# Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results

- Ishak Altun
^{1}Email author and - Dorel Miheţ
^{2}

**2010**:782680

https://doi.org/10.1155/2010/782680

© I. Altun and D. Miheţ. 2010

**Received: **2 July 2009

**Accepted: **9 February 2010

**Published: **14 February 2010

## Abstract

In the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order -contractive type mappings. Then a common fixed point theorem is given for noncontractive type mappings. Kirk's problem on an extension of Caristi's theorem is also discussed.

## 1. Introduction and Preliminaries

After the definition of the concept of fuzzy metric space by some authors [1–3], the fixed point theory on these spaces has been developing (see, e.g., [4–9]). Generally, this theory on fuzzy metric space is done for contractive or contractive-type mappings (see [2, 10–13] and references therein). In this paper we introduce the concept of fuzzy order -contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order -contractive type mappings. Then, using an idea in [14], we will provide a common fixed point theorem for weakly increasing single-valued mappings in a complete fuzzy metric space endowed with a partial order induced by an appropriate function. Some fixed point results on ordered probabilistic metric spaces can be found in [15].

For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper.

Definition 1.1 (see [16]).

A binary operation is called a continuous -norm if is an Abelian topological monoid with the unit such that whenever and for all .

A continuous *t*-norm
is of *Hadžić-type* if there exists a strictly increasing sequence
such that
for all

Definition 1.2 (see [3]).

A fuzzy metric space (in the sense of Kramosil and Michálek) is a triple , where is a nonempty set, is a continuous -norm and is a fuzzy set on , satisfying the following properties:

(KM-2) , for all if and only if

(KM-4) is left continuous, for all

then the triple
is called a *non-Archimedean fuzzy metric space*. It is easy to check that the triangular inequality (NA) implies (KM-5), that is, every non-Archimedean fuzzy metric space is itself a fuzzy metric space.

Example 1.3.

for all . It is easy to see that is a non-Archimedean fuzzy metric space.

Let
be a fuzzy metric space. A sequence
in
is called an *M*-Cauchy sequence, if for each
and
there exists
such that
for all
. A sequence
in a fuzzy metric space
is said to be convergent to
if
for all
. A fuzzy metric space
is called *M*-complete if every
-Cauchy sequence is convergent.

Definition 1.5 (see [7]).

for all
The space
is called *G-* complete if every *G*-Cauchy sequence is convergent.

Lemma 1.6 (see [11]).

Each M -complete non-Archimedean fuzzy metric space with of Hadžić-type is G-complete.

Theorem 2.10in the next section is related to a partial order on a fuzzy metric space under the
ukasiewicz *t*-norm. We will refer to [14].

Lemma 1.7 (see [14]).

Then is a (partial) order on named the partial order induced by .

## 2. Main Results

The first two theorems in this section are related to Theorem in [17]. We begin by giving the following definitions.

Definition 2.1.

Let be an order relation on . A mapping is called nondecreasing w.r.t if implies .

Definition 2.2.

Theorem 2.3.

for each , then has a fixed point.

Proof.

for all
that is,
is *G*-Cauchy. Since
is
-complete (Lemma 1.6), then there exists
such that
.

Now, if is continuous then it is clear that , while if the condition (2.3) hold then, for all ,

Theorem 2.4.

for all , then has a fixed point.

Proof.

*M*-Cauchy sequence. Supposing this is not true, then there are and such that for each there exist with and

*M*-Cauchy sequence. Since is -complete, then there exists such that

Example 2.5.

*M*-complete non-Archimedean fuzzy metric space (see [18]) satisfying for all . Define a self map of as follows:

Now, it is easy to see that is continuous and nondecreasing w.r.t . Also, for we have . Now we can see that is fuzzy order -contractive with .

Indeed, let with . Now if , then

Therefore is fuzzy order -contractive with . Hence all conditions of Theorem 2.4 are satisfied and so has a fixed point on .

In order to state our next theorem, we give the concept of weakly comparable mappings on an ordered space.

Definition 2.6.

Let be an ordered space. Two mappings are said to be weakly comparable if and for all .

Note that two weakly comparable mappings need not to be nondecreasing.

Example 2.7.

Then it is obvious that and for all . Thus and are weakly comparable mappings. Note that both and are not nondecreasing.

Example 2.8.

Let and be coordinate-wise ordering, that is, and . Let be defined by and , then and . Thus and are weakly comparable mappings.

Example 2.9.

then and for all Thus and are weakly comparable mappings. Note that, but then is not nondecreasing. Similarly is not nondecreasing.

Theorem 2.10.

Let be an M -complete non-Archimedean fuzzy metric space with be a bounded-from-above function, and let be the partial order induced by If are two continuous and weakly comparable mappings, then and have a common fixed point in

Proof.

This shows that the sequence
is *M*-Cauchy. Since
is *M*-complete, it converges to a point
. As
and
, by the continuity of
and
we get
.

Corollary 2.11 ([Caristi fixed point theorem in non-Archimedean fuzzy metric spaces]).

for all and Then has a fixed point in

Proof.

We take in the above theorem and note that the weak comparability of and reduces to (2.36).

The generalization suggested by Kirk of Caristi's fixed point theorem [19] is well known. A similar theorem in the setting of non-Archimedean fuzzy metric spaces is stated in the final part of our paper.

In what follows is nondecreasing, subadditive mapping (i.e., for all ), with

Theorem 2.12.

Then " " is a (partial) order on

Proof.

Let be such that and Then for all

implying that for all that is, . Thus " " is antisymmetric.

Now for , let and . Then, for given ,

On the other hand, from the triangular inequality (NA), the inequality

This shows that , that is, " " is transitive.

From the above theorem we can immediately obtain the following generalization of Corollary 2.11.

Corollary 2.13.

The reader is referred to the nice paper [20] for some discussion of Kirk's problem on an extension of Caristi's fixed point theorem.

## Authors’ Affiliations

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