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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 890678, 18 pages
http://dx.doi.org/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-Cheng Liou

Copyright © 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 Rodríguez-López and Romaguera, and we introduce a new 𝑀-fuzzy metric, where 𝑀-fuzzy metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under 𝜙-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 [413] 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, 1416]. Besides that, a number of metrics are used on subspaces of fuzzy sets. For example, the sendograph metric [1719] and the 𝑑-metric for fuzzy sets [2025] induced by the Hausdorff-Pompeiu metric have been studied most frequently, where 𝑑-metric is an ordinary metric between two fuzzy sets. Combining fuzzy metric (in the sense of George and Veeramani) and Hausdorff-Pompeiu metric, Rodríguez-Lópezand 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 𝑀-fuzzy metric on the fuzzy sets whose 𝜆-cut are nonempty compact for each 𝜆[0,1]. In particular, we explore several properties of 𝑀-fuzzy metric. Then, under 𝜙-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 [0,1]×[0,1][0,1] is called a continuous 𝑡-norm if ([0,1],) is an Abelian topological semigroups with unit 1 such that 𝑎𝑏𝑐𝑑 whenever 𝑎𝑐 and 𝑏𝑑 for all 𝑎,𝑏,𝑐,𝑑[0,1].

Definition 2.1 (see [6]). The 3-tuple (𝑋,𝑀,) is said to be a fuzzy metric space if 𝑋 is an arbitrary set, is a continuous 𝑡-norm, and 𝑀 is a fuzzy set on 𝑋2×(0,) satisfying the following conditions, for all 𝑥,𝑦,𝑧𝑋,𝑡,𝑠>0:(i)𝑀(𝑥,𝑦,𝑡)>0;(ii)𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦;(iii)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡);(iv)𝑀(𝑥,𝑧,𝑡+𝑠)𝑀(𝑥,𝑦,𝑡)𝑀(𝑧,𝑦,𝑠);(v)𝑀(𝑥,𝑦,)(0,)[0,1] is continuous.
If (𝑋,𝑀,) is a fuzzy metric space, it will be said that (𝑀,) is a fuzzy metric on 𝑋.

A simply but useful fact [7] is that 𝑀(𝑥,𝑦,) is nondecreasing for all 𝑥,𝑦𝑋. Let (𝑋,𝑑) be a metric space. Denote by 𝑎𝑏 the usual multiplication for all 𝑎,𝑏[0,1], and let 𝑀𝑑 be the fuzzy set defined on 𝑋×𝑋×(0,) by𝑀𝑑(𝑡𝑥,𝑦,𝑡)=.𝑡+𝑑(𝑥,𝑦)(2.1) Then, (𝑋,𝑀𝑑,) is a fuzzy metric space, and (𝑀𝑑,) is called the standard fuzzy metric induced by 𝑑 [8].

George and Veeramani [6] proved that every fuzzy metric (𝑀,) on 𝑋 generates a topology 𝜏𝑀 on 𝑋 which has a base the family of open sets of the form:𝐵𝑀(𝑥,𝜀,𝑡)𝑥𝑋,0<𝜀<1,𝑡>0,(2.2) where 𝐵𝑀(𝑥,𝜀,𝑡)={𝑦𝑋𝑀(𝑥,𝑦,𝑡)>1𝜀} for all 𝜀(0,1) and 𝑡>0. They proved that (𝑋,𝜏𝑀) 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 𝜏𝑀𝑑 generated by the induced fuzzy metric (𝑀𝑑,) (see [8]).

Lemma 2.2 (see [6]). Let (𝑋,𝑀,) be a fuzzy metric space and let 𝜏 be the topology induced by the fuzzy metric. Then, for a sequence {𝑥𝑛}𝑛 in 𝑋, 𝑥𝑛𝑥 if and only if 𝑀(𝑥𝑛,𝑥,𝑡)1 as 𝑛 for all 𝑡>0.

Definition 2.3 (see [6]). A sequence {𝑥𝑛}𝑛 in a fuzzy metric space (𝑋,𝑀,) is called a Cauchy sequence if and only if for each 1>𝜀>0,𝑡>0, there exists 𝑛0 such that 𝑀(𝑥𝑛,𝑥𝑚,𝑡)>1𝜀 for all 𝑛,𝑚𝑛0. 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,). For 𝑎𝑋 and 𝑡>0, 𝑀(𝑎,𝐴,𝑡)=sup{𝑀(𝑎,𝑦,𝑡)𝑦𝐴,𝑡>0}.

Lemma 2.5 (see [28]). Let 𝐺 be a set and let {𝐺𝛼𝛼[0,1]} be a family of subsets of 𝐺 such that(1)𝐺0=𝐺;(2)𝛼𝛽 implies 𝐺𝛽𝐺𝛼;(3)𝛼1𝛼2, lim𝑛𝛼𝑛=𝛼 implies 𝐺𝛼=𝑛=1𝐺𝛼𝑛.Then, the function 𝜑𝐺[0,1] defined by 𝜑(𝑥)=sup{𝛼[0,1]𝑥𝐺𝛼} has the property that {𝑥𝐺𝜑(𝑥)𝛼}=𝐺𝛼  for every 𝛼[0,1].

Next, we recall some pertinent concepts on Hausdorff fuzzy metric. Denote by 𝐶0(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 𝐶0(𝑋)×𝐶0(𝑋) by𝐻𝑑(𝐴,𝐵)=maxsup𝑎𝐴𝑑(𝑎,𝐵),sup𝑏𝐵,𝑑(𝐴,𝑏)(2.3) for all 𝐴,𝐵𝐶0(𝑋), is a metric on 𝐶0(𝑋) called the Hausdorff-Pompeiu metric. In [30], it is proved that the metric (𝐶(𝑋),𝐻) is complete provided 𝑋 is complete.

Let 𝐶(𝑋) be the set of all nonempty compact subsets of a fuzzy metric space (𝑋,𝑀,), 𝐴,𝐵𝐶(𝑋), 𝑡>0, according to [26], the Hausdorff fuzzy metric 𝐻𝑀 on 𝐶(𝑋)×𝐶(𝑋)×(0,) is defined as𝐻𝑀(𝐴,𝐵,𝑡)=mininf𝑎𝐴𝑀(𝑎,𝐵,𝑡),inf𝑏𝐵𝑀(𝐴,𝑏,𝑡)=min{𝜌(𝐴,𝐵,𝑡),𝜌(𝐵,𝐴,𝑡)},(2.4) where 𝜌(𝐴,𝐵,𝑡)=inf𝑎𝐴𝑀(𝑎,𝐵,𝑡), and (𝐻𝑀,) is a fuzzy metric on 𝐶(𝑋). It is shown that 𝜌(𝐴,𝐵,𝑡)=1 if and only if 𝐴𝐵, and 𝐻𝑀(𝐴,𝐵,𝑡)=1 if and only if 𝐴=𝐵.

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

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

3. On 𝑀-Fuzzy Metric

Let (𝑋,𝑀,) be a fuzzy metric space. Denote by 𝒞(𝑋) the totality of fuzzy sets:[]𝜇𝑋0,1=𝐼,(3.1) which satisfy that, for each 𝜆𝐼, the 𝜆-cut of 𝜇,[𝜇]𝜆={𝑥𝑋𝜇(𝑥)𝜆},(3.2) is nonempty compact in 𝑋.

Definition 3.1. Let (𝑋,𝑀,) be a fuzzy metric space. The 𝑀-fuzzy metric between two fuzzy sets is induced by the Hausdorff fuzzy metric 𝐻𝑀 as 𝑀𝜇1,𝜇2𝜌,𝑡=min𝜇1,𝜇2,𝑡,𝜌𝜇2,𝜇1,𝑡,(3.3) where 𝜇1,𝜇2𝒞(𝑋),𝑡>0, and 𝜌𝜇1,𝜇2,𝑡=inf0𝜆1𝜌𝜇1𝜆,𝜇2𝜆,𝑡(3.4) is the fuzzy separation of 𝜇1 from 𝜇2.

Lemma 3.2. Let (𝑋,𝑀,) be a fuzzy metric space, 𝜇1,𝜇2,𝜇3𝒞(𝑋), 𝑠,𝑡>0. Then one has(1)𝑀(𝜇1,𝜇2,𝑡)(0,1],(2)𝑀(𝜇1,𝜇2,𝑡)=𝑀(𝜇2,𝜇1,𝑡),(3)𝜌(𝜇1,𝜇2,𝑡)=1 if and only if 𝜇1𝜇2,(4)𝑀(𝜇1,𝜇2,𝑡)=1 if and only if 𝜇1=𝜇2,(5)if 𝜇1𝜇2, then 𝜌(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇2,𝜇3,𝑡),(6)𝜌(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇1,𝜇2,𝑡)𝜌(𝜇2,𝜇3,𝑠),(7)𝑀(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇1,𝜇2,𝑡)𝑀(𝜇2,𝜇3,𝑠), (8)𝑀(𝜇1,𝜇2,)(0,)[0,1] is continuous.

Proof. For (1), by the definition of the 𝜆-cut [𝜇1]𝜆, for every 𝜆𝐼, [𝜇1]𝜆 is nonempty compact in 𝑋. By the theorem of nested intervals, there exists a point 𝑎0 in [𝜇1]𝜆 for every 𝜆𝐼, likewise, there exists a points 𝑏0 in [𝜇2]𝜆 for every 𝜆𝐼. Thus, 𝑀(𝜇1,𝜇2,𝑡)>0. Moreover, it is clear that 𝐴=𝐵𝐻𝑀(𝐴,𝐵,𝑡)=1𝑀(𝜇1,𝜇2,𝑡)=1.
For (2), it is clear that 𝑀(𝜇1,𝜇2,𝑡)=𝑀(𝜇2,𝜇1,𝑡).
For (3), since 𝜌(𝜇1,𝜇2,𝑡)=1 if and only if 𝜌([𝜇1]𝜆,[𝜇2]𝜆,𝑡)=1 for all 𝜆𝐼, which implies [𝜇1]𝜆[𝜇2]𝜆 for all 𝜆𝐼, we have that 𝜌(𝜇1,𝜇2,𝑡)=1 if and only if 𝜇1𝜇2.
For (4), it follows from (3).
For (5), for every 𝜆𝐼, any 𝑥[𝜇1]𝜆, 𝑦[𝜇2]𝜆 and 𝑧[𝜇3]𝜆, by the proof of Theorem  1 in [26], we have 𝑀(𝑥,𝑧,𝑡+𝑠)𝑀(𝑥,𝑦,𝑡)𝑀(𝑦,𝑧,𝑠)(3.5) with all 𝑥,𝑦,𝑧𝑋, which implies 𝑀𝜇𝑥,3𝜆𝜇,𝑡+𝑠𝑀(𝑥,𝑦,𝑡)𝑀𝑦,3𝜆,𝑠(3.6) for all 𝑥[𝜇1]𝜆 and all 𝑦[𝜇2]𝜆. Since 𝜇1𝜇2, then 𝜌([𝜇1]𝜆,[𝜇2]𝜆,𝑠)=1. By (iv) of Definition 2.1 and the arbitrariness of 𝑥 and 𝑦, we have 𝜌𝜇1𝜆,𝜇3𝜆,𝑡+𝑠=inf𝑥[𝜇1]𝜆𝑀𝜇𝑥,3𝜆,𝑡+𝑠inf𝑥[𝜇1]𝜆𝑀𝜇𝑥,2𝜆,𝑠inf𝑦[𝜇2]𝜆𝑀𝜇𝑦,3𝜆𝜇,𝑡=𝜌1𝜆,𝜇2𝜆𝜇,𝑠𝜌2𝜆,𝜇3𝜆𝜇,𝑡=𝜌2𝜆,𝜇3𝜆,𝑡𝐻𝑀𝜇2𝜆,𝜇3𝜆,,𝑡(3.7) which implies inf0𝜆1𝜌𝜇1𝜆,𝜇3𝜆,𝑡+𝑠inf0𝜆1𝐻𝑀𝜇2𝜆,𝜇3𝜆,𝑡.(3.8) Consequently, 𝜌(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇2,𝜇3,𝑡).
For (6), for every 𝜆𝐼, by the proof of (5) and (iv) of Definition 2.1, we have inf0𝜆1𝜌𝜇1𝜆,𝜇3𝜆,𝑡+𝑠inf0𝜆1𝜌𝜇1𝜆,𝜇2𝜆𝜇,𝑡𝜌2𝜆,𝜇3𝜆,𝑠inf0𝜆1𝐻𝑀𝜇1𝜆,𝜇2𝜆𝜇,𝑡𝜌2𝜆,𝜇3𝜆.,𝑠(3.9) Consequently, 𝜌(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇1,𝜇2,𝑡)𝜌(𝜇2,𝜇3,𝑠).
For (7), for every 𝜆𝐼, by the proof of (6), we have inf0𝜆1𝜌𝜇1𝜆,𝜇3𝜆,𝑡+𝑠inf0𝜆1𝜌𝜇1𝜆,𝜇2𝜆𝜇,𝑡𝜌2𝜆,𝜇3𝜆,𝑠inf0𝜆1𝜌𝜇1𝜆,𝜇2𝜆,𝑡inf0𝜆1𝜌𝜇2𝜆,𝜇3𝜆.,𝑠(3.10) Similarly, it can be shown that inf0𝜆1𝜌𝜇3𝜆,𝜇1𝜆,𝑡+𝑠inf0𝜆1𝜌𝜇3𝜆,𝜇2𝜆,𝑠inf0𝜆1𝜌𝜇2𝜆,𝜇1𝜆,𝑡.(3.11) Hence, 𝑀(𝜇1,𝜇3,𝑡+𝑠)𝑀(𝜇1,𝜇2,𝑡)𝑀(𝜇2,𝜇3,𝑠).
For (8), by the continuity on (0,) of the function 𝑡𝐻𝑀(𝐴,𝐵,𝑡), it is clear that 𝑀(𝜇1,𝜇2,)(0,)[0,1] is continuous.

Theorem 3.3. Let (𝑋,𝑀,) be a fuzzy metric space. Then, (𝒞(𝑋),𝑀,) is a fuzzy metric space, where 𝑀 is a fuzzy set on the 𝒞(𝑋)×𝒞(𝑋)×(0,+).

Proof. It is easily proved by Lemma 3.2.

Example 3.4. Let 𝑑 be the Euclidean metric on , and let 𝐴=[𝑎1,𝑎2] and let 𝐵=[𝑏1,𝑏2] be two compact intervals. Then, 𝐻𝑑(𝐴,𝐵)=max{|𝑎1𝑏1|,|𝑎2𝑏2|}. Let (,𝑀𝑑,) be a fuzzy metric space, where 𝑎𝑏 the usual multiplication for all 𝑎,𝑏[0,1], and 𝑀𝑑 is defined on ××(0,) by 𝑀𝑑(𝑡𝑥,𝑦,𝑡)=𝑡+𝑑(𝑥,𝑦).(3.12) Denote by 𝒞() the totality of fuzzy sets 𝜇[0,1] which satisfy that for each 𝜆𝐼, the 𝜆-cut of 𝜇[𝜇]𝜆={𝑥𝜇(𝑥)𝜆} is a nonempty compact interval. For any 𝜆-cuts of fuzzy sets 𝜇1,𝜇2𝒞() and for all 𝑡>0, by a simple calculation, we have 𝐻𝑀𝜇1𝜆,𝜇2𝜆=𝑡,𝑡𝑡+𝐻𝑑𝜇1𝜆,𝜇2𝜆.(3.13) So by Definition 3.1, we get 𝑀𝜇1,𝜇2,𝑡=inf0𝜆1𝑡𝑡+𝐻𝑑𝜇1𝜆,𝜇2𝜆.(3.14)

4. Properties of the 𝑀-Fuzzy Metric

Definition 4.1. Let (𝒞(𝑋),𝑀,) be a fuzzy metric space. For 𝑡(0,+), define 𝐵(𝜇,𝑟,𝑡) with center a fuzzy set 𝜇𝒞(𝑋) and radius 𝑟,0<𝑟<1,𝑡>0 as 𝐵(𝜇,𝑟,𝑡)=𝛾𝒞(𝑋)𝑀(𝜇,𝛾,𝑡)>1𝑟.(4.1)

Proposition 4.2. Every 𝐵(𝜇,𝑟,𝑡) is an open set.

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

Proposition 4.3. Let (𝒞(𝑋),𝑀,) be a fuzzy metric space. Define 𝜏𝑀={𝒜𝒞(𝑋)𝜇𝒜ifandonlyifthereexist𝑡>0and𝑟,0<𝑟<1suchthat𝐵(𝜇,𝑟,𝑡)𝒜}.
Then, 𝜏𝑀 is a topology on 𝒞(𝑋).

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

Definition 4.4. A sequence {𝜇𝑛} in a fuzzy metric space (𝒞(𝑋),𝑀,) is a Cauchy sequence if and only if for each 𝜀>0, 𝑡>0, there exists 𝑛0 such that 𝑀(𝜇𝑛,𝜇𝑚,𝑡)>1𝜀 for all 𝑛,𝑚𝑛0.

Lemma 4.5. Let (𝒞(𝑋),𝑀,) be a fuzzy metric space on fuzzy metric 𝑀 and let 𝜏 be the topology induced by the fuzzy metric 𝑀. Then, for a sequence {𝜇𝑛} in 𝒞(𝑋), 𝜇𝑛𝜇 if and only if 𝑀(𝜇,𝜇𝑛,𝑡)1 as 𝑛.

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

Theorem 4.6. The fuzzy metric space (𝒞(𝑋),𝑀,) is complete provided (𝑋,𝑀,) is complete.

Proof. Let (𝑋,𝑀,) be a complete fuzzy metric space and let a sequence {𝜇𝑛,𝑛1} be a Cauchy sequence in 𝒞(𝑋). Consider a fixed 0<𝜆<1. Then, {[𝜇𝑛]𝜆,𝑛1} is a Cauchy sequence in (𝐶(𝑋),𝐻𝑀,), where 𝐶(𝑋) denotes all nonempty compact subsets of (𝑋,𝑀,).
Since (𝐶(𝑋),𝐻𝑀,) is complete by Lemma 2.6, it follows that [𝜇𝑛]𝜆𝜇𝜆𝐶(𝑋). Actually, from the definition of 𝑀 and the continuity of 𝐻𝑀, it is easy to see that [𝜇𝑛]𝜆𝜇𝜆, uniformly in 𝜆[0,1].
Now, consider the family {𝜇𝜆𝜆[0,1]}, where 𝜇0=𝑋. Take 𝜆𝛽, we have 𝜌𝜇𝛽,𝜇𝜆𝜇,𝑡𝜌𝛽,𝜇𝑛𝛽,𝑡3𝜇𝜌𝑛𝛽,𝜇𝑛𝜆,𝑡3𝜇𝜌𝑛𝜆,𝜇𝜆,𝑡3.(4.2) Since [𝜇𝑛]𝛽[𝜇𝑛]𝜆, it follows that 𝜌([𝜇𝑛]𝛽,[𝜇𝑛]𝜆,𝑡/3)=1. Thus, for each 0<𝜀<1, 𝜌(𝜇𝛽,𝜇𝜆,𝑡)𝜌(𝜇𝛽,[𝜇𝑛]𝛽,𝑡/3)𝜌([𝜇𝑛]𝜆,𝜇𝜆,𝑡/3) if 𝑛 is large enough. Hence, 𝜌(𝜇𝛽,𝜇𝜆,𝑡)=1, and by Lemma 3.2, we have 𝜇𝛽𝜇𝜆.
Now, take 𝜆𝑛 and lim𝑛𝜆𝑛=𝜆. We have to show that 𝜇𝜆=𝑛=1𝜇𝜆𝑛. It is clear that 𝜇𝜆𝑛=1𝜇𝜆𝑛.() On the other hand, we have 𝜌𝑛=1𝜇𝜆𝑛,𝜇𝜆,𝑡𝜌𝑛=1𝜇𝜆𝑛,𝑛=1𝜇𝑗𝜆𝑛,𝑡3𝜌𝑛=1𝜇𝑗𝜆𝑛,𝜇𝑗𝜆,𝑡3𝜇𝜌𝑗𝜆,𝜇𝜆,𝑡3,(4.3) for fixed 𝑗. However, 𝜌𝑛=1𝜇𝑗𝜆𝑛,𝜇𝑗𝜆,𝑡3=1.(4.4) Consequently, for every 0<𝜀<1, there exists 0<𝜀0<𝜀<1 such that (1𝜀0)(1𝜀0)(1𝜀0)>1𝜀. For given 𝜀0, since [𝜇𝑗]𝜆𝜇𝜆, there exists 𝑗𝜀0 such that 𝜌𝑛=1𝜇𝜆𝑛,𝜇𝜆,𝑡𝜌𝑛=1𝜇𝜆𝑛,𝑛=1𝜇𝑗𝜆𝑛,𝑡31𝜀0,(4.5) for 𝑗𝑗𝜀0. Now, 𝜌𝑛=1𝜇𝜆𝑛,𝑛=1𝜇𝑗𝜆𝑛,𝑡3𝜌𝑛=1𝜇𝜆𝑛,𝜇𝜆𝑝,𝑡9𝜇𝜌𝜆𝑝,𝜇𝑗𝜆𝑛,𝑡9𝜇𝜌𝑗𝜆𝑝,𝑛=1𝜇𝑗𝜆𝑛,𝑡9,(4.6) for any 𝑝1. Since 𝑛=1𝜇𝜆𝑛𝜇𝜆𝑝, we obtain 𝜌𝑛=1𝜇𝜆𝑛,𝑛=1𝜇𝑗𝜆𝑛𝜇,𝑡𝜌𝜆𝑝,𝜇𝑗𝜆𝑝,𝑡2𝜇𝜌𝑗𝜆𝑝,𝑛=1𝜇𝑗𝜆𝑛,𝑡2.(4.7) Now, 𝜌(𝜇𝜆𝑝,[𝜇𝑗]𝜆𝑝,𝑡/2)>1𝜀0 for 𝑗𝑗0 and all 𝑡>0. Note that (since the convergence [𝜇𝑗]𝜆𝜇𝜆 is uniform in 𝜆) 𝑗0 does not depend on 𝑝. Since {[𝜇𝑗]𝜆𝑝,𝑝1} decreases to 𝑛=1[𝜇𝑗]𝜆𝑛, if follows that 𝜌([𝜇𝑗]𝜆𝑝0,𝑛=1[𝜇𝑗]𝜆𝑛,𝑡/2)>1𝜀0 for some 𝑝0 (depending on 𝑗).
Thus, 𝜌(𝑛=1𝜇𝜆𝑛,𝑛=1[𝜇𝑗]𝜆𝑛,𝑡/3)(1𝜀0)(1𝜀0), if 𝑗 is large.
Finally, by taking 𝑗 large enough, we obtain 𝜌𝑛=1𝜇𝜆𝑛,𝜇𝜆,𝑡1𝜀01𝜀01𝜀01𝜀,(4.8) that is, 𝑛=1𝜇𝜆𝑛𝜇𝜆.() From (4.3) and (4.9), it yields 𝑛=1𝜇𝜆𝑛=𝜇𝜆. Thus, Lemma 2.5 is applicable and there exists 𝜇𝒞(𝑋) for every 𝜆[0,1] such that [𝜇𝑛]𝜆𝜇𝜆. It remains to show that 𝜇𝑛𝜇 in (𝒞(𝑋),𝑀,).
Let 𝜀>0. Then, since {𝜇𝑛} is a Cauchy sequence, there exists 𝑛𝜀 such that 𝑛,𝑚>𝑛𝜀 implies 𝑀(𝜇𝑛,𝜇𝑚,𝑡)>1𝜀.
Let 𝑛(>𝑛𝜀) be fixed. Then, 𝐻𝑀𝜇𝑛𝜆,[𝜇]𝜆,𝑡=lim𝑚𝐻𝑀𝜇𝑛𝜆,𝜇𝑚𝜆,𝑡lim𝑚inf0𝜆1𝐻𝑀𝜇𝑛𝜆,𝜇𝑚𝜆=,𝑡lim𝑚𝑀𝜇𝑛,𝜇𝑚,𝑡>1𝜀.(4.9) Thus, 𝜇𝑛𝜇 in the 𝑀-fuzzy metric. The proof is completed.

Lemma 4.7. Let (𝑋,𝑀,) be a compact fuzzy metric space and compact subsets 𝐴,𝐵𝐶(𝑋). Then, for each 𝑥𝐴 and 𝑡>0, there exists a 𝑦𝐵 such that 𝑀(𝑥,𝑦,𝑡)𝐻𝑀(𝐴,𝐵,𝑡).

Proof. Suppose there exists a 𝑥0𝐴 such that 𝑀(𝑥0,𝑦,𝑡)<𝐻𝑀(𝐴,𝐵,𝑡) for any 𝑦𝐵 and 𝑡>0. Then, sup𝑦𝐵𝑀𝑥0,𝑦,𝑡<𝐻𝑀(𝐴,𝐵,𝑡),(4.10) that is, sup𝑦𝐵𝑀𝑥0,𝑦,𝑡<mininf𝑥𝐵sup𝑦𝐴𝑀(𝑥,𝑦,𝑡),inf𝑥𝐴sup𝑦𝐵𝑀(𝑥,𝑦,𝑡).(4.11) So, sup𝑦𝐵𝑀𝑥0,𝑦,𝑡<inf𝑥𝐴sup𝑦𝐵𝑀(𝑥,𝑦,𝑡).(4.12) This is a contradiction with 𝑥𝐴.

Lemma 4.8. Let (𝑋,𝑀,) be a compact fuzzy metric space, 𝑡>0 and 𝐴,𝐵𝐶(𝑋). Then, for any compact set 𝐴1𝐴, there exists a compact set 𝐵1𝐵 such that 𝐻𝑀(𝐴1,𝐵1,𝑡)𝐻𝑀(𝐴,𝐵,𝑡).

Proof. Let 𝐶={𝑦 there exists a 𝑥𝐴1 such that 𝑀(𝑥,𝑦,𝑡)𝐻𝑀(𝐴,𝐵,𝑡)} and let 𝐵1𝐵=𝐶. For any 𝑥𝐴1𝐴,𝑡>0, by Lemma 4.7, there exists a 𝑦𝐵 such that 𝑀(𝑥,𝑦,𝑡)𝐻𝑀(𝐴,𝐵,𝑡).(4.13) Thus, 𝐵1, moreover, 𝐵1 is compact since it is closed in 𝑋 and 𝐵1𝐵.
Now, for any 𝑥𝐴1, 𝑡>0, there exists a 𝑦𝐵1 such that 𝑀(𝑥,𝑦,𝑡)𝐻𝑀(𝐴,𝐵,𝑡).(4.14) Thus, we have 𝑀(𝑥,𝐵1,𝑡)𝐻𝑀(𝐴,𝐵,𝑡), which implies that 𝜌𝐴1,𝐵1,𝑡=inf𝑥𝐴1𝑀𝑥,𝐵1,𝑡𝐻𝑀(𝐴,𝐵,𝑡).(4.15) Similarly, it can be shown that 𝜌(𝐴1,𝐵1,𝑡)𝐻𝑀(𝐴,𝐵,𝑡).
Hence, 𝐻𝑀(𝐴1,𝐵1,𝑡)𝐻𝑀(𝐴,𝐵,𝑡). This completes the proof.

Theorem 4.9. Let (𝑋,𝑀,) be a compact fuzzy metric space and 𝜇1,𝜇2𝒞(𝑋), 𝑡>0. Then, for any 𝜇3𝒞(𝑋) satisfying 𝜇3𝜇1, there exists a 𝜇4𝒞(𝑋) such that 𝜇4𝜇2 and 𝑀𝜇3,𝜇4,𝑡𝑀𝜇1,𝜇2,𝑡.(4.16)

Proof. Since 𝜇1,𝜇2, and 𝜇3 are normal, we have [𝜇3]𝜆[𝜇1]𝜆 and [𝜇2]𝜆 for all 𝜆𝐼. Let 𝐶𝜆=𝜇𝑦thereexistsa𝑥3𝜆suchthat𝑀(𝑥,𝑦,𝑡)𝑀𝜇1,𝜇2,𝑡,(4.17) and let 𝐵𝜆=𝐶𝜆[𝜇2]𝜆. For any 𝑥[𝜇3]𝜆[𝜇1]𝜆, by Lemma 4.7, there exists a 𝑦[𝜇2]𝜆 such that 𝑀(𝑥,𝑦,𝑡)𝐻𝑀𝜇1𝜆,𝜇2𝜆,𝑡𝑀𝜇1,𝜇2.,𝑡(4.18) Thus, 𝐵𝜆 is nonempty compact in 𝑋, moreover, 𝐵𝜆𝐵𝛾 if 0𝛾𝜆1.
From the proof of Lemma 4.8, we have 𝐻𝑀𝜇3𝜆,𝐵𝜆,𝑡𝑀𝜇1,𝜇2.,𝑡(4.19) By Lemma  3.1 in [28], there exists a fuzzy set 𝜇4 with the property that [𝜇4]𝜆=𝐵𝜆 for 𝜆𝐼. Since 𝐵𝜆 are nonempty compact for all 𝜆𝐼, we have 𝜇4𝒞(𝑋). Consequently, 𝑀𝜇3,𝜇4,𝑡𝑀𝜇1,𝜇2,𝑡.(4.20) This completes the proof.

Definition 4.10 (see [24]). Let 𝑋,𝑌 be any fuzzy metric space. is said to be a fuzzy mapping if and only if is a mapping from the space 𝒞(𝑋) into 𝒞(𝑌), that is, (𝜇)𝒞(𝑌) for each 𝜇𝒞(𝑋).

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

Theorem 5.1. Let (𝑋,𝑀,) be a compact fuzzy metric space and let {𝑖}𝑖=1 be a sequence of fuzzy self-mappings of 𝒞(𝑋). Let 𝜙[0,1][0,1] be a nondecreasing function satisfying the following condition: 𝜙 is continuous from the left and 𝜙()𝜙2()𝜙𝑛]()1as𝑛,(0,1,(5.1) where 𝜙𝑛 denote the 𝑛th iterative function of 𝜙. Suppose that for each 𝜇1,𝜇2𝒞(𝑋), and for arbitrary positive integers 𝑖 and 𝑗,𝑖𝑗,𝑡>0, 𝑀𝑖𝜇1,𝑗𝜇2𝑀,𝑡𝜙inf𝜇1,𝜇2,𝑡,𝜌𝜇1,𝑖𝜇1,2𝑡,𝜌𝜇2,𝑗𝜇2,1,2𝑡2𝜌𝜇2,𝑖𝜇1,4𝑡+𝜌𝜇1,𝑗𝜇2,,4𝑡(5.2) then there exists 𝜇𝒞(𝑋) such that 𝜇𝑖(𝜇) for all 𝑖𝑍+.

Proof. Let 𝜇0𝒞(𝑋) and 𝜇11(𝜇0). By Theorem 4.9, for any 𝑡>0, there exists 𝜇2𝒞(𝑋) such that 𝜇22(𝜇1) and 𝑀𝜇1,𝜇2,𝑡𝑀1𝜇0,2𝜇1,𝑡.(5.3) Again by Theorem 4.9, for any 𝑡>0, we can find 𝜇3𝒞(𝑋) such that 𝜇33(𝜇2) and 𝑀𝜇2,𝜇3,𝑡𝑀2𝜇1,3𝜇2,𝑡.(5.4) By induction, we produce a sequence {𝜇𝑛} of points of 𝒞(𝑋) such that 𝜇𝑛+1𝑛+1𝜇𝑛𝑀,𝑛=0,1,2,;𝜇𝑛,𝜇𝑛+1,𝑡𝑀𝑛𝜇𝑛1,𝑛+1𝜇𝑛.,𝑡(5.5) Now, we prove that {𝜇𝑛} is a Cauchy sequence in 𝒞(𝑋). In fact, for arbitrary positive integer 𝑛, by the inequality (5.2), Lemma 3.2, and the formula (5.5), we have 𝑀𝜇𝑛,𝜇𝑛+1,𝑡𝑀𝑛𝜇𝑛1,𝑛+1𝜇𝑛𝑀,𝑡𝜙inf𝜇𝑛1,𝜇𝑛,𝑡,𝜌𝜇𝑛1,𝑛𝜇𝑛1,2𝑡,𝜌𝜇𝑛,𝑛+1𝜇2,1,2𝑡2𝜌𝜇𝑛1,𝑛+1𝜇𝑛,4𝑡+𝜌𝜇𝑛,𝑛𝜇𝑛1,𝑀,4𝑡𝜙inf𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑀,𝑡𝜇𝑛,𝜇𝑛+1,1,2𝑡2𝑀𝜇𝑛1,𝜇𝑛+1,𝑀,2𝑡+1𝜙inf𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛,𝜇𝑛+1,𝑀,𝑡𝜇𝑛,𝜇𝑛+1,1,2𝑡2𝑀𝜇𝑛1,𝜇𝑛,𝑡+1𝑀𝜇𝑛,𝜇𝑛+1,,𝑡+1(5.6) where 𝜇𝑛𝑛(𝜇𝑛1) implies 𝜌(𝜇𝑛,𝑛(𝜇𝑛1),2𝑡)=1, by (3) of Lemma 3.2. In addition, it is easy to get that 𝜙()> for all (0,1). In fact, suppose that there exists some 𝑡0(0,1) such that 𝜙(0)0. Since 𝜙 is nondecreasing, we have 𝜙𝑛0𝜙𝑛10𝜙00.(5.7) Since 𝜙()𝜙2()𝜙𝑛()1 as 𝑛, for all (0,1), then we have 𝜙𝑛(0)1 as 𝑛. From the inequality (5.7), we have 10. This is a contradiction which implies 𝜙()> for all (0,1). We can prove that 𝑀(𝜇𝑛1,𝜇𝑛,𝑡)𝑀(𝜇𝑛,𝜇𝑛+1,𝑡). In fact, if 𝑀(𝜇𝑛1,𝜇𝑛,𝑡)>𝑀(𝜇𝑛,𝜇𝑛+1,𝑡), then from the inequality (5.6), we get 𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝜙𝜇𝑛,𝜇𝑛+1,𝑡>𝑀𝜇𝑛,𝜇𝑛+1,𝑡,(5.8) which is a contradiction. Thus, from the inequality (5.6), we have 𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝜙𝜇𝑛1,𝜇𝑛,𝑡𝜙𝑛𝑀𝜇0,𝜇1,𝑡.(5.9) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 1𝑀𝜇𝑘,𝜇𝑘+𝑚,𝑡𝑀𝜇𝑘,𝜇𝑘+1,𝑡𝑚𝑀𝜇𝑘+1,𝜇𝑘+2,𝑡𝑚𝑀𝜇𝑘+𝑚1,𝜇𝑘+𝑚,𝑡𝑚𝜙𝑘𝑀𝜇0,𝜇1,𝑡𝑚𝜙𝑘+1𝑀𝜇0,𝜇1,𝑡𝑚𝜙𝑘+𝑚1𝑀𝜇0,𝜇1,𝑡𝑚,(5.10) and 𝜙()𝜙2()𝜙𝑛()1 as 𝑛, for all (0,1), it follows that 𝜙𝑘𝑀𝜇0,𝜇1,𝑡𝑚𝜙𝑘+1𝑀𝜇0,𝜇1,𝑡𝑚𝜙𝑘+𝑚1𝑀𝜇0,𝜇1,𝑡𝑚(5.11) is convergent, which implies that {𝜇𝑛} is a Cauchy sequence in 𝒞(𝑋). Since 𝑋 is a compact fuzzy metric space, it follows 𝑋 is complete. By Theorem 4.6, 𝒞(𝑋) is complete. Let 𝜇𝑛𝜇. Next, we show that 𝜇𝑖(𝜇) for all 𝑖𝑍+. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖𝑗, by Theorem 4.9, we have 𝜌𝜇,𝑖𝜇,𝑡𝑀𝜇,𝜇𝑗,𝑡4𝜌𝜇𝑗,𝑖𝜇,3𝑡4𝑀𝜇,𝜇𝑗,𝑡4𝑀𝑗𝜇𝑗1,𝑖𝜇,𝑡2𝑀𝜇,𝜇𝑗,𝑡4𝑀𝜙inf𝜇𝑗1,𝜇,𝑡2,𝜌𝜇𝑗1,𝑗𝜇𝑗1,𝑡,𝜌𝜇,𝑖𝜇,1,𝑡2𝜌𝜇,𝑗𝜇𝑗1,2𝑡+𝜌𝜇𝑗1,𝑖𝜇,2𝑡𝑀𝜇,𝜇𝑗,𝑡4𝑀𝜙inf𝜇𝑗1,𝜇,𝑡2,𝑀𝜇𝑗1,𝜇𝑗,𝑡2,𝜌𝜇,𝑖𝜇,1,𝑡2𝑀𝜇,𝜇𝑗,𝑡+𝑀𝜇,𝜇𝑗1,𝑡𝜌𝜇,𝑖𝜇,,𝑡(5.12) where 𝜇𝑗𝑗(𝜇𝑗1) implies 𝜌(𝜇𝑗,𝑗(𝜇𝑗1),𝑡)=1. Letting 𝑛,𝑀(𝜇𝑛,𝜇,𝑡)=1, and using the left continuity of 𝜙, we have 𝜌𝜇,𝑖𝜇𝜌,𝑡𝜙𝜇,𝑖𝜇,𝑡,(5.13) which implies 𝜌(𝜇,𝑖(𝜇),𝑡)=1. Hence, by Lemma 3.2, it follows that 𝜇𝑖(𝜇). Then, the proof is completed.

Theorem 5.2. Let (𝑋,𝑀,) be a compact fuzzy metric space and let {𝑖}𝑖=1 be a sequence of fuzzy self-mappings of 𝒞(𝑋). Suppose that for each 𝜇1,𝜇2𝒞(𝑋), and for arbitrary positive integers 𝑖 and 𝑗, 𝑖𝑗, 𝑡>0, 𝑀𝑖𝜇1,𝑗𝜇2𝑀,𝑡𝜙𝜇1,𝜇2,𝑡,𝜌𝜇1,𝑖𝜇1,𝜌,2𝑡𝜇2,𝑗𝜇2,2𝑡,𝜌𝜇1,𝑗𝜇2,4𝑡,𝜌𝜇2,𝑖𝜇1,,𝑡(5.14) where 𝜙(1,2,3,4,5)(0,1]5[0,1] is nondecreasing and continuous from the left for each variable. Denote 𝛾()=𝜙(,,,𝑎,𝑏), where (𝑎,𝑏){(,1),(1,)}. If 𝛾()𝛾2()𝛾𝑛]()1as𝑛,(0,1,(5.15) where 𝛾𝑛 denote the 𝑛th iterative function of 𝛾, then there exists 𝜇𝒞(𝑋) such that 𝜇𝑖(𝜇) for all 𝑖𝑍+.

Proof. Let 𝜇0𝒞(𝑋) and 𝜇11(𝜇0). By Theorem 4.9, for any 𝑡>0, there exists 𝜇2𝒞(𝑋) such that 𝜇22(𝜇1) and 𝑀𝜇1,𝜇2,𝑡𝑀1𝜇0,2𝜇1,𝑡.(5.16) Again by Theorem 4.9, for any 𝑡>0, we can find 𝜇3𝒞(𝑋) such that 𝜇33(𝜇2) and 𝑀𝜇2,𝜇3,𝑡𝑀2𝜇1,3𝜇2,𝑡.(5.17) By induction, we produce a sequence {𝜇𝑛} of points of 𝒞(𝑋) such that 𝜇𝑛+1𝑛+1𝜇𝑛𝑀,𝑛=0,1,2,;𝜇𝑛,𝜇𝑛+1,𝑡𝑀𝑛𝜇𝑛1,𝑛+1𝜇𝑛.,𝑡(5.18) Now, we prove that {𝜇𝑛} is a Cauchy sequence in 𝒞(𝑋). In fact, for arbitrary positive integer 𝑛, by the inequality (5.14), Lemma 3.2, and the formula (5.18), we have 𝑀𝜇𝑛,𝜇𝑛+1,𝑡𝑀𝑛𝜇𝑛1,𝑛+1𝜇𝑛𝑀,𝑡𝜙𝜇𝑛1,𝜇𝑛,𝑡,𝜌𝜇𝑛1,𝑛𝜇𝑛1,𝜌,2𝑡𝜇𝑛,𝑛+1𝜇𝑛,2𝑡,𝜌𝜇𝑛1,𝑛+1𝜇𝑛,4𝑡,𝜌𝜇𝑛,𝑛𝜇𝑛1𝑀,𝑡𝜙𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛,𝜇𝑛+1,𝑡,𝑀𝜇𝑛1,𝜇𝑛+1𝑀,2𝑡,1𝜙𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑀,𝑡𝜇𝑛,𝜇𝑛+1,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑡𝑀𝜇𝑛,𝜇𝑛+1,,𝑡,1(5.19) where 𝜇𝑛𝑛(𝜇𝑛1) implies 𝜌(𝜇𝑛,𝑛(𝜇𝑛1),2𝑡)=1 by (3) in Lemma 3.2 Likewise, we have 𝛾()> for all (0,1), 𝑡>0. If 𝑀(𝜇𝑛1,𝜇𝑛,𝑡)>𝑀(𝜇𝑛,𝜇𝑛+1,𝑡), then from the inequality (5.19), we obtain 𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝛾𝜇𝑛,𝜇𝑛+1,𝑡>𝑀𝜇𝑛,𝜇𝑛+1,𝑡,(5.20) which is a contradiction. Thus, from the inequality (5.19), we have 𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝛾𝜇𝑛1,𝜇𝑛,𝑡𝛾𝑛𝑀𝜇0,𝜇1,𝑡.(5.21) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝜙𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑀,𝑡𝜇𝑛,𝜇𝑛+1,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑡𝑀𝜇𝑛,𝜇𝑛+1𝑀,𝑡,1𝜙𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑀,𝑡𝜇𝑛1,𝜇𝑛,𝑡,𝑀𝜇𝑛1,𝜇𝑛,𝑡𝑀𝜇𝑛1,𝜇𝑛𝑀,𝑡,1=𝛾𝜇𝑛1,𝜇𝑛,𝑀,𝑡𝜇𝑛,𝜇𝑛+1𝑀,𝑡𝛾𝜇𝑛1,𝜇𝑛𝑀,𝑡𝛾𝜇0,𝜇1.,𝑡(5.22) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 1𝑀𝜇𝑘,𝜇𝑘+𝑚,𝑡𝑀𝜇𝑘,𝜇𝑘+1,𝑡𝑚𝑀𝜇𝑘+1,𝜇𝑘+2,𝑡𝑚𝑀𝜇𝑘+𝑚1,𝜇𝑘+𝑚,𝑡𝑚𝛾𝑘𝑀𝜇0,𝜇1,𝑡𝑚𝛾𝑘+1𝑀𝜇0,𝜇1,𝑡𝑚𝛾𝑘+𝑚1𝑀𝜇0,𝜇1,𝑡𝑚.(5.23) Since 𝜙()𝜙2()𝜙𝑛()1 as 𝑛, for all (0,1), it follows that 𝛾𝑘𝑀𝜇0,𝜇1,𝑡𝑚𝛾𝑘+1𝑀𝜇0,𝜇1,𝑡𝑚𝛾𝑘+𝑚1𝑀𝜇0,𝜇1,𝑡𝑚(5.24) is convergent, this implies that {𝜇𝑛} is a Cauchy sequence in 𝒞(𝑋). Since 𝑋 is a compact fuzzy metric space, it follows that 𝑋 is complete. By Theorem 4.6, 𝒞(𝑋) is complete. Let 𝜇𝑛𝜇. Now, we show that 𝜇𝑖(𝜇) for all 𝑖𝑍+. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖𝑗, by Theorem 4.9, we have 𝜌𝜇,𝑖𝜇,𝑡𝑀𝜇,𝜇𝑗,𝑡4𝜌𝜇𝑗,𝑖𝜇,3𝑡4𝑀𝜇,𝜇𝑗,𝑡4𝑀𝑗𝜇𝑗1,𝑖𝜇,𝑡2𝑀𝜇,𝜇𝑗,𝑡4𝑀𝜙𝜇𝑗1,𝜇,𝑡2,𝜌𝜇𝑗1,𝑗𝜇𝑗1,𝜌,𝑡𝜇,𝑖𝜇,𝑡,𝜌𝜇𝑗1,𝑖𝜇,2𝑡,𝜌𝜇,𝑗𝜇𝑗1,𝑡𝑀𝜇,𝜇𝑗,𝑡4𝑀𝜙𝜇𝑗1,𝜇,𝑡2,𝑀𝜇𝑗1,𝜇𝑗,𝑡2,𝜌𝜇,𝑖𝜇,𝑀,𝑡𝜇𝑗1,𝜇,𝑡𝜌𝜇,𝑖𝜇,𝑡,𝑀𝜇,𝜇𝑗,𝑡2,(5.25) where 𝜇𝑗𝑗(𝜇𝑗1) implies 𝜌(𝜇𝑗,𝑗(𝜇𝑗1),𝑡)=1. Letting 𝑛,𝑀(𝜇𝑛,𝜇,𝑡)=1, and using the left continuity of 𝜙, we have 𝜌𝜇,𝑖𝜇,𝑡𝜙1,1,𝜌𝜇,𝑖𝜇,𝑡,𝜌𝜇,𝑖𝜇𝜌,𝑡,1𝛾𝜇,𝑖𝜇,𝑡,(5.26) which implies 𝜌(𝜇,𝑖(𝜇),𝑡)=1. Hence, by Lemma 3.2, it follows that 𝜇𝑖(𝜇), 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 (𝒞(𝑋),𝑀,) be a fuzzy metric space, where 𝑋=[1,1], 𝑀𝑑,𝐻𝑀, and 𝑀 are the same as in Example 3.4. Then, (𝒞(𝑋),𝑀,) is a compact metric space.
Now, define 𝜙[0,1][0,1] as 𝜙(𝑥)=𝑥, and define {𝑖}𝑖=1 a sequence of fuzzy self-mappings of 𝒞(𝑋) as 𝑖1(𝜇)=2𝑖𝜇,forany𝜇𝒞(𝑋).(5.27)
For arbitrary positive integers 𝑖 and 𝑗, without loss of generality, suppose 𝑖<𝑗. For each 𝜇1,𝜇2𝒞(𝑋), by a routine calculation, we have 𝑀𝑖𝜇1,𝑗𝜇2,𝑡=𝑀12𝑖𝜇1,12𝑗𝜇2,𝑡=𝑀𝜇1,12𝑗𝑖𝜇2,2𝑖𝑡𝑀𝜇1,𝜇2,2𝑖𝑡𝑀𝜙𝜇1,𝜇2𝑀,𝑡𝜙inf𝜇1,𝜇2,𝜌𝜇1,𝑖𝜇1,2𝑡,𝜌𝜇2,𝑗𝜇2,1,2𝑡2𝜌𝜇2,𝑖𝜇1,4𝑡+𝜌𝜇1,𝑗𝜇2.,4𝑡(5.28) Therefore, by Theorem 5.1, we assert that the sequence of fuzzy self-mappings {𝑖}𝑖=1 has a common fixed point 𝜇 in 𝒞(𝑋). In fact, it is easy to check that 𝜇(𝑥)=1,if𝑥=(0,0,),0,otherwise.(5.29)

6. Conclusion

So far many authors have made a great deal of work in the Hausdorff-Pompeiu metric [2025]. To describe the degree of nearness between two crisp sets, Rodrguez-López and Romaguera have defined Hausdorff fuzzy metric. In this paper, we define a new 𝑀-fuzzy metric, which describes the degree of nearness between two fuzzy sets. Then, some properties on 𝑀-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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