Research Article | Open Access

# Fixed Points of -Fuzzy Mappings in Ordered -Metric Spaces

**Academic Editor:**Tomonari Suzuki

#### Abstract

We prove common fixed point theorems for weakly commuting and occasionally coincidentally idempotent -fuzzy mappings in ordered -metric spaces. We also obtain common fixed point for pair of mapping satisfying property. An application to integral type and usual contractive condition is given.

#### 1. Introduction

In 1967, Goguen [1] introduced the notion of -fuzzy sets as a generalization of fuzzy sets [2]. Afterward, Heilpern [3] gave the concept of fuzzy mappings and proved fixed point theorems for fuzzy contractive mappings in metric linear spaces as a generalization of Nadler [4] contraction principle. Subsequently, several authors studied the generalizations/extensions and applications of these results; in many papers sufficient conditions for the existence of fixed point for fuzzy and -fuzzy contractive mappings in metric spaces and -metric spaces are obtained (see [5â€“8]). Rashid et al. [9] proved common -fuzzy fixed point theorem in complete metric spaces. They extended the results of Heilpern [3] into -fuzzy mappings in metric spaces. In this paper, we prove some common fixed point theorems for -fuzzy mappings in ordered -metric spaces. An application to integral type and usual contractive condition is also given, we will generalize and extend results [10, 11].

#### 2. Preliminaries

*Definition 1 (see [12]). *Let be a nonempty set. A mapping is called -metric if there exists a real number such that, for every , we have(),(),().In this case, the pair is called a -metric space.

*Definition 2 (see [13]). *Let be a -metric space. A sequence in is called(i)convergent if and only if there exists such that as ,(ii)Cauchy if and only if as .A -metric space is said to be complete if and only if each Cauchy sequence in this space is convergent.

Let be a -metric space, let denote the collection of all nonempty closed subsets of , let be the set of all fuzzy sets of , where its -level sets are nonempty compact subsets of , and let be the set of nonnegative real numbers; define the metrics and as and . Notice that and , for each and . Let , , and be the set of the coincidence points of .

Lemma 3 (see [14]). *Let be a -metric space, ; then for all .*

*Definition 4 (see [15]). *A partially ordered set consists of a set and a binary relation on which satisfies the following conditions for all :(1) (reflexivity).(2)If and , then (antisymmetry).(3)If and , then (transitivity).A set with a partial order is called a partially ordered set. Let be a partially ordered set and . Elements and are said to be comparable elements of if either or .

*Definition 5. *Let and be two nonempty subsets of ; the relation between and is defined as : if for every there exists such that .

*Definition 6 (see [1]). *A function is said to be an -fuzzy set if is a mapping from a nonempty set into a complete distributive lattice

*Definition 7 (see [9, 10, 16]). *Let be an -fuzzy set in . The -level set of (denoted by ) is defined as where denotes the closure of the set .

*Definition 8 (see [9]). *Let and be two arbitrary nonempty sets and denote the collection of all -fuzzy sets in . A mapping is called -fuzzy mapping if is a mapping from into . An -fuzzy mapping is an -fuzzy subset on with membership function . The function is the grade of membership of in .

*Definition 9 (see [9]). *Let be -fuzzy mappings from an arbitrary nonempty set into . A point is called an -fuzzy fixed point of if , where . The point is called a common -fuzzy fixed point of and if .

*Remark 10 (see [10, 16]). *If then the -fuzzy fixed point is just the fixed point for the -fuzzy mapping.

*Definition 11 (see [17]). *Let be a metric space, , and . Then the pair is said to have property if there exists a sequence in and such that , with , for some .

*Definition 12 (see [18]). *Let be a metric space, , , and . Two mappings are said to be occasionally coincidentally idempotent if for some .

*Definition 13 (see [19]). *Let be a metric space, , and . Let ; if there exists a sequence in such that then is called the orbit for the mappings .

*Definition 14 (see [19]). *Metric space is called joint orbitally complete, if every Cauchy sequence of each orbit at is convergent in .

*Definition 15 (see [20]). *Let be a mapping from a subset of a metric space into and ; then is said to be -weakly commuting at if .

#### 3. Definitions in -Metric Spaces

Let be a -metric space, , and are two -fuzzy mappings from into such that, for each , , and are nonempty closed subsets of . Similar to [11], first we rewrite some notions in -metric spaces as follows.

*Definition 16. *The pairs and are said to have property if there exist two sequences and in and such that with , for some .

*Definition 17. *Two mappings and are said to be occasionally coincidentally idempotent if for some .

*Definition 18. *Two mappings and are said to be -weakly commuting at if .

Let be the family of all continuous mappings satisfying the following properties:() is nondecreasing in the variable and nonincreasing in the , , , and coordinate variables.()There exists such that for every with or implies .

*Example 19. *(i) .

(ii) .

(iii) .

Let be the family of continuous mappings , which is nondecreasing in the first coordinate and satisfying the following condition for each : if or , then .

*Example 20. *(i) .

(ii) .

(iii) .

#### 4. Common Fixed Points in Ordered -Metric Spaces

Theorem 21. *Let be a -metric space, , and be a partial order defined on , such that and are closed. Suppose that are two -fuzzy mappings from into such that for each and , and are nonempty closed subsets of which satisfy the following conditions:** and .** or implies .**If , then for all .** and are weakly commuting and occasionally coincidentally idempotent.**For all comparable elements and , there exists such that*â€‰*where **One of or is joint orbitally complete for some ;** then and have a common fixed point, while and have a common -fuzzy fixed point*

*Proof. *Let and ; then by (1), there exist such that and , from (2), . Now, ; then but then from the property which implies , there exists such that . Again, , , , and ; then Also then from the property which implies , there exists such that . By induction we obtain . Continuing in this way, we obtain an orbit such that and . Further, Therefore, . Hence, is a Cauchy sequence. As is a Cauchy sequence in , and is joint orbitally complete, therefore, there exists such that , for . Next, we show that . Since where then As By which implies , we have . Thus, . As , therefore, there exists such that . Similarly, . Since the pair are weakly commuting and occasionally coincidentally idempotent and , then ; therefore, . Also ; then ; therefore, .

*Example 22. *Let and ; then is a -metric space with . Define the partial order as for each and as follows: for each , there exists such that . Let and define the maps on as , , Define the sequences , , and in as , , and , ; then and . Now, and ; then and . Further, . Let ; then we have where Finally, and ; that is, and are weakly commuting and occasionally coincidentally idempotent and are weakly commuting and occasionally coincidentally idempotent. Now, satisfy all conditions of Theorem 21 and is a common fixed point.

Corollary 23. *Let be a complete -metric space, , and be a partial order defined on , such that and are closed. Suppose that are two -fuzzy mappings from into such that, for each and , and are nonempty closed subsets of which satisfy conditions ((1)â€“(5)); then and have a common fixed point, while and have a common -fuzzy fixed point.*

*Remark 24. * Theorem 21 is a generalization of Theorem [6], Theorem [9], and Theorem [8].

In Theorem 21, one may put one of or or both; in this case, condition (4) has the following versions:

Theorem 25. *Let be a -metric space, , and be a partial order defined on , such that and are closed. Suppose that is a sequence of -fuzzy mappings from into such that, for each and , are nonempty closed subsets of which satisfy the following conditions:** and .** or implies .**If , then for all .** and are weakly commuting and occasionally coincidentally idempotent.**For all comparable elements , and , and , there exists such that*â€‰*where **One of or is joint orbitally complete for some ,** then and have a common fixed point while and have a common -fuzzy fixed point.*

*Proof. *Put and in Theorem 21. This concludes the proof.

#### 5. Common Fixed Points with (JCLR) Property

Theorem 26. *Let be a -metric space, , and be a partial order defined on , such that and are closed. Suppose that are two -fuzzy mappings from into such that, for each and , and are nonempty closed subsets of which satisfy the following conditions:** and satisfy (JCLR) property.** and are weakly commuting and occasionally coincidentally idempotent.**For all comparable elements and there exists such that*â€‰*where **then and have a common fixed point, while and have a common -fuzzy fixed point.*

*Proof. *Since and satisfy property, there exist two sequences in and such that With and , for some , we show that , since where As , which gives From the property which implies , there exists such that , so that . Now, ; to prove this, since , we show that . As where when , we have that So that From the property which implies , there exists such that ; this gives ; then , and . By a similar way, one can find for . Further, and so that . Also, and imply . Then have a common fixed point.

Theorem 27. *In Theorem 26, we may replace condition (3) by another: for all and there exist such that then and have a common fixed point, while and have a common -fuzzy fixed point.*

*Proof. *Since and satisfy (JCLR) property, there exist two sequences in and such that