- Research Article
- Open Access

# Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space

- Narin Petrot
^{1, 2}, - Kriengsak Wattanawitoon
^{3, 4}and - Poom Kumam
^{2, 3}Email author

**2009**:483497

https://doi.org/10.1155/2009/483497

© Narin Petrot et al. 2009

**Received:**17 March 2009**Accepted:**12 September 2009**Published:**11 October 2009

## Abstract

We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su's result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures.

## Keywords

- Hilbert Space
- Banach Space
- Nonexpansive Mapping
- Real Banach Space
- Nonempty Closed Convex Subset

## 1. Introduction

Let
be a nonempty closed convex subset of a real Banach space
. A mapping
is said to be *nonexpansive* if
for all
We denote by
the set of fixed points of
, that is
. A mapping
is said to be *quasi-nonexpansive* if
and
for all
and
. It is easy to see that if
is nonexpansive with
, then it is quasi-nonexpansive. Some iterative processes are often used to approximate a fixed point of a nonexpansive mapping. The Mann's iterative algorithm was introduced by Mann [1] in 1953. This iterative process is now known as Mann's iterative process, which is defined as

where the initial guess is taken in arbitrarily and the sequence is in the interval .

In 1976, Halpern [2] first introduced the following iterative scheme:

see also Browder [3]. He pointed out that the conditions and are necessary in the sence that, if the iteration (1.2) converges to a fixed point of , then these conditions must be satisfied.

In 1974, Ishikawa [4] introduced a new iterative scheme, which is defined recursively by

where the initial guess is taken in arbitrarily and the sequences and are in the interval .

Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, [5]. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [6]).

Zhang and Su [7] introduced the following implicit hybrid method for a finite family of nonexpansive mappings in a real Hilbert space:

where , and are sequences in and for some and for some .

In 2008, Nakprasit et al. [8] established weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. In the same year, Cho et al. [9] introduced the normal Mann's iterative process and proved some strong convergence theorems for a finite family nonexpansive mapping in the framework Banach spaces.

To find a common fixed point of a family of nonexpansive mappings, Aoyama et al. [10] introduced the following iterative sequence. Let and

for all , where is a nonempty closed convex subset of a Banach space, is a sequence of and is a sequence of nonexpansive mappings. Then they proved that, under some suitable conditions, the sequence defined by (1.5) converges strongly to a common fixed point of .

In 2008, by using a (new) hybrid method, Takahashi et al. [11] proved the following theorem.

Theorem 1.1 (Takahashi et al. [11]).

where for all and is said to satisfy the NST-condition with if for each bounded sequence , implies that for all . Then, converges strongly to .

Note that, recently, many authors try to extend the above result from Hilbert spaces to a Banach space setting.

Let
be a real Banach space with dual
. Denote by
the duality product. The *normalized duality mapping*
from
to
is defined by
for all
. The function
is defined by

A mapping
is said to be *hemi-relatively nonexpansive* (see [12]) if
and

A point
in
is said to be an *asymptotic* fixed point of
[13] if
contains a sequence
which converges weakly to
such that the strong
. The set of asymptotic fixed points of
will be denoted by
. A hemi-relatively nonexpansive mapping
from
into itself is called *relatively nonexpansive* if
; see [14–16]) for more details.

On the other hand, Matsushita and Takahashi [17] introduced the following iteration. A sequence defined by

where the initial guess element is arbitrary, is a real sequence in , is a relatively nonexpansive mapping, and denotes the generalized projection from onto a closed convex subset of . Under some suitable conditions, they proved that the sequence converges weakly to a fixed point of .

Recently, Kohsaka and Takahashi [18] extended iteration (1.9) to obtain a weak convergence theorem for common fixed points of a finite family of relatively nonexpansive mappings by the following iteration:

where and with , for all . Moreover, Matsushita and Takahashi [14] proposed the following modification of iteration (1.9) in a Banach space :

and proved that the sequence converges strongly to .

Qin and Su [15] showed that the sequence , which is generated by relatively nonexpansive mappings in a Banach space , as follows:

Moreover, they also showed that the sequence , which is generated by

In 2008, Nilsrakoo and Saejung [19] used the following Mann's iterative process:

and showed that the sequence converges strongly to a common fixed point of a countable family of relatively nonexpansive mappings.

Recently, Su et al. [12] extended the results of Qin and Su [15], Matsushita and Takahashi [14] to a class of closed hemi-relatively nonexpansive mapping. Note that, since the hybrid iterative methods presented by Qin and Su [15] and Matsushita and Takahashi [14] cannot be used for hemi-relatively nonexpansive mappings. Thus, as we know, Su et al. [12] showed their results by using the method as a monotone (CQ) hybrid method.

In this paper, motivated by Qin and Su [15], Nilsrakoo and Saejung [19], we consider the modified Ishikawa iterative (1.12) and Halpern iterative processes (1.13), which is different from those of (1.12)–(1.14), for countable hemi-relatively nonexpansive mappings. By using the shrinking projection method, some strong convergence theorems in a uniformly convex and uniformly smooth Banach space are provided. Our results extend and improve the recent results by Nilsrakoo and Saejung's result [19], Qin and Su [15], Su et al. [12], Takahashi et al.'s theorem [11], and many others.

## 2. Preliminaries

In this section, we will recall some basic concepts and useful well-known results.

A Banach space
is said to be *strictly convex* if

for all
with
and
. It is said to be *uniformly convex* if for any two sequences
in
such that
and

Let
be the unit sphere of
. Then the Banach space
is said to be *smooth* if

exists for each
It is said to be *uniformly smooth* if the limit is attained uniformly for
. In this case, the norm of
is said to be *Gâteaux differentiable*. The space
is said to have *uniformly Gâteaux differentiable* if for each
, the limit (2.3) is attained uniformly for
. The norm of
is said to be *uniformly Fréchet differentiable* (and
is said to be uniformly smooth) if the limit (2.3) is attained uniformly for
.

In our work, the concept duality mapping is very important. Here, we list some known facts, related to the duality mapping , as follows.

(a) ( , resp.) is uniformly convex if and only if ( , resp.) is uniformly smooth.

For more information, the readers may consult [20, 21].

If
is a nonempty closed convex subset of a real Hilbert space
and
is the *metric projection*, then
is nonexpansive. Alber [22] has recently introduced a *generalized projection* operator
in a Banach space
which is an analogue representation of the metric projection in Hilbert spaces.

The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem

Notice that the existence and uniqueness of the operator is followed from the properties of the functional and strict monotonicity of the mapping , and moreover, in the Hilbert spaces setting we have . It is obvious from the definition of the function that

Remark 2.1.

If is a strictly convex and a smooth Banach space, then for all , if and only if , see Matsushita and Takahashi [14].

To obtain our results, following lemmas are important.

Lemma 2.2 (Kamimura and Takahashi [23]).

Lemma 2.3 (Kamimura and Takahashi [23]).

Let be a uniformly convex and smooth real Banach space and let be two sequences of . If and either or is bounded, then .

Lemma 2.4 (Alber [22]).

Lemma 2.5 (Alber [22]).

Lemma 2.6 (Matsushita and Takahashi [14]).

Let be a strictly convex and smooth real Banach space, let be a closed convex subset of E, and let T be a hemi-relatively nonexpansive mapping from C into itself. Then F(T) is closed and convex.

Let be a subset of a Banach space and let be a family of mappings from into . For a subset of , one says that

(a) satisfies condition AKTT if

(b) satisfies condition AKTT if

For more information, see Aoyama et al. [10].

Lemma 2.7 (Aoyama et al. [10]).

Inspired by Lemma 2.7, Nilsrakoo and Saejung [19] prove the following results.

Lemma 2.8 (Nilsrakoo and Saejung [19]).

Lemma 2.9 (Nilsrakoo and Saejung [19]).

## 3. Modified Ishikawa Iterative Scheme

In this section, we establish the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. It is worth mentioning that our main theorem generalizes recent theorems by Su et al. [12] from relatively nonexpansive mappings to a more general concept. Moreover, our results also improve and extend the corresponding results of Nilsrakoo and Saejung [19]. In order to prove the main result, we recall a concept as follows. An operator in a Banach space is closed if and , then .

Theorem 3.1.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Proof.

for all . Therefore, is nondecreasing.

Based on the hypothesis, we now consider the following two cases.

Case 1.

Case 2.

for all . Therefore, This completes the proof.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we obtain the following result for a countable family of relatively nonexpansive mappings of modified Ishikawa iterative process.

Corollary 3.2.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Theorem 3.3.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

In Theorem 3.1, if for all then (3.1) reduced to (3.28).

Corollary 3.4.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Notice that every uniformly continuous mapping must be a continuous and closed mapping. Then setting for all , in Theorems 3.1 and 3.3, we immediately obtain the following results.

Corollary 3.5.

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Corollary 3.6.

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Proof.

Since a closed relatively nonexpansive mapping is a closed hemi-relatively one, Corollary 3.6 is implied by Corollary 3.5.

Corollary 3.7.

for , where is the single-valued duality mapping on . Then converges strongly to .

Corollary 3.8.

for , where is the single-valued duality mapping on . Then converges strongly to .

Similarly, as in the proof of Theorem 3.1, we obtain the following results.

Theorem 3.9.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Corollary 3.10.

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Theorem 3.11.

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

Putting , for all , in Theorem 3.9 we immediately obtain Theorem 3.11.

Corollary 3.12.

for , where is the single-valued duality mapping on . Then converges strongly to .

Remark 3.13.

Our results extend and improve the corresponding results in the following senses.

- (i)
Corollary 3.10 improves Theorem 2.1 of Qin and Su [15] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings.

- (ii)
Theorem 3.11 improves the algorithm in Theorem 3.1 of Nilsakoo and Saejung [19] from the Mann iteration process to modify Ishikawa iteration process and from countable relatively nonexpansive mappings to more general countable hemi-relatively nonexpansive mappings; that is, we relax the strong restriction . From (i) and (ii), it means that we relax the strongly restriction as from the assumption.

- (iii)
Corollary 3.12 improves Theorem 3.1 of Matsushita and Takahashi [14] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings in a Banach space.

## 4. Halpern Iterative Scheme

In this section, we prove the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings, which can be viewed as a generalization of the recently result of [15, Theorem 2.2].

Theorem 4.1.

Proof.

As in the proof of Theorem 3.1, we have that is closed and convex for each .

From the definition of , since , we have This implies . Using the Kadec-Klee property ([24]) of the space , we obtain that converges strongly to . Since is an arbitrary weakly convergent sequence of , we can conclude that convergence strongly to

Corollary 4.2.

for , where is the single-valued duality mapping on . Then converges strongly to .

Proof.

By setting for all , we immediately obtain the result.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we immediately obtain the following corollaries.

Corollary 4.3.

Corollary 4.4.

for , where is the single-valued duality mapping on . Then converges strongly to .

Similarly, as in the proof of Theorem 4.1, we obtain the following result.

Theorem 4.5.

If , then Theorem 4.5 reduces to the following corollary.

Corollary 4.6 (see [15, Theorem 2.2]).

for , where is the single-valued duality mapping on . Then converges strongly to .

## 5. Some Applications to Hilbert Spaces

It is well known that, in the Hilbert space setting, the concepts of hemi-relatively nonexpansive mappings and quasi-nonexpansive mappings are the equivalent. Thus, the following results can be obtained.

Theorem 5.1.

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to .

Proof.

for every and Hence, is quasi-nonexpansive if and only if is hemi-relatively nonexpansive. Then, by Theorem 3.1, we obtain the result.

Theorem 5.2.

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

In Theorem 5.1 setting for all , then (5.1) reduces to (5.4).

Theorem 5.3.

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

## Declarations

### Acknowledgments

The authors would like to thank the referees for the valuable suggestions which helped to improve this manuscript. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

## Authors’ Affiliations

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