- Research Article
- Open access
- Published:
Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric Spaces with
-Distance
Fixed Point Theory and Applications volume 2010, Article number: 134897 (2010)
Abstract
We introduce the concept of a -compatible mapping to obtain a coupled coincidence point and a coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces equipped with
-distances. Related coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend, and unify several well-known comparable results in the literature.
1. Introduction and Preliminaries
In 1996, Kada et al. [1] introduced the notion of -distance. They elaborated, with the help of examples, that the concept of
-distance is general than that of metric on a nonempty set. They also proved a generalization of Caristi fixed point theorem employing the definition of
-distance on a complete metric space. Recently, Ilić and Rakočević [2] obtained fixed point and common fixed point theorems in terms of
-distance on complete metric spaces (see also [3–9]).
Definition 1.1.
Let be a metric space. A mapping
is called a
-distance on
if the following are satisfied:
(w1) for all
,
(w2) for any ,
is lower semicontinuous,
(w3) for any there exists
such that
and
imply
, for any
.
The metric is a
-distance on
. For more examples of
-distances, we refer to [10].
Definition 1.2.
Let be a nonempty set with a
-distance on
. Ones denotes the
-closure of a subset
of
by
which is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ1_HTML.gif)
The next Lemma is crucial in the proof of our results.
Lemma 1.3 (see [1]).
Let be a metric space, and let
be a
-distance on
. Let
and
be sequences in
, let
and
be sequences in
converging to 0, and let
. Then the following hold.
(1)If and
for any
, then
. In particular, if
,
then
.
(2)If and
for any
, then
converges to
.
(3)If for any
with
, then
is a Cauchy sequence.
(4)If for any
, then
is a Cauchy sequence.
Bhaskar and Lakshmikantham in [11] introduced the concept of coupled fixed point of a mapping and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in [12] introduced this concept in cone metric spaces. They investigated some coupled fixed point theorems in cone metric spaces. Recently, Lakshmikantham and Ćirić [13] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces which extend the coupled fixed point theorem given in [11]. The following are some other definitions needed in the sequel.
Definition 1.4 . (see [12]).
Let be any nonempty set. Let
and
be two mappings. An ordered pair
is called
(1)a coupled fixed point of a mapping if
and
,
(2)a coupled coincidence point of hybrid pair if
and
and
is called coupled point of coincidence,
(3)a common coupled fixed point of hybrid pair if
and
.
Note that if is a coupled fixed point of
, then
is also a coupled fixed point of the mapping
.
Definition 1.5.
Let be any nonempty set. Mappings
and
are called
-compatible if
whenever
and
.
Definition 1.6.
Let be a metric space with
-distance
. A mapping
is said to be
-continuous at a point
with respect to mapping
if for every
there exists a
such that
implies that
for all
.
Definition 1.7.
Let be a partially ordered set. Mapping
is called strictly monotone increasing mapping if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ2_HTML.gif)
Definition 1.8.
Let be a partially ordered set. A mapping
is said to be a mixed monotone if
is monotone nondecreasing in
and monotone nonincreasing in
, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ3_HTML.gif)
Kada et al. [1] gave an example to show that is not symmetric in general. We denote by
and
, respectively, the class of all
-distances on
and the class of all
-distances on
which are symmetric for comparable elements in
. Also in the sequel, we will consider that
and
are comparable with respect to ordering in
if
and
.
2. Coupled Coincidence Point
In this section, we prove coincidence point results in the frame work of partially ordered metric spaces in terms of a -distance.
Theorem 2.1.
Let be a partially ordered metric space with a
-distance
and
a strictly monotone increasing mapping. Suppose that a mixed monotone mapping
is
-continuous with respect to
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ4_HTML.gif)
for all with
or
and
. Let
and
whenever
, for some
. If
is complete and there exist
such that
and
, then
and
have a coupled coincidence point.
Proof.
Let and
for some
; this can be done since
. Following the same arguments, we obtain
and
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ5_HTML.gif)
Similarly for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ6_HTML.gif)
Since is strictly monotone increasing and
has the mixed monotone property, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ7_HTML.gif)
Similarly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ8_HTML.gif)
Now for all , using (2.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ9_HTML.gif)
From (2.6),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ10_HTML.gif)
where . Continuing, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ11_HTML.gif)
if is odd, where
. Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ12_HTML.gif)
if is even, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ13_HTML.gif)
Let ; then for every
in
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ14_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ15_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ16_HTML.gif)
For , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ17_HTML.gif)
which further implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ18_HTML.gif)
Lemma 1.3(3) implies that and
are Cauchy sequences in
. Since
is complete, there exist
such that
and
. Since
is lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ19_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ20_HTML.gif)
Similarly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ21_HTML.gif)
Let be given. Since
is
-continuous at
with respect to
, there exists
such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ22_HTML.gif)
Since and
, for
, there exists
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ23_HTML.gif)
Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ24_HTML.gif)
implies that . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ25_HTML.gif)
using Lemma 1.3(1), we obtain . Similarly, we can prove that
. Hence
is coupled coincidence point of
and
.
Theorem 2.2.
Let be a partially ordered metric space with a
-distance
having the following properties.
(1)If is in
with
for all
and
for some
, then
for all
.
(2)If is in
with
for all
and
for some
, then
for all
.
Let be a mixed monotone and
a strict monotone increasing mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ26_HTML.gif)
for all with
or
and
. Let
and
whenever
, for some
. If
is complete and there exist
such that
and
, then
and
have a coupled coincidence point.
Proof.
Construct two sequences and
such that
and
for all
and
and
for some
, as given in the proof of Theorem 2.1. Now, we need to show that
and
. Let
. Since
and
, there exists
such that, for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ27_HTML.gif)
Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ28_HTML.gif)
which implies that . Also, from Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ29_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ30_HTML.gif)
implies that . Similarly, we can prove that
. Hence
is coupled coincidence point of
and
.
3. Coupled Common Fixed Point
In this section, using the concept of -compatible maps, we obtain a unique coupled common fixed point of two mappings.
Theorem 3.1.
Let all the hypotheses of Theorem 2.1 (resp., Theorem 2.2) hold with . If for every
there exists
that is comparable to
and
with respect to ordering in
, then there exists a unique coupled point of coincidence of
and
. Moreover if
and
are
-compatible, then
and
have a unique coupled common fixed point.
Proof.
Let be another coupled coincidence point of
and
. We will discuss the following two cases.
Case 1.
If is comparable to
with respect to ordering in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ31_HTML.gif)
implies that . Hence
. Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ32_HTML.gif)
gives that . The result follows using Lemma 1.3(1).
Case 2.
If is not comparable to
, then there exists an upper bound or lower bound
of
. Again since
is strictly monotone increasing mapping and
satisfies mixed monotone property, therefore, for all
,
is comparable to
and
. Following similar arguments to those given in the proof of Theorem 2.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ33_HTML.gif)
where and
. On taking limit as
on both sides of (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ34_HTML.gif)
and . By the same lines as in Case 1, we prove that
. Again Lemma 1.3(1) implies that
and
. Hence
is unique coupled point of coincidence of
and
. Note that if
is a coupled point of coincidence of
and
, then
are also a coupled points of coincidence of
and
. Then
and therefore
is unique coupled point of coincidence of
and
. Let
. Since
and
are w-compatible, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ35_HTML.gif)
Consequently . Therefore
. Hence
is a coupled common fixed point of
and
.
Remark 3.2.
If in addition to the hypothesis of Theorem 2.1 (resp., Theorem 2.2) we suppose that ,
and
are comparable, then
.
Proof.
Recall that . Now, if
, then
. We claim that, for all
,
. Since
is strictly monotone increasing and
satisfies mixed monotone property, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ36_HTML.gif)
Assuming that , since
is strictly monotone increasing, so
. By the mixed monotone property of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ37_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ38_HTML.gif)
Letting , there exists an
such that
and
for all
. Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F134897/MediaObjects/13663_2010_Article_1203_Equ39_HTML.gif)
implies that . Since
, therefore
. Similarly we can prove that
. Hence by Lemma 1.3(1), we have
. Similarly, if
, we can show that
for each
and
.
References
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.
Ilić D, Rakočević V: Common fixed points for maps on metric space with w -distance. Applied Mathematics and Computation 2008,199(2):599–610. 10.1016/j.amc.2007.10.016
Guran L: Fixed points for multivalued operators with respect to a w -distance on metric spaces. Carpathian Journal of Mathematics 2007,23(1–2):89–92.
Lin L-J, Du W-S: Some equivalent formulations of the generalized Ekeland's variational principle and their applications. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):187–199. 10.1016/j.na.2006.05.006
Lin L-J, Du W-S: Systems of equilibrium problems with applications to new variants of Ekeland's variational principle, fixed point theorems and parametric optimization problems. Journal of Global Optimization 2008,40(4):663–677. 10.1007/s10898-007-9146-0
Morales JR: Generalizations of Some Fixed Point Theorems. Notas de mathematica, 1999. Pre-Print, no. 199
Morales JR: Fixed point's theorems for --contractions. Notas de mathematica, 2004. Pre-Print, no. 230
Ansari QH: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. Journal of Mathematical Analysis and Applications 2007,334(1):561–575. 10.1016/j.jmaa.2006.12.076
Ume J-S: Fixed point theorems related to Ćirić's contraction principle. Journal of Mathematical Analysis and Applications 1998,225(2):630–640. 10.1006/jmaa.1998.6030
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis: Theory, Methods and Applications 2006,65(7):1379–1393. 10.1016/j.na.2005.10.017
Sabetghadam F, Masiha HP, Sanatpour AH: Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009, 2009:-8.
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(12):4341–4349. 10.1016/j.na.2008.09.020
Acknowledgment
The present version of the paper owes much to the precise and kind remarks of the learned referees.
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
Abbas, M., Ilić, D. & Khan, M. Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric Spaces with -Distance.
Fixed Point Theory Appl 2010, 134897 (2010). https://doi.org/10.1155/2010/134897
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/134897