Research Article | Open | Published:

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

*Fixed Point Theory and Applications***volume 2010**, Article number: 572838 (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 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:

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.

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:

Lemma 1.2 (see [10]).

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

## 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:

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

is equivalent to

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

This implies that

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

From , we have

Since for all , one has

This implies that is bounded. For each fixed , by (2.1) we have (noting that )

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

On the other hand, observing that for all , we have

for all . Since , we have

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

It follows from (2.11) that

Since the limit of exists, we get

It follows that is a Cauchy sequence. Since is a Hilbert space and is closed and convex, one can assume that

By taking in (2.12), one arrives that

that is,

Noticing that , we get

This implies that . Since each , we conclude that

From (2.16) and (2.18), we get

By and , we have

This implies that

Finally, we prove that . From and , one gets

Taking the limit in (2.22) and noting that as , we get that

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:

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:

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

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

and hence

for each

Observe that , that is,

It follows from (2.28) and (2.29) that

Since is strictly decreasing, and as , we get

for each Since each is nonexpansive, one has and hence

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.

## References

- 1.
Mann WR:

**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3 - 2.
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.2003.07.023 - 3.
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/s00013-008-2545-8 - 4.
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.2005.02.007 - 5.
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)00458-4 - 6.
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)00682-2 - 7.
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.1997.5398 - 8.
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.2007.11.031 - 9.
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.2006.08.036 - 10.
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.2006.06.055

## Acknowledgment

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

## Author information

## Rights and permissions

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- Hilbert Space
- Banach Space
- Differential Geometry
- Convex Subset
- Iteration Method