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New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.

In recent years many authors have worked on domain theory in order to equip semantics domain with a notion of distance. In particular, Matthews [

Many generalizations of Banach's contractive condition have been introduced in order to obtain more general fixed point results in metric spaces and their generalizations. We mention here conditions introduced by Geraghty [

The notion of a coupled fixed point was introduced and studied by Bhaskar and Lakshmikantham in [

In this paper, we further develop the method of Berinde and obtain new coupled coincidence and coupled fixed point results in ordered partial metric spaces, under the contractive conditions of Geraghty, Rakotch, and Branciari types. Examples show that these results are distinct from the known ones. In particular, they show that using the order and/or the partial metric enables conclusions which cannot be obtained in the classical case.

The following definitions and details can be seen in [

A partial metric on a nonempty set

It is clear that if _{1}) and (P_{2})

Each partial metric

If

A basic example of a partial metric space is the pair

Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function

Let

a sequence

the space

Let

The space

Let

We will say that the space

Let

A point

Let

(i) Let

(ii) If

(iii) If

(i) Relation _{1}) and (P_{4}) are nontrivial.

(P_{1}) If _{2}) that _{1}) of partial metric

(P_{4}) Let

(ii, iii) The proofs of these assertions are straightforward.

Let

It is easy to see that (using notation as in the previous lemma), the mappings

Assertions similar to the following lemma (see, e.g., [

Let

As a corollary (applying Lemma

Let

Let

Let

Subsequently, several authors proved such results, including the very recent paper of Ðukić et al. [

We begin with the following auxiliary result.

Let

The proof follows the lines of proof of [

Take

Under this assumption, we get that

We will prove first that in this case the sequence

For each

In order to prove that _{2}),

Thus,

(i) Suppose that

(ii) If _{4}) and the contractive condition to obtain

Now, we are in the position to prove the main result of this section.

Let

there exists

there exist

Let relation

there exists

there exists

Putting

Let

there exists

there exist

If

Let

Take arbitrary

Let

Let

We will prove the respective result for the existence of a coupled coincidence point in the frame of partial metric spaces. We begin with the following auxiliary result, which may be of interest on its own.

Let

The proof is similar to the proof of Lemma

With

The following example shows that the existence of order may be crucial.

Let

On the other hand, consider the same problem, but without order. Then the contractive condition does not hold and the conclusion about the coincidence point cannot be obtained in this way. Indeed, take any

Now, we are in the position to give

Let

there exists

The proof is analogous to the proof of Theorem

Putting

Let

there exists

Denote by

Let

Subsequently, several authors proved such results, including the very recent paper of Liu et al. [

We begin with the following.

Let

Take

Denote by

Now we prove that

(i) Suppose that

(ii) If _{4}) and the contractive condition to obtain

Putting

Let

On the other hand, consider the same problem in an (ordered) partial metric space, with the partial metric given by

The following theorem is obtained from Lemma

Let

there exists

Putting

Let

there exists

The authors are thankful to the referees for very useful suggestions that helped to improve the paper. They are also thankful to the Ministry of Science and Technological Development of Serbia.