Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals

Satit Saejung

doi:10.1155/2008/745010

Research Article

Open Access

Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals

Satit Saejung¹Email author

Fixed Point Theory and Applications20082008:745010

https://doi.org/10.1155/2008/745010

Received: 28 November 2007

Accepted: 30 January 2008

Published: 5 February 2008

Abstract

We prove a convergence theorem by the new iterative method introduced by Takahashi et al. (2007). Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen (2007).

Keywords

Hilbert SpaceMathematical ProgrammingConvergence TheoremHybrid MethodPositive Real Number

1. Introduction

Let

be a real Hilbert space with the inner product

and the norm

. 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:

(1)

for all ;
(2)

for all ;
(3)

for each the mapping is continuous;
(4)

for all and .

Motivated by Suzuki's result [1] and Nakajo-Takahashi's results [2], He and Chen [3] recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. However, their proof of the main result ([3, Theorem 2.3]) is very questionable. Indeed, the existence of the subsequence

such that (2.16) of [3] are satisfied, that is,

(1.1)

needs to be proved precisely. So, the aim of this short paper is to correct He-Chen's result and also to give a new result by using the method recently introduced by Takahashi et al.

We need the following lemma proved by Suzuki [4, Lemma 1].

Lemma 1.1.

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 .

2. Results

2.1. The Shrinking Projection Method

The following method is introduced by Takahashi et al. in [5]. We use this method to approximate a common fixed point of a nonexpansive semigroup without Bochner integrals as was the case in [5, Theorem 4.4].

Theorem 2.1.

Let

be a closed convex subset of a real Hilbert space

. Let

be a nonexpansive semigroup on

with a nonempty common fixed point

, that is,

. Suppose that

is a sequence iteratively generated by the following scheme:

(2.1)

where , , , and . Then

Proof.

It is well known that

is closed and convex. We first show that the iterative scheme is well defined. To see that each

is nonempty, it suffices to show that

. The proof is by induction. Clearly,

. Suppose that

. Then, for

,

(2.2)

That is, as required.

Notice that

(2.3)

is convex since

(2.4)

This implies that each subset is convex. It is also clear that is closed. Hence the first claim is proved.

Next, we prove that

is bounded. As

,

(2.5)

In particular, for for all , the sequence is bounded and hence so is .

Next, we show that

is a Cauchy sequence. As

and

,

(2.6)

Moreover, since the sequence

is bounded,

(2.7)

Note that

(2.8)

In particular, since

for all

,

(2.9)

It then follows from the existence of

that

is a Cauchy sequence. In fact, for

, there exists a natural number

such that, for all

,

(2.10)

where

. In particular, if

and

, then

(2.11)

Moreover,

(2.12)

We now assume that

for some

. Now since

for all

and

,

(2.13)

The last convergence follows from (2.12). We choose a sequence

of positive real number such that

(2.14)

We now show that how such a special subsequence can be constructed. First we fix

such that

(2.15)

From (2.13), there exists

such that

for all

. By Lemma 1.1,

is a cluster point of

. In particular, there exists

such that

. Next, we choose

such that

for all

. Again, by Lemma 1.1,

is a cluster point of

and this implies that there exists

such that

. Continuing in this way, we obtain a subsequence

of

satisfying

(2.16)

Consequently, (2.14) is satisfied.

We next show that

. To see this, we fix

,

(2.17)

As and (2.14), we have and so .

Finally, we show that

. Since

and

,

(2.18)

But

; we have

(2.19)

Hence as required. This completes the proof.

2.2. The Hybrid Method

We consider the iterative scheme computing by the hybrid method (some authors call the CQ-method). The following result is proved by He and Chen [3]. However, the important part of the proof seems to be overlooked. Here we present the correction under some additional restriction on the parameter .

Theorem 2.2.

Let

be a closed convex subset of a real Hilbert space

. Let

be a nonexpansive semigroup on

with a nonempty common fixed point

, that is,

. Suppose that

is a sequence iteratively generated by the following scheme:

(2.20)

where , , , and . Then .

Proof.

For the sake of clarity, we give the whole sketch proof even though some parts of the proof are the same as [3]. To see that the scheme is well defined, it suffices to show that both

and

are closed and convex, and

for all

. It follows easily from the definition that

and

are just the intersection of

and the half-spaces, respectively,

(2.21)

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.

We next show that

. To see this, we first prove that

(2.22)

As

and

,

(2.23)

For fixed

. It follows from

for all

that

(2.24)

This implies that sequence

is bounded and

(2.25)

Notice that

(2.26)

This implies that

(2.27)

It then follows from

that

and hence

(2.28)

As in Theorem 2.1, we can choose a subsequence

of

such that

(2.29)

Consequently, for any

,

(2.30)

This implies that

(2.31)

In virtue of Opial's condition of

, we have

for all

, that is,

. Next, we observe that

(2.32)

This implies that

(2.33)

Consequently,

(2.34)

Hence the whole sequence must converge to , as required.

Declarations

Acknowledgments

The author would like to thank the referee(s) for his comments and suggestions on the manuscript. This work is supported by the Commission on Higher Education and the Thailand Research Fund (Grant MRG4980022).

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, Thailand

References

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