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

## 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 [143]. Recently Nieto and Rodríguez-López [2729] and Ran and Reurings [33] presented some new results for contractions in partially ordered metric spaces. The main idea in [2733] 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 [2733] and Ran and Reurings [33] is the following fixed point theorem.

Theorem 1.1.

Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose that is a nondecreasing mapping with

(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, 4648]. 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 .

Lemma (see [13, 14]).

Consider the set and the operation defined by

(2.1)

, 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

(2.2)

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

(2.3)

for and .

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

(2.4)

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

(2.5)

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.

If the -fuzzy metric space satisfies the condition:

(2.6)

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

(2.7)

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]).

Let be a metric space. Denote for all and in and let and be fuzzy sets on defined as follows:

(2.8)

Then is an intuitionistic fuzzy metric space.

Example.

Let . Define for all and in , and let on be defined as follows:

(2.9)

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.

A sequence in an -fuzzy metric space is called a Cauchy sequence, if for each and , there exists such that for all ,

(2.10)

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.

Let be an -fuzzy metric space. is said to be continuous on if

(2.11)

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.

If an -fuzzy metric space satisfies the following condition:

(2.12)

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]).

Let be an -fuzzy metric space in which is Hadži type. Suppose

(2.13)

for some and . Then is a Cauchy sequence.

## 3. Main Results

Definition.

Suppose that is a partially ordered set and are mappings of into itself. We say that is -nondecreasing if for ,

(3.1)

Now we present the main result in this paper.

Theorem.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be two self-mappings of such that there exist and such that is a -nondecreasing mapping and

(3.2)

for all for which and all

Also suppose that

(3.3)

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.

Let be such that Since we can choose such that Again from we can choose such that Continuing this process we can choose a sequence in such that

(3.4)

Since and we have Then from (3.1),

(3.5)

that is, by (3.4), Again from (3.1),

(3.6)

that is, Continuing we obtain

(3.7)

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

(3.8)

Since from (3.4) and (3.7) we have from (3.1) with and

(3.9)

So by (3.4),

(3.10)

Since by (d) of Definition 2.5

(3.11)

we have

(3.12)

As -norm is continuous, letting we get

(3.13)

Consequently,

(3.14)

By repeating the above inequality, we obtain

(3.15)

Since as it follows that

(3.16)

Thus we proved (3.7). By repeating the above inequality (3.7), we get

(3.17)

Since as and , letting in (3.17) we get

(3.18)

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

(3.19)

Let and be arbitrary. For any we have

(3.20)

Since is nondecreasing with respect to , for all in ,

(3.21)

and hence, by (d) of Definition 2.5,

(3.22)

From (3.17) it follows that

(3.23)

From (3.23) with we get

(3.24)

Thus by (3.22),

(3.25)

Hence we get

(3.26)

From (3.26) and (3.17),

(3.27)

Hence we conclude, as as and , that there exists such that

(3.28)

Thus we proved that is a Cauchy sequence.

Since is closed and as , there is some such that

(3.29)

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

(3.30)

Letting we get

(3.31)

for all Therefore,

(3.32)

Hence we get

(3.33)

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

(3.34)

Since from (3.3) we have and as and from (3.2) we get

(3.35)

Letting we get

(3.36)

Hence, similarly as above, we get

(3.37)

Hence we conclude that Since we have

(3.38)

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.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be a nondecreasing self-mappings of such that there exist and such that

(3.39)

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

(3.40)

Corollary 3.5.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be a nondecreasing self-mappings of such that there exist and such that

(3.41)

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.

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## 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

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Shakeri, S., Ćirić, L. & Saadati, R. Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces. Fixed Point Theory Appl 2010, 125082 (2010). https://doi.org/10.1155/2010/125082

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• DOI: https://doi.org/10.1155/2010/125082

### Keywords

• Open Subset
• Approximation Result
• Wide Class
• Fixed Point Theorem
• Point Theory