Open Access

New Hybrid Iterative Schemes for an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

Fixed Point Theory and Applications20102010:572838

https://doi.org/10.1155/2010/572838

Received: 20 November 2009

Accepted: 4 February 2010

Published: 14 February 2010

Abstract

We propose some new iterative schemes for finding common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the proposed schemes. Our results extend and improve ones of Nakajo and Takahashi (2003).

1. Introduction and Preliminaries

Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonlinear mapping of into itself. We use and to denote the set of fixed points of and the metric projection from onto , respectively.

Recall that is said to be nonexpansive if

(1.1)

for all .

For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann [1] in 1953 introduced a famous iterative scheme as follows:

(1.2)

where is a nonexpansive mapping of into itself and is a sequence in . It is well known that defined in (1.2) converges weakly to a fixed point of .

Attempts to modify the normal Mann iteration method (1.2) for nonexpansive mappings so that strong convergence is guaranteed have recently been made; see, for example, [29].

Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2) for a single nonexpansive mapping in a Hilbert space :

(1.3)

where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one, then the sequence generated by (1.3) converges strongly to .

In this paper, we introduce some new iterative schemes for infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the algorithms. Our results extend and improve the corresponding one of Nakajo and Takahashi [5].

The following two lemmas will be used for the main results of this paper.

Lemma 1.1.

Let be a closed convex subset of a real Hilbert space and let be the metric projection from onto (i.e., for , is the only point in such that . Given and , then if and only if there holds the following relation:
(1.4)

Lemma 1.2 (see [10]).

Let be a real Hilbert space. Then the following equation holds:
(1.5)

2. Main Results

Theorem 2.1.

Let be a nonempty closed convex subset of a Hilbert space . Let be an infinite family of nonexpansive mappings such that . Let be a sequence generated by the following manner:
(2.1)

where is a sequence in satisfying and is a sequence in satisfying . Then defined by (2.1) converges strongly to .

Proof.

We first show that is closed and convex. By Lemma 1.2, one observes that
(2.2)
is equivalent to
(2.3)

for all . So, is closed and convex for all and hence is also closed and convex for all . This implies that is well defined.

Next, we show that for all . To end this, we need to prove that for all . Indeed, for each , we have
(2.4)
This implies that
(2.5)

Therefore, and is nonempty for all . On the other hand, from the definition of , we see that for all .

From , we have
(2.6)
Since for all , one has
(2.7)
This implies that is bounded. For each fixed , by (2.1) we have (noting that )
(2.8)

for all . Since is bounded, is bounded for each .

On the other hand, observing that for all , we have
(2.9)
for all . Since , we have
(2.10)

for all . It follows from (2.7) and (2.10) that the limit of exists.

Since and for all and , by Lemma 1.1 one has
(2.11)
It follows from (2.11) that
(2.12)
Since the limit of exists, we get
(2.13)
It follows that is a Cauchy sequence. Since is a Hilbert space and is closed and convex, one can assume that
(2.14)
By taking in (2.12), one arrives that
(2.15)
that is,
(2.16)
Noticing that , we get
(2.17)
This implies that . Since each , we conclude that
(2.18)
From (2.16) and (2.18), we get
(2.19)
By and , we have
(2.20)
This implies that
(2.21)
Finally, we prove that . From and , one gets
(2.22)
Taking the limit in (2.22) and noting that as , we get that
(2.23)

In view of Lemma 1.1, one sees that . This completes the proof.

Corollary 2.2.

Let be a nonempty closed convex subset of a Hilbert space . Let be a nonexpansive mapping such that . Let be a sequence generated by the following manner:
(2.24)

where is a sequence in satisfying that . Then defined by (2.24) converges strongly to .

Proof.

Set for all , and for all in Theorem 2.1. By Theorem 2.1, we obtain the desired result.

Theorem 2.3.

Let be a nonempty closed convex subset of a Hilbert space . Let be an infinite family of nonexpansive mappings such that . Let be a sequence generated by the following manner:
(2.25)

where is a strictly decreasing sequence in and set . Then defined by (2.25) converges strongly to .

Proof.

Obviously, is closed and convex for all and hence is also closed and convex for all . Next, we prove that for all . For any , we have
(2.26)

This shows that for all . Therefore, for all . It follows that for all .

By using the method of Theorem 2.1, we can conclude that is bounded, , and as . This implies that as .

Next, we show that . To end this, we see a fact. For and , we have
(2.27)
and hence
(2.28)

for each

Observe that , that is,
(2.29)
It follows from (2.28) and (2.29) that
(2.30)
Since is strictly decreasing, and as , we get
(2.31)
for each Since each is nonexpansive, one has and hence
(2.32)

Finally, by using the method of Theorem 2.1, we can conclude that . This completes the proof.

Remark 2.4.

In this paper, we extend result of Nakajo and Takahashi [5] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

Remark 2.5.

The iterative schemes introduced in this paper are new and of independent interest.

Remark 2.6.

It is of interest to extend the algorithm (2.25) to certain Banach spaces.

Declarations

Acknowledgment

The work was supported by Youth Foundation of North China Electric Power University.

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University

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Copyright

© B. Guo and S.Wang. 2010

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