1. Introduction

The famous and probably the well known fixed point theorem is the Banach Contraction Principle. It has been extended and improved by many athematicians. Its significance lies in its vast applicability in a number of branches of mathematics. Recently, W. Zhong et al ^[11] give the existence and uniqueness of solutions to the Cauchy problem for the local fractional differential equation with fractal conditions in a generalized Banach space.

In 2006, Bhaskar and Lakshmikantham ^[3] established some coupled fixed point theorem on ordered metric spaces and give some application in the existence and uniqueness of a solution for periodic boundary value problem. Ciric and Lakshmikantham ^[5] later on investigated some more coupled _xed point theorems in partially ordered sets. Also, many researchers have obtained coupled fixed point results for mappings under various contractive conditions in the framework of partial metric spaces ^{[1, 2, 4, 6]}.

In this paper, we prove some unique coupled fixed point theorems in a complete metric space endowed with a partial order. At the end of this paper we give an example to support our main theorem.

The organization of this paper is as follows. In section 2, the preliminary result on partial metric space is discussed. In section 3, we investigated the necessary condition for the uniqueness of coupled fixed point of the given mapping in partially ordered metric space and give an example to illustrate our main theorem.

2. Preliminaries

In this section, we give some definitions, lemma which are useful for main result in this paper.

Definition 2.1. ^{[3, 5]} An element is said to be coupled fixed point of the mapping if

Definition 2.2. ^[3] Let be a partially ordered set and We say that has the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in that is, for any

and

Definition 2.3. ^{[7, 8, 9]} Let be a non empty set. A partial metric on is a function such that for all

(P₁)

(P₂)

(P₃)

(P₄)

A partial metric space is a pair such that is a non empty set and is a partial metric on .

If is a partial metric on , then the function given by

is a metric on .

Definition 2.4. ^{[7, 8, 9]} Let be a partial metric space. Then:

(a) a sequence in partial metric space converges to a point if and only if

(b) a sequence in partial metric space converges to a point if and only if if and only if

(c) a sequence in partial metric space is called a cauchy sequence if there exists (and is finite)

(d) a partial metric space is said to be complete if every cauchy sequence in converges to a point , that is

Lemma 2.5. ^{[7, 8]} Let be partial metric space;

a. is cauchy sequence in if and only if it is Cauchy sequence in the metric space

(b) a partial metric space is complete if and only if the metric space is complete; furthermore, iff

3. Main Theorem

Theorem 3.1. Let be a partially ordered set and let p be a partial metric on such that is complete. Suppose the mapping satisfies the following condition for all we have

1) is continuous or

2) has the following properties,

(a) if a non-decreasing sequence in converges to some point then

(b) if a non-increasing sequence in converges to some point then

3) such that and

4) is a continuous and non decreasing function such that it is positive in and

(3.1)

Then has a coupled fixed point

Proof: Choose and set and Repeating this process, set and Then by (3.1), we have

(3.2)

and similarly,

(3.3)

By adding, we have

(3.4)

Let

If such that then and is fixed point of and the proof is finished. In other case for all Then by using assumption on , we have,

(3.5)

is a non - negative sequence and hence posses a limit Taking limit when , we get,

and consequently By our assumption on , we conclude ie.

(3.6)

Next, we prove that , are cauchy sequences. Suppose that at least one or be not a cauchy sequence. Then and two subsequence of integers with such that

(3.7)

Further, corresponding to we can choose in such a way that it is smallest integer with satisfying equation (3.7), we have

(3.8)

Using (3.7) and (3.8) and triangle inequality, we get

(3.9)

Letting and using (3.6), we have

Now, we get

(3.10)

similarly,

(3.11)

Using(3.10) and (3.11), we get

(3.12)

taking of both sides of equation (3.12) from it follows that

which is a contraction. Therefore and are cauchy sequences. By lemma (2.5), and are cauchy sequence in . Since is complete, hence is also complete, so such that

By lemma, we have

By condition and equation, we get

It follows that

Similarly,

We now prove that . We shall distinguish the cases (1), 2(a) and 2(b) of the Theorem 3.1.

Since is a complete metric space, such that We now show that if the assumption (1) holds, then is coupled fixed point of

As, we have

and

Suppose now that the condition 2(a) and 2(b) of the theorem holds.

The sequence ,

Letting , we have

This implies that similarly, we can show that This completes the theorem.

Theorem 3.2. Let the hypotheses of Theorem 3.1 hold. In addition, suppose that there exists which is comparable to and for all . Then has a unique coupled fixed point.

Proof: Suppose that there exists are coupled fixed points of Consider the following two cases:

Case1: and are compareable. We have

similarly,

It follows that

So, The proof is complete.

Case 2: Suppose now that and are not compareable. Choose an element compareable with both of them.

So, The proof is complete.

Example 3.3. Let endowed with the usual partial metric p defined by with The partial metric space is complete because is complete for any

Thus is Euclidean metric space which is complete.

Consider the mapping defined by

Let us take such that

Clearly is continous and has the mixed monotone property. Also there are in such that

Then it is obvious that is the coupled fixed point of

Now, we have following possibilities for values of and such that

Thus all the conditions of theorem 3.1 are satisfied.

Therefore has a coupled fixed point in

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