#### Abstract

Coupled fixed point theorems for a map satisfying mixed monotone property and a nonlinear, rational type contractive condition are established in a partially ordered -metric space. The conditions for uniqueness of the coupled fixed point are discussed. We also present results for the existence of coupled coincidence points of two maps.

#### 1. Introduction

The idea of weakening the contractive condition in a metric space by introducing partial order in the space and considering monotone functions satisfying contractive conditions was first developed by Ran and Reurings [1]. Later, this was extended by Bhaskar and Lakshmikantham [2] to prove a coupled fixed point theorem for functions satisfying mixed monotone property. Since then, there has been considerable interest in the development of coupled fixed point theorems in partially ordered metric spaces with a variety of contractive conditions [318].

Nonlinear contractive conditions were considered in [4, 6, 19]. In particular, a rational type contractive condition was considered by Jaggi [19] in a complete metric space and this was extended to a partially ordered complete metric space by Harjani et al. [6] to prove some fixed point theorems.

Some coupled fixed point theorems in partially ordered, complete metric spaces were developed by Choudhury and Maity [8] and Saadati et al. [9]. The contractive conditions used in [8] were extensions of that used by Bhaskar and Lakshmikantham [2] into a metric space. A new concept of an distance was introduced in [9].

In this paper we develop a coupled fixed point theorem using a rational type, nonlinear contractive condition in a partially ordered complete metric space. The condition is similar to the rational type contractive condition of iri et al. [3] and may be considered as a generalization of the condition given in [3]. We also find conditions for the uniqueness of the coupled fixed point. Finally we consider the conditions for existence of coupled coincidence points.

We begin by introducing the basic definitions and notions used in the paper.

#### 2. General Preliminaries

Throughout this work will denote a partial order relation on some given set. For any two elements in some partially ordered set endowed with the partial order relation , and are equivalent. Also by we mean and .

Definition 1 (see [20]). Let be a nonempty set and let   : be a function satisfying the following properties:(1) if ,(2) for all with ,(3) for all with ,(4) (symmetry in all three variables),(5) for all (rectangle inequality).
Then is called a generalized metric or more specifically a metric on and the pair is called a metric space.

Theorem 2 (see [20]). Let be a metric space; then for any it follows that(1)if then ;(2);(3);(4).

Definition 3 (see [20]). Let be a metric space. The sequence is convergent to if for any arbitrary there is a positive integer such that for , that is, if .

Theorem 4 (see [20]). Let be a metric space; then for a sequence and a point the following are equivalent:(1) is convergent to ;(2) as ;(3) as ;(4) as .

Theorem 5 (see [20]). Let be a metric space. Then the function is jointly continuous in all three of its variables.

Remark 6. This means if , , and are sequences in such that , , and , then as .

Definition 7 (see [20]). Let be a metric space. Then sequence is said to be Cauchy if, for every , there exists a positive integer such that for all .

Theorem 8 (see [20]). In a metric space the following are equivalent:(1)the sequence is Cauchy;(2)for every , there exists a positive integer such that , for all .

Definition 9 (see [20]). A metric space is said to be complete if and only if every Cauchy sequence is convergent in .

Definition 10 (see [2]). Let be a partially ordered set and . Then is said to have mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in . That is, for all ,

Definition 11 (see [4]). Let be a partially ordered set and and . We say has the mixed -monotone property if   is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument. That is, for all ,

Definition 12 (see [2]). An element is called a coupled fixed point of a map if and .

Definition 13 (see [4]). An element is called a coupled coincidence point of the maps and if and .

Definition 14 (see [4]). The maps and are said to be commutative if .

Definition 15. For a map , by , we mean . Similarly we define .

#### 3. Main Results

Our main results are presented in this section. We first develop a rational type contractive condition on a partially ordered, complete metric space and give a coupled fixed point theorem for a map satisfying this condition.

We start with a partially ordered set and suppose that there is a metric on so that is a complete metric space. We induce partial ordering on by demanding that, for any , , .

Theorem 16. Let be a partially ordered set and let be a generalized metric on such that is a complete metric space. Suppose is a continuous mapping on having the mixed monotone property. Suppose also that for all with where . If there exists such that and , then has a coupled fixed point . That is, satisfies .

Proof. Suppose that there exists such that and . We write , and define , . From the conditions of the theorem and the mixed monotone property it easily follows that This gives
Similarly proceeding with we find Considering the sequence and using (3) we have
Now using inequality from Theorem 2, . Setting and in this we find Using this (8) becomes where in the last step we have used inequality from Theorem 2. Rearranging and simplifying this we get Evidently the condition that (11) is contractive is , that is, Similarly, considering , ,   and arguing as above we find With condition (12), we find (13) is contractive.
Let and . Using (11) we get If , then . But this means . Similarly writing we find If in addition to , we deduce similarly that . So if , is a coupled fixed point.
However if , for , using inequality of Definition 1 we get So is a -Cauchy sequence in . Next, considering and arguing as above we can show that is also a -Cauchy sequence in . Completeness of now implies that there are points such that and as .
We next show that is a coupled fixed point of . Using the fact that is continuous on and as a metric is continuous in each of its variables, we have But this means . Similarly by considering and repeating the arguments used to derive (17) we can show that . This proves is a coupled fixed point of .

The map was introduced by Chen [21] and a subclass of functions was defined by Chakrabarti in [22].

Definition 17 (see [22]). We call a function of class if there is a such that , and the following conditions are satisfied: (1) for all and ,(2) for all .

Using Definition 17 we obtain the following as a generalization of Theorem 16.

Theorem 18. Let be a partially ordered set and let be a metric on such that is a complete metric space. Suppose is a continuous mapping on having the mixed monotone property. For some given , let and where . Suppose also that for all with If there exists such that and , then has a coupled fixed point . That is, satisfies .

Proof. Since and , it follows from Definition 17 that and for all . Inequality (18) now becomes equivalent to inequality (3) of Theorem 16 and the proof is immediate.

Example 19. Let and consider the function defined by Then is a complete metric space [23]. We define a partial order on by the following: for any , if . Also let be defined by Suppose satisfy with nonzero , , , but are otherwise arbitrary. Then we have . So the left side of (3) is . The right side of (3) If and then and . So all conditions of Theorem 16 are satisfied. Easily we find that is a coupled fixed point of . Similarly is a coupled fixed point.

In the next theorem, we provide conditions under which the coupled fixed point of the map established in Theorem 16 is unique.

Theorem 20. Suppose that the conditions of Theorem 16 are valid. In addition suppose that for each there is a which is comparable to and . Then has a unique coupled fixed point.

Proof. Suppose that are coupled fixed points.
Case 1. If and are comparable, This is equivalent to . However this is a contradiction since . So we must have . Similarly, considering we easily show that . This shows that , so the coupled fixed point is unique.
Case 2. If and are not comparable, by the condition of the theorem there is a comparable to and . If there is a positive integer such that , then So for and hence as .
On the other hand, if no such exists, we have that, for any , where we have used the fact that for any . Since is a coupled fixed point of , for all and from (24), we now deduce that since . This proves as . Similarly we can show that as . Replacing with and with and repeating the above arguments we can deduce and as . But this means and . So the and the coupled fixed point is unique.

We next establish the conditions under which two maps and have a coupled coincidence point.

Theorem 21. Let be a partially ordered set and let be a metric on such that is a complete metric space. Let and be continuous mapping on such that has the mixed monotone property. Suppose that , commutes with , and, for with and , , where . If there exists such that and , then has a coupled coincidence point . That is satisfies .

Proof. Since we can choose such that , . For similar reasons, can be found such that , . Due to the mixed monotone property of , we have and . In general, it can be shown that, [4] for , Now by the same arguments used to deduce (10) we have This gives Since , . Writing we find from (28) that If , then giving , so is a coincidence point. However if , we have for , since . This shows that is a Cauchy sequence in and completeness of ensures a point such that as .
Replacing by for all we get the analogue of (29): Next writing we find as in (30) that As before, if , , giving so that is a coincidence point. If in addition , we have and . So is a coupled coincidence point.
Proceeding as in (31) we can show further that is a Cauchy sequence in and due to completeness of there is a point such that as .
Finally we prove that is a coupled coincident point. Since and commute, we have Taking limits as in (34) and noting that and are, respectively, continuous on and , we get Next we observe that as metric is continuous in all its variables. This finally leads to So . Similarly, we show that . This proves is a coupled coincidence point.

#### 4. Conclusion

To summarize, we have introduced a rational type contractive condition in a metric space and proved some coupled fixed point theorems for maps satisfying mixed monotone property. We established the conditions for uniqueness of the coupled fixed point. Conditions for the existence of coupled coincidence points of two maps are also deduced.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.