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.