Open Access

On Uniformly Generalized Lipschitzian Mappings

Fixed Point Theory and Applications20102010:692401

https://doi.org/10.1155/2010/692401

Received: 30 November 2009

Accepted: 11 April 2010

Published: 17 May 2010

Abstract

We consider another class of generalized Lipschitzian type mappings and utilize the same to prove fixed point theorems for asymptotically regular and uniformly generalized Lipschitzian one-parameter semigroups of self-mappings defined on bounded metric spaces equipped with uniform normal structure which yield corresponding results in respect of semigroup of iterates of a self-mapping as corollaries. Our results also generalize some relevant results due to the work of Lim and Xu (1995), Yao and Zeng (2007), and Soliman (2009).

1. Introduction

In 1989, Khamsi [1] defined normal structure and uniform normal structure for metric spaces and utilized the same to prove that nonexpansive mappings on a complete bounded metric space equipped with uniform normal structure have fixed point property and further satisfy a kind of intersection property which extends results of Maluta [2] to metric spaces. In 1995, Lim and Xu [3] proved a fixed point theorem for uniformly Lipschitzian mappings in metric spaces equipped with both property and uniform normal structure which, in turn, extends a result of Khamsi [1]. This result is indeed the metric space version of a result contained in the work of Casini and Maluta [4]. Recently, Yao and Zeng [5] established a fixed point theorem for an asymptotically regular semigroup of uniformly Lipschitzian mappings in a complete bounded metric space equipped with uniform normal structure and the property which is an improvement over certain relevant results contained in Lim and Xu [3]. Here, it may be pointed out that there exists extensive literature on weak and strong convergence theorems via iterative procedures in respect of semigroups of nonexpansive operators in Banach spaces (e.g., [68]).

In this paper, we introduce yet another class of uniformly generalized Lipschitzian one-parameter semigroups of self mappings defined on bounded metric spaces equipped with uniform normal structure and utilize the same to prove our results. Our results are generalizations of certain results due to Yao and Zeng [5] and also Soliman [9].

2. Preliminaries

Throughout this paper, (also ) stands for a metric space. In what follows, we recall some relevant definitions and results in respect of uniformly Lipschitzian mappings and uniformly generalized Lipschitzian mappings in metric spaces.

Definition 2.1 (see [4]).

A mapping is said to be a Lipschitzian mapping if for each integer , there exists a constant , such that
(2.1)

If , then is called uniformly Lipschitzian, and if , then is called nonexpansive.

In 2001, Jung and Thakur [10] introduced and studied the following class of mappings.

Definition 2.2 (see [10]).

A mapping is said to be generalized Lipschitzian mapping (in short G1-Lipschitzian) if
(2.2)

for each and , where are nonnegative constants such that there exists an integer such that for all . Here it may be pointed out that this class of generalized Lipschitzian mappings is relatively larger than the classes of nonexpansive, asymptotically nonexpansive, Lipschitzian, and uniformly k-Lipschitzian mappings. The earlier mentioned facts can be realized by choosing constants , , and suitably.

Recently, in 2009, Soliman [9] defined another class of generalized Lipschitzian mappings on metric spaces as follows.

Definition 2.3.

A mapping is said to be a generalized Lipschitzian (in short G2-Lipschitzian) mapping if for each integer there exists a constant (depending on ) such that
(2.3)

for every If , then is called uniformly G2-Lipschitzian.

In the following (motivated by Khan and Imdad [11]), we define yet another class of generalized Lipschitzian mappings on metric spaces.

Definition 2.4.

A mapping is said to be a generalized Lipschitzian (in short G3-Lipschitzian) mapping if for each integer there exists a constant (depending on ) such that
(2.4)

for every If , then is called uniformly G3-Lipschitzian.

Definition 2.5 (see [12]).

A mapping is called asymptotically regular, if
(2.5)
Let be a sub-semigroup of with addition "+" such that
(2.6)

Notice that the foregoing condition is satisfied if we take or , the set of nonnegative integers.

Let be a family of self mappings on Then is called a (one-parameter) semigroup on if the following conditions are satisfied:

(i)

(ii) and

(iii)    a mapping from into is continuous when has the relative topology of

(iv)for each is continuous.

A semigroup on is said to be asymptotically regular at a point if
(2.7)

If is asymptotically regular at each , then is called an asymptotically regular semigroup on

Definition 2.6.

A semigroup of self mappings defined on is called a uniformly G1-Lipschitzian semigroup if
(2.8)

for each , where are nonnegative constants, , and with

The simplest uniformly G1-Lipschitzian semigroup is a semigroup of iterates of a mapping whenever and with

The following definition is introduced by Soliman [9].

Definition 2.7.

A semigroup of self mappings defined on is called a uniformly G2-Lipschitzian semigroup if
(2.9)
whenever
(2.10)

for each and

Definition 2.8.

A semigroup of self mappings defined on is called a uniformly G3-Lipschitzian semigroup if
(2.11)
whenever
(2.12)

for each and , , ( 0

The simplest uniformly G3-Lipschitzian semigroup is a semigroup of iterates of a mapping with

Here it may be pointed out that the different terms, namely, uniformly k-Lipschitzian semigroups of self mappings, uniformly G1-Lipschitzian semigroups of self mappings, uniformly G2-Lipschitzian semigroups of self mappings, uniformly G3-Lipschitzian semigroups of self mappings, and uniformly k-Lipschitzian semigroups of self mappings are adopted to facilitate the statements of our results.

Remark 2.9.

Notice that the class of uniformly G3-Lipschitzian semigroups is relatively larger than the other classes, namely, uniformly G1-Lipschitzian semigroups, uniformly G2-Lipschitzian semigroups, and also uniformly k-Lipschitzian semigroups.

In a metric space , let stand for a nonempty family of subsets of Following Khamsi [1], we say that defines a convexity structure on if is stable under intersection. Also, we say that has Property (R) if any decreasing sequence of closed bounded nonempty subsets of with has a nonvoid intersection. Recall that a subset of is said to be admissible (cf. [13]) if it can be expressed as an intersection of closed balls. We denote by the family of all admissible subsets of It is obvious that defines a convexity structure on Throughout this paper any other convexity structure on is always assumed to contain Let be a bounded subset of whereas stands for the closed ball centered at with radius Following Lim and Xu [3], we will adopt the following conventions and notations:
(2.13)
For a bounded subset of , we define the admissible hull of as the intersection of all those admissible subsets of which contain and is denoted by , that is,
(2.14)

Proposition 2.10 (see [3]).

For a point and a bounded subset of , one has
(2.15)

Definition 2.11 (see [1]).

A metric space is said to have normal (resp., uniform normal) structure if there exists a convexity structure on such that (resp., for some constant for all which is bounded and consists of more than one point. In this case is said to be normal (resp., uniformly normal) in

We define the normal structure coefficient of (with respect to a given convexity structure ) as the number
(2.16)

where the supremum is taken over all bounded with It is said that has uniform normal structure if and only if

Khamsi [1] proved the following result which will be very handy in the proof of our main theorem.

Proposition 2.12 (see [1]).

Let be a complete bounded metric space and let be a convexity structure of with uniform normal structure. Then has the property

Definition 2.13 . (see [3]).

A metric space is said to have property if given any two bounded sequences and in , one can find some such that .

The following lemma due to Lim and Xu [3] will be utilized in proving our results.

Lemma 2.14 (see [3]).

Let be a complete bounded metric space equipped with uniform normal structure and the property (P). Then for any sequence and constant , the normal structure coefficient with respect to a given convexity structure , there exists some satisfying the following properties:

(i) ;

(ii) for all

Definition 2.15 (see [5]).

Let be a metric space and a semigroup of self mappings on Let one write the set
(2.17)

Definition 2.16 (see [5]).

Let be a complete bounded metric space and a semigroup of self mappings defined on Then is said to have the property if for each and each , the following conditions are satisfied:

(a)the sequence is bounded;

(b)for any sequence in there exists some such that
(2.18)

Yao and Zeng [5] proved the following result which will be used in the proof of our results.

Lemma 2.17.

Let be a complete bounded metric space with uniform normal structure and a semigroup of self mappings defined on equipped with property (*). Then for each , each , and any constant (where stands for the normal structure coefficient with respect to the given convexity structure ), there exists some satisfying the following properties:

(I) , where

(II) for all

3. Common Fixed Point Theorems

Our first result is a fixed point theorem for uniformly G1-Lipschitzian semigroups of self mappings defined on bounded metric spaces with uniform normal structure.

Theorem 3.1.

Let be a complete bounded metric space equipped with uniform normal structure. If is an asymptotically regular and uniformly G1-Lipschitzian semigroup of self mappings on which satisfies the property (*) with , then there exists some such that for all

Proof.

Choose a constant such that and We can pick a sequence such that Observe that
(3.1)

for each and

Now fix an Then by Lemma 2.17, we can inductively construct a sequence such that for each integer

(III) , where

(IV) for all

Let
(3.2)
Now for each , using (III) and the asymptotic regularity of on , we have
(3.3)
whereas
(3.4)
or
(3.5)
so that
(3.6)
Similarly, one can also show that
(3.7)
Now making use of (3.6) and (3.7) in (3.3), we have
(3.8)
which implies that for each ,
(3.9)
By taking the limit of both the sides of (3.9) as with each ( ), we have
(3.10)
where , , , , , and
(3.11)
Hence by using (III) and (3.8), we have
(3.12)
Thus and henceforth
(3.13)
By taking the limit of both the sides of (3.13) as , we have
(3.14)
which shows that is a Cauchy sequence and is convergent as is complete. Let Then we have
(3.15)
that is, Hence for each , we deduce that
(3.16)

yielding thereby , that is, for each This concludes the proof.

Our second result is a fixed point theorem for uniformly G3-Lipschitzian semigroups of self mappings defined on bounded metric spaces equipped with uniform normal structure.

Theorem 3.2.

Let be a complete bounded metric space equipped with uniform normal structure. If is an asymptotically regular and uniformly G3-Lipschitzian semigroup of self mappings defined on with which also satisfies the property (*), then there exists some such that for all

Proof.

Choose a constant such that and We can select a sequence such that and , where Observe that
(3.17)

for each and

Now fix an Then by Lemma 2.17, we can inductively construct a sequence such that for each integer

() , where

() for all

Let
(3.18)
Observe that for each , using (IV) and the asymptotic regularity of on , we have
(3.19)
(3.20)
Now making use of (3.19) in (3.20), we get
(3.21)
which implies that for each
(3.22)
Hence by using (III) and (3.22), we have
(3.23)
Hence by the asymptotic regularity of on , we have for each integer
(3.24)
It follows from (3.23) that
(3.25)
Thus, we have Consequently is Cauchy and hence convergent as is complete. Let Then we have
(3.26)
that is, Hence for each , we deduce that
(3.27)

Then we have , that is, for each

Since the class of uniformly -Lipschitzian semigroups of self mappings is contained in the class of uniformly G2-Lipschitzian semigroups of self mappings, therefore Theorem 3.1 yields the following.

Corollary 3.3 (see [5]).

Let be a complete bounded metric space with uniform normal structure. If is an asymptotically regular and uniformly -Lipschitzian semigroup of self mappings defined on equipped with the property (*) which also satisfies
(3.28)

then there exists some such that for all

Again, as the class of uniformly G3-Lipschitzian semigroups is larger than the class of uniformly G2-Lipschitzian semigroups, therefore using Theorem 3.2, one immediately derives the following result due to Soliman [9].

Corollary 3.4 (see [9]).

Let be a complete bounded metric space with uniform normal structure. If is an asymptotically regular and uniformly G2-Lipschitzian semigroup of self mappings defined on with which also satisfies the property (*). Then there exists some such that for all

If one replaces the respective one parameter semigroups of generalized Lipschitzian mappings in Theorems 3.1 and 3.2 with respective semigroups of iterates of generalized Lipschitzian mappings, then one can immediately derive the following two corollaries.

Corollary 3.5.

Let be a complete bounded metric space equipped with uniform normal structure and the property (P). If is a self-mapping whose set of iterates is an asymptotically regular semigroup of G1-Lipschitzian mappings satisfying the condition (2.2), then there exists some such that .

Corollary 3.6.

Let be a complete bounded metric space equipped with uniform normal structure and the property . If is a self-mapping whose set of iterates is an asymptotically regular semigroup of G3-Lipschitzian mappings satisfying the condition (2.4), then there exists some such that .

Remark 3.7.

It will be interesting to establish Theorems 3.1 and 3.2 for left reversible semigroup of self mappings defined on a complete bounded metric space equipped with uniform normal structure following the lines of Holmes and Lau [14], Lau and Takahashi [15], and Lau [16].

Declarations

Acknowledgments

The second author is very grateful to Professor Dr. Anthony To-Ming Lau (University of Alberta, Edmonton, Alberta, Canada) with whom he had fruitful discussions on the theme of this work. Thanks are also due to an anonymous referee for his fruitful comments.

Authors’ Affiliations

(1)
Department of Mathematics, Aligarh Muslim University
(2)
Department of Mathematics, Faculty of Science, Al-Azhar University

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Copyright

© M. Imdad and A. H. Soliman. 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.