Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 986748 | https://doi.org/10.1155/2011/986748

K. P. R. Rao, I. Altun, S. Hima Bindu, "Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces", Advances in Fuzzy Systems, vol. 2011, Article ID 986748, 6 pages, 2011. https://doi.org/10.1155/2011/986748

Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces

Academic Editor: E. E. Kerre
Received09 Aug 2011
Accepted02 Nov 2011
Published10 Dec 2011

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 [1–3] and Naidu et al. [4] demonstrated that most of the claims concerning the fundamental topological structure of 𝐷-metric introduced by Dhage [5–8] 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, 9–15]. 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 𝐺∶𝑋×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,𝑦∈𝑋 with 𝑥≠𝑦,(G3)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋 with 𝑦≠𝑧,(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,ğ‘Ž,ğ‘Ž)+𝐺(ğ‘Ž,𝑦,𝑧) for all 𝑥,𝑦,𝑧,ğ‘Žâˆˆğ‘‹.
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,𝑦∈𝑋.

Definition 3 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space and let {𝑥𝑛} be a sequence in 𝑋. A point 𝑥∈𝑋 is said to be limit of {𝑥𝑛} if and only if lim𝑛,ğ‘šâ†’âˆžğº(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,ğ‘šâ†’âˆžğº(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 exists𝑁∈𝐍 such that 𝐺(𝑥𝑛,𝑥𝑚,𝑥𝑚)<𝜖, for all 𝑛,𝑚≥𝑁.

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,ğ‘Žâˆˆğ‘‹, 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,ğ‘Ž,𝑧)+𝐺(ğ‘Ž,𝑦,𝑧),(v)𝐺(𝑥,𝑦,𝑧)≤(2/3)[𝐺(𝑥,ğ‘Ž,ğ‘Ž)+𝐺(𝑦,ğ‘Ž,ğ‘Ž)+𝐺(𝑧,ğ‘Ž,ğ‘Ž)].

Proposition 8 (see [2]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, for a sequence {𝑥𝑛}⊆𝑋 and a point 𝑥∈𝑋, the following are equivalent:(i){𝑥𝑛} is 𝐺-convergent to 𝑥,(ii)𝐺(𝑥𝑛,𝑥𝑛,𝑥)→0 as ğ‘›â†’âˆž,(iii)𝐺(𝑥𝑛,𝑥,𝑥)→0 as ğ‘›â†’âˆž,(iv)𝐺(𝑥𝑚,𝑥𝑛,𝑥)→0 as 𝑚,ğ‘›â†’âˆž.

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,𝑦∈𝑋 with 𝑥≠𝑦,(ii)𝐺(𝑥,𝑥,𝑦,𝑡)≥𝐺(𝑥,𝑦,𝑧,𝑡) for all 𝑥,𝑦,𝑧∈𝑋 with 𝑦≠𝑧,(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(ğ‘Ž,𝑦,𝑧,𝑡)∗𝐺(𝑥,ğ‘Ž,ğ‘Ž,𝑠) for all 𝑥,𝑦,𝑧,ğ‘Žâˆˆğ‘‹,(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,𝑦∈𝑋 and for each 𝑡>0.

Example 11. Let 𝑋 be a nonempty set and let 𝐺 be a 𝐺-metric on 𝑋. Denote ğ‘Žâˆ—ğ‘=ğ‘Žğ‘ for all ğ‘Ž,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 𝑥∈𝑋, the set 𝐵𝐺(𝑥,𝑟,𝑡)={𝑦∈𝑋∶𝐺(𝑥,𝑦,𝑦,𝑡)>1−𝑟} is called an open ball with center 𝑥 and radius 𝑟.

A subset 𝐴 of 𝑋 is called an open set if for each 𝑥∈𝑋,  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 𝑥∈𝑋 if 𝐺(𝑥𝑛,𝑥𝑛,𝑥,𝑡)→1 as ğ‘›â†’âˆž for each 𝑡>0. It is called a 𝐺-Cauchy sequence if 𝐺(𝑥𝑛,𝑥𝑛,𝑥𝑚,𝑡)→1 as 𝑛,ğ‘šâ†’âˆž 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,𝑧∈𝑋.

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:limğ‘¡â†’âˆžğº(𝑥,𝑦,𝑧,𝑡)=1∀𝑥,𝑦,𝑧∈𝑋.(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,𝑤∈𝑋 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 𝐹∶𝑋×𝑋→𝑋 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 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥),(ii)a common coupled fixed point of 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝑔𝑥=𝐹(𝑥,𝑦) and 𝑦=𝑔𝑦=𝐹(𝑦,𝑥).

Definition 17 (see [18]). Let 𝑋 be a nonempty set. The mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 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 ğ‘Žâˆ—ğ‘=min{ğ‘Ž,𝑏} for all ğ‘Ž,𝑏∈[0,1] and 𝑆∶𝑋×𝑋→𝑋 and let 𝑓∶𝑋→𝑋 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,𝑣∈𝑋, where 0≤𝑘<1, 𝑆(𝑋×𝑋)⊆𝑓(𝑋)ğ‘Žğ‘›ğ‘‘ğ‘“(𝑋)ğ‘–ğ‘ ğ‘Žğ‘ğ‘œğ‘šğ‘ğ‘™ğ‘’ğ‘¡ğ‘’ğ‘ ğ‘¢ğ‘ğ‘ ğ‘ğ‘Žğ‘ğ‘’ğ‘œğ‘“ğ‘‹,ğ‘¡â„Žğ‘’ğ‘ğ‘Žğ‘–ğ‘Ÿ(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,𝑦0∈𝑋 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+1𝑧(𝑘𝑡)=𝐺𝑛+1,𝑧𝑛+2,𝑧𝑛+2𝑆𝑥,𝑘𝑡=𝐺𝑛+1,𝑦𝑛+1𝑥,𝑆𝑛+2,𝑦𝑛+2𝑥,𝑆𝑛+2,𝑦𝑛+2𝐺𝑧,𝑘𝑡≥min𝑛,𝑧𝑛+1,𝑧𝑛+1𝑝,𝑡,𝐺𝑛,𝑝𝑛+1,𝑝𝑛+1𝑑,𝑡≥min𝑛(𝑡),𝑒𝑛(.𝑡)(4) Also, 𝑒𝑛+1𝑝(𝑘𝑡)=𝐺𝑛+1,𝑝𝑛+2,𝑝𝑛+2𝑆𝑦,𝑘𝑡=𝐺𝑛+1,𝑥𝑛+1𝑦,𝑆𝑛+2,𝑥𝑛+2𝑦,𝑆𝑛+2,𝑥𝑛+2𝐺𝑝,𝑘𝑡≥min𝑛,𝑝𝑛+1,𝑝𝑛+1𝑧,𝑡,𝐺𝑛,𝑧𝑛+1,𝑧𝑛+1𝑒,𝑡≥min𝑛(𝑡),𝑑𝑛(.𝑡)(5) Thus, min{𝑑𝑛+1(𝑘𝑡),𝑒𝑛+1(𝑘𝑡)}≥min{𝑑𝑛(𝑡),𝑒𝑛(𝑡)}. Hence, 𝑑min𝑛(𝑡),𝑒𝑛𝑑(𝑡)≥min𝑛−1𝑡𝑘,𝑒𝑛−1𝑡𝑘𝑑≥min𝑛−2𝑡𝑘2,𝑒𝑛−2𝑡𝑘2⋮𝑑≥min0𝑡𝑘𝑛,𝑒0𝑡𝑘𝑛𝐺𝑧=min0,𝑧1,𝑧1,𝑡𝑘𝑛𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑘𝑛.(6) For any positive integer 𝑛 and fixed positive integer 𝑝, we have 𝐺𝑧𝑛,𝑧𝑛+𝑝,𝑧𝑛+𝑝𝑧,𝑡≥𝐺𝑛+𝑝−1,𝑧𝑛+𝑝,𝑧𝑛+𝑝,𝑡𝑝𝑧∗𝐺𝑛+𝑝−2,𝑧𝑛+𝑝−1,𝑧𝑛+𝑝−1,𝑡𝑝𝑧∗⋯∗𝐺𝑛,𝑧𝑛+1,𝑧𝑛+1,𝑡𝑝𝐺𝑧≥min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛+𝑝−1𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛+𝑝−1𝐺𝑧∗min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛+𝑝−2𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛+𝑝−2𝐺𝑧∗⋯∗min0,𝑧1,𝑧1,𝑡𝑝𝑘𝑛𝑝,𝐺0,𝑝1,𝑝1,𝑡𝑝𝑘𝑛.(7) Letting ğ‘›â†’âˆž and using (P), we get limğ‘›â†’âˆžğºî€·ğ‘§ğ‘›,𝑧𝑛+𝑝,𝑧𝑛+𝑝,𝑡≥1∗1∗⋯∗1=1.(8) Hence, limğ‘›â†’âˆžğº(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: 𝐺𝑧𝑛𝑆𝑥,𝑆(𝑥,𝑦),𝑆(𝑥,𝑦),𝑘𝑡=𝐺𝑛,𝑦𝑛𝐺𝑧,𝑆(𝑥,𝑦),𝑆(𝑥,𝑦),𝑘𝑡≥min𝑛−1𝑝,𝑓𝑥,𝑓𝑥,𝑡,𝐺𝑛−1.,𝑓𝑦,𝑓𝑦,𝑡(9) Letting ğ‘›â†’âˆž, 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 𝐺𝑧𝑓𝛼=𝑓𝑓𝑥=𝑓(𝑆(𝑥,𝑦))=𝑆(𝑓𝑥,𝑓𝑦)=𝑆(𝛼,𝛽),𝑓𝛽=𝑓𝑓𝑦=𝑓(𝑆(𝑦,𝑥))=𝑆(𝑓𝑦,𝑓𝑥)=𝑆(𝛽,𝛼).𝑛𝑆𝑥,𝑓𝛼,𝑓𝛼,𝑘𝑡=𝐺𝑛,𝑦𝑛𝐺𝑧,𝑆(𝛼,𝛽),𝑆(𝛼,𝛽),𝑘𝑡≥min𝑛−1𝑝,𝑓𝛼,𝑓𝛼,𝑡,𝐺𝑛−1.,𝑓𝛽,𝑓𝛽,𝑡(11) Letting ğ‘›â†’âˆž, 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:𝐺𝛼,𝛼1,𝛼1𝑆𝛼,𝑘𝑡=𝐺(𝛼,𝛽),𝑆1,𝛽1𝛼,𝑆1,𝛽1𝐺,𝑘𝑡≥min𝛼,𝛼1,𝛼1,𝑡,𝐺𝛽,𝛽1,𝛽1.,𝑡(15) Similarly, 𝐺𝛽,𝛽1,𝛽1𝑆𝛽,𝑘𝑡=𝐺(𝛽,𝛼),𝑆1,𝛼1𝛽,𝑆1,𝛼1𝐺,𝑘𝑡≥min𝛼,𝛼1,𝛼1,𝑡,𝐺𝛽,𝛽1,𝛽1.,𝑡(16) Thus, 𝐺min𝛼,𝛼1,𝛼1,𝑘𝑡,𝐺𝛽,𝛽1,𝛽1𝐺,𝑘𝑡≥min𝛼,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: 𝐺𝛼1𝑆𝛼,𝛼,𝛼,𝑡=𝐺1,𝛼1𝐺𝛼,𝑆(𝛼,𝛼),𝑆(𝛼,𝛼),𝑡≥min1𝑡,𝛼,𝛼,𝑘𝛼,𝐺1𝑡,𝛼,𝛼,𝑘𝛼≥𝐺1𝑡,𝛼,𝛼,𝑘2⋮𝛼≥𝐺1𝑡,𝛼,𝛼,𝑘𝑛⟶1asğ‘›âŸ¶âˆž.(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 ğ‘Žâˆ—ğ‘=min{ğ‘Ž,𝑏} for all ğ‘Ž,𝑏∈[0,1] and let 𝑆,𝑇,𝑅∶𝑋×𝑋→𝑋 be mappings satisfying 𝐺(𝑆(𝑥,𝑦),𝑇(𝑢,𝑣),𝑅(𝑝,ğ‘ž),𝑘𝑡)≥min{𝐺(𝑥,𝑢,𝑝,𝑡),𝐺(𝑦,𝑣,ğ‘ž,𝑡),𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡),𝐺(𝑢,𝑢,𝑇(𝑢,𝑣),𝑡),𝐺(𝑝,𝑝,𝑅(𝑝,ğ‘ž),𝑡)}(21)forall𝑥,𝑦,𝑢,𝑣,𝑝,ğ‘žâˆˆğ‘‹, where 0≤𝑘<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,ğ‘Žğ‘›ğ‘‘ğ‘…â„Žğ‘Žğ‘£ğ‘’ğ‘Žğ‘¢ğ‘›ğ‘–ğ‘žğ‘¢ğ‘’ğ‘ğ‘œğ‘šğ‘šğ‘œğ‘›ğ‘ğ‘œğ‘¢ğ‘ğ‘™ğ‘’ğ‘‘ğ‘“ğ‘–ğ‘¥ğ‘’ğ‘‘ğ‘ğ‘œğ‘–ğ‘›ğ‘¡ğ‘œğ‘“ğ‘¡â„Žğ‘’ğ‘“ğ‘œğ‘Ÿğ‘š(𝑥,𝑥)𝑖𝑛𝑋×𝑋.(24)

Proof. Let 𝑥0,𝑦0∈𝑋. 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 𝑥𝑛≠𝑥𝑛+1 and 𝑦𝑛≠𝑦𝑛+1 for all 𝑛. Write 𝑑𝑛(𝑡)=𝐺(𝑥𝑛,𝑥𝑛+1,𝑥𝑛+2,𝑡) and 𝑒𝑛(𝑡)=𝐺(𝑦𝑛,𝑦𝑛+1,𝑦𝑛+2,𝑡): 𝑑3𝑛𝑥(𝑘𝑡)=𝐺3𝑛,𝑥3𝑛+1,𝑥3𝑛+2𝑆𝑥,𝑘𝑡=𝐺3𝑛,𝑦3𝑛𝑥,𝑇3𝑛+1,𝑦3𝑛+1𝑥,𝑅3𝑛−1,𝑦3𝑛−1𝑑,𝑘𝑡≥min3𝑛−1(𝑡),𝑒3𝑛−1𝑥(𝑡),𝐺3𝑛,𝑥3𝑛,𝑥3𝑛+1,𝐺𝑥,𝑡3𝑛+1,𝑥3𝑛+1,𝑥3𝑛+2𝑥,𝑡,𝐺3𝑛−1,𝑥3𝑛−1,𝑥3𝑛𝑑,𝑡≥min3𝑛−1(𝑡),𝑒3𝑛−1(𝑡),𝑑3𝑛(𝑡),𝑑3𝑛(𝑡),𝑑3𝑛−1.(𝑡)(28) Thus, 𝑑3𝑛(𝑘𝑡)≥min{𝑑3𝑛−1(𝑡),𝑒3𝑛−1(𝑡)}. Similarly, we have 𝑒3𝑛(𝑘𝑡)≥min𝑑3𝑛−1(𝑡),𝑒3𝑛−1(𝑡).
Thus,𝑑min3𝑛(𝑘𝑡),𝑒3𝑛𝑑(𝑘𝑡)≥min3𝑛−1(𝑡),𝑒3𝑛−1(𝑡).(29) Similarly, we can show that 𝑑min3𝑛+1(𝑘𝑡),𝑒3𝑛+1𝑑(𝑘𝑡)≥min3𝑛(𝑡),𝑒3𝑛,𝑑(𝑡)min3𝑛+2(𝑘𝑡),𝑒3𝑛+2(𝑑𝑘𝑡)≥min3𝑛+1(𝑡),𝑒3𝑛+1(.𝑡)(30) Thus, 𝑑min𝑛+1(𝑘𝑡),𝑒𝑛+1𝑑(𝑘𝑡)≥min𝑛(𝑡),𝑒𝑛(𝑡).(31) Hence 𝑑min𝑛(𝑡),𝑒𝑛𝑑(𝑡)≥min𝑛𝑡𝑘,𝑒𝑛𝑡𝑘𝑑≥min𝑛𝑡𝑘2,𝑒𝑛𝑡𝑘2⋮𝑑≥min0𝑡𝑘𝑛,𝑒0𝑡𝑘𝑛𝐺𝑥=min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,𝐺0,𝑦1,𝑦2,𝑡𝑘𝑛.(32) Thus, 𝐺𝑥𝑛,𝑥𝑛+1,𝑥𝑛+2𝐺𝑥,𝑡≥min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,𝐺0,𝑦1,𝑦2,𝑡𝑘𝑛.(33) From (𝐺3), we have 𝐺𝑥𝑛,𝑥𝑛,𝑥𝑛+1𝑥,𝑡≥𝐺𝑛,𝑥𝑛+1,𝑥𝑛+2𝐺𝑥,𝑡≥min0,𝑥1,𝑥2,𝑡𝑘𝑛𝑦,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,𝑦∈𝑋 such that 𝑥𝑛→𝑥 and 𝑦𝑛→𝑦∶𝐺𝑆(𝑥,𝑦),𝑥3𝑛+2,𝑥3𝑛+3𝑥,𝑘𝑡=𝐺𝑆(𝑥,𝑦),𝑇3𝑛+1,𝑦3𝑛+1𝑥,𝑅3𝑛+2,𝑦3𝑛+2𝐺,𝑘𝑡≥min𝑥,𝑥3𝑛+1,𝑥3𝑛+2,𝑡,𝐺𝑦,𝑦3𝑛+1,𝑦3𝑛+2,𝑥,𝑡𝐺(𝑥,𝑥,𝑆(𝑥,𝑦),𝑡),𝐺3𝑛+1,𝑥3𝑛+1,𝑥3𝑛+2,𝐺𝑥,𝑡3𝑛+2,𝑥3𝑛+2,𝑥3𝑛+3.,𝑡(35) Letting ğ‘›â†’âˆž, 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 𝐺𝑥,𝑥,𝑥1𝑥,𝑘𝑡=𝐺𝑆(𝑥,𝑦),𝑇(𝑥,𝑦),𝑅1,𝑦1𝐺,𝑘𝑡≥min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑡,1,1,1=min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1.,𝑡(37) Also, 𝐺𝑦,𝑦,𝑦1𝑦,𝑘𝑡=𝐺𝑆(𝑦,𝑥),𝑇(𝑦,𝑥),𝑅1,𝑥1𝐺,𝑘𝑡≥min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑡,1,1,1=min𝑥,𝑥,𝑥1,𝑡,𝐺𝑦,𝑦,𝑦1.,𝑡(38) Thus, 𝐺min𝑥,𝑥,𝑥1,𝑘𝑡,𝐺𝑦,𝑦,𝑦1𝐺,𝑘𝑡≥min𝑥,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.

References

  1. Z. Mustafa and B. Sims, “Some remarks concerninig D-metric spaces,” in Proceedings of the Internatinal Conferences on Fixed Point Theory and Applications, pp. 189–198, Valencia, Spain, July 2003. View at: Google Scholar
  2. Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006. View at: Google Scholar
  3. Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009. View at: Google Scholar
  4. S. V. R. Naidu, K. P. R. Rao, and N. Srinivasa Rao, “On convergent sequences and fixed point theorems in D-metric spaces,” International Journal of Mathematics and Mathematical Sciences, no. 12, pp. 1969–1988, 2005. View at: Publisher Site | Google Scholar
  5. B. C. Dhage, “Generalized metric spaces and mapping with fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 84, pp. 329–336, 1992. View at: Google Scholar
  6. B. C. Dhage, “On generalized metric spaces and topological structure II,” Pure and Applied Mathematika Sciences, vol. 40, no. 1-2, pp. 37–41, 1994. View at: Google Scholar
  7. B. C. Dhage, “A common fixed point principle in D-metric spaces,” Bulletin of the Calcutta Mathematical Society, vol. 91, no. 6, pp. 475–480, 1999. View at: Google Scholar
  8. B. C. Dhage, “Generalized metric spaces and topological structure. I,” Annalele Stiintifice ale Universitatii Al.I.Cuza, vol. 46, no. 1, pp. 3–24, 2000. View at: Google Scholar
  9. M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 262–269, 2009. View at: Publisher Site | Google Scholar
  10. R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, “Property P in G-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 401684, 12 pages, 2010. View at: Google Scholar
  11. Z. Mustafa, F. Awawdeh, and W. Shatanawi, “Fixed point theorem for expansive mappings in G-metric spaces,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 49–52, pp. 2463–2472, 2010. View at: Google Scholar
  12. Z. Mustafa and H. Obiedat, “A fixed point theorem of Reich in G-metric spaces,” Cubo A Mathematical Journal, vol. 12, no. 1, pp. 83–93, 2010. View at: Google Scholar
  13. Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008. View at: Google Scholar
  14. Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results in G-metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 283028, 10 pages, 2009. View at: Google Scholar
  15. W. Shatanawi, “Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 181650, 9 pages, 2010. View at: Google Scholar
  16. G. Sun and K. Yang, “Generalized fuzzy metric spaces with properties,” Research journal of Applied Sciences, Engineering and Technology, vol. 2, no. 7, pp. 673–678, 2010. View at: Google Scholar
  17. T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006. View at: Publisher Site | Google Scholar
  18. M. Abbas, M. Ali Khan, and S. Radenović, “Common coupled fixed point theorems in cone metric spaces for W-compatible mappings,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 195–202, 2010. View at: Publisher Site | Google Scholar

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