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
(11)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
(12)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 
,
(13)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
(21)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
(22)Similarly for all 
,
(23)Since 
is strictly monotone increasing and 
has the mixed monotone property, we have
(24)Similarly
(25)Now for all 
, using (2.1), we get
(26)From (2.6),
(27)where 
. Continuing, we conclude that
(28)if 
is odd, where 
. Also,
(29)if 
is even, where
(210)Let 
; then for every 
in 
we have
(211)where
(212)Hence,
(213)For 
, we get
(214)which further implies that
(215)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
(216)which implies that
(217)Similarly
(218)Let 
be given. Since 
is 
-continuous at 
with respect to 
, there exists 
such that for each 
(219)Since 
and 
, for 
, there exists 
such that, for all 
,
(220)Now,
(221)implies that 
. Since
(222)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
(223)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
(224)Consider
(225)which implies that 
. Also, from Theorem 2.1, we have
(226)Therefore,
(227)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
(31)implies that 
. Hence 
. Also,
(32)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
(33)where 
and 
. On taking limit as 
on both sides of (3.3), we have
(34)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
(35)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
(36)Assuming that 
, since 
is strictly monotone increasing, so 
. By the mixed monotone property of 
, we have
(37)Therefore,
(38)Letting 
, there exists an 
such that 
and 
for all 
. Now,
(39)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 
.
Acknowledgment
The present version of the paper owes much to the precise and kind remarks of the learned referees.
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