Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces

Abstract

We prove some common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using the new property and give some examples. Our results improve and generalize the main results of Mihet in (Mihet, 2010) and many fixed point theorems in fuzzy metric spaces.

1. Introduction and Preliminaries

The notion of fuzzy sets was introduced by Zadeh [1] in 1965. Since that time a substantial literature has developed on this subject; see, for example, [2–4]. Fixed point theory is one of the most famous mathematical theories with application in several branches of science, especially in chaos theory, game theory, nonlinear programming, economics, theory of differential equations, and so forth. The works noted in [5–10] are some examples from this line of research.

Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern [11]. He introduced the concept of fuzzy mappings and proved some fixed point theorems for fuzzy contraction mappings in metric linear space, which is a fuzzy extension of the Banach's contraction principle. Subsequently several authors [12–20] have studied existence of fixed points of fuzzy mappings. Butnariu [21] also proved some useful fixed point results for fuzzy mappings. Badshah and Joshi [22] studied and proved a common fixed point theorem for six mappings on fuzzy metric spaces by using notion of semicompatibility and reciprocal continuity of mappings satisfying an implicit relation.

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

Definition 1.1 (Schweizer and Sklar [23]). A continuous 𝑡-norm is a binary operation ∗ on [0,1] satisfying the following conditions: (i)∗ is commutative and associative; (ii)𝑎∗1=𝑎 for all 𝑎∈[0,1]; (iii)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑(𝑎,𝑏,𝑐,𝑑∈[0,1]); (iv) the mapping ∗∶[0,1]×[0,1]→[0,1] is continuous.

Example 1.2. The following examples are classical examples of a continuous 𝑡-norms. (TL)(the Lukasiewicz 𝑡-norm). A mapping 𝑇_𝐿∶[0,1]×[0,1]→[0,1] which defined through 𝑇_𝐿(𝑎,𝑏)=max{𝑎+𝑏−1,0}. (1.1)(TP)(the product 𝑡-norm). A mapping 𝑇_𝑃∶[0,1]×[0,1]→[0,1] which defined through 𝑇_𝑃(𝑎,𝑏)=𝑎𝑏. (1.2)(TM)(the minimum 𝑡-norm). A mapping 𝑇_𝑀∶[0,1]×[0,1]→[0,1] which defined through 𝑇_𝑀(𝑎,𝑏)=min{𝑎,𝑏}. (1.3)

In 1975, Kramosil and Michalek [4] gave a notion of fuzzy metric space which could be considered as a reformulation, in the fuzzy context, of the notion of probabilistic metric space due to Menger [24].

Definition 1.3 (Kramosil and Michalek [4]). A fuzzy metric space is a triple (𝑋,𝑀,∗) where 𝑋 is a nonempty set, ∗ is a continuous 𝑡-norm and 𝑀 is a fuzzy set on 𝑋²×[0,1] such that the following axioms hold: (KM-1)𝑀(𝑥,𝑦,0)=0 for all 𝑥,𝑦∈𝑋; (KM-2)𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋 where 𝑡>0⇔𝑥=𝑦; (KM-3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡) for all 𝑥,𝑦∈𝑋; (KM-4)𝑀(𝑥,𝑦,⋅)∶[0,∞)→[0,1] is left continuous for all 𝑥,𝑦∈𝑋; (KM-5)𝑀(𝑥,𝑧,𝑡+𝑠)≥𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠) for all 𝑥,𝑦,𝑧∈𝑋 and for all 𝑠,𝑡>0.

We will refer to these spaces as KM-fuzzy metric spaces.

Lemma 1.4 (Grabiec [15]). For every 𝑥,𝑦∈𝑋, the mapping 𝑀(𝑥,𝑦,⋅) is nondecreasing on [0,∞].

George and Veeramani [2, 25] introduced and studied a notion of fuzzy metric space which constitutes a modification of the one due to Kramosil and Michalek.

Definition 1.5 (George and Veeramani [2, 25]). A fuzzy metric space is a triple (𝑋,𝑀,∗) where 𝑋 is a nonempty set, ∗ is a continuous 𝑡-norm and 𝑀 is a fuzzy set on 𝑋²×[0,1] and the following conditions are satisfied for all 𝑥,𝑦∈𝑋 and 𝑡,𝑠>0: (GV-1)𝑀(𝑥,𝑦,𝑡)>0; (GV-2)𝑀(𝑥,𝑦,𝑡)=1⇔𝑥=𝑦; (GV-3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡); (GV-4)𝑀(𝑥,𝑦,⋅)∶(0,∞)→[0,1] is continuous; (GV-5)𝑀(𝑥,𝑧,𝑡+𝑠)≥𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠).

From (GV-1) and (GV-2), it follows that if 𝑥≠𝑦, then 0<𝑀(𝑥,𝑦,𝑡)<1 for all 𝑡>0. In what follows, fuzzy metric spaces in the sense of George and Veeramani will be called GV-fuzzy metric spaces.

From now on, by fuzzy metric we mean a fuzzy metric in the sense of George and Veeramani. Several authors have contributed to the development of this theory, for instance [26–29].

Example 1.6. Let (𝑋,𝑑) be a metric space, 𝑎∗𝑏=𝑇_𝑀(𝑎,𝑏) and, for all 𝑥,𝑦∈𝑋 and 𝑡>0, 𝑀=𝑡𝑥,𝑦,𝑡𝑡+𝑑𝑥,𝑦. (1.4) Then (𝑋,𝑀,∗) is a GV-fuzzy metric space, called standard fuzzy metric space induced by (𝑋,𝑑).

Definition 1.7. Let (𝑋,𝑀,∗) be a (KM- or GV-) fuzzy metric space. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥∈𝑋 if lim_𝑛→∞𝑀𝑥_𝑛,𝑥,𝑡=1 (1.5) for all 𝑡>0.

Definition 1.8. Let (𝑋,𝑀,∗) be a (KM- or GV-) fuzzy metric space. A sequence {𝑥_𝑛} in 𝑋 is said to be 𝐺-Cauchy sequence if lim_𝑛→∞𝑀𝑥_𝑛,𝑥_𝑛+𝑚,𝑡=1 (1.6) for all 𝑡>0 and 𝑚∈ℕ.

Definition 1.9. A fuzzy metric space (𝑋,𝑀,∗) is called 𝐺-complete if every 𝐺-Cauchy sequence converges to a point in 𝑋.

Lemma 1.10 (Schweizer and Sklar [23]). If (𝑋,𝑀,∗) is a KM-fuzzy metric space and {𝑥_𝑛}, {𝑦_𝑛} are sequences in 𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥,lim_𝑛→∞𝑦_𝑛=𝑦, (1.7) then lim_𝑛→∞𝑀𝑥_𝑛,𝑦_𝑛,𝑡=𝑀𝑥,𝑦,𝑡 (1.8) for every continuity point 𝑡 of 𝑀(𝑥,𝑦,⋅).

Definition 1.11 (Jungck and Rhoades [30]). Let 𝑋 be a nonempty set. Two mappings 𝑓,𝑔∶𝑋→𝑋 are said to be weakly compatible if 𝑓𝑔𝑥=𝑔𝑓𝑥 for all 𝑥 which 𝑓𝑥=𝑔𝑥.

In 1995, Subrahmanyam [31] gave a generalization of Jungck's [32] common fixed point theorem for commuting mappings in the setting of fuzzy metric spaces. Even if in the recent literature weaker conditions of commutativity, as weakly commuting mappings, compatible mappings, 𝑅-weakly commuting mappings, weakly compatible mappings and several authors have been utilizing, the existence of a common fixed point requires some conditions on continuity of the maps, 𝐺-completeness of the space, or containment of ranges.

The concept of E.A. property in metric spaces has been recently introduced by Aamri and El Moutawakil [33].

Definition 1.12 (Aamri and El Moutawakil [33]). Let 𝑓 and 𝑇 be self-mapping of a metric space (𝑋,𝑑). We say that 𝑓 and 𝑇 satisfy E.A. property if there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑡 (1.9) for some 𝑡∈𝑋.

The class of E.A. mappings contains the class of noncompatible mappings.

In a similar mode, it is said that two self-mappings of 𝑓 and 𝑇 of a fuzzy metric space (𝑋,𝑀,∗) satisfy E.A. property, if there exists a sequence {𝑥_𝑛} in 𝑋 such that 𝑓𝑥_𝑛 and 𝑔𝑥_𝑛 converge to 𝑡 for some 𝑡∈𝑋 in the sense of Definition 1.7.

The concept of E.A. property allows to replace the completeness requirement of the space with a more natural condition of closeness of the range.

Recently, Mihet [34] proved two common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using E.A. property.

Let Φ be class of all mappings 𝜑∶[0,1]→[0,1] satisfying the following properties:(φ1)φ is continuous and nondecreasing on [0,1]; (φ2)𝜑(𝑥)>𝑥 for all 𝑥∈(0,1).

Theorem 1.13 (see [34, Theorem  2.1]). Let (𝑋,𝑀,∗) be a KM-fuzzy metric space satisfying the following property: ∀𝑥,𝑦∈𝑋,𝑥≠𝑦,∃𝑡>0∶0<𝑀𝑥,𝑦,𝑡<1, (1.10) and let 𝑓,𝑔 be weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (1.11) for all 𝑥,𝑦∈𝑋 where 𝑡>0. If 𝑓 and 𝑔 satisfy E.A. property and the range of 𝑔 is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point.

Theorem 1.14 (see [34, Theorem  3.1]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space and 𝑓,𝑔 weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ and some 𝑠>0, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑠≥𝜑min𝑔𝑥,𝑔𝑦,𝑠,𝑀𝑓𝑥,𝑔𝑥,𝑠𝑓𝑦,𝑔𝑦,𝑠,𝑀𝑓𝑦,𝑔𝑥,𝑠,𝑀𝑓𝑥,𝑔𝑦,𝑠 (1.12) for all 𝑥,𝑦∈𝑋. If 𝑓 and 𝑔 satisfy E.A. property and the range of 𝑔 is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point.

We obtain that Theorems 1.13 and 1.14 require special condition, that is, the range of 𝑔 is a closed subspace of 𝑋. Sometimes, the range of 𝑔 maybe is not a closed subspace of 𝑋. Therefore Theorems 1.13 and 1.14 cannot be used for this case.

The aim of this work is to introduce the new property which is so called “common limit in the range” for two self-mappings 𝑓,𝑔 and give some examples of mappings which satisfy this property. Moreover, we establish some new existence of a common fixed point theorem for generalized contractive mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using new property and give some examples. Ours results does not require condition of closeness of range and so our theorems generalize, unify, and extend many results in literature.

2. Common Fixed Point in KM and GV-Fuzzy Metric Spaces

We first introduce the concept of new property.

Definition 2.1. Suppose that (𝑋,𝑑) is a metric space and 𝑓,𝑔∶𝑋→𝑋. Two mappings 𝑓 and 𝑔 are said to satisfy the common limit in the range of g property if lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑥 (2.1) for some 𝑥∈𝑋.

In what follows, the common limit in the range of 𝑔 property will be denoted by the (CLRg) property.

Next, we show examples of mappings 𝑓 and 𝑔 which are satisfying the (CLRg) property.

Example 2.2. Let 𝑋=[0,∞) be the usual metric space. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=𝑥/4 and 𝑔𝑥=3𝑥/4 for all 𝑥∈𝑋. We consider the sequence {𝑥_𝑛}={1/𝑛}. Since lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=0=𝑔0, (2.2) therefore 𝑓 and 𝑔 satisfy the (CLRg) property.

Example 2.3. Let 𝑋=[0,∞) be the usual metric space. Define 𝑓,𝑔∶𝑋→𝑋 by 𝑓𝑥=𝑥+1 and 𝑔𝑥=2𝑥 for all 𝑥∈𝑋. Consider the sequence {𝑥_𝑛}={1+1/𝑛}. Since lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=2=𝑔1, (2.3) therefore 𝑓 and 𝑔 satisfy the (CLRg) property.

In a similar mode, two self-mappings 𝑓 and 𝑔 of a fuzzy metric space (𝑋,𝑀,∗) satisfy the (CLRg) property, if there exists a sequence {𝑥_𝑛} in 𝑋 such that 𝑓𝑥_𝑛 and 𝑔𝑥_𝑛 converge to 𝑔𝑥 for some 𝑥∈𝑋 in the sense of Definition 1.7.

Theorem 2.4. Let (𝑋,𝑀,∗) be a KM-fuzzy metric space satisfying the following property: ∀𝑥,𝑦∈𝑋,𝑥≠𝑦,∃𝑡>0∶0<𝑀𝑥,𝑦,𝑡<1, (2.4) and let 𝑓,𝑔 be weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.5) for all 𝑥,𝑦∈𝑋, where 𝑡>0. If 𝑓 and 𝑔 satisfy the (CLRg) property, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Since 𝑓 and 𝑔 satisfy the (CLRg) property, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑥 (2.6) for some 𝑥∈𝑋. Let 𝑡 be a continuity point of (𝑋,𝑀,∗). Then 𝑀𝑓𝑥_𝑛,𝑓𝑥,𝑡≥𝜑min𝑀𝑔𝑥_𝑛,𝑔𝑥,𝑡,𝑀𝑓𝑥_𝑛,𝑔𝑥_𝑛,𝑀,𝑡𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑥_𝑛,𝑡,𝑀𝑓𝑥_𝑛,𝑔𝑥,𝑡 (2.7) for all 𝑛∈ℕ. By making 𝑛→∞, we have 𝑀𝑀,𝑀𝑀𝑔𝑥,𝑓𝑥,𝑡≥𝜑min𝑔𝑥,𝑔𝑥,𝑡,𝑀𝑔𝑥,𝑔𝑥,𝑡𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑔𝑥,𝑔𝑥,𝑡=𝜑min1,1,𝑀𝑔𝑥,𝑓𝑥,𝑡,𝑀𝑔𝑥,𝑓𝑥,𝑡,1=𝜑𝑔𝑥,𝑓𝑥,𝑡 (2.8) for every 𝑡>0. We claim that 𝑔𝑥=𝑓𝑥. If not, then ∃𝑡₀>0∶0<𝑀𝑔𝑥,𝑓𝑥,𝑡₀<1. (2.9) It follows from the condition of (𝜑2) that 𝜑(𝑀(𝑔𝑥,𝑓𝑥,𝑡₀))>𝑀(𝑔𝑥,𝑓𝑥,𝑡₀), which is a contradiction. Therefore 𝑔𝑥=𝑓𝑥.
Next, we let 𝑧∶=𝑓𝑥=𝑔𝑥. Since 𝑓 and 𝑔 are weakly compatible mappings, 𝑓𝑔𝑥=𝑔𝑓𝑥 which implies that 𝑓𝑧=𝑓𝑔𝑥=𝑔𝑓𝑥=𝑔𝑧. (2.10) We claim that 𝑓𝑧=𝑧. Assume not, then by (2.4), it implies that 0<𝑀(𝑓𝑧,𝑧,𝑡₁)<1 for some 𝑡₁>0. By condition of (𝜑2), we have 𝜑(𝑀(𝑓𝑧,𝑧,𝑡₁))>𝑀(𝑓𝑧,𝑧,𝑡₁). Using condition (2.5) again, we get 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑓𝑧,𝑧,𝑡=𝑀𝑓𝑧,𝑓𝑥,𝑡≥𝜑min𝑔𝑧,𝑔𝑥,𝑡,𝑀𝑓𝑧,𝑔𝑧,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑥,𝑡=𝜑min𝑔𝑧,𝑔𝑥,𝑡,1,1,𝑀𝑓𝑥,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑥,𝑡=𝜑min𝑓𝑧,𝑓𝑥,𝑡,1,1,𝑀𝑓𝑥,𝑓𝑧,𝑡,𝑀𝑓𝑧,𝑓𝑥,𝑡=𝜑min𝑓𝑧,𝑓𝑥,𝑡,1,1,𝑀𝑓𝑧,𝑓𝑥,𝑡,𝑀𝑓𝑧,𝑓𝑥,𝑡=𝜑𝑓𝑧,𝑓𝑥,𝑡=𝜑𝑓𝑧,𝑧,𝑡 (2.11) for all 𝑡>0, which is a contradiction. Hence 𝑓𝑧=𝑧, that is, 𝑧=𝑓𝑧=𝑔𝑧. Therefore 𝑧 is a common fixed point of 𝑓 and 𝑔.
For the uniqueness of a common fixed point, we suppose that 𝑤 is another common fixed point in which 𝑤≠𝑧. It follows from condition (2.4) that there exists 𝑡₂>0 such that 0<𝑀(𝑤,𝑧,𝑡₂)<1. Since 𝑀(𝑤,𝑧,𝑡₂)∈(0,1), we have 𝜑(𝑀(𝑤,𝑧,𝑡₂))>𝑀(𝑤,𝑧,𝑡₂) by virtue of (𝜑₂). From (2.5), we have 𝑀()𝑀,𝑀({𝑀()()((𝑀(𝑧,𝑤,𝑡=𝑀𝑓𝑧,𝑓𝑤,𝑡≥𝜑min𝑔𝑧,𝑔𝑤,𝑡,𝑀𝑓𝑧,𝑔𝑧,𝑡𝑓𝑤,𝑔𝑤,𝑡,𝑀𝑓𝑤,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑤,𝑡=𝜑min𝑧,𝑤,𝑡,1,1,𝑀𝑤,𝑧,𝑡,𝑀𝑧,𝑤,𝑡)})=𝜑𝑧,𝑤,𝑡)) (2.12) for all 𝑡>0, which is a contradiction. Therefore, it must be the case that 𝑤=𝑧 which implies that 𝑓 and 𝑔 have a unique a common fixed point. This finishes the proof.

Next, we will give example which cannot be used [34, Theorem  2.1]. However, we can apply Theorem 2.4 for this case.

Example 2.5. Let 𝑋=(0,∞) and, for each 𝑥,𝑦∈𝑋 and 𝑡>0, 𝑀𝑥,𝑦,𝑡=min𝑥,𝑦maxx,𝑦. (2.13) It is well known (see [2]) that (𝑋,𝑀,𝑇_𝑝) is a GV-fuzzy metric space. If the mappings 𝑓,𝑔∶𝑋→𝑋 are defined on 𝑋 through 𝑓𝑥=𝑥^1/4 and 𝑔𝑥=𝑥^1/2, then the range of 𝑔 is (0,∞) which is not a closed subspace of 𝑋. So Theorem  2.1 of Mihet in [34] cannot be used for this case. It is easy to see that the mappings 𝑓 and 𝑔 satisfy the (CLRg) property with a sequence {𝑥_𝑛}={1+1/𝑛}. Therefore all hypothesis of the above theorem holds, with 𝜑(𝑡)=𝑡 for 𝑡∈[0,1]. Their common fixed point is 𝑥=1.

Corollary 2.6 ([34, Theorem  2.1]). Let (𝑋,𝑀,∗) be a KM-fuzzy metric space satisfying the following property: ∀𝑥,𝑦∈𝑋,𝑥≠𝑦,∃𝑡>0∶0<𝑀𝑥,𝑦,𝑡<1, (2.14) and let 𝑓,𝑔 be weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.15) for all 𝑥,𝑦∈𝑋, where 𝑡>0. If 𝑓 and 𝑔 satisfy E.A. property and the range of 𝑔 is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Since 𝑓 and 𝑔 satisfy E.A. property, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑢 (2.16) for some 𝑢∈𝑋. It follows from 𝑔𝑋 being a closed subspace of 𝑋 that 𝑢=𝑔𝑥 for some 𝑥∈𝑋 and then 𝑓 and 𝑔 satisfy the (CLRg) property. By Theorem 2.4, we get that 𝑓 and 𝑔 have a unique common fixed point.

Corollary 2.7. Let (𝑋,𝑀,∗) be a KM-fuzzy metric space satisfying the following property: ∀𝑥,𝑦∈𝑋,𝑥≠𝑦,∃𝑡>0∶0<𝑀𝑥,𝑦,𝑡<1, (2.17) and let 𝑓,𝑔 be weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑(𝑀) (2.18) for all 𝑥,𝑦∈𝑋, where 𝑡>0 and 𝑀∈𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡. (2.19) If 𝑓 and 𝑔 satisfy the (CLRg) property, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. As 𝜑 is nondecreasing and 𝑀≥min𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.20) for 𝑀∈{𝑀(𝑔𝑥,𝑔𝑦,𝑡),𝑀(𝑓𝑥,𝑔𝑥,𝑡),𝑀(𝑓𝑦,𝑔𝑦,𝑡),𝑀(𝑓𝑦,𝑔𝑥,𝑡),𝑀(𝑓𝑥,𝑔𝑦,𝑡)}, we have 𝜑(𝑀)≥𝜑min𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡. (2.21) So inequality (2.18) implies that 𝑀𝑀,𝑀.𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.22) By Theorem 2.4, we get 𝑓 and 𝑔 have a unique common fixed point.

If (𝑋,𝑀,∗) is a fuzzy metric space in the sense of George and Veeramani, then some of the hypotheses in the preceding theorem can be relaxed.

Theorem 2.8. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space and 𝑓,𝑔 weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.23) for all 𝑥,𝑦∈𝑋, where 𝑡>0. If 𝑓 and g satisfy the (CLRg) property, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. It follows from 𝑓 and 𝑔 satisfying the (CLRg) property that we can find a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑥 (2.24) for some 𝑥∈𝑋.
Let 𝑡 be a continuity point of (𝑋,𝑀,∗). Then 𝑀𝑓𝑥_𝑛,𝑓𝑥,𝑡≥𝜑min𝑀𝑔𝑥_𝑛,𝑔𝑥,𝑡,𝑀𝑓𝑥_𝑛,𝑔𝑥_𝑛,𝑀,𝑡𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑥_𝑛,𝑡,𝑀𝑓𝑥_𝑛,𝑔𝑥,𝑡 (2.25) for all 𝑛∈ℕ. By taking the limit as 𝑛 tends to infinity in (2.25), we have 𝑀𝑀,𝑀𝑀𝑔𝑥,𝑓𝑥,𝑡≥𝜑min𝑔𝑥,𝑔𝑥,𝑡,𝑀𝑔𝑥,𝑔𝑥,𝑡𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑔𝑥,𝑔𝑥,𝑡=𝜑min1,1,𝑀𝑔𝑥,𝑓𝑥,𝑡,𝑀𝑔𝑥,𝑓𝑥,𝑡,1=𝜑𝑔𝑥,𝑓𝑥,𝑡 (2.26) for every 𝑡>0. Now, we show that 𝑔𝑥=𝑓𝑥. If 𝑔𝑥≠𝑓𝑥, then from (GV-1) and (GV-2), 0<𝑀𝑔𝑥,𝑓𝑥,𝑡<1 (2.27) for all 𝑡>0. From condition of (𝜑2),𝜑(𝑀(𝑔𝑥,𝑓𝑥,𝑡))>𝑀(𝑔𝑥,𝑓𝑥,𝑡), which is a contradiction. Hence 𝑔𝑥=𝑓𝑥.
Similarly in the proof of Theorem 2.4, by denoting a point 𝑓𝑥(=𝑔𝑥) by 𝑧. Since 𝑓 and 𝑔 are weakly compatible mappings, 𝑓𝑔𝑥=𝑔𝑓𝑥 which implies that 𝑓𝑧=𝑔𝑧.
Next, we will show that 𝑓𝑧=𝑧. We will suppose that 𝑓𝑧≠𝑧. By (GV-1) and (GV-2), it implies that 0<𝑀(𝑓𝑧,𝑧,𝑡)<1 for all 𝑡>0. By (𝜑2), we know that 𝜑(𝑀(𝑓𝑧,𝑧,𝑡))>𝑀(𝑓𝑧,𝑧,𝑡). It follows from condition (2.23) that 𝑀𝑀,𝑀𝑀𝑀𝑀𝑀𝑀𝑓𝑧,𝑧,𝑡=𝑀𝑓𝑧,𝑓𝑥,𝑡≥𝜑min𝑔𝑧,𝑔𝑥,𝑡,𝑀𝑓𝑧,𝑔𝑧,𝑡𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑥,𝑡=𝜑min𝑔𝑧,𝑔𝑥,𝑡,1,1,𝑀𝑓𝑥,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑥,𝑡=𝜑min𝑓𝑧,𝑓𝑥,𝑡,1,1,𝑀𝑓𝑥,𝑓𝑧,𝑡,𝑀𝑓𝑧,𝑓𝑥,𝑡=𝜑min𝑓𝑧,𝑓𝑥,𝑡,1,1,𝑀𝑓𝑧,𝑓𝑥,𝑡,𝑀𝑓𝑧,𝑓𝑥,𝑡=𝜑𝑓𝑧,𝑓𝑥,𝑡=𝜑𝑓𝑧,𝑧,𝑡 (2.28) for all 𝑡>0, which is contradicting the above inequality. Therefore 𝑓𝑧=𝑧, and then 𝑧=𝑓𝑧=𝑔𝑧. Consequently, 𝑓 and 𝑔 have a common fixed point that is 𝑧.
Finally, we will prove that a common fixed point of 𝑓 and 𝑔 is unique. Let us suppose that 𝑤 is a common fixed point of 𝑓 and 𝑔 in which 𝑤≠𝑧. It follows from condition of (GV-1) and (GV-2) that for every 𝑡>0, we have 𝑀(𝑤,𝑧,𝑡)∈(0,1) which implies that 𝜑(𝑀(𝑤,𝑧,𝑡))>𝑀(𝑤,𝑧,𝑡). On the other hand, we know that 𝑀()𝑀,𝑀({𝑀()()((𝑀(𝑧,𝑤,𝑡=𝑀𝑓𝑧,𝑓𝑤,𝑡≥𝜑min𝑔𝑧,𝑔𝑤,𝑡,𝑀𝑓𝑧,𝑔𝑧,𝑡𝑓𝑤,𝑔𝑤,𝑡,𝑀𝑓𝑤,𝑔𝑧,𝑡,𝑀𝑓𝑧,𝑔𝑤,𝑡=𝜑min𝑧,𝑤,𝑡,1,1,𝑀𝑤,𝑧,𝑡,𝑀𝑧,𝑤,𝑡)})=𝜑𝑧,𝑤,𝑡)) (2.29) for all 𝑡>0, which is contradiction. Hence we conclude that 𝑤=𝑧. It finishes the proof of this theorem.

Corollary 2.9 ([34, Theorem  3.1]). Let (𝑋,𝑀,∗) be a GV-fuzzy metric space and 𝑓,𝑔 weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑀,𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.30) for all 𝑥,𝑦∈𝑋, where 𝑡>0. If 𝑓 and 𝑔 satisfy E.A. property and the range of 𝑔 is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Since 𝑓 and 𝑔 satisfy E.A. property, there exists a sequence {𝑥_𝑛} in 𝑋 satisfies lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑢 (2.31) for some 𝑢∈𝑋. It follows from 𝑔𝑋 being a closed subspace of 𝑋 that there exists 𝑥∈𝑋 in which 𝑢=𝑔𝑥. Therefore 𝑓 and 𝑔 satisfy the (CLRg) property. It follows from Theorem 2.8 that there exists a unique common fixed point of 𝑓 and 𝑔.

Corollary 2.10. Let (𝑋,𝑀,∗) be a GV-fuzzy metric space and 𝑓,𝑔 weakly compatible self-mappings of 𝑋 such that, for some 𝜑∈Φ, 𝑀𝑓𝑥,𝑓𝑦,𝑡≥𝜑(𝑀) (2.32) for all 𝑥,𝑦∈𝑋, where 𝑡>0 and 𝑀∈𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡. (2.33) If 𝑓 and 𝑔 satisfy the (CLRg) property, then 𝑓 and 𝑔 have a unique common fixed point.

Proof. Since 𝜑 is nondecreasing and 𝑀≥min𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡, (2.34) where 𝑀∈𝑀𝑔𝑥,𝑔𝑦,t,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡, (2.35) we get 𝜑(𝑀)≥𝜑min𝑀𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡,𝑀𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡. (2.36) Now, we know that inequality (2.32) implies that 𝑀𝑀,𝑀.𝑓𝑥,𝑓𝑦,𝑡≥𝜑min𝑔𝑥,𝑔𝑦,𝑡,𝑀𝑓𝑥,𝑔𝑥,𝑡𝑓𝑦,𝑔𝑦,𝑡,𝑀𝑓𝑦,𝑔𝑥,𝑡,𝑀𝑓𝑥,𝑔𝑦,𝑡 (2.37) It follows from Theorem 2.8 that 𝑓 and 𝑔 have a unique common fixed point.

Acknowledgments

The authors would like to thank the reviewer, who have made a number of valuable comments and suggestions which have improved the paper greatly. The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this paper for the Ph.D. Program at KMUTT. Moreover, they also would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (NRU-CSEC Project no. 54000267). This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.