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Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example. The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

In recent times coupled fixed point theory has experienced a rapid growth in partially ordered metric spaces. The speciality of this line of research is that the problems herein utilize both order theoretic and analytic methods. References [

A function

Let

Recently, Berinde and Borcut [

Let

They also extended the mixed monotone property to functions with three arguments.

Let

Our purpose here is to establish tripled coincidence point results in metric spaces with partial ordering. For that purpose we define mixed

Let

Coupled coincidence point was defined by Lakshmikantham and Ćirić [

Let

We extend the concept of commuting mappings given by Lakshmikantham and Ćirić [

Let

The following is the definition of compatible mappings which is an extension of the compatibility defined by Choudhury and Kundu in [

Let

Let

if a nondecreasing sequence

if a nonincreasing sequence

If there exist

By a condition of the theorem, there exist

We presume that (

If for some

Then

Again, for all

Since

Since

For all

Similarly, by using (

Thus we have proved that

Suppose that the assumption

This completes the proof of the theorem.

Let

if a nondecreasing sequence

if a nonincreasing sequence

If there exist

Since a commuting pair is also a compatible pair, the result of the Corollary

Later, by an example, we will show that the Corollary

Let

if a nondecreasing sequence

if a nonincreasing sequence

If there exist

Taking

Let

if a nondecreasing sequence

if a nonincreasing sequence

If there exist

Taking

The following corollary is the result of Berinde and Borcut in [

Let

if a nondecreasing sequence

if a nonincreasing sequence

If there exist

The proof follows from Corollary

The method used in the proof of Corollary

Next we discuss an example.

Let

Let

Let

Let

Let

Let,

Therefore,

Also,

We next verify inequality (

Let

Then

Thus it is verified that the functions

It is observed that in Example