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

*Fixed Point Theory and Applications***volume 2010**, Article number: 782680 (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-1), for all

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

(KM-3) for all and

(KM-4) is left continuous, for all

(KM-5) for all for all

If, in the above definition, the triangular inequality (KM-5) is replaced by

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.

Let be an ordinary metric space and let be a nondecreasing and continuous function from into such that . Some examples of these functions are , and . Let for all . For each , define

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]).

Let be a fuzzy metric space. A sequence in is called *G-Cauchy* if

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]).

Let be a non-Archimedean fuzzy metric space with and Define the relation "" on as follows:

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.

Let be a partially ordered set, let be a fuzzy metric space, and let be a function from to . A mapping is called a fuzzy order -contractive mapping if the following implication holds:

Theorem 2.3.

Let be a partially ordered set and be an -complete non-Archimedean fuzzy metric space with of Hadžić-type. Let be a continuous, nondecreasing function and let be a fuzzy order -contractive and nondecreasing mapping w.r.t . Suppose that either

or

hold. If there exists such that

for each , then has a fixed point.

Proof.

Let for . Since and is nondecreasing w.r.t , we have

Then, it immediately follows by induction that

hence

By taking the limit as we obtain

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 ,

and letting it follows

hence

Theorem 2.4.

Let be a partially ordered set, let be an -complete non-Archimedean fuzzy metric space, and let be a continuous mapping such that for all . Also, let be a nondecreasing mapping w.r.t , with the property

Suppose that either (2.2) or (2.3) holds. If there exists such that

for all , then has a fixed point.

Proof.

Let for . Then, as in the proof of the preceding theorem we can prove that

Therefore, for every , is a nondecreasing sequence of numbers in . Let, for fixed , Then we have , since . Also, since

and is continuous, we have . This implies and therefore, for all

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

Let, for each , be the least integer exceeding satisfying the inequality (2.16), that is,

Then, for each ,

Letting and using (2.15), we have, for ,

Then, since , we have

Letting and using (2.15) and (2.19), we obtain

which is a contradiction. Thus is an *M*-Cauchy sequence. Since is -complete, then there exists such that

If is continuous, then from it follows that Also, if (2.3) holds, then (since ) we have

Letting , we obtain that

hence .

Example 2.5.

Let . Consider the following relation on :

It is easy to see that is a partial order on . Let and

Then is an *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

If with , 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.

Let and be usual ordering. Let defined by

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.

Let and be lexicographical ordering, that is, or if then . Let be defined by

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.

Let be an arbitrary point of and let us define a sequence in as follows:

Note that, since and are weakly comparable, we have

By continuing this process we get

that is, the sequence is nondecreasing. By the definition of we have for all , that is, for even , the sequence is a nondecreasing sequence in . Since is bounded from above, is convergent and hence it is Cauchy. Then, for all there exists such that for all and we have . Therefore, since , we have

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]).

Let be an M -complete non-Archimedean fuzzy metric space with let be a bounded-from-above function and be a continuous mapping, such that

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.

Let be a non-Archimedean fuzzy metric space with and Define the relation "" on through

Then "" is a (partial) order on

Proof.

Since , then for all and ,

that is, "" is reflexive.

Let be such that and Then for all

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

Now for , let and . Then, for given ,

By using (2.40) and (2.41) we get

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

holds. This implies

As is nondecreasing, it follows that

and therefore

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.

Let be an M -complete non-Archimedean fuzzy metric space with let be a bounded-from-above function and be a continuous mapping, such that

for all and If satisfies the property

then has a fixed point in

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.

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### Keywords

- Partial Order
- Fixed Point Theorem
- Cauchy Sequence
- Contractive Mapping
- Comparable Mapping