We prove some common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using the new property and give some examples. Our results improve and generalize the main results of Mihet in (Mihet, 2010) and many fixed point theorems in fuzzy metric spaces.

The notion of fuzzy sets was introduced by Zadeh [

Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern [

For the reader's convenience we recall some terminologies from the theory of fuzzy metric spaces, which will be used in what follows.

A continuous

the mapping

The following examples are classical examples of a continuous

(the Lukasiewicz

(the product

(the minimum

In 1975, Kramosil and Michalek [

A fuzzy metric space is a triple

We will refer to these spaces as KM-fuzzy metric spaces.

For every

George and Veeramani [

A fuzzy metric space is a triple

From (GV-1) and (GV-2), it follows that if

From now on, by fuzzy metric we mean a fuzzy metric in the sense of George and Veeramani. Several authors have contributed to the development of this theory, for instance [

Let

Let

Let

A fuzzy metric space

If

Let

In 1995, Subrahmanyam [

The concept of E.A. property in metric spaces has been recently introduced by Aamri and El Moutawakil [

Let

The class of E.A. mappings contains the class of noncompatible mappings.

In a similar mode, it is said that two self-mappings of

The concept of E.A. property allows to replace the completeness requirement of the space with a more natural condition of closeness of the range.

Recently, Mihet [

Let

Let

Let

We obtain that Theorems

The aim of this work is to introduce the new property which is so called “common limit in the range” for two self-mappings

We first introduce the concept of new property.

Suppose that

In what follows, the common limit in the range of

Next, we show examples of mappings

Let

Let

In a similar mode, two self-mappings

Let

Since

Next, we let

For the uniqueness of a common fixed point, we suppose that

Next, we will give example which cannot be used [

Let

Let

Since

Let

As

If

Let

It follows from

Let

Similarly in the proof of Theorem

Next, we will show that

Finally, we will prove that a common fixed point of

Let

Since

Let

Since

The authors would like to thank the reviewer, who have made a number of valuable comments and suggestions which have improved the paper greatly. The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this paper for the Ph.D. Program at KMUTT. Moreover, they also would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (NRU-CSEC Project no. 54000267). This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.