- Research Article
- Open access
- Published:
On Uniformly Generalized Lipschitzian Mappings
Fixed Point Theory and Applications volume 2010, Article number: 692401 (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., [6–8]).
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ4_HTML.gif)
for every If
, then
is called uniformly G3-Lipschitzian.
Definition 2.5 (see [12]).
A mapping is called asymptotically regular, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ5_HTML.gif)
Let be a sub-semigroup of
with addition "+" such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ9_HTML.gif)
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ10_HTML.gif)
for each and
Definition 2.8.
A semigroup of self mappings defined on
is called a uniformly G3-Lipschitzian semigroup if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ11_HTML.gif)
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ12_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ13_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ14_HTML.gif)
Proposition 2.10 (see [3]).
For a point and a bounded subset
of
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ20_HTML.gif)
Now for each , using (III) and the asymptotic regularity of
on
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ21_HTML.gif)
whereas
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ22_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ23_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ24_HTML.gif)
Similarly, one can also show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ25_HTML.gif)
Now making use of (3.6) and (3.7) in (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ26_HTML.gif)
which implies that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ27_HTML.gif)
By taking the limit of both the sides of (3.9) as with each (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ28_HTML.gif)
where ,
,
,
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ29_HTML.gif)
Hence by using (III) and (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ30_HTML.gif)
Thus and henceforth
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ31_HTML.gif)
By taking the limit of both the sides of (3.13) as , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ32_HTML.gif)
which shows that is a Cauchy sequence and is convergent as
is complete. Let
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ33_HTML.gif)
that is, Hence for each
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ36_HTML.gif)
Observe that for each , using (IV) and the asymptotic regularity of
on
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ38_HTML.gif)
Now making use of (3.19) in (3.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ39_HTML.gif)
which implies that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ40_HTML.gif)
Hence by using (III) and (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ41_HTML.gif)
Hence by the asymptotic regularity of on
, we have for each integer
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ42_HTML.gif)
It follows from (3.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ43_HTML.gif)
Thus, we have Consequently
is Cauchy and hence convergent as
is complete. Let
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ44_HTML.gif)
that is, Hence for each
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F692401/MediaObjects/13663_2009_Article_1323_Equ46_HTML.gif)
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].
References
Khamsi MA: On metric spaces with uniform normal structure. Proceedings of the American Mathematical Society 1989,106(3):723–726. 10.1090/S0002-9939-1989-0972234-4
Maluta E: Uniformly normal structure and related coefficients. Pacific Journal of Mathematics 1984,111(2):357–369.
Lim T-C, Xu HK: Uniformly Lipschitzian mappings in metric spaces with uniform normal structure. Nonlinear Analysis: Theory, Methods & Applications 1995,25(11):1231–1235. 10.1016/0362-546X(94)00243-B
Casini E, Maluta E: Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure. Nonlinear Analysis: Theory, Methods & Applications 1985,9(1):103–108. 10.1016/0362-546X(85)90055-0
Yao J-C, Zeng L-C: Fixed point theorem for asymptotically regular semigroups in metric spaces with uniform normal structure. Journal of Nonlinear and Convex Analysis 2007,8(1):153–163.
Atsushiba S, Takahashi W: Weak and strong convergence theorems for nonexpansive semigroups in Banach spaces. Fixed Point Theory and Applications 2005,2005(3):343–354. 10.1155/FPTA.2005.343
He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:-8.
Saejung S: Strong convergence theorems for nonexpansive semigroups without Bochner integrals. Fixed Point Theory and Applications 2008, 2008:-7.
Soliman AH: A fixed point theorem for semigroups on metric spaces with uniform normal structure. Scientiae Mathematicae Japonicae 2009,69(3):323–328.
Jung JS, Thakur BS: Fixed point theorems for generalized Lipschitzian semigroups. International Journal of Mathematics and Mathematical Sciences 2001,28(1):41–50. 10.1155/S0161171201007426
Khan MS, Imdad M: Fixed points of certain involutions in Banach spaces. Journal of the Australian Mathematical Society. Series A 1984,37(2):169–177. 10.1017/S144678870002200X
Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345
Dunford N, Schwartz JT: Linear Operators. I. General Theory, Pure and Applied Mathematics. Volume 7. Interscience, New York, NY, USA; 1958:xiv+858.
Holmes RD, Lau AT: Non-expansive actions of topological semigroups and fixed points. Journal of the London Mathematical Society 1972, 5: 330–336. 10.1112/jlms/s2-5.2.330
Lau AT, Takahashi W: Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure. Journal of Functional Analysis 1996,142(1):79–88. 10.1006/jfan.1996.0144
Lau AT: Invariant means on almost periodic functions and fixed point properties. The Rocky Mountain Journal of Mathematics 1973,3(1):69–75. 10.1216/RMJ-1973-3-1-69
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Imdad, M., Soliman, A. On Uniformly Generalized Lipschitzian Mappings. Fixed Point Theory Appl 2010, 692401 (2010). https://doi.org/10.1155/2010/692401
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/692401