Abstract
In this paper, we prove the coupled coincidence point theorems for a
MSC: 47H10, 54H25.
Keywords:
cone metric spaces; common coupled fixed point; coupled coincidence point;1 Introduction
The famous Banach contraction principle states that if
Following this trend, the problem of existence and uniqueness of fixed points in partially ordered sets has been studied thoroughly because of its interesting nature. In 1986, Turinici [1] presented the first result in this direction. Afterward, Ran and Reurings [2] gave some applications of Turinici’s theorem to matrix equations. The results of Ran and Reurings were further extended to ordered cone metric spaces in [35]. In 2005, Nieto and RodríguezLópez [6] extended Ran and Reurings’s theorems for nondecreasing mappings and obtained a unique solution for a firstorder ordinary differential equation with periodic boundary conditions.
The notion of coupled fixed points was introduced by Guo and Lakshmikantham [7]. Since then, the concept has been of interest to many researchers in metrical fixed point theory. In 2006, Bhaskar and Lakshmikantham [8] introduced the concept of a mixed monotone property (see further Definition 2.4). They proved classical coupled fixed point theorems for mappings satisfying the mixed monotone property and also discussed an application of their result by investigating the existence and uniqueness of a solution of the periodic boundary value problem. Following this result, Harjani et al.[9] (see also [10,11]) studied the existence and uniqueness of solutions of a nonlinear integral equation as an application of coupled fixed points. Very recently, motivated by the work of Caballero et al.[12], Jleli and Samet [13] discussed the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
where
Since their important role in the study of the existence and uniqueness of a solution of the periodic boundary value problem, a nonlinear integral equation, and the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem, a wide discussion on coupled fixed point theorems aimed the interest of many scientists.
In 2009, Lakshmikantham and Ćirić [14] extended the concept of a mixed monotone property to a mixed gmonotone mapping and proved coupled coincidence point and common coupled fixed point theorems which are more general than the result of Bhaskar and Lakshmikantham in [8]. A number of articles on coupled fixed point, coupled coincidence point, and common coupled fixed point theorems have been dedicated to the improvement; see [1530] and the references therein.
On the other hand, in 2007, Huang and Zhang [31] have reintroduced the concept of a cone metric space which is replacing the set of real numbers by an ordered Banach space E. They went further and defined the convergence via interior points of the cone by which the order in E is defined. This approach allows the investigation of cone spaces in the case when the cone is not necessarily normal. They also continued with results concerned with the normal cones only. One of the main results from [31] is the Banach contraction principle in the setting of normal cone spaces. Afterward, many authors generalized their fixed point theorems in cone spaces with normal cones. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal but just has interior that is nonempty. In this case only, proper generalizations of results from the ordinary metric spaces can be obtained. In 2011, Janković et al.[32] gave some examples showing that theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces, when the cone is nonnormal.
Recently, Nashine et al.[33] established common coupled fixed point theorems for mixed gmonotone and
In this work, we show that the mixed gmonotone property in common coupled fixed point theorems in ordered cone metric spaces can be replaced by another property due to Ðorić et al.[37]. This property is automatically satisfied in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Our results generalize and extend many wellknown comparable results in the literature. An illustrative example is presented in this work when our results can be used in proving the existence of a common coupled fixed point, while the results of Nashine et al.[33] cannot.
2 Preliminaries
In this section, we give some notations and a property that are useful for our main
results. Let E be a real Banach space with respect to a given norm
1. P is closed and
2.
3.
Given a cone
The cone P is said to be normal if there exists a real number
The least positive number K satisfying the above statement is called a normal constant of P. In 2008, Rezapour and Hamlbarani [38] showed that there are no normal cones with a normal constant
In what follows, we always suppose that E is a real Banach space with cone P satisfying
Definition 2.1 ([31])
Let X be a nonempty set and
1.
2.
3.
Then d is called a cone metric on X and
Definition 2.2 ([31])
Let
1. If for every
2. If for every
3. If every Cauchy sequence in X is convergent in X, then
Let
(
(
(
(
Definition 2.3 Let X be a nonempty set. Then
(i)
(ii)
Let
Let
and
hold. If in the previous relations g is the identity mapping, then it is said that F has a mixed monotone property.
Let X be a nonempty set and
(C_{1}) a coupled fixed point of F if
(C_{2}) a coupled coincidence point of mappings g and F if
and in this case
(C_{3}) a common coupled fixed point of mappings g and F if
Definition 2.6 ([36])
Let X be a nonempty set. Mappings
(
(
It is easy to see that wcompatible implies
Example 2.7 Let
It is easy to see that
However,
For elements x, y of a partially ordered set
Next, we give a new property due to Ðorić et al.[37].
Let X be a nonempty set and let
In particular, when
Remark 2.8 We obtain that the conditions (2.3) and (2.4) are trivially satisfied if
The following examples show that the condition (2.3) ((2.4), resp.) may be satisfied when F does not have the mixed gmonotone property (monotone property, resp.).
Example 2.9 Let
for all
(1) For each
(2) For each
Example 2.10 Let
for all
(1) For each
(2) For each
(3) The other two cases are trivial.
3 Coupled coincidence point theorems lacking the mixed gmonotone property
In this section, we give the existence of coupled coincidence point theorems in ordered cone metric spaces lacking the mixed gmonotone property. Our first main result is the following theorem.
Theorem 3.1Let
(i)
(ii) gandFsatisfy property (2.3);
(iii) there exist
(iv) there exists
holds;
(v) if
Then there exist
that is, Fandghave a coupled coincidence point
Proof Starting from
for all
Proceeding by induction, we get that
which implies that
Similarly, starting with
Combining (3.3) and (3.4), we obtain that
Now, starting from
Similarly, starting from
Again adding up, we obtain that
Finally, adding up (3.5) and (3.6), it follows that
with
since
From the relation (3.7), we have
If
For any
Since
From (
By (
Since
and
then by (
By (v), we have
If
which further implies that
Since
for all
Now, according to (
So, we suppose that
which further implies that
Since
Now, according to (
Remark 3.2 In Theorem 3.1, the condition (ii) is a substitution for the mixed gmonotone property that has been used in most of the coupled coincidence point theorems so far. Therefore, Theorem 3.1 improves the results of Nashine et al.[33]. Moreover, it is an ordered version extension of the results of Abbas et al.[36].
Corollary 3.3Let
(i)
(ii) gandFsatisfy property (2.3);
(iii) there exist
(iv) there exist
holds;
(v) if
Then there exist
that is, Fandghave a coupled coincidence point
Putting
Corollary 3.4Let
(i) Xis complete;
(ii) gandFsatisfy property (2.4);
(iii) there exist
(iv) there exists
holds;
(v) if
Then there exist
that is, Fhas a coupled fixed point
Our second main result is the following.
Theorem 3.5Let
(i)
(ii) gandFsatisfy property (2.3);
(iii) there exist
(iv) there is some
such that
(v) if
Then there exist
that is, Fandghave a coupled coincidence point
Proof Since
for all
By repeating this process, we have
Since
such that
Similarly, one can show that there exists
such that
Now, denote
Case 1.
Case 2.
Case 3.
Case 4.
Thus, in all cases, we get
and by the same argument as in Theorem 3.1, it is proved that
From (v), we get
If
where
By property (
Now, it follows that for n sufficiently large,
Therefore, again by property (
Then, we suppose that
where
By property (
Now, it follows that for n sufficiently large,
Thus, again by property (
Similarly,
Remark 3.6 It would be interesting to relate our Theorem 3.5 with Theorem 2.1 of Long et al.[39].
Putting
Corollary 3.7Let
(i) Xis complete;
(ii) Fsatisfies property (2.4);
(iii) there exist
(iv) there is some
such that
(v) if
Then there exist
that is, Fhas a coupled fixed point
4 Common coupled fixed point theorems lacking the mixed monotone property
Some questions arise naturally from Theorems 3.1 and 3.5. For example, one may ask if there are necessary conditions for the existence and uniqueness of a common coupled fixed point of F and g?
The next theorem provides a positive answer to this question with additional hypotheses to Theorems 3.1 and 3.5.
For the given partial order ⪯ on the set X, we will denote also by ⪯ the order on
Theorem 4.1In addition to the hypotheses of Theorem 3.1, suppose that for every
and
IfFandgare
Proof From Theorem 3.1, the set of coupled coincidence points of F and g is nonempty. Suppose
By assumption, there exists
and
Put
for all n. Further, set
and
as
Since
we have
for all
that is,
In the same way, starting from
Thus,
In a similar way, starting from
Finally, adding up (4.3) and (4.4), we obtain that
where λ is determined as in (3.8), and hence
By inequality (4.5) n time, we have
It follows from
for all
it follows by (
Note that if
Next, we show that F and g have a common coupled fixed point. Let
Thus,
Finally, we show the uniqueness of a common coupled fixed point of F and g. Let
Then
Next, we give some illustrative example which supports Theorem 4.1, while the results of Nashine et al.[33] do not.
Example 4.2 Let
Let
It is well known (see, e.g., [40]) that the cone P is not normal. Let
for all
Let
Consider
So, the mapping F does not satisfy the mixed gmonotone property. Therefore, Theorems 3.1 and 3.2 of Nashine et al.[33] cannot be used to reach this conclusion.
Now, we show that Theorem 4.1 can be used for this case.
Take
For
Next, we show that F and g are
Therefore, F and g are
Moreover, other conditions in Theorem 4.1 are also satisfied. Now, we can apply Theorem 4.1
to conclude the existence of a unique common coupled fixed point of F and g that is a point
The following uniqueness result corresponding to Theorem 3.5 can be proved in the same way as Theorem 4.1.
Theorem 4.3In addition to the hypotheses of Theorem 3.5, suppose that for every
and
IfFandgare
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Professor Hichem BenElMechaiekh and the referee for valuable comments. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund and KMUTT under Grant No. MRG5580213 for financial support during the preparation of this manuscript.
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