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

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## 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 Open image in new window -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.

## Keywords

Partial Order Fixed Point Theorem Cauchy Sequence Contractive Mapping Comparable Mapping## 1. Introduction and Preliminaries

After the definition of the concept of fuzzy metric space by some authors [1, 2, 3], the fixed point theory on these spaces has been developing (see, e.g., [4, 5, 6, 7, 8, 9]). Generally, this theory on fuzzy metric space is done for contractive or contractive-type mappings (see [2, 10, 11, 12, 13] and references therein). In this paper we introduce the concept of fuzzy order Open image in new window -contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order Open image in new window -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 Open image in new window is called a continuous Open image in new window -norm if Open image in new window is an Abelian topological monoid with the unit Open image in new window such that Open image in new window whenever Open image in new window and Open image in new window for all Open image in new window .

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

Definition 1.2 (see [3]).

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

(KM-1) Open image in new window , for all Open image in new window

(KM-2) Open image in new window , for all Open image in new window if and only if Open image in new window

(KM-3) Open image in new window for all Open image in new window and Open image in new window

(KM-4) Open image in new window is left continuous, for all Open image in new window

(KM-5) Open image in new window for all Open image in new window for all Open image in new window

then the triple Open image in new window 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 Open image in new window . It is easy to see that Open image in new window is a non-Archimedean fuzzy metric space.

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

Definition 1.5 (see [7]).

*G-Cauchy*if

for all Open image in new window The space Open image in new window is called *G-* complete if every *G*-Cauchy sequence is convergent.

Lemma 1.6 (see [11]).

Each M -complete non-Archimedean fuzzy metric space Open image in new window with Open image in new window 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 Open image in new window ukasiewicz *t*-norm. We will refer to [14].

Lemma 1.7 (see [14]).

Then Open image in new window is a (partial) order on Open image in new window named the partial order induced by Open image in new window .

## 2. Main Results

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

Definition 2.1.

Let Open image in new window be an order relation on Open image in new window . A mapping Open image in new window is called nondecreasing w.r.t Open image in new window if Open image in new window implies Open image in new window .

Definition 2.2.

Theorem 2.3.

for each Open image in new window , then Open image in new window has a fixed point.

Proof.

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

Now, if Open image in new window is continuous then it is clear that Open image in new window , while if the condition (2.3) hold then, for all Open image in new window ,

hence Open image in new window

Theorem 2.4.

for all Open image in new window , then Open image in new window has a fixed point.

Proof.

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

*M*-Cauchy sequence. Since Open image in new window is Open image in new window -complete, then there exists Open image in new window such that

hence Open image in new window .

Example 2.5.

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

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

Indeed, let Open image in new window with Open image in new window . Now if Open image in new window , then

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

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

Definition 2.6.

Let Open image in new window be an ordered space. Two mappings Open image in new window are said to be weakly comparable if Open image in new window and Open image in new window for all Open image in new window .

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

Example 2.7.

Then it is obvious that Open image in new window and Open image in new window for all Open image in new window . Thus Open image in new window and Open image in new window are weakly comparable mappings. Note that both Open image in new window and Open image in new window are not nondecreasing.

Example 2.8.

Let Open image in new window and Open image in new window be coordinate-wise ordering, that is, Open image in new window and Open image in new window . Let Open image in new window be defined by Open image in new window and Open image in new window , then Open image in new window and Open image in new window . Thus Open image in new window and Open image in new window are weakly comparable mappings.

Example 2.9.

then Open image in new window and Open image in new window for all Open image in new window Thus Open image in new window and Open image in new window are weakly comparable mappings. Note that, Open image in new window but Open image in new window then Open image in new window is not nondecreasing. Similarly Open image in new window is not nondecreasing.

Theorem 2.10.

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

Proof.

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

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

for all Open image in new window and Open image in new window Then Open image in new window has a fixed point in Open image in new window

Proof.

We take in the above theorem Open image in new window and note that the weak comparability of Open image in new window and Open image in new window 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 Open image in new window is nondecreasing, subadditive mapping (i.e., Open image in new window for all Open image in new window ), with Open image in new window

Theorem 2.12.

Then " Open image in new window " is a (partial) order on Open image in new window

Proof.

that is, " Open image in new window " is reflexive.

Let Open image in new window be such that Open image in new window and Open image in new window Then for all Open image in new window

implying that Open image in new window for all Open image in new window that is, Open image in new window . Thus " Open image in new window " is antisymmetric.

Now for Open image in new window , let Open image in new window and Open image in new window . Then, for given Open image in new window ,

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

This shows that Open image in new window , that is, " Open image in new window " is transitive.

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

Corollary 2.13.

then Open image in new window has a fixed point in Open image in new window

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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