- Research Article
- Open Access

# Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of -Strict Pseudocontractions in -Uniformly Smooth Banach Spaces

- Yanlai Song
^{1}Email author and - Changsong Hu
^{1}

**2010**:354202

https://doi.org/10.1155/2010/354202

© Yanlai Song and Changsong Hu. 2010

**Received:**9 August 2010**Accepted:**14 November 2010**Published:**30 November 2010

## Abstract

We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions in -uniformly smooth Banach spaces. We also discuss the strong convergence theorems for the new iterative scheme in -uniformly smooth Banach space. Our results improve and extend the corresponding results announced by many others.

## Keywords

- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Duality Mapping
- Convex Banach Space

## 1. Introduction

Throughout this paper, we denote by and a real Banach space and the dual space of , respectively. Let be a subset of , and lrt be a non-self-mapping of . We use to denote the set of fixed points of .

exists for all , on the unit sphere . If, for each , the limit (1.1) is uniformly attained for , then the norm of is said to be uniformly Gâteaux differentiable. The norm of is said to be Fréchet differentiable if, for each , the limit (1.1) is attained uniformly for . The norm of is said to be uniformly Fréchet differentiable (or uniformly smooth) if the limit (1.1) is attained uniformly for .

A Banach space is said to be uniformly smooth if as . Let . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that . It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable. If is -uniformly smooth, then and is uniformly smooth, and hence the norm of is uniformly Fréchet differentiable, in particular, the norm of is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for every .

Then it is known that is the subdifferential of the convex function at . It is well known that if is smooth, then is single valued, which is denoted by .

The following famous theorem is referred to as the Banach contraction principle.

Theorem 1.1 (Banach [4]).

Let be a complete metric space and let be a contraction on , that is, there exists such that for all . Then has a unique fixed point.

Theorem 1.2 (Meir and Keeler [5]).

Let be a complete metric space and let be a Meir-Keeler contraction (MKC, for short) on , that is, for every , there exists such that implies for all . Then has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

where is the identity mapping and is the normalized duality mapping.

Attempts to modify the normal Mann's iteration method for nonexpansive mappings and -strictly pseudocontractions so that strong convergence is guaranteed have recently been made; see, for example, [6–11] and the references therein.

where is a nonexpansive mapping of into itself is a given point. They proved the sequence defined by (1.10) converges strongly to a fixed point of , provided the control sequences and satisfy appropriate conditions.

where is non-self- -strictly pseudocontraction, is a contraction and is a strong positive linear bounded operator in Banach space. They have proved, under certain appropriate assumptions on the sequences , and , that defined by (1.11) converges strongly to a common fixed point of a finite family of -strictly pseudocontractions, which solves some variational inequality.

Question 1.

Can Theorem 3.1 of Zhou [8], Theorem 2.2 of Hu and Cai [12] and so on be extended from finite -strictly pseudocontraction to infinite -strictly pseudocontraction?

Question 2.

We know that the Meir-Keeler contraction (MKC, for short) is more general than the contraction. What happens if the contraction is replaced by the Meir-Keeler contraction?

where is non-self -strictly pseudocontraction, is a MKC contraction and is a strong positive linear bounded operator in Banach space. Under certain appropriate assumptions on the sequences , , , and , that defined by (1.12) converges strongly to a common fixed point of an infinite family of -strictly pseudocontractions, which solves some variational inequality.

## 2. Preliminaries

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

Lemma 2.1 (see [13]).

Let be bounded sequences in a Banach space and be a sequence in which satisfies the following condition: . Suppose that for all and . Then, .

Lemma 2.2 (see Xu [14]).

Assume that is a sequence of nonnegative real numbers such that , where is a sequence in (0, 1) and is a sequence in such that

Lemma 2.3 (see [15] demiclosedness principle).

Let be a nonempty closed convex subset of a reflexive Banach space which satisfies Opial's condition, and suppose is nonexpansive. Then the mapping is demiclosed at zero, that is, , implies .

Lemma 2.4 (see [16, Lemmas 3.1, 3.3]).

Let be real smooth and strictly convex Banach space, and be a nonempty closed convex subset of which is also a sunny nonexpansive retraction of . Assume that is a nonexpansive mapping and is a sunny nonexpansive retraction of onto , then .

Lemma 2.5 (see [17, Lemma 2.2]).

Let be a nonempty convex subset of a real -uniformly smooth Banach space and be a -strict pseudocontraction. For , we define . Then, as , , is nonexpansive such that .

Lemma 2.6 (see [12, Remark 2.6]).

When is non-self-mapping, the Lemma 2.5 also holds.

Lemma 2.7 (see [12, Lemma 2.8]).

Lemma 2.8 (see [18, Lemma 2.3]).

Lemma 2.9.

Proof.

Therefore and the uniqueness is proved. Below, we use to denote the unique solution of (2.3).

We observe that is bounded. Indeed, we may assume, with no loss of generality, , for all , fixed , for each .

In this case, we can see easily that is bounded.

therefore, . This implies the is bounded.

Using that the duality map is single valued and weakly sequentially continuous from to , by (2.15), we get that . It is a contradiction. Hence, we have .

Now replacing in (2.19) with and letting , noticing for , we obtain . That is, is a solution of (2.3); Hence by uniqueness. In a summary, we have shown that each cluster point of (at ) equals , therefore, as .

Lemma 2.10 . (see, e.g., Mitrinović [19, page 63]).

for arbitrary positive real numbers , .

Lemma 2.11.

Let be a -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping from to and be a nonempty convex subset of . Assume that is a countable family of -strict pseudocontraction for some and such that . Assume that is a positive sequence such that . Then is a -strict pseudocontraction with and .

Proof.

which shows that is a -strict pseudocontraction with . By the same way, our proof method easily carries over to the general finite case.

## 3. Main Results

Lemma 3.1.

Let be a real -uniformly smooth, strictly convex Banach space and be a closed convex subset of such that . Let be also a sunny nonexpansive retraction of . Let be a MKC. Let be a strongly positive linear bounded operator with the coefficient such that and be -strictly pseudo-contractive non-self-mapping such that . Let . Let be a sequence of generated by (1.12) with the sequences , and in , assume for each , be an infinity sequence of positive number such that for all and . The following control conditions are satisfied

Proof.

The rest of the proof will now be split into two parts.

Step 1.

which gives that the sequence is bounded, so are and .

Step 2.

Theorem 3.2.

Proof.

Case 1.

Fixed ( ), if for some such that , and for the other such that .

Case 2.

Apply Lemma 2.2 to (3.36) to conclude as . It contradict the . This completes the proof.

Corollary 3.3.

Remark 3.4.

We conclude the paper with the following observations.

(i)Theorem 3.2 improve and extends Theorem 3.1 of Zhang and Su [17], Theorem 1 of Yao et al. [11], and Theorem 2.2 of Cai and Hu [12]. Corollary 3.3 also improve and extend Theorem 2.1 of Choa et al. [20], Theorem 2.1 of Jung [21], Theorem 2.1 of Qin et al. [22] and includes those results as special cases. Especially, Our results extends above results form contractions to more general Meir-Keeler contraction (MKC, for short). Our iterative scheme studied in present paper can be viewed as a refinement and modification of the iterative methods in [12, 13, 17, 22]. On the other hand, our iterative schemes concern an infinite countable family of -strict pseudocontractions mappings, in this respect, they can be viewed as an another improvement.

(ii)The advantage of the results in this paper is that less restrictions on the parameters , , and are imposed. Our results unify many recent results including the results in [12, 17, 22].

(iii)It is worth noting that we obtained two strong convergence results concerning an infinite countable family of -strict pseudocontractions mappings. Our result is new and the proofs are simple and different from those in [11, 12, 17, 19–25].

## Declarations

### Acknowledgments

The authors are extremely grateful to the referee and the editor for their useful comments and suggestions which helped to improve this paper. This work was supported by the National Science Foundation of China under Grant (no. 10771175).

## Authors’ Affiliations

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