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Fixed Point Theory for Contractive Mappings Satisfying
-Maps in
-Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 181650 (2010)
Abstract
We prove some fixed point results for self-mapping in a complete
-metric space
under some contractive conditions related to a nondecreasing map
with
for all
. Also, we prove the uniqueness of such fixed point, as well as studying the
-continuity of such fixed point.
1. Introduction
The fixed point theorems in metric spaces are playing a major role to construct methods in mathematics to solve problems in applied mathematics and sciences. So the attraction of metric spaces to a large numbers of mathematicians is understandable. Some generalizations of the notion of a metric space have been proposed by some authors. In 2006, Mustafa in collaboration with Sims introduced a new notion of generalized metric space called -metric space [1]. In fact, Mustafa et al. studied many fixed point results for a self-mapping in
-metric space under certain conditions; see[1–5]. In the present work, we study some fixed point results for self-mapping in a complete
-metric space
under some contractive conditions related to a nondecreasing map
with
for all
.
2. Basic Concepts
In this section, we present the necessary definitions and theorems in -metric spaces.
Definition 2.1 (see [1]).
Let be a nonempty set and let
be a function satisfying the following properties:
(1)  
  if
;
(2),  for all
with
;
(3)   for all
with
;
(4), symmetry in all three variables;
(5)   for all
.
Then the function is called a generalized metric, or, more specifically, a
-metric on
, and the pair
is called a
-metric space.
Definition 2.2 (see [1]).
Let be a
-metric space, and let
be a sequence of points of
, a point
is said to be the limit of the sequence
, if
, and we say that the sequence
is
-convergent to
or
  
-converges to
.
Thus, in a
-metric space
if for any
, there exists
such that
for all
.
Proposition 2.3 (see [1]).
Let be a
-metric space. Then the following are equivalent.
(1) is
-convergent to
.
(2) as
.
(3) as
.
(4) as
.
Definition 2.4 (see [1]).
Let be a
-metric space; a sequence
is called
-Cauchy if for every
, there is
such that
, for all
; that is,
as
Proposition 2.5 (see [3]).
Let be a
-metric space. Then the following are equivalent.
(1)The sequence is
-Cauchy.
(2)For every , there is
such that
, for all
.
Definition 2.6 (see [1]).
Let and
be
-metric spaces, and let
be a function. Then
is said to be
-continuous at a point
if and only if for every
, there is
such that
and
implies
. A function
is
-continuous at
if and only if it is
-continuous at all
.
Proposition 2.7 (see [1]).
Let and
be
-metric spaces. Then
is
-continuous at
if and only if it is
-sequentially continuous at
; that is, whenever
is
-convergent to
,
is
-convergent to
.
Proposition 2.8 (see [1]).
Let be a
-metric space. Then the function
is jointly continuous in all three of its variables.
The following are examples of -metric spaces.
Example 2.9 (see [1]).
Let be the usual metric space. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ1_HTML.gif)
for all . Then it is clear that
is a
-metric space.
Example 2.10 (see [1]).
Let . Define
on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ2_HTML.gif)
and extend to
by using the symmetry in the variables. Then it is clear that
is a
-metric space.
Definition 2.11 (see [1]).
A -metric space
is called
-complete if every
-Cauchy sequence in
is
-convergent in
.
3. Main Results
Following to Matkowski [6], let be the set of all functions
such that
be a nondecreasing function with
for all
. If
, then
is called
-map. If
is
-map, then it is an easy matter to show that
(1) for all
;
(2).
From now unless otherwise stated we mean by the
-map. Now, we introduce and prove our first result.
Theorem 3.1.
Let be a complete
-metric space. Suppose the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ3_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Choose . Let
,
. Assume
, for each
. Claim
is a
-Cauchy sequence in
: for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ4_HTML.gif)
given , since
and
, there is an integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ5_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ6_HTML.gif)
For with
, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ7_HTML.gif)
We prove Inequality (3.5) by induction on . Inequality (3.5) holds for
by using Inequality (3.4) and the fact that
. Assume Inequality (3.5) holds for
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ8_HTML.gif)
By induction on , we conclude that Inequality (3.5) holds for all
. So
is
-Cauchy and hence
is
-convergent to some
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ9_HTML.gif)
Letting , and using the fact that
is continuous on its variable, we get that
. Hence
. So
is a fixed point of
. Now, let
be another fixed point of
with
. Since
is a
-map, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ10_HTML.gif)
which is a contradiction. So , and hence
has a unique fixed point. To Show that
is
-continuous at
, let
be any sequence in
such that
is
-convergent to
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ11_HTML.gif)
Letting , we get
. Hence
is
-convergent to
. So
is
-continuous at u.
As an application of Theorem 3.1, we have the following results.
Corollary 3.2.
Let be a complete
-metric space. Suppose that the map
satisfies for
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ12_HTML.gif)
for all . Then
has a unique fixed point (say
).
Proof.
From Theorem 3.1, we conclude that has a unique fixed point say
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ13_HTML.gif)
we have that is also a fixed point to
. By uniqueness of
, we get
.
Corollary 3.3.
Let be a complete
-metric space. Suppose that the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ14_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
follows from Theorem 3.1 by taking .
Corollary 3.4.
Let be a complete
-metric space. Suppose there is
such that the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ15_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Define by
. Then it is clear that
is a nondecreasing function with
for all
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ16_HTML.gif)
the result follows from Theorem 3.1.
The above corollary has been stated in [7, Theorem  5.1.7], and proved by a different way.
Corollary 3.5.
Let be a complete
-metric space. Suppose the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ17_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Define by
. Then it is clear that
is a nondecreasing function with
for all
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ18_HTML.gif)
the result follows from Theorem 3.1.
Theorem 3.6.
Let be a complete
-metric space. Suppose that the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ19_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Choose . Let
,
. Assume
, for each
. Thus for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ20_HTML.gif)
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ21_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ22_HTML.gif)
which is impossible. So it must be the case that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ23_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ24_HTML.gif)
Thus for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ25_HTML.gif)
The same argument is similar to that in proof of Theorem 3.1; one can show that is a
-Cauchy sequence. Since
is
-complete, we conclude that
is
-convergent to some
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ26_HTML.gif)
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ27_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ28_HTML.gif)
Letting , we conclude that
, and hence
.
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ29_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ30_HTML.gif)
Letting , we conclude that
, and hence
.
Case 3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ31_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ32_HTML.gif)
Letting , we conclude that
, and hence
. In all cases, we conclude that
is a fixed point of
. Let
be any other fixed point of
such that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ33_HTML.gif)
which is a contradiction since . Therefore,
and hence
. To show that
is
-continuous at
, let
be any sequence in
such that
is
-convergent to
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ34_HTML.gif)
Let , we get that
is
-convergent to
. Hence
is
-continuous at
.
As an application to Theorem 3.6, we have the following results.
Corollary 3.7.
Let be a complete
-metric space. Suppose there is
such that the map
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ35_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Define by
. Then it is clear that
is a nondecreasing function with
for all
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ36_HTML.gif)
for all the result follows from Theorem 3.6.
Corollary 3.8.
Let be a complete
-metric space. Suppose that the map
satisfies:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F181650/MediaObjects/13663_2010_Article_1215_Equ37_HTML.gif)
for all . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
It follows from Theorem 3.6 by replacing .
References
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Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete
-metric spaces Fixed Point Theory and Applications 2008, 2008:-12.
Mustafa Z, Sims B: Some remarks concerning -metric spaces. In Proceedings of the International Conference on Fixed Point Theory and Applications, 2004, Yokohama, Japan. Yokohama; 189–198.
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete
-metric spaces Fixed Point Theory and Applications 2009, 2009:-10.
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in
-metric spaces International Journal of Mathematics and Mathematical Sciences 2009, 2009:-10.
Matkowski J: Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society 1977,62(2):344–348. 10.1090/S0002-9939-1977-0436113-5
Mustafa Z: A new structure for generalized metric spaces with applications to fixed point theory, Ph.D. thesis. University of Newcastle, Newcastle, UK; 2005.
Acknowledgments
The author would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper. This paper is financially supported by the Deanship of the Academic Research at the Hashemite University, Zarqa, Jordan.
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Shatanawi, W. Fixed Point Theory for Contractive Mappings Satisfying -Maps in
-Metric Spaces.
Fixed Point Theory Appl 2010, 181650 (2010). https://doi.org/10.1155/2010/181650
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DOI: https://doi.org/10.1155/2010/181650