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A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation
Fixed Point Theory and Applications volume 2013, Article number: 38 (2013)
Abstract
In this manuscript, we discuss the existence of a coupled coincidence point for mappings and , where F has the mixed g-monotone property, in the context of partially ordered metric spaces with an implicit relation. Our main theorem improves and extends various results in the literature. We also state some examples to illustrate our work.
MSC:47H10, 54H25, 46J10, 46J15.
1 Introduction and preliminaries
It is well known that fixed point theory is one of the crucial and very efficient tools in nonlinear functional analysis. This is because its ever-growing use in this field is very extensive in applications. In particular, the effects of fixed point theory are most apparent in fields like economy, computer sciences and engineering including many branches of mathematics. Historically, in 1886, Poincaré initiated first fixed point results. Then, in 1912, Brouwer published a result in this field, which was equivalent to Poincaré’s theorem, which in the simplest terms states that a continuous function from a disk D to itself has a fixed point. But most of the substantial advances in fixed point theory started after the celebrated fixed point result of Banach, known as Banach’s contraction mapping principle, in 1922. This principle can be stated as follows: any contraction in a complete metric space has a unique fixed point. When compared to Browder’s fixed point theorem, the power of Banach’s principle comes from the fact that it guarantees the uniqueness of a fixed point and gives a method to determine the fixed point. These two strengths of Banach’s contraction mapping principle have attracted attention of many prominent mathematicians who aim to broaden the applications of nonlinear functional analysis via fixed point theory in various quantitative sciences.
In the light of these developments, Guo and Lakshmikantham [1] defined the notion of a coupled fixed point in 1987. Later, Gnana-Bhaskar and Lakshmikantham [2] improved the idea of a coupled fixed point in the category of partially ordered metric spaces by introducing the notion of a mixed monotone mapping and presented certain applications on the solution of periodic boundary value problems. Interested readers may refer to [3–5] and the references therein to follow the development of fixed point theory on partially ordered metric spaces.
For the sake of completeness, we will review the basic definitions and fundamental results.
Definition 1 (See [2])
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,
and
Definition 2 (See [2])
An element is called a coupled fixed point of the mapping if
We state now the main results of Gnana-Bhaskar and Lakshmikantham in [2].
Theorem 3 (See [2])
Let be a partially ordered set and suppose there exists a metric d on X such that is a complete metric space. Assume that there exists a with
for all and . Let either
-
(a)
be a continuous mapping having the mixed monotone property on X, or
-
(b)
X have the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
-
(i)
If there exist two elements with
then there exist such that
Following this theorem, several coupled coincidence/fixed point theorems and their applications to integral equations, matrix equations and a periodic boundary value problem have been reported (see, e.g., [6–19] and references therein). In particular, Lakshmikantham and Ćirić [6] established coupled coincidence and coupled fixed point theorems for two mappings and , where F has the mixed g-monotone property and the functions F and g commute, as an extension of the fixed point results in [2]. For the sake of completeness, we recall these characterizations.
Definition 4 (See [6])
Let be a partially ordered set and let and be two mappings. We say F has the mixed g-monotone property if is g-non-decreasing in its first argument and is g-non-increasing in its second argument, that is, for any ,
and
Definition 5 (See [6])
An element is called a coupled coincident point of the mappings and if
Definition 6 [7]
The mappings F and g, where , , are said to be compatible if
and
where and are sequences in X such that and as for all are satisfied.
Definition 7 (See [20])
Two self-mappings A and B are said to be weakly compatible if they commute at their coincidence points, i.e., whenever , .
Choudhury and Kundu in [7] defined the notion of compatibility and showed the result established in [6] with a different set of conditions. In other words, the authors constructed their result by assuming that F and g are compatible mappings. Later, Luong and Thuan [13] slightly improved the notion of compatible mappings on partially ordered metric spaces, namely O-compatible mappings. In this paper [13], the authors proved some coupled coincidence point theorems for O-compatible type mappings in the context of partially ordered generalized metric spaces.
We recall the concept of O-compatible mappings as follows.
Definition 8 (cf. [13])
Let be a partially ordered metric space. Two mappings and are said to be O-compatible if
and
where and are sequences in X such that , are monotone and
and
are satisfied for some .
Let be a partially ordered metric space. If and are compatible, then they are O-compatible. However, the converse is not true. The following example shows that there exist mappings which are O-compatible but not compatible.
Example 9 Let with the usual metric for all . We consider the following order relation on X:
Let be given by
and be defined by
Then F and g are O-compatible but not compatible. To discern this, let and be two sequences in X such that and are monotone and
and
for some . It is easy to see that since . It is not possible to have . Assume otherwise. There exists a positive integer N such that and for all , since both and are monotone. Then we have for all , which implies that and for all , a contradiction. Thus, . In this case, we derive that and for all for some positive integer M. Then, we get and for all . As a result, we obtain
, and . Therefore, F and g are O-compatible.
On the other hand, let
for . Observe that
and
as n approaches to ∞, where and . But we have
which does not approach to 0 as n approaches to ∞. Hence, F and g are not compatible.
Remark 10 In Example 9, if we let , we obtain the example presented in [13].
In nonlinear analysis, especially in fixed point theory, implicit relations on metric spaces have been investigated heavily in many articles (see, e.g., [21–24] and references therein). In this paper, by using the following implicit relation, we examine the existence of a coupled coincidence point theorem for mappings and in the context of a partial metric space, where F has the mixed g-monotone property and F, g are O-compatible.
Let denote the set of all nonnegative real numbers. Also, let Φ denote the collection of all functions which satisfy
-
(i)
φ is continuous and non-decreasing,
-
(ii)
for each and .
We reserve ℍ for the class of all continuous functions satisfying
-
(H1)
is non-increasing in and ,
-
(H2)
if , then , where .
Example 11 The following functions lie in ℍ:
-
, where α, β, γ, θ are nonnegative real numbers satisfying .
-
, where .
-
, where .
-
, where .
-
, where .
-
, where α, β, γ are nonnegative real numbers satisfying .
-
, where .
In this paper, we prove a coupled coincidence point theorem for mappings satisfying such implicit relations.
2 Main result
We start by stating our primary theorem:
Theorem 12 Let be a partially ordered complete metric space. Suppose that and are two mappings such that F has the mixed g-monotone property. Assume that there exists such that
for all with and . Suppose also that and g is continuous on X and O-compatible with F. Additionally, suppose that either
-
(a)
F is continuous, or
-
(b)
X has the properties
-
(i)
if a non-decreasing sequence , then for all n, and
-
(ii)
if a non-increasing sequence , then for all n.
-
(i)
If there exist two elements with
then F and g have a coupled coincidence point in X.
Proof Let be such that and . We will construct the iterative sequences and in X as follows:
The sequences and are well defined since . Using the mathematical induction and the fact that F has the mixed g-monotone property, we obtain
for all . If there is a number such that and , then and . In this case, the theorem follows since is a coupled coincidence point of F and g.
Assume that or for all . Since and , we have
by (2.1). Using the definitions of and in 2.2, we obtain
By combining the property (H1) of H with the triangle inequality, inequalities in (2.4) turn into
Then, the property (H2) of H implies that
where .
Set . Thus, the inequality (2.5) can be written as
We will show that the sequences and are Cauchy. Without loss of generality, we may assume that . Then, by the triangle inequality, we have
Letting in (2.6), we have
Consequently, and . Hence the sequences and are Cauchy. Since X is complete, there exist such that
and
Thus, we obtain
as . Since F and g are O-compatible, we have
and
by (2.9).
Now, assume that (a) holds, i.e., F is continuous. By taking the limit in the following inequality:
as , we obtain
by (2.7), (2.10) and the continuity of F and g. Similarly, we can show that . As a result, F and g have a coupled coincidence point in X, i.e., and .
Assume that (b) holds. Since is a non-decreasing sequence and , we have for all by (i). Similarly, since is a non-increasing sequence and , we also have for all . Since g is continuous, using (2.7), (2.10) and (2.11), we derive that
and
By the hypothesis of the theorem, we know that
for every . In the inequality above, we let and use (2.12) and (2.13) to obtain
which implies that by (H2) for . Finally, we find that and completing the proof. □
Example 13 Let , F and g be defined as in Example 9. We see that
-
X is complete and X has the properties
-
(i)
if a non-decreasing sequence , then for all ,
-
(ii)
if a non-increasing sequence , then for all .
-
(i)
-
.
-
g is continuous and g and F are O-compatible.
-
There exist and such that and .
-
F has the mixed g-monotone property, which can be proved as follows: Let such that . There are two cases to consider:
-
(1)
If , then and or or or . Thus, if and and or . Otherwise, .
-
(2)
If , then and , i.e., and . Thus, if and if .
Therefore, F is g-non-decreasing in its first argument. Similarly, it can be shown that F is g-non-increasing in its second argument.
-
(1)
-
There exists such that (2.1) holds. To discern this, for any with and , we need to show that . Indeed,
-
(1)
if and , then and . Thus
-
(2)
if and , then and . Either , or . In any case,
Otherwise, we get
Similarly, if and , then ;
-
(3)
if and , then both x, u are in one of the sets , , or and both y, v are also in one of the sets , , or . Thus
If or and or , otherwise,
-
(1)
Therefore, all of the conditions of Theorem 12 are satisfied. Therefore, we conclude that F and g have a coupled coincidence point.
Note that we cannot apply the result of Choudhury and Kundu [7], the result of Choudhury, Metiya and Kundu [25] as well as the result of Lakshmikantham and Ciric [6] to this example.
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Gülyaz, S., Karapınar, E. & Yüce, İ.S. A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation. Fixed Point Theory Appl 2013, 38 (2013). https://doi.org/10.1186/1687-1812-2013-38
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DOI: https://doi.org/10.1186/1687-1812-2013-38