- Research Article
- Open access
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Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q-Function
Fixed Point Theory and Applications volume 2011, Article number: 703938 (2011)
Abstract
Using the concept of a mixed g-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009) and many others.
1. Introduction
The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions (cf. [1–31]). Recently, Bhaskar and Lakshmikantham [8], Nieto and Rodríguez-López [28, 29], Ran and Reurings [30], and Agarwal et al. [1] presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [8] noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book [22].
Recently, Al-Homidan et al. [2] introduced the concept of a -function defined on a quasi-metric space which generalizes the notions of a
-function and a
-distance and establishes the existence of the solution of equilibrium problem (see also [3–7]). The aim of this paper is to extend the results of Lakshmikantham and Ćirić [24] for a mixed monotone nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a
-function
. We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to Lakshmikantham and Ćirić [24] and many others.
Recall that if is a partially ordered set and
such that for
implies
, then a mapping
is said to be nondecreasing. Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham [8] introduced the following notions of a mixed monotone mapping and a coupled fixed point.
Definition 1.1 (Bhaskar and Lakshmikantham [8]).
Let be a partially ordered set and
. The mapping
is said to have the mixed monotone property if
is nondecreasing monotone in its first argument and is nonincreasing monotone in its second argument, that is, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ1_HTML.gif)
Definition 1.2 (Bhaskar and Lakshmikantham [8]).
An element is called a coupled fixed point of the mapping
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ2_HTML.gif)
The main theoretical result of Lakshmikantham and Ćirić in [24] is the following coupled fixed point theorem.
Theorem 1.3 (Lakshmikantham and Ćirić [24, Theorem ]).
Let be a partially ordered set, and suppose, there is a metric
on
such that
is a complete metric space. Assume there is a function
with
and
for each
, and also suppose that
and
such that
has the mixed
-monotone property and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ3_HTML.gif)
for all for which
and
Suppose that
and
is continuous and commutes with
, and also suppose that either
(a) is continuous or
(b) has the following property:
(i) if a nondecreasing sequence ,then
for all
(ii) if a nonincreasing sequence ,then
for all
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ4_HTML.gif)
then there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ5_HTML.gif)
that is, and
have a coupled coincidence.
Definition 1.4.
Let be a nonempty set. A real-valued function
is said to be quasi-metric on
if
for all
if and only if
for all
.
The pair is called a quasi-metric space.
Definition 1.5.
Let be a quasi-metric space. A mapping
is called a
-function on
if the following conditions are satisfied:
for all
if and
is a sequence in
such that it converges to a point
(with respect to the quasi-metric) and
for some
then
;
for any , there exists
such that
, and
implies that
Remark 1.6 (see [2]).
If is a metric space, and in addition to
the following condition is also satisfied:
for any sequence in
with
and if there exists a sequence
in
such that
then
then a -function is called a
-function, introduced by Lin and Du [27]. It has been shown in [27]that every
-distance or
-function, introduced and studied by Kada et al. [21], is a
-function. In fact, if we consider
as a metric space and replace
by the following condition:
for any , the function
is lower semicontinuous,
then a -function is called a
-distance on
. Several examples of
-distance are given in [21]. It is easy to see that if
is lower semicontinuous, then
holds. Hence, it is obvious that every
-function is a
-function and every
-function is a
-function, but the converse assertions do not hold.
Example 1.7 (see [2]).
-
(a)
Let
. Define
by
(16)
and by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ7_HTML.gif)
Then one can easily see that is a quasi-metric and
is a
-function on
, but
is neither a
-function nor a
-function.
-
(b)
Let
Define
by
(18)
and by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ9_HTML.gif)
Then is a
-function on
However,
is neither a
-function nor a
-function, because
is not a metric space.
The following lemma lists some properties of a -function on
which are similar to that of a
-function (see [21]).
Lemma 1.8 (see [2]).
Let be a
-function on
Let
and
be sequences in
, and let
and
be such that they converge to
and
Then, the following hold:
(1) if and
for all
, then
. In particular, if
and
, then
;
(2) if and
for all
, then
converges to
;
(3) if for all
with
, then
is a Cauchy sequence;
(4) if for all
, then
is a Cauchy sequence;
(5) if are
-functions on
, then
is also a
-function on
.
2. Main Results
Analogous with Definition 1.1, Lakshmikantham and Ćirić [24] introduced the following concept of a mixed -monotone mapping.
Definition 2.1 (Lakshmikantham and Ćirić [24]).
Let be a partially ordered set, and
and
We say
has the mixed
-monotone property if
is nondecreasing
-monotone in its first argument and is nondecreasing
-monotone in its second argument, that is, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ10_HTML.gif)
Note that if is the identity mapping, then Definition 2.1 reduces to Definition 1.1.
Definition 2.2 (see [24]).
An element is called a coupled coincidence point of a mapping
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ11_HTML.gif)
Definition 2.3 (see [24]).
Let be a nonempty set and
and
one says
and
are commutative if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ12_HTML.gif)
for all
Following theorem is the main result of this paper.
Theorem 2.4.
Let be a partially ordered complete quasi-metric space with a
-function
on
. Assume that the function
is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ13_HTML.gif)
Further, suppose that and
are such that
has the mixed
-monotone property and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ14_HTML.gif)
for all for which
and
Suppose that
and
is continuous and commutes with
, and also suppose that either
(a) is continuous or
(b) has the following property:
(i) if a nondecreasing sequence , then
for all
(ii) if a nonincreasing sequence , then
for all
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ15_HTML.gif)
then there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ16_HTML.gif)
that is, and
have a coupled coincidence.
Proof.
Choose to be such that
and
Since
we can choose
such that
and
Again from
, we can choose
such that
and
Continuing this process, we can construct sequences
and
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ17_HTML.gif)
We will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ19_HTML.gif)
We will use the mathematical induction. Let Since
and
and as
and
we have
and
Thus, (2.9) and (2.10) hold for
Suppose now that (2.9) and (2.10) hold for some fixed
Then, since
and
and as
has the mixed
-monotone property, from (2.8) and (2.9),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ20_HTML.gif)
and from (2.8) and (2.10),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ21_HTML.gif)
Now from (2.11) and (2.12), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ22_HTML.gif)
Thus, by the mathematical induction, we conclude that (2.9) and (2.10) hold for all. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ23_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ24_HTML.gif)
We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ25_HTML.gif)
Since and
from (2.11) and (2.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ26_HTML.gif)
Similarly, from (2.11) and (2.5), as and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ27_HTML.gif)
Adding (2.17) and (2.18), we obtain (2.16). Since for
it follows, from (2.16), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ28_HTML.gif)
and so, by squeezing, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ29_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ30_HTML.gif)
Now, we prove that and
are Cauchy sequences. For
and since
for each
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ31_HTML.gif)
This means that for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ32_HTML.gif)
Therefore, by Lemma 1.8, and
are Cauchy sequences. Since
is complete, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ33_HTML.gif)
and (2.24) combined with the continuity of yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ34_HTML.gif)
From (2.11) and commutativity of and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ35_HTML.gif)
We now show that and
Case 1.
Suppose that the assumption (a) holds. Taking the limit as in (2.26), and using the continuity of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ36_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ37_HTML.gif)
Case 2.
Suppose that the assumption (b) holds. Let . Now, since
is continuous,
is nondecreasing with
for all
, and
is nonincreasing with
for all
, so
is nondecreasing, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ38_HTML.gif)
with,
for all
, and
is nonincreasing, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ39_HTML.gif)
with ,
for all
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ40_HTML.gif)
Then replacing by
and
by
in (2.16), we get
such that
We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ41_HTML.gif)
In , replacing
by
and
by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ42_HTML.gif)
that is, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ43_HTML.gif)
or for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ44_HTML.gif)
Let , and
Then, since
, and
by axiom
of the
-function, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ45_HTML.gif)
Therefore, by the triangle inequality and ( ), we have for
Case 3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ46_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ47_HTML.gif)
Case 4.
Also, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ48_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ49_HTML.gif)
That is, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ50_HTML.gif)
Hence, by Lemma 1.8, and
Thus,
and
have a coupled coincidence point.
The following example illustrates Theorem 2.4.
Example 2.5.
Let with the usual partial order
Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ51_HTML.gif)
and by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ52_HTML.gif)
Then is a quasi-metric and
is a
-function on
Thus,
is a partially ordered complete quasi-metric space with a
-function
on
Let
for
Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ53_HTML.gif)
and by
, where
Then,
has the mixed
-monotone property with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ54_HTML.gif)
and ,
are both continuous on their domains and
. Let
be such that
and
There are four possibilities for (2.5) to hold. We first compute expression on the left of (2.5) for these cases:
(i) and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ55_HTML.gif)
(ii) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ56_HTML.gif)
(iii) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ57_HTML.gif)
(iv) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ58_HTML.gif)
On the other hand, (in all the above four cases), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ59_HTML.gif)
Thus, satisfies the contraction condition (2.5) of Theorem 2.4. Now, suppose that
be, respectively, nondecreasing and nonincreasing sequences such that
and
, then by Theorem 2.4,
and
for all
Let Then, this point satisfies the relations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ60_HTML.gif)
Therefore, by Theorem 2.4, there exists such that
and
Corollary 2.6.
Let be a partially ordered complete quasi-metric space with a
-function
on
. Suppose
and
are such that
has the mixed
-monotone property and assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ61_HTML.gif)
for all for which
and
Suppose that
and
is continuous and commutes with
, and also suppose that either
(a) is continuous or
(b) has the following properties:
(i) if a nondecreasing sequence , then
for all
(ii) if a nonincreasing sequence , then
for all
.
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ62_HTML.gif)
then there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ63_HTML.gif)
that is, and
have a coupled coincidence.
Proof.
Taking in Theorem 2.4, we obtain Corollary 2.6.
Now, we will prove the existence and uniqueness theorem of a coupled common fixed point. Note that if is a partially ordered set, then we endow the product
with the following partial order:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ64_HTML.gif)
From Theorem 2.4, it follows that the set of coupled coincidences is nonempty.
Theorem 2.7.
The hypothesis of Theorem 2.4 holds. Suppose that for every there exists a
such that
is comparable to
and
Then,
and
have a unique coupled common fixed point; that is, there exist a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ65_HTML.gif)
Proof.
By Theorem, 2.1 . Let
. We show that if
and
,
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ66_HTML.gif)
By assumption there is such that
is comparable with
and
Put
,
and choose
so that
and
Then, as in the proof of Theorem 2.4, we can inductively define sequences
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ67_HTML.gif)
Further, set ,
,
,
, and, as above, define the sequences
and
Then it is easy to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ68_HTML.gif)
for all Since
and
are comparable; therefore
and
It is easy to show that
and
are comparable, that is,
and
for all
From (2.5) and properties of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ69_HTML.gif)
where From this, it follows that, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ70_HTML.gif)
Similarly, one can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ71_HTML.gif)
where Thus by Lemma 1.8,
and
. Since
and
, by commutativity of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ72_HTML.gif)
Denote Then from (2.61),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ73_HTML.gif)
Thus, is a coupled coincidence point. Then, from (2.55), with
and
, it follows that
and
; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ74_HTML.gif)
From (2.62) and (2.63),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ75_HTML.gif)
Therefore, is a coupled common fixed point of
and
To prove the uniqueness, assume that
is another coupled common fixed point. Then, by (2.55), we have
and
Corollary 2.8.
Let be a partially ordered complete quasi-metric space with a
-function
on
. Assume that the function
is such that
for each
Let
and let
be a mapping having the mixed monotone property on
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ76_HTML.gif)
Also suppose that either
(a) is continuous or
(b) has the following properties:
(i) if a nondecreasing sequence , then
for all
(ii) if a non-increasing sequence , then
for all
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ77_HTML.gif)
then, there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ78_HTML.gif)
Furthermore, if are comparable, then
that is,
Proof.
Following the proof of Theorem 2.4 with (the identity mapping on
), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ79_HTML.gif)
We show that Let us suppose that
We will show that
are comparable for all
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ80_HTML.gif)
where ,
Suppose that (2.69) holds for some fixed
Then, by mixed monotone property of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ81_HTML.gif)
and (2.69) follows. Now from (2.69), (2.65), and properties of we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ82_HTML.gif)
where Similarly, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ83_HTML.gif)
where . Hence, by Lemma 1.8,
that is,
Corollary 2.9.
Let be a partially ordered complete quasi-metric space with a
-function
on
. Let
be a mapping having the mixed monotone property on
. Assume that there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ84_HTML.gif)
Also, suppose that either
(a) is continuous or
(b) has the following properties:
(i) if a nondecreasing sequence , then
for all
(ii) if a nonincreasing sequence , then
for all
If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ85_HTML.gif)
then, there exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F703938/MediaObjects/13663_2010_Article_1425_Equ86_HTML.gif)
Furthermore, if are comparable, then
that is,
Proof.
Taking in Corollary 2.8, we obtain Corollary 2.9.
Remark 2.10.
As an application of fixed point results, the existence of a solution to the equilibrium problem was considered in [2–7]. It would be interesting to solve Ekeland-type variational principle, Ky Fan type best approximation problem and equilibrium problem utilizing recent results on coupled fixed points and coupled coincidence points.
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The first and third author are grateful to DSR, King Abdulaziz University for supporting research project no. (3-74/430).
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Hussain, N., Shah, M. & Kutbi, M. Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q-Function. Fixed Point Theory Appl 2011, 703938 (2011). https://doi.org/10.1155/2011/703938
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DOI: https://doi.org/10.1155/2011/703938