Open Access

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

Fixed Point Theory and Applications20102010:632137

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

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.

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.

Throughout this paper, we denote the single valued duality mapping by and denote the set of fixed points of a nonlinear mapping by
(1.1)

Definition 1.1.

A mapping with domain and range in is called

(i)pseudocontractive [1] if, for all , there exists such that
(1.2)
(ii) -strictly pseudocontractive [2] if for all , there exist and such that
(1.3)
or equivalently
(1.4)
(iii)L-Lipschitzian if, for all , there exists a constant such that
(1.5)

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.

Let be a nonempty subset of a real Hilbert space and a mapping. Then is said to be -strictly pseudocontractive [2] if, for all , there exists such that
(1.6)
It is well know that (1.6) is equivalent to the following:
(1.7)

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.

In 1953, Mann [5] introduced the iteration as follows: a sequence defined by and
(1.8)

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.

In 2001, Osilike-Udomene [11] proved the convergence theorems of the Mann [5] and Ishikawa [12] iteration methods in the framework of -uniformly smooth and uniformly convex Banach spaces. They also obtained that a sequence defined by (1.8) converges weakly to a fixed point of under suitable control conditions. However, the sequence excluded the canonical choice . This was a motivation for Zhang-Guo [13] to improve the results in the same space. Observe that the results of Osilike-Udomene [11] and Zhang-Guo [13] hold under the assumption that
(1.9)

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

Lemma 1.5 (see [1416]).

Let be a uniformly smooth real Banach space. Then there exists a nondecreasing continuous function with and for such that, for all , the following inequality holds:
(1.10)
Recently, Chidume-Shahzad [17] extended the results of Osilike-Udomene [11] and Zhang-Guo [13] by using Reich's inequality (1.10) to the much more general real Banach spaces which are uniformly smooth and uniformly convex. Under the assumption that
(1.11)

for some , they proved the following theorem.

Theorem 1.6 (see [17]).

Let be a uniformly smooth real Banach space which is also uniformly convex and a nonempty closed convex subset of . Let be a -strict pseudocontraction for some with . For a fixed , define a sequence by
(1.12)

where is a real sequence in satisfying the following conditions:

(i) ;

(ii) .

Then, converges weakly to a fixed point of .

However, we would like to point out that the results of Chidume-Shahzad [17] do not hold in real Hilbert spaces. Indeed, we know from Chidume [14] that
(1.13)
If is a real Hilbert space, then we have
(1.14)
On the other hand, by assumption (1.11), we see that
(1.15)

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 [1822] 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.

Motivated and inspired by Marino-Xu [7], Osilike-Udomene [11], Zhou [8], Zhang-Guo [13], and Chidume-Shahzad [17], we consider the following Mann-type iteration: and
(1.16)

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:

(i) for weak convergence and for strong convergence.

(ii) denotes the weak -limit set of .

2. Preliminaries

A Banach space is said to be 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
(2.1)
for all . is uniformly convex if , and for all . It is known that every uniformly convex Banach space is strictly convex and reflexive. Let . Then the norm of is said to be Gâteaux differentiable if
(2.2)
exists for each . In this case is called 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
(2.3)
for all . In this case the norm is Gâteaux differentiable and
(2.4)
for all . On the other hand,
(2.5)

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 .

Let be the modulus of smoothness of defined by
(2.6)
A Banach space is said to be 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
(2.7)

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:

(i) for all , where ;

(ii) for all , where .

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 , .

Lemma 2.5 (see [17, 27]).

Let and , be sequences of nonnegative real numbers satisfying the inequality
(2.8)

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

To deal with a family of mappings, the following conditions are introduced. Let be a subset of a real Banach space , and let be a family of mappings of such that . Then is said to satisfy the AKTT-condition [28] if for each bounded subset of ,
(2.9)

Lemma 2.6 (see [28]).

Let be a nonempty and closed subset of a Banach space , and let be a family of mappings of into itself which satisfies the AKTT-condition, then the mapping defined by
(2.10)
satisfies
(2.11)

for each bounded subset of .

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

Lemma 2.7 (see [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;

(2) .

Lemma 2.8 (see [30]).

Let be a closed and convex subset of a smooth Banach space . Suppose that is a countable family of -strictly pseudocontractive mappings of into itself with . For each , define by
(2.12)

where is a family of nonnegative numbers satisfying

(i) for all ;

(ii) for all ;

(iii) .

Then

(1)each is a -strictly pseudocontractive mapping;

(2) satisfies AKTT-condition;

(3)If is defined by

(2.13)

then and .

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.

Let be a real Banach space, and let be a nonempty, closed, and convex subset of . Let be a family of -strict pseudocontractions for some such that . Define a sequence by ,
(3.1)

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

(i) exists for all ;

(ii) .

Proof.

Let , and put . First, we observe that
(3.2)
Since is a -strict pseudocontraction, there exists . By Lemma 2.1 we have
(3.3)
This implies that
(3.4)
Hence, by , we have from Lemma 2.5 that exists; consequently, is bounded. Moreover, by (3.3), we also have
(3.5)
where . It follows that . Since is bounded,
(3.6)

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

Lemma 3.2.

Let be a real Banach space with the Fréchet differentiable norm. For , let be defined for by
(3.7)
Then, , and
(3.8)

for all .

Proof.

Let . Since has the Fréchet differentiable norm, it follows that
(3.9)
Then , and hence
(3.10)
which implies that
(3.11)
Suppose that . Put and . By (3.11), we have
(3.12)

This completes the proof.

Remark 3.3.

In a real Hilbert space, we see that for .

In our more general setting, throughout this paper we will assume that
(3.13)

where is a function appearing in (3.8).

So we obtain the following results.

Lemma 3.4.

Let be a real Banach space with the Fréchet differentiable norm, and let be a nonempty, closed, and convex subset of . Let be a family of -strict pseudocontractions for some such that . Define a sequence by ,
(3.14)

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

Proof.

Let , and put . Then by (3.8) and (3.13) we have
(3.15)
It follows that
(3.16)
Observe that
(3.17)
By (3.17), we have
(3.18)
Since , , and , it follows from Lemma 2.5 that exists. Hence, by Lemma 3.1(ii), we can conclude that . Since
(3.19)

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

Now, we prove our main result.

Theorem 3.5.

Let be a real uniformly convex Banach space with the Fréchet differentiable norm, and let be a nonempty, closed, and convex subset of . Let be a family of -strict pseudocontractions for some such that . Define a sequence by ,
(3.20)

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

Proof.

Let , and define by
(3.21)
Then . By (3.8), we have for bounded that
(3.22)

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 .

Finally, we will show that is a singleton. Suppose that . Hence . By Lemma 2.4, exists. Suppose that and are subsequences of such that and . Then
(3.23)

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.

Let be a real uniformly convex Banach space with the Fréchet differentiable norm, and let be a nonempty, closed, and convex subset of . Let be a sequence of -strict pseudocontractions of into itself such that and . Define a sequence by ,
(3.24)

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

Remark 3.7.
  1. (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.

Let be a real Fréchet smooth and reflexive Banach space which satisfies Opial's condition, and let be a nonempty, closed, and convex subset of . Let be a family of -strict pseudocontractions for some such that . Define a sequence by ,
(3.25)

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

Proof.

Let . By Lemma 3.1(i), we know that exists. Since has the Fréchet differentiable norm, by Lemma 3.4, we know that . It follows from Lemma 2.3 that . Finally, we show that is a singleton. Let , and let and be subsequences of chosen so that and . If , then Opial's condition of implies that
(3.26)

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

Theorem 3.9.

Let be a real Fréchet smooth and reflexive Banach space which satisfies Opial's condition, and let be a nonempty, closed, and convex subset of . Let be a sequence of -strict pseudocontractions of into itself such that and . Define a sequence by ,
(3.27)

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

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University
(2)
Centre of Excellence in Mathematics, CHE

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© Prasit Cholamjiak and Suthep Suantai. 2010

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.