Common Fixed Point Theorems of New Contractive Conditions in Fuzzy Metric Spaces

Abstract

Some new limit contractive conditions in fuzzy metric spaces are introduced, by using property (E.A), some common fixed point theorems for four maps are proved in GV-fuzzy metric spaces. As an application of our results, some new contractive conditions are presented, and some common fixed point theorems are proved under these contractive conditions. The contractive conditions presented in this paper contain or generalize many contractive conditions that appeared in the literatures. Some examples are given to illustrate that our results are real generalizations for the results in the references and to show that our limit contractive conditions are important for the existence of fixed point.

1. Introduction

The theory of fuzzy sets was first introduced by Zadeh [1], after many authors introduced the notion of fuzzy metric spaces in different ways (see [2–5]). In particular, Kramosil and Michálek [4] generalized the concept of probabilistic metric space given by Menger [6] to the fuzzy framework. Later on, George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michálek and defined the Hausdorff and first countable topology on the modified fuzzy metric space. Actually, this topology can also be constructed on each fuzzy metric space in the sense of Kramosil and Michálek and it is metrizbale [2, 7]. Other recent contributions to the study of fuzzy metric spaces in the sense of [2] may be found in [8, 9]. Since then, many authors have proved fixed point and common fixed point theorems in fuzzy metric spaces in the sense of [2]. Especially, we want to emphasize that some common fixed point theorems for 𝜑-type contraction maps in fuzzy metric spaces have been recently obtained in [10–16].

Quite recently, Miheţ [13] proved some existence theorems of common fixed point for two self-mappings 𝑓, 𝑔 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡𝑓𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡 (1) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝜑∶[0,1]→[0,1] is continuous and nondecreasing on [0,1], and 𝜑(𝑥)>𝑥 for all 𝑥∈[0,1].

C. Vetro and P. Vetro [15] proved some existence theorems of common fixed point for two self-mappings 𝑓, 𝑔 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 1𝑀1𝑓𝑥,𝑓𝑦,𝑡−1≤𝑟min𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡−1 (2) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝑟∶[0,+∞)→[0,+∞) with 𝑟(𝜏)<𝜏 for every 𝜏>0, an upper semicontinuous function.

Gopal et al. [10] proved some existence theorems of common fixed point for four self-mappings 𝐴, 𝐵, 𝑆 and 𝑇 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 1𝑀1𝐴𝑥,𝐵𝑦,𝑡−1≤𝑟min𝑀𝑆𝑥,𝑇𝑦,𝑡,𝑀(𝐴𝑥,𝑆𝑥,𝑡),𝑀𝐵𝑦,𝑇𝑦,𝑡−1 (3) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝑟∶[0,+∞)→[0,+∞) with 𝑟(𝜏)<𝜏 for every 𝜏>0, an upper semicontinuous function.

Imdad and Ali [11], Vijayaraju and Sajath [16] proved some existence theorems of common fixed point for four self-mappings 𝐴, 𝐵, 𝑆, and 𝑇 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 𝑀𝑀()𝐴𝑥,𝐵𝑦,𝑡≥𝜙min𝑆𝑥,𝑇𝑦,𝑡,𝑀𝑆𝑥,𝐴𝑥,𝑡,𝑀𝐵𝑦,𝑇𝑦,𝑡 (4) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝜙∶[0,1]→[0,1] with 𝜙(𝑠)>𝑠 whenever 0<𝑠<1 is a continuous or increasing and left-continuous function.

Imdad et al. [12] proved some existence theorems of common fixed point for four self-mappings 𝑓, 𝑔, 𝑆, and 𝑇 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 𝑀𝑀,𝑀𝑓𝑥,𝑔𝑦,𝑡≥𝜑min𝑆𝑥,𝑇𝑦,𝑡,𝑀𝑓𝑥,𝑆𝑥,𝑡𝑓𝑦,𝑇𝑦,𝑡,𝑀𝑓𝑥,𝑇𝑦,𝑡,𝑀𝑓𝑦,𝑆𝑥,𝑡 (5) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝜑∶[0,1]→[0,1] is continuous and nondecreasing on [0,1], and 𝜑(𝑥)>𝑥 for all 𝑥∈[0,1].

Shen et al. [14] proved an existence theorem of fixed point for self-mapping 𝑇 of a fuzzy metric space (𝑋,𝑀,∗) under the following contractive condition: 𝜑𝑀𝑇𝑥,𝑇𝑦,𝑡≤𝑘(𝑡)⋅𝜑𝑀𝑥,𝑦,𝑡 (6) for all 𝑥,𝑦∈𝑋 and 𝑡>0, where 𝜑∶[0,1]→[0,1] satisfies the following properties:(P1)𝜑 is strictly decreasing and left continuous;(P2)𝜑(𝜆)=0 if and only if 𝜆=1.

Furthermore, let 𝑘 be a function from (0,∞) into (0,1).

The purpose of this paper is to present limit contractive conditions to unify all of these 𝜑-type nonlinear contractive conditions. Then, by using property (𝐸.𝐴), some common fixed point theorems for four maps are proved in GV-fuzzy metric spaces. As an application of our limit contraction condition, we present some new 𝜑-type integral contractive conditions and some common fixed point theorems for four maps in GV-fuzzy metric spaces under these contractive conditions. Our results generalize the corresponding results in [10–16]. Some examples are given to illustrate that our results are real generalizations for the results in the references and show that our limit contractive conditions are important for the existence of fixed point.

For the reader’s convenience, we recall some terminologies from the theory of fuzzy metric spaces, which will be used in what follows.

Definition 1 (see [4]). A continuous 𝑡-norm in the sense of Kramosil and Michálek is a binary operation ∗ on [0,1] satisfying the following conditions:(1)∗ is associative and commutative,(2)𝑎∗1=𝑎 for all 𝑎∈[0,1],(3)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1],(4)the mapping ∗∶[0,1]×[0,1]→[0,1] is continuous.

Three typical examples of continuous 𝑡-norm are 𝑎∗𝑏=max{𝑎+𝑏−1,0}, 𝑎∗𝑏=𝑎𝑏, and 𝑎∗𝑏=min{𝑎,𝑏}.

Definition 2 (see [2]). A fuzzy metric space in the sense of George and Veeramani is a triple (𝑋,𝑀,∗), where 𝑋 is a nonempty set, 𝑀 is a fuzzy set on 𝑋²×(0,∞), and ∗ is a continuous 𝑡-norm such that the following conditions are satisfied for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0(GV-1)𝑀(𝑥,𝑦,𝑡)>0; (GV-2)𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦;(GV-3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡); (GV-4)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠); (GV-5)𝑀(𝑥,𝑦,⋅)∶(0,∞)→[0,1] is continuous.

In what follows, fuzzy metric spaces in the sense of George and Veeramani will be called GV-fuzzy metric spaces.

Lemma 3 (see [17]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then 𝑀(𝑥,𝑦,𝑡) is non-decreasing with respect to 𝑡 for all 𝑥,𝑦∈𝑋.

Definition 4 (see [13]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then one has the following:(1)a sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥∈𝑋 if lim_𝑛→∞𝑀(𝑥_𝑛,𝑥,𝑡)=1 for all 𝑡>0.(2)a sequence {𝑥_𝑛} in 𝑋 is said to be Cauchy sequence if lim_𝑛→∞𝑀(𝑥_𝑛,𝑥_𝑛+𝑝,𝑡)=1 for all 𝑡>0 and 𝑝∈ℕ.(3)a fuzzy metric space is called complete if every Cauchy sequence converges in 𝑋.

Lemma 5 (see [5]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then 𝑀 is a continuous function on 𝑋²×(0,∞).

Definition 6 (see [18]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then two self-mappings 𝐴 and 𝑆 of (𝑋,𝑀,∗) satisfy property (𝐸.𝐴) if there exists a sequence {𝑥_𝑛} in 𝑋 and 𝑧 in 𝑋 such that {𝐴𝑥_𝑛} and {𝑆𝑥_𝑛} converge to 𝑧 that is, for any 𝑡>0, lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞𝑀𝑆𝑥_𝑛,𝑧,𝑡=1. (7)

Definition 7 (see [18]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then two pairs of self-mappings 𝐴,𝑆 and 𝐵,𝑇 of (𝑋,𝑀,∗) are said to share common property (𝐸.𝐴) if there exist sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that, for any 𝑡>0, lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞𝑀𝑆𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞𝑀𝐵𝑦_𝑛,𝑧,𝑡=lim_𝑛→∞𝑀𝑇𝑦_𝑛,𝑧,𝑡=1 (8) for some 𝑧∈𝑋.

Definition 8 (see [19]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space. Then two self-mappings 𝐴 and 𝑆 of (𝑋,𝑀,∗) are said to be weak compatible if they commute at their coincidence point; that is, 𝐴𝑥=𝑆𝑥implies𝑆𝐴𝑥=𝐴𝑆𝑥. (9)

2. Main Results

Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗). For any 𝑥,𝑦∈𝑋 and 𝑡>0, we define 𝑀(),𝑀.Min𝑥,𝑦,𝑡=min𝑆𝑥,𝑇𝑦,𝑡,𝑀𝑆𝑥,𝐴𝑥,𝑡,𝑀𝑇𝑦,𝐵𝑦,𝑡𝑆𝑥,𝐵𝑦,𝑡,𝑀𝑇𝑦,𝐴𝑥,𝑡 (10)

Consider that (𝐶1)0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1 implies that lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)>𝐿(𝑡) for all 𝑡>0 and any sequence {𝑥_𝑛} and {𝑦_𝑛} in 𝑋.

Theorem 9. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following:(1)(𝐶1) holds;(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy property (𝐸.𝐴) or 𝐵, 𝑇 satisfy property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆, or 𝑇 is a closed subspace of 𝑋.Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point.

Proof. Suppose that 𝐵,𝑇 satisfy the property (𝐸.𝐴). Then there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑀𝐵𝑥_𝑛,𝑧,𝑡=lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝑧,𝑡=1 (11) for all 𝑡>0 and some 𝑧∈𝑋.
Since 𝐵𝑋⊂𝑆𝑋, there exists a sequence {𝑦_𝑛} in 𝑋 such that 𝐵𝑥_𝑛=𝑆𝑦_𝑛. Hence lim_𝑛→∞𝑀(𝑆𝑦_𝑛,𝑧,𝑡)=1 for all 𝑡>0.
Suppose that 𝑆𝑋 is a closed subspace of 𝑋. Then 𝑧=𝑆𝑢 for some 𝑢∈𝑋. Subsequently, we have that lim_𝑛→∞𝑀𝐵𝑥_𝑛,𝑆𝑢,𝑡=lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝑆𝑢,𝑡=lim_𝑛→∞𝑀𝑆𝑦_𝑛,𝑆𝑢,𝑡=1 (12) for all 𝑡>0. Then by using Lemma 5 we have that lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝐵𝑥_𝑛,𝑡=1,lim_𝑛→∞𝑀𝑇𝑥_𝑛,𝐴𝑢,𝑡=𝑀(𝑆𝑢,𝐴𝑢,𝑡). (13) Thus, we have that lim_𝑛→∞Min𝑢,𝑥_𝑛,𝑡=lim_𝑛→∞min𝑀𝑆𝑢,𝑇𝑥_𝑛(),𝑀,𝑡,𝑀𝑆𝑢,𝐴𝑢,𝑡𝑇𝑥_𝑛,𝐵𝑥_𝑛,𝑡,𝑀𝑆𝑢,𝐵𝑥_𝑛,𝑀,𝑡𝑇𝑥_𝑛(),𝐴𝑢,𝑡=𝑀𝑆𝑢,𝐴𝑢,𝑡=lim_𝑛→∞𝑀𝐵𝑥_𝑛,𝐴𝑢,𝑡 (14) for any 𝑡>0; that is, lim_𝑛→∞Min𝑢,𝑥_𝑛,𝑡=lim_𝑛→∞𝑀𝐴𝑢,𝐵𝑥_𝑛,𝑡=𝑀(𝑆𝑢,𝐴𝑢,𝑡). (15) If 𝑆𝑢≠𝐴𝑢, then there exists 𝑡>0 such that 0<𝑀(𝑆𝑢,𝐴𝑢,𝑡)<1. This and (𝐶1) imply that lim_𝑛→∞𝑀𝐴𝑢,𝐵𝑥_𝑛,𝑡>lim_𝑛→∞Min𝑢,𝑥_𝑛,𝑡. (16) This is a contradiction. Thus, we have that 𝑆𝑢=𝐴𝑢. The weak compatibility of 𝐴 and 𝑆 implies that 𝐴𝑆𝑢=𝑆𝐴𝑢, and then 𝐴𝐴𝑢=𝐴𝑆𝑢=𝑆𝐴𝑢=𝑆𝑆𝑢.
On the other hand, since 𝐴𝑋⊂𝑇𝑋, there exists 𝑣∈𝑋 such that 𝐴𝑢=𝑇𝑣. Since for any 𝑡>0(){𝑀()()(),𝑀()((),Min𝑢,𝑣,𝑡=min𝑆𝑢,𝑇𝑣,𝑡,𝑀𝑆𝑢,𝐴𝑢,𝑡,𝑀𝑇𝑣,𝐵𝑣,𝑡𝑆𝑢,𝐵𝑣,𝑡,𝑀𝑇𝑣,𝐴𝑢,𝑡)}=𝑀𝐴𝑢,𝐵𝑣,𝑡 (17) by (𝐶1), we get  𝑀(𝐴𝑢,𝐵𝑣,𝑡)=1, which implies that 𝐴𝑢=𝐵𝑣; that is, 𝐴𝑢=𝑆𝑢=𝑇𝑣=𝐵𝑣. The weak compatibility of 𝐵 and 𝑇 implies that 𝐵𝑇𝑣=𝑇𝐵𝑣 and 𝑇𝑇𝑣=𝑇𝐵𝑣=𝐵𝑇𝑣=𝐵𝐵𝑣.
Let us show that 𝐴𝑢 is a common fixed point of 𝐴, 𝐵, 𝑇, and 𝑆. Since (){𝑀()(),𝑀()()({𝑀()(),𝑀((),Min𝐴𝑢,𝑣,𝑡=min𝑆𝐴𝑢,𝑇𝑣,𝑡,𝑀𝑆𝐴𝑢,𝐴𝐴𝑢,𝑡𝑇𝑣,𝐵𝑣,𝑡,𝑀𝑆𝐴𝑢,𝐵𝑣,𝑡,𝑀𝑇𝑣,𝐴𝐴𝑢,𝑡)}=min𝐴𝐴𝑢,𝐵𝑣,𝑡,1,1,𝑀𝐴𝐴𝑢,𝐵𝑣,𝑡𝑇𝑣,𝐴𝐴𝑢,𝑡)}=𝑀𝐴𝐴𝑢,𝐵𝑣,𝑡 (18) by (𝐶1), we get that 𝑀(𝐴𝐴𝑢,𝐵𝑣,𝑡)=1, which implies that 𝐴𝐴𝑢=𝐵𝑣. Therefore, 𝐴𝐴𝑢=𝑆𝐴𝑢=𝐵𝑣=𝐴𝑢, and 𝐴𝑢 is a common fixed point of 𝐴 and 𝑆. Similarly, we can prove that 𝐵𝑣 is a common fixed point of 𝐵 and 𝑇. Noting that 𝐴𝑢=𝐵𝑣, we conclude that 𝐴𝑢 is a common fixed point 𝐴,𝐵,𝑇, and 𝑆. The proof is similar when 𝑇𝑋 is assumed to be a closed subspace of 𝑋. The cases in which 𝐴𝑋 or 𝐵𝑋 is a closed subspace of 𝑋 are similar to the cases in which 𝑇𝑋 or 𝑆𝑋, respectively, is closed since 𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋. If 𝐴𝑢=𝐵𝑢=𝑇𝑢=𝑆𝑢=𝑢 and 𝐴𝑣=𝐵𝑣=𝑇𝑣=𝑆𝑢=𝑣, then (){𝑀()()(),𝑀()({𝑀()(),𝑀(().Min𝑢,𝑣,𝑡=min𝑆𝑢,𝑇𝑣,𝑡,𝑀𝑆𝑢,𝐴𝑢,𝑡,𝑀𝑇𝑣,𝐵𝑣,𝑡𝑆𝑢,𝐵𝑣,𝑡,𝑀𝑇𝑣,𝐴𝑢,𝑡)}=min𝐴𝑢,𝑇𝑣,𝑡,1,1,𝑀𝑆𝑢,𝐵𝑣,𝑡𝑇𝑣,𝐴𝑢,𝑡)}=𝑀𝐴𝑢,𝐵𝑣,𝑡 (19) Therefore, by (𝐶1), we have 𝑢=𝑣; that is, the common fixed point is unique. This completes the proof.

To introduce some integral contractive conditions, let ℎ∶[0,1]→ℝ₊ be nonnegative, Lebesgue integrable, and satisfy ^𝜀₀ℎ(𝑡)𝑑𝑡>0 (20) for each 0<𝜀≤1. We denote 𝑑=∫¹₀ℎ(𝑡)𝑑𝑡.

Theorem 10. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗). If one of the following conditions is satisfied
(𝐶2) There exists a function 𝜔∶[0,𝑑]→[0,𝑑] such that for any 0<𝑠<𝑑, 𝜔(𝑠)<𝑠, lim_𝑢→𝑠𝜔(𝑢)<𝑠 and for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that 𝜔₀^{𝑀(𝐴𝑥,𝐵𝑦,𝑡)}ℎ(𝑠)𝑑𝑠≥^Min₀^{(𝑥,𝑦,𝑡)}ℎ(𝑠)𝑑𝑠. (21)
(𝐶3) There exists a function 𝜙∶[0,𝑑]→[0,𝑑] such that, for any 0<𝑠<𝑑, 𝜙(𝑠)>𝑠, lim_𝑡→𝑠𝜙(𝑡)>𝑠, and for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that ₀^{𝑀(𝐴𝑥,𝐵𝑦,𝑡)}ℎ(𝑠)𝑑𝑠≥𝜙𝑛^Min₀^{(𝑥,𝑦,𝑡)}ℎ(𝑠)𝑑𝑠. (22) Then (𝐶1) holds.

Proof. (𝐶2)⇒(𝐶1). Assume that (𝐶2) holds. If {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 and 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1, then lim_𝑛→∞𝜔^{𝑀(𝐴𝑥𝑛,𝐵𝑦𝑛}₀^,𝑡)ℎ(𝜏)≥𝑑𝜏lim_𝑛→∞^{Min(𝑥𝑛,𝑦𝑛}₀^,𝑡)ℎ(𝜏)≥𝑑𝜏_{lim𝑛→∞Min(𝑥𝑛,𝑦𝑛0,𝑡)}ℎ(𝜏)𝑑𝜏=₀^𝐿(𝑡)ℎ(𝜏)𝑑𝜏. (23) If lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)=1, then lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)>𝐿(𝑡). If lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)=𝑟(𝑡)<1, then there exists a subsequence {𝑀(𝐴𝑥_𝑛𝑖,𝐵𝑦_𝑛𝑖,𝑡)} such that lim_𝑖→∞𝑀𝐴𝑥_𝑛𝑖,𝐵𝑦_𝑛𝑖,𝑡=lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡=𝑟(𝑡). (24) Thus, lim_𝑛→∞^{𝑀(𝐴𝑥𝑛,𝐵𝑦𝑛}₀^,𝑡)ℎ(𝜏)𝑑𝜏=lim_𝑖→∞^{𝑀(𝐴𝑥𝑛𝑖,𝐵𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)=𝑑𝜏_0𝑟(𝑡)ℎ(𝜏)𝑑𝜏>lim_{𝑢→∫0𝑟(𝑡)ℎ(𝜏)𝑑𝜏}𝜔(𝑢)≥lim_𝑖→∞𝜔^{Min(𝑥𝑛𝑖,𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)𝑑𝜏≥₀^𝐿(𝑡)ℎ(𝜏)𝑑𝜏. (25) This implies that lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡=𝑟(𝑡)>𝐿(𝑡). (26) Thus, (𝐶1) holds.
(𝐶3)⇒(𝐶1). Assume that (𝐶3) holds. If {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 and 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1, then there exists a subsequence {Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)} such that lim_𝑖→∞Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)=𝐿(𝑡). This implies that lim_𝑖→∞𝜙^{Min(𝑥𝑛𝑖,𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)𝑑𝜏≥lim_{𝑡→∫0𝐿(𝑡)ℎ(𝜏)𝑑𝜏}𝜙(𝑡)>₀^𝐿(𝑡)ℎ(𝜏)𝑑𝜏=lim_𝑖→∞^{Min(𝑥𝑛𝑖,𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)𝑑𝜏. (27) It follows from ₀^{𝑀(𝐴𝑥,𝐵𝑦,𝑡)}ℎ(𝜏)𝑑𝜏≥𝜙₀^{Min(𝑥,𝑦,𝑡)}ℎ(𝜏)𝑑𝜏 (28) that we can get lim_𝑛→∞^{𝑀(𝐴𝑥𝑛,𝐵𝑦𝑛}₀^,𝑡)ℎ(𝜏)=𝑑𝜏_{lim𝑛→∞𝑀(𝐴𝑥𝑛,𝐵𝑦𝑛0,𝑡)}ℎ(𝜏)≥𝑑𝜏lim_𝑛→∞𝜙^{Min(𝑥𝑛,𝑦𝑛}₀^,𝑡)ℎ(𝜏)≥𝑑𝜏lim_𝑖→∞𝜙^{Min(𝑥𝑛𝑖,𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)𝑑𝜏≥lim_𝑖→∞𝜙^{Min(𝑥𝑛𝑖,𝑦𝑛𝑖}₀^,𝑡)ℎ(𝜏)𝑑𝜏>₀^𝐿(𝑡)ℎ(𝜏)𝑑𝜏. (29) This implies that lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)>𝐿(𝑡); that is, (𝐶1) holds.

It follows from Theorems 9 and 10 that we have the following fixed point theorems for integral type contractive mappings.

Theorem 11. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following(1)one of (𝐶2)-(𝐶3) holds;(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy the property (𝐸.𝐴) or 𝐵, 𝑇 satisfy the property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆, or 𝑇 is a closed subspace of 𝑋. Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point.

In (𝐶2) and (𝐶3), by taking ℎ(𝑡)≡1, we have the following contractive conditions.

(𝐶4) There exists a function 𝜔∶[0,1]→[0,1] such that for any 0<𝑠<1, 𝜔(𝑠)<𝑠, lim_𝑢→𝑠𝜔(𝑢)<𝑠 and for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that 𝜔𝑀𝑥,𝑦,𝑡≥Min𝑥,𝑦,𝑡. (30)

(𝐶5) There exists a function 𝜙∶[0,1]→[0,1] such that, for any 0<𝑠<1,𝜙(𝑠)>𝑠,lim_𝑡→𝑠𝜙(𝑡)>𝑠,and for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that 𝑀𝐴𝑥,𝐵𝑦,𝑡≥𝜙Min𝑥,𝑦,𝑡. (31)

Corollary 12. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following:(1)one of (𝐶4)-(𝐶5) holds;(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy property (𝐸.𝐴) or 𝐵, 𝑇 satisfy the property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆, or 𝑇 is a closed subspace of 𝑋. Then 𝐴, 𝐵, 𝑆 and 𝑇 have a unique common fixed point.

Remark 13. As a special case of (𝐶4), we can take function 𝜔∶[0,1]→[0,1] as one of the following:(1)𝜔 is nonincreasing, for any 0<𝑠<1, 𝜔(𝑠)<𝑠, lim_{𝑢→𝑠−}𝜔(𝑢)<𝑠;(2)𝜔 is an upper semicontinuous function such that, for any 0<𝑠<1, 𝜔(𝑠)<𝑠;(3)𝜔 is non-increasing and left-upper semicontinuous such that 𝜔(𝑠)<𝑠 for any 0<𝑠<1.
As a special case of (𝐶5), we can take function 𝜙∶[0,1]→[0,1] as one of the follows:(1)𝜙 is a nondecreasing and left-continuous function such that, for any 0<𝑠<1, 𝜙(𝑠)>𝑠;(2)𝜙 is a a lower semi-continuous function such that for any 0<𝑠<1, 𝜙(𝑠)>𝑠;(3)𝜙(𝑠)=𝑠+𝜓(𝑠), where 𝜓∶[0,1]→[0,1] is a continuous function with for any 0<𝑠<1, 𝜓(𝑠)>0.

From Corollary 12, we have the following corollaries.

Corollary 14. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following:(1)there exists an upper semicontinuous function 𝑟∶[0,+∞)→[0,+∞) with 𝑟(𝑠)<𝑠 for any 𝑠>0 such that for all 𝑥,𝑦∈𝑋 and 𝑡>01𝑀1𝐴𝑥,𝐵𝑦,𝑡−1≤𝑟Min𝑥,𝑦,𝑡−1; (32)(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy property (𝐸.𝐴) or 𝐵, 𝑇 satisfy property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆 or 𝑇 is a closed subspace of 𝑋. Then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point.

Proof. Define function 𝜙∶[0,1]→[0,1] by 𝜙(𝑡)=10if𝑡=0;))(].1+𝑟((1/𝑡−1if𝑡∈0,1 (33) Then, for any 𝑠>0, lim_𝑡→𝑠𝜙(𝑡)=11+lim_𝑡→𝑠𝑟))=1((1/𝑡−1))>11+𝑟((1/𝑠−11+(1/𝑠)−1=𝑠, (34) and (32) can be rewritten as: for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that 𝑀𝐴𝑥,𝐵𝑦,𝑡≥𝜙Min𝑥,𝑦,𝑡. (35) That is, (𝐶5) holds. Then the conclusion can be deduced from Corollary 12. This completes the proof.

Corollary 15. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and 𝐴,𝐵,𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following:(1)there exists a strictly decreasing and left continuous function 𝜑∶[0,1]→[0,1] with 𝜑(𝜆)=0 if and only if 𝜆=1 and function 𝑘∶(0,∞)→(0,1) such that for all 𝑥,𝑦∈𝑋(𝑥≠𝑦) and 𝑡>0𝜑𝑀𝐴𝑥,𝐵𝑦,𝑡≤𝑘(𝑡)⋅𝜑Min𝑥,𝑦,𝑡; (36)(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy property (𝐸.𝐴) or 𝐵, 𝑇 satisfy property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆, or 𝑇 is a closed subspace of 𝑋.Then, 𝐴, 𝐵, 𝑆 and 𝑇 have a unique common fixed point.

Proof. Define function 𝜙∶[0,1]×(0,+∞)→[0,1] by 𝜙(𝑠,𝑡)=𝜑⁻¹𝑘(𝑡)𝜑(𝑠). (37) Since 𝜑 is strictly decreasing and left continuous, we have that 𝜑⁻¹ is strictly decreasing and right continuous and 𝜙(𝑠,𝑡) is increasing in 𝑠. Then we have that lim_{𝑢→𝑠−}𝜙(𝑢,𝑡)=𝜑⁻¹𝑘(𝑡)𝜑(𝑠). (38) Also we have that 𝜙(𝑢,𝑡)≥𝜙(𝑠,𝑡) for 𝑢>𝑠; this shows that lim_𝑢→𝑠+𝜙(𝑢,𝑡)≥𝜑⁻¹𝑘(𝑡)𝜑(𝑠). (39) That is, we can get that lim_𝑢→𝑠𝜙(𝑢,𝑡)≥𝜙(𝑠,𝑡)=𝜑⁻¹𝑘(𝑡)𝜑(𝑠)>𝜑⁻¹𝜑(𝑠)=𝑠, (40) and (36) can be rewritten as follows: for any 𝑥,𝑦∈𝑋, 𝑡∈(0,∞), 0<Min(𝑥,𝑦,𝑡)<1 implies that 𝑀𝐴𝑥,𝐵𝑦,𝑡≥𝜙Min𝑥,𝑦,𝑡,𝑡. (41) If {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 and 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1, then there exists a subsequence {Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)} such that lim_𝑖→∞Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)=𝐿(𝑡). This shows that lim_𝑖→∞𝜙Min𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡,𝑡≥lim_{(𝑡)𝑠→𝐿}𝜙(𝑠,𝑡)>𝐿(𝑡)=lim_𝑖→∞Min𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡. (42) It follows from 𝑀(𝐴𝑥,𝐵𝑦,𝑡)≥𝜙(Min(𝑥,𝑦,𝑡),𝑡) that we can get lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛≥,𝑡lim_𝑛→∞𝜙Min𝑥_𝑛,𝑦_𝑛≥,𝑡,𝑡lim_𝑖→∞𝜙Min𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡,𝑡≥lim_𝑖→∞𝜙Min𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡,𝑡>𝐿(𝑡). (43) Thus, we have that (𝐶1) holds. The conclusion can be deduced from Theorem 9. This completes the proof.

Remark 16. The main result Theorems 3.1 and 2.1 in [13] are the special cases of our Theorem 9 and Corollary 14 for 𝐴=𝐵=𝑓 and 𝑆=𝑇=𝑔. Especially, it follows from Corollary 12 that the condition “𝜑 is nondecreasing” is not needed. Therefore, our results improve and generalize the results in [13].

Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of (𝑋,𝑀,∗). For any 𝑥,𝑦∈𝑋 and 𝑡>0, we define 𝑀().Min𝑥,𝑦,𝑡=min𝑆𝑥,𝑇𝑦,𝑡,𝑀𝑆𝑥,𝐴𝑥,𝑡,𝑀𝑇𝑦,𝐵𝑦,𝑡 (44)

Consider that (𝐶1) 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1 implies lim_𝑛→∞𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)>𝐿(𝑡) for all 𝑡>0 and any sequence {𝑥_𝑛} and {𝑦_𝑛} in 𝑋.

By (44), we can write the conditions (𝐶2)–(𝐶5) which correspond to the conditions (𝐶2)--(𝐶5) in Theorem 9, Theorem 10 and Corollary 12 by replacing Min(𝑥,𝑦,𝑡) with Min(𝑥,𝑦,𝑡). It is similar to the proof of Theorem 10, we can know that one of (𝐶4)-(𝐶5) can imply (𝐶1) holds.

Corollary 17. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space, and 𝐴, 𝐵, 𝑆 and 𝑇 be self-mappings of (𝑋,𝑀,∗) such that one has the following:(1)(𝐶1) or one of (𝐶4)-(𝐶5) holds;(2)𝐴, 𝑆 are weakly compatible and 𝐵, 𝑇 are weakly compatible;(3)𝐴, 𝑆 satisfy property (𝐸.𝐴) or 𝐵, 𝑇 satisfy property (𝐸.𝐴);(4)𝐴𝑋⊂𝑇𝑋 and 𝐵𝑋⊂𝑆𝑋;(5)one of the range of the mappings 𝐴, 𝐵, 𝑆, or 𝑇 is a closed subspace of 𝑋. Then 𝐴,𝐵,𝑆, and 𝑇 have a unique common fixed point.

Proof. It is clear that we only need to prove Corollary 17 for (𝐶1). Assume that sequence {𝑥_𝑛} and {𝑦_𝑛} in 𝑋, 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1. Then lim_𝑛→∞Min𝑥_𝑛,𝑦_𝑛,𝑡=𝐿(𝑡)≥lim_𝑛→∞Min𝑥_𝑛,𝑦_𝑛,𝑡=𝐿(𝑡). (45) If lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1, then by (𝐶1) we have that lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡>𝐿(𝑡)≥𝐿(𝑡). (46) If lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)=1, then, for any 𝑛, either Min(𝑥_𝑛,𝑦_𝑛,𝑡)<1 or Min(𝑥_𝑛,𝑦_𝑛,𝑡)=1. If Min(𝑥_𝑛,𝑦_𝑛,𝑡)<1, then by (𝐶1) we have that 𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)>Min(𝑥_𝑛,𝑦_𝑛,𝑡). If Min(𝑥_𝑛,𝑦_𝑛,𝑡)=1, then by (44) we have that 𝐴𝑥_𝑛=𝐵𝑦_𝑛. By (GV-2) we get that 𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)=1=Min(𝑥_𝑛,𝑦_𝑛,𝑡). Thus, (𝐶1) implies that 𝑀(𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡)≥Min(𝑥_𝑛,𝑦_𝑛,𝑡) for any 𝑛. This implies that lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡≥lim_𝑛→∞Min𝑥_𝑛,𝑦_𝑛,𝑡=𝐿(𝑡)=1>𝐿(𝑡). (47) Therefore, (𝐶1) implies that (𝐶1) holds. Then by Theorem 9 we know that 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point. This completes the proof.

Remark 18. In Theorem 9 and Corollaries 12–17, conditions (3), (4), and (5) can be replaced by the following conditions:(3^′)𝐴, 𝑆 and 𝐵, 𝑇 share the common property (𝐸.𝐴);(4^′) the range, of the mappings 𝑆 and 𝑇 are closed subspaces of 𝑋.
The proof can be got by properly modifying the proof of Theorem 9, Corollaries 12–17. Thus, from Theorem 9, Corollaries 12–17 and Remark 13, we can see that our results generalize and improve the results in [10–16].

Example 19. Let 𝑋=[0,1], 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+|𝑥−𝑦|) for every 𝑥,𝑦∈𝑋 and 𝑡>0 and 𝑎∗𝑏=min{𝑎,𝑏} for all 𝑎,𝑏∈[0,1]. Then (𝑋,𝑀,∗) is a fuzzy metric space. Define 𝐴=𝐵 and 𝑇=𝑆∶𝑋→𝑋 by 𝐴𝑥=sin(𝑥/(1+𝑥)) and 𝑇𝑥=sin𝑥 for all 𝑥∈𝑋. Then we have the following:(1)𝐴 and 𝑆 satisfy the property (𝐸.𝐴) for the sequence 𝑥_𝑛=1/𝑛,𝑛=1,2,…,(2)𝐴 and 𝑆 are weakly compatible,(3)𝐴𝑋=[0,sin(1/2)]⊂𝑇𝑋=[0,sin1], and 𝐴𝑋,𝑇𝑋 are closed;(4)(𝐶1) holds. In fact, if {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 and 0<lim_𝑛→∞Min(𝑥_𝑛,𝑦_𝑛,𝑡)=𝐿(𝑡)<1, then there exists a subsequence {Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)} such that lim_𝑖→∞Min(𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡)=𝐿(𝑡). Since {𝑥_𝑛} and {𝑦_𝑛} in 𝑋, {𝑥_𝑛} and {𝑦_𝑛} have convergent subsequence. With out of generality, assume that lim_𝑖→∞𝑥_𝑛𝑖=𝑥₀, and lim_𝑖→∞𝑦_𝑛𝑖=𝑦₀. Then we have that lim_𝑖→∞𝑀𝐴𝑥_𝑛𝑖,𝐵𝑦_𝑛𝑖=𝑡,𝑡𝑡+sin𝑥₀/1+𝑥₀−sin𝑦₀/1+𝑦₀,lim_𝑖→∞Min𝑥_𝑛𝑖,𝑦_𝑛𝑖𝑡,𝑡=min𝑡+sin𝑥₀−sin𝑦₀,𝑡𝑡+sin𝑥₀−sin𝑥₀/1+𝑥₀,𝑡𝑡+sin𝑦₀−sin𝑦₀/1+𝑦₀,𝑡𝑡+sin𝑥₀−sin𝑦₀/1+𝑦₀,𝑡𝑡+sin𝑦₀−sin𝑥₀/1+𝑥₀=𝐿(𝑡). (48)
If 0≤𝑥₀<𝑦₀, then 𝐿(𝑡)𝑡=min𝑡+sin𝑦₀−sin𝑥₀,𝑡𝑡+sin𝑥₀−sin𝑥₀/1+𝑥₀,𝑡𝑡+sin𝑦₀−sin𝑦₀/1+𝑦₀,𝑡𝑡+sin𝑥₀−sin𝑦₀/1+𝑦₀,𝑡𝑡+sin𝑦₀−sin𝑥₀/1+𝑥₀. (49) It is clear that sin𝑦₀𝑥−sin⁰1+𝑥₀>sin𝑦₀−sin𝑥₀,sin𝑦₀𝑥−sin⁰1+𝑥₀>sin𝑥₀𝑥−sin⁰1+𝑥₀,sin𝑦₀𝑥−sin⁰1+𝑥₀>sin𝑦₀𝑦−sin⁰1+𝑦₀. (50) It follows from sin𝑦₀+sin𝑥₀𝑥>sin⁰1+𝑥₀𝑦+sin⁰1+𝑦₀ (51) that sin𝑦₀𝑥−sin⁰1+𝑥₀𝑦>sin⁰1+𝑦₀−sin𝑥₀. (52) On the other hand, sin𝑦₀𝑥−sin⁰1+𝑥₀>sin𝑥₀𝑥−sin⁰1+𝑥₀>sin𝑥₀𝑦−sin⁰1+𝑦₀. (53) Thus, we have that sin𝑦₀𝑥−sin⁰1+𝑥₀>sin𝑥₀𝑦−sin⁰1+𝑦₀. (54) It follows from those inequalities that 𝐿(𝑡)=𝑡𝑡+sin𝑦₀−sin𝑥₀/1+𝑥₀. (55) Similarly, if 0≤𝑦₀<𝑥₀, then we can get that 𝐿(𝑡)=𝑡𝑡+sin𝑥₀−sin𝑦₀/1+𝑦₀, (56) and if 𝑥₀=𝑦₀>0, then we have that 𝐿(𝑡)=𝑡𝑡+sin𝑥₀−sin𝑥₀/1+𝑥₀. (57) Thus, we have that 𝐿(𝑡)=𝑡𝑡+sin𝑦₀−sin𝑥₀/1+𝑥₀if0≤𝑥₀<𝑦₀;𝑡𝑡+sin𝑥₀−sin𝑦₀/1+𝑦₀if0≤𝑦₀<𝑥₀;𝑡𝑡+sin𝑥₀−sin𝑥₀/1+𝑥₀if𝑥₀=𝑦₀>0. (58) It is clear that 𝑡𝑡+sin𝑦₀/1+𝑦₀−sin𝑥₀/1+𝑥₀>𝑡𝑡+sin𝑦₀−sin𝑥₀/1+𝑥₀if0≤𝑥₀<𝑦₀;𝑡𝑡+sin𝑥₀/1+𝑥₀−sin𝑦₀/1+𝑦₀>𝑡𝑡+sin𝑥₀−sin𝑦₀/1+𝑦₀if0≤𝑦₀<𝑥₀;𝑡1>𝑡+sin𝑥₀−sin𝑥₀/1+𝑥₀if𝑥₀=𝑦₀>0. (59)(60)(61) This shows that lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛,𝑡≥lim_𝑖→∞𝑀𝐴𝑥_𝑛𝑖,𝐵𝑦_𝑛𝑖,𝑡>𝐿(𝑡)=lim_𝑖→∞Min𝑥_𝑛𝑖,𝑦_𝑛𝑖,𝑡=lim_𝑛→∞Min𝑥_𝑛,𝑦_𝑛,𝑡. (62) Thus, (𝐶1) holds.

(5) By Theorem 9, 𝐴 and 𝑇 have common fixed point.

Example 20. Let 𝑋=[0,1], 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+|𝑥−𝑦|) for every 𝑥,𝑦∈𝑋 and 𝑡>0 and 𝑎∗𝑏=min{𝑎,𝑏} for all 𝑎,𝑏∈[0,1]. Then (𝑋,𝑀,∗) is a fuzzy metric space. Define 𝐴=𝐵 and 𝑇=𝑆∶𝑋→𝑋 by 1𝐴𝑥=2𝑥if𝑥=0;0if𝑥=1;2if𝑥∈(0,1); (63) and 𝑇𝑥=𝑥 for all 𝑥∈𝑋. Then we have the following:(1)𝐴 and 𝑆 satisfy the property (𝐸.𝐴) for the sequence 𝑥_𝑛=1/𝑛, 𝑛=1,2,…;(2)𝐴 and 𝑆 are weakly compatible;(3)𝐴𝑋=[0,1/2]⊂𝑇𝑋=[0,1], and 𝐴𝑋, 𝑇𝑋 are closed;(4)(𝐶1) does not hold. In fact, for 𝑥_𝑛=1/𝑛 and 𝑦_𝑛=0, 𝑛=1,2,…, we have that Min𝑥_𝑛,𝑦_𝑛𝑀1,𝑡=min𝑛1,0,𝑡,𝑀𝑛,112𝑛,𝑡,𝑀0,2,𝑀1,𝑡𝑛,121,𝑡,𝑀0,𝑛𝑡,𝑡=min(),𝑡𝑡+1/𝑛(),𝑡𝑡+1/2𝑛(),𝑡𝑡+1/2()−(),𝑡𝑡+1/21/𝑛()→𝑡𝑡+1/𝑛()(),𝑡+1/2𝑛→∞lim_𝑛→∞𝑀𝐴𝑥_𝑛,𝐵𝑦_𝑛=,𝑡lim_𝑛→∞𝑀1,12𝑛2=,𝑡lim_𝑛→∞𝑡()−()=𝑡𝑡+1/21/2𝑛().𝑡+1/2 (64) Thus, (𝐶1) does not hold.
(5) 𝐴 and 𝑇 have no common fixed point.

Example 19 does not satisfy the 𝜑-type contractive conditions used in [10–16]. Example 20 shows that if (𝐶1) does not hold, then 𝐴 and 𝑇 may have no common fixed point. Thus, (𝐶1) is important for the existence of common fixed point.

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This research was supported by the NNSF of China (fund no. 11171286).