^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

The aim of this paper is to extend the notions of E.A. property and

The concept of fuzzy set was introduced by Zadeh [

Bhaskar and Lakshmikantham [

Sedghi et al. [

Aamri and Moutawakil [

In this paper, we give the concept of E.A. property and (

Before we give our main result, we need the following preliminaries.

A fuzzy set

A binary operation

Some examples are below:

Let

The

The 3-tuple

In present paper, we consider

Let

convergent to a point

a Cauchy sequence, if for all

A fuzzy metric space

We note that

Let

Define

An element

a coupled fixed point of the mapping

a coupled coincidence point of the mappings

a common coupled fixed point of the mappings

An element

The mappings

commutative if

compatible if

The maps

We note that the maps

There exist pair of mappings that are neither compatible nor weakly compatible, as shown in the following example.

Let

We next show that the pair

Consider the sequences

Hence the pair (

We note that, if

Let

Now we introduce our notions.

Aamri and El Moutawakil [

Let

Now we extend this notion for a pair of coupled maps as follows.

Let

In a similar mode, we state E.A. property for coupled mappings in fuzzy metric spaces as follows.

Let

Let

It is to be noted that property E.A. need not imply compatibility, since in Example

Recently, Sintunavarat and Kuman [

Let

Now we extend this notion for a pair of coupled mappings as follows.

Let

Similarly, we state (

Let

Let

In the next example, we show that the maps satisfying

Let

We next show that the pair of maps satisfying

Let

Define the maps

Consider the sequences

Hence, the pair

For convenience, we denote

Hu [

Let

We now give our main result which provides a generalization of Theorem

Let

the pair

the pair

Since

Similarly,

Since

Since

Then for any

This implies

Since

Also since

Hence

We still get a unique common fixed point if weakly compatible notion is replaced by w-compatible notion.

Now we give another generalization of Theorem

Let

the pair

Since

It follows from

Let

Suppose that

It follows immediately from Corollary

Taking

Let

there exists sequences

Then, there exists a unique