- Open Access
A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation
© Gülyaz et al.; licensee Springer. 2013
- Received: 7 September 2012
- Accepted: 6 February 2013
- Published: 25 February 2013
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  defined the notion of a coupled fixed point in 1987. Later, Gnana-Bhaskar and Lakshmikantham  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 )
Definition 2 (See )
We state now the main results of Gnana-Bhaskar and Lakshmikantham in .
Theorem 3 (See )
be a continuous mapping having the mixed monotone property on X, or
- (b)X have the following property:
if a non-decreasing sequence , then for all n,
if a non-increasing sequence , then for all n.
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  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 . For the sake of completeness, we recall these characterizations.
Definition 4 (See )
Definition 5 (See )
Definition 6 
where and are sequences in X such that and as for all are satisfied.
Definition 7 (See )
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  defined the notion of compatibility and showed the result established in  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  slightly improved the notion of compatible mappings on partially ordered metric spaces, namely O-compatible mappings. In this paper , 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. )
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.
, and . Therefore, F and g are O-compatible.
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 .
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.
φ is continuous and non-decreasing,
for each and .
is non-increasing in and ,
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:
F is continuous, or
- (b)X has the properties
if a non-decreasing sequence , then for all n, and
if a non-increasing sequence , then for all n.
then F and g have a coupled coincidence point in X.
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.
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 .
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
if a non-decreasing sequence , then for all ,
if a non-increasing sequence , then for all .
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:
If , then and or or or . Thus, if and and or . Otherwise, .
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.
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 . ThusIf or and or , otherwise,
Therefore, all of the conditions of Theorem 12 are satisfied. Therefore, we conclude that F and g have a coupled coincidence point.
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