- Research Article
- Open Access

# Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups

- Kriengsak Wattanawitoon
^{1}and - Poom Kumam
^{1}Email author

**2010**:914702

https://doi.org/10.1155/2010/914702

© K. Wattanawitoon and P. Kumam. 2010

**Received:**26 September 2009**Accepted:**24 November 2009**Published:**25 November 2009

## Abstract

We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung (2008) and some others.

## Keywords

- Hilbert Space
- Convergence Theorem
- Hybrid Method
- Nonexpansive Mapping
- Strong Convergence

## 1. Introduction

Let be a subset of real Hilbert spaces with the inner product and the norm . is called a nonexpansive mapping if

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

Let be a family of mappings from a subset of into itself. We call it a nonexpansive semigroup on if the following conditions are satisfied:

(i)

(ii) for all

(iii)for each the mapping is continuous;

(iv) for all and

where the initial guess is taken in arbitrarily and the sequence is in the interval .

In 1967, Halpern [2] first introduced the following iterative scheme:

see also Browder [3]. He pointed out that the conditions and are necessary in the sence that, if the iteration (1.3) converges to a fixed point of , then these conditions must be satisfied.

On the other hand, in 2002, Suzuki [4] was the first to introduce the following implicit iteration process in Hilbert spaces:

for the nonexpansive semigroup. In 2005, Xu [5] established a Banach space version of the sequence (1.4) of Suzuki [4].

In 2007, Chen and He [6] studied the viscosity approximation process for a nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive semigroup in Banach spaces, which is defined by

where is a fixed contractive mapping.

Recently He and Chen [7] is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very recently, Saejung [8] proved a convergence theorem by the new iterative method introduced by Takahashi et al. [9] without Bochner integrals for a nonexpansive semigroup with in Hilbert spaces:

where denotes the metric projection from onto a closed convex subset of .

In 1974, Ishikawa [10] introduced a new iterative scheme, which is defined recursively by

where the initial guess is taken in arbitrarily and the sequences and are in the interval .

In this paper, motivated by the iterative sequences (1.6) given by Saejung in [8] and Ishikawa [10], we introduce the modified Ishikawa iterative scheme for two nonexpansive semigroups in Hilbert spaces. Further, we obtain strong convergence theorems by using the hybrid methods. This result extends and improves the result of Saejung [8] and some others.

## 2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

It is known that every Hilbert space satisfies the Opial's condition [11], that is,

Recall that the metric (nearest point) projection from a Hilbert space to a closed convex subset of is defined as follows. Given is the only point in with the property

is characterized as follows.

Lemma 2.1.

Lemma 2.2.

for all and

Lemma 2.3 (see [12, Lemma ]).

Let be a real sequence and let be a real number such that . Suppose that either of the following holds:

(i) or

(ii) ,

then is a cluster point of . Moreover, for , there exists such that for every integer with

## 3. Main Results

### 3.1. The Shrinking Projection Method

In this section, we prove strong convergence of an iterative sequence generated by the shrinking hybrid projection method in mathematical programming.

Theorem 3.1.

then converges strongly to

Proof.

Therefore, is nondecreasing. From , we also have , for all

Since , we get

Thus, , for all and . Then exists and is bounded.

Next, we show that as . From (3.6) we have

We now show that .

For , we have This implies that and hence Moreover, since

As and (3.19), we obtain and so Similarly, we have Thus .

Finally, we show that Since and

Hence as required. This completes the proof.

Corollary 3.2.

then converges strongly to

Proof.

Putting , in Theorem 3.1, we obtain the conclusion immediately.

Corollary 3.3 (see [8, Theorem ]).

then

Proof.

If for all and for every in Theorem 3.1 then (3.1) reduced to (3.25). By using Theorem 3.1, we get the following conclusion.

### 3.2. The CQ Hybrid Method

In this section, we consider the modified Ishikawa iterative scheme computing by the CQ hybrid method [13–15]. We use the same idea as Saejung's Theorem in [8] and our Theorem 3.1 to obtain the following result and the proof is omitted.

Theorem 3.4.

then converges strongly to

Proof.

First, we show that both and are closed and convex, and for all . It follows easily from the definition that and are just intersection of and the half-spaces see also [9]. As in the proof of the preceding theorem, we have for all . Clearly, . Suppose that for some , we have . In particular, that is, . It follows from the induction that for all . This proves the claim.

Next, we show that and

We first claim that Indeed, as and ,

By using the same argument of Saejung [8, Theorem , page 6] and in the proof of Theorem 3.1, we have and . And we can choose a subsequence of such that , , and as .

From (3.21), we obtain

By the Opial's condition of , we have and for all , that is, .

We note that

Hence the whole sequence must converge to , as required. This completes the proof.

Corollary 3.5.

then converges strongly to

Proof.

If for all , in Theorem 3.4 then (3.26) reduced to (3.36). So, we obtain the result immediately.

We also deduce the following corollary.

Corollary 3.6 (see [8, Theorem ]).

then

## Declarations

### Acknowledgments

The authors would like to thank the editors and the anonymous referees for their valuable suggestions which help to improve this paper. This research was supported by the Computational Science and Engineering Research Cluster, King Mongkut's University of Technology Thonburi (KMUTT) (National Research Universities under CSEC Project no. E01008).

## Authors’ Affiliations

## References

- Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Fixed-point theorems for noncompact mappings in Hilbert space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**53:**1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.***Proceedings of the American Mathematical Society*2002,**131**(7):2133–2136.View ArticleMathSciNetMATHGoogle Scholar - Xu H-K:
**A strong convergence theorem for contraction semigroups in Banach spaces.***Bulletin of the Australian Mathematical Society*2005,**72**(3):371–379. 10.1017/S000497270003519XMathSciNetView ArticleMATHGoogle Scholar - Chen R, He H:
**Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space.***Applied Mathematics Letters*2007,**20**(7):751–757. 10.1016/j.aml.2006.09.003MathSciNetView ArticleMATHGoogle Scholar - He H, Chen R:
**Strong convergence theorems of the CQ method for nonexpansive semigroups.***Fixed Point Theory and Applications*2007,**2007:**-8.Google Scholar - Saejung S:
**Strong convergence theorems for nonexpansive semigroups without Bochner integrals.***Fixed Point Theory and Applications*2008,**2008:**-7.Google Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar - Haugazeau Y:
*Sur les Inéquations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. thesis*. Université Paris, Paris, France; 1968.Google Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Shimoji K, Takahashi W:
**Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces.***Taiwanese Journal of Mathematics*2006,**10**(2):339–360.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.