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Fixed Point Theory and Applications

, 2008:824607 | Cite as

Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions

Open Access
Research Article

Abstract

This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems.

Keywords

Strong Convergence Real Banach Space Common Fixed Point Nonempty Closed Convex Subset Real Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a real Banach space and Open image in new window a nonempty closed subset of Open image in new window . A mapping Open image in new window is said to be pseudocontractive (see, e.g., [1]) if
holds for all Open image in new window . Open image in new window is said to be strictly pseudocontractive if, for all Open image in new window , there exists a constant Open image in new window such that
Denote by Open image in new window the set of fixed points of Open image in new window . A map Open image in new window is called hemicontractive if Open image in new window and for all Open image in new window , Open image in new window , the following inequality holds:

It is easy to see that the class of pseudocontractive mappings with fixed points is a subset of the class of hemicontractions.

There are many papers in the literature dealing with the approximation of fixed points for several classes of nonlinear mappings (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], and the reference therein). In these works, there are two iterative methods to be used to find a point in Open image in new window . One is explicit and one is implicit.

The explicit one is the following well-known Mann iteration.

Let Open image in new window be a nonempty closed convex subset of Open image in new window . For any Open image in new window , the sequence Open image in new window is defined by

where Open image in new window is a real sequence in Open image in new window satisfying some assumptions.

It has been applied to many classes of nonlinear mappings to find a fixed point. However, for hemicontractive mappings and strictly pseudocontractive mappings, the iteration process of convergence is in general not strong (see a counterexample given by Chidume and Mutangadura [3]). Most recently, Marino and Xu [6] proved that the Mann iterative sequence Open image in new window converges weakly to a fixed point for strictly pseudocontractive mappings in a Hilbert space, while the real sequence Open image in new window satisfying (i) Open image in new window and (ii) Open image in new window .

In order to get strong convergence for fixed points of hemicontractive mappings and strictly pseudocontractive mappings, the following Mann-type implicit iteration scheme is introduced.

Let Open image in new window be a nonempty closed convex subset of Open image in new window with Open image in new window . For any Open image in new window , the sequence Open image in new window is generated by

where Open image in new window is a real sequence in Open image in new window satisfying suitable conditions.

Recently, in the setting of a Hilbert space, Rafiq [12] proved that the Mann-type implicit iterative sequence Open image in new window converges strongly to a fixed point for hemicontractive mappings, under the assumption that the domain Open image in new window of Open image in new window is a compact convex subset of a Hilbert space, and Open image in new window for some Open image in new window .

In this paper, we will study the strong convergence of the generalized Mann-type iteration scheme (see Definition 2.1) for hemicontractive and, respectively, pseudocontractive mappings. As we will see, our theorems extend the corresponding results in [12] in four aspects. (1) The space setting is a more general one: uniformly convex Banach space, which could not be a Hilbert space. (2) The requirement of the compactness on the domain of the mapping is dropped. (3) A single mapping is replaced by a family of mappings. (4) The Mann-type implicit iteration is replaced by the generalized Mann iteration. Moreover, we give answers to a question asked in [13].

2. Preliminaries and Lemmas

Definition 2.1 (generalized Mann iteration).

Let Open image in new window be a fixed integer, Open image in new window , and Open image in new window a nonempty closed convex subset of Open image in new window satisfying the condition Open image in new window . Let Open image in new window be a family of mappings. For each Open image in new window , the sequence Open image in new window is defined by

where Open image in new window , Open image in new window , Open image in new window and Open image in new window are three sequences in Open image in new window with Open image in new window and Open image in new window is bounded.

The modulus of convexity of Open image in new window is the function Open image in new window defined by
Open image in new window is called uniformly convex if and only if, for all Open image in new window such that Open image in new window . Open image in new window is called Open image in new window -uniformly convex if there exists a constant Open image in new window , such that Open image in new window . It is well known (see [10]) that

Let Open image in new window be a Banach space, Open image in new window and Open image in new window . Then, we denote Open image in new window .

Definition 2.2 (see [4]).

Let Open image in new window be a nondecreasing function with Open image in new window and Open image in new window , for all Open image in new window .

  1. (i)

    A mapping Open image in new window with Open image in new window is said to satisfy condition (A) on Open image in new window if there is a function Open image in new window such that for all Open image in new window , Open image in new window .

     
  2. (ii)

    A finite family of mappings Open image in new window with Open image in new window are said to satisfy condition ( Open image in new window ) if there exists a function Open image in new window , such that Open image in new window holds for all Open image in new window .

     

Lemma 2.3 (see [8]).

Let Open image in new window be a real uniformly convex Banach space with the modulus of convexity of power type Open image in new window . Then, for all Open image in new window in Open image in new window and Open image in new window , there exists a constant Open image in new window such that

where Open image in new window .

Remark 2.4.

If Open image in new window in the previous lemma, then we denote Open image in new window .

Lemma 2.5.

Let Open image in new window be a real Banach space and Open image in new window the normalized duality mapping. Then for any Open image in new window in Open image in new window and Open image in new window , such that

Lemma 2.6 (see [7]).

Let Open image in new window and Open image in new window be three nonnegative real sequences, satisfying

with Open image in new window and Open image in new window . Then, Open image in new window exists. In addition, if Open image in new window has a subsequence converging to zero, then Open image in new window .

Proposition 2.7.

If Open image in new window is a strict pseudocontraction, then Open image in new window satisfies the Lipschitz condition

Proof.

By the definition of the strict pseudocontraction, we have

A simple computation shows the conclusion.

3. Main Results

Lemma 3.1.

Let Open image in new window be a uniformly convex Banach space with the convex modulus of power type Open image in new window , Open image in new window a nonempty closed convex subset of Open image in new window satisfying Open image in new window , and Open image in new window hemicontractive mappings with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window be the sequences in (II) and

where Open image in new window is the constant in Remark 2.4. Then,

(1) Open image in new window exists for all Open image in new window ,

Proof. (1) Let Open image in new window . By the boundedness assumption on Open image in new window , there exists a constant Open image in new window , for any Open image in new window , such that Open image in new window . From the definition of hemicontractive mappings, we have
Using Lemmas 2.3, 2.5, and (3.2), we obtain
It follows from (II) and Lemma 2.5 that
By the condition Open image in new window , we may assume that
Therefore,
Substituting (3.7) into (3.4), we get
Assumptions (i) and (ii) imply that there exists a positive integer Open image in new window such that for every Open image in new window ,
From (3.9) and conditions (i) and (ii), it follows that

By Lemma 2.6, we see that Open image in new window exists and the sequence Open image in new window is bounded.

(2) It is easy to verify that Open image in new window exists.

(3) By the boundedness of Open image in new window , there exists a constant Open image in new window such that Open image in new window , for all Open image in new window . From (3.10), we get, for Open image in new window ,
which implies
It implies that
Therefore, by (3.7), we have
Using (II), we obtain
By a combination with the continuity of Open image in new window ( Open image in new window , we get
It is clear that for each Open image in new window , there exists Open image in new window such that Open image in new window . Consequently,

This completes the proof.

Theorem 3.2.

Let the assumptions of Lemma 3.1 hold, and let Open image in new window be continuous. Then, Open image in new window converges strongly to a common fixed point of Open image in new window if and only if Open image in new window .

Proof.

The necessity is obvious.

Now, we prove the sufficiency. Since Open image in new window , it follows from Lemma 3.1 that Open image in new window .

Hence, we get
So, Open image in new window is a Cauchy sequence in Open image in new window . By the closedness of Open image in new window , we get that the sequence Open image in new window converges strongly to Open image in new window . Let a sequence Open image in new window , for some Open image in new window , be such that Open image in new window converges strongly to Open image in new window . By the continuity of Open image in new window , we obtain

Therefore, Open image in new window . This implies that Open image in new window is closed. Therefore, Open image in new window is closed. By Open image in new window , we get Open image in new window . This completes the proof.

Theorem 3.3.

Let the assumptions of Lemma 3.1 hold. Let Open image in new window be continuous and Open image in new window satisfy condition Open image in new window . Then, Open image in new window converges strongly to a common fixed point of Open image in new window .

Proof.

Since Open image in new window satisfies condition Open image in new window , and Open image in new window for each Open image in new window , it follows from the existence of Open image in new window that Open image in new window . Applying the similar arguments as in the proof of Theorem 3.2, we conclude that Open image in new window converges strongly to a common fixed point of Open image in new window . This completes the proof.

As a direct consequence of Theorem 3.3, we get the following result.

Corollary 3.4 (see [12, Theorem 3]).

Let H be a real Hilbert space, Open image in new window a nonempty closed convex subset of Open image in new window satisfying Open image in new window , and Open image in new window continuous hemicontractive mapping which satisfies condition (A). Let Open image in new window be a real sequence in Open image in new window with Open image in new window . For any Open image in new window , the sequence Open image in new window is defined by

Then, Open image in new window converges strongly to a fixed point of Open image in new window .

Proof.

Employing the similar proof method of Lemma 3.1, we obtain by (3.10)
This implies

By Open image in new window , we have Open image in new window . Equation (3.7) implies that Open image in new window Open image in new window . Since Open image in new window satisfies condition (A) and the limit Open image in new window exists, we get Open image in new window . The rest of the proof follows now directly from Theorem 3.2. This completes the proof.

Remark 3.5.

Theorems 3.2 and 3.3 extend [12, Theorem 3] essentially since the following hold.

  1. (i)

    Hilbert spaces are extended to uniformly convex Banach spaces.

     
  2. (ii)

    The requirement of compactness on domain Open image in new window on [12, Theorem 3] is dropped.

     
  3. (iii)

    A single mapping is replaced by a family of mappings.

     
  4. (iv)

    The Mann-type implicit iteration is replaced by the generalized Mann iteration. So the restrictions of Open image in new window with Open image in new window for some Open image in new window are relaxed to Open image in new window . The error term is also considered in the iteration (II).

     

Moreover, if Open image in new window , then Open image in new window is well defined by (II). Hence, Theorems 3.2 and 3.3 are also answers to the question proposed by Qing [13].

Theorem 3.6.

Let Open image in new window and Open image in new window be as the assumptions of Lemma 3.1. Let Open image in new window be strictly pseudocontractive mappings with Open image in new window being nonempty. Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be the sequences in (II) and

where Open image in new window is the constant in Remark 2.4. Then,

(1) Open image in new window converges strongly to a common fixed point of Open image in new window if and only if Open image in new window .
  1. (2)

    If Open image in new window satisfies condition ( Open image in new window ) , then Open image in new window converges strongly to a common fixed point of Open image in new window .

     

Remark 3.7.

Theorem 3.6 extends the corresponding result [6, Theorem 3.1].

Notes

Acknowledgments

The authors would like to thank the referees very much for helpful comments and suggestions. The work was supported partly by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.

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Copyright information

© Liang-Gen Hu et al. 2008

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.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina
  3. 3.Department of MathematicsShanghai Jiaotong UniversityShanghaiChina

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