- Research Article
- Open Access

# Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces

- Prasit Cholamjiak
^{1}and - Suthep Suantai
^{1, 2}Email author

**2010**:632137

https://doi.org/10.1155/2010/632137

© Prasit Cholamjiak and Suthep Suantai. 2010

**Received: **2 June 2010

**Accepted: **16 September 2010

**Published: **20 September 2010

## Abstract

We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.

## Keywords

- Banach Space
- Convex Subset
- Real Hilbert Space
- Real Banach Space
- Convex Banach Space

## 1. Introduction

Let and be a real Banach space and the dual space of , respectively. Let be a nonempty subset of . Let denote the normalized duality mapping from into given by , for all , where denotes the duality pairing between and . If is smooth or is strictly convex, then is single-valued.

Definition 1.1.

A mapping with domain and range in is called

Remark 1.2.

It is obvious by the definition that

(1)every strictly pseudocontractive mapping is pseudocontractive,

(2)every -strictly pseudocontractive mapping is -Lipschitzian; see [3].

Remark 1.3.

It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. Moreover, we know from [4] that the class of pseudocontractions also includes properly the class of strict pseudocontractions. A mapping
is called *accretive* if, for all
, there exists
such that
. It is also known that
is accretive if and only if
is pseudocontractive. Hence, a solution of the equation
is a solution of the fixed point of
. Note that if
, then
is
-strictly accretive if and only if
is
-strictly pseudocontractive.

where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.8) converges weakly to a fixed point of ( this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [6] ). However, if is a Lipschitzian pseudocontractive mapping, then Mann iteration defined by (1.8) may fail to converge in a Hilbert space; see [4].

In 1967, Browder-Petryshyn [2] introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann's iteration (1.8) with a constant sequence for all . Respectively, Marino-Xu [7] and Zhou [8] extended the results of Browder-Petryshyn [2] to Mann's iteration process (1.8). To be more precise, they proved the following theorem.

Theorem 1.4 (see [7]).

Let be a closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction for some , and assume that admits a fixed point in . Let a sequence be the sequence generated by Mann's algorithm (1.8). Assume that the control sequence is chosen so that for all and . Then converges weakly to a fixed point of .

Meanwhile, Marino, and Xu raised the open question: whether Theorem 1.4 can be extended to Banach spaces which are uniformly convex and have a Fréchet differentiable norm. Later, Zhou [9] and Zhang-Su [10], respectively, extended the result above to -uniformly smooth and -uniformly smooth Banach spaces which are uniformly convex or satisfy Opial's condition.

for some and is a constant depending on the geometry of the space.

for some , they proved the following theorem.

Theorem 1.6 (see [17]).

where is a real sequence in satisfying the following conditions:

Then, converges weakly to a fixed point of .

which is a contradiction.

It is known that one can extend his result from a single strict pseudocontraction to a finite family of strict pseudocontractions by replacing the convex combination of these mappings in the iteration under suitable conditions. The construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors; see also [18–22] and the references therein.

Our motivation in this paper is the following:

(1)to modify the normal Mann iteration process for finding common fixed points of an infinitely countable family of strict pseudocontractions,

(2)to improve and extend the results of Chidume-Shahzad [17] from a real uniformly smooth and uniformly convex Banach space to a real uniformly convex Banach space which has the Fréchet differentiable norm.

where is a real sequence in and is a countable family of strict pseudocontractions on a closed and convex subset of a real Banach space .

In this paper, we prove the weak convergence of a Mann-type iteration process (1.16) in a uniformly convex Banach space which has the Fréchet differentiable norm for a countable family of strict pseudocontractions under some appropriate conditions. The results obtained in this paper improve and extend the results of Chidume-Shahzad [17], Marino-Xu [7], Osilike-Udomene [11], Zhou [8], and Zhang-Guo [13] in some aspects.

We will use the following notation:

## 2. Preliminaries

*strictly convex*if for all with and . A Banach space is called

*uniformly convex*if for each there is a such that, for with , and holds. The modulus of convexity of is defined by

*Gâteaux differentiable*if

*smooth*. The norm of is said to be

*Fréchet differentiable*or is

*Fréchet smooth*if, for each , the limit is attained uniformly for . In other words, there exists a function with as such that

for all
, where
is a function defined on
such that
. The norm of
is called *uniformly Fréchet differentiable* if the limit is attained uniformly for
.

*uniformly smooth*if as . Let , then is said to be

*-uniformly smooth*if there exists such that . It is easy to see that if is -uniformly smooth, then is uniformly smooth. It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable, and hence the norm of is Fréchet differentiable, and it is also known that if is Fréchet smooth, then is smooth. Moreover, every uniformly smooth Banach space is reflexive. For more details, we refer the reader to [14, 23]. A Banach space is said to satisfy

*Opial's condition*[24] if and ; then

In the sequel, we will need the following lemmas.

Lemma 2.1 (see [23]).

Let be a Banach space and the duality mapping. Then one has the following:

Lemma 2.2 (see [25]).

Let be a real uniformly convex Banach space, a nonempty, closed, and convex subset of , and a continuous pseudocontractive mapping. Then, is demiclosed at zero, that is, for all sequence with and it follows that .

Lemma 2.3 (see [25]).

Let be a real reflexive Banach space which satisfies Opial's condition, a nonempty, closed and convex subset of and a continuous pseudocontractive mapping. Then, is demiclosed at zero.

Lemma 2.4 (see [26]).

Let be a real uniformly convex Banach space with a Fréchet differentiable norm. Let be a closed and convex subset of and a family of -Lipschitzian self-mappings on such that and . For arbitrary , define for all . Then for every , exists, in particular, for all and , .

If and , then exists. If, in addition, has a subsequence converging to 0, then .

Lemma 2.6 (see [28]).

So we have the following results proved by Boonchari-Saejung [29, 30].

Let be a closed and convex subset of a smooth Banach space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then the following conclusions hold:

(1) is a -strictly pseudocontractive mapping;

Lemma 2.8 (see [30]).

where is a family of nonnegative numbers satisfying

Then

(1)each is a -strictly pseudocontractive mapping;

For convenience, we will write that satisfies the AKTT-condition if satisfies the AKTT-condition and is defined by Lemma 2.6 with .

## 3. Main Results

Lemma 3.1.

where satisfying and . If satisfies the AKTT-condition, then

Proof.

Since satisfies the AKTT-condition, it follows that . This completes the proof of (i) and (ii).

Lemma 3.2.

Proof.

This completes the proof.

Remark 3.3.

In a real Hilbert space, we see that for .

where is a function appearing in (3.8).

So we obtain the following results.

Lemma 3.4.

where satisfying and . If satisfies the AKTT-condition, then .

Proof.

it follows from Lemma 2.6 that . This completes the proof.

Now, we prove our main result.

Theorem 3.5.

where satisfying and . If satisfies the AKTT-condition, then converges weakly to a common fixed point of .

Proof.

This implies that is nonexpansive. By Lemma 3.1(i), we know that is bounded. By Lemma 3.4, we also know that . Applying Lemma 2.2, we also have .

Hence ; consequently, as . This completes the proof.

As a direct consequence of Theorem 3.5, Lemmas 2.7 and 2.8 we also obtain the following results.

Theorem 3.6.

where satisfying and and satisfies conditions (i)–(iii) of Lemma 2.8. Then, converges weakly to a common fixed point of .

- (i)
Theorems 3.5 and 3.6 extend and improve Theorems and of Chidume-Shahzad [17] in the following senses:

(i)from real uniformly smooth and uniformly convex Banach spaces to real uniformly convex Banach spaces with Fréchet differentiable norms;

(ii)from finite strict pseudocontractions to infinite strict pseudocontractions.

Using Opial's condition, we also obtain the following results in a real reflexive Banach space.

Theorem 3.8.

where satisfying and . If satisfies the AKTT-condition, then converges weakly to a common fixed point of .

Proof.

This is a contradiction, and thus the proof is complete.

Theorem 3.9.

where satisfying and and satisfies conditions (i)–(iii) of Lemma 2.8. Then, converges weakly to a common fixed point of .

## Declarations

### Acknowledgments

The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund, Thailand. The first author is supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

## Authors’ Affiliations

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