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First, we prove a common fixed point theorem using weakly compatible maps in 2-Menger space with

The theory of probabilistic metric spaces is an important part of stochastic analysis, and so it is of interest to develop the fixed point theory in such spaces. The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha-Reid [

A probabilistic 2-metric space is an ordered pair

for distinct points

if

In 2003, Shi et al. [

A mapping

For

Basic examples of

A special class of

Let

We say that the

The family

A trivial example of

Every

There is a nice characterization of continuous

If there exists a strictly increasing sequence

If

If

Let

A sequence

converge with limit

Cauchy sequence in

complete if every Cauchy sequence in

Two maps

In 2002, Aamri and Moutawakil [

Let

Now in a similar mode, we can state the E.A. property in 2-Menger space as follows.

A pair of self-mappings

Let

Although E.A property is generalization of the concept of noncompatible maps, yet it requires either completeness of the whole space or any of the range spaces or continuity of maps. But on the contrary, the new notion of the CLR property (common limit range property) recently given by Sintunavarat and Kumam [

Two maps

Let

Now we state a lemma which is useful in our study.

Let

Let

the pairs

there exists

one of the subsets

Suppose that

Define

Then one can see that (2.1)–(2.4) are satisfied.

Conversely, assume that there exist two self-mappings

Also, letting

We now show that

Let

From (

Let

Since

From (

Subsequently, we have

For this purpose, we put

Taking limit as

Hence

Since

Next we claim that

Putting

That is,

By Lemma

Thus

Since the pairs

Now we prove that

For this purpose, puting

By Lemma

Hence

Then, for all

Hence

This completes the proof of the theorem.

Let

pair

there exists

one of the subspaces

We can easily prove the theorem by setting

Let

there exists

Proof easily follows by setting

Now we prove our main result for weakly compatible maps along with the E.A. property as follows.

Let

pair

If the pair

Since

Suppose that

Next, we claim that

For this purpose, we put

Taking limit as

By Lemma

Hence

Since

Next we claim that

Putting

By Lemma

Thus

Since the pairs

Now we prove that

For this purpose, we put

By Lemma

Hence

Then, for all

Hence

Let

Then

Now we prove our main result for weakly compatible maps along with the CLR_{S} property as follows.

Let

pair _{S} property;

one of the subspaces

If the pair _{S} property then there exists a sequence

Therefore,

Thus in all, we have

Now we are required to show that

Putting

Let

By Lemma

Hence

Uniqueness follows from Theorem

Let

Then

As an application of Theorem

Let

A and

B and

The conclusions (i) and (ii) are immediate as

By setting

Let

One of the authors (S. Kumar) would like to acknowledge the UGC for providing financial grant through Major Research Project under Ref. 39-41/2010 (SR). I am also thankful to the referee for careful reading of the paper and giving useful comments to improve it.