On Convergence of Fixed Points in Fuzzy Metric Spaces

Abstract

We mainly focus on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in fuzzy metric spaces. Our results provide a novel research direction for fixed point theory in fuzzy metric spaces as well as a substantial extension of several important results from classical metric spaces.

1. Introduction

Fixed point theory of classical metric spaces plays an important role in general topology. In 1988, Grabiec [1] first extended fixed point theorems of Banach and Edelstein to fuzzy metric spaces in the sense of Kramosil and Michalek. Since then, many authors had dedicated themselves to the study of fixed point theory in fuzzy metric spaces [2–18]. Besides, some authors extended fixed point theory to other types of fuzzy metric spaces in recent years. For instance, Alaca et al. [19] extended the well-known fixed point theorems of Banach and Edelstein to intuitionistic fuzzy metric spaces with the help of Grabiec's work. Simultaneously, Mohamad [20] and Razani [21] proved the existence of fixed point for a nonexpansive mapping of intuitionistic fuzzy metric spaces and the intuitionistic Banach fixed point theorem in complete intuitionistic fuzzy metric spaces, respectively. Later, Ćirić et al. [22] investigated the existence of fixed points for a class of asymptotically nonexpansive mappings in an arbitrary intuitionistic fuzzy metric space. On the other hand, Adibi et al. [23] extended a common fixed point theorem to 𝐿-fuzzy metric spaces and proved a coincidence point theorem and a fixed point theorem for compatible mappings of type (𝑃) in these spaces. In 2008, Ješić and Babačev [24] further studied some common fixed point theorems for a pair of 𝑅-weakly commuting mappings with nonlinear contractive condition in intuitionistic fuzzy metric spaces and 𝐿-fuzzy metric spaces. In the same year, Park et al. [25] extended some common fixed point theorems for five mappings to 𝑀-fuzzy metric spaces. Up to now, one can see that the majority of papers mainly focus on the existence of fixed points for different mappings in different fuzzy metric spaces. However, the aim of this paper is to show that the convergence of the sequence of fixed points to some sequences of contraction mappings or fuzzy metrics satisfies certain conditions in fuzzy metric spaces.

2. Preliminaries

Now, we begin with some basic concepts and lemmas. Let ℕ denote the set of all positive integers.

Definition 1 (Schweizer and Sklar [26]). A binary operation ∗∶[0,1]×[0,1]→[0,1] is called a continuous triangular norm (shortly, continuous 𝑡-norm) if it satisfies the following conditions:(TN-1)∗ is commutative and associative;(TN-2)∗ is continuous;(TN-3)𝑎∗1=𝑎 for every 𝑎∈[0,1];(TN-4)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐, 𝑏≤𝑑, and 𝑎,𝑏,𝑐,𝑑∈[0,1].

In particular, a t-norm ∗ is said to be positive [27] if 𝑎∗𝑏>0 whenever 𝑎,𝑏∈(0,1].

We redefine the notion of a fuzzy metric space by appending the following condition (FM-6) based on the one in the sense of George and Veeramani [2].

Definition 2. A fuzzy metric space is an ordered triple (𝑋,𝑀,∗) such that 𝑋 is a (nonempty) set, ∗ is a continuous t-norm, and 𝑀 is a fuzzy set on 𝑋×𝑋×(0,∞) satisfying the following conditions, for all 𝑥,𝑦,𝑧∈𝑋,  𝑠,𝑡>0: (FM-1) 𝑀(𝑥,𝑦,𝑡)>0; (FM-2) 𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦; (FM-3) 𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡); (FM-4) 𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠); (FM-5) 𝑀(𝑥,𝑦,⋅)∶(0,∞)→(0,1] is continuous; (FM-6) lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1.

Definition 3 (Grabiec [1] and Vasuki and Veeramani [17]). Let (𝑋,𝑀,∗) be a fuzzy metric space. Then(a)a sequence {𝑥_𝑛} is said to converge to 𝑥 in 𝑋, denoted by 𝑥_𝑛→𝑥, if and only if lim_𝑛→∞𝑀(𝑥_𝑛,𝑥,𝑡)=1 for all 𝑡>0; that is, for each 𝑟∈(0,1) and 𝑡>0, there exists an 𝑛₀∈ℕ such that 𝑀(𝑥_𝑛,𝑥,𝑡)>1−𝑟 for all 𝑛≥𝑛₀;(b)a sequence {𝑥_𝑛} in 𝑋 is a G-Cauchy sequence if and only if lim_𝑛→∞𝑀(𝑥_𝑛+𝑝,𝑥_𝑛,𝑡)=1 for any 𝑝>0 and 𝑡>0;(c)the fuzzy metric space (𝑋,𝑀,∗) is called G-complete if every G-Cauchy sequence is convergent.

Definition 4 (Grabiec [1]). Let (𝑋,𝑀,∗) be a fuzzy metric space. A mapping 𝑇∶𝑋→𝑋 is called a contraction mapping if there exists 𝑘∈(0,1) such that 𝑀𝑇𝑥,𝑇𝑦,𝑘𝑡≥𝑀𝑥,𝑦,𝑡 (1) for every 𝑥,𝑦∈𝑋 and 𝑡>0.

According to fuzzy Banach contraction theorem of complete fuzzy metric space in the sense of Grabiec [1], we can obtain the following lemma.

Lemma 5. Let (𝑋,𝑀,∗) be a G-complete fuzzy metric space. If 𝑇∶𝑋→𝑋 is a contraction mapping, then 𝑇 has a unique fixed point.

Definition 6. Let (𝑋,𝑀,∗) be a fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋. 𝑇₀∶𝑋→𝑋 is a given mapping. The sequence {𝑇_𝑛} is said to converge pointwise to 𝑇₀ if for each 𝑟∈(0,1) and 𝑥₀∈𝑋, there exists an 𝑛₀∈ℕ such that 𝑀𝑇_𝑛𝑥₀,𝑇₀𝑥₀,𝑡>1−𝑟 (2) for all 𝑛≥𝑛₀ and 𝑡>0.

Definition 7. Let (𝑋,𝑀,∗) be a fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋. 𝑇₀∶𝑋→𝑋 is a given mapping. The sequence {𝑇_𝑛} is said to converge uniformly to 𝑇₀ if for each 𝑟∈(0,1) and 𝑡>0, there exists an 𝑛₀∈ℕ such that 𝑀𝑇_𝑛𝑥,𝑇₀𝑥,𝑡>1−𝑟 (3) for all 𝑛≥𝑛₀ and 𝑥∈𝑋.

Definition 8. Let (𝑋,𝑀,∗) be a fuzzy metric space. A sequence of self-mappings {𝑇_𝑛} is uniformly equicontinuous if for each 𝑟∈(0,1), there exists an 𝜖∈(0,1) such that 𝑀(𝑥,𝑦,𝑠)>1−𝜖 implies 𝑀(𝑇_𝑛𝑥,𝑇_𝑛𝑦,𝑡)>1−𝑟 for every 𝑥,𝑦∈𝑋,  𝑛∈ℕ, and 𝑠,𝑡>0.

Definition 9 (George and Veeramani [2]). Let (𝑋,𝑀,∗) be a fuzzy metric space. The open ball 𝐵(𝑥,𝑟,𝑡) and closed ball 𝐵[𝑥,𝑟,𝑡] with center 𝑥∈𝑋 and radius 𝑟, 0<𝑟<1, 𝑡>0, respectively, are defined as follows: 𝐵()=,𝐵[]=.𝑥,𝑟,𝑡𝑦∈𝑋∶𝑀𝑥,𝑦,𝑡>1−𝑟𝑥,𝑟,𝑡𝑦∈𝑋∶𝑀𝑥,𝑦,𝑡≥1−𝑟 (4)

Lemma 10 (George and Veeramani [2]). Every open (closed) ball is an open (a closed) set.

Definition 11 (Gregori and Romaguera [3]). A fuzzy metric space (𝑋,𝑀,∗) is a compact space if (𝑋,𝜏_𝑀) is a compact topological space, where 𝜏_𝑀 is a topology induced by the fuzzy metric 𝑀.

Based on the corresponding conclusions stated in [2], we can easily obtain the following lemma.

Lemma 12. Every closed subset 𝐴 of a compact fuzzy metric space (𝑋,𝑀,∗) is compact.

Lemma 13. Let (𝑋,𝑀,∗) be a fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋. 𝑇₀∶𝑋→𝑋 is a contraction mapping of 𝑋; that is, there exists 𝑘∈(0,1) such that 𝑀(𝑇₀𝑥,𝑇₀𝑦,𝑘𝑡)≥𝑀(𝑥,𝑦,𝑡) for every 𝑥,𝑦∈𝑋. 𝐴 is a compact subset of 𝑋. If {𝑇_𝑛} converges pointwise to 𝑇₀ in 𝐴 and it is a uniformly equicontinuous sequence, then the sequence {𝑇_𝑛} converges uniformly to 𝑇₀ in 𝐴.

Proof. For each 𝑟∈(0,1), we may choose an appropriate 𝑟 such that (1−𝑟)∗(1−𝑟)∗(1−𝑟)>1−𝑟. Since {𝑇_𝑛} is uniformly equicontinuous, there exists 𝜖∈(0,1)(𝜖≤𝑟) such that 𝑀(𝑥,𝑦,𝑠)>1−𝜖⇒𝑀(𝑇_𝑛𝑥,𝑇_𝑛𝑦,𝑡)>1−𝑟 for every 𝑥,𝑦∈𝑋, 𝑠,𝑡>0, and 𝑛∈ℕ. For the foregoing 𝜖, we fix 𝑠>0. Define 𝒞={𝐵(𝑥,𝜖,𝑠)∶𝑥∈𝐴}. By Lemma 10, 𝒞 is a family of open sets of 𝐴. Obviously, 𝒞 constitutes an open covering of 𝐴; that is, 𝐴⊂⋃𝐵(𝑥,𝜖,𝑠). Since 𝐴 is compact, there exist 𝑥₁,𝑥₂,…,𝑥_𝑚∈𝐴 such that 𝐴⊂⋃^𝑚_𝑖=1𝐵(𝑥_𝑖,𝜖,𝑠). For every 𝑥_𝑖∈𝐴 (𝑖=1,2,…,𝑚), since {𝑇_𝑛} converges pointwise to 𝑇₀ in 𝐴, for 𝑟∈(0,1), there exist 𝑛_𝑖∈ℕ (𝑖=1,2,…,𝑚) such that 𝑀(𝑇_𝑛𝑥_𝑖,𝑇₀𝑥_𝑖,𝑡)>1−𝑟 for all 𝑛≥𝑛_𝑖. Set 𝑛^∗=max{𝑛_𝑖∶𝑖=1,2,…,𝑚}. Clearly, 𝑛^∗ depends only on 𝑟. For every 𝑥∈𝑋, there is an 𝑖₀∈{1,2,…,𝑚} such that 𝑥∈𝐵(𝑥_𝑖0,𝜖,𝑠). Then we have 𝑀(𝑥,𝑥_𝑖0,𝑠)>1−𝜖⇒𝑀(𝑇_𝑛𝑥,𝑇_𝑛𝑥_𝑖0,𝑡)>1−𝑟 for all 𝑛∈ℕ. Thus, for all 𝑛≥𝑛^∗, 𝑀𝑇_𝑛𝑥,𝑇₀()𝑇𝑥,2𝑡+𝑘𝑠≥𝑀_𝑛𝑥,𝑇_𝑛𝑥_𝑖0,𝑡∗𝑀𝑇_𝑛𝑥_𝑖0,𝑇₀𝑇𝑥,𝑡+𝑘𝑠≥𝑀_𝑛𝑥,𝑇_𝑛𝑥_𝑖0,𝑡∗𝑀𝑇_𝑛𝑥_𝑖0,𝑇₀𝑥_𝑖0𝑇,𝑡∗𝑀₀𝑥_𝑖0,𝑇₀𝑇𝑥,𝑘𝑠≥𝑀_𝑛𝑥,𝑇_𝑛𝑥_𝑖0,𝑡∗𝑀𝑇_𝑛𝑥_𝑖0,𝑇₀𝑥_𝑖0,𝑡∗𝑀𝑥_𝑖0≥()∗()∗()≥()∗()∗(),𝑥,𝑠1−𝑟1−𝑟1−𝜖1−𝑟1−𝑟1−𝑟>1−𝑟. (5) Hence, the sequence {𝑇_𝑛} converges uniformly to 𝑇₀ in 𝐴.

Definition 14. A fuzzy metric space (𝑋,𝑀,∗) in which every point has a compact neighborhood is called locally compact.

Definition 15. Let (𝑋,𝑀₀,∗) be a fuzzy metric space and let {𝑀_𝑛} be a sequence of fuzzy metrics on 𝑋. The sequence {𝑀_𝑛} is said to upper semiconverge uniformly to 𝑀₀ if for each 𝑟∈(0,1) and 𝑡>0, there exists an 𝑛₀∈ℕ such that 𝑀_𝑛(𝑥,𝑦,𝑡)≥𝑀₀(𝑥,𝑦,𝑡) and 𝑀₀(𝑥,𝑦,𝑡)/𝑀_𝑛(𝑥,𝑦,𝑡)>1−𝑟 for all 𝑛≥𝑛₀, 𝑥,𝑦∈𝑋.

3. Main Results

Theorem 16. Let (𝑋,𝑀,∗) be a G-complete fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋 where t-norm 𝑎∗𝑏=min{𝑎,𝑏}. 𝑇₀ is a contraction mapping of 𝑋; that is, there exists 𝑘₀∈(0,1) such that 𝑀(𝑇₀𝑥,𝑇₀𝑦,𝑘₀𝑡)≥𝑀(𝑥,𝑦,𝑡) for all 𝑥,𝑦∈𝑋, 𝑡>0, and satisfying 𝑇₀𝑥₀=𝑥₀. If there exists at least a fixed point 𝑥_𝑛 for each 𝑇_𝑛 (𝑛∈ℕ) and the sequence {𝑇_𝑛} converges uniformly to 𝑇₀, then 𝑥_𝑛→𝑥₀.

Proof. Suppose that 𝑥_𝑛↛𝑥₀; namely, there exist 𝑡₀>0 and 𝑟₀∈(0,1) such that for any 𝑛∈ℕ there is a 𝑘(𝑛)>𝑛 satisfying 𝑀(𝑥_𝑘(𝑛),𝑥₀,𝑡₀)<1−𝑟₀. Fix a number ℎ∈(𝑘₀,1). According to the condition (FM-6) of Definition 2, for 𝑡₀>0, we can find an appropriate 𝑝∈ℕ such that 𝑀(𝑥_𝑛,𝑥₀,𝑡₀(ℎ/𝑘₀)^𝑝)>1−𝑟₀ for any 𝑛∈ℕ. Since the sequence {𝑇_𝑛} converges uniformly to 𝑇₀, we can make 𝑛₀ sufficiently large such that 𝑀(𝑇_𝑛𝑥_𝑛,𝑇₀𝑥,𝑡)>1−𝑟₀ for all 𝑛≥𝑛₀, 𝑡>0. Now for 𝑛≥𝑛₀, we have 1−𝑟₀>𝑀𝑥_𝑘(𝑛),𝑥₀,𝑡₀=𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥₀,𝑡₀≥𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),(1−ℎ)𝑡₀∗𝑀𝑇₀𝑥_𝑘(𝑛),𝑇₀𝑥₀,ℎ𝑡₀≥𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),(1−ℎ)𝑡₀∗𝑀𝑥_𝑘(𝑛),𝑥₀,𝑡₀ℎ𝑘₀≥𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),(1−ℎ)𝑡₀∗𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),𝑡₀(1−ℎ)ℎ𝑘₀∗𝑀𝑥_𝑘(𝑛),𝑥₀,𝑡₀ℎ𝑘₀²≥𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),(1−ℎ)𝑡₀∗𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),𝑡₀(1−ℎ)ℎ𝑘₀∗⋯∗𝑀𝑇_𝑘(𝑛)𝑥_𝑘(𝑛),𝑇₀𝑥_𝑘(𝑛),𝑡₀()ℎ1−ℎ𝑘₀^𝑝−1∗𝑀𝑥_𝑘(𝑛),𝑥₀,𝑡₀ℎ𝑘₀^𝑝≥1−𝑟₀∗1−𝑟₀∗⋯∗1−𝑟_0𝑝∗1−𝑟₀=1−𝑟₀. (6) This leads to a contradiction. Hence, 𝑥_𝑛→𝑥₀.

Theorem 17. Let (𝑋,𝑀,∗) be a G-complete fuzzy metric space where t-norm is positive. If 𝑇₀∶𝑋→𝑋 is a self-mapping of 𝑋 and 𝑇^𝑚₀ is a contraction mapping for a certain positive integer 𝑚, then 𝑇₀ has a unique fixed point.

Proof. First of all, if 𝑚=1, the theorem is evident. In addition, if 𝑚≥2, according to Lemma 5, we need only to prove that 𝑇₀ is a contraction mapping. Since 𝑇^𝑚₀ is a contraction mapping, there is 𝑘₀∈(0,1) such that 𝑀(𝑇^𝑚₀𝑥,𝑇^𝑚₀𝑦,𝑘^𝑚₀𝑡)≥𝑀(𝑥,𝑦,𝑡) for every 𝑥,𝑦∈𝑋, and 𝑡>0. Define another fuzzy metric 𝑀(𝑥,𝑦,𝑡) on 𝑋 using 𝑀(𝑥,𝑦,𝑡) as follows: 𝑀𝑇𝑥,𝑦,𝑡=𝑀𝑥,𝑦,𝑡∗𝑀₀𝑥,𝑇₀𝑇𝑦,𝑘𝑡∗𝑀²₀𝑥,𝑇²₀𝑦,𝑘²𝑡∗⋯∗𝑀𝑇₀^𝑚−1𝑥,𝑇₀^𝑚−1𝑦,𝑘^𝑚−1𝑡. (7) Actually, it is easy to verify that the foregoing two fuzzy metrics are equivalent. Meantime, we claim that 𝑇₀ is a contraction mapping with respect to the fuzzy metric 𝑀(𝑥,𝑦,𝑡), since 𝑀𝑇₀𝑥,𝑇₀𝑦,𝑘₀𝑡=𝑀𝑇₀𝑥,𝑇₀𝑦,𝑘𝑡∗𝑀𝑇²₀𝑥,𝑇²₀𝑦,𝑘²𝑡∗𝑀𝑇³₀𝑥,𝑇³₀𝑦,𝑘³𝑡∗⋯∗𝑀𝑇^𝑚₀𝑥,𝑇^𝑚₀𝑦,𝑘^𝑚𝑡≥𝑀𝑇₀𝑥,𝑇₀𝑦,𝑘𝑡∗𝑀𝑇²₀𝑥,𝑇²₀𝑦,𝑘²𝑡∗𝑀𝑇³₀𝑥,𝑇³₀𝑦,𝑘³𝑡∗⋯∗𝑀𝑥,𝑦,𝑡=𝑀𝑥,𝑦,𝑡. (8)

Corollary 18. Let (𝑋,𝑀,∗) be a G-complete fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋 where t-norm 𝑎∗𝑏=min{𝑎,𝑏}. 𝑇₀∶𝑋→𝑋 is a self-mapping of 𝑋, and 𝑇^𝑚₀ is a contraction mapping for a certain positive integer 𝑚. If there exists at least a fixed point 𝑥_𝑛 for each 𝑇_𝑛(𝑛∈ℕ) and the sequence {𝑇_𝑛} converges uniformly to 𝑇₀, then 𝑥_𝑛→𝑥₀=𝑇₀𝑥₀.

Proof. It follows from Theorems 16 and 17.

Theorem 19. Let (𝑋,𝑀,∗) be a locally compact fuzzy metric space and let {𝑇_𝑛} be a sequence of self-mappings on 𝑋. 𝑇₀∶𝑋→𝑋 is a contraction mapping; that is, there exists a 𝑘₀∈(0,1) such that 𝑀(𝑇₀𝑥,𝑇₀𝑦,𝑘₀𝑡)≥𝑀(𝑥,𝑦,𝑡) for all 𝑥,𝑦∈𝑋, 𝑡>0. If the following conditions are satisfied:(i)𝑇^𝑚_𝑛 is a contraction mapping for a certain 𝑚=𝑚(𝑛),(ii){𝑇_𝑛} converges pointwise to 𝑇₀ and {𝑇_𝑛} is a uniformly equicontinuous sequence,(iii)𝑇_𝑛𝑥_𝑛=𝑥_𝑛, 𝑛=0,1,2,3,…,then the sequence {𝑥_𝑛} converges to 𝑥₀; that is, 𝑥_𝑛→𝑥₀.

Proof. For each 𝜖∈(0,1), we choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)≥1−𝜖. If given 𝑥₀∈𝑋, we may assume that 𝑟 is sufficiently small such that 𝐾(𝑥₀,𝑟)={𝑥∶𝑀(𝑥,𝑥₀,𝑡)≥1−𝑟} is a compact subset of 𝑋. By Lemma 13, since {𝑇_𝑛} is uniformly equicontinuous and pointwise convergent on 𝐾(𝑥₀,𝑟), we know that {𝑇_𝑛} converges uniformly to 𝑇₀ on the compact subset 𝐾(𝑥₀,𝑟). Then, for the foregoing 𝑟, there exists 𝑛_𝜖∈ℕ such that 𝑀(𝑇_𝑛𝑥,𝑇₀𝑥,(1−𝑘₀)𝑡)>1−𝑟 for all 𝑛≥𝑛_𝜖, 𝑡>0, and 𝑥∈𝐾(𝑥₀,𝑟). In addition, since 𝑇₀ is a contraction mapping, we have 𝑀(𝑇₀𝑥,𝑇₀𝑦,𝑘₀𝑡)≥𝑀(𝑥,𝑦,𝑡) for all 𝑥,𝑦∈𝐾(𝑥₀,𝑟). Thus, for all 𝑛≥𝑛_𝜖 and 𝑥∈𝐾(𝑥₀,𝑟), we can obtain 𝑀𝑇_𝑛𝑥,𝑥₀,𝑡=𝑀𝑇_𝑛𝑥,𝑇₀𝑥₀𝑇,𝑡≥𝑀_𝑛𝑥,𝑇₀𝑥,1−𝑘₀𝑡∗𝑀𝑇₀𝑥,𝑇₀𝑥₀,𝑘₀𝑡≥𝑀𝑇_𝑛𝑥,𝑇₀𝑥,1−𝑘₀𝑡∗𝑀𝑥,𝑥₀≥()∗(),𝑡1−𝑟1−𝑟≥1−𝜖. (9) Therefore, for all 𝑛≥𝑛_𝜖, 𝐾(𝑥₀,𝑟) is an invariant set for 𝑇_𝑛. Since 𝑇^𝑚_𝑛 is a contraction mapping for a certain positive integer 𝑚=𝑚(𝑛), it follows that the fixed point 𝑥_𝑛 of 𝑇_𝑛 is contained in the set 𝐾(𝑥₀,𝑟), when 𝑛≥𝑛_𝜖. By the definition of 𝐾(𝑥₀,𝑟), we have 𝑀(𝑥_𝑛,𝑥₀,𝑡)≥1−𝑟 for all 𝑛≥𝑛_𝜖. In fact, although 𝑟 should satisfy the foregoing condition, it may be sufficiently small. Hence, we can obtain 𝑥_𝑛→𝑥₀.

In addition, if t-norm 𝑎∗𝑏=𝑎⋅𝑏, then we can obtain the following some important conclusions.

Lemma 20. Let (𝑋,𝑀₀,∗) be a G-complete fuzzy metric space and let 𝐴 be a compact subset of 𝑋 where t-norm 𝑎∗𝑏=𝑎⋅𝑏. {𝑀_𝑛} and {𝑇_𝑛} are a sequence of fuzzy metrics and a sequence of self-mappings on 𝑋, respectively. If they satisfy the following conditions:(i){𝑀_𝑛} upper semiconverges uniformly to 𝑀₀,(ii)𝑇_𝑛 is a contraction mapping for the fuzzy metric 𝑀_𝑛, 𝑛=0,1,2,…,(iii){𝑇_𝑛} converges pointwise to 𝑇₀,then {𝑇_𝑛} converges uniformly to 𝑇₀ in 𝐴 with regard to the fuzzy metric 𝑀₀.

Proof. For each 𝜖∈(0,1), choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)>1−𝜖. Since {𝑀_𝑛} upper semiconverges uniformly to 𝑀₀, there exists 𝑛_𝑟∈ℕ such that 𝑀_𝑛(𝑥,𝑦,𝑡)≥𝑀₀(𝑥,𝑦,𝑡) and 𝑀₀(𝑥,𝑦,𝑡)/𝑀_𝑛(𝑥,𝑦,𝑡)>1−𝑟 for all 𝑛≥𝑛_𝑟, 𝑡>0. Choose 𝑥,𝑦 in 𝑋 such that 𝑀₀(𝑥,𝑦,𝑡)>1−𝑟 for each 𝑡>0. Then, for all 𝑛≥𝑛_𝑟, we have 𝑀₀𝑇_𝑛𝑥,𝑇_𝑛=𝑀𝑦,𝑡⁰𝑇_𝑛𝑥,𝑇_𝑛𝑦,𝑡𝑀_𝑛𝑇_𝑛𝑥,𝑇_𝑛𝑦,𝑡∗𝑀_𝑛𝑇_𝑛𝑥,𝑇_𝑛≥()𝑦,𝑡1−𝑟∗𝑀_𝑛𝑇_𝑛𝑥,𝑇_𝑛≥()𝑦,𝑡1−𝑟∗𝑀_𝑛𝑡𝑥,𝑦,𝑘_𝑛𝑘_𝑛∈()≥()0,11−𝑟∗𝑀₀𝑡𝑥,𝑦,𝑘_𝑛≥(1−𝑟)∗(1−𝑟)>1−𝜖. (10) Therefore, the sequence {𝑇_𝑛}(𝑛≥𝑛_𝑟) is uniformly equicontinuous in 𝐴 with regard to the fuzzy metric 𝑀₀. Since {𝑇_𝑛} is pointwise convergent and 𝐴 is a compact subset of 𝑋, according to Lemma 13, it follows that the subsequence {𝑇_𝑛} (𝑛≥𝑛_𝑟) converges uniformly to 𝑇₀ in 𝐴. Hence, {𝑇_𝑛} converges uniformly to 𝑇₀ in 𝐴.

Theorem 21. Let (𝑋,𝑀₀,∗) be a locally compact fuzzy metric space where t-norm 𝑎∗𝑏=𝑎⋅𝑏. If {𝑀_𝑛} and {𝑇_𝑛} satisfy the following conditions:(i){𝑀_𝑛} upper semiconverges uniformly to 𝑀₀,(ii)𝑇_𝑛 is a contraction mapping for the fuzzy metric 𝑀_𝑛, 𝑛=0,1,2,…,(iii){𝑇_𝑛} converges pointwise to 𝑇₀,(iv)𝑇_𝑛𝑥_𝑛=𝑥_𝑛, 𝑛=0,1,2,…,then the sequence of fixed points {𝑥_𝑛} of {𝑇_𝑛} converges to the fixed point 𝑥₀ of 𝑇₀; that is, 𝑥_𝑛→𝑥₀.

Proof. For each 𝜖∈(0,1), choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)≥1−𝜖. Meantime, for 𝑥₀∈𝑋, we may make 𝑟 sufficiently small such that 𝐾(𝑥₀,𝑟)={𝑥∶𝑀(𝑥,𝑥₀,𝑡)≥1−𝑟} is compact in 𝑋 for each 𝑡>0. By Lemma 20, we know that {𝑇_𝑛} converges uniformly to 𝑇₀ in 𝐾(𝑥₀,𝑟) with respect to the fuzzy metric 𝑀₀. Then, for every 𝑥∈𝑋, there exists an 𝑛_𝑟∈ℕ such that 𝑀₀(𝑇_𝑛𝑥,𝑇₀𝑥,𝑡)>1−𝑟 for all 𝑛≥𝑛_𝑟, 𝑡>0. Thus, when 𝑛≥𝑛_𝑟, for all 𝑥∈𝐾(𝑥₀,𝑟), we have 𝑀₀𝑇_𝑛𝑥,𝑥₀,1+𝑘₀𝑡≥𝑀₀𝑇_𝑛𝑥,𝑇₀𝑥,𝑡∗𝑀₀𝑇₀𝑥,𝑥₀,𝑘₀𝑡≥𝑀₀𝑇_𝑛𝑥,𝑇₀𝑥,𝑡∗𝑀₀𝑇₀𝑥,𝑇₀𝑥₀,𝑘₀𝑡≥𝑀₀𝑇_𝑛𝑥,𝑇₀𝑥,𝑡∗𝑀₀𝑥,𝑥₀≥()∗(),𝑡1−𝑟1−𝑟≥1−𝜖. (11) Therefore, 𝐾(𝑥₀,𝑟) is an invariant set in 𝑋 with regard to 𝑀₀. Since 𝑇_𝑛 is still a contraction mapping restricted to 𝐾(𝑥₀,𝑟) concerning on 𝑀_𝑛, one can see that the fixed point is also included in 𝐾(𝑥₀,𝑟). Apparently, for all 𝑛≥𝑛_𝑟, we can obtain 𝑀(𝑥_𝑛,𝑥₀,𝑡)≥1−𝑟. Since 𝑟 is sufficiently small, it can easily be shown that {𝑥_𝑛} converges to 𝑥₀; that is, 𝑥_𝑛→𝑥₀. This completes the proof.

Theorem 22. Let (𝑋,𝑀₀,∗) be a compact fuzzy metric space where t-norm 𝑎∗𝑏=𝑎⋅𝑏. The sequences {𝑀_𝑛} and {𝑇_𝑛} satisfy the following conditions:(i){𝑀_𝑛} upper semiconverges uniformly to 𝑀₀;(ii)𝑇_𝑛 is a contraction mapping for the fuzzy metric 𝑀_𝑛, 𝑛=0,1,2,…;(iii){𝑇_𝑛} converges pointwise to 𝑇₀.
If every mapping 𝑇_𝑛 (𝑛∈ℕ) has a fixed point 𝑥_𝑛 and there is a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} which converges to 𝑥₀, then 𝑇₀𝑥₀=𝑥₀.

Proof. Let 𝐾 denote the closure of the set {𝑥_𝑛𝑘}. By Lemma 12, we can easily know that 𝐾 is a compact set. According to Lemma 20, it follows that the subsequence {𝑇_𝑛𝑘} converges uniformly to 𝑇₀ in 𝐾 with regard to 𝑀₀. Obviously, {𝑇_𝑛𝑘𝑥_𝑛𝑘} converges to 𝑇₀𝑥₀. Hence, 𝑇₀𝑥₀=𝑥₀.

Theorem 23. Let (𝑋,𝑀,∗) be a fuzzy metric space. {𝑇_𝑛} is a sequence of contraction mappings and satisfying 𝑇_𝑛𝑥_𝑛=𝑥_𝑛 (𝑛=1,2,3,…). 𝑇₀∶𝑋→𝑋 is a contraction mapping. If {𝑇_𝑛} is a pointwise convergent sequence with respect to 𝑇₀ and the subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} converges to 𝑥₀, then 𝑇₀ has a fixed point 𝑥₀=𝑇₀𝑥₀.

Proof. For each 𝜖∈(0,1), choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)≥1−𝜖. Since {𝑥_𝑛𝑘} is a convergent subsequence and {𝑇_𝑛} is a pointwise convergent sequence, for a given 𝑥₀, we may choose 𝐾_𝑟∈ℕ such that 𝑀(𝑥_𝑛𝑘,𝑥₀,𝑡)≥1−𝑟 and 𝑀(𝑇_𝑛𝑘𝑥₀,𝑇₀𝑥₀,𝑡)≥1−𝑟 for all 𝑘≥𝐾_𝑟, 𝑡>0. For every 𝑛∈ℕ∪{0}, we denote by 𝑙_𝑛 (𝑙_𝑛∈(0,1)) the contraction constant of 𝑇_𝑛. Thus, for all 𝑘≥𝐾_𝑟, we have 𝑀𝑥_𝑛𝑘,𝑇₀𝑥₀,𝑙_𝑛𝑘𝑡𝑇+1=𝑀_𝑛𝑘𝑥_𝑛𝑘,𝑇₀𝑥₀,𝑙_𝑛𝑘+1𝑡𝑙_𝑛𝑘∈()𝑇0,1≥𝑀_𝑛𝑘𝑥_𝑛𝑘,𝑇_𝑛𝑘𝑥₀,𝑙_𝑛𝑘𝑡∗𝑀𝑇_𝑛𝑘𝑥₀,𝑇₀𝑥₀𝑥,𝑡≥𝑀_𝑛𝑘,𝑥₀,𝑡∗𝑀𝑇_𝑛𝑘𝑥₀,𝑇₀𝑥₀≥()∗(),𝑡1−𝑟1−𝑟≥1−𝜖. (12) Therefore, the subsequence {𝑥_𝑛𝑘} converges to 𝑇₀𝑥₀. According to the uniqueness of limit, it follows that 𝑥₀=𝑇₀𝑥₀; that is, 𝑥₀ is a fixed point of 𝑇₀.

Acknowledgments

This work was supported by “Qing Lan” Talent Engineering Funds by Tianshui Normal University. Dong Qiu acknowledges the support of the National Natural Science Foundation of China (Grant no. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001). Wei Chen acknowledges the support of the Beijing Municipal Education Commission Foundation of China (no. KM201210038001) and the Humanity and Social Science Youth Foundation of Ministry of Education of China (no. 13YJC630012).