Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 961210 | 13 pages | https://doi.org/10.1155/2012/961210

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

Academic Editor: Yansheng Liu
Received26 May 2012
Revised06 Jul 2012
Accepted08 Jul 2012
Published12 Aug 2012

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 πœ™-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 βˆ—βˆΆ[0,1]Γ—[0,1]β†’[0,1] is continuous 𝑑-norm, if ([0,1],βˆ—) is a topological abelian monoid with unit 1 such that π‘Žβˆ—π‘β‰€π‘βˆ—π‘‘ whenever π‘Žβ‰€π‘ and 𝑏≀𝑑 for all π‘Ž,𝑏,𝑐,π‘‘βˆˆ[0,1].
Some examples are below: (i)βˆ—(π‘Ž,𝑏)=π‘Žπ‘, (ii)βˆ—(π‘Ž,𝑏)=min(π‘Ž,𝑏).

Definition 2.3 (see [11]). Let supπ‘‘βˆˆ(0,1)Ξ”(𝑑,𝑑)=1. A 𝑑-norm Ξ” is said to be of 𝐻-type if the family of functions {Ξ”π‘š(𝑑)}βˆžπ‘š=1 is equicontinuous at 𝑑=1, where Ξ”1(𝑑)=𝑑,Ξ”(Ξ”π‘š)=Ξ”π‘š+1(𝑑)=𝑑.(2.1) A 𝑑-norm Ξ” is an 𝐻-type 𝑑-norm if and only if for any πœ†βˆˆ(0,1), there exists 𝛿(πœ†)∈(0,1) such that Ξ”π‘š(𝑑)>(1βˆ’πœ†) for all π‘šβˆˆβ„•, when 𝑑>(1βˆ’π›Ώ).
The 𝑑-norm Δ𝑀=min is an example of 𝑑-norm, of 𝐻-type.

Definition 2.4 (see [2]). The 3-tuple (𝑋,𝑀,βˆ—) is said to be a fuzzy metric space if 𝑋 is an arbitrary set, βˆ— is a continuous 𝑑-norm and 𝑀 is a fuzzy set on 𝑋2Γ—[0,∞) satisfying the following conditions: (FM-1)𝑀(π‘₯,𝑦,0)>0 for all π‘₯,π‘¦βˆˆπ‘‹,(FM-2)𝑀(π‘₯,𝑦,𝑑)=1 if and only if π‘₯=𝑦, for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0,(FM-3)𝑀(π‘₯,𝑦,𝑑)=𝑀(𝑦,π‘₯,𝑑) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0,(FM-4)𝑀(π‘₯,𝑦,𝑑)βˆ—π‘€(𝑦,𝑧,𝑠)≀𝑀(π‘₯,𝑧,𝑑+𝑠) for all π‘₯,𝑦,π‘§βˆˆπ‘‹ and 𝑑,𝑠>0,(FM-5)𝑀(π‘₯,𝑦,β‹…)∢[0,∞)β†’[0,1] is continuous for all π‘₯,π‘¦βˆˆπ‘‹.

In present paper, we consider 𝑀 to be fuzzy metric space with, the following condition:(FM-6)limπ‘‘β†’βˆžπ‘€(π‘₯,𝑦,𝑑)=1,  for  all  π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0.

Definition 2.5 (see [2]). Let (𝑋,𝑀,βˆ—) be a fuzzy metric space. A sequence {π‘₯𝑛}βˆˆπ‘‹ is said to be: (i)convergent to a point π‘₯βˆˆπ‘‹, if for all 𝑑>0, limπ‘›β†’βˆžπ‘€ξ€·π‘₯𝑛,π‘₯,𝑑=1,(2.2)(ii)a Cauchy sequence, if for all 𝑑>0 and 𝑝>0, limπ‘›β†’βˆžπ‘€ξ€·π‘₯𝑛+𝑝,π‘₯𝑛,𝑑=1.(2.3)

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

We note that 𝑀(π‘₯,𝑦,β‹…) is nondecreasing for all π‘₯,π‘¦βˆˆπ‘‹.

Lemma 2.6 (see [12]). Let π‘₯𝑛→π‘₯ and 𝑦𝑛→𝑦, then for all 𝑑>0: (i)limπ‘›β†’βˆžπ‘€(π‘₯𝑛,𝑦𝑛,𝑑)β‰₯𝑀(π‘₯,𝑦,𝑑), (ii)limπ‘›β†’βˆžπ‘€(π‘₯𝑛,𝑦𝑛,𝑑)=𝑀(π‘₯,𝑦,𝑑) if 𝑀(π‘₯,𝑦,𝑑) is continuous.

Definition 2.7 (see [7]). Define Ξ¦={πœ™βˆΆβ„+→ℝ+}, and each πœ™βˆˆΞ¦ satisfies the following conditions:(πœ™-1)πœ™ is nondecreasing;(πœ™-2)πœ™ is upper semicontinuous from the right;(πœ™-3)βˆ‘βˆžπ‘›=0πœ™π‘›(𝑑)<+∞ for all 𝑑>0, where πœ™π‘›+1(𝑑)=πœ™(πœ™π‘›(𝑑)),π‘›βˆˆβ„•.Clearly, if πœ™βˆˆΞ¦, then πœ™(𝑑)<𝑑 for all 𝑑>0.

Definition 2.8 (see [4]). An element (π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹ is called: (i)a coupled fixed point of the mapping π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ if 𝑓(π‘₯,𝑦)=π‘₯, 𝑓(𝑦,π‘₯)=𝑦,(ii)a coupled coincidence point of the mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ if 𝑓(π‘₯,𝑦)=𝑔(π‘₯),  𝑓(𝑦,π‘₯)=𝑔(𝑦),(iii)a common coupled fixed point of the mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ if π‘₯=𝑓(π‘₯,𝑦)=𝑔(π‘₯),  𝑦=𝑓(𝑦,π‘₯)=𝑔(𝑦).

Definition 2.9 (see [6]). An element π‘₯βˆˆπ‘‹ is called a common fixed point of the mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ if π‘₯=𝑓(π‘₯,π‘₯)=𝑔(π‘₯).

Definition 2.10 (see [6]). The mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are called: (i)commutative if 𝑔𝑓(π‘₯,𝑦)=𝑓(𝑔π‘₯,𝑔𝑦) for all π‘₯,π‘¦βˆˆπ‘‹,(ii)compatible if limπ‘›β†’βˆžπ‘€ξ€·π‘”π‘“ξ€·π‘₯𝑛,𝑦𝑛,𝑓𝑔π‘₯𝑛,𝑔𝑦𝑛,𝑑=1,limπ‘›β†’βˆžπ‘€ξ€·π‘”π‘“ξ€·π‘¦π‘›,π‘₯𝑛,𝑓𝑔𝑦𝑛,𝑔π‘₯𝑛,𝑑=1,(2.4) for all 𝑑>0 whenever {π‘₯𝑛} and {𝑦𝑛} are sequences in 𝑋, such that  limπ‘›β†’βˆžπ‘“(π‘₯𝑛,𝑦𝑛)=limπ‘›β†’βˆžπ‘”(π‘₯𝑛)=π‘₯, and limπ‘›β†’βˆžπ‘“(𝑦𝑛,π‘₯𝑛)=limπ‘›β†’βˆžπ‘”(𝑦𝑛)=𝑦, for some π‘₯,π‘¦βˆˆπ‘‹.

Definition 2.11 (see [13]). The maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are called 𝑀-compatible if 𝑔𝑓(π‘₯,𝑦)=𝑓(𝑔π‘₯,𝑔𝑦) whenever 𝑓(π‘₯,𝑦)=𝑔(π‘₯),𝑓(𝑦,π‘₯)=𝑔(𝑦).
We note that the maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are called weakly compatible if 𝑓(π‘₯,𝑦)=𝑔(π‘₯),𝑓(𝑦,π‘₯)=𝑔(𝑦),(2.5) implies 𝑔𝑓(π‘₯,𝑦)=𝑓(𝑔π‘₯,𝑔𝑦),  𝑔𝑓(𝑦,π‘₯)=𝑓(𝑔𝑦,𝑔π‘₯), for all π‘₯,π‘¦βˆˆπ‘‹.

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

Example 2.12. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space, βˆ— being a continuous norm with 𝑋=[0,1). Define 𝑀(π‘₯,𝑦,𝑑)=𝑑/(𝑑+|π‘₯βˆ’π‘¦|) for all 𝑑>0,  π‘₯,π‘¦βˆˆπ‘‹. Also define the maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=(π‘₯2/2)+(𝑦2/2) and 𝑔(π‘₯)=π‘₯/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 {π‘₯𝑛}={(1/2)+(1/𝑛)} and {𝑦𝑛}={(1/2)βˆ’(1/𝑛)}, 𝑛β‰₯3, then limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=14=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=14=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ,(2.6) but 𝑀𝑓𝑔π‘₯𝑛,𝑔𝑦𝑛,𝑔𝑓π‘₯𝑛,𝑦𝑛,𝑑=𝑑𝑑+||𝑓𝑔π‘₯𝑛,π‘”π‘¦π‘›ξ€Έβˆ’π‘”π‘“ξ€·π‘₯𝑛,𝑦𝑛||=𝑑𝑑+(1/8)ξ€·(1/2)+ξ€·2/𝑛2ξ€Έξ€Έ,(2.7) which is not convergent to 1 as π‘›β†’βˆž.
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,βˆ—) be a fuzzy metric space, βˆ— being a continuous norm with 𝑋=[2,20]. Define 𝑀(π‘₯,𝑦,𝑑)=𝑑/(𝑑+|π‘₯βˆ’π‘¦|) for all 𝑑>0,  π‘₯,π‘¦βˆˆπ‘‹. Define the maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=ξ‚»2,ifπ‘₯=2orπ‘₯>5,π‘¦βˆˆπ‘‹,6,if2<π‘₯≀5,π‘¦βˆˆπ‘‹,𝑔(π‘₯)=⎧βŽͺ⎨βŽͺ⎩2,ifπ‘₯=2,12,if2<π‘₯≀5,π‘₯βˆ’3,π‘₯>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 {π‘₯𝑛}={𝑦𝑛}={5+(1/𝑛)}, 𝑛β‰₯1 we have 𝑓(π‘₯𝑛,𝑦𝑛)=2, 𝑔(π‘₯𝑛)β†’2, 𝑓(𝑦𝑛,π‘₯𝑛)=2, 𝑔(𝑦𝑛)β†’2, 𝑀(𝑓(𝑔π‘₯𝑛,𝑔𝑦𝑛),𝑔(𝑓(π‘₯𝑛,𝑦𝑛)),𝑑)=𝑑/(𝑑+4)↛1 as π‘›β†’βˆž. 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 π‘“βˆΆπ‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are said to satisfy E.A. property if there exists a sequence {π‘₯𝑛}βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛=𝑑(2.9) for some π‘‘βˆˆπ‘‹.

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 π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are said to satisfy E.A. property if there exists sequences {π‘₯𝑛},{𝑦𝑛}βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛=π‘₯,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ=𝑦,(2.10) for some π‘₯,π‘¦βˆˆπ‘‹.

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

Let (𝑋,𝑀,βˆ—) be a FM space. Two maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ satisfy E.A. property if there exists sequences {π‘₯𝑛} and {𝑦𝑛}βˆˆπ‘‹ such that 𝑓(π‘₯𝑛,𝑦𝑛),   𝑔(π‘₯𝑛) converges to π‘₯ and 𝑓(𝑦𝑛,π‘₯𝑛),  𝑔(𝑦𝑛) converges to 𝑦 in the sense of Definition 2.5.

Example 2.15. Let (βˆ’βˆž,∞) be a usual metric space. Define mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=π‘₯2+𝑦2 and 𝑔(π‘₯)=2π‘₯ for all π‘₯,π‘¦βˆˆπ‘‹. Consider the sequences {π‘₯𝑛}={1/𝑛} and {𝑦𝑛}={βˆ’1/𝑛}. Since limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘“ξ‚€1𝑛,βˆ’1𝑛=0=limπ‘›β†’βˆžπ‘”ξ‚€1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘“ξ‚€βˆ’1𝑛,1𝑛=0=limπ‘›β†’βˆžπ‘”ξ‚€βˆ’1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ,(2.11) therefore, 𝑓 and 𝑔 satisfy E.A. property, since 0βˆˆπ‘‹.

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 {π‘₯𝑛}={(1/2)+(1/𝑛)} and {π‘₯𝑛}={(1/2)βˆ’(1/𝑛)} we have limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=14=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=14=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ,(2.12) since 1/4βˆˆπ‘‹.

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 π‘“βˆΆπ‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are said to satisfy (CLRg) property if there exists a sequence {π‘₯𝑛}βˆˆπ‘‹β€‰β€‰such that   limπ‘›β†’βˆžπ‘“(π‘₯𝑛)=limπ‘›β†’βˆžπ‘”(π‘₯𝑛)=𝑔(𝑝)  for someβ€‰β€‰π‘βˆˆπ‘‹.

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 π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ are said to satisfy (CLRg) property if there exists sequences {π‘₯𝑛},{𝑦𝑛}βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛=𝑔(𝑝),limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ=𝑔(π‘ž),(2.13) for some 𝑝,π‘žβˆˆπ‘‹.

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

Let (𝑋,𝑀,βˆ—) be an FM space. Two maps π‘“βˆΆπ‘‹Γ—X→𝑋 and π‘”βˆΆπ‘‹β†’π‘‹ satisfy (CLRg) property if there exists sequences {π‘₯𝑛},{𝑦𝑛}βˆˆπ‘‹ such that 𝑓(π‘₯𝑛,𝑦𝑛),𝑔(π‘₯𝑛) converge to 𝑔(𝑝) and 𝑓(𝑦𝑛,π‘₯𝑛), 𝑔(𝑦𝑛) converge to 𝑔(π‘ž), in the sense of Definition 2.5.

Example 2.19. Let 𝑋=[0,∞) be a metric space under usual metric. Define mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=π‘₯+𝑦+2 and 𝑔(π‘₯)=2(1+π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹. We consider the sequences {π‘₯𝑛}={1+(1/𝑛)} and {π‘₯𝑛}={1βˆ’(1/𝑛)}. Since limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘“ξ‚€1+1𝑛,1βˆ’1𝑛=4=𝑔(1)=limπ‘›β†’βˆžπ‘”ξ‚€1+1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘“ξ‚€1βˆ’1𝑛,1+1𝑛=4=𝑔(1)=limπ‘›β†’βˆžπ‘”ξ‚€1βˆ’1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ,(2.14) therefore, the maps 𝑓 and 𝑔 satisfy (𝐢𝐿𝑅𝑔) property.

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

Example 2.20. Let 𝑋=[0,∞) be a metric space under usual metric. Define mappings π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=⎧⎨⎩π‘₯+𝑦,ifπ‘₯∈[0,1),π‘¦βˆˆπ‘‹,π‘₯+𝑦2,ifπ‘₯∈[1,∞),π‘¦βˆˆπ‘‹,𝑔(π‘₯)=⎧⎨⎩1+π‘₯,ifπ‘₯∈[0,1),π‘₯2,ifπ‘₯∈[1,∞).(2.15) We consider the sequences {π‘₯𝑛}={1/𝑛} and {𝑦𝑛}={1+(1/𝑛)}. Since limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘“ξ‚€1𝑛,1+1𝑛=1=𝑔(0)=limπ‘›β†’βˆžπ‘”ξ‚€1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘“ξ‚€1+1𝑛,1𝑛=12=𝑔(1)=limπ‘›β†’βˆžπ‘”ξ‚€1+1𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ,(2.16) therefore, the maps 𝑓 and 𝑔satisfy (𝐢𝐿𝑅𝑔) property but the maps are not continuous.

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

Example 2.21. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space, βˆ— being a continuous norm,𝑋=[0,1/2), and 𝑀(π‘₯,𝑦,𝑑)=𝑑/(𝑑+|π‘₯βˆ’π‘¦|) for all π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0.
Define the maps π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ by 𝑓(π‘₯,𝑦)=(π‘₯2+𝑦2)/2 and 𝑔(π‘₯)=π‘₯/3, respectively.
Consider the sequences {π‘₯𝑛}={(1/3)+(1/𝑛)} and {𝑦𝑛}={(1/3)βˆ’(1/𝑛)},𝑛>7. Then limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=19=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=19=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ.(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 (𝐢𝐿𝑅𝑔) property. But, 𝑀𝑓𝑔π‘₯𝑛,𝑔𝑦𝑛,𝑔𝑓π‘₯𝑛,𝑦𝑛,𝑑=𝑑𝑑+||𝑓𝑔π‘₯𝑛,π‘”π‘¦π‘›ξ€Έβˆ’π‘”π‘“ξ€·π‘₯𝑛,𝑦𝑛||=𝑑𝑑+(1/18)ξ€·(1/9)+ξ€·1/𝑛2ξ€Έξ€Έ(2.18) does not converge to 1 as π‘›β†’βˆž.
Hence, the pair (𝑓,𝑔) is not compatible.

3. Main Results

For convenience, we denote(1)[𝑀(π‘₯,𝑦,𝑑)]𝑛=𝑀(π‘₯,𝑦,𝑑)βˆ—π‘€(π‘₯,𝑦,𝑑)βˆ—β‹―βˆ—π‘€(π‘₯,𝑦,𝑑)𝑛,(3.1) for all π‘›βˆˆβ„•.

Hu [7] proved the following result.

Theorem 3.1. Let (𝑋,𝑀,βˆ—) be a complete fuzzy metric space where βˆ— is a continuous t-norm of H-type. Let π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ be two mappings and there exists πœ™βˆˆΞ¦ such that(2)𝑀(𝑓(π‘₯,𝑦),𝑓(𝑒,𝑣),πœ™(𝑑))β‰₯𝑀(𝑔π‘₯,𝑔𝑒,𝑑)βˆ—π‘€(𝑔𝑦,𝑔𝑣,𝑑),(3.2) for all π‘₯,𝑦,𝑒,π‘£βˆˆπ‘‹ and 𝑑>0. Suppose that 𝑓(𝑋×𝑋)βŠ†π‘”(𝑋), 𝑔 is continuous, 𝑓 and 𝑔 are compatible maps. Then there exists a unique point π‘₯βˆˆπ‘‹ such that π‘₯=𝑔(π‘₯)=𝑓(π‘₯,π‘₯), 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,βˆ—) be a Fuzzy Metric Space, βˆ— being continuous t-norm of H-type. Let π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ be two mappings and there exists πœ™βˆˆΞ¦ satisfying (2) with the following conditions: (3) the pair (𝑓,𝑔) is weakly compatible, (4) the pair (𝑓,𝑔) satisfy (𝐢𝐿𝑅𝑔) property. Then 𝑓 and 𝑔 have a coupled coincidence point in 𝑋. Moreover, there exists a unique point π‘₯βˆˆπ‘‹ such that π‘₯=𝑓(π‘₯,π‘₯)=𝑔(π‘₯).

Proof . Since 𝑓 and 𝑔 satisfy (𝐢𝐿𝑅𝑔) property, there exists sequences {π‘₯𝑛} and {𝑦𝑛} in 𝑋 such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛=𝑔(𝑝),limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ=𝑔(π‘ž),(3.3) for some 𝑝,π‘žβˆˆπ‘‹.
Step  1. To show that 𝑓 and 𝑔 have a coupled coincidence point. From (2), 𝑀𝑓π‘₯𝑛,𝑦𝑛,𝑓(𝑝,π‘ž),𝑑β‰₯𝑀𝑓π‘₯𝑛,𝑦𝑛,𝑓(𝑝,π‘ž),πœ™(𝑑)ξ€Έβ‰₯𝑀𝑔π‘₯𝑛,𝑔(𝑝),π‘‘ξ€Έβˆ—π‘€ξ€·π‘”π‘¦π‘›,𝑔(π‘ž),𝑑.(3.4) Taking limit π‘›β†’βˆž, we get 𝑀(𝑔(𝑝),𝑓(𝑝,π‘ž),𝑑)=1, that is, 𝑓(𝑝,π‘ž)=𝑔(𝑝)=π‘₯.
Similarly, 𝑓(π‘ž,𝑝)=𝑔(π‘ž)=𝑦.
Since 𝑓 and 𝑔 are weakly compatible, so that 𝑓(𝑝,π‘ž)=𝑔(𝑝)=π‘₯(say) and 𝑓(π‘ž,𝑝)=𝑔(π‘ž)=𝑦(say) implies 𝑔𝑓(𝑝,π‘ž)=𝑓(𝑔(𝑝),𝑔(π‘ž)) and 𝑔𝑓(π‘ž,𝑝)=𝑓(𝑔(π‘ž),𝑔(𝑝)), that is, 𝑔(π‘₯)=𝑓(π‘₯,𝑦) and 𝑔(𝑦)=𝑓(𝑦,π‘₯). Hence 𝑓 and 𝑔 have a coupled coincidence point.
Step  2. To show that 𝑔(π‘₯)=π‘₯, and  𝑔(𝑦)=𝑦. Since βˆ— is a 𝑑-norm of 𝐻-type, for any πœ–>0, there exists 𝛿>0 such that (1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘β‰₯(1βˆ’πœ–),(3.5) for all π‘βˆˆβ„•.
Since limπ‘‘β†’βˆžπ‘€(π‘₯,𝑦,𝑑)=1 for all π‘₯,π‘¦βˆˆπ‘‹, there exists 𝑑0>0 such that 𝑀𝑔π‘₯,π‘₯,𝑑0ξ€Έβ‰₯(1βˆ’π›Ώ),𝑀𝑔𝑦,𝑦,𝑑0ξ€Έβ‰₯(1βˆ’π›Ώ).(3.6) Also since πœ™βˆˆΞ¦ using condition (πœ™βˆ’3), we have βˆ‘βˆžπ‘›=1πœ™π‘›(𝑑0)<∞.
Then for any 𝑑>0, there exists 𝑛0βˆˆβ„• such that 𝑑>βˆ‘βˆžπ‘˜=𝑛0πœ™π‘˜(𝑑0). From (2), we have 𝑀𝑔π‘₯,π‘₯,πœ™ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑓(π‘₯,𝑦),𝑓(𝑝,π‘ž),πœ™ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔π‘₯,𝑔𝑝,𝑑0ξ€Έβˆ—π‘€ξ€·π‘”π‘¦,π‘”π‘ž,𝑑0ξ€Έ=𝑀𝑔π‘₯,π‘₯,𝑑0ξ€Έβˆ—π‘€ξ€·π‘”π‘¦,𝑦,𝑑0ξ€Έ,𝑀𝑔𝑦,𝑦,πœ™ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑓(𝑦,π‘₯),𝑓(π‘ž,𝑝),πœ™ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔𝑦,π‘”π‘ž,𝑑0ξ€Έβˆ—π‘€ξ€·π‘”π‘₯,𝑔𝑝,𝑑0ξ€Έ=𝑀𝑔𝑦,𝑦,𝑑0ξ€Έβˆ—π‘€ξ€·π‘”π‘₯,π‘₯,𝑑0ξ€Έ.(3.7) Similarly, we can also get 𝑀𝑔π‘₯,π‘₯,πœ™2𝑑0ξ€Έξ€Έ=𝑀𝑓(π‘₯,𝑦),𝑓(𝑝,π‘ž),πœ™2𝑑0ξ€Έξ€Έβ‰₯𝑀𝑔π‘₯,𝑔𝑝,πœ™ξ€·π‘‘0ξ€Έξ€Έβˆ—π‘€ξ€·π‘”π‘¦,π‘”π‘ž,πœ™ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑔π‘₯,π‘₯,πœ™ξ€·π‘‘0ξ€Έξ€Έβˆ—π‘€ξ€·π‘”π‘¦,𝑦,πœ™ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔π‘₯,π‘₯,𝑑0ξ€Έξ€»2βˆ—ξ€Ίπ‘€ξ€·π‘”π‘¦,𝑦,𝑑0ξ€Έξ€»2,𝑀𝑔𝑦,𝑦,πœ™2𝑑0ξ€Έξ€Έ=𝑀𝑓(𝑦,π‘₯),𝑓(π‘ž,𝑝),πœ™2𝑑0ξ€Έξ€Έβ‰₯𝑀𝑔𝑦,𝑦,𝑑0ξ€Έξ€»2βˆ—ξ€Ίπ‘€ξ€·π‘”π‘₯,π‘₯,𝑑0ξ€Έξ€»2.(3.8) Continuing in the same way, we can get for all π‘›βˆˆβ„•, 𝑀𝑔π‘₯,π‘₯,πœ™π‘›ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑔π‘₯,π‘₯,πœ™π‘›βˆ’1𝑑0ξ€Έξ€Έβˆ—π‘€ξ€·π‘”π‘¦,𝑦,πœ™π‘›βˆ’1𝑑0ξ€Έξ€Έβ‰₯𝑀𝑔π‘₯,π‘₯,𝑑0ξ€Έ2π‘›βˆ’1βˆ—π‘€ξ€·π‘”π‘¦,𝑦,𝑑0ξ€Έ2π‘›βˆ’1,𝑀𝑔𝑦,𝑦,πœ™π‘›ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔𝑦,𝑦,𝑑0ξ€Έξ€»2π‘›βˆ’1βˆ—ξ€Ίπ‘€ξ€·π‘”π‘₯,π‘₯,𝑑0ξ€Έξ€»2π‘›βˆ’1.(3.9) Then, we have 𝑀(𝑔π‘₯,π‘₯,𝑑)β‰₯π‘€βŽ›βŽœβŽπ‘”π‘₯,π‘₯,βˆžξ“π‘˜=𝑛0πœ™π‘˜ξ€·π‘‘0ξ€ΈβŽžβŽŸβŽ β‰₯𝑀𝑔π‘₯,π‘₯,πœ™π‘›0𝑑0ξ€Έβ‰₯𝑀(𝑔π‘₯,π‘₯,𝑑0)ξ€»2𝑛0βˆ’1βˆ—ξ€Ίπ‘€(𝑔𝑦,𝑦,𝑑0)ξ€»2𝑛0βˆ’1β‰₯(1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ2𝑛0β‰₯(1βˆ’πœ–).(3.10) So, for any πœ–>0, we have 𝑀(𝑔π‘₯,π‘₯,𝑑)β‰₯(1βˆ’πœ–) for all 𝑑>0.
This implies 𝑔(π‘₯)=π‘₯. Similarly, 𝑔(𝑦)=𝑦.
Step  3. Next we shall show that π‘₯=𝑦. Since βˆ— is a 𝑑-norm of 𝐻-type, for any πœ–>0 there exists 𝛿>0 such that (1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘β‰₯(1βˆ’πœ–),(3.11) for all π‘βˆˆβ„•.
Since limπ‘‘β†’βˆžπ‘€(π‘₯,𝑦,𝑑)=1 for all π‘₯,π‘¦βˆˆπ‘‹, there exists 𝑑0>0 such that 𝑀(π‘₯,𝑦,𝑑0)β‰₯(1βˆ’π›Ώ).
Also since πœ™βˆˆΞ¦, using condition (πœ™-3), we have βˆ‘βˆžπ‘›=1πœ™π‘›(𝑑0)<∞. Then for any 𝑑>0, there exists 𝑛0βˆˆβ„• such that 𝑑>βˆžξ“π‘˜=𝑛0πœ™π‘˜ξ€·π‘‘0ξ€Έ.(3.12) Using condition (2), we have 𝑀π‘₯,𝑦,πœ™ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑓(𝑝,π‘ž),𝑓(π‘ž,𝑝),πœ™ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔𝑝,π‘”π‘ž,𝑑0ξ€Έβˆ—π‘€ξ€·π‘”π‘ž,𝑔𝑝,𝑑0ξ€Έ=𝑀π‘₯,𝑦,𝑑0ξ€Έβˆ—π‘€ξ€·π‘¦,π‘₯,𝑑0ξ€Έ.(3.13) Continuing in the same way, we can get for all 𝑛0βˆˆβ„•, 𝑀π‘₯,𝑦,πœ™π‘›ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀(π‘₯,𝑦,𝑑0)ξ€»2𝑛0βˆ’1βˆ—ξ€Ίπ‘€(𝑦,π‘₯,𝑑0)ξ€»2𝑛0βˆ’1.(3.14) Then we have 𝑀(π‘₯,𝑦,𝑑)β‰₯π‘€βŽ›βŽœβŽπ‘₯,𝑦,βˆžξ“π‘˜=𝑛0πœ™π‘˜ξ€·π‘‘0ξ€ΈβŽžβŽŸβŽ β‰₯𝑀π‘₯,𝑦,πœ™π‘›0𝑑0ξ€Έβ‰₯𝑀(π‘₯,𝑦,𝑑0)ξ€»2𝑛0βˆ’1βˆ—ξ€Ίπ‘€(𝑦,π‘₯,𝑑0)ξ€»2𝑛0βˆ’1β‰₯(1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ2𝑛0β‰₯(1βˆ’πœ–),(3.15) which implies that π‘₯=𝑦. Thus, we have proved that 𝑓 and 𝑔 have a common fixed point π‘₯βˆˆπ‘‹.
Step  4. We now prove the uniqueness of π‘₯. Let 𝑧 be any point in 𝑋 such that 𝑧≠π‘₯ with 𝑔(𝑧)=𝑧=𝑓(𝑧,𝑧). Since βˆ— is a 𝑑-norm of 𝐻-type, for any πœ–>0, there exists 𝛿>0 such that (1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘β‰₯(1βˆ’πœ–),(3.16) for all π‘βˆˆβ„•. Since limπ‘‘β†’βˆžπ‘€(π‘₯,𝑦,𝑑)=1 for all π‘₯,π‘¦βˆˆπ‘‹, there exists 𝑑0>0 such that 𝑀(π‘₯,𝑧,𝑑0)β‰₯(1βˆ’π›Ώ). Also since πœ™βˆˆΞ¦ and using condition (πœ™-3), we have βˆ‘βˆžπ‘›=1πœ™π‘›(𝑑0)<∞. Then for any 𝑑>0, there exists 𝑛0βˆˆβ„• such that 𝑑>βˆžξ“π‘˜=𝑛0πœ™π‘˜ξ€·π‘‘0ξ€Έ.(3.17) Using condition (2), we have 𝑀π‘₯,𝑧,πœ™ξ€·π‘‘0ξ€Έξ€Έ=𝑀𝑓(π‘₯,π‘₯),𝑓(𝑧,𝑧),πœ™ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀𝑔(π‘₯),𝑔(𝑧),𝑑0ξ€Έβˆ—π‘€ξ€·π‘”(π‘₯),𝑔(𝑧),𝑑0ξ€Έβ‰₯𝑀π‘₯,𝑧,𝑑0ξ€Έβˆ—π‘€ξ€·π‘₯,𝑧,𝑑0ξ€Έ=𝑀π‘₯,𝑧,𝑑0ξ€Έξ€»2.(3.18) Continuing in the same way, we can get for all π‘›βˆˆβ„•, 𝑀π‘₯,𝑧,πœ™π‘›ξ€·π‘‘0ξ€Έξ€Έβ‰₯𝑀π‘₯,𝑧,𝑑0ξ€Έξ€»2𝑛0βˆ’12.(3.19) Then we have 𝑀(π‘₯,𝑧,𝑑)β‰₯π‘€βŽ›βŽœβŽπ‘₯,𝑧,βˆžξ“π‘˜=𝑛0πœ™π‘˜ξ€·π‘‘0ξ€ΈβŽžβŽŸβŽ β‰₯𝑀π‘₯,𝑧,πœ™π‘›0𝑑0ξ€Έξ€Έβ‰₯𝑀(π‘₯,𝑧,𝑑0)ξ€»2𝑛0βˆ’12=𝑀(π‘₯,𝑧,𝑑0)ξ€»2𝑛0β‰₯(1βˆ’π›Ώ)βˆ—β‹―βˆ—(1βˆ’π›Ώ)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ2𝑛0β‰₯(1βˆ’πœ–),(3.20) which implies that π‘₯=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,βˆ—) be a fuzzy metric space where βˆ— is a continuous t-norm of H-type. Let π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ be two mappings and there exists πœ™βˆˆΞ¦ 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 {π‘₯𝑛} and {𝑦𝑛} in 𝑋 such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘₯𝑛=π‘₯,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘”ξ€·π‘¦π‘›ξ€Έ=𝑦,(3.21) for some π‘₯,π‘¦βˆˆπ‘‹.
It follows from 𝑔(𝑋) being a closed subspace of 𝑋 that π‘₯=𝑔(𝑝), 𝑦=𝑔(π‘ž) for some 𝑝,π‘žβˆˆπ‘‹ and then 𝑓 and 𝑔 satisfy the (𝐢𝐿𝑅𝑔) 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,βˆ—) be a fuzzy metric space where βˆ— is a continuous 𝑑-norm of 𝐻-type. Let π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ be two mappings and there exists πœ™βˆˆΞ¦ satisfying (2), (3), and (5).
Suppose that 𝑓(𝑋×𝑋)βŠ†π‘”(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=𝐼𝑋 in Theorem 3.2, the Corollary 3.6 follows immediately the following.

Corollary 3.6. Let (𝑋,𝑀,βˆ—) be a fuzzy metric space where βˆ— is a continuous 𝑑-norm of 𝐻-type. Let π‘“βˆΆπ‘‹Γ—π‘‹β†’π‘‹ and π‘”βˆΆπ‘‹β†’π‘‹ be two mappings and there exists πœ™βˆˆΞ¦ satisfying the following conditions, for all π‘₯,𝑦,𝑒,π‘£βˆˆπ‘‹ and 𝑑>0:(6)𝑀(𝑓(π‘₯,𝑦),𝑓(𝑒,𝑣),πœ™(𝑑))β‰₯𝑀(π‘₯,𝑒,𝑑)βˆ—π‘€(𝑦,𝑣,𝑑), (7) there exists sequences {π‘₯𝑛} and {𝑦𝑛} in 𝑋 such that limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛,𝑦𝑛=limπ‘›β†’βˆžπ‘₯𝑛=π‘₯,limπ‘›β†’βˆžπ‘“ξ€·π‘¦π‘›,π‘₯𝑛=limπ‘›β†’βˆžπ‘¦π‘›=𝑦,(3.22) for some π‘₯,π‘¦βˆˆπ‘‹.

Then, there exists a unique π‘§βˆˆπ‘‹ such that 𝑧=𝑓(𝑧,𝑧).

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Copyright © 2012 Manish Jain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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