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We establish some common fixed point results for a new class of pair of contraction mappings having functions as contractive parameters, and satisfying minimal noncommutative operators property.

The metric fixed point theory has a vast literature; the Banach-Caccioppoli contraction principle is one of the most outstanding results in this theory. Since its appearance, several generalizations of this result have appeared in the literature. In 1976, Jungck [

Here, we are going to establish existence and uniqueness results of common fixed point for a pair of contractive type of mappings whose contractive parameters are nonconstants and its contractive inequality is controlled by a positive function satisfying a stability condition at 0 (see (

In order to establish our results the following notions will be needed. A pair of self-mappings

A pair of self-mappings

If

If

A point

A pair of mappings

We would like to show that weak compatibility is a necessary, hence minimal, condition for the existence of common fixed points of contractive type mapping pairs. Suppose

This shows that weak compatibility is a necessary, hence minimal, condition for the existence of common fixed points of contractive type mapping pairs.

The following result due to Babu and Sailaja in [

Let

In order to introduce the class of mappings which will be the focus of study of this paper, as in [

Now, we introduce the following class of pair of contraction type of mappings.

Let

Let

Let

Let

To prove (1), let

To prove (2), we are going to suppose that

In this way we have

In this section we prove our main results concerning the existence and uniqueness of common fixed points for a

Let

the pair

Then,

the pair

if

Let

Compatibility and orbital continuity of

On the other hand, since the pair

Let

the pair

Then,

the pair

if the pair

Let

Theorem

Let

the pair

Then,

the pair

if the pair

Notice that in Theorem

Let

Since the pair

Since two noncompatible self-mappings on a metric space

In the next theorem we drop closeness of the range of mapping and replace property (E.A.) with

Let

Since the pair

The rest of the proof of the theorem follows easily.

In 1976, Delbosco [

A function

By

Since every nondecreasing map

On the other hand, in 2002, Branciari [

By

for each

A relation between these two classes of functions

For each

In this way, additionally to the class of

Notice that due to the minor restrictions on the functions involved in the definition of the class of

Next, we are going to show some examples in support of our results.

Let

Notice that

As in the example before,

Let

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are thankful to the referees for the very constructive comments and suggestions that led to an improvement of the paper. E. M. Rojas is sponsored by Pontificia Universidad Javeriana under Grant no. 000000000005781.