [24]).

Assume that a Banach space has a weakly continuous duality mapping with gauge .

## 3. Main Results

For simplicity we will write for provided no confusion occurs. Next, we will prove the following lemma.

Lemma 3.1.

Proof.

So, is a solution of the variational inequality (3.2), and hence by the uniqueness. In a summary, we have shown that each cluster point of (at ) equals . Therefore, as . This completes the proof.

Theorem 3.2.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a family of uniformly -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Proof.

Apply Lemma 2.7 to (3.53) to conclude as ; that is, as . This completes the proof.

If is a family of nonexpansive mappings, then we obtain the following results.

Corollary 3.3.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a family of nonexpansive mappings such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Corollary 3.4.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a -strict pseudo-contraction with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Theorem 3.5.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a family of uniformly -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly to a common fixed point of .

Proof.

Hence applying in Lemma 2.7 to (3.67), we conclude that .

Corollary 3.6.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a family of nonexpansive mappings such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly to a common fixed point of .

Corollary 3.7.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

Then the sequence converges strongly to a common fixed point of .

## 4. Some Applications for Accretive Operators

By the proof of Theorem in [3], we have the following lemma.

Lemma 4.1.

Let be a Banach space and a nonempty closed convex subset of Let be an accretive operator such that and Suppose that is a sequence of such that and Then

(i)The sequence satisfies the AKTT-condition.

By Corollary 3.3, we obtain the following result.

Theorem 4.2.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let is an -accretive operator in such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(iv) is a sequence of such that and

By Corollary 3.6, we obtain the following result.

Theorem 4.3.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of . Let is an -accretive operator in such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

## Declarations

### Acknowledgments

The first author is supported by the Thailand Research Fund under Grant TRG5280011 and the second author is supported by grant from the program of Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand.

## Authors’ Affiliations

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