# Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces

- S Shakeri
^{1}, - LJB Ćirić
^{2}and - R Saadati
^{3}Email author

**2010**:125082

**DOI: **10.1155/2010/125082

© S. Shakeri et al. 2010

**Received: **29 October 2009

**Accepted: **27 January 2010

**Published: **23 February 2010

## Abstract

We introduce partially ordered -fuzzy metric spaces and prove a common fixed point theorem in these spaces.

## 1. Introduction

The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions [1–43]. Recently Nieto and Rodríguez-López [27–29] and Ran and Reurings [33] presented some new results for contractions in partially ordered metric spaces. The main idea in [27–33] involves combining the ideas of iterative technique in the contraction mapping principle with those in the monotone technique.

Recall that if is a partially ordered set and is such that for implies , then a mapping is said to be nondecreasing. The main result of Nieto and Rodríguez-López [27–33] and Ran and Reurings [33] is the following fixed point theorem.

Theorem 1.1.

for all where Also suppose the following.

(a) is continuous.

(b)If is a nondecreasing sequence with in

then for all hold.

If there exists an with , then has a fixed point.

The works of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] have motivated Agarwal et al. [1], Bhaskar and Lakshmikantham [3], and Lakshmikantham and Ćirić [23] to undertake further investigation of fixed points in the area of ordered metric spaces. We prove the existence and approximation results for a wide class of contractive mappings in intuitionistic metric space. Our results are an extension and improvement of the results of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] to more general class of contractive type mappings and include several recent developments.

## 2. Preliminaries

The notion of fuzzy sets was introduced by Zadeh [44]. Various concepts of fuzzy metric spaces were considered in [15, 16, 22, 45]. Many authors have studied fixed point theory in fuzzy metric spaces; see, for example, [7, 8, 25, 26, 39, 46–48]. In the sequel, we will adopt the usual terminology, notation, and conventions of -fuzzy metric spaces introduced by Saadati et al. [36] which are a generalization of fuzzy metric sapces [49] and intuitionistic fuzzy metric spaces [32, 37].

Definition (see [46]).

Let be a complete lattice, and a nonempty set called a universe. An -fuzzy set on is defined as a mapping . For each in , represents the degree (in ) to which satisfies .

, and , for every . Then is a complete lattice.

Classically, a triangular norm on is defined as an increasing, commutative, associative mapping satisfying , for all . These definitions can be straightforwardly extended to any lattice . Define first and .

Definition.

A negation on is any strictly decreasing mapping satisfying and . If , for all , then is called an involutive negation.

In this paper the negation is fixed.

Definition.

A triangular norm ( -norm) on is a mapping satisfying the following conditions:

(i) for all (boundary condition);

(ii) for all (commutativity);

(iii) for all (associativity);

(iv) for all and (monotonicity).

A -norm on is said to be continuous if for any and any sequences and which converge to and we have

For example, and are two continuous -norms on . A -norm can also be defined recursively as an -ary operation ( ) by and

for and .

A
-norm
is said to be of *Hadžić type* if the family
is equicontinuous at
, that is,

is a trivial example of a -norm of Hadžić type, but there exist -norms of Hadžić type weaker than [50] where

Definition.

The 3-tuple
is said to be an
*-fuzzy metric space* if
is an arbitrary (nonempty) set,
is a continuous
-norm on
and
is an
-fuzzy set on
satisfying the following conditions for every
in
and
in
:

(a) ;

(b) for all if and only if ;

(c) ;

(d) ;

(e) is continuous.

then
is said to be *Menger*
*-fuzzy metric space* or for short a
-fuzzy metric space.

Let
be an
-fuzzy metric space. For
, we define the *open ball*
with center
and radius
, as

A subset
is called *open* if for each
, there exist
and
such that
. Let
denote the family of all open subsets of
. Then
is called the *topology induced by the*
*-fuzzy metric*
.

Example (see [38]).

Then is an intuitionistic fuzzy metric space.

Example.

for all and . Then is an -fuzzy metric space.

Lemma (see [49]).

Let be an -fuzzy metric space. Then, is nondecreasing with respect to , for all in .

Definition.

*Cauchy sequence*, if for each and , there exists such that for all ,

The sequence
is said to be *convergent* to
in the
-fuzzy metric space
(denoted by
) if
whenever
for every
. A
-fuzzy metric space is said to be *complete* if and only if every Cauchy sequence is convergent.

Definition.

whenever a sequence in converges to a point , that is, and .

Lemma.

Let be an -fuzzy metric space. Then is continuous function on .

Proof.

The proof is the same as that for fuzzy spaces (see [35, Proposition ]).

Lemma.

then one has and .

Proof.

Let for all . Then by of Definition 2.5, we have and by of Definition 2.5, we conclude that .

Lemma (see [50]).

for some and . Then is a Cauchy sequence.

## 3. Main Results

Definition.

Now we present the main result in this paper.

Theorem.

for all for which and all

Also suppose that

Also suppose that is closed. If there exists an with , then and have a coincidence. Further, if and commute at their coincidence points, then and have a common fixed point.

Proof.

Now we will show that a sequence converges to for each . If for some and for each , then it is easily to show that for all . So we suppose that for all We show that for each

Now we will prove that is a Cauchy sequence which means that for every and there exists such that

Thus we proved that is a Cauchy sequence.

Since is closed and as , there is some such that

Now we show that is a coincidence of and Since from (3.3) and (3.29) we have for all then from (3.2) and by (d) of Definition 2.5 we have

Hence we conclude that for all Then by (b) of Definition 2.5 we have Thus we proved that and have a coincidence.

Suppose now that and commute at . Set Then

Thus we proved that and have a common fixed point.

Remark.

Note that is -nondecreasing and can be replaced by which is -non-increasing in Theorem 3.2 provided that is replaced by in Theorem 3.2.

Corollary 3.4.

for all for which and all Also suppose the following.

(i)If is a nondecreasing sequence with in , then for all hold.

(ii) is continuous.

If there exists an with , then has a fixed point.

Proof.

Taking ( = the identity mapping) in Theorem 3.2, then (3.3) reduces to the hypothesis

Suppose now that is continuous. Since from (3.4) we have for all and as from (3.29), then

Corollary 3.5.

for all for which and all Also suppose the following.

(i)If is a nondecreasing sequence with in , then for all hold.

(ii) is continuous.

If there exists an with , then has a fixed point.

## Declarations

### Acknowledgments

This research is supported by Young research Club, Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The authors would like to thank Professor J. J. Nieto for giving useful suggestions for the improvement of this paper.

## Authors’ Affiliations

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