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Advances in Fuzzy Systems
Volume 2011 (2011), Article ID 986748, 6 pages
http://dx.doi.org/10.1155/2011/986748
Research Article

Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces

1Department of Mathematics, Acharya Nagarjuna University, Dr. M.R. Appa Row Campus, Nuzvid 521 201, India
2Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Turkey
3Department of Mathematics, CH. S.D. St. Theresa’s Junior College for Women, Eluru 534 001, India

Received 9 August 2011; Accepted 2 November 2011

Academic Editor: E. E. Kerre

Copyright © 2011 K. P. R. Rao 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.

Abstract

We prove two unique common coupled fixed-point theorems for self maps in symmetric G-fuzzy metric spaces.

1. Introduction and Preliminaries

Mustafa and Sims [13] and Naidu et al. [4] demonstrated that most of the claims concerning the fundamental topological structure of 𝐷-metric introduced by Dhage [58] and hence all theorems are incorrect. Alternatively, Mustafa and Sims [1, 2] introduced a 𝐺-metric space and obtained some fixed-point theorems in it. Some interesting references in 𝐺-metric spaces are [3, 915]. In this paper, we prove two unique common coupled fixed-point theorems for Jungck type and for three mappings in symmetric 𝐺-fuzzy metric spaces.

Before giving our main results, we recall some of the basic concepts and results in 𝐺-metric spaces and 𝐺-fuzzy metric spaces.

Definition 1 (see [2]). Let 𝑋 be a nonempty set and let 𝐺𝑋×𝑋×𝑋[0,) be a function satisfying the following properties:(G1)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧,(G2)0<𝐺(𝑥,𝑥,𝑦) for all 𝑥,𝑦𝑋 with 𝑥𝑦,(G3)𝐺(𝑥,𝑥,𝑦)𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧𝑋 with 𝑦𝑧,(G4)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=, symmetry in all three variables,(G5)𝐺(𝑥,𝑦,𝑧)𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧) for all 𝑥,𝑦,𝑧,𝑎𝑋.
Then, the function 𝐺 is called a generalized metric or a 𝐺-metric on 𝑋 and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Definition 2 (see [2]). The 𝐺-metric space (𝑋,𝐺) is called symmetric if 𝐺(𝑥,𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦) for all 𝑥,𝑦𝑋.

Definition 3 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space and let {𝑥𝑛} be a sequence in 𝑋. A point 𝑥𝑋 is said to be limit of {𝑥𝑛} if and only if lim𝑛,𝑚𝐺(𝑥,𝑥𝑛,𝑥𝑚)=0. In this case, the sequence {𝑥𝑛} is said to be 𝐺-convergent to 𝑥.

Definition 4 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space and let {𝑥𝑛} be a sequence in 𝑋. {𝑥𝑛} is called 𝐺-Cauchy if and only if lim𝑙,𝑛,𝑚𝐺(𝑥𝑙,𝑥𝑛,𝑥𝑚)=0. (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in (𝑋,𝐺).

Proposition 5 (see [2]). In a 𝐺-metric space (𝑋,𝐺), the following are equivalent.(i)The sequence {𝑥𝑛} is 𝐺-Cauchy.(ii)For every 𝜖>0,there exists𝑁𝐍 such that 𝐺(𝑥𝑛,𝑥𝑚,𝑥𝑚)<𝜖, for all 𝑛,𝑚𝑁.

Proposition 6 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

Proposition 7 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, for any 𝑥,𝑦,𝑧,𝑎𝑋, it follows that(i)if 𝐺(𝑥,𝑦,𝑧)=0, then 𝑥=𝑦=𝑧,(ii)𝐺(𝑥,𝑦,𝑧)𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧),(iii)𝐺(𝑥,𝑦,𝑦)2𝐺(𝑥,𝑥,𝑦),(iv)𝐺(𝑥,𝑦,𝑧)𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧),(v)𝐺(𝑥,𝑦,𝑧)(2/3)[𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎)].

Proposition 8 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, for a sequence {𝑥𝑛}𝑋 and a point 𝑥𝑋, the following are equivalent:(i){𝑥𝑛} is 𝐺-convergent to 𝑥,(ii)𝐺(𝑥𝑛,𝑥𝑛,𝑥)0 as 𝑛,(iii)𝐺(𝑥𝑛,𝑥,𝑥)0 as 𝑛,(iv)𝐺(𝑥𝑚,𝑥𝑛,𝑥)0 as 𝑚,𝑛.

Recently, Sun and Yang [16] introduced the concept of 𝐺-fuzzy metric spaces and proved two common fixed-point theorems for four mappings.

Definition 9 (see [16]). A 3-tuple (𝑋,𝐺,) is called a 𝐺-fuzzy metric space if 𝑋 is an arbitrary nonempty set, is a continuous 𝑡-norm, and 𝐺 is a fuzzy set on 𝑋3×(0,) satisfying the following conditions for each 𝑡,𝑠>0:(i)𝐺(𝑥,𝑥,𝑦,𝑡)>0 for all 𝑥,𝑦𝑋 with 𝑥𝑦,(ii)𝐺(𝑥,𝑥,𝑦,𝑡)𝐺(𝑥,𝑦,𝑧,𝑡) for all 𝑥,𝑦,𝑧𝑋 with 𝑦𝑧,(iii)𝐺(𝑥,𝑦,𝑧,𝑡)=1 if and only if 𝑥=𝑦=𝑧,(iv)𝐺(𝑥,𝑦,𝑧,𝑡)=𝐺(𝑝(𝑥,𝑦,𝑧),𝑡), where 𝑝 is a permutation function,(v)𝐺(𝑥,𝑦,𝑧,𝑡+𝑠)𝐺(𝑎,𝑦,𝑧,𝑡)𝐺(𝑥,𝑎,𝑎,𝑠) for all 𝑥,𝑦,𝑧,𝑎𝑋,(vi)𝐺(𝑥,𝑦,𝑧,)(0,)[0,1] is continuous.

Definition 10 (see [16]). A 𝐺-fuzzy metric space (𝑋,𝐺,) is said to be symmetric if 𝐺(𝑥,𝑥,𝑦,𝑡)=𝐺(𝑥,𝑦,𝑦,𝑡) for all 𝑥,𝑦𝑋 and for each 𝑡>0.

Example 11. Let 𝑋 be a nonempty set and let 𝐺 be a 𝐺-metric on 𝑋. Denote 𝑎𝑏=𝑎𝑏 for all 𝑎,𝑏[0,1]. For each 𝑡>0, 𝐺(𝑥,𝑦,𝑧,𝑡)=𝑡/(𝑡+𝐺(𝑥,𝑦,𝑧)) is a 𝐺-fuzzy metric on 𝑋.

Let (𝑋,𝐺,) be a 𝐺-fuzzy metric space. For 𝑡>0,0<𝑟<1, and 𝑥𝑋, the set 𝐵𝐺(𝑥,𝑟,𝑡)={𝑦𝑋𝐺(𝑥,𝑦,𝑦,𝑡)>1𝑟} is called an open ball with center 𝑥 and radius 𝑟.

A subset 𝐴 of 𝑋 is called an open set if for each 𝑥𝑋,  there exist 𝑡>0 and 0<𝑟<1 such that 𝐵𝐺(𝑥,𝑟,𝑡)𝐴.

A sequence {𝑥𝑛} in 𝐺-fuzzy metric space 𝑋 is said to be 𝐺-convergent to 𝑥𝑋 if 𝐺(𝑥𝑛,𝑥𝑛,𝑥,𝑡)1 as 𝑛 for each 𝑡>0. It is called a 𝐺-Cauchy sequence if 𝐺(𝑥𝑛,𝑥𝑛,𝑥𝑚,𝑡)1 as 𝑛,𝑚 for each 𝑡>0. 𝑋 is called 𝐺-complete if every 𝐺-Cauchy sequence in 𝑋 is 𝐺-convergent in 𝑋.

Lemma 12 (see [16]). Let (𝑋,𝐺,) be a 𝐺-fuzzy metric space. Then, 𝐺(𝑥,𝑦,𝑧,𝑡) is nondecreasing with respect to 𝑡 for all 𝑥,𝑦,𝑧𝑋.

Lemma 13 (see [16]). Let (𝑋,𝐺,) be a 𝐺-fuzzy metric space. Then, 𝐺 is a continuous function on 𝑋3×(0,).

Now onwards, we assume the following condition:lim𝑡𝐺(𝑥,𝑦,𝑧,𝑡)=1𝑥,𝑦,𝑧𝑋.(P) Using (P), one can prove the following lemma.

Lemma 14. Let (𝑋,𝐺,) be a 𝐺-fuzzy metric space. If there exists 𝑘(0,1) such that min{𝐺(𝑥,𝑦,𝑧,𝑘𝑡),𝐺(𝑢,𝑣,𝑤,𝑘𝑡)}min{𝐺(𝑥,𝑦,𝑧,𝑡),𝐺(𝑢,𝑣,𝑤,𝑡)}(1) for all 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤𝑋 and 𝑡>0, then 𝑥=𝑦=𝑧 and 𝑢=𝑣=𝑤.

Definition 15 (see [17]). Let 𝑋 be a nonempty set. An element (𝑥,𝑦)𝑋×𝑋 is called a coupled fixed point of the mapping 𝐹𝑋×𝑋𝑋 if 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).

Definition 16 (see [18]). Let 𝑋 be a nonempty set. An element (𝑥,𝑦)𝑋×𝑋 is called(i)a coupled coincidence point of 𝐹𝑋×𝑋𝑋 and 𝑔𝑋𝑋 if 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥),(ii)a common coupled fixed point of 𝐹𝑋×𝑋𝑋 and 𝑔𝑋𝑋 if 𝑥=𝑔𝑥=𝐹(𝑥,𝑦) and 𝑦=𝑔𝑦=𝐹(𝑦,𝑥).

Definition 17 (see [18]). Let 𝑋 be a nonempty set. The mappings 𝐹𝑋×𝑋𝑋 and 𝑔𝑋𝑋 are called 𝑊-compatible if 𝑔(𝐹(𝑥,𝑦))=𝐹(𝑔𝑥,𝑔𝑦) and 𝑔(𝐹(𝑦,𝑥))=𝐹(𝑔𝑦,𝑔𝑥) whenever 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥) for some (𝑥,𝑦)𝑋×𝑋.

Now, we give our main results.

2. Main Results

Theorem 18. Let (𝑋,𝐺,) be a 𝐺-fuzzy metric space with 𝑎𝑏=min{𝑎,𝑏} for all 𝑎,𝑏[0,1] and 𝑆𝑋×𝑋𝑋 and let 𝑓𝑋𝑋 be mappings satisfying 𝐺(𝑆(𝑥,𝑦),𝑆(𝑢,𝑣),𝑆(𝑢,𝑣),𝑘𝑡)min{𝐺(𝑓𝑥,𝑓𝑢,𝑓𝑢,𝑡),𝐺(𝑓𝑦,𝑓𝑣,𝑓𝑣,𝑡)}(2)forall𝑥,𝑦,𝑢,𝑣𝑋, where 0𝑘<1, 𝑆(𝑋×𝑋)𝑓(𝑋)𝑎𝑛𝑑𝑓(𝑋)𝑖𝑠𝑎𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒𝑜𝑓𝑋,𝑡𝑒𝑝𝑎𝑖𝑟(𝑓,𝑆)𝑖𝑠𝑊-compatible.(3)
Then 𝑆 and 𝑓 have a unique common coupled fixed point of the form (𝛼,𝛼) in 𝑋×𝑋.

Proof. Let 𝑥0,𝑦0𝑋 and denote 𝑧𝑛=𝑆(𝑥𝑛,𝑦𝑛)=𝑓𝑥𝑛+1,𝑝𝑛=𝑆(𝑦𝑛,𝑥𝑛)=𝑓𝑦𝑛+1,𝑛=0,1,2,. Let 𝑑𝑛(𝑡)=𝐺(𝑧𝑛,𝑧𝑛+1,𝑧𝑛+1,𝑡), 𝑒𝑛(𝑡)=𝐺(𝑝𝑛,𝑝𝑛+1,𝑝𝑛+1,𝑡). From (2), we have 𝑑𝑛+1𝑧(𝑘𝑡)=𝐺𝑛+1,𝑧𝑛+2,𝑧𝑛+2𝑆𝑥,𝑘𝑡=𝐺𝑛+1,𝑦𝑛+1𝑥,𝑆𝑛+2,𝑦𝑛+2𝑥,𝑆𝑛+2,𝑦𝑛+2𝐺𝑧,𝑘𝑡min𝑛,𝑧𝑛+1,𝑧𝑛+1𝑝,𝑡,𝐺𝑛,𝑝𝑛+1,𝑝𝑛+1𝑑,𝑡min𝑛(𝑡),𝑒𝑛(.𝑡)(4) Also, 𝑒𝑛+1𝑝(𝑘𝑡)=𝐺𝑛+1,𝑝𝑛+2,𝑝𝑛+2𝑆𝑦,𝑘𝑡=𝐺𝑛+1,𝑥𝑛+1𝑦,𝑆𝑛+2,𝑥𝑛+2𝑦,𝑆𝑛+2,𝑥𝑛+2𝐺𝑝,𝑘𝑡min𝑛,𝑝𝑛+1,𝑝𝑛+1𝑧,𝑡,𝐺𝑛,𝑧𝑛+1,𝑧𝑛+1𝑒,𝑡min𝑛(𝑡),𝑑𝑛(.𝑡)(5) Thus, min{𝑑𝑛+1(𝑘𝑡),𝑒𝑛+1(𝑘𝑡)}min{𝑑𝑛(𝑡),𝑒𝑛(𝑡)}. Hence, 𝑑min𝑛(𝑡),𝑒𝑛𝑑(𝑡)min𝑛1𝑡𝑘,𝑒𝑛1𝑡𝑘𝑑min𝑛2𝑡𝑘2,𝑒𝑛2𝑡𝑘2𝑑min0𝑡𝑘𝑛,𝑒0𝑡𝑘𝑛𝐺𝑧=min0,𝑧1,𝑧1,𝑡𝑘𝑛𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑘𝑛.(6) For any positive integer 𝑛 and fixed positive integer 𝑝, we have 𝐺𝑧𝑛,𝑧𝑛+𝑝,𝑧𝑛+𝑝𝑧,𝑡𝐺𝑛+𝑝1,𝑧𝑛+𝑝,𝑧𝑛+𝑝,𝑡𝑝𝑧𝐺𝑛+𝑝2,𝑧𝑛+𝑝1,𝑧𝑛+𝑝1,𝑡𝑝𝑧𝐺𝑛,𝑧𝑛+1,𝑧𝑛+1,𝑡𝑝𝐺𝑧min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛+𝑝1𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛+𝑝1𝐺𝑧min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛+𝑝2𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛+𝑝2𝐺𝑧min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛.(7) Letting 𝑛 and using (P), we get lim𝑛𝐺𝑧𝑛,𝑧𝑛+𝑝,𝑧𝑛+𝑝,𝑡111=1.(8) Hence, lim𝑛𝐺(𝑧𝑛,𝑧𝑛+𝑝,𝑧𝑛+𝑝,𝑡)=1. Thus, {𝑧𝑛} is 𝐺-Cauchy in 𝑋. Similarly, we can show that {𝑝𝑛} is 𝐺-Cauchy in 𝑋. Since 𝑓(𝑋) is 𝐺-complete, {𝑧𝑛} and {𝑝𝑛} converge to some 𝛼 and 𝛽 in 𝑓(𝑋), respectively. Hence, there exist 𝑥 and 𝑦 in 𝑋 such that 𝛼=𝑓𝑥,𝛽=𝑓𝑦: 𝐺𝑧𝑛𝑆𝑥,𝑆(𝑥,𝑦),𝑆(𝑥,𝑦),𝑘𝑡=𝐺𝑛,𝑦𝑛𝐺𝑧,𝑆(𝑥,𝑦),𝑆(𝑥,𝑦),𝑘𝑡min𝑛1𝑝,𝑓𝑥,𝑓𝑥,𝑡,𝐺𝑛1.,𝑓𝑦,𝑓𝑦,𝑡(9) Letting 𝑛, we get G(𝑓𝑥,𝑆(𝑥,𝑦),𝑆(𝑥,𝑦),𝑘𝑡)min{1,1}=1.(10) Hence, 𝑆(𝑥,𝑦)=𝑓𝑥. Similarly, it can be shown that 𝑆(𝑦,𝑥)=𝑓𝑦. Since (𝑓,𝑆) is 𝑊-compatible, we have 𝐺𝑧𝑓𝛼=𝑓𝑓𝑥=𝑓(𝑆(𝑥,𝑦))=𝑆(𝑓𝑥,𝑓𝑦)=𝑆(𝛼,𝛽),𝑓𝛽=𝑓𝑓𝑦=𝑓(𝑆(𝑦,𝑥))=𝑆(𝑓𝑦,𝑓𝑥)=𝑆(𝛽,𝛼).𝑛𝑆𝑥,𝑓𝛼,𝑓𝛼,𝑘𝑡=𝐺𝑛,𝑦𝑛𝐺𝑧,𝑆(𝛼,𝛽),𝑆(𝛼,𝛽),𝑘𝑡min𝑛1𝑝,𝑓𝛼,𝑓𝛼,𝑡,𝐺𝑛1.,𝑓𝛽,𝑓𝛽,𝑡(11) Letting 𝑛, we get 𝐺(𝛼,𝑓𝛼,𝑓𝛼,𝑘𝑡)min{𝐺(𝛼,𝑓𝛼,𝑓𝛼,𝑡),𝐺(𝛽,𝑓𝛽,𝑓𝛽,𝑡)}.(12) Similarly, we can show that 𝐺(𝛽,𝑓𝛽,𝑓𝛽,𝑘𝑡)min{𝐺(𝛼,𝑓𝛼,𝑓𝛼,𝑡),𝐺(𝛽,𝑓𝛽,𝑓𝛽,𝑡)}.(13) Thus, min{𝐺(𝛼,𝑓𝛼,𝑓𝛼,𝑘𝑡),𝐺(𝛽,𝑓𝛽,𝑓𝛽,𝑘𝑡)}min{𝐺(𝛼,𝑓𝛼,𝑓𝛼,𝑡),𝐺(𝛽,𝑓𝛽,𝑓𝛽,𝑡)}.(14) From Lemma 14, we have 𝑓𝛼=𝛼 and 𝑓𝛽=𝛽. Thus, 𝛼=𝑓𝛼=𝑆(𝛼,𝛽) and 𝛽=𝑓𝛽=𝑆(𝛽,𝛼). Hence, (𝛼,𝛽) is a common coupled fixed point of 𝑆 and 𝑓.
Suppose (𝛼1,𝛽1) is another common coupled fixed point of 𝑆 and 𝑓:𝐺𝛼,𝛼1,𝛼1𝑆𝛼,𝑘𝑡=𝐺(𝛼,𝛽),𝑆1,𝛽1𝛼,𝑆1,𝛽1𝐺,𝑘𝑡min𝛼,𝛼1,𝛼1,𝑡,𝐺𝛽,𝛽1,𝛽1.,𝑡(15) Similarly, 𝐺𝛽,𝛽1,𝛽1𝑆𝛽,𝑘𝑡=𝐺(𝛽,𝛼),𝑆1,𝛼1𝛽,𝑆1,𝛼1𝐺,𝑘𝑡min𝛼,𝛼1,𝛼1,𝑡,𝐺𝛽,𝛽1,𝛽1.,𝑡(16) Thus, 𝐺min𝛼,𝛼1,𝛼1,𝑘𝑡,𝐺𝛽,𝛽1,𝛽1𝐺,𝑘𝑡min𝛼,𝛼1,𝛼1,𝑡,𝐺𝛽,𝛽1,𝛽1.,𝑡(17) From Lemma 14, 𝛼1=𝛼 and 𝛽1=𝛽. Thus, (𝛼,𝛽) is the unique common coupled fixed point of 𝑆 and 𝑓. Now, we will show that 𝛼=𝛽: 𝐺(𝛼,𝛼,𝛽,𝑘𝑡)=𝐺(𝑆(𝛼,𝛽),𝑆(𝛼,𝛽),𝑆(𝛽,𝛼),𝑘𝑡)min{𝐺(𝛼,𝛼,𝛽,𝑡),𝐺(𝛽,𝛽,𝛼,𝑡)},𝐺(𝛼,𝛽,𝛽,𝑘𝑡)=𝐺(𝑆(𝛼,𝛽),𝑆(𝛽,𝛼),𝑆(𝛽,𝛼),𝑘𝑡)min{𝐺(𝛼,𝛽,𝛽,𝑡),𝐺(𝛽,𝛼,𝛼,𝑡)}.(18) Thus, min{𝐺(𝛼,𝛼,𝛽,𝑘𝑡),𝐺(𝛼,𝛽,𝛽,𝑘𝑡)}min{𝐺(𝛼,𝛼,𝛽,𝑡),𝐺(𝛼,𝛽,𝛽,𝑡)}.(19) From Lemma 14, we have 𝛼=𝛽. Thus, 𝛼 is a common fixed point of 𝑆 and 𝑓, that is, 𝛼=𝑓𝛼=𝑆(𝛼,𝛼). Suppose 𝛼1 is another common fixed point of 𝑆 and 𝑓: 𝐺𝛼1𝑆𝛼,𝛼,𝛼,𝑡=𝐺1,𝛼1𝐺𝛼,𝑆(𝛼,𝛼),𝑆(𝛼,𝛼),𝑡min1𝑡,𝛼,𝛼,𝑘𝛼,𝐺1𝑡,𝛼,𝛼,𝑘𝛼𝐺1𝑡,𝛼,𝛼,𝑘2𝛼𝐺1𝑡,𝛼,𝛼,𝑘𝑛1as𝑛.(20) Hence, 𝛼1=𝛼. Thus, 𝑆 and 𝑓 have a unique common coupled fixed point of the form (𝛼,𝛼).

Finally, we prove a common coupled fixed-point theorem for three mappings in symmetric 𝐺-fuzzy metric spaces.

Theorem 19. Let (𝑋,𝐺,) be a symmetric 𝐺-complete fuzzy metric space with 𝑎𝑏=min{𝑎,𝑏} for all 𝑎,𝑏[0,1] and let 𝑆,𝑇,𝑅𝑋×𝑋𝑋 be mappings satisfying 𝐺(𝑆(𝑥,𝑦),𝑇(𝑢,𝑣),𝑅(𝑝,𝑞),𝑘𝑡)min{𝐺(𝑥,𝑢,𝑝,𝑡),𝐺(𝑦,𝑣,𝑞,𝑡),𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡),𝐺(𝑢,𝑢,𝑇(𝑢,𝑣),𝑡),𝐺(𝑝,𝑝,𝑅(𝑝,𝑞),𝑡)}(21)forall𝑥,𝑦,𝑢,𝑣,𝑝,𝑞𝑋, where 0𝑘<1. Then, there exists (𝑥,𝑦)𝑋×𝑋 such that 𝑥=𝑆(𝑥,𝑦)=𝑇(𝑥,𝑦)=𝑅(𝑥,𝑦),(22)𝑦=𝑆(𝑦,𝑥)=𝑇(𝑦,𝑥)=𝑅(𝑦,𝑥).(23) Or 𝑆,𝑇,𝑎𝑛𝑑𝑅𝑎𝑣𝑒𝑎𝑢𝑛𝑖𝑞𝑢𝑒𝑐𝑜𝑚𝑚𝑜𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑𝑓𝑖𝑥𝑒𝑑𝑝𝑜𝑖𝑛𝑡𝑜𝑓𝑡𝑒𝑓𝑜𝑟𝑚(𝑥,𝑥)𝑖𝑛𝑋×𝑋.(24)

Proof. Let 𝑥0,𝑦0𝑋. Define the sequences {𝑥𝑛} and {𝑦𝑛} in 𝑋 as follows: 𝑥3𝑛+1=𝑆(𝑥3𝑛,𝑦3𝑛),𝑦3𝑛+1=𝑆(𝑦3𝑛,𝑥3𝑛); 𝑥3𝑛+2=𝑇(𝑥3𝑛+1,𝑦3𝑛+1), 𝑦3𝑛+2=𝑇(𝑦3𝑛+1,𝑥3𝑛+1); 𝑥3𝑛+3=𝑅(𝑥3𝑛+2,𝑦3𝑛+2), 𝑦3𝑛+3=𝑅(𝑦3𝑛+2,𝑥3𝑛+2), 𝑛=0,1,2,. Suppose 𝑥3𝑛+1=𝑥3𝑛 for some 𝑛. Then, 𝑆(𝑥,𝑦)=𝑥, where 𝑥=𝑥3𝑛,𝑦=𝑦3𝑛. Suppose 𝑇(𝑥,𝑦)𝑅(𝑥,𝑦). Then, 𝐺(𝑥,𝑇(𝑥,𝑦),𝑅(𝑥,𝑦),𝑘𝑡)=𝐺(𝑆(𝑥,𝑦),𝑇(𝑥,𝑦),𝑅(𝑥,𝑦),𝑘𝑡)min{1,1,1,𝐺(𝑥,𝑥,𝑇(𝑥,𝑦),𝑡),𝐺(𝑥,𝑥,𝑅(𝑥,𝑦),𝑡)}𝐺(𝑥,𝑇(𝑥,𝑦),𝑅(𝑥,𝑦),𝑡).(25) It is a contradiction. Hence, 𝑇(𝑥,𝑦)=𝑅(𝑥,𝑦). From (25) and since 𝑋 is symmetric, 𝐺(𝑥,𝑇(𝑥,𝑦),𝑇(𝑥,𝑦),𝑘𝑡)𝐺(𝑥,𝑥,𝑇(𝑥,𝑦),𝑡)=𝐺(𝑥,𝑇(𝑥,𝑦),𝑇(𝑥,𝑦),𝑡).(26) From Lemma 14, we have 𝑇(𝑥,𝑦)=𝑥. Thus, 𝑆(𝑥,𝑦)=𝑇(𝑥,𝑦)=𝑅(𝑥,𝑦)=𝑥. Similarly, if 𝑥3𝑛+1=𝑥3𝑛+2 or 𝑥3𝑛+2=𝑥3𝑛+3, then also we can show that 𝑆(𝑥,𝑦)=𝑇(𝑥,𝑦)=𝑅(𝑥,𝑦)=𝑥 for some 𝑥, 𝑦 in 𝑋. Similarly, it can be shown that if 𝑦3𝑛=𝑦3𝑛+1 or 𝑦3𝑛+1=𝑦3𝑛+2 or 𝑦3𝑛+2=𝑦3𝑛+3 then there exists (𝑥,𝑦)𝑋×𝑋 such that 𝑆(𝑦,𝑥)=𝑇(𝑦,𝑥)=𝑅(𝑦,𝑥)=𝑦.(27) Now, assume that 𝑥𝑛𝑥𝑛+1 and 𝑦𝑛𝑦𝑛+1 for all 𝑛. Write 𝑑𝑛(𝑡)=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+2,𝑡) and 𝑒𝑛(𝑡)=𝐺(𝑦𝑛,𝑦𝑛+1,𝑦𝑛+2,𝑡): 𝑑3𝑛𝑥(𝑘𝑡)=𝐺3𝑛,𝑥3𝑛+1,𝑥3𝑛+2𝑆𝑥,𝑘𝑡=𝐺3𝑛,𝑦3𝑛𝑥,𝑇3𝑛+1,𝑦3𝑛+1𝑥,𝑅3𝑛1,𝑦3𝑛1𝑑,𝑘𝑡min3𝑛1(𝑡),𝑒3𝑛1𝑥(𝑡),𝐺3𝑛,𝑥3𝑛,𝑥3𝑛+1,𝐺𝑥,𝑡3𝑛+1,𝑥3𝑛+1,𝑥3𝑛+2𝑥,𝑡,𝐺3𝑛1,𝑥3𝑛1,𝑥3𝑛𝑑,𝑡min3𝑛1(𝑡),𝑒3𝑛1(𝑡),𝑑3𝑛(𝑡),𝑑3𝑛(𝑡),𝑑3𝑛1.(𝑡)(28) Thus, 𝑑3𝑛(𝑘𝑡)min{𝑑3𝑛1(𝑡),𝑒3𝑛1(𝑡)}. Similarly, we have 𝑒3𝑛(𝑘𝑡)min𝑑3𝑛1(𝑡),𝑒3𝑛1(𝑡).
Thus,𝑑min3𝑛(𝑘𝑡),𝑒3𝑛𝑑(𝑘𝑡)min3𝑛1(𝑡),𝑒3𝑛1(𝑡).(29) Similarly, we can show that 𝑑min3𝑛+1(𝑘𝑡),𝑒3𝑛+1𝑑(𝑘𝑡)min3𝑛(𝑡),𝑒3𝑛,𝑑(𝑡)min3𝑛+2(𝑘𝑡),𝑒3𝑛+2(𝑑𝑘𝑡)min3𝑛+1(𝑡),𝑒3𝑛+1(.𝑡)(30) Thus, 𝑑min𝑛+1(𝑘𝑡),𝑒𝑛+1𝑑(𝑘𝑡)min𝑛(𝑡),𝑒𝑛(𝑡).(31) Hence 𝑑min𝑛(𝑡),𝑒𝑛𝑑(𝑡)min𝑛𝑡𝑘,𝑒𝑛𝑡𝑘𝑑min𝑛𝑡𝑘2,𝑒𝑛𝑡𝑘2𝑑min0𝑡𝑘𝑛,𝑒0𝑡𝑘𝑛𝐺𝑥=min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,𝐺0,𝑦1,𝑦2,𝑡𝑘𝑛.(32) Thus, 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+2𝐺𝑥,𝑡min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,𝐺0,𝑦1,𝑦2,𝑡𝑘𝑛.(33) From (𝐺3), we have 𝐺𝑥𝑛,𝑥𝑛,𝑥𝑛+1𝑥,𝑡𝐺𝑛,𝑥𝑛+1,𝑥𝑛+2𝐺𝑥,𝑡min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,𝐺0,𝑦1,𝑦2,𝑡𝑘𝑛.(34) As in Theorem 18, we can show that {𝑥𝑛} and {𝑦𝑛} are 𝐺-Cauchy sequences in 𝑋. Since 𝑋 is 𝐺-complete, there exist 𝑥,𝑦𝑋 such that 𝑥𝑛𝑥 and 𝑦𝑛𝑦𝐺𝑆(𝑥,𝑦),𝑥3𝑛+2,𝑥3𝑛+3𝑥,𝑘𝑡=𝐺𝑆(𝑥,𝑦),𝑇3𝑛+1,𝑦3𝑛+1𝑥,𝑅3𝑛+2,𝑦3𝑛+2𝐺,𝑘𝑡min𝑥,𝑥3𝑛+1,𝑥3𝑛+2,𝑡,𝐺𝑦,𝑦3𝑛+1,𝑦3𝑛+2,𝑥,𝑡𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡),𝐺3𝑛+1,𝑥3𝑛+1,𝑥3𝑛+2,𝐺𝑥,𝑡3𝑛+2,𝑥3𝑛+2,𝑥3𝑛+3.,𝑡(35) Letting 𝑛, 𝐺(𝑆(𝑥,𝑦),𝑥,𝑥,𝑘𝑡)min{1,1,𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡),1,1}=𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡).(36) From this, we have 𝑆(𝑥,𝑦)=𝑥. As in the first part of proof, we can show that 𝑆(𝑥,𝑦)=𝑇(𝑥,𝑦)=𝑅(𝑥,𝑦)=𝑥. Similarly, it can be shown that 𝑆(𝑦,𝑥)=𝑇(𝑦,𝑥)=𝑅(𝑦,𝑥)=𝑦. Thus, (𝑥,𝑦) is a common coupled fixed point of 𝑆, 𝑇, and 𝑅. Suppose (𝑥1,𝑦1) is another common coupled fixed point of 𝑆, 𝑇, and 𝑅. Consider 𝐺𝑥,𝑥,𝑥1𝑥,𝑘𝑡=𝐺𝑆(𝑥,𝑦),𝑇(𝑥,𝑦),𝑅1,𝑦1𝐺,𝑘𝑡min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑡,1,1,1=min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1.,𝑡(37) Also, 𝐺𝑦,𝑦,𝑦1𝑦,𝑘𝑡=𝐺𝑆(𝑦,𝑥),𝑇(𝑦,𝑥),𝑅1,𝑥1𝐺,𝑘𝑡min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑡,1,1,1=min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1.,𝑡(38) Thus, 𝐺min𝑥,𝑥,𝑥1,𝑘𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑘𝑡min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1.,𝑡(39) From Lemma 14, we have 𝑥1=𝑥 and 𝑦1=𝑦. Thus, (𝑥,𝑦) is the unique common coupled fixed point of 𝑆, 𝑇, and 𝑅. Now, we will show that 𝑥=𝑦. Consider 𝐺(𝑥,𝑥,𝑦,𝑘𝑡)=𝐺(𝑆(𝑥,𝑦),𝑇(𝑥,𝑦),𝑅(𝑦,𝑥),𝑘𝑡)min{𝐺(𝑥,𝑥,𝑦,𝑡)𝐺(𝑦,𝑦,𝑥,𝑡),1,1,1}=𝐺(𝑥,𝑥,𝑦,𝑡).(40) Hence, 𝑥=𝑦. Thus, 𝑆, 𝑇, and 𝑅 have a unique common coupled fixed point of the form (𝑥,𝑥).

Acknowledgment

The authors are thankful to the referee for his valuable suggestions.

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