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We prove the existence of common fixed points for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space using orbital continuity of one of the involved maps. We furnish a suitable example to demonstrate the validity of the hypotheses of our results.

Browder and Petryshyn introduced the concept of asymptotic regularity of a self-map at a point in a metric space.

A self-map

Recall that the set

A metric space

Here, it can be pointed out that every complete metric space is

A self-map

Clearly, every continuous self-mapping of a metric space is orbitally continuous, but not conversely.

Sastry et al. [

Let

If for a point

The space

The map

The pair

On the other side, Khan et al. [

A function

Simple examples of generalized altering distance functions with four variables are

On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [

In this paper, an attempt has been made to derive some common fixed point theorems for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space, using orbital continuity of one of the involved maps and conditions involving a generalized altering distance function. The presented theorems generalize, extend, and improve some recent results given in [

First, we introduce some further notations and definitions that will be used later.

If

Let

[

[

[

[

Note that none of two weakly increasing mappings need to be nondecreasing. There exist some examples to illustrate this fact in [

Let

Let

Let

Let

Let

Let

The first main result is as follows.

Let

for a nondecreasing sequence

Since

From (

Finally, we prove the existence of a common fixed point of the three mappings

We have

Similarly, the result follows when condition (b) holds.

Now, suppose that the set of common fixed points of

Now, it is easy to state a corollary of Theorem

Let

If we take

Other results could be derived for other choices of

As consequences of Theorem

Let

Then

Let

Then

We present an example showing the usage of our results.

Let the set

It was shown by examples in [

if the contractive condition is satisfied just on

under the given hypotheses (common), fixed point might not be unique in the whole space

The authors thank the referees for their careful reading of the text and for suggestions that helped to improve the exposition of the paper. The second and third authors are thankful to the Ministry of Science and Technological Development of Serbia.