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

Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces

K. P. R. Rao,1 I. Altun,2 and S. Hima Bindu3

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 [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 , 𝑎 , 𝑎 ) + 𝐺 ( 𝑎 , 𝑦 , 𝑧 ) for all 𝑥 , 𝑦 , 𝑧 , 𝑎 ∈ 𝑋 .
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 { 𝑥 𝑛 } be a sequence in 𝑋 . A point 𝑥 ∈ 𝑋 is said to be limit of { 𝑥 𝑛 } if and only if l i m 𝑛 , 𝑚 → ∞ 𝐺 ( 𝑥 , 𝑥 𝑛 , 𝑥 𝑚 ) = 0 . In this case, the sequence { 𝑥 𝑛 } is said to be 𝐺 -convergent to 𝑥 .

Definition 4 (see [2]). Let ( 𝑋 , 𝐺 ) be a 𝐺 -metric space and let { 𝑥 𝑛 } be a sequence in 𝑋 . { 𝑥 𝑛 } is called 𝐺 -Cauchy if and only if l i m 𝑙 , 𝑛 , 𝑚 → ∞ 𝐺 ( 𝑥 𝑙 , 𝑥 𝑛 , 𝑥 𝑚 ) = 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 { 𝑥 𝑛 } is 𝐺 -Cauchy.(ii)For every 𝜖 > 0 , there exists 𝑁 ∈ 𝐍 such that 𝐺 ( 𝑥 𝑛 , 𝑥 𝑚 , 𝑥 𝑚 ) < 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 , 𝑎 ∈ 𝑋 , 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 ) ≤ 2 𝐺 ( 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 { 𝑥 𝑛 } ⊆ 𝑋 and a point 𝑥 ∈ 𝑋 , the following are equivalent:(i) { 𝑥 𝑛 } is 𝐺 -convergent to 𝑥 ,(ii) 𝐺 ( 𝑥 𝑛 , 𝑥 𝑛 , 𝑥 ) → 0 as 𝑛 → ∞ ,(iii) 𝐺 ( 𝑥 𝑛 , 𝑥 , 𝑥 ) → 0 as 𝑛 → ∞ ,(iv) 𝐺 ( 𝑥 𝑚 , 𝑥 𝑛 , 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 𝑋 3 × ( 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 ( 𝑎 , 𝑦 , 𝑧 , 𝑡 ) ∗ 𝐺 ( 𝑥 , 𝑎 , 𝑎 , 𝑠 ) for all 𝑥 , 𝑦 , 𝑧 , 𝑎 ∈ 𝑋 ,(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 𝑎 ∗ 𝑏 = 𝑎 𝑏 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 𝑥 ∈ 𝑋 , the set 𝐵 𝐺 ( 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 𝐵 𝐺 ( 𝑥 , 𝑟 , 𝑡 ) ⊆ 𝐴 .

A sequence { 𝑥 𝑛 } in 𝐺 -fuzzy metric space 𝑋 is said to be 𝐺 -convergent to 𝑥 ∈ 𝑋 if 𝐺 ( 𝑥 𝑛 , 𝑥 𝑛 , 𝑥 , 𝑡 ) → 1 as 𝑛 → ∞ for each 𝑡 > 0 . It is called a 𝐺 -Cauchy sequence if 𝐺 ( 𝑥 𝑛 , 𝑥 𝑛 , 𝑥 𝑚 , 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 , 𝑧 ∈ 𝑋 .

Lemma 13 (see [16]). Let ( 𝑋 , 𝐺 , ∗ ) be a 𝐺 -fuzzy metric space. Then, 𝐺 is a continuous function on 𝑋 3 × ( 0 , ∞ ) .

Now onwards, we assume the following condition: l i m 𝑡 → ∞ 𝐺 ( 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 m i n { 𝐺 ( 𝑥 , 𝑦 , 𝑧 , 𝑘 𝑡 ) , 𝐺 ( 𝑢 , 𝑣 , 𝑤 , 𝑘 𝑡 ) } ≥ m i n { 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 𝑔 𝑥 = 𝐹 ( 𝑥 , 𝑦 ) and 𝑔 𝑦 = 𝐹 ( 𝑦 , 𝑥 ) ,(ii)a common coupled fixed point of 𝐹 ∶ 𝑋 × 𝑋 → 𝑋 and 𝑔 ∶ 𝑋 → 𝑋 if 𝑥 = 𝑔 𝑥 = 𝐹 ( 𝑥 , 𝑦 ) and 𝑦 = 𝑔 𝑦 = 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 ( 𝑔 𝑥 , 𝑔 𝑦 ) and 𝑔 ( 𝐹 ( 𝑦 , 𝑥 ) ) = 𝐹 ( 𝑔 𝑦 , 𝑔 𝑥 ) whenever 𝑔 𝑥 = 𝐹 ( 𝑥 , 𝑦 ) and 𝑔 𝑦 = 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 𝑎 ∗ 𝑏 = m i n { 𝑎 , 𝑏 } for all 𝑎 , 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 ) , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝑓 𝑥 , 𝑓 𝑢 , 𝑓 𝑢 , 𝑡 ) , 𝐺 ( 𝑓 𝑦 , 𝑓 𝑣 , 𝑓 𝑣 , 𝑡 ) } ( 2 ) f o r a l l 𝑥 , 𝑦 , 𝑢 , 𝑣 ∈ 𝑋 , where 0 ≤ 𝑘 < 1 , 𝑆 ( 𝑋 × 𝑋 ) ⊆ 𝑓 ( 𝑋 ) 𝑎 𝑛 𝑑 𝑓 ( 𝑋 ) 𝑖 𝑠 𝑎 𝑐 𝑜 𝑚 𝑝 𝑙 𝑒 𝑡 𝑒 𝑠 𝑢 𝑏 𝑠 𝑝 𝑎 𝑐 𝑒 𝑜 𝑓 𝑋 , 𝑡 ℎ 𝑒 𝑝 𝑎 𝑖 𝑟 ( 𝑓 , 𝑆 ) 𝑖 𝑠 𝑊 - c o m p a t i b l e . ( 3 )
Then 𝑆 and 𝑓 have a unique common coupled fixed point of the form ( 𝛼 , 𝛼 ) in 𝑋 × 𝑋 .

Proof. Let 𝑥 0 , 𝑦 0 ∈ 𝑋 and denote 𝑧 𝑛 = 𝑆 ( 𝑥 𝑛 , 𝑦 𝑛 ) = 𝑓 𝑥 𝑛 + 1 , 𝑝 𝑛 = 𝑆 ( 𝑦 𝑛 , 𝑥 𝑛 ) = 𝑓 𝑦 𝑛 + 1 , 𝑛 = 0 , 1 , 2 , … . Let 𝑑 𝑛 ( 𝑡 ) = 𝐺 ( 𝑧 𝑛 , 𝑧 𝑛 + 1 , 𝑧 𝑛 + 1 , 𝑡 ) , 𝑒 𝑛 ( 𝑡 ) = 𝐺 ( 𝑝 𝑛 , 𝑝 𝑛 + 1 , 𝑝 𝑛 + 1 , 𝑡 ) . From (2), we have 𝑑 𝑛 + 1 𝑧 ( 𝑘 𝑡 ) = 𝐺 𝑛 + 1 , 𝑧 𝑛 + 2 , 𝑧 𝑛 + 2 𝑆 𝑥 , 𝑘 𝑡 = 𝐺 𝑛 + 1 , 𝑦 𝑛 + 1 𝑥 , 𝑆 𝑛 + 2 , 𝑦 𝑛 + 2 𝑥 , 𝑆 𝑛 + 2 , 𝑦 𝑛 + 2 𝐺 𝑧 , 𝑘 𝑡 ≥ m i n 𝑛 , 𝑧 𝑛 + 1 , 𝑧 𝑛 + 1 𝑝 , 𝑡 , 𝐺 𝑛 , 𝑝 𝑛 + 1 , 𝑝 𝑛 + 1 𝑑 , 𝑡 ≥ m i n 𝑛 ( 𝑡 ) , 𝑒 𝑛 ( . 𝑡 ) ( 4 ) Also, 𝑒 𝑛 + 1 𝑝 ( 𝑘 𝑡 ) = 𝐺 𝑛 + 1 , 𝑝 𝑛 + 2 , 𝑝 𝑛 + 2 𝑆 𝑦 , 𝑘 𝑡 = 𝐺 𝑛 + 1 , 𝑥 𝑛 + 1 𝑦 , 𝑆 𝑛 + 2 , 𝑥 𝑛 + 2 𝑦 , 𝑆 𝑛 + 2 , 𝑥 𝑛 + 2 𝐺 𝑝 , 𝑘 𝑡 ≥ m i n 𝑛 , 𝑝 𝑛 + 1 , 𝑝 𝑛 + 1 𝑧 , 𝑡 , 𝐺 𝑛 , 𝑧 𝑛 + 1 , 𝑧 𝑛 + 1 𝑒 , 𝑡 ≥ m i n 𝑛 ( 𝑡 ) , 𝑑 𝑛 ( . 𝑡 ) ( 5 ) Thus, m i n { 𝑑 𝑛 + 1 ( 𝑘 𝑡 ) , 𝑒 𝑛 + 1 ( 𝑘 𝑡 ) } ≥ m i n { 𝑑 𝑛 ( 𝑡 ) , 𝑒 𝑛 ( 𝑡 ) } . Hence, 𝑑 m i n 𝑛 ( 𝑡 ) , 𝑒 𝑛 𝑑 ( 𝑡 ) ≥ m i n 𝑛 − 1 𝑡 𝑘 , 𝑒 𝑛 − 1 𝑡 𝑘 𝑑 ≥ m i n 𝑛 − 2 𝑡 𝑘 2 , 𝑒 𝑛 − 2 𝑡 𝑘 2 ⋮ 𝑑 ≥ m i n 0 𝑡 𝑘 𝑛 , 𝑒 0 𝑡 𝑘 𝑛 𝐺 𝑧 = m i n 0 , 𝑧 1 , 𝑧 1 , 𝑡 𝑘 𝑛 𝑝 , 𝐺 0 , 𝑝 1 , 𝑝 1 , 𝑡 𝑘 𝑛 . ( 6 ) For any positive integer 𝑛 and fixed positive integer 𝑝 , we have 𝐺 𝑧 𝑛 , 𝑧 𝑛 + 𝑝 , 𝑧 𝑛 + 𝑝 𝑧 , 𝑡 ≥ 𝐺 𝑛 + 𝑝 − 1 , 𝑧 𝑛 + 𝑝 , 𝑧 𝑛 + 𝑝 , 𝑡 𝑝 𝑧 ∗ 𝐺 𝑛 + 𝑝 − 2 , 𝑧 𝑛 + 𝑝 − 1 , 𝑧 𝑛 + 𝑝 − 1 , 𝑡 𝑝 𝑧 ∗ ⋯ ∗ 𝐺 𝑛 , 𝑧 𝑛 + 1 , 𝑧 𝑛 + 1 , 𝑡 𝑝 𝐺 𝑧 ≥ m i n 0 , 𝑧 1 , 𝑧 1 , 𝑡 𝑝 𝑘 𝑛 + 𝑝 − 1 𝑝 , 𝐺 0 , 𝑝 1 , 𝑝 1 , 𝑡 𝑝 𝑘 𝑛 + 𝑝 − 1 𝐺 𝑧 ∗ m i n 0 , 𝑧 1 , 𝑧 1 , 𝑡 𝑝 𝑘 𝑛 + 𝑝 − 2 𝑝 , 𝐺 0 , 𝑝 1 , 𝑝 1 , 𝑡 𝑝 𝑘 𝑛 + 𝑝 − 2 𝐺 𝑧 ∗ ⋯ ∗ m i n 0 , 𝑧 1 , 𝑧 1 , 𝑡 𝑝 𝑘 𝑛 𝑝 , 𝐺 0 , 𝑝 1 , 𝑝 1 , 𝑡 𝑝 𝑘 𝑛 . ( 7 ) Letting 𝑛 → ∞ and using (P), we get l i m 𝑛 → ∞ 𝐺 𝑧 𝑛 , 𝑧 𝑛 + 𝑝 , 𝑧 𝑛 + 𝑝 , 𝑡 ≥ 1 ∗ 1 ∗ ⋯ ∗ 1 = 1 . ( 8 ) Hence, l i m 𝑛 → ∞ 𝐺 ( 𝑧 𝑛 , 𝑧 𝑛 + 𝑝 , 𝑧 𝑛 + 𝑝 , 𝑡 ) = 1 . Thus, { 𝑧 𝑛 } is 𝐺 -Cauchy in 𝑋 . Similarly, we can show that { 𝑝 𝑛 } is 𝐺 -Cauchy in 𝑋 . Since 𝑓 ( 𝑋 ) is 𝐺 -complete, { 𝑧 𝑛 } and { 𝑝 𝑛 } converge to some 𝛼 and 𝛽 in 𝑓 ( 𝑋 ) , respectively. Hence, there exist 𝑥 and 𝑦 in 𝑋 such that 𝛼 = 𝑓 𝑥 , 𝛽 = 𝑓 𝑦 : 𝐺 𝑧 𝑛 𝑆 𝑥 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑆 ( 𝑥 , 𝑦 ) , 𝑘 𝑡 = 𝐺 𝑛 , 𝑦 𝑛 𝐺 𝑧 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑆 ( 𝑥 , 𝑦 ) , 𝑘 𝑡 ≥ m i n 𝑛 − 1 𝑝 , 𝑓 𝑥 , 𝑓 𝑥 , 𝑡 , 𝐺 𝑛 − 1 . , 𝑓 𝑦 , 𝑓 𝑦 , 𝑡 ( 9 ) Letting 𝑛 → ∞ , we get G ( 𝑓 𝑥 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑆 ( 𝑥 , 𝑦 ) , 𝑘 𝑡 ) ≥ m i n { 1 , 1 } = 1 . ( 1 0 ) Hence, 𝑆 ( 𝑥 , 𝑦 ) = 𝑓 𝑥 . Similarly, it can be shown that 𝑆 ( 𝑦 , 𝑥 ) = 𝑓 𝑦 . Since ( 𝑓 , 𝑆 ) is 𝑊 -compatible, we have 𝐺 𝑧 𝑓 𝛼 = 𝑓 𝑓 𝑥 = 𝑓 ( 𝑆 ( 𝑥 , 𝑦 ) ) = 𝑆 ( 𝑓 𝑥 , 𝑓 𝑦 ) = 𝑆 ( 𝛼 , 𝛽 ) , 𝑓 𝛽 = 𝑓 𝑓 𝑦 = 𝑓 ( 𝑆 ( 𝑦 , 𝑥 ) ) = 𝑆 ( 𝑓 𝑦 , 𝑓 𝑥 ) = 𝑆 ( 𝛽 , 𝛼 ) . 𝑛 𝑆 𝑥 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑘 𝑡 = 𝐺 𝑛 , 𝑦 𝑛 𝐺 𝑧 , 𝑆 ( 𝛼 , 𝛽 ) , 𝑆 ( 𝛼 , 𝛽 ) , 𝑘 𝑡 ≥ m i n 𝑛 − 1 𝑝 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑡 , 𝐺 𝑛 − 1 . , 𝑓 𝛽 , 𝑓 𝛽 , 𝑡 ( 1 1 ) Letting 𝑛 → ∞ , we get 𝐺 ( 𝛼 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝛼 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑡 ) , 𝐺 ( 𝛽 , 𝑓 𝛽 , 𝑓 𝛽 , 𝑡 ) } . ( 1 2 ) Similarly, we can show that 𝐺 ( 𝛽 , 𝑓 𝛽 , 𝑓 𝛽 , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝛼 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑡 ) , 𝐺 ( 𝛽 , 𝑓 𝛽 , 𝑓 𝛽 , 𝑡 ) } . ( 1 3 ) Thus, m i n { 𝐺 ( 𝛼 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑘 𝑡 ) , 𝐺 ( 𝛽 , 𝑓 𝛽 , 𝑓 𝛽 , 𝑘 𝑡 ) } ≥ m i n { 𝐺 ( 𝛼 , 𝑓 𝛼 , 𝑓 𝛼 , 𝑡 ) , 𝐺 ( 𝛽 , 𝑓 𝛽 , 𝑓 𝛽 , 𝑡 ) } . ( 1 4 ) From Lemma 14, we have 𝑓 𝛼 = 𝛼 and 𝑓 𝛽 = 𝛽 . Thus, 𝛼 = 𝑓 𝛼 = 𝑆 ( 𝛼 , 𝛽 ) and 𝛽 = 𝑓 𝛽 = 𝑆 ( 𝛽 , 𝛼 ) . Hence, ( 𝛼 , 𝛽 ) is a common coupled fixed point of 𝑆 and 𝑓 .
Suppose ( 𝛼 1 , 𝛽 1 ) is another common coupled fixed point of 𝑆 and 𝑓 : 𝐺 𝛼 , 𝛼 1 , 𝛼 1 𝑆 𝛼 , 𝑘 𝑡 = 𝐺 ( 𝛼 , 𝛽 ) , 𝑆 1 , 𝛽 1 𝛼 , 𝑆 1 , 𝛽 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝛼 , 𝛼 1 , 𝛼 1 , 𝑡 , 𝐺 𝛽 , 𝛽 1 , 𝛽 1 . , 𝑡 ( 1 5 ) Similarly, 𝐺 𝛽 , 𝛽 1 , 𝛽 1 𝑆 𝛽 , 𝑘 𝑡 = 𝐺 ( 𝛽 , 𝛼 ) , 𝑆 1 , 𝛼 1 𝛽 , 𝑆 1 , 𝛼 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝛼 , 𝛼 1 , 𝛼 1 , 𝑡 , 𝐺 𝛽 , 𝛽 1 , 𝛽 1 . , 𝑡 ( 1 6 ) Thus, 𝐺 m i n 𝛼 , 𝛼 1 , 𝛼 1 , 𝑘 𝑡 , 𝐺 𝛽 , 𝛽 1 , 𝛽 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝛼 , 𝛼 1 , 𝛼 1 , 𝑡 , 𝐺 𝛽 , 𝛽 1 , 𝛽 1 . , 𝑡 ( 1 7 ) From Lemma 14, 𝛼 1 = 𝛼 and 𝛽 1 = 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 , 𝑘 𝑡 ) = 𝐺 ( 𝑆 ( 𝛼 , 𝛽 ) , 𝑆 ( 𝛼 , 𝛽 ) , 𝑆 ( 𝛽 , 𝛼 ) , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝛼 , 𝛼 , 𝛽 , 𝑡 ) , 𝐺 ( 𝛽 , 𝛽 , 𝛼 , 𝑡 ) } , 𝐺 ( 𝛼 , 𝛽 , 𝛽 , 𝑘 𝑡 ) = 𝐺 ( 𝑆 ( 𝛼 , 𝛽 ) , 𝑆 ( 𝛽 , 𝛼 ) , 𝑆 ( 𝛽 , 𝛼 ) , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝛼 , 𝛽 , 𝛽 , 𝑡 ) , 𝐺 ( 𝛽 , 𝛼 , 𝛼 , 𝑡 ) } . ( 1 8 ) Thus, m i n { 𝐺 ( 𝛼 , 𝛼 , 𝛽 , 𝑘 𝑡 ) , 𝐺 ( 𝛼 , 𝛽 , 𝛽 , 𝑘 𝑡 ) } ≥ m i n { 𝐺 ( 𝛼 , 𝛼 , 𝛽 , 𝑡 ) , 𝐺 ( 𝛼 , 𝛽 , 𝛽 , 𝑡 ) } . ( 1 9 ) From Lemma 14, we have 𝛼 = 𝛽 . Thus, 𝛼 is a common fixed point of 𝑆 and 𝑓 , that is, 𝛼 = 𝑓 𝛼 = 𝑆 ( 𝛼 , 𝛼 ) . Suppose 𝛼 1 is another common fixed point of 𝑆 and 𝑓 : 𝐺 𝛼 1 𝑆 𝛼 , 𝛼 , 𝛼 , 𝑡 = 𝐺 1 , 𝛼 1 𝐺 𝛼 , 𝑆 ( 𝛼 , 𝛼 ) , 𝑆 ( 𝛼 , 𝛼 ) , 𝑡 ≥ m i n 1 𝑡 , 𝛼 , 𝛼 , 𝑘 𝛼 , 𝐺 1 𝑡 , 𝛼 , 𝛼 , 𝑘 𝛼 ≥ 𝐺 1 𝑡 , 𝛼 , 𝛼 , 𝑘 2 ⋮ 𝛼 ≥ 𝐺 1 𝑡 , 𝛼 , 𝛼 , 𝑘 𝑛 ⟶ 1 a s 𝑛 ⟶ ∞ . ( 2 0 ) Hence, 𝛼 1 = 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 𝑎 ∗ 𝑏 = m i n { 𝑎 , 𝑏 } for all 𝑎 , 𝑏 ∈ [ 0 , 1 ] and let 𝑆 , 𝑇 , 𝑅 ∶ 𝑋 × 𝑋 → 𝑋 be mappings satisfying 𝐺 ( 𝑆 ( 𝑥 , 𝑦 ) , 𝑇 ( 𝑢 , 𝑣 ) , 𝑅 ( 𝑝 , 𝑞 ) , 𝑘 𝑡 ) ≥ m i n { 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 ) } ( 2 1 ) f o r a l l 𝑥 , 𝑦 , 𝑢 , 𝑣 , 𝑝 , 𝑞 ∈ 𝑋 , where 0 ≤ 𝑘 < 1 . Then, there exists ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑋 such that 𝑥 = 𝑆 ( 𝑥 , 𝑦 ) = 𝑇 ( 𝑥 , 𝑦 ) = 𝑅 ( 𝑥 , 𝑦 ) , ( 2 2 ) 𝑦 = 𝑆 ( 𝑦 , 𝑥 ) = 𝑇 ( 𝑦 , 𝑥 ) = 𝑅 ( 𝑦 , 𝑥 ) . ( 2 3 ) Or 𝑆 , 𝑇 , 𝑎 𝑛 𝑑 𝑅 ℎ 𝑎 𝑣 𝑒 𝑎 𝑢 𝑛 𝑖 𝑞 𝑢 𝑒 𝑐 𝑜 𝑚 𝑚 𝑜 𝑛 𝑐 𝑜 𝑢 𝑝 𝑙 𝑒 𝑑 𝑓 𝑖 𝑥 𝑒 𝑑 𝑝 𝑜 𝑖 𝑛 𝑡 𝑜 𝑓 𝑡 ℎ 𝑒 𝑓 𝑜 𝑟 𝑚 ( 𝑥 , 𝑥 ) 𝑖 𝑛 𝑋 × 𝑋 . ( 2 4 )

Proof. Let 𝑥 0 , 𝑦 0 ∈ 𝑋 . Define the sequences { 𝑥 𝑛 } and { 𝑦 𝑛 } in 𝑋 as follows: 𝑥 3 𝑛 + 1 = 𝑆 ( 𝑥 3 𝑛 , 𝑦 3 𝑛 ) , 𝑦 3 𝑛 + 1 = 𝑆 ( 𝑦 3 𝑛 , 𝑥 3 𝑛 ) ; 𝑥 3 𝑛 + 2 = 𝑇 ( 𝑥 3 𝑛 + 1 , 𝑦 3 𝑛 + 1 ) , 𝑦 3 𝑛 + 2 = 𝑇 ( 𝑦 3 𝑛 + 1 , 𝑥 3 𝑛 + 1 ) ; 𝑥 3 𝑛 + 3 = 𝑅 ( 𝑥 3 𝑛 + 2 , 𝑦 3 𝑛 + 2 ) , 𝑦 3 𝑛 + 3 = 𝑅 ( 𝑦 3 𝑛 + 2 , 𝑥 3 𝑛 + 2 ) , 𝑛 = 0 , 1 , 2 , … . Suppose 𝑥 3 𝑛 + 1 = 𝑥 3 𝑛 for some 𝑛 . Then, 𝑆 ( 𝑥 , 𝑦 ) = 𝑥 , where 𝑥 = 𝑥 3 𝑛 , 𝑦 = 𝑦 3 𝑛 . Suppose 𝑇 ( 𝑥 , 𝑦 ) ≠ 𝑅 ( 𝑥 , 𝑦 ) . Then, 𝐺 ( 𝑥 , 𝑇 ( 𝑥 , 𝑦 ) , 𝑅 ( 𝑥 , 𝑦 ) , 𝑘 𝑡 ) = 𝐺 ( 𝑆 ( 𝑥 , 𝑦 ) , 𝑇 ( 𝑥 , 𝑦 ) , 𝑅 ( 𝑥 , 𝑦 ) , 𝑘 𝑡 ) ≥ m i n { 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 ) . ( 2 5 ) 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 ) , 𝑘 𝑡 ) ≥ 𝐺 ( 𝑥 , 𝑥 , 𝑇 ( 𝑥 , 𝑦 ) , 𝑡 ) = 𝐺 ( 𝑥 , 𝑇 ( 𝑥 , 𝑦 ) , 𝑇 ( 𝑥 , 𝑦 ) , 𝑡 ) . ( 2 6 ) From Lemma 14, we have 𝑇 ( 𝑥 , 𝑦 ) = 𝑥 . Thus, 𝑆 ( 𝑥 , 𝑦 ) = 𝑇 ( 𝑥 , 𝑦 ) = 𝑅 ( 𝑥 , 𝑦 ) = 𝑥 . Similarly, if 𝑥 3 𝑛 + 1 = 𝑥 3 𝑛 + 2 or 𝑥 3 𝑛 + 2 = 𝑥 3 𝑛 + 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 𝑦 3 𝑛 = 𝑦 3 𝑛 + 1 or 𝑦 3 𝑛 + 1 = 𝑦 3 𝑛 + 2 or 𝑦 3 𝑛 + 2 = 𝑦 3 𝑛 + 3 then there exists ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑋 such that 𝑆 ( 𝑦 , 𝑥 ) = 𝑇 ( 𝑦 , 𝑥 ) = 𝑅 ( 𝑦 , 𝑥 ) = 𝑦 . ( 2 7 ) Now, assume that 𝑥 𝑛 ≠ 𝑥 𝑛 + 1 and 𝑦 𝑛 ≠ 𝑦 𝑛 + 1 for all 𝑛 . Write 𝑑 𝑛 ( 𝑡 ) = 𝐺 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 , 𝑥 𝑛 + 2 , 𝑡 ) and 𝑒 𝑛 ( 𝑡 ) = 𝐺 ( 𝑦 𝑛 , 𝑦 𝑛 + 1 , 𝑦 𝑛 + 2 , 𝑡 ) : 𝑑 3 𝑛 𝑥 ( 𝑘 𝑡 ) = 𝐺 3 𝑛 , 𝑥 3 𝑛 + 1 , 𝑥 3 𝑛 + 2 𝑆 𝑥 , 𝑘 𝑡 = 𝐺 3 𝑛 , 𝑦 3 𝑛 𝑥 , 𝑇 3 𝑛 + 1 , 𝑦 3 𝑛 + 1 𝑥 , 𝑅 3 𝑛 − 1 , 𝑦 3 𝑛 − 1 𝑑 , 𝑘 𝑡 ≥ m i n 3 𝑛 − 1 ( 𝑡 ) , 𝑒 3 𝑛 − 1 𝑥 ( 𝑡 ) , 𝐺 3 𝑛 , 𝑥 3 𝑛 , 𝑥 3 𝑛 + 1 , 𝐺 𝑥 , 𝑡 3 𝑛 + 1 , 𝑥 3 𝑛 + 1 , 𝑥 3 𝑛 + 2 𝑥 , 𝑡 , 𝐺 3 𝑛 − 1 , 𝑥 3 𝑛 − 1 , 𝑥 3 𝑛 𝑑 , 𝑡 ≥ m i n 3 𝑛 − 1 ( 𝑡 ) , 𝑒 3 𝑛 − 1 ( 𝑡 ) , 𝑑 3 𝑛 ( 𝑡 ) , 𝑑 3 𝑛 ( 𝑡 ) , 𝑑 3 𝑛 − 1 . ( 𝑡 ) ( 2 8 ) Thus, 𝑑 3 𝑛 ( 𝑘 𝑡 ) ≥ m i n { 𝑑 3 𝑛 − 1 ( 𝑡 ) , 𝑒 3 𝑛 − 1 ( 𝑡 ) } . Similarly, we have 𝑒 3 𝑛 ( 𝑘 𝑡 ) ≥ m i n 𝑑 3 𝑛 − 1 ( 𝑡 ) , 𝑒 3 𝑛 − 1 ( 𝑡 ) .
Thus, 𝑑 m i n 3 𝑛 ( 𝑘 𝑡 ) , 𝑒 3 𝑛 𝑑 ( 𝑘 𝑡 ) ≥ m i n 3 𝑛 − 1 ( 𝑡 ) , 𝑒 3 𝑛 − 1 ( 𝑡 ) . ( 2 9 ) Similarly, we can show that 𝑑 m i n 3 𝑛 + 1 ( 𝑘 𝑡 ) , 𝑒 3 𝑛 + 1 𝑑 ( 𝑘 𝑡 ) ≥ m i n 3 𝑛 ( 𝑡 ) , 𝑒 3 𝑛 , 𝑑 ( 𝑡 ) m i n 3 𝑛 + 2 ( 𝑘 𝑡 ) , 𝑒 3 𝑛 + 2 ( 𝑑 𝑘 𝑡 ) ≥ m i n 3 𝑛 + 1 ( 𝑡 ) , 𝑒 3 𝑛 + 1 ( . 𝑡 ) ( 3 0 ) Thus, 𝑑 m i n 𝑛 + 1 ( 𝑘 𝑡 ) , 𝑒 𝑛 + 1 𝑑 ( 𝑘 𝑡 ) ≥ m i n 𝑛 ( 𝑡 ) , 𝑒 𝑛 ( 𝑡 ) . ( 3 1 ) Hence 𝑑 m i n 𝑛 ( 𝑡 ) , 𝑒 𝑛 𝑑 ( 𝑡 ) ≥ m i n 𝑛 𝑡 𝑘 , 𝑒 𝑛 𝑡 𝑘 𝑑 ≥ m i n 𝑛 𝑡 𝑘 2 , 𝑒 𝑛 𝑡 𝑘 2 ⋮ 𝑑 ≥ m i n 0 𝑡 𝑘 𝑛 , 𝑒 0 𝑡 𝑘 𝑛 𝐺 𝑥 = m i n 0 , 𝑥 1 , 𝑥 2 , 𝑡 𝑘 𝑛 𝑦 , 𝐺 0 , 𝑦 1 , 𝑦 2 , 𝑡 𝑘 𝑛 . ( 3 2 ) Thus, 𝐺 𝑥 𝑛 , 𝑥 𝑛 + 1 , 𝑥 𝑛 + 2 𝐺 𝑥 , 𝑡 ≥ m i n 0 , 𝑥 1 , 𝑥 2 , 𝑡 𝑘 𝑛 𝑦 , 𝐺 0 , 𝑦 1 , 𝑦 2 , 𝑡 𝑘 𝑛 . ( 3 3 ) From ( 𝐺 3 ) , we have 𝐺 𝑥 𝑛 , 𝑥 𝑛 , 𝑥 𝑛 + 1 𝑥 , 𝑡 ≥ 𝐺 𝑛 , 𝑥 𝑛 + 1 , 𝑥 𝑛 + 2 𝐺 𝑥 , 𝑡 ≥ m i n 0 , 𝑥 1 , 𝑥 2 , 𝑡 𝑘 𝑛 𝑦 , 𝐺 0 , 𝑦 1 , 𝑦 2 , 𝑡 𝑘 𝑛 . ( 3 4 ) As in Theorem 18, we can show that { 𝑥 𝑛 } and { 𝑦 𝑛 } are 𝐺 -Cauchy sequences in 𝑋 . Since 𝑋 is 𝐺 -complete, there exist 𝑥 , 𝑦 ∈ 𝑋 such that 𝑥 𝑛 → 𝑥 and 𝑦 𝑛 → 𝑦 ∶ 𝐺 𝑆 ( 𝑥 , 𝑦 ) , 𝑥 3 𝑛 + 2 , 𝑥 3 𝑛 + 3 𝑥 , 𝑘 𝑡 = 𝐺 𝑆 ( 𝑥 , 𝑦 ) , 𝑇 3 𝑛 + 1 , 𝑦 3 𝑛 + 1 𝑥 , 𝑅 3 𝑛 + 2 , 𝑦 3 𝑛 + 2 𝐺 , 𝑘 𝑡 ≥ m i n 𝑥 , 𝑥 3 𝑛 + 1 , 𝑥 3 𝑛 + 2 , 𝑡 , 𝐺 𝑦 , 𝑦 3 𝑛 + 1 , 𝑦 3 𝑛 + 2 , 𝑥 , 𝑡 𝐺 ( 𝑥 , 𝑥 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑡 ) , 𝐺 3 𝑛 + 1 , 𝑥 3 𝑛 + 1 , 𝑥 3 𝑛 + 2 , 𝐺 𝑥 , 𝑡 3 𝑛 + 2 , 𝑥 3 𝑛 + 2 , 𝑥 3 𝑛 + 3 . , 𝑡 ( 3 5 ) Letting 𝑛 → ∞ , 𝐺 ( 𝑆 ( 𝑥 , 𝑦 ) , 𝑥 , 𝑥 , 𝑘 𝑡 ) ≥ m i n { 1 , 1 , 𝐺 ( 𝑥 , 𝑥 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑡 ) , 1 , 1 } = 𝐺 ( 𝑥 , 𝑥 , 𝑆 ( 𝑥 , 𝑦 ) , 𝑡 ) . ( 3 6 ) 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 ( 𝑥 1 , 𝑦 1 ) is another common coupled fixed point of 𝑆 , 𝑇 , and 𝑅 . Consider 𝐺 𝑥 , 𝑥 , 𝑥 1 𝑥 , 𝑘 𝑡 = 𝐺 𝑆 ( 𝑥 , 𝑦 ) , 𝑇 ( 𝑥 , 𝑦 ) , 𝑅 1 , 𝑦 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 𝐺 , 𝑡 , 1 , 1 , 1 = m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 . , 𝑡 ( 3 7 ) Also, 𝐺 𝑦 , 𝑦 , 𝑦 1 𝑦 , 𝑘 𝑡 = 𝐺 𝑆 ( 𝑦 , 𝑥 ) , 𝑇 ( 𝑦 , 𝑥 ) , 𝑅 1 , 𝑥 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 𝐺 , 𝑡 , 1 , 1 , 1 = m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 . , 𝑡 ( 3 8 ) Thus, 𝐺 m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑘 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 𝐺 , 𝑘 𝑡 ≥ m i n 𝑥 , 𝑥 , 𝑥 1 , 𝑡 , 𝐺 𝑦 , 𝑦 , 𝑦 1 . , 𝑡 ( 3 9 ) From Lemma 14, we have 𝑥 1 = 𝑥 and 𝑦 1 = 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 , 𝑘 𝑡 ) = 𝐺 ( 𝑆 ( 𝑥 , 𝑦 ) , 𝑇 ( 𝑥 , 𝑦 ) , 𝑅 ( 𝑦 , 𝑥 ) , 𝑘 𝑡 ) ≥ m i n { 𝐺 ( 𝑥 , 𝑥 , 𝑦 , 𝑡 ) 𝐺 ( 𝑦 , 𝑦 , 𝑥 , 𝑡 ) , 1 , 1 , 1 } = 𝐺 ( 𝑥 , 𝑥 , 𝑦 , 𝑡 ) . ( 4 0 ) 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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