Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy Metric Spaces

Abstract

The aim of this paper is to extend the notions of E.A. property and CLRg property for coupled mappings and use these notions to generalize the recent results of Xin-Qi Hu (2011). The main result is supported by a suitable example.

1. Introduction and Preliminaries

The concept of fuzzy set was introduced by Zadeh [1] and after his work there has been a great endeavor to obtain fuzzy analogues of classical theories. This problem has been searched by many authors from different points of view. In 1994, George and Veeramani [2] introduced and studied the notion of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space.

Bhaskar and Lakshmikantham [3] introduced the notion of coupled fixed points and proved some coupled fixed point results in partially ordered metric spaces. The work [3] was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem. These results were further extended and generalized by Lakshmikantham and Cirić [4] to coupled coincidence and coupled common fixed point results for nonlinear contractions in partially ordered metric spaces.

Sedghi et al. [5] proved some coupled fixed point theorems under contractive conditions in fuzzy metric spaces. The results proved by Fang [6] for compatible and weakly compatible mappings under 𝜙-contractive conditions in Menger spaces that provide a tool to Hu [7] for proving fixed points results for coupled mappings and these results are the genuine generalization of the result of [5].

Aamri and Moutawakil [8] introduced the concept of E.A. property in a metric space. Recently, Sintunavarat and Kuman [9] introduced a new concept of (CLRg). The importance of CLRg property ensures that one does not require the closeness of range subspaces.

In this paper, we give the concept of E.A. property and (CLRg) property for coupled mappings and prove a result which provides a generalization of the result of [7].

2. Preliminaries

Before we give our main result, we need the following preliminaries.

Definition 2.1 (see [1]). A fuzzy set 𝐴 in 𝑋 is a function with domain 𝑋 and values in [0,1].

Definition 2.2 (see [10]). A binary operation ∗∶[0,1]×[0,1]→[0,1] is continuous 𝑡-norm, if ([0,1],∗) is a topological abelian monoid with unit 1 such that 𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for all 𝑎,𝑏,𝑐,𝑑∈[0,1].
Some examples are below: (i)∗(𝑎,𝑏)=𝑎𝑏, (ii)∗(𝑎,𝑏)=min(𝑎,𝑏).

Definition 2.3 (see [11]). Let sup_𝑡∈(0,1)Δ(𝑡,𝑡)=1. A 𝑡-norm Δ is said to be of 𝐻-type if the family of functions {Δ^𝑚(𝑡)}^∞_𝑚=1 is equicontinuous at 𝑡=1, where Δ¹(𝑡)=𝑡,Δ(Δ^𝑚)=Δ^𝑚+1(𝑡)=𝑡.(2.1) A 𝑡-norm Δ is an 𝐻-type 𝑡-norm if and only if for any 𝜆∈(0,1), there exists 𝛿(𝜆)∈(0,1) such that Δ^𝑚(𝑡)>(1−𝜆) for all 𝑚∈ℕ, when 𝑡>(1−𝛿).
The 𝑡-norm Δ_𝑀=min is an example of 𝑡-norm, of 𝐻-type.

Definition 2.4 (see [2]). 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: (FM-1)𝑀(𝑥,𝑦,0)>0 for all 𝑥,𝑦∈𝑋,(FM-2)𝑀(𝑥,𝑦,𝑡)=1 if and only if 𝑥=𝑦, for all 𝑥,𝑦∈𝑋 and 𝑡>0,(FM-3)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡) for all 𝑥,𝑦∈𝑋 and 𝑡>0,(FM-4)𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠)≤𝑀(𝑥,𝑧,𝑡+𝑠) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0,(FM-5)𝑀(𝑥,𝑦,⋅)∶[0,∞)→[0,1] is continuous for all 𝑥,𝑦∈𝑋.

In present paper, we consider 𝑀 to be fuzzy metric space with, the following condition:(FM-6)lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1,  for  all  𝑥,𝑦∈𝑋 and 𝑡>0.

Definition 2.5 (see [2]). Let (𝑋,𝑀,∗) be a fuzzy metric space. A sequence {𝑥_𝑛}∈𝑋 is said to be: (i)convergent to a point 𝑥∈𝑋, if for all 𝑡>0, lim_𝑛→∞𝑀𝑥_𝑛,𝑥,𝑡=1,(2.2)(ii)a Cauchy sequence, if for all 𝑡>0 and 𝑝>0, lim_𝑛→∞𝑀𝑥_𝑛+𝑝,𝑥_𝑛,𝑡=1.(2.3)

A fuzzy metric space (𝑋,𝑀,∗) is said to be complete if and only if every Cauchy sequence in 𝑋 is convergent.

We note that 𝑀(𝑥,𝑦,⋅) is nondecreasing for all 𝑥,𝑦∈𝑋.

Lemma 2.6 (see [12]). Let 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦, then for all 𝑡>0: (i)lim_𝑛→∞𝑀(𝑥_𝑛,𝑦_𝑛,𝑡)≥𝑀(𝑥,𝑦,𝑡), (ii)lim_𝑛→∞𝑀(𝑥_𝑛,𝑦_𝑛,𝑡)=𝑀(𝑥,𝑦,𝑡) if 𝑀(𝑥,𝑦,𝑡) is continuous.

Definition 2.7 (see [7]). Define Φ={𝜙∶ℝ⁺→ℝ⁺}, and each 𝜙∈Φ satisfies the following conditions:(𝜙-1)𝜙 is nondecreasing;(𝜙-2)𝜙 is upper semicontinuous from the right;(𝜙-3)∑^∞_𝑛=0𝜙^𝑛(𝑡)<+∞ for all 𝑡>0, where 𝜙^𝑛+1(𝑡)=𝜙(𝜙^𝑛(𝑡)),𝑛∈ℕ.Clearly, if 𝜙∈Φ, then 𝜙(𝑡)<𝑡 for all 𝑡>0.

Definition 2.8 (see [4]). An element (𝑥,𝑦)∈𝑋×𝑋 is called: (i)a coupled fixed point of the mapping 𝑓∶𝑋×𝑋→𝑋 if 𝑓(𝑥,𝑦)=𝑥, 𝑓(𝑦,𝑥)=𝑦,(ii)a coupled coincidence point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑓(𝑥,𝑦)=𝑔(𝑥),  𝑓(𝑦,𝑥)=𝑔(𝑦),(iii)a common coupled fixed point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝑓(𝑥,𝑦)=𝑔(𝑥),  𝑦=𝑓(𝑦,𝑥)=𝑔(𝑦).

Definition 2.9 (see [6]). An element 𝑥∈𝑋 is called a common fixed point of the mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝑓(𝑥,𝑥)=𝑔(𝑥).

Definition 2.10 (see [6]). The mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called: (i)commutative if 𝑔𝑓(𝑥,𝑦)=𝑓(𝑔𝑥,𝑔𝑦) for all 𝑥,𝑦∈𝑋,(ii)compatible if lim_𝑛→∞𝑀𝑔𝑓𝑥_𝑛,𝑦_𝑛,𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑡=1,lim_𝑛→∞𝑀𝑔𝑓𝑦_𝑛,𝑥_𝑛,𝑓𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑡=1,(2.4) for all 𝑡>0 whenever {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝑋, such that  lim_𝑛→∞𝑓(𝑥_𝑛,𝑦_𝑛)=lim_𝑛→∞𝑔(𝑥_𝑛)=𝑥, and lim_𝑛→∞𝑓(𝑦_𝑛,𝑥_𝑛)=lim_𝑛→∞𝑔(𝑦_𝑛)=𝑦, for some 𝑥,𝑦∈𝑋.

Definition 2.11 (see [13]). The maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called 𝑤-compatible if 𝑔𝑓(𝑥,𝑦)=𝑓(𝑔𝑥,𝑔𝑦) whenever 𝑓(𝑥,𝑦)=𝑔(𝑥),𝑓(𝑦,𝑥)=𝑔(𝑦).
We note that the maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called weakly compatible if 𝑓(𝑥,𝑦)=𝑔(𝑥),𝑓(𝑦,𝑥)=𝑔(𝑦),(2.5) implies 𝑔𝑓(𝑥,𝑦)=𝑓(𝑔𝑥,𝑔𝑦),  𝑔𝑓(𝑦,𝑥)=𝑓(𝑔𝑦,𝑔𝑥), for all 𝑥,𝑦∈𝑋.

There exist pair of mappings that are neither compatible nor weakly compatible, as shown in the following example.

Example 2.12. Let (𝑋,𝑀,∗) be a fuzzy metric space, ∗ being a continuous norm with 𝑋=[0,1). Define 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+|𝑥−𝑦|) for all 𝑡>0,  𝑥,𝑦∈𝑋. Also define the maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=(𝑥²/2)+(𝑦²/2) and 𝑔(𝑥)=𝑥/2, respectively. Note that (0, 0) is the coupled coincidence point of 𝑓 and 𝑔 in 𝑋. It is clear that the pair (𝑓,𝑔) is weakly compatible on 𝑋.
We next show that the pair (𝑓,𝑔) is not compatible.
Consider the sequences {𝑥_𝑛}={(1/2)+(1/𝑛)} and {𝑦_𝑛}={(1/2)−(1/𝑛)}, 𝑛≥3, then lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=14=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=14=lim_𝑛→∞𝑔𝑦_𝑛,(2.6) but 𝑀𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑓𝑥_𝑛,𝑦_𝑛,𝑡=𝑡𝑡+||𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛−𝑔𝑓𝑥_𝑛,𝑦_𝑛||=𝑡𝑡+(1/8)(1/2)+2/𝑛²,(2.7) which is not convergent to 1 as 𝑛→∞.
Hence the pair (𝑓,𝑔) is not compatible.

We note that, if 𝑓 and 𝑔 are compatible then they are weakly compatible. But the converse need not be true, as shown in the following example.

Example 2.13. Let (𝑋,𝑀,∗) be a fuzzy metric space, ∗ being a continuous norm with 𝑋=[2,20]. Define 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+|𝑥−𝑦|) for all 𝑡>0,  𝑥,𝑦∈𝑋. Define the maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=2,if𝑥=2or𝑥>5,𝑦∈𝑋,6,if2<𝑥≤5,𝑦∈𝑋,𝑔(𝑥)=⎧⎪⎨⎪⎩2,if𝑥=2,12,if2<𝑥≤5,𝑥−3,𝑥>5.(2.8) The only coupled coincidence point of the pair (𝑓,𝑔) is (2,2). The mappings 𝑓 and 𝑔 are noncompatible, since for the sequences {𝑥_𝑛}={𝑦_𝑛}={5+(1/𝑛)}, 𝑛≥1 we have 𝑓(𝑥_𝑛,𝑦_𝑛)=2, 𝑔(𝑥_𝑛)→2, 𝑓(𝑦_𝑛,𝑥_𝑛)=2, 𝑔(𝑦_𝑛)→2, 𝑀(𝑓(𝑔𝑥_𝑛,𝑔𝑦_𝑛),𝑔(𝑓(𝑥_𝑛,𝑦_𝑛)),𝑡)=𝑡/(𝑡+4)↛1 as 𝑛→∞. But they are weakly compatible since they commute at their coupled coincidence point (2,2).

Now we introduce our notions.

Aamri and El Moutawakil [8] introduced the concept of E.A. property in a metric space as follows.

Let (𝑋,𝑑) be a metric space. Self mappings 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to satisfy E.A. property if there exists a sequence {𝑥_𝑛}∈𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑡(2.9) for some 𝑡∈𝑋.

Now we extend this notion for a pair of coupled maps as follows.

Definition 2.14. Let (𝑋,𝑑) be a metric space. Two mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to satisfy E.A. property if there exists sequences {𝑥_𝑛},{𝑦_𝑛}∈𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑦,(2.10) for some 𝑥,𝑦∈𝑋.

In a similar mode, we state E.A. property for coupled mappings in fuzzy metric spaces as follows.

Let (𝑋,𝑀,∗) be a FM space. Two maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 satisfy E.A. property if there exists sequences {𝑥_𝑛} and {𝑦_𝑛}∈𝑋 such that 𝑓(𝑥_𝑛,𝑦_𝑛),   𝑔(𝑥_𝑛) converges to 𝑥 and 𝑓(𝑦_𝑛,𝑥_𝑛),  𝑔(𝑦_𝑛) converges to 𝑦 in the sense of Definition 2.5.

Example 2.15. Let (−∞,∞) be a usual metric space. Define mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=𝑥²+𝑦² and 𝑔(𝑥)=2𝑥 for all 𝑥,𝑦∈𝑋. Consider the sequences {𝑥_𝑛}={1/𝑛} and {𝑦_𝑛}={−1/𝑛}. Since lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑓1𝑛,−1𝑛=0=lim_𝑛→∞𝑔1𝑛=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑓−1𝑛,1𝑛=0=lim_𝑛→∞𝑔−1𝑛=lim_𝑛→∞𝑔𝑦_𝑛,(2.11) therefore, 𝑓 and 𝑔 satisfy E.A. property, since 0∈𝑋.

Remark 2.16. It is to be noted that property E.A. need not imply compatibility, since in Example 2.12, the maps 𝑓 and 𝑔 defined are not compatible, but satisfy property E.A., since for the sequences {𝑥_𝑛}={(1/2)+(1/𝑛)} and {𝑥_𝑛}={(1/2)−(1/𝑛)} we have lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=14=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=14=lim_𝑛→∞𝑔𝑦_𝑛,(2.12) since 1/4∈𝑋.

Recently, Sintunavarat and Kuman [9] introduced a new concept of the common limit in the range of g, (CLRg) property, as follows.

Definition 2.17. Let (𝑋,𝑑) be a metric space. Two mappings 𝑓∶𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to satisfy (CLRg) property if there exists a sequence {𝑥_𝑛}∈𝑋  such that   lim_𝑛→∞𝑓(𝑥_𝑛)=lim_𝑛→∞𝑔(𝑥_𝑛)=𝑔(𝑝)  for some  𝑝∈𝑋.

Now we extend this notion for a pair of coupled mappings as follows.

Definition 2.18. Let (𝑋,𝑑) be a metric space. Two mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to satisfy (CLRg) property if there exists sequences {𝑥_𝑛},{𝑦_𝑛}∈𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔(𝑝),lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑔(𝑞),(2.13) for some 𝑝,𝑞∈𝑋.

Similarly, we state (CLRg) property for coupled mappings in fuzzy metric spaces.

Let (𝑋,𝑀,∗) be an FM space. Two maps 𝑓∶𝑋×X→𝑋 and 𝑔∶𝑋→𝑋 satisfy (CLRg) property if there exists sequences {𝑥_𝑛},{𝑦_𝑛}∈𝑋 such that 𝑓(𝑥_𝑛,𝑦_𝑛),𝑔(𝑥_𝑛) converge to 𝑔(𝑝) and 𝑓(𝑦_𝑛,𝑥_𝑛), 𝑔(𝑦_𝑛) converge to 𝑔(𝑞), in the sense of Definition 2.5.

Example 2.19. Let 𝑋=[0,∞) be a metric space under usual metric. Define mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=𝑥+𝑦+2 and 𝑔(𝑥)=2(1+𝑥) for all 𝑥,𝑦∈𝑋. We consider the sequences {𝑥_𝑛}={1+(1/𝑛)} and {𝑥_𝑛}={1−(1/𝑛)}. Since lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑓1+1𝑛,1−1𝑛=4=𝑔(1)=lim_𝑛→∞𝑔1+1𝑛=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑓1−1𝑛,1+1𝑛=4=𝑔(1)=lim_𝑛→∞𝑔1−1𝑛=lim_𝑛→∞𝑔𝑦_𝑛,(2.14) therefore, the maps 𝑓 and 𝑔 satisfy (𝐶𝐿𝑅𝑔) property.

In the next example, we show that the maps satisfying (𝐶𝐿𝑅𝑔) property need not be continuous, that is, continuity is not the necessary condition for self maps to satisfy (𝐶𝐿𝑅𝑔) property.

Example 2.20. Let 𝑋=[0,∞) be a metric space under usual metric. Define mappings 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=⎧⎨⎩𝑥+𝑦,if𝑥∈[0,1),𝑦∈𝑋,𝑥+𝑦2,if𝑥∈[1,∞),𝑦∈𝑋,𝑔(𝑥)=⎧⎨⎩1+𝑥,if𝑥∈[0,1),𝑥2,if𝑥∈[1,∞).(2.15) We consider the sequences {𝑥_𝑛}={1/𝑛} and {𝑦_𝑛}={1+(1/𝑛)}. Since lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑓1𝑛,1+1𝑛=1=𝑔(0)=lim_𝑛→∞𝑔1𝑛=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑓1+1𝑛,1𝑛=12=𝑔(1)=lim_𝑛→∞𝑔1+1𝑛=lim_𝑛→∞𝑔𝑦_𝑛,(2.16) therefore, the maps 𝑓 and 𝑔satisfy (𝐶𝐿𝑅𝑔) property but the maps are not continuous.

We next show that the pair of maps satisfying (𝐶𝐿𝑅𝑔) property may not be compatible.

Example 2.21. Let (𝑋,𝑀,∗) be a fuzzy metric space, ∗ being a continuous norm,𝑋=[0,1/2), and 𝑀(𝑥,𝑦,𝑡)=𝑡/(𝑡+|𝑥−𝑦|) for all 𝑥,𝑦∈𝑋 and 𝑡>0.
Define the maps 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 by 𝑓(𝑥,𝑦)=(𝑥²+𝑦²)/2 and 𝑔(𝑥)=𝑥/3, respectively.
Consider the sequences {𝑥_𝑛}={(1/3)+(1/𝑛)} and {𝑦_𝑛}={(1/3)−(1/𝑛)},𝑛>7. Then lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=19=lim_𝑛→∞𝑔𝑥_𝑛,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=19=lim_𝑛→∞𝑔𝑦_𝑛.(2.17) Further there exists the point 1/3 in 𝑋 such that 𝑔(1/3)=1/9, so that the pair (𝑓,𝑔) satisfies (𝐶𝐿𝑅𝑔) property. But, 𝑀𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑓𝑥_𝑛,𝑦_𝑛,𝑡=𝑡𝑡+||𝑓𝑔𝑥_𝑛,𝑔𝑦_𝑛−𝑔𝑓𝑥_𝑛,𝑦_𝑛||=𝑡𝑡+(1/18)(1/9)+1/𝑛²(2.18) does not converge to 1 as 𝑛→∞.
Hence, the pair (𝑓,𝑔) is not compatible.

3. Main Results

For convenience, we denote(1)[𝑀(𝑥,𝑦,𝑡)]^𝑛=𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑥,𝑦,𝑡)∗⋯∗𝑀(𝑥,𝑦,𝑡)𝑛,(3.1) for all 𝑛∈ℕ.

Hu [7] proved the following result.

Theorem 3.1. Let (𝑋,𝑀,∗) be a complete fuzzy metric space where ∗ is a continuous t-norm of H-type. Let 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings and there exists 𝜙∈Φ such that(2)𝑀(𝑓(𝑥,𝑦),𝑓(𝑢,𝑣),𝜙(𝑡))≥𝑀(𝑔𝑥,𝑔𝑢,𝑡)∗𝑀(𝑔𝑦,𝑔𝑣,𝑡),(3.2) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0. Suppose that 𝑓(𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous, 𝑓 and 𝑔 are compatible maps. Then there exists a unique point 𝑥∈𝑋 such that 𝑥=𝑔(𝑥)=𝑓(𝑥,𝑥), that is, 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

We now give our main result which provides a generalization of Theorem 3.1.

Theorem 3.2. Let (𝑋,𝑀,∗) be a Fuzzy Metric Space, ∗ being continuous t-norm of H-type. Let 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings and there exists 𝜙∈Φ satisfying (2) with the following conditions: (3) the pair (𝑓,𝑔) is weakly compatible, (4) the pair (𝑓,𝑔) satisfy (𝐶𝐿𝑅𝑔) property. Then 𝑓 and 𝑔 have a coupled coincidence point in 𝑋. Moreover, there exists a unique point 𝑥∈𝑋 such that 𝑥=𝑓(𝑥,𝑥)=𝑔(𝑥).

Proof . Since 𝑓 and 𝑔 satisfy (𝐶𝐿𝑅𝑔) property, there exists sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑔(𝑝),lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑔(𝑞),(3.3) for some 𝑝,𝑞∈𝑋.
Step  1. To show that 𝑓 and 𝑔 have a coupled coincidence point. From (2), 𝑀𝑓𝑥_𝑛,𝑦_𝑛,𝑓(𝑝,𝑞),𝑡≥𝑀𝑓𝑥_𝑛,𝑦_𝑛,𝑓(𝑝,𝑞),𝜙(𝑡)≥𝑀𝑔𝑥_𝑛,𝑔(𝑝),𝑡∗𝑀𝑔𝑦_𝑛,𝑔(𝑞),𝑡.(3.4) Taking limit 𝑛→∞, we get 𝑀(𝑔(𝑝),𝑓(𝑝,𝑞),𝑡)=1, that is, 𝑓(𝑝,𝑞)=𝑔(𝑝)=𝑥.
Similarly, 𝑓(𝑞,𝑝)=𝑔(𝑞)=𝑦.
Since 𝑓 and 𝑔 are weakly compatible, so that 𝑓(𝑝,𝑞)=𝑔(𝑝)=𝑥(say) and 𝑓(𝑞,𝑝)=𝑔(𝑞)=𝑦(say) implies 𝑔𝑓(𝑝,𝑞)=𝑓(𝑔(𝑝),𝑔(𝑞)) and 𝑔𝑓(𝑞,𝑝)=𝑓(𝑔(𝑞),𝑔(𝑝)), that is, 𝑔(𝑥)=𝑓(𝑥,𝑦) and 𝑔(𝑦)=𝑓(𝑦,𝑥). Hence 𝑓 and 𝑔 have a coupled coincidence point.
Step  2. To show that 𝑔(𝑥)=𝑥, and  𝑔(𝑦)=𝑦. Since ∗ is a 𝑡-norm of 𝐻-type, for any 𝜖>0, there exists 𝛿>0 such that (1−𝛿)∗⋯∗(1−𝛿)_{𝑝}≥(1−𝜖),(3.5) for all 𝑝∈ℕ.
Since lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀𝑔𝑥,𝑥,𝑡₀≥(1−𝛿),𝑀𝑔𝑦,𝑦,𝑡₀≥(1−𝛿).(3.6) Also since 𝜙∈Φ using condition (𝜙−3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞.
Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>∑^∞_𝑘=𝑛0𝜙^𝑘(𝑡₀). From (2), we have 𝑀𝑔𝑥,𝑥,𝜙𝑡₀=𝑀𝑓(𝑥,𝑦),𝑓(𝑝,𝑞),𝜙𝑡₀≥𝑀𝑔𝑥,𝑔𝑝,𝑡₀∗𝑀𝑔𝑦,𝑔𝑞,𝑡₀=𝑀𝑔𝑥,𝑥,𝑡₀∗𝑀𝑔𝑦,𝑦,𝑡₀,𝑀𝑔𝑦,𝑦,𝜙𝑡₀=𝑀𝑓(𝑦,𝑥),𝑓(𝑞,𝑝),𝜙𝑡₀≥𝑀𝑔𝑦,𝑔𝑞,𝑡₀∗𝑀𝑔𝑥,𝑔𝑝,𝑡₀=𝑀𝑔𝑦,𝑦,𝑡₀∗𝑀𝑔𝑥,𝑥,𝑡₀.(3.7) Similarly, we can also get 𝑀𝑔𝑥,𝑥,𝜙²𝑡₀=𝑀𝑓(𝑥,𝑦),𝑓(𝑝,𝑞),𝜙²𝑡₀≥𝑀𝑔𝑥,𝑔𝑝,𝜙𝑡₀∗𝑀𝑔𝑦,𝑔𝑞,𝜙𝑡₀=𝑀𝑔𝑥,𝑥,𝜙𝑡₀∗𝑀𝑔𝑦,𝑦,𝜙𝑡₀≥𝑀𝑔𝑥,𝑥,𝑡₀²∗𝑀𝑔𝑦,𝑦,𝑡₀²,𝑀𝑔𝑦,𝑦,𝜙²𝑡₀=𝑀𝑓(𝑦,𝑥),𝑓(𝑞,𝑝),𝜙²𝑡₀≥𝑀𝑔𝑦,𝑦,𝑡₀²∗𝑀𝑔𝑥,𝑥,𝑡₀².(3.8) Continuing in the same way, we can get for all 𝑛∈ℕ, 𝑀𝑔𝑥,𝑥,𝜙^𝑛𝑡₀=𝑀𝑔𝑥,𝑥,𝜙^𝑛−1𝑡₀∗𝑀𝑔𝑦,𝑦,𝜙^𝑛−1𝑡₀≥𝑀𝑔𝑥,𝑥,𝑡₀^2𝑛−1∗𝑀𝑔𝑦,𝑦,𝑡₀^2𝑛−1,𝑀𝑔𝑦,𝑦,𝜙^𝑛𝑡₀≥𝑀𝑔𝑦,𝑦,𝑡₀^2𝑛−1∗𝑀𝑔𝑥,𝑥,𝑡₀^2𝑛−1.(3.9) Then, we have 𝑀(𝑔𝑥,𝑥,𝑡)≥𝑀⎛⎜⎝𝑔𝑥,𝑥,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀⎞⎟⎠≥𝑀𝑔𝑥,𝑥,𝜙^𝑛0𝑡₀≥𝑀(𝑔𝑥,𝑥,𝑡₀)^2𝑛0−1∗𝑀(𝑔𝑦,𝑦,𝑡₀)^2𝑛0−1≥(1−𝛿)∗⋯∗(1−𝛿)_{2𝑛0}≥(1−𝜖).(3.10) So, for any 𝜖>0, we have 𝑀(𝑔𝑥,𝑥,𝑡)≥(1−𝜖) for all 𝑡>0.
This implies 𝑔(𝑥)=𝑥. Similarly, 𝑔(𝑦)=𝑦.
Step  3. Next we shall show that 𝑥=𝑦. Since ∗ is a 𝑡-norm of 𝐻-type, for any 𝜖>0 there exists 𝛿>0 such that (1−𝛿)∗⋯∗(1−𝛿)_{𝑝}≥(1−𝜖),(3.11) for all 𝑝∈ℕ.
Since lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝑥,𝑦,𝑡₀)≥(1−𝛿).
Also since 𝜙∈Φ, using condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>^∞_𝑘=𝑛0𝜙^𝑘𝑡₀.(3.12) Using condition (2), we have 𝑀𝑥,𝑦,𝜙𝑡₀=𝑀𝑓(𝑝,𝑞),𝑓(𝑞,𝑝),𝜙𝑡₀≥𝑀𝑔𝑝,𝑔𝑞,𝑡₀∗𝑀𝑔𝑞,𝑔𝑝,𝑡₀=𝑀𝑥,𝑦,𝑡₀∗𝑀𝑦,𝑥,𝑡₀.(3.13) Continuing in the same way, we can get for all 𝑛₀∈ℕ, 𝑀𝑥,𝑦,𝜙^𝑛𝑡₀≥𝑀(𝑥,𝑦,𝑡₀)^2𝑛0−1∗𝑀(𝑦,𝑥,𝑡₀)^2𝑛0−1.(3.14) Then we have 𝑀(𝑥,𝑦,𝑡)≥𝑀⎛⎜⎝𝑥,𝑦,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀⎞⎟⎠≥𝑀𝑥,𝑦,𝜙^𝑛0𝑡₀≥𝑀(𝑥,𝑦,𝑡₀)^2𝑛0−1∗𝑀(𝑦,𝑥,𝑡₀)^2𝑛0−1≥(1−𝛿)∗⋯∗(1−𝛿)_{2𝑛0}≥(1−𝜖),(3.15) which implies that 𝑥=𝑦. Thus, we have proved that 𝑓 and 𝑔 have a common fixed point 𝑥∈𝑋.
Step  4. We now prove the uniqueness of 𝑥. Let 𝑧 be any point in 𝑋 such that 𝑧≠𝑥 with 𝑔(𝑧)=𝑧=𝑓(𝑧,𝑧). Since ∗ is a 𝑡-norm of 𝐻-type, for any 𝜖>0, there exists 𝛿>0 such that (1−𝛿)∗⋯∗(1−𝛿)_{𝑝}≥(1−𝜖),(3.16) for all 𝑝∈ℕ. Since lim_𝑡→∞𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that 𝑀(𝑥,𝑧,𝑡₀)≥(1−𝛿). Also since 𝜙∈Φ and using condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then for any 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑡>^∞_𝑘=𝑛0𝜙^𝑘𝑡₀.(3.17) Using condition (2), we have 𝑀𝑥,𝑧,𝜙𝑡₀=𝑀𝑓(𝑥,𝑥),𝑓(𝑧,𝑧),𝜙𝑡₀≥𝑀𝑔(𝑥),𝑔(𝑧),𝑡₀∗𝑀𝑔(𝑥),𝑔(𝑧),𝑡₀≥𝑀𝑥,𝑧,𝑡₀∗𝑀𝑥,𝑧,𝑡₀=𝑀𝑥,𝑧,𝑡₀².(3.18) Continuing in the same way, we can get for all 𝑛∈ℕ, 𝑀𝑥,𝑧,𝜙^𝑛𝑡₀≥𝑀𝑥,𝑧,𝑡₀^2𝑛0−1².(3.19) Then we have 𝑀(𝑥,𝑧,𝑡)≥𝑀⎛⎜⎝𝑥,𝑧,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀⎞⎟⎠≥𝑀𝑥,𝑧,𝜙^𝑛0𝑡₀≥𝑀(𝑥,𝑧,𝑡₀)^2𝑛0−1²=𝑀(𝑥,𝑧,𝑡₀)^2𝑛0≥(1−𝛿)∗⋯∗(1−𝛿)_{2𝑛0}≥(1−𝜖),(3.20) which implies that 𝑥=𝑧.
Hence 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

Remark 3.3. We still get a unique common fixed point if weakly compatible notion is replaced by w-compatible notion.

Now we give another generalization of Theorem 3.1.

Corollary 3.4. Let (𝑋,𝑀,∗) be a fuzzy metric space where ∗ is a continuous t-norm of H-type. Let 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings and there exists 𝜙∈Φ satisfying (2) and (3) with the following condition:(5) the pair (𝑓,𝑔) satisfy E.A. property.If 𝑔(𝑋) is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

Proof. Since 𝑓 and 𝑔 satisfy E.A. property, there exists sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑦,(3.21) for some 𝑥,𝑦∈𝑋.
It follows from 𝑔(𝑋) being a closed subspace of 𝑋 that 𝑥=𝑔(𝑝), 𝑦=𝑔(𝑞) for some 𝑝,𝑞∈𝑋 and then 𝑓 and 𝑔 satisfy the (𝐶𝐿𝑅𝑔) property. By Theorem 3.2, we get that 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

Corollary 3.5. Let (𝑋,𝑀,∗) be a fuzzy metric space where ∗ is a continuous 𝑡-norm of 𝐻-type. Let 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings and there exists 𝜙∈Φ satisfying (2), (3), and (5).
Suppose that 𝑓(𝑋×𝑋)⊆𝑔(𝑋), if range of one of the maps 𝑓 or 𝑔 is a closed subspace of 𝑋, then 𝑓 and 𝑔 have a unique common fixed point in 𝑋.

Proof. It follows immediately from Corollary 3.5.

Taking 𝑔=𝐼_𝑋 in Theorem 3.2, the Corollary 3.6 follows immediately the following.

Corollary 3.6. Let (𝑋,𝑀,∗) be a fuzzy metric space where ∗ is a continuous 𝑡-norm of 𝐻-type. Let 𝑓∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings and there exists 𝜙∈Φ satisfying the following conditions, for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0:(6)𝑀(𝑓(𝑥,𝑦),𝑓(𝑢,𝑣),𝜙(𝑡))≥𝑀(𝑥,𝑢,𝑡)∗𝑀(𝑦,𝑣,𝑡), (7) there exists sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑥_𝑛=𝑥,lim_𝑛→∞𝑓𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑦_𝑛=𝑦,(3.22) for some 𝑥,𝑦∈𝑋.

Then, there exists a unique 𝑧∈𝑋 such that 𝑧=𝑓(𝑧,𝑧).