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An original coupled coincidence point result for a pair of mappings without MMP
Journal of Inequalities and Applications volume 2014, Article number: 61 (2014)
Abstract
The purpose of this paper is to establish a coupled coincidence point theorem for a pair of mappings without MMP (mixed monotone property) in metric spaces endowed with partial order, which is not an immediate consequence of a well-known theorem in the literature. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend some of the results of Bhaskar and Lakshmikantham (Nonlinear Anal. 65:1379-1393, 2006), Choudhury, Metiya and Kundu (Ann. Univ. Ferrara 57:1-16, 2011), Harjani, Lopez and Sadarangani (Nonlinear Anal. 74:1749-1760, 2011) and of Luong and Thuan (Bull. Math. Anal. Appl. 2:16-24, 2010) for the mappings having no MMP. We introduce an example that there exists a common coupled fixed point of the mappings g and F such that F does not satisfy the g-mixed monotone property, and also g and F do not commute.
MSC:41A50, 47H10, 54H25.
1 Introduction and preliminaries
Fixed point theory is one of the famous and traditional theories in mathematics and has a large number of applications. The Banach contraction mapping is one of the pivotal results of analysis. It is a very popular tool for solving existence problems in many different fields of mathematics. There are a lot of generalizations of the Banach contraction principle in the literature. Ran and Reurings [1] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and Rodŕiguez-López [2] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [3] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results on a first-order differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results in cone metric spaces, fuzzy metric spaces, intuitionistic fuzzy normed spaces, partially ordered metric spaces and others (see [1–25]).
Definition 1.1 Let be a metric space and and , F and g are said to commute if for all .
Definition 1.2 Let be a metric space and let , . The mappings g and F are said to be compatible if and hold whenever and are sequences in X such that and .
Definition 1.3 Let be a partially ordered set and . The mapping F is said to be non-decreasing if for , implies and non-increasing if for , implies .
Definition 1.4 Let be a partially ordered set and and . The mapping F is said to have the mixed g-monotone property if is monotone g-non-decreasing in x and monotone g-non-increasing in y, that is, for any ,
and
If g= identity mapping in Definition 1.4, then the mapping F is said to have the mixed monotone property.
Recently, Ðoric et al. [12] showed that the mixed monotone property in coupled fixed point results for mappings in ordered metric spaces can be replaced by another property which is often easy to check. In particular, it is automatically satisfied in the case of a totally ordered space, the case which is important in applications. Hence, these results can be applied in a much wider class of problems.
If elements x, y of a partially ordered set are comparable (i.e., or holds) we will write . Let and . We will consider the following condition:
If g is an identity mapping, for all x, y, v, if then .
Ðoric et al. [12] gave some examples that these conditions may be satisfied when F does not have the g-mixed monotone property.
Definition 1.5 An element is called a coupled coincidence point of the mappings and if and .
If g= identity mapping in Definition 1.5, then is called a coupled fixed point.
The purpose of this paper is to establish some coupled coincidence point results in partially ordered metric spaces for a pair of mappings without mixed monotone property satisfying a contractive condition. Also, we present a result on the existence and uniqueness of coupled common fixed points. Also, we give an example to illustrate the main result in this paper. The results proved generalize some of the results of Bhaskar and Lakshmikantham [3], Choudhury et al. [10], Luong and Thuan [17] and Harjani et al. [13] for the mappings having no mixed monotone property.
2 Main results
2.1 Coupled common fixed point theorems
In this section, we prove some coupled common fixed point theorems in the context of ordered metric spaces.
We denote by Φ the set of functions satisfying:
-
(i)
ϕ is continuous;
-
(ii)
for all and if and only if .
Theorem 2.1 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that the following conditions hold:
-
(i)
g is continuous and is closed;
-
(ii)
and g and F are compatible;
-
(iii)
for all , if , then ;
-
(iv)
there exist such that and ;
-
(v)
there exists a non-negative real number L such that
(2.1)
for all with and , where ;
-
(vi)
(a) F is continuous or (b) , when in X, then for sufficiently large n.
Then there exist such that and , that is, F and g have a coupled coincidence point .
Proof Using conditions (ii) and (iv), construct sequences and in X satisfying and for .
By (iv), and condition (iii) implies that . Proceeding by induction, we get that , and similarly, for each .
Now from the contractive condition (2.1), we have
which implies that .
Similarly, we have .
Therefore, from the above two inequalities we have
Since for all and if and only if , from (2.3) we have
Set , then is a non-increasing sequence of positive real numbers. Thus, there is such .
Suppose that , letting two sides of (2.3) and using the properties of ϕ, we have
which is a contradiction. Hence , i.e.,
Now, we shall prove that and are Cauchy sequences. Suppose, to the contrary, that at least one of or is not a Cauchy sequence. This means that there exists an for which we can find subsequences , of and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (2.6). Then
Using the triangle inequality and (2.7), we have
and
From (2.6), (2.8) and (2.9), we have
Letting in the inequalities above and using (2.5), we get
By the triangle inequality,
and
From the last two inequalities and (2.6), we have
Again, by the triangle inequality,
and
Therefore,
Taking in (2.11) and (2.12) and using (2.5), (2.10), we have
Since , and . Then from (2.1) we have
Similarly,
From (2.14) and (2.15), we have
Letting in the above inequality and using (2.5), (2.10), (2.13) and the properties of ϕ, we have
which is a contradiction. Therefore, and are Cauchy sequences and since is closed in a complete metric space (condition (i)), there exist such that and .
Compatibility of F and g (condition (ii)) implies that
and
Consider the two possibilities given in condition (vi).
-
(a)
Suppose that F is continuous. Using the triangle inequality, we get that
By taking limit and using the continuity of F and g, we have , i.e., and, in a similar way, we have . Thus F and g have a coupled coincidence point.
-
(b)
In this case and for some and n sufficiently large. For such n, using (2.1) we get
Taking in the above inequality and using the compatibility of F and g and the properties of ϕ, we have . Hence . Similarly, one can show that . Hence the result. □
Remark 2.1 Very recently, using the equivalence of the three basic metrics, Samet et al. [19] show that many of the coupled fixed point theorems are immediate consequences of well-known fixed point theorems in the literature.
In our Theorem 2.1, it is easy to see that if there is no equivalence and this theorem is not a consequence of a known fixed point theorem.
Remark 2.2 In the above theorem, condition (iii) is a substitution for the mixed monotone property that has been used in most of the coupled fixed point results so far. Note that this condition is trivially satisfied if the order ⪯ on X is total, which is the case in most of the examples in articles mentioned in the references.
If g is an identity mapping in the above theorem, we have the following result.
Corollary 2.2 Let be a complete partially ordered metric space and let . Suppose that the following hold:
-
(i)
for all , if , then ;
-
(ii)
there exist such that and ;
-
(iii)
there exists a non-negative real number L such that
for all with and , where ;
-
(iv)
(a) F is continuous or (b) , when in X, then for sufficiently large n.
Then there exist such that and , that is, F has a coupled fixed point .
Remark 2.3 Letting , in inequality (2.1), for all , , , we have
where for all is in Φ. Hence Theorem 2.1 generalizes the corresponding coupled fixed point results of Bhaskar and Lakshmikantham [3], Choudhury et al. [10], Luong and Thuan [17] and Harjani et al. [13] for the mappings having no mixed monotone property.
Taking , we have the following result.
Corollary 2.3 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that the following conditions hold:
-
(i)
g is continuous and is closed;
-
(ii)
and g and F are compatible;
-
(iii)
for all , if , then ;
-
(iv)
there exist such that and ;
-
(v)
F and g satisfy
(2.16)
for all with and , where ;
-
(vi)
(a) F is continuous or (b) , when in X, then for sufficiently large n.
Then there exist such that and , that is, F and g have a coupled coincidence point .
Corollary 2.4 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that following conditions hold:
-
(i)
g is continuous and is closed;
-
(ii)
and g and F are compatible;
-
(iii)
for all , if , then ;
-
(iv)
there exist such that and ;
-
(v)
there exist non-negative real numbers α, β with such that
(2.17)
for all with and ;
-
(vi)
(a) F is continuous or (b) , when in X, then for sufficiently large n.
Then there exist such that and , that is, F and g have a coupled coincidence point .
Taking , we have the following result.
Corollary 2.5 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that following conditions hold:
-
(i)
g is continuous and is closed;
-
(ii)
and g and F are compatible;
-
(iii)
for all , if , then ;
-
(iv)
there exist such that and ;
-
(v)
there exists such that
(2.18)
for all with and ;
-
(vi)
(a) F is continuous or (b) , when in X, then for sufficiently large n.
Then there exist such that and , that is, F and g have a coupled coincidence point .
Now, we shall prove the existence and uniqueness of a coupled common fixed point. Note that if is a partially ordered set, then we endow the product space with the following partial order relation:
Theorem 2.6 In addition to hypotheses of Theorem 2.1, suppose that
-
(vii)
for every , there exists such that is comparable to and .
Then F and g have a unique coupled common fixed point, that is, there exists a unique such that and .
Proof From Theorem 2.1, there exists such that and . Suppose that there is also such that and . We will prove that and . Condition (vii) implies that there exists such that is comparable to both and . Put , and, analogously to the proof of Theorem 2.1, choose sequences , satisfying
for . Starting from , and , , choose sequences , and , , satisfying , and , for , taking into account properties of coincidence points, it is easy to see that this can be done so that , and , , i.e.,
Since and are comparable, then and , and, in a similar way, we have and . Thus from (2.1) we have
which implies that .
Similarly, we can prove that .
Therefore, from the above two inequalities we have
Since for all , from (2.20) we have
Hence the sequence defined by is non-negative and decreasing and so for some .
Now, we show that . Assume that , letting two sides of (2.20) and using the properties of ϕ, we have
which is a contradiction. Hence , i.e.,
Similarly, we can prove that
Using relations (2.22) and (2.23), together with the triangle inequality, we have and and so and .
Denote and . So, we have that
By definition of the sequences and we have
and so
as well as
Compatibility of g and F implies that
i.e., . This together with (2.24) implies that and, in a similar way, . Thus, we have another coincidence, and by the property we have just proved, it follows that and . In other words, and , and is a common coupled fixed point of g and F.
To prove the uniqueness, assume that is another coupled common fixed point. Then by (2.24) we have and . Hence we get the result. □
Example 2.7 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for all . Define a mapping by and a mapping by
Then it is easy to check all the conditions of Theorems 2.1 and 2.6. In particular, we will check that g and F are compatible.
Let and be two sequences in X such that
Then and , where from it follows that . Then
and similarly .
Now, we verify inequality (2.1) of Theorem 2.1 for , and for all with and .
Thus there exists a common coupled fixed point of the mappings g and F. Note that F does not satisfy the g-mixed monotone property. Also, g and F do not commute.
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Chandok, S., Tas, K. An original coupled coincidence point result for a pair of mappings without MMP. J Inequal Appl 2014, 61 (2014). https://doi.org/10.1186/1029-242X-2014-61
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DOI: https://doi.org/10.1186/1029-242X-2014-61