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

- Baohua Guo
^{1}and - Shenghua Wang
^{1}Email author

**2010**:572838

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

© B. Guo and S.Wang. 2010

**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

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

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, [2–9].

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

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.

Lemma 1.2 (see [10]).

## 2. Main Results

Theorem 2.1.

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

Proof.

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

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

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

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

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

Corollary 2.2.

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.

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

Proof.

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 .

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

## 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 - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces.***Archiv der Mathematik*2008,**91**(2):166–177. 10.1007/s00013-008-2545-8MathSciNetView ArticleMATHGoogle Scholar - Matsushita S-Y, Takahashi W:
**A strong convergence theorem for relatively nonexpansive mappings in a Banach space.***Journal of Approximation Theory*2005,**134**(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleMATHGoogle 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 - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*1998,**34**(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar - Shimizu T, Takahashi W:
**Strong convergence to common fixed points of families of nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1997,**211**(1):71–83. 10.1006/jmaa.1997.5398MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Zembayashi K:
**Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleMATHGoogle Scholar - Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle 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.