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Journal of Function Spaces
Volume 2018 (2018), Article ID 5650242, 9 pages
https://doi.org/10.1155/2018/5650242
Research Article

Fixed Points of -Fuzzy Mappings in Ordered -Metric Spaces

1Lahore School of Economics, Lahore, Pakistan
2Assiut University, Assiut, Egypt
3Port Said University, Port Said, Egypt

Correspondence should be addressed to Ismat Beg; kp.ude.pcu@gebi

Received 26 October 2017; Revised 20 December 2017; Accepted 11 January 2018; Published 1 March 2018

Academic Editor: Tomonari Suzuki

Copyright © 2018 Ismat Beg et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [58]). 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 thatwhere 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)–(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 thatwhere 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 thatwhere 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 , with and , for some . Now, we show that ; otherwise, sinceWhen , we have . But implies ; then By , this gives which is a contradiction of ; then and . Also, for . We prove this by contradiction; otherwise, sinceLetting , we have , but implies ; then From , this gives which is a contradiction; then . Further, being weakly commuting and occasionally coincidentally idempotent imply that, respectively, and , so that . Also, and imply . Then have a common fixed point.

6. An Application to Integral Contractive Condition

Suppose that are summable nonnegative Lebesgue integrable functions such that for each , and . As a simple application of Theorems 21, 25, and 26, we give the following result.

Theorem 28. Let be a -metric space, , and let 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 and satisfying the following conditions(1) and sa