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Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappings on 0-complete partial metric spaces to obtain fixed point results, thus generalizing several known results. Various examples are presented showing how the obtained theorems can be used and that they are proper extensions of the known ones.

The celebrated Banach contraction principle has been generalized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics.

Cyclic representations and cyclic contractions were introduced by Kirk et al. [

On the other hand, Matthews [

Khan et al. [

In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a generalized control function (with two variables) for mappings on

In 2003, Kirk et al. introduced the following notion of cyclic representation.

Let

They proved the following fixed point result.

Let

In 2010, Păcurar and Rus introduced the following notion of cyclic weaker

Let

there exists a continuous, nondecreasing function

They proved the following result.

Suppose that

This was generalized by Karapınar in [

Khan et al. introduced the following notion.

A function

Choudhury introduced a generalization of Chatterjea type contraction as follows.

A self-mapping

In [

The following definitions and details can be seen, for example, in [

A partial metric on a nonempty set

(_{1}

(_{2}

(_{3}

(_{4}

It is clear that, if

Each partial metric

A sequence

(1) A paradigmatic example of a partial metric space is the pair

(2) Let

Let

A sequence

(see [

Let

(a) (see [

(b) (see [

The converse assertion of (b) does not hold as the following easy example shows.

The space

It is easy to see that every closed subset of a

In this section, we will prove some fixed point theorems for self-mappings defined on a

Let

for any

Our main result is the following.

Let

Let

Without loss of the generality, we may assume that

Putting

Next, we claim that

Using (

Since

Now, we shall prove that

We claim that there is a unique fixed point of

If we take

Let

Corollary

By taking

Let

for any

As a special case of Corollary

Let

for any

The following example shows how Theorem

Consider the partial metric space

Let

The case

The case

We conclude that all conditions of Theorem

Here is another example showing the use of Theorem

Let

Thus, all the conditions of Theorem

We state a more involved example that is inspired with the one from [

Let

Consider the mapping

Let us check the contractive condition (NZ2) of Theorem

Finally, we present an example showing that in certain situations the existence of a fixed point can be concluded under partial metric conditions, while the same cannot be obtained using the standard metric.

Let

On the other hand, consider the same problem in the standard metric

The results of this paper are obtained under the assumption that the given partial metric space is

The authors thank the referees for valuable suggestions that helped them to improve the text. The second author is thankful to the Ministry of Science and Technological Development of Serbia.