Common Fixed Point Theorems in a New Fuzzy Metric Space

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 [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 𝑑_∞-metric for fuzzy sets [20–25] 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 𝐻_𝑀, 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 𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 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,𝑧∈𝑋,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,𝑦∈𝑋. Let (𝑋,𝑑) be a metric space. Denote by 𝑎⋅𝑏 the usual multiplication for all 𝑎,𝑏∈[0,1], and let 𝑀_𝑑 be the fuzzy set defined on 𝑋×𝑋×(0,∞) by𝑀_𝑑(𝑥,𝑦,𝑡)=𝑡𝑡+𝑑(𝑥,𝑦).(2.1) Then, (𝑋,𝑀_𝑑,⋅) is a fuzzy metric space, and (𝑀_𝑑,⋅) 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 𝜏_𝑀 on 𝑋 which has a base the family of open sets of the form:𝐵_𝑀(𝑥,𝜀,𝑡)∶𝑥∈𝑋,0<𝜀<1,𝑡>0,(2.2) where 𝐵_𝑀(𝑥,𝜀,𝑡)={𝑦∈𝑋∶𝑀(𝑥,𝑦,𝑡)>1−𝜀} for all 𝜀∈(0,1) and 𝑡>0. They proved that (𝑋,𝜏_𝑀) 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 𝜏_𝑀𝑑 generated by the induced fuzzy metric (𝑀_𝑑,∗) (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 {𝑥_𝑛}_𝑛∈ℕ in 𝑋, 𝑥_𝑛→𝑥 if and only if 𝑀(𝑥_𝑛,𝑥,𝑡)→1 as 𝑛→∞ for all 𝑡>0.

Definition 2.3 (see [6]). A sequence {𝑥_𝑛}_𝑛∈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∈ℕ such that 𝑀(𝑥_𝑛,𝑥_𝑚,𝑡)>1−𝜀 for all 𝑛,𝑚≥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,∗). For 𝑎∈𝑋 and 𝑡>0, 𝑀(𝑎,𝐴,𝑡)=sup{𝑀(𝑎,𝑦,𝑡)∣𝑦∈𝐴,𝑡>0}.

Lemma 2.5 (see [28]). Let 𝐺 be a set and let {𝐺_𝛼∶𝛼∈[0,1]} be a family of subsets of 𝐺 such that(1)𝐺₀=𝐺;(2)𝛼≤𝛽 implies 𝐺_𝛽⊆𝐺_𝛼;(3)𝛼₁≤𝛼₂≤⋯, lim_𝑛→∞𝛼_𝑛=𝛼 implies 𝐺_𝛼=⋂^∞𝑛=1𝐺_𝛼𝑛.Then, the function 𝜑∶𝐺→[0,1] defined by 𝜑(𝑥)=sup{𝛼∈[0,1]∶𝑥∈𝐺_𝛼} has the property that {𝑥∈𝐺∶𝜑(𝑥)≥𝛼}=𝐺_𝛼  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 𝐻_𝑑 defined on 𝐶₀(𝑋)×𝐶₀(𝑋) by𝐻_𝑑(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝐴,𝑏),(2.3) for all 𝐴,𝐵∈𝐶₀(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,𝐵∈𝐶(𝑋), 𝑡>0, according to [26], the Hausdorff fuzzy metric 𝐻_𝑀 on 𝐶(𝑋)×𝐶(𝑋)×(0,∞) is defined as𝐻_𝑀(𝐴,𝐵,𝑡)==mmiinn{𝜌i(n𝐴f_𝑎∈𝐴,𝑀𝐵,(𝑡)𝑎,,𝜌𝐵(,𝐵𝑡),,𝐴in,f_𝑏∈𝐵𝑡)𝑀},(𝐴,𝑏,𝑡)(2.4) where 𝜌(𝐴,𝐵,𝑡)=inf_𝑎∈𝐴𝑀(𝑎,𝐵,𝑡), and (𝐻_𝑀,∗) is a fuzzy metric on 𝐶(𝑋). It is shown that 𝜌(𝐴,𝐵,𝑡)=1 if and only if 𝐴⊆𝐵, and 𝐻_𝑀(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),𝐻_𝑀,∗) 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 (𝐻_𝑀𝑑,⋅) of the standard fuzzy metric (𝑀_𝑑,⋅) coincides with standard fuzzy metric (𝑀_𝐻𝑑,⋅) of the Hausdorff metric 𝐻_𝑑 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:𝜇∶𝑋⟶[0,1]=𝐼,(3.1) which satisfy that, for each 𝜆∈𝐼, the 𝜆-cut of 𝜇,[𝜇]_𝜆={𝑥∈𝑋∶𝜇(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 𝐻_𝑀 as 𝑀_∞𝜇₁,𝜇₂,𝑡=min𝜌_∞𝜇₁,𝜇₂,𝑡,𝜌_∞𝜇₂,𝜇₁,𝑡,(3.3) where 𝜇₁,𝜇₂∈𝒞(𝑋),𝑡>0, and 𝜌_∞𝜇₁,𝜇₂,𝑡=inf_0≤𝜆≤1𝜌𝜇₁_𝜆,𝜇₂_𝜆,𝑡(3.4) is the fuzzy separation of 𝜇₁ from 𝜇₂.

Lemma 3.2. Let (𝑋,𝑀,∗) be a fuzzy metric space, 𝜇₁,𝜇₂,𝜇₃∈𝒞(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 𝜇₁⊆𝜇₂,(4)𝑀_∞(𝜇₁,𝜇₂,𝑡)=1 if and only if 𝜇₁=𝜇₂,(5)if 𝜇₁⊆𝜇₂, 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 𝜆∈𝐼, [𝜇₁]_𝜆 is nonempty compact in 𝑋. By the theorem of nested intervals, there exists a point 𝑎₀ in [𝜇₁]_𝜆 for every 𝜆∈𝐼, likewise, there exists a points 𝑏₀ in [𝜇₂]_𝜆 for every 𝜆∈𝐼. Thus, 𝑀_∞(𝜇₁,𝜇₂,𝑡)>0. Moreover, it is clear that 𝐴=𝐵⇔𝐻_𝑀(𝐴,𝐵,𝑡)=1⇔𝑀_∞(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 𝜆∈𝐼, which implies [𝜇₁]_𝜆⊆[𝜇₂]_𝜆 for all 𝜆∈𝐼, we have that 𝜌_∞(𝜇₁,𝜇₂,𝑡)=1 if and only if 𝜇₁⊆𝜇₂.
For (4), it follows from (3).
For (5), for every 𝜆∈𝐼, 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,𝑧∈𝑋, which implies 𝑀𝑥,𝜇₃_𝜆,𝑡+𝑠≥𝑀(𝑥,𝑦,𝑡)∗𝑀𝑦,𝜇₃_𝜆,𝑠(3.6) for all 𝑥∈[𝜇₁]_𝜆 and all 𝑦∈[𝜇₂]_𝜆. Since 𝜇₁⊆𝜇₂, then 𝜌([𝜇₁]_𝜆,[𝜇₂]_𝜆,𝑠)=1. By (iv) of Definition 2.1 and the arbitrariness of 𝑥 and 𝑦, we have 𝜌𝜇₁_𝜆,𝜇₃_𝜆,𝑡+𝑠=≥iinnff_{𝑥∈[𝜇1]𝜆𝑥∈[𝜇1]𝜆}𝑀𝑥,𝜇₃_𝜆,𝑡+𝑠=𝜌𝜇₁𝑀𝑥,𝜇₂_𝜆,𝑠∗inf_{𝑦∈[𝜇2]𝜆}𝑀𝑦,𝜇₃_𝜆,𝑡=𝜌𝜇₂_𝜆,𝜇₂_𝜆,𝑠∗𝜌𝜇₂_𝜆,𝜇₃_𝜆,𝑡≥𝐻_𝑀_𝜆,𝜇₃_𝜆,𝑡𝜇₂_𝜆,𝜇₃_𝜆,𝑡,(3.7) which implies inf_0≤𝜆≤1𝜌𝜇₁_𝜆,𝜇₃_𝜆,𝑡+𝑠≥inf_0≤𝜆≤1𝐻_𝑀𝜇₂_𝜆,𝜇₃_𝜆,𝑡.(3.8) Consequently, 𝜌_∞(𝜇₁,𝜇₃,𝑡+𝑠)≥𝑀_∞(𝜇₂,𝜇₃,𝑡).
For (6), for every 𝜆∈𝐼, by the proof of (5) and (iv) of Definition 2.1, we have inf_0≤𝜆≤1𝜌𝜇₁_𝜆,𝜇₃_𝜆,𝑡+𝑠≥≥iinnff_{0≤𝜆≤10≤𝜆≤1}𝜌𝜇₁_𝜆,𝜇₂_𝜆,𝑡∗𝜌𝜇₂_𝜆,𝜇₃_𝜆,𝑠𝐻_𝑀𝜇₁_𝜆,𝜇₂_𝜆,𝑡∗𝜌𝜇₂_𝜆,𝜇₃_𝜆,𝑠.(3.9) Consequently, 𝜌_∞(𝜇₁,𝜇₃,𝑡+𝑠)≥𝑀_∞(𝜇₁,𝜇₂,𝑡)∗𝜌_∞(𝜇₂,𝜇₃,𝑠).
For (7), for every 𝜆∈𝐼, by the proof of (6), we have inf_0≤𝜆≤1𝜌𝜇₁_𝜆,𝜇₃_𝜆,𝑡+𝑠≥≥infin_0≤𝜆≤1f_0≤𝜆≤1𝜌𝜇₁_𝜆,𝜇₂_𝜆,𝑡∗𝜌𝜇₂_𝜆,𝜇₃_𝜆,𝑠𝜌𝜇₁_𝜆,𝜇₂_𝜆,𝑡∗inf_0≤𝜆≤1𝜌𝜇₂_𝜆,𝜇₃_𝜆,𝑠.(3.10) Similarly, it can be shown that inf_0≤𝜆≤1𝜌𝜇₃_𝜆,𝜇₁_𝜆,𝑡+𝑠≥inf_0≤𝜆≤1𝜌𝜇₃_𝜆,𝜇₂_𝜆,𝑠∗inf_0≤𝜆≤1𝜌𝜇₂_𝜆,𝜇₁_𝜆,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 𝑡↦𝐻_𝑀(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 ℝ, and let 𝐴=[𝑎₁,𝑎₂] and let 𝐵=[𝑏₁,𝑏₂] be two compact intervals. Then, 𝐻_𝑑(𝐴,𝐵)=max{|𝑎₁−𝑏₁|,|𝑎₂−𝑏₂|}. Let (ℝ,𝑀_𝑑,∗) be a fuzzy metric space, where 𝑎∗𝑏 the usual multiplication for all 𝑎,𝑏∈[0,1], and 𝑀_𝑑 is defined on ℝ×ℝ×(0,∞) by 𝑀_𝑑(𝑥,𝑦,𝑡)=𝑡𝑡+𝑑(𝑥,𝑦).(3.12) Denote by 𝒞(ℝ) the totality of fuzzy sets 𝜇∶ℝ→[0,1] which satisfy that for each 𝜆∈𝐼, the 𝜆-cut of 𝜇[𝜇]_𝜆={𝑥∈ℝ∶𝜇(𝑥)≥𝜆} is a nonempty compact interval. For any 𝜆-cuts of fuzzy sets 𝜇₁,𝜇₂∈𝒞(ℝ) and for all 𝑡>0, by a simple calculation, we have 𝐻_𝑀𝜇₁_𝜆,𝜇₂_𝜆,𝑡=𝑡𝑡+𝐻_𝑑𝜇₁_𝜆,𝜇₂_𝜆.(3.13) So by Definition 3.1, we get 𝑀_∞𝜇₁,𝜇₂,𝑡=inf_0≤𝜆≤1𝑡𝑡+𝐻_𝑑𝜇₁_𝜆,𝜇₂_𝜆.(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 𝜇∈𝒞(𝑋) and radius 𝑟,0<𝑟<1,𝑡>0 as 𝐵(𝜇,𝑟,𝑡)=𝛾∈𝒞(𝑋)∣𝑀_∞(𝜇,𝛾,𝑡)>1−𝑟.(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 𝜏_𝑀∞={𝒜⊂𝒞(𝑋)∣𝜇∈𝒜ifandonlyifthereexist𝑡>0and𝑟,0<𝑟<1suchthat𝐵(𝜇,𝑟,𝑡)⊂𝒜}.
Then, 𝜏_𝑀∞ is a topology on 𝒞(𝑋).

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

Definition 4.4. A sequence {𝜇_𝑛} 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∈ℕ such that 𝑀_∞(𝜇_𝑛,𝜇_𝑚,𝑡)>1−𝜀 for all 𝑛,𝑚≥𝑛₀.

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 {𝜇_𝑛} in 𝒞(𝑋), 𝜇_𝑛→𝜇 if and only if 𝑀_∞(𝜇,𝜇_𝑛,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 {𝜇_𝑛,𝑛≥1} be a Cauchy sequence in 𝒞(𝑋). Consider a fixed 0<𝜆<1. Then, {[𝜇_𝑛]_𝜆,𝑛≥1} is a Cauchy sequence in (𝐶(𝑋),𝐻_𝑀,∗), where 𝐶(𝑋) denotes all nonempty compact subsets of (𝑋,𝑀,∗).
Since (𝐶(𝑋),𝐻_𝑀,∗) is complete by Lemma 2.6, it follows that [𝜇_𝑛]_𝜆→𝜇_𝜆∈𝐶(𝑋). Actually, from the definition of 𝑀_∞ and the continuity of 𝐻_𝑀, it is easy to see that [𝜇_𝑛]_𝜆→𝜇_𝜆, uniformly in 𝜆∈[0,1].
Now, consider the family {𝜇_𝜆∶𝜆∈[0,1]}, where 𝜇₀=𝑋. Take 𝜆≤𝛽, we have 𝜌𝜇_𝛽,𝜇_𝜆,𝑡≥𝜌𝜇_𝛽,𝜇_𝑛_𝛽,𝑡3∗𝜌𝜇^𝑛_𝛽,𝜇_𝑛_𝜆,𝑡3∗𝜌𝜇^𝑛_𝜆,𝜇_𝜆,𝑡3.(4.2) Since [𝜇_𝑛]_𝛽⊆[𝜇_𝑛]_𝜆, it follows that 𝜌([𝜇_𝑛]_𝛽,[𝜇_𝑛]_𝜆,𝑡/3)=1. Thus, for each 0<𝜀<1, 𝜌(𝜇_𝛽,𝜇_𝜆,𝑡)≥𝜌(𝜇_𝛽,[𝜇_𝑛]_𝛽,𝑡/3)∗𝜌([𝜇_𝑛]_𝜆,𝜇_𝜆,𝑡/3) if 𝑛 is large enough. Hence, 𝜌(𝜇_𝛽,𝜇_𝜆,𝑡)=1, and by Lemma 3.2, we have 𝜇_𝛽⊆𝜇_𝜆.
Now, take 𝜆_𝑛↑ and lim_𝑛→∞𝜆_𝑛=𝜆. We have to show that 𝜇_𝜆=⋂^∞𝑛=1𝜇_𝜆𝑛. It is clear that 𝜇_𝜆⊆^∞_𝑛=1𝜇_𝜆𝑛.(∗) On the other hand, we have 𝜌^∞_𝑛=1𝜇_𝜆𝑛,𝜇_𝜆,𝑡≥𝜌^∞_𝑛=1𝜇_𝜆𝑛,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡3∗𝜌^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝜇_𝑗_𝜆,𝑡3∗𝜌𝜇^𝑗_𝜆,𝜇_𝜆,𝑡3,(4.3) for fixed 𝑗. However, 𝜌^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝜇_𝑗_𝜆,𝑡3=1.(4.4) Consequently, for every 0<𝜀<1, there exists 0<𝜀₀<𝜀<1 such that (1−𝜀₀)∗(1−𝜀₀)∗(1−𝜀₀)>1−𝜀. For given 𝜀₀, since [𝜇_𝑗]_𝜆→𝜇_𝜆, there exists 𝑗_𝜀0 such that 𝜌^∞_𝑛=1𝜇_𝜆𝑛,𝜇_𝜆,𝑡≥𝜌^∞_𝑛=1𝜇_𝜆𝑛,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡3∗1−𝜀⁰,(4.5) for 𝑗≥𝑗_𝜀0. Now, 𝜌^∞_𝑛=1𝜇_𝜆𝑛,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡3≥𝜌^∞_𝑛=1𝜇_𝜆𝑛,𝜇_𝜆𝑝,𝑡9∗𝜌𝜇^𝜆𝑝,𝜇_𝑗_𝜆𝑛,𝑡9∗𝜌𝜇^𝑗_𝜆𝑝,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡9,(4.6) for any 𝑝≥1. Since ⋂^∞𝑛=1𝜇_𝜆𝑛⊆𝜇_𝜆𝑝, we obtain 𝜌^∞_𝑛=1𝜇_𝜆𝑛,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡≥𝜌𝜇_𝜆𝑝,𝜇_𝑗_𝜆𝑝,𝑡2∗𝜌𝜇^𝑗_𝜆𝑝,^∞_𝑛=1𝜇_𝑗_𝜆𝑛,𝑡2.(4.7) Now, 𝜌(𝜇_𝜆𝑝,[𝜇_𝑗]_𝜆𝑝,𝑡/2)>1−𝜀₀ for 𝑗≥𝑗₀ and all 𝑡>0. Note that (since the convergence [𝜇_𝑗]_𝜆→𝜇_𝜆 is uniform in 𝜆) 𝑗₀ does not depend on 𝑝. Since {[𝜇_𝑗]_𝜆𝑝,𝑝≥1} decreases to ⋂^∞𝑛=1[𝜇_𝑗]_𝜆𝑛, if follows that 𝜌([𝜇_𝑗]_𝜆𝑝0,⋂^∞𝑛=1[𝜇_𝑗]_𝜆𝑛,𝑡/2)>1−𝜀₀ for some 𝑝₀ (depending on 𝑗).
Thus, 𝜌(⋂^∞𝑛=1𝜇_𝜆𝑛,⋂^∞𝑛=1[𝜇_𝑗]_𝜆𝑛,𝑡/3)≥(1−𝜀₀)∗(1−𝜀₀), if 𝑗 is large.
Finally, by taking 𝑗 large enough, we obtain 𝜌^∞_𝑛=1𝜇_𝜆𝑛,𝜇_𝜆,𝑡≥1−𝜀₀∗1−𝜀₀∗1−𝜀₀≥1−𝜀,(4.8) that is, ^∞_𝑛=1𝜇_𝜆𝑛⊆𝜇_𝜆.(∗∗) From (4.3) and (4.9), it yields ⋂^∞𝑛=1𝜇_𝜆𝑛=𝜇_𝜆. Thus, Lemma 2.5 is applicable and there exists 𝜇∈𝒞(𝑋) for every 𝜆∈[0,1] such that [𝜇_𝑛]_𝜆→𝜇_𝜆. It remains to show that 𝜇_𝑛→𝜇 in (𝒞(𝑋),𝑀_∞,∗).
Let 𝜀>0. Then, since {𝜇_𝑛} is a Cauchy sequence, there exists 𝑛_𝜀 such that 𝑛,𝑚>𝑛_𝜀 implies 𝑀_∞(𝜇_𝑛,𝜇_𝑚,𝑡)>1−𝜀.
Let 𝑛(>𝑛_𝜀) be fixed. Then, 𝐻_𝑀𝜇_𝑛_𝜆,[𝜇]_𝜆,𝑡==lim_𝑚→∞𝐻_𝑀𝜇_𝑛_𝜆,𝜇_𝑚_𝜆,𝑡≥lim_𝑚→∞inf_0≤𝜆≤1𝐻_𝑀𝜇_𝑛_𝜆,𝜇_𝑚_𝜆,𝑡lim_𝑚→∞𝑀_∞𝜇_𝑛,𝜇_𝑚,𝑡>1−𝜀.(4.9) Thus, 𝜇_𝑛→𝜇 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,𝐵∈𝐶(𝑋). Then, for each 𝑥∈𝐴 and 𝑡>0, there exists a 𝑦∈𝐵 such that 𝑀(𝑥,𝑦,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡).

Proof. Suppose there exists a 𝑥₀∈𝐴 such that 𝑀(𝑥₀,𝑦,𝑡)<𝐻_𝑀(𝐴,𝐵,𝑡) for any 𝑦∈𝐵 and 𝑡>0. Then, sup_𝑦∈𝐵𝑀𝑥₀,𝑦,𝑡<𝐻_𝑀(𝐴,𝐵,𝑡),(4.10) that is, sup_𝑦∈𝐵𝑀𝑥₀,𝑦,𝑡<mininf_𝑥∈𝐵sup_𝑦∈𝐴𝑀(𝑥,𝑦,𝑡),inf_𝑥∈𝐴sup_𝑦∈𝐵𝑀(𝑥,𝑦,𝑡).(4.11) So, sup_𝑦∈𝐵𝑀𝑥₀,𝑦,𝑡<inf_𝑥∈𝐴sup_𝑦∈𝐵𝑀(𝑥,𝑦,𝑡).(4.12) This is a contradiction with 𝑥∈𝐴.

Lemma 4.8. Let (𝑋,𝑀,∗) be a compact fuzzy metric space, 𝑡>0 and 𝐴,𝐵∈𝐶(𝑋). Then, for any compact set 𝐴₁⊆𝐴, there exists a compact set 𝐵₁⊆𝐵 such that 𝐻_𝑀(𝐴₁,𝐵₁,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡).

Proof. Let 𝐶={𝑦∶ there exists a 𝑥∈𝐴₁ such that 𝑀(𝑥,𝑦,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡)} and let 𝐵₁=𝐶⋂𝐵. For any 𝑥∈𝐴₁⊆𝐴,𝑡>0, by Lemma 4.7, there exists a 𝑦∈𝐵 such that 𝑀(𝑥,𝑦,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡).(4.13) Thus, 𝐵₁≠∅, moreover, 𝐵₁ is compact since it is closed in 𝑋 and 𝐵₁⊆𝐵.
Now, for any 𝑥∈𝐴₁, 𝑡>0, there exists a 𝑦∈𝐵₁ such that 𝑀(𝑥,𝑦,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡).(4.14) Thus, we have 𝑀(𝑥,𝐵₁,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡), which implies that 𝜌𝐴₁,𝐵₁,𝑡=inf_𝑥∈𝐴1𝑀𝑥,𝐵₁,𝑡≥𝐻_𝑀(𝐴,𝐵,𝑡).(4.15) Similarly, it can be shown that 𝜌(𝐴₁,𝐵₁,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡).
Hence, 𝐻_𝑀(𝐴₁,𝐵₁,𝑡)≥𝐻_𝑀(𝐴,𝐵,𝑡). This completes the proof.

Theorem 4.9. Let (𝑋,𝑀,∗) be a compact fuzzy metric space and 𝜇₁,𝜇₂∈𝒞(𝑋), 𝑡>0. Then, for any 𝜇₃∈𝒞(𝑋) satisfying 𝜇₃⊆𝜇₁, there exists a 𝜇₄∈𝒞(𝑋) such that 𝜇₄⊆𝜇₂ and 𝑀_∞𝜇₃,𝜇₄,𝑡≥𝑀_∞𝜇₁,𝜇₂,𝑡.(4.16)

Proof. Since 𝜇₁,𝜇₂, and 𝜇₃ are normal, we have ∅≠[𝜇₃]_𝜆⊆[𝜇₁]_𝜆 and ∅≠[𝜇₂]_𝜆 for all 𝜆∈𝐼. Let 𝐶_𝜆=𝑦∶thereexistsa𝑥∈𝜇₃_𝜆suchthat𝑀(𝑥,𝑦,𝑡)≥𝑀_∞𝜇₁,𝜇₂,𝑡,(4.17) and let 𝐵_𝜆=𝐶_𝜆⋂[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)≥𝐻_𝑀𝜇₁_𝜆,𝜇₂_𝜆,𝑡≥𝑀_∞𝜇₁,𝜇₂,𝑡.(4.18) Thus, 𝐵_𝜆 is nonempty compact in 𝑋, moreover, 𝐵_𝜆⊆𝐵𝛾 if 0≤𝛾≤𝜆≤1.
From the proof of Lemma 4.8, we have 𝐻_𝑀𝜇₃_𝜆,𝐵_𝜆,𝑡≥𝑀_∞𝜇₁,𝜇₂,𝑡.(4.19) By Lemma  3.1 in [28], there exists a fuzzy set 𝜇₄ with the property that [𝜇₄]_𝜆=𝐵_𝜆 for 𝜆∈𝐼. Since 𝐵_𝜆 are nonempty compact for all 𝜆∈𝐼, we have 𝜇₄∈𝒞(𝑋). Consequently, 𝑀_∞𝜇₃,𝜇₄,𝑡≥𝑀_∞𝜇₁,𝜇₂,𝑡.(4.20) This completes the proof.

Definition 4.10 (see [24]). Let 𝑋,𝑌 be any fuzzy metric space. ℱ is said to be a fuzzy mapping if and only if ℱ is a mapping from the space 𝒞(𝑋) into 𝒞(𝑌), that is, ℱ(𝜇)∈𝒞(𝑌) for each 𝜇∈𝒞(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 {ℱ_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(ℎ)∗𝜙²(ℎ)∗⋯∗𝜙^𝑛(ℎ)⟶1as𝑛⟶∞,∀ℎ∈(0,1],(5.1) where 𝜙^𝑛 denote the 𝑛th iterative function of 𝜙. Suppose that for each 𝜇₁,𝜇₂∈𝒞(𝑋), and for arbitrary positive integers 𝑖 and 𝑗,𝑖≠𝑗,𝑡>0, 𝑀_∞ℱ_𝑖𝜇₁,ℱ_𝑗𝜇₂,𝑡≥𝜙in1f𝑀_∞𝜇₁,𝜇₂,𝑡,𝜌_∞𝜇₁,ℱ_𝑖𝜇₁,2𝑡,𝜌_∞𝜇₂,ℱ_𝑗𝜇₂,2𝑡,2𝜌^∞𝜇₂,ℱ_𝑖𝜇₁,4𝑡+𝜌_∞𝜇₁,ℱ_𝑗𝜇₂,4𝑡,(5.2) then there exists 𝜇_∗∈𝒞(𝑋) such that 𝜇_∗⊆ℱ_𝑖(𝜇_∗) for all 𝑖∈𝑍₊.

Proof. Let 𝜇₀∈𝒞(𝑋) and 𝜇₁⊆ℱ₁(𝜇₀). By Theorem 4.9, for any 𝑡>0, there exists 𝜇₂∈𝒞(𝑋) such that 𝜇₂⊆ℱ₂(𝜇₁) and 𝑀_∞𝜇₁,𝜇₂,𝑡≥𝑀_∞ℱ₁𝜇₀,ℱ₂𝜇₁,𝑡.(5.3) Again by Theorem 4.9, for any 𝑡>0, we can find 𝜇₃∈𝒞(𝑋) such that 𝜇₃⊆ℱ₃(𝜇₂) and 𝑀_∞𝜇₂,𝜇₃,𝑡≥𝑀_∞ℱ₂𝜇₁,ℱ₃𝜇₂,𝑡.(5.4) By induction, we produce a sequence {𝜇_𝑛} of points of 𝒞(𝑋) such that 𝜇𝑀_𝑛+1⊆ℱ_𝑛+1𝜇_𝑛,𝑛=0,1,2,…;_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝑀_∞ℱ_𝑛𝜇_𝑛−1,ℱ_𝑛+1𝜇_𝑛,𝑡.(5.5) Now, we prove that {𝜇_𝑛} 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 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥≥𝑀𝜙_∞inℱf_𝑛𝑀𝜇_𝑛−1∞,ℱ_𝑛+1𝜇_𝑛,𝑡1𝜇_𝑛−1,𝜇_𝑛,𝑡,𝜌_∞𝜇_𝑛−1,ℱ_𝑛𝜇_𝑛−1,2𝑡,𝜌_∞𝜇_𝑛,ℱ_𝑛+1𝜇₂,2𝑡,2𝜌^∞𝜇_𝑛−1≥𝜙inf𝑀_∞,ℱ_𝑛+1𝜇_𝑛,4𝑡+𝜌_∞𝜇_𝑛,ℱ_𝑛𝜇_𝑛−1,4𝑡,𝑀𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡_∞,≥𝜙inf𝑀𝜇_𝑛∞,𝜇_𝑛+1,2𝑡,12𝑀^∞𝜇_𝑛−1,𝜇_𝑛+1,2𝑡+1,𝑀𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡_∞,𝜇_𝑛,𝜇_𝑛+1,2𝑡,12𝑀^∞𝜇_𝑛−1,𝜇_𝑛,𝑡+1∗𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡+1,(5.6) where 𝜇_𝑛⊆ℱ_𝑛(𝜇_𝑛−1) implies 𝜌_∞(𝜇_𝑛,ℱ_𝑛(𝜇_𝑛−1),2𝑡)=1, by (3) of Lemma 3.2. In addition, it is easy to get that 𝜙(ℎ)>ℎ for all ℎ∈(0,1). In fact, suppose that there exists some 𝑡₀∈(0,1) such that 𝜙(ℎ₀)≤ℎ₀. Since 𝜙 is nondecreasing, we have 𝜙^𝑛ℎ₀≤𝜙^𝑛−1ℎ₀≤⋯≤𝜙ℎ₀≤ℎ₀.(5.7) Since 𝜙(ℎ)∗𝜙²(ℎ)∗⋯∗𝜙^𝑛(ℎ)→1 as 𝑛→∞, for all ℎ∈(0,1), then we have 𝜙^𝑛(ℎ₀)→1 as 𝑛→∞. From the inequality (5.7), we have 1≤ℎ₀. This is a contradiction which implies 𝜙(ℎ)>ℎ for all ℎ∈(0,1). We can prove that 𝑀_∞(𝜇_𝑛−1,𝜇_𝑛,𝑡)≤𝑀_∞(𝜇_𝑛,𝜇_𝑛+1,𝑡). In fact, if 𝑀_∞(𝜇_𝑛−1,𝜇_𝑛,𝑡)>𝑀_∞(𝜇_𝑛,𝜇_𝑛+1,𝑡), then from the inequality (5.6), we get 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝜙𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡>𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡,(5.8) which is a contradiction. Thus, from the inequality (5.6), we have 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝜙𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡≥⋯≥𝜙^𝑛𝑀_∞𝜇₀,𝜇₁,𝑡.(5.9) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 1≥≥𝑀𝑀_∞∞𝜇_𝑘,𝜇_𝑘+𝑚,𝑡≥𝜙_𝑘𝜇^𝑘,𝜇_𝑘+1,𝑚𝑡∗𝑀^∞𝜇^𝑘+1,𝜇_𝑘+2,𝑚𝑡∗⋯∗𝑀^∞𝜇^{𝑘+𝑚−1},𝜇_𝑘+𝑚,𝑚𝑡𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡∗𝜙^𝑘+1𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡∗⋯∗𝜙^{𝑘+𝑚−1}𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡,(5.10) and 𝜙(ℎ)∗𝜙²(ℎ)∗⋯∗𝜙^𝑛(ℎ)→1 as 𝑛→∞, for all ℎ∈(0,1), it follows that 𝜙^𝑘𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚∗𝜙^𝑘+1𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚∗⋯∗𝜙^{𝑘+𝑚−1}𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚(5.11) is convergent, which implies that {𝜇_𝑛} 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 𝜇_𝑛→𝜇_∗. Next, we show that 𝜇_∗⊆ℱ_𝑖(𝜇_∗) for all 𝑖∈𝑍₊. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, by Theorem 4.9, we have 𝜌_∞𝜇≥_∗,𝑀ℱ_𝑖∞𝜇_∗,𝑡≥𝑀_∞𝜇^∗,𝜇_𝑗,4𝑡∗𝜌^∞𝜇^𝑗,ℱ_𝑖𝜇_∗,34𝑡≥𝑀_∞𝜇^∗,𝜇_𝑗,4𝑡∗𝑀^∞ℱ^𝑗𝜇_𝑗−1,ℱ_𝑖𝜇_∗,2𝑡𝜇^∗,𝜇_𝑗,4𝑡1∗𝜙inf𝑀^∞𝜇^𝑗−1,𝜇_∗,2𝑡,𝜌^∞𝜇_𝑗−1,ℱ_𝑗𝜇_𝑗−1,𝑡,𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,2𝜌^∞𝜇_∗≥𝑀_∞,ℱ_𝑗𝜇_𝑗−1,2𝑡+𝜌_∞𝜇_𝑗−1,ℱ_𝑖𝜇_∗,2𝑡𝜇^∗,𝜇_𝑗,4𝑡1∗𝜙inf𝑀^∞𝜇^𝑗−1,𝜇_∗,2𝑡,𝑀^∞𝜇^𝑗−1,𝜇_𝑗,2𝑡,𝜌^∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,2𝑀^∞𝜇_∗,𝜇_𝑗,𝑡+𝑀_∞𝜇_∗,𝜇_𝑗−1,𝑡∗𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,(5.12) where 𝜇_𝑗⊆ℱ_𝑗(𝜇_𝑗−1) implies 𝜌_∞(𝜇_𝑗,ℱ_𝑗(𝜇_𝑗−1),𝑡)=1. Letting 𝑛→∞,𝑀_∞(𝜇_𝑛,𝜇_∗,𝑡)=1, and using the left continuity of 𝜙, we have 𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡≥𝜙𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,(5.13) which implies 𝜌_∞(𝜇_∗,ℱ_𝑖(𝜇_∗),𝑡)=1. Hence, by Lemma 3.2, it follows that 𝜇_∗⊆ℱ_𝑖(𝜇_∗). Then, the proof is completed.

Theorem 5.2. Let (𝑋,𝑀,∗) be a compact fuzzy metric space and let {ℱ_𝑖}^∞𝑖=1 be a sequence of fuzzy self-mappings of 𝒞(𝑋). Suppose that for each 𝜇₁,𝜇₂∈𝒞(𝑋), and for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, 𝑡>0, 𝑀_∞ℱ_𝑖𝜇₁,ℱ_𝑗𝜇₂,𝑡≥𝜌𝜙𝑀_∞𝜇₁,𝜇₂,𝑡,𝜌_∞𝜇₁,ℱ_𝑖𝜇₁,2𝑡_∞,𝜇₂,ℱ_𝑗𝜇₂,2𝑡,𝜌_∞𝜇₁,ℱ_𝑗𝜇₂,4𝑡,𝜌_∞𝜇₂,ℱ_𝑖𝜇₁,𝑡,(5.14) where 𝜙(ℎ₁,ℎ₂,ℎ₃,ℎ₄,ℎ₅)∶(0,1]⁵→[0,1] is nondecreasing and continuous from the left for each variable. Denote 𝛾(ℎ)=𝜙(ℎ,ℎ,ℎ,𝑎,𝑏), where (𝑎,𝑏)∈{(ℎ∗ℎ,1),(1,ℎ∗ℎ)}. If 𝛾(ℎ)∗𝛾²(ℎ)∗⋯∗𝛾^𝑛(ℎ)⟶1as𝑛⟶∞,∀ℎ∈(0,1],(5.15) where 𝛾^𝑛 denote the 𝑛th iterative function of 𝛾, then there exists 𝜇_∗∈𝒞(𝑋) such that 𝜇_∗⊆ℱ_𝑖(𝜇_∗) for all 𝑖∈𝑍₊.

Proof. Let 𝜇₀∈𝒞(𝑋) and 𝜇₁⊆ℱ₁(𝜇₀). By Theorem 4.9, for any 𝑡>0, there exists 𝜇₂∈𝒞(𝑋) such that 𝜇₂⊆ℱ₂(𝜇₁) and 𝑀_∞𝜇₁,𝜇₂,𝑡≥𝑀_∞ℱ₁𝜇₀,ℱ₂𝜇₁,𝑡.(5.16) Again by Theorem 4.9, for any 𝑡>0, we can find 𝜇₃∈𝒞(𝑋) such that 𝜇₃⊆ℱ₃(𝜇₂) and 𝑀_∞𝜇₂,𝜇₃,𝑡≥𝑀_∞ℱ₂𝜇₁,ℱ₃𝜇₂,𝑡.(5.17) By induction, we produce a sequence {𝜇_𝑛} of points of 𝒞(𝑋) such that 𝜇𝑀_𝑛+1⊆ℱ_𝑛+1𝜇_𝑛,𝑛=0,1,2,…;_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝑀_∞ℱ_𝑛𝜇_𝑛−1,ℱ_𝑛+1𝜇_𝑛,𝑡.(5.18) Now, we prove that {𝜇_𝑛} 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 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥≥𝑀𝜙_∞𝑀_∞ℱ_𝑛𝜇_𝑛−1,ℱ_𝑛+1𝜇_𝑛,𝑡𝜌𝜇_𝑛−1,𝜇_𝑛,𝑡,𝜌_∞𝜇_𝑛−1,ℱ_𝑛𝜇_𝑛−1,2𝑡_∞,≥𝜙𝑀𝜇_𝑛∞,ℱ_𝑛+1𝜇_𝑛,2𝑡,𝜌_∞𝜇_𝑛−1,ℱ_𝑛+1𝜇_𝑛,4𝑡,𝜌_∞𝜇_𝑛,ℱ_𝑛𝜇_𝑛−1,𝑡≥𝜙𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛+1,2𝑡,1𝑀𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡_∞,𝜇_𝑛,𝜇_𝑛+1,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡∗𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡,1,(5.19) where 𝜇_𝑛⊆ℱ_𝑛(𝜇_𝑛−1) implies 𝜌_∞(𝜇_𝑛,ℱ_𝑛(𝜇_𝑛−1),2𝑡)=1 by (3) in Lemma 3.2 Likewise, we have 𝛾(ℎ)>ℎ for all ℎ∈(0,1), 𝑡>0. If 𝑀_∞(𝜇_𝑛−1,𝜇_𝑛,𝑡)>𝑀_∞(𝜇_𝑛,𝜇_𝑛+1,𝑡), then from the inequality (5.19), we obtain 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝛾𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡>𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡,(5.20) which is a contradiction. Thus, from the inequality (5.19), we have 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝛾𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡≥⋯≥𝛾^𝑛𝑀_∞𝜇₀,𝜇₁,𝑡.(5.21) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝑀𝜙𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡_∞,≥𝜙𝑀𝜇_𝑛∞,𝜇_𝑛+1,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡∗𝑀_∞𝜇_𝑛,𝜇_𝑛+1,𝑡,1𝑀𝜇_𝑛−1,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡_∞,=𝛾𝑀𝜇_𝑛−1∞,𝜇_𝑛,𝑡,𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡∗𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡,1𝑀𝜇_𝑛−1,𝜇_𝑛,𝑡_∞,𝜇_𝑛,𝜇_𝑛+1,𝑡≥𝛾𝑀_∞𝜇_𝑛−1,𝜇_𝑛,𝑡≥⋯≥𝛾𝑀_∞𝜇₀,𝜇₁,𝑡.(5.22) Furthermore, for arbitrary positive integers 𝑚 and 𝑘, we have 1≥≥𝑀𝑀_∞∞𝜇_𝑘,𝜇_𝑘+𝑚,𝑡≥𝛾_𝑘𝜇^𝑘,𝜇_𝑘+1,𝑚𝑡∗𝑀^∞𝜇^𝑘+1,𝜇_𝑘+2,𝑚𝑡∗⋯∗𝑀^∞𝜇^{𝑘+𝑚−1},𝜇_𝑘+𝑚,𝑚𝑡𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡∗𝛾^𝑘+1𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡∗⋯∗𝛾^{𝑘+𝑚−1}𝑀^∞𝜇⁰,𝜇₁,𝑚𝑡.(5.23) Since 𝜙(ℎ)∗𝜙²(ℎ)∗⋯∗𝜙^𝑛(ℎ)→1 as 𝑛→∞, for all ℎ∈(0,1), it follows that 𝛾^𝑘𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚∗𝛾^𝑘+1𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚∗⋯∗𝛾^{𝑘+𝑚−1}𝑀^∞𝜇⁰,𝜇₁,𝑡𝑚(5.24) is convergent, this implies that {𝜇_𝑛} 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 𝜇_𝑛→𝜇_∗. Now, we show that 𝜇_∗⊆ℱ_𝑖(𝜇_∗) for all 𝑖∈𝑍₊. In fact, for arbitrary positive integers 𝑖 and 𝑗, 𝑖≠𝑗, by Theorem 4.9, we have 𝜌_∞𝜇≥_∗,𝑀ℱ_𝑖∞𝜇_∗,𝑡≥𝑀_∞𝜇^∗,𝜇_𝑗,4𝑡∗𝜌^∞𝜇^𝑗,ℱ_𝑖𝜇_∗,34𝑡≥𝑀_∞𝜇^∗,𝜇_𝑗,4𝑡∗𝑀^∞ℱ^𝑗𝜇_𝑗−1,ℱ_𝑖𝜇_∗,2𝑡𝜇^∗,𝜇_𝑗,4𝑡𝜌∗𝜙𝑀^∞𝜇^𝑗−1,𝜇_∗,2𝑡,𝜌^∞𝜇_𝑗−1,ℱ_𝑗𝜇_𝑗−1,𝑡_∞,≥𝑀_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,𝜌_∞𝜇_𝑗−1,ℱ_𝑖𝜇_∗,2𝑡,𝜌_∞𝜇_∗,ℱ_𝑗𝜇_𝑗−1,𝑡𝜇^∗,𝜇_𝑗,4𝑡𝑀∗𝜙𝑀^∞𝜇^𝑗−1,𝜇_∗,2𝑡,𝑀^∞𝜇^𝑗−1,𝜇_𝑗,2𝑡,𝜌^∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡_∞,𝜇_𝑗−1,𝜇_∗,𝑡∗𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,𝑀_∞𝜇^∗,𝜇_𝑗,2𝑡,(5.25) where 𝜇_𝑗⊆ℱ_𝑗(𝜇_𝑗−1) implies 𝜌_∞(𝜇_𝑗,ℱ_𝑗(𝜇_𝑗−1),𝑡)=1. Letting 𝑛→∞,𝑀_∞(𝜇_𝑛,𝜇_∗,𝑡)=1, and using the left continuity of 𝜙, we have 𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡≥𝜙1,1,𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,1≥𝛾𝜌_∞𝜇_∗,ℱ_𝑖𝜇_∗,𝑡,(5.26) which implies 𝜌_∞(𝜇_∗,ℱ_𝑖(𝜇_∗),𝑡)=1. Hence, by Lemma 3.2, it follows that 𝜇_∗⊆ℱ_𝑖(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], 𝑀_𝑑,𝐻_𝑀, 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 {ℱ_𝑖}^∞𝑖=1 a sequence of fuzzy self-mappings of 𝒞(𝑋) as ℱ_𝑖(𝜇)=12^𝑖𝜇,forany𝜇∈𝒞(𝑋).(5.27)
For arbitrary positive integers 𝑖 and 𝑗, without loss of generality, suppose 𝑖<𝑗. For each 𝜇₁,𝜇₂∈𝒞(𝑋), by a routine calculation, we have 𝑀_∞ℱ_𝑖𝜇₁,ℱ_𝑗𝜇₂,𝑡=𝑀_∞21^𝑖𝜇₁,12^𝑗=𝑀_∞𝜇₂,𝑡𝜇¹,12^𝑗−𝑖≥𝑀_∞𝜇₂,2^𝑖𝑡≥𝜙𝑀_∞𝜇₁,𝜇₂,2^𝑖𝑡≥𝜙inf𝜇𝑀₁,_∞𝜇₂,𝑡1𝜇₁,𝜇₂,𝜌_∞𝜇₁,ℱ_𝑖𝜇₁,2𝑡,𝜌_∞𝜇₂,ℱ_𝑗𝜇₂,2𝑡,2𝜌^∞𝜇₂,ℱ_𝑖𝜇₁,4𝑡+𝜌_∞𝜇₁,ℱ_𝑗𝜇₂,4𝑡.(5.28) Therefore, by Theorem 5.1, we assert that the sequence of fuzzy self-mappings {ℱ_𝑖}^∞𝑖=1 has a common fixed point 𝜇_∗ in 𝒞(𝑋). In fact, it is easy to check that 𝜇_∗(𝑥)=10,,iofth𝑥er=w(i0se,.0,…),(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-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).