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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 890678, 18 pages
doi:10.1155/2012/890678
Research Article

Common Fixed Point Theorems in a New Fuzzy Metric Space

1School of Information Engineering, Guangdong Medical College, Dongguan, Guangdong 523808, China
2College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Nanan, Chongqing 400065, China

Received 26 October 2011; Accepted 1 December 2011

Academic Editor: Yeong-ChengΒ Liou

Copyright Β© 2012 Weiquan Zhang 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 generalize the Hausdorff fuzzy metric in the sense of RodrΓ­guez-LΓ³pez and Romaguera, and we introduce a new π‘€βˆž-fuzzy metric, where π‘€βˆž-fuzzy metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under πœ™-contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.

1. Introduction

The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. After that, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application [2, 3]. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of fuzzy metric space. This problem was investigated by many authors [4–13] from different points of view. George and Veeramani’s fuzzy metric space [6] has been widely accepted as an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. Further, it is necessary to mention that this fuzzy metric space has very important application in studying fixed point theorems for contraction-type mappings [7, 14–16]. Besides that, a number of metrics are used on subspaces of fuzzy sets. For example, the sendograph metric [17–19] and the π‘‘βˆž-metric for fuzzy sets [20–25] induced by the Hausdorff-Pompeiu metric have been studied most frequently, where π‘‘βˆž-metric is an ordinary metric between two fuzzy sets. Combining fuzzy metric (in the sense of George and Veeramani) and Hausdorff-Pompeiu metric, RodrΓ­guez-LΓ³pezand Romaguera [26] construct a Hausdorff fuzzy metric, where Hausdorff fuzzy metric can be thought of as the degree of nearness between two crisp nonempty compact sets with respect to any positive real number.

In this present investigation, considering the Hausdorff-Pompeiu metric and theories on fuzzy metric spaces (in the sense of George and Veeramani) together, we study the degree of nearness between two fuzzy sets as a natural generalization of the degree of nearness between two crisp sets, in turn, it helps in studying new problems in fuzzy topology. Based on the Hausdorff fuzzy metric 𝐻𝑀, we introduce a suitable notion for the π‘€βˆž-fuzzy metric on the fuzzy sets whose πœ†-cut are nonempty compact for each πœ†βˆˆ[0,1]. In particular, we explore several properties of π‘€βˆž-fuzzy metric. Then, under πœ™-contraction condition, we give some common fixed point theorems in the fuzzy metric space on fuzzy sets.

2. Preliminaries

According to [27], a binary operation βˆ—βˆΆ[0,1]Γ—[0,1]β†’[0,1] is called a continuous 𝑑-norm if ([0,1],βˆ—) is an Abelian topological semigroups with unit 1 such that π‘Žβˆ—π‘β‰€π‘βˆ—π‘‘ whenever π‘Žβ‰€π‘ and 𝑏≀𝑑 for all π‘Ž,𝑏,𝑐,π‘‘βˆˆ[0,1].

Definition 2.1 (see [6]). The 3-tuple (𝑋,𝑀,βˆ—) is said to be a fuzzy metric space if 𝑋 is an arbitrary set, βˆ— is a continuous 𝑑-norm, and 𝑀 is a fuzzy set on 𝑋2Γ—(0,∞) satisfying the following conditions, for all π‘₯,𝑦,π‘§βˆˆπ‘‹,𝑑,𝑠>0:(i)𝑀(π‘₯,𝑦,𝑑)>0;(ii)𝑀(π‘₯,𝑦,𝑑)=1 if and only if π‘₯=𝑦;(iii)𝑀(π‘₯,𝑦,𝑑)=𝑀(𝑦,π‘₯,𝑑);(iv)𝑀(π‘₯,𝑧,𝑑+𝑠)β‰₯𝑀(π‘₯,𝑦,𝑑)βˆ—π‘€(𝑧,𝑦,𝑠);(v)𝑀(π‘₯,𝑦,βˆ’)∢(0,∞)β†’[0,1] is continuous.
If (𝑋,𝑀,βˆ—) is a fuzzy metric space, it will be said that (𝑀,βˆ—) is a fuzzy metric on 𝑋.

A simply but useful fact [7] is that 𝑀(π‘₯,𝑦,βˆ’) is nondecreasing for all π‘₯,π‘¦βˆˆπ‘‹. Let (𝑋,𝑑) be a metric space. Denote by π‘Žβ‹…π‘ the usual multiplication for all π‘Ž,π‘βˆˆ[0,1], and let 𝑀𝑑 be the fuzzy set defined on 𝑋×𝑋×(0,∞) by𝑀𝑑(𝑑π‘₯,𝑦,𝑑)=.𝑑+𝑑(π‘₯,𝑦)(2.1) Then, (𝑋,𝑀𝑑,β‹…) is a fuzzy metric space, and (𝑀𝑑,β‹…) is called the standard fuzzy metric induced by 𝑑 [8].

George and Veeramani [6] proved that every fuzzy metric (𝑀,βˆ—) on 𝑋 generates a topology πœπ‘€ on 𝑋 which has a base the family of open sets of the form:𝐡𝑀(π‘₯,πœ€,𝑑)∢π‘₯βˆˆπ‘‹,0<πœ€<1,𝑑>0,(2.2) where 𝐡𝑀(π‘₯,πœ€,𝑑)={π‘¦βˆˆπ‘‹βˆΆπ‘€(π‘₯,𝑦,𝑑)>1βˆ’πœ€} for all πœ€βˆˆ(0,1) and 𝑑>0. They proved that (𝑋,πœπ‘€) is a Hausdorff first countable topological space. Moreover, if (𝑋,𝑑) is a metric space, then the topology generated by 𝑑 coincides with the topology πœπ‘€π‘‘ generated by the induced fuzzy metric (𝑀𝑑,βˆ—) (see [8]).

Lemma 2.2 (see [6]). Let (𝑋,𝑀,βˆ—) be a fuzzy metric space and let 𝜏 be the topology induced by the fuzzy metric. Then, for a sequence {π‘₯𝑛}π‘›βˆˆβ„• in 𝑋, π‘₯𝑛→π‘₯ if and only if 𝑀(π‘₯𝑛,π‘₯,𝑑)β†’1 as π‘›β†’βˆž for all 𝑑>0.

Definition 2.3 (see [6]). A sequence {π‘₯𝑛}π‘›βˆˆβ„• in a fuzzy metric space (𝑋,𝑀,βˆ—) is called a Cauchy sequence if and only if for each 1>πœ€>0,𝑑>0, there exists 𝑛0βˆˆβ„• such that 𝑀(π‘₯𝑛,π‘₯π‘š,𝑑)>1βˆ’πœ€ for all 𝑛,π‘šβ‰₯𝑛0. A fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

Definition 2.4 (see [13]). Let 𝐴 be a nonempty subset of a fuzzy metric space (𝑋,𝑀,βˆ—). For π‘Žβˆˆπ‘‹ and 𝑑>0, 𝑀(π‘Ž,𝐴,𝑑)=sup{𝑀(π‘Ž,𝑦,𝑑)βˆ£π‘¦βˆˆπ΄,𝑑>0}.

Lemma 2.5 (see [28]). Let 𝐺 be a set and let {πΊπ›ΌβˆΆπ›Όβˆˆ[0,1]} be a family of subsets of 𝐺 such that(1)𝐺0=𝐺;(2)𝛼≀𝛽 implies πΊπ›½βŠ†πΊπ›Ό;(3)𝛼1≀𝛼2≀⋯, limπ‘›β†’βˆžπ›Όπ‘›=𝛼 implies 𝐺𝛼=β‹‚βˆžπ‘›=1𝐺𝛼𝑛.Then, the function πœ‘βˆΆπΊβ†’[0,1] defined by πœ‘(π‘₯)=sup{π›Όβˆˆ[0,1]∢π‘₯βˆˆπΊπ›Ό} has the property that {π‘₯βˆˆπΊβˆΆπœ‘(π‘₯)β‰₯𝛼}=𝐺𝛼  for every π›Όβˆˆ[0,1].

Next, we recall some pertinent concepts on Hausdorff fuzzy metric. Denote by 𝐢0(𝑋) the set of nonempty closed and bounded subsets of a metric space (𝑋,𝑑). It is well known (see, e.g., [29]) that the function 𝐻𝑑 defined on 𝐢0(𝑋)×𝐢0(𝑋) by𝐻𝑑(𝐴,𝐡)=maxsupπ‘Žβˆˆπ΄π‘‘(π‘Ž,𝐡),supπ‘βˆˆπ΅ξ‚Ό,𝑑(𝐴,𝑏)(2.3) for all 𝐴,𝐡∈𝐢0(𝑋), is a metric on 𝐢0(𝑋) called the Hausdorff-Pompeiu metric. In [30], it is proved that the metric (𝐢(𝑋),𝐻) is complete provided 𝑋 is complete.

Let 𝐢(𝑋) be the set of all nonempty compact subsets of a fuzzy metric space (𝑋,𝑀,βˆ—), 𝐴,𝐡∈𝐢(𝑋), 𝑑>0, according to [26], the Hausdorff fuzzy metric 𝐻𝑀 on 𝐢(𝑋)×𝐢(𝑋)Γ—(0,∞) is defined as𝐻𝑀(𝐴,𝐡,𝑑)=mininfπ‘Žβˆˆπ΄π‘€(π‘Ž,𝐡,𝑑),infπ‘βˆˆπ΅ξ‚Όπ‘€(𝐴,𝑏,𝑑)=min{𝜌(𝐴,𝐡,𝑑),𝜌(𝐡,𝐴,𝑑)},(2.4) where 𝜌(𝐴,𝐡,𝑑)=infπ‘Žβˆˆπ΄π‘€(π‘Ž,𝐡,𝑑), and (𝐻𝑀,βˆ—) is a fuzzy metric on 𝐢(𝑋). It is shown that 𝜌(𝐴,𝐡,𝑑)=1 if and only if π΄βŠ†π΅, and 𝐻𝑀(𝐴,𝐡,𝑑)=1 if and only if 𝐴=𝐡.

Lemma 2.6 (see [26]). Let (𝑋,𝑀,βˆ—) be a fuzzy metric space. Then, (𝐢(𝑋),𝐻𝑀,βˆ—) is complete if and only if (𝑋,𝑀,βˆ—) is complete.

Lemma 2.7 (see [26]). Let (𝑋,𝑑) be a metric space. Then, the Hausdorff fuzzy metric (𝐻𝑀𝑑,β‹…) of the standard fuzzy metric (𝑀𝑑,β‹…) coincides with standard fuzzy metric (𝑀𝐻𝑑,β‹…) of the Hausdorff metric 𝐻𝑑 on 𝐢(𝑋).

3. On π‘€βˆž-Fuzzy Metric

Let (𝑋,𝑀,βˆ—) be a fuzzy metric space. Denote by π’ž(𝑋) the totality of fuzzy sets:[]πœ‡βˆΆπ‘‹βŸΆ0,1=𝐼,(3.1) which satisfy that, for each πœ†βˆˆπΌ, the πœ†-cut of πœ‡,[πœ‡]πœ†={π‘₯βˆˆπ‘‹βˆΆπœ‡(π‘₯)β‰₯πœ†},(3.2) is nonempty compact in 𝑋.

Definition 3.1. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space. The π‘€βˆž-fuzzy metric between two fuzzy sets is induced by the Hausdorff fuzzy metric 𝐻𝑀 as π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έξ€½πœŒ,𝑑=minβˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑,πœŒβˆžξ€·πœ‡2,πœ‡1,𝑑,(3.3) where πœ‡1,πœ‡2βˆˆπ’ž(𝑋),𝑑>0, and πœŒβˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑=inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έ,𝑑(3.4) is the fuzzy separation of πœ‡1 from πœ‡2.

Lemma 3.2. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space, πœ‡1,πœ‡2,πœ‡3βˆˆπ’ž(𝑋), 𝑠,𝑑>0. Then one has(1)π‘€βˆž(πœ‡1,πœ‡2,𝑑)∈(0,1],(2)π‘€βˆž(πœ‡1,πœ‡2,𝑑)=π‘€βˆž(πœ‡2,πœ‡1,𝑑),(3)𝜌∞(πœ‡1,πœ‡2,𝑑)=1 if and only if πœ‡1βŠ†πœ‡2,(4)π‘€βˆž(πœ‡1,πœ‡2,𝑑)=1 if and only if πœ‡1=πœ‡2,(5)if πœ‡1βŠ†πœ‡2, then 𝜌∞(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡2,πœ‡3,𝑑),(6)𝜌∞(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡1,πœ‡2,𝑑)βˆ—πœŒβˆž(πœ‡2,πœ‡3,𝑠),(7)π‘€βˆž(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡1,πœ‡2,𝑑)βˆ—π‘€βˆž(πœ‡2,πœ‡3,𝑠), (8)π‘€βˆž(πœ‡1,πœ‡2,βˆ’)∢(0,∞)β†’[0,1] is continuous.

Proof. For (1), by the definition of the πœ†-cut [πœ‡1]πœ†, for every πœ†βˆˆπΌ, [πœ‡1]πœ† is nonempty compact in 𝑋. By the theorem of nested intervals, there exists a point π‘Ž0 in [πœ‡1]πœ† for every πœ†βˆˆπΌ, likewise, there exists a points 𝑏0 in [πœ‡2]πœ† for every πœ†βˆˆπΌ. Thus, π‘€βˆž(πœ‡1,πœ‡2,𝑑)>0. Moreover, it is clear that 𝐴=𝐡⇔𝐻𝑀(𝐴,𝐡,𝑑)=1β‡”π‘€βˆž(πœ‡1,πœ‡2,𝑑)=1.
For (2), it is clear that π‘€βˆž(πœ‡1,πœ‡2,𝑑)=π‘€βˆž(πœ‡2,πœ‡1,𝑑).
For (3), since 𝜌∞(πœ‡1,πœ‡2,𝑑)=1 if and only if 𝜌([πœ‡1]πœ†,[πœ‡2]πœ†,𝑑)=1 for all πœ†βˆˆπΌ, which implies [πœ‡1]πœ†βŠ†[πœ‡2]πœ† for all πœ†βˆˆπΌ, we have that 𝜌∞(πœ‡1,πœ‡2,𝑑)=1 if and only if πœ‡1βŠ†πœ‡2.
For (4), it follows from (3).
For (5), for every πœ†βˆˆπΌ, any π‘₯∈[πœ‡1]πœ†, π‘¦βˆˆ[πœ‡2]πœ† and π‘§βˆˆ[πœ‡3]πœ†, by the proof of Theorem  1 in [26], we have 𝑀(π‘₯,𝑧,𝑑+𝑠)β‰₯𝑀(π‘₯,𝑦,𝑑)βˆ—π‘€(𝑦,𝑧,𝑠)(3.5) with all π‘₯,𝑦,π‘§βˆˆπ‘‹, which implies π‘€ξ€·ξ€Ίπœ‡π‘₯,3ξ€»πœ†ξ€Έξ€·ξ€Ίπœ‡,𝑑+𝑠β‰₯𝑀(π‘₯,𝑦,𝑑)βˆ—π‘€π‘¦,3ξ€»πœ†ξ€Έ,𝑠(3.6) for all π‘₯∈[πœ‡1]πœ† and all π‘¦βˆˆ[πœ‡2]πœ†. Since πœ‡1βŠ†πœ‡2, then 𝜌([πœ‡1]πœ†,[πœ‡2]πœ†,𝑠)=1. By (iv) of Definition 2.1 and the arbitrariness of π‘₯ and 𝑦, we have πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑+𝑠=infπ‘₯∈[πœ‡1]πœ†π‘€ξ€·ξ€Ίπœ‡π‘₯,3ξ€»πœ†ξ€Έ,𝑑+𝑠β‰₯infπ‘₯∈[πœ‡1]πœ†π‘€ξ€·ξ€Ίπœ‡π‘₯,2ξ€»πœ†ξ€Έ,π‘ βˆ—infπ‘¦βˆˆ[πœ‡2]πœ†π‘€ξ€·ξ€Ίπœ‡π‘¦,3ξ€»πœ†ξ€Έπœ‡,𝑑=πœŒξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έπœ‡,π‘ βˆ—πœŒξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έπœ‡,𝑑=πœŒξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑β‰₯π»π‘€πœ‡ξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,,𝑑(3.7) which implies inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑+𝑠β‰₯inf0β‰€πœ†β‰€1π»π‘€πœ‡ξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑.(3.8) Consequently, 𝜌∞(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡2,πœ‡3,𝑑).
For (6), for every πœ†βˆˆπΌ, by the proof of (5) and (iv) of Definition 2.1, we have inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑+𝑠β‰₯inf0β‰€πœ†β‰€1ξ€½πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έπœ‡,π‘‘βˆ—πœŒξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†,𝑠β‰₯inf0β‰€πœ†β‰€1ξ€½π»π‘€πœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έπœ‡,π‘‘βˆ—πœŒξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†.,𝑠(3.9) Consequently, 𝜌∞(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡1,πœ‡2,𝑑)βˆ—πœŒβˆž(πœ‡2,πœ‡3,𝑠).
For (7), for every πœ†βˆˆπΌ, by the proof of (6), we have inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έ,𝑑+𝑠β‰₯inf0β‰€πœ†β‰€1ξ€½πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έπœ‡,π‘‘βˆ—πœŒξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†β‰₯ξ‚»,𝑠inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έξ‚Όβˆ—ξ‚»,𝑑inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡3ξ€»πœ†ξ€Έξ‚Ό.,𝑠(3.10) Similarly, it can be shown that inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί3ξ€»πœ†,ξ€Ίπœ‡1ξ€»πœ†ξ€Έβ‰₯ξ‚»,𝑑+𝑠inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί3ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έξ‚Όβˆ—ξ‚»,𝑠inf0β‰€πœ†β‰€1πœŒπœ‡ξ€·ξ€Ί2ξ€»πœ†,ξ€Ίπœ‡1ξ€»πœ†ξ€Έξ‚Ό,𝑑.(3.11) Hence, π‘€βˆž(πœ‡1,πœ‡3,𝑑+𝑠)β‰₯π‘€βˆž(πœ‡1,πœ‡2,𝑑)βˆ—π‘€βˆž(πœ‡2,πœ‡3,𝑠).
For (8), by the continuity on (0,∞) of the function 𝑑↦𝐻𝑀(𝐴,𝐡,𝑑), it is clear that π‘€βˆž(πœ‡1,πœ‡2,βˆ’)∢(0,∞)β†’[0,1] is continuous.

Theorem 3.3. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space. Then, (π’ž(𝑋),π‘€βˆž,βˆ—) is a fuzzy metric space, where π‘€βˆž is a fuzzy set on the π’ž(𝑋)Γ—π’ž(𝑋)Γ—(0,+∞).

Proof. It is easily proved by Lemma 3.2.

Example 3.4. Let 𝑑 be the Euclidean metric on ℝ, and let 𝐴=[π‘Ž1,π‘Ž2] and let 𝐡=[𝑏1,𝑏2] be two compact intervals. Then, 𝐻𝑑(𝐴,𝐡)=max{|π‘Ž1βˆ’π‘1|,|π‘Ž2βˆ’π‘2|}. Let (ℝ,𝑀𝑑,βˆ—) be a fuzzy metric space, where π‘Žβˆ—π‘ the usual multiplication for all π‘Ž,π‘βˆˆ[0,1], and 𝑀𝑑 is defined on ℝ×ℝ×(0,∞) by 𝑀𝑑(𝑑π‘₯,𝑦,𝑑)=𝑑+𝑑(π‘₯,𝑦).(3.12) Denote by π’ž(ℝ) the totality of fuzzy sets πœ‡βˆΆβ„β†’[0,1] which satisfy that for each πœ†βˆˆπΌ, the πœ†-cut of πœ‡[πœ‡]πœ†={π‘₯βˆˆβ„βˆΆπœ‡(π‘₯)β‰₯πœ†} is a nonempty compact interval. For any πœ†-cuts of fuzzy sets πœ‡1,πœ‡2βˆˆπ’ž(ℝ) and for all 𝑑>0, by a simple calculation, we have π»π‘€πœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έ=𝑑,𝑑𝑑+π»π‘‘πœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έ.(3.13) So by Definition 3.1, we get π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑=inf0β‰€πœ†β‰€1𝑑𝑑+π»π‘‘πœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έ.(3.14)

4. Properties of the π‘€βˆž-Fuzzy Metric

Definition 4.1. Let (π’ž(𝑋),π‘€βˆž,βˆ—) be a fuzzy metric space. For π‘‘βˆˆ(0,+∞), define 𝐡(πœ‡,π‘Ÿ,𝑑) with center a fuzzy set πœ‡βˆˆπ’ž(𝑋) and radius π‘Ÿ,0<π‘Ÿ<1,𝑑>0 as 𝐡(πœ‡,π‘Ÿ,𝑑)=π›Ύβˆˆπ’ž(𝑋)βˆ£π‘€βˆžξ€Ύ(πœ‡,𝛾,𝑑)>1βˆ’π‘Ÿ.(4.1)

Proposition 4.2. Every 𝐡(πœ‡,π‘Ÿ,𝑑) is an open set.

Proof. It is identical with the proof in [6].

Proposition 4.3. Let (π’ž(𝑋),π‘€βˆž,βˆ—) be a fuzzy metric space. Define πœπ‘€βˆž={π’œβŠ‚π’ž(𝑋)βˆ£πœ‡βˆˆπ’œifandonlyifthereexist𝑑>0andπ‘Ÿ,0<π‘Ÿ<1suchthat𝐡(πœ‡,π‘Ÿ,𝑑)βŠ‚π’œ}.
Then, πœπ‘€βˆž is a topology on π’ž(𝑋).

Proof. It is identical with the proof in [6].

Definition 4.4. A sequence {πœ‡π‘›} in a fuzzy metric space (π’ž(𝑋),π‘€βˆž,βˆ—) is a Cauchy sequence if and only if for each πœ€>0, 𝑑>0, there exists 𝑛0βˆˆβ„• such that π‘€βˆž(πœ‡π‘›,πœ‡π‘š,𝑑)>1βˆ’πœ€ for all 𝑛,π‘šβ‰₯𝑛0.

Lemma 4.5. Let (π’ž(𝑋),π‘€βˆž,βˆ—) be a fuzzy metric space on fuzzy metric π‘€βˆž and let 𝜏 be the topology induced by the fuzzy metric π‘€βˆž. Then, for a sequence {πœ‡π‘›} in π’ž(𝑋), πœ‡π‘›β†’πœ‡ if and only if π‘€βˆž(πœ‡,πœ‡π‘›,𝑑)β†’1 as π‘›β†’βˆž.

Proof. It is identical with the proof of Theorem  3.11 in [6].

Theorem 4.6. The fuzzy metric space (π’ž(𝑋),π‘€βˆž,βˆ—) is complete provided (𝑋,𝑀,βˆ—) is complete.

Proof. Let (𝑋,𝑀,βˆ—) be a complete fuzzy metric space and let a sequence {πœ‡π‘›,𝑛β‰₯1} be a Cauchy sequence in π’ž(𝑋). Consider a fixed 0<πœ†<1. Then, {[πœ‡π‘›]πœ†,𝑛β‰₯1} is a Cauchy sequence in (𝐢(𝑋),𝐻𝑀,βˆ—), where 𝐢(𝑋) denotes all nonempty compact subsets of (𝑋,𝑀,βˆ—).
Since (𝐢(𝑋),𝐻𝑀,βˆ—) is complete by Lemma 2.6, it follows that [πœ‡π‘›]πœ†β†’πœ‡πœ†βˆˆπΆ(𝑋). Actually, from the definition of π‘€βˆž and the continuity of 𝐻𝑀, it is easy to see that [πœ‡π‘›]πœ†β†’πœ‡πœ†, uniformly in πœ†βˆˆ[0,1].
Now, consider the family {πœ‡πœ†βˆΆπœ†βˆˆ[0,1]}, where πœ‡0=𝑋. Take πœ†β‰€π›½, we have πœŒξ€·πœ‡π›½,πœ‡πœ†ξ€Έξ‚€πœ‡,𝑑β‰₯πœŒπ›½,ξ€Ίπœ‡π‘›ξ€»π›½,𝑑3ξ‚ξ‚€ξ€Ίπœ‡βˆ—πœŒπ‘›ξ€»π›½,ξ€Ίπœ‡π‘›ξ€»πœ†,𝑑3ξ‚ξ‚€ξ€Ίπœ‡βˆ—πœŒπ‘›ξ€»πœ†,πœ‡πœ†,𝑑3.(4.2) Since [πœ‡π‘›]π›½βŠ†[πœ‡π‘›]πœ†, it follows that 𝜌([πœ‡π‘›]𝛽,[πœ‡π‘›]πœ†,𝑑/3)=1. Thus, for each 0<πœ€<1, 𝜌(πœ‡π›½,πœ‡πœ†,𝑑)β‰₯𝜌(πœ‡π›½,[πœ‡π‘›]𝛽,𝑑/3)βˆ—πœŒ([πœ‡π‘›]πœ†,πœ‡πœ†,𝑑/3) if 𝑛 is large enough. Hence, 𝜌(πœ‡π›½,πœ‡πœ†,𝑑)=1, and by Lemma 3.2, we have πœ‡π›½βŠ†πœ‡πœ†.
Now, take πœ†π‘›β†‘ and limπ‘›β†’βˆžπœ†π‘›=πœ†. We have to show that πœ‡πœ†=β‹‚βˆžπ‘›=1πœ‡πœ†π‘›. It is clear that πœ‡πœ†βŠ†βˆžξ™π‘›=1πœ‡πœ†π‘›.(βˆ—) On the other hand, we have πœŒξƒ©βˆžξ™π‘›=1πœ‡πœ†π‘›,πœ‡πœ†ξƒͺ,𝑑β‰₯πœŒβˆžξ™π‘›=1πœ‡πœ†π‘›,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑3ξƒͺξƒ©βˆ—πœŒβˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,ξ€Ίπœ‡π‘—ξ€»πœ†,𝑑3ξƒͺξ‚€ξ€Ίπœ‡βˆ—πœŒπ‘—ξ€»πœ†,πœ‡πœ†,𝑑3,(4.3) for fixed 𝑗. However, πœŒξƒ©βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,ξ€Ίπœ‡π‘—ξ€»πœ†,𝑑3ξƒͺ=1.(4.4) Consequently, for every 0<πœ€<1, there exists 0<πœ€0<πœ€<1 such that (1βˆ’πœ€0)βˆ—(1βˆ’πœ€0)βˆ—(1βˆ’πœ€0)>1βˆ’πœ€. For given πœ€0, since [πœ‡π‘—]πœ†β†’πœ‡πœ†, there exists π‘—πœ€0 such that πœŒξƒ©βˆžξ™π‘›=1πœ‡πœ†π‘›,πœ‡πœ†ξƒͺ,𝑑β‰₯πœŒβˆžξ™π‘›=1πœ‡πœ†π‘›,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑3ξƒͺβˆ—ξ€·1βˆ’πœ€0ξ€Έ,(4.5) for 𝑗β‰₯π‘—πœ€0. Now, πœŒξƒ©βˆžξ™π‘›=1πœ‡πœ†π‘›,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑3ξƒͺβ‰₯πœŒβˆžξ™π‘›=1πœ‡πœ†π‘›,πœ‡πœ†π‘,𝑑9ξƒͺξ‚€πœ‡βˆ—πœŒπœ†π‘,ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑9ξ‚ξƒ©ξ€Ίπœ‡βˆ—πœŒπ‘—ξ€»πœ†π‘,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑9ξƒͺ,(4.6) for any 𝑝β‰₯1. Since β‹‚βˆžπ‘›=1πœ‡πœ†π‘›βŠ†πœ‡πœ†π‘, we obtain πœŒξƒ©βˆžξ™π‘›=1πœ‡πœ†π‘›,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›ξƒͺξ‚€πœ‡,𝑑β‰₯πœŒπœ†π‘,ξ€Ίπœ‡π‘—ξ€»πœ†π‘,𝑑2ξ‚ξƒ©ξ€Ίπœ‡βˆ—πœŒπ‘—ξ€»πœ†π‘,βˆžξ™π‘›=1ξ€Ίπœ‡π‘—ξ€»πœ†π‘›,𝑑2ξƒͺ.(4.7) Now, 𝜌(πœ‡πœ†π‘,[πœ‡π‘—]πœ†π‘,𝑑/2)>1βˆ’πœ€0 for 𝑗β‰₯𝑗0 and all 𝑑>0. Note that (since the convergence [πœ‡π‘—]πœ†β†’πœ‡πœ† is uniform in πœ†) 𝑗0 does not depend on 𝑝. Since {[πœ‡π‘—]πœ†π‘,𝑝β‰₯1} decreases to β‹‚βˆžπ‘›=1[πœ‡π‘—]πœ†π‘›, if follows that 𝜌([πœ‡π‘—]πœ†π‘0,β‹‚βˆžπ‘›=1[πœ‡π‘—]πœ†π‘›,𝑑/2)>1βˆ’πœ€0 for some 𝑝0 (depending on 𝑗).
Thus, β‹‚πœŒ(βˆžπ‘›=1πœ‡πœ†π‘›,β‹‚βˆžπ‘›=1[πœ‡π‘—]πœ†π‘›,𝑑/3)β‰₯(1βˆ’πœ€0)βˆ—(1βˆ’πœ€0), if 𝑗 is large.
Finally, by taking 𝑗 large enough, we obtain πœŒξƒ©βˆžξ™π‘›=1πœ‡πœ†π‘›,πœ‡πœ†ξƒͺβ‰₯ξ€·,𝑑1βˆ’πœ€0ξ€Έβˆ—ξ€·1βˆ’πœ€0ξ€Έβˆ—ξ€·1βˆ’πœ€0ξ€Έβ‰₯1βˆ’πœ€,(4.8) that is, βˆžξ™π‘›=1πœ‡πœ†π‘›βŠ†πœ‡πœ†.(βˆ—βˆ—) From (4.3) and (4.9), it yields β‹‚βˆžπ‘›=1πœ‡πœ†π‘›=πœ‡πœ†. Thus, Lemma 2.5 is applicable and there exists πœ‡βˆˆπ’ž(𝑋) for every πœ†βˆˆ[0,1] such that [πœ‡π‘›]πœ†β†’πœ‡πœ†. It remains to show that πœ‡π‘›β†’πœ‡ in (π’ž(𝑋),π‘€βˆž,βˆ—).
Let πœ€>0. Then, since {πœ‡π‘›} is a Cauchy sequence, there exists π‘›πœ€ such that 𝑛,π‘š>π‘›πœ€ implies π‘€βˆž(πœ‡π‘›,πœ‡π‘š,𝑑)>1βˆ’πœ€.
Let 𝑛(>π‘›πœ€) be fixed. Then, π»π‘€πœ‡ξ€·ξ€Ίπ‘›ξ€»πœ†,[πœ‡]πœ†ξ€Έ,𝑑=limπ‘šβ†’βˆžπ»π‘€πœ‡ξ€·ξ€Ίπ‘›ξ€»πœ†,ξ€Ίπœ‡π‘šξ€»πœ†ξ€Έβ‰₯,𝑑limπ‘šβ†’βˆžinf0β‰€πœ†β‰€1π»π‘€πœ‡ξ€·ξ€Ίπ‘›ξ€»πœ†,ξ€Ίπœ‡π‘šξ€»πœ†ξ€Έ=,𝑑limπ‘šβ†’βˆžπ‘€βˆžξ€·πœ‡π‘›,πœ‡π‘šξ€Έ,𝑑>1βˆ’πœ€.(4.9) Thus, πœ‡π‘›β†’πœ‡ in the π‘€βˆž-fuzzy metric. The proof is completed.

Lemma 4.7. Let (𝑋,𝑀,βˆ—) be a compact fuzzy metric space and compact subsets 𝐴,𝐡∈𝐢(𝑋). Then, for each π‘₯∈𝐴 and 𝑑>0, there exists a π‘¦βˆˆπ΅ such that 𝑀(π‘₯,𝑦,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑).

Proof. Suppose there exists a π‘₯0∈𝐴 such that 𝑀(π‘₯0,𝑦,𝑑)<𝐻𝑀(𝐴,𝐡,𝑑) for any π‘¦βˆˆπ΅ and 𝑑>0. Then, supπ‘¦βˆˆπ΅π‘€ξ€·π‘₯0ξ€Έ,𝑦,𝑑<𝐻𝑀(𝐴,𝐡,𝑑),(4.10) that is, supπ‘¦βˆˆπ΅π‘€ξ€·π‘₯0ξ€Έξƒ―,𝑦,𝑑<mininfπ‘₯∈𝐡supπ‘¦βˆˆπ΄π‘€(π‘₯,𝑦,𝑑),infπ‘₯∈𝐴supπ‘¦βˆˆπ΅ξƒ°π‘€(π‘₯,𝑦,𝑑).(4.11) So, supπ‘¦βˆˆπ΅π‘€ξ€·π‘₯0ξ€Έ,𝑦,𝑑<infπ‘₯∈𝐴supπ‘¦βˆˆπ΅π‘€(π‘₯,𝑦,𝑑).(4.12) This is a contradiction with π‘₯∈𝐴.

Lemma 4.8. Let (𝑋,𝑀,βˆ—) be a compact fuzzy metric space, 𝑑>0 and 𝐴,𝐡∈𝐢(𝑋). Then, for any compact set 𝐴1βŠ†π΄, there exists a compact set 𝐡1βŠ†π΅ such that 𝐻𝑀(𝐴1,𝐡1,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑).

Proof. Let 𝐢={π‘¦βˆΆ there exists a π‘₯∈𝐴1 such that 𝑀(π‘₯,𝑦,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑)} and let 𝐡1⋂𝐡=𝐢. For any π‘₯∈𝐴1βŠ†π΄,𝑑>0, by Lemma 4.7, there exists a π‘¦βˆˆπ΅ such that 𝑀(π‘₯,𝑦,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑).(4.13) Thus, 𝐡1β‰ βˆ…, moreover, 𝐡1 is compact since it is closed in 𝑋 and 𝐡1βŠ†π΅.
Now, for any π‘₯∈𝐴1, 𝑑>0, there exists a π‘¦βˆˆπ΅1 such that 𝑀(π‘₯,𝑦,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑).(4.14) Thus, we have 𝑀(π‘₯,𝐡1,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑), which implies that πœŒξ€·π΄1,𝐡1ξ€Έ,𝑑=infπ‘₯∈𝐴1𝑀π‘₯,𝐡1ξ€Έ,𝑑β‰₯𝐻𝑀(𝐴,𝐡,𝑑).(4.15) Similarly, it can be shown that 𝜌(𝐴1,𝐡1,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑).
Hence, 𝐻𝑀(𝐴1,𝐡1,𝑑)β‰₯𝐻𝑀(𝐴,𝐡,𝑑). This completes the proof.

Theorem 4.9. Let (𝑋,𝑀,βˆ—) be a compact fuzzy metric space and πœ‡1,πœ‡2βˆˆπ’ž(𝑋), 𝑑>0. Then, for any πœ‡3βˆˆπ’ž(𝑋) satisfying πœ‡3βŠ†πœ‡1, there exists a πœ‡4βˆˆπ’ž(𝑋) such that πœ‡4βŠ†πœ‡2 and π‘€βˆžξ€·πœ‡3,πœ‡4ξ€Έ,𝑑β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑.(4.16)

Proof. Since πœ‡1,πœ‡2, and πœ‡3 are normal, we have βˆ…β‰ [πœ‡3]πœ†βŠ†[πœ‡1]πœ† and βˆ…β‰ [πœ‡2]πœ† for all πœ†βˆˆπΌ. Let πΆπœ†=ξ€½ξ€Ίπœ‡π‘¦βˆΆthereexistsaπ‘₯∈3ξ€»πœ†suchthat𝑀(π‘₯,𝑦,𝑑)β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2,𝑑,(4.17) and let π΅πœ†=πΆπœ†β‹‚[πœ‡2]πœ†. For any π‘₯∈[πœ‡3]πœ†βŠ†[πœ‡1]πœ†, by Lemma 4.7, there exists a π‘¦βˆˆ[πœ‡2]πœ† such that 𝑀(π‘₯,𝑦,𝑑)β‰₯π»π‘€πœ‡ξ€·ξ€Ί1ξ€»πœ†,ξ€Ίπœ‡2ξ€»πœ†ξ€Έ,𝑑β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ.,𝑑(4.18) Thus, π΅πœ† is nonempty compact in 𝑋, moreover, π΅πœ†βŠ†π΅π›Ύ if 0β‰€π›Ύβ‰€πœ†β‰€1.
From the proof of Lemma 4.8, we have π»π‘€πœ‡ξ€·ξ€Ί3ξ€»πœ†,π΅πœ†ξ€Έ,𝑑β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ.,𝑑(4.19) By Lemma  3.1 in [28], there exists a fuzzy set πœ‡4 with the property that [πœ‡4]πœ†=π΅πœ† for πœ†βˆˆπΌ. Since π΅πœ† are nonempty compact for all πœ†βˆˆπΌ, we have πœ‡4βˆˆπ’ž(𝑋). Consequently, π‘€βˆžξ€·πœ‡3,πœ‡4ξ€Έ,𝑑β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑.(4.20) This completes the proof.

Definition 4.10 (see [24]). Let 𝑋,π‘Œ be any fuzzy metric space. β„± is said to be a fuzzy mapping if and only if β„± is a mapping from the space π’ž(𝑋) into π’ž(π‘Œ), that is, β„±(πœ‡)βˆˆπ’ž(π‘Œ) for each πœ‡βˆˆπ’ž(𝑋).

5. Common Fixed Point Theorems in the Fuzzy Metric Space on Fuzzy Sets

Theorem 5.1. Let (𝑋,𝑀,βˆ—) be a compact fuzzy metric space and let {ℱ𝑖}βˆžπ‘–=1 be a sequence of fuzzy self-mappings of π’ž(𝑋). Let πœ™βˆΆ[0,1]β†’[0,1] be a nondecreasing function satisfying the following condition: πœ™ is continuous from the left and πœ™(β„Ž)βˆ—πœ™2(β„Ž)βˆ—β‹―βˆ—πœ™π‘›](β„Ž)⟢1asπ‘›βŸΆβˆž,βˆ€β„Žβˆˆ(0,1,(5.1) where πœ™π‘› denote the 𝑛th iterative function of πœ™. Suppose that for each πœ‡1,πœ‡2βˆˆπ’ž(𝑋), and for arbitrary positive integers 𝑖 and 𝑗,𝑖≠𝑗,𝑑>0, π‘€βˆžξ€·β„±π‘–ξ€·πœ‡1ξ€Έ,β„±π‘—ξ€·πœ‡2𝑀,𝑑β‰₯πœ™infβˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑,πœŒβˆžξ€·πœ‡1,β„±π‘–ξ€·πœ‡1ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡2,β„±π‘—ξ€·πœ‡2ξ€Έξ€Έ,1,2𝑑2ξ€ΊπœŒβˆžξ€·πœ‡2,β„±π‘–ξ€·πœ‡1ξ€Έξ€Έ,4𝑑+πœŒβˆžξ€·πœ‡1,β„±π‘—ξ€·πœ‡2ξ€Έ,,4𝑑(5.2) then there exists πœ‡βˆ—βˆˆπ’ž(𝑋) such that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—) for all π‘–βˆˆπ‘+.

Proof. Let πœ‡0βˆˆπ’ž(𝑋) and πœ‡1βŠ†β„±1(πœ‡0). By Theorem 4.9, for any 𝑑>0, there exists πœ‡2βˆˆπ’ž(𝑋) such that πœ‡2βŠ†β„±2(πœ‡1) and π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±1ξ€·πœ‡0ξ€Έ,β„±2ξ€·πœ‡1ξ€Έξ€Έ,𝑑.(5.3) Again by Theorem 4.9, for any 𝑑>0, we can find πœ‡3βˆˆπ’ž(𝑋) such that πœ‡3βŠ†β„±3(πœ‡2) and π‘€βˆžξ€·πœ‡2,πœ‡3ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±2ξ€·πœ‡1ξ€Έ,β„±3ξ€·πœ‡2ξ€Έξ€Έ,𝑑.(5.4) By induction, we produce a sequence {πœ‡π‘›} of points of π’ž(𝑋) such that πœ‡π‘›+1βŠ†β„±π‘›+1ξ€·πœ‡π‘›ξ€Έπ‘€,𝑛=0,1,2,…;βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έ,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έ.,𝑑(5.5) Now, we prove that {πœ‡π‘›} is a Cauchy sequence in π’ž(𝑋). In fact, for arbitrary positive integer 𝑛, by the inequality (5.2), Lemma 3.2, and the formula (5.5), we have π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έ,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έξ‚€ξ‚†π‘€,𝑑β‰₯πœ™infβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,πœŒβˆžξ€·πœ‡π‘›βˆ’1,β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡π‘›,ℱ𝑛+1ξ€·πœ‡2ξ€Έξ€Έ,1,2𝑑2ξ€ΊπœŒβˆžξ€·πœ‡π‘›βˆ’1,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έ,4𝑑+πœŒβˆžξ€·πœ‡π‘›,β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έ,𝑀,4𝑑β‰₯πœ™infβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,1,2𝑑2ξ€Ίπ‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›+1ξ€Έξ€»,𝑀,2𝑑+1β‰₯πœ™infβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,1,2𝑑2ξ€Ίπ‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑+1βˆ—π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έξ€»,,𝑑+1(5.6) where πœ‡π‘›βŠ†β„±π‘›(πœ‡π‘›βˆ’1) implies 𝜌∞(πœ‡π‘›,ℱ𝑛(πœ‡π‘›βˆ’1),2𝑑)=1, by (3) of Lemma 3.2. In addition, it is easy to get that πœ™(β„Ž)>β„Ž for all β„Žβˆˆ(0,1). In fact, suppose that there exists some 𝑑0∈(0,1) such that πœ™(β„Ž0)β‰€β„Ž0. Since πœ™ is nondecreasing, we have πœ™π‘›ξ€·β„Ž0ξ€Έβ‰€πœ™π‘›βˆ’1ξ€·β„Ž0ξ€Έξ€·β„Žβ‰€β‹―β‰€πœ™0ξ€Έβ‰€β„Ž0.(5.7) Since πœ™(β„Ž)βˆ—πœ™2(β„Ž)βˆ—β‹―βˆ—πœ™π‘›(β„Ž)β†’1 as π‘›β†’βˆž, for all β„Žβˆˆ(0,1), then we have πœ™π‘›(β„Ž0)β†’1 as π‘›β†’βˆž. From the inequality (5.7), we have 1β‰€β„Ž0. This is a contradiction which implies πœ™(β„Ž)>β„Ž for all β„Žβˆˆ(0,1). We can prove that π‘€βˆž(πœ‡π‘›βˆ’1,πœ‡π‘›,𝑑)β‰€π‘€βˆž(πœ‡π‘›,πœ‡π‘›+1,𝑑). In fact, if π‘€βˆž(πœ‡π‘›βˆ’1,πœ‡π‘›,𝑑)>π‘€βˆž(πœ‡π‘›,πœ‡π‘›+1,𝑑), then from the inequality (5.6), we get π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯πœ™βˆžξ€·πœ‡π‘›,πœ‡π‘›+1,𝑑>π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑,(5.8) which is a contradiction. Thus, from the inequality (5.6), we have π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›,𝑑β‰₯β‹―β‰₯πœ™π‘›ξ€·π‘€βˆžξ€·πœ‡0,πœ‡1,𝑑.(5.9) Furthermore, for arbitrary positive integers π‘š and π‘˜, we have 1β‰₯π‘€βˆžξ€·πœ‡π‘˜,πœ‡π‘˜+π‘šξ€Έ,𝑑β‰₯π‘€βˆžξ‚€πœ‡π‘˜,πœ‡π‘˜+1,π‘‘π‘šξ‚βˆ—π‘€βˆžξ‚€πœ‡π‘˜+1,πœ‡π‘˜+2,π‘‘π‘šξ‚βˆ—β‹―βˆ—π‘€βˆžξ‚€πœ‡π‘˜+π‘šβˆ’1,πœ‡π‘˜+π‘š,π‘‘π‘šξ‚β‰₯πœ™π‘˜ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—πœ™π‘˜+1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—β‹―βˆ—πœ™π‘˜+π‘šβˆ’1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘š,(5.10) and πœ™(β„Ž)βˆ—πœ™2(β„Ž)βˆ—β‹―βˆ—πœ™π‘›(β„Ž)β†’1 as π‘›β†’βˆž, for all β„Žβˆˆ(0,1), it follows that πœ™π‘˜ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—πœ™π‘˜+1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—β‹―βˆ—πœ™π‘˜+π‘šβˆ’1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚(5.11) is convergent, which implies that {πœ‡π‘›} is a Cauchy sequence in π’ž(𝑋). Since 𝑋 is a compact fuzzy metric space, it follows 𝑋 is complete. By Theorem 4.6, π’ž(𝑋) is complete. Let πœ‡π‘›β†’πœ‡βˆ—. Next, we show that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—) for all π‘–βˆˆπ‘+. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, by Theorem 4.9, we have πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑑β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚βˆ—πœŒβˆžξ‚€πœ‡π‘—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,3𝑑4β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚βˆ—π‘€βˆžξ‚€β„±π‘—ξ€·πœ‡π‘—βˆ’1ξ€Έ,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,𝑑2β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚ξ‚€ξ‚†π‘€βˆ—πœ™infβˆžξ‚€πœ‡π‘—βˆ’1,πœ‡βˆ—,𝑑2,πœŒβˆžξ€·πœ‡π‘—βˆ’1,β„±π‘—ξ€·πœ‡π‘—βˆ’1ξ€Έξ€Έ,𝑑,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,1,𝑑2ξ€ΊπœŒβˆžξ€·πœ‡βˆ—,β„±π‘—ξ€·πœ‡π‘—βˆ’1ξ€Έξ€Έ,2𝑑+πœŒβˆžξ€·πœ‡π‘—βˆ’1,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,2𝑑β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚ξ‚€ξ‚†π‘€βˆ—πœ™infβˆžξ‚€πœ‡π‘—βˆ’1,πœ‡βˆ—,𝑑2,π‘€βˆžξ‚€πœ‡π‘—βˆ’1,πœ‡π‘—,𝑑2,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,1,𝑑2ξ€Ίπ‘€βˆžξ€·πœ‡βˆ—,πœ‡π‘—ξ€Έ,𝑑+π‘€βˆžξ€·πœ‡βˆ—,πœ‡π‘—βˆ’1ξ€Έ,π‘‘βˆ—πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,,𝑑(5.12) where πœ‡π‘—βŠ†β„±π‘—(πœ‡π‘—βˆ’1) implies 𝜌∞(πœ‡π‘—,ℱ𝑗(πœ‡π‘—βˆ’1),𝑑)=1. Letting π‘›β†’βˆž,π‘€βˆž(πœ‡π‘›,πœ‡βˆ—,𝑑)=1, and using the left continuity of πœ™, we have πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έξ€·πœŒ,𝑑β‰₯πœ™βˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,𝑑,(5.13) which implies 𝜌∞(πœ‡βˆ—,ℱ𝑖(πœ‡βˆ—),𝑑)=1. Hence, by Lemma 3.2, it follows that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—). Then, the proof is completed.

Theorem 5.2. Let (𝑋,𝑀,βˆ—) be a compact fuzzy metric space and let {ℱ𝑖}βˆžπ‘–=1 be a sequence of fuzzy self-mappings of π’ž(𝑋). Suppose that for each πœ‡1,πœ‡2βˆˆπ’ž(𝑋), and for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, 𝑑>0, π‘€βˆžξ€·β„±π‘–ξ€·πœ‡1ξ€Έ,β„±π‘—ξ€·πœ‡2𝑀,𝑑β‰₯πœ™βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑,πœŒβˆžξ€·πœ‡1,β„±π‘–ξ€·πœ‡1ξ€Έξ€Έ,𝜌,2π‘‘βˆžξ€·πœ‡2,β„±π‘—ξ€·πœ‡2ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡1,β„±π‘—ξ€·πœ‡2ξ€Έξ€Έ,4𝑑,πœŒβˆžξ€·πœ‡2,β„±π‘–ξ€·πœ‡1ξ€Έ,,𝑑(5.14) where πœ™(β„Ž1,β„Ž2,β„Ž3,β„Ž4,β„Ž5)∢(0,1]5β†’[0,1] is nondecreasing and continuous from the left for each variable. Denote 𝛾(β„Ž)=πœ™(β„Ž,β„Ž,β„Ž,π‘Ž,𝑏), where (π‘Ž,𝑏)∈{(β„Žβˆ—β„Ž,1),(1,β„Žβˆ—β„Ž)}. If 𝛾(β„Ž)βˆ—π›Ύ2(β„Ž)βˆ—β‹―βˆ—π›Ύπ‘›](β„Ž)⟢1asπ‘›βŸΆβˆž,βˆ€β„Žβˆˆ(0,1,(5.15) where 𝛾𝑛 denote the 𝑛th iterative function of 𝛾, then there exists πœ‡βˆ—βˆˆπ’ž(𝑋) such that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—) for all π‘–βˆˆπ‘+.

Proof. Let πœ‡0βˆˆπ’ž(𝑋) and πœ‡1βŠ†β„±1(πœ‡0). By Theorem 4.9, for any 𝑑>0, there exists πœ‡2βˆˆπ’ž(𝑋) such that πœ‡2βŠ†β„±2(πœ‡1) and π‘€βˆžξ€·πœ‡1,πœ‡2ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±1ξ€·πœ‡0ξ€Έ,β„±2ξ€·πœ‡1ξ€Έξ€Έ,𝑑.(5.16) Again by Theorem 4.9, for any 𝑑>0, we can find πœ‡3βˆˆπ’ž(𝑋) such that πœ‡3βŠ†β„±3(πœ‡2) and π‘€βˆžξ€·πœ‡2,πœ‡3ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±2ξ€·πœ‡1ξ€Έ,β„±3ξ€·πœ‡2ξ€Έξ€Έ,𝑑.(5.17) By induction, we produce a sequence {πœ‡π‘›} of points of π’ž(𝑋) such that πœ‡π‘›+1βŠ†β„±π‘›+1ξ€·πœ‡π‘›ξ€Έπ‘€,𝑛=0,1,2,…;βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έ,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έ.,𝑑(5.18) Now, we prove that {πœ‡π‘›} is a Cauchy sequence in π’ž(𝑋). In fact, for arbitrary positive integer 𝑛, by the inequality (5.14), Lemma 3.2, and the formula (5.18), we have π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑β‰₯π‘€βˆžξ€·β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έ,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έξ€·π‘€,𝑑β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,πœŒβˆžξ€·πœ‡π‘›βˆ’1,β„±π‘›ξ€·πœ‡π‘›βˆ’1ξ€Έξ€Έ,𝜌,2π‘‘βˆžξ€·πœ‡π‘›,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡π‘›βˆ’1,ℱ𝑛+1ξ€·πœ‡π‘›ξ€Έξ€Έ,4𝑑,πœŒβˆžξ€·πœ‡π‘›,β„±π‘›ξ€·πœ‡π‘›βˆ’1𝑀,𝑑β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›+1𝑀,2𝑑,1β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,π‘‘βˆ—π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έξ€Έ,,𝑑,1(5.19) where πœ‡π‘›βŠ†β„±π‘›(πœ‡π‘›βˆ’1) implies 𝜌∞(πœ‡π‘›,ℱ𝑛(πœ‡π‘›βˆ’1),2𝑑)=1 by (3) in Lemma 3.2 Likewise, we have 𝛾(β„Ž)>β„Ž for all β„Žβˆˆ(0,1), 𝑑>0. If π‘€βˆž(πœ‡π‘›βˆ’1,πœ‡π‘›,𝑑)>π‘€βˆž(πœ‡π‘›,πœ‡π‘›+1,𝑑), then from the inequality (5.19), we obtain π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯π›Ύβˆžξ€·πœ‡π‘›,πœ‡π‘›+1,𝑑>π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑,(5.20) which is a contradiction. Thus, from the inequality (5.19), we have π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯π›Ύβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›,𝑑β‰₯β‹―β‰₯π›Ύπ‘›ξ€·π‘€βˆžξ€·πœ‡0,πœ‡1,𝑑.(5.21) Furthermore, for arbitrary positive integers π‘š and π‘˜, we have π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘›,πœ‡π‘›+1ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,π‘‘βˆ—π‘€βˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑,1β‰₯πœ™βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,𝑑,π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έ,π‘‘βˆ—π‘€βˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€Έξ€Έξ€·π‘€,𝑑,1=π›Ύβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›,𝑀,π‘‘ξ€Έξ€Έβˆžξ€·πœ‡π‘›,πœ‡π‘›+1𝑀,𝑑β‰₯π›Ύβˆžξ€·πœ‡π‘›βˆ’1,πœ‡π‘›ξ€·π‘€,𝑑β‰₯β‹―β‰₯π›Ύβˆžξ€·πœ‡0,πœ‡1.,𝑑(5.22) Furthermore, for arbitrary positive integers π‘š and π‘˜, we have 1β‰₯π‘€βˆžξ€·πœ‡π‘˜,πœ‡π‘˜+π‘šξ€Έ,𝑑β‰₯π‘€βˆžξ‚€πœ‡π‘˜,πœ‡π‘˜+1,π‘‘π‘šξ‚βˆ—π‘€βˆžξ‚€πœ‡π‘˜+1,πœ‡π‘˜+2,π‘‘π‘šξ‚βˆ—β‹―βˆ—π‘€βˆžξ‚€πœ‡π‘˜+π‘šβˆ’1,πœ‡π‘˜+π‘š,π‘‘π‘šξ‚β‰₯π›Ύπ‘˜ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—π›Ύπ‘˜+1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—β‹―βˆ—π›Ύπ‘˜+π‘šβˆ’1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘š.(5.23) Since πœ™(β„Ž)βˆ—πœ™2(β„Ž)βˆ—β‹―βˆ—πœ™π‘›(β„Ž)β†’1 as π‘›β†’βˆž, for all β„Žβˆˆ(0,1), it follows that π›Ύπ‘˜ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—π›Ύπ‘˜+1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚βˆ—β‹―βˆ—π›Ύπ‘˜+π‘šβˆ’1ξ‚€π‘€βˆžξ‚€πœ‡0,πœ‡1,π‘‘π‘šξ‚ξ‚(5.24) is convergent, this implies that {πœ‡π‘›} is a Cauchy sequence in π’ž(𝑋). Since 𝑋 is a compact fuzzy metric space, it follows that 𝑋 is complete. By Theorem 4.6, π’ž(𝑋) is complete. Let πœ‡π‘›β†’πœ‡βˆ—. Now, we show that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—) for all π‘–βˆˆπ‘+. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, by Theorem 4.9, we have πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑑β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚βˆ—πœŒβˆžξ‚€πœ‡π‘—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,3𝑑4β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚βˆ—π‘€βˆžξ‚€β„±π‘—ξ€·πœ‡π‘—βˆ’1ξ€Έ,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,𝑑2β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚ξ‚€π‘€βˆ—πœ™βˆžξ‚€πœ‡π‘—βˆ’1,πœ‡βˆ—,𝑑2,πœŒβˆžξ€·πœ‡π‘—βˆ’1,β„±π‘—ξ€·πœ‡π‘—βˆ’1ξ€Έξ€Έ,𝜌,π‘‘βˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑑,πœŒβˆžξ€·πœ‡π‘—βˆ’1,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘—ξ€·πœ‡π‘—βˆ’1,𝑑β‰₯π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑4ξ‚ξ‚€π‘€βˆ—πœ™βˆžξ‚€πœ‡π‘—βˆ’1,πœ‡βˆ—,𝑑2,π‘€βˆžξ‚€πœ‡π‘—βˆ’1,πœ‡π‘—,𝑑2,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑀,π‘‘βˆžξ€·πœ‡π‘—βˆ’1,πœ‡βˆ—ξ€Έ,π‘‘βˆ—πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑑,π‘€βˆžξ‚€πœ‡βˆ—,πœ‡π‘—,𝑑2,(5.25) where πœ‡π‘—βŠ†β„±π‘—(πœ‡π‘—βˆ’1) implies 𝜌∞(πœ‡π‘—,ℱ𝑗(πœ‡π‘—βˆ’1),𝑑)=1. Letting π‘›β†’βˆž,π‘€βˆž(πœ‡π‘›,πœ‡βˆ—,𝑑)=1, and using the left continuity of πœ™, we have πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έξ€·,𝑑β‰₯πœ™1,1,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έ,𝑑,πœŒβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έξ€Έξ€Έξ€·πœŒ,𝑑,1β‰₯π›Ύβˆžξ€·πœ‡βˆ—,β„±π‘–ξ€·πœ‡βˆ—ξ€Έ,𝑑,(5.26) which implies 𝜌∞(πœ‡βˆ—,ℱ𝑖(πœ‡βˆ—),𝑑)=1. Hence, by Lemma 3.2, it follows that πœ‡βˆ—βŠ†β„±π‘–(πœ‡βˆ—), then the proof is completed.

Now, we give an example to illustrate the validity of the results in fixed point theory. For simplicity, we only exemplify Theorem 5.1, while the example may be similarly constructed for Theorem 5.2.

Example 5.3. Let (π’ž(𝑋),π‘€βˆž,βˆ—) be a fuzzy metric space, where 𝑋=[βˆ’1,1], 𝑀𝑑,𝐻𝑀, and π‘€βˆž are the same as in Example 3.4. Then, (π’ž(𝑋),π‘€βˆž,βˆ—) is a compact metric space.
Now, define πœ™βˆΆ[0,1]β†’[0,1] as βˆšπœ™(π‘₯)=π‘₯, and define {ℱ𝑖}βˆžπ‘–=1 a sequence of fuzzy self-mappings of π’ž(𝑋) as ℱ𝑖1(πœ‡)=2π‘–πœ‡,foranyπœ‡βˆˆπ’ž(𝑋).(5.27)
For arbitrary positive integers 𝑖 and 𝑗, without loss of generality, suppose 𝑖<𝑗. For each πœ‡1,πœ‡2βˆˆπ’ž(𝑋), by a routine calculation, we have π‘€βˆžξ€·β„±π‘–ξ€·πœ‡1ξ€Έ,β„±π‘—ξ€·πœ‡2ξ€Έξ€Έ,𝑑=π‘€βˆžξ‚΅12π‘–πœ‡1,12π‘—πœ‡2ξ‚Ά,𝑑=π‘€βˆžξ‚΅πœ‡1,12π‘—βˆ’π‘–πœ‡2,2𝑖𝑑β‰₯π‘€βˆžξ€·πœ‡1,πœ‡2,2𝑖𝑑𝑀β‰₯πœ™βˆžξ€·πœ‡1,πœ‡2𝑀,𝑑β‰₯πœ™infβˆžξ€·πœ‡1,πœ‡2ξ€Έ,πœŒβˆžξ€·πœ‡1,β„±π‘–ξ€·πœ‡1ξ€Έξ€Έ,2𝑑,πœŒβˆžξ€·πœ‡2,β„±π‘—ξ€·πœ‡2ξ€Έξ€Έ,1,2𝑑2ξ€ΊπœŒβˆžξ€·πœ‡2,β„±π‘–ξ€·πœ‡1ξ€Έξ€Έ,4𝑑+πœŒβˆžξ€·πœ‡1,β„±π‘—ξ€·πœ‡2ξ€Έ.,4𝑑(5.28) Therefore, by Theorem 5.1, we assert that the sequence of fuzzy self-mappings {ℱ𝑖}βˆžπ‘–=1 has a common fixed point πœ‡βˆ— in π’ž(𝑋). In fact, it is easy to check that πœ‡βˆ—ξ‚»(π‘₯)=1,ifπ‘₯=(0,0,…),0,otherwise.(5.29)

6. Conclusion

So far many authors have made a great deal of work in the Hausdorff-Pompeiu metric [20–25]. To describe the degree of nearness between two crisp sets, Rodrguez-LΓ³pez and Romaguera have defined Hausdorff fuzzy metric. In this paper, we define a new π‘€βˆž-fuzzy metric, which describes the degree of nearness between two fuzzy sets. Then, some properties on π‘€βˆž-fuzzy metric are discussed. In addition, in this new circumstances, we give some fixed point theorems which are the important generalizations of contraction mapping principle in functional analysis.

The results of the present paper may be applied in different settings. In terms of topology, one can make use of topology in data analysis and knowledge acquisition [31]. For another, topologies corresponding to fuzzy sets are used to detect dependencies of attributes in information systems with respect to gradual rules as in [32]. Furthermore, fuzzy fixed point theory can be used in existence and continuity theorems for dynamical systems with some vague parameters [33, 34]. In addition, this work offers a new tool for the description and analysis of fuzzy metric spaces. It would be possible to obtain more topological properties on the new fuzzy metric space. So, we hope our results contribute to dealing with some problems in practical applications for future study.

Acknowledgments

The authors thank the anonymous reviewers for their valuable comments. This work was supported by National Natural Science Foundation (NSFC) of China (Grant no. 61170320), Mathematical Tianyuan Foundation of China (Grant no. 11126087), Foundation of Guangdong Natural Science (no. S2011040002981), and Science and Technology Research Program of Chongqing Municipal Educational Committee (Grant no. KJ100518).

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