# A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

- Urailuk Singthong
^{1}and - Suthep Suantai
^{1, 2}Email author

**2010**:262691

**DOI: **10.1155/2010/262691

© Urailuk Singthong and Suthep Suantai. 2010

**Received: **10 February 2010

**Accepted: **15 July 2010

**Published: **2 August 2010

## Abstract

We introduce a new general iterative method by using the -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.

## 1. Introduction

*nonexpansive*if for all A point is called a fixed point of provided that . We denote by the set of fixed points of (i.e., ). Recall that a self-mapping is a contraction on , if there exists a constant such that for all A bounded linear operator on is called

*strongly positive*with coefficient if there is a constant with the property

where the initial guess is taken in arbitrarily, and the sequence is in the interval . But Mann's iteration process has only weak convergence, even in a Hilbert space setting. In general for example, Reich [2] showed that if is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence is such that , then the sequence generated by process (1.2) converges weakly to a point in . Therefore, many authors try to modify Mann's iteration process to have strong convergence.

They proved in a uniformly smooth Banach space that the sequence defined by (1.3) converges strongly to a fixed point of under some appropriate conditions on and .

In 2008, Yao et al. [4] alsomodified Mann's iterative scheme 1.2 to get a strong convergence theorem.

Let be a finite family of nonexpansive mappings with There are many authors introduced iterative method for finding an element of which is an optimal point for the minimization problem. For , is understood as with the mod function taking values in . Let be a fixed element of

where This mapping is called the mapping generated by and .

In 2000, Takahashi and Shimoji [7] proved that if is strictly convex Banach space, then , where .

where is a contraction, and is a linear bounded operator.

Note that the iterative scheme (1.8) is not well-defined, because may not lie in , so is not defined. However, if , the iterative scheme (1.8) is well-defined and Theorem [8] is obtained. In the case , we have to modify the iterative scheme (1.8) in order to make it well-defined.

The mapping
is called the *K-mapping* generated by
and
.

where is a contraction, and is a bounded linear operator. We prove, under certain appropriate conditions on the sequences and that defined by (1.10) converges strongly to a common fixed point of the finite family of nonexpansive mappings , which solves a variational inequaility problem.

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

Lemma 1.1.

Lemma 1.2 (see [11]).

Then

Lemma 1.3 (see [5]).

Assume that is a sequence of nonnegative real numbers such that , where and is a sequence in such that

(i) ,

(ii) or .

Then .

Lemma 1.4 (see [10]).

Let be a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 1.5 (see [10]).

Lemma 1.6 (see [1]).

Demiclosedness principle. Assume that is nonexpansive self-mapping of closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here, is identity mapping of .

Lemma 1.7 (see [9]).

Let be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpansive mappings of into itself with , and let be real numbers such that for every and Let be the -mapping of into itself generated by and . Then .

By using the same argument as in [9, Lemma ], we obtain the following lemma.

Lemma 1.8.

Let be a nonempty closed convex subset of Banach space. Let be a finite family of nonexpanxive mappings of into itself and sequences in such that Moreover, for every , let and be the K -mappings generated by and , and and , respectively. Then, for every bounded sequence , one has

Let be real Hilbert space with inner product , a nonempty closed convex subset of . Recall that the metric (nearest point) projection from a real Hilbert space to a closed convex subset of is defined as follows. Given that , is the only point in with the property . Below Lemma 1.9 can be found in any standard functional analysis book.

Lemma 1.9.

Let be a closed convex subset of a real Hilbert space . Given that and then

(i) if and only if the inequality for all ,

(ii) is nonexpansive,

(iii) for all ,

(iv) for all and .

## 2. Main Result

In this section, we prove strong convergence of the sequences defined by the iteration scheme (1.10).

Theorem 2.1.

Let be a Hilbert space, a closed convex nonempty subset of . Let be a strongly positive linear bounded operator with coefficient , and let Let be a finite family of nonexpansive mappings of into itself, and let be defined by (1.9). Assume that and . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4) and , where ;

(C5)

(C6) .

Proof.

Since , we may assume that for all . By Lemma 1.4, we have for all .

where .

it implies that as .

which implies that .

where . By , we get . By applying Lemma 1.3 to (2.23), we can conclude that . This completes the proof.

If and in Theorem 2.1, we obtain the following result.

Corollary 2.2.

Let be a Hilbert space, a closed convex nonempty subset of , and let . Let be a finite family of nonexpansive mappings of into itself, and let be defined by (1.9). Assume that . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4) and , where ;

(C5)

(C6) .

If , , , and is a constant in Theorem 2.1, we get the results of Kim and Xu [3].

Corollary 2.3.

Let be a Hilbert space, a closed convex nonempty subset of , and let . Let be a nonexpansive mapping of into itself. . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4)

(C5) .

## Declarations

### Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and thank the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate School of Chiang Mai University for financial support.

## Authors’ Affiliations

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