Common Fixed Point Theorems for Six Mappings in Generalized Metric Spaces

Abstract

By using the weakly commutative and weakly compatible conditions of self-mapping pairs, we prove some new common fixed point theorems for six self-mappings in the framework of generalized metric spaces. An example is provided to support our result. The results presented in this paper generalize the well-known comparable results in the literature due to Abbas, Nazir, Saadati, Mustafa, and Sims.

1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain conditions has been at the center of vigorous research activity. In 2006, Mustafa and Sims [1] introduced a new structure of generalized metric spaces, which are called 𝐺-metric spaces as follows.

Definition 1.1 (see [1]). Let 𝑋 be a nonempty set and let 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ 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 𝑧≠𝑦;(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 (rectangle inequality). Then the function 𝐺 is called a generalized metric, or, more specifically, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Since then the fixed point theory in 𝐺-metric spaces has been studied and developed by authors (see [2–43]). Fixed point problems have also been considered in partially ordered 𝐺-metric spaces (see [44–56]).

The purpose of this paper is to use the concept of weakly commuting mappings and weakly compatible mappings to discuss some new common fixed point problem for six self-mappings in 𝐺-metric spaces. The results presented in this paper extend and improve the corresponding results of Abbas et al. [2] and Mustafa and Sims [3].

We now recall some of the basic concepts and results in 𝐺-metric spaces.

Proposition 1.2 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, then the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in three of its variables.

Definition 1.3 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, and let (𝑥_𝑛) be a sequence of points of 𝑋. A point 𝑥∈𝑋 is said to be the limit of the sequence (𝑥_𝑛), if lim_{𝑛,𝑚→+∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and we say that the sequence (𝑥_𝑛) is 𝐺-convergent to 𝑥 or  (𝑥_𝑛)𝐺-convergent to 𝑥, that is, for any 𝜖>0, there exists 𝑁∈ℕ such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜖 for all 𝑚,𝑛≥𝑁(throughout this paper we mean by ℕ the set of all natural numbers).

Proposition 1.4 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, then 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_𝑛,𝑥_𝑚,𝑥)→0 as 𝑛,𝑚→+∞.

Definition 1.5 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. A sequence (𝑥_𝑛) is called 𝐺-cauchy if for every 𝜖>0, there is 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜖 for all 𝑚,𝑛,𝑙≥𝑁, that is 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→+∞.

Proposition 1.6 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, then the following are equivalent:(i)the sequence (𝑥_𝑛) is 𝐺-cauchy;(ii)for every 𝜖>0, there is 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜖 for all 𝑚,𝑛≥𝑁.

Definition 1.7 (see [1]). A 𝐺-metric space (𝑋,𝐺) is 𝐺-complete if every 𝐺-cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in 𝑋.

Definition 1.8 (see [1]). Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be 𝐺-metric spaces, and let 𝑓∶(𝑋,𝐺)→(𝑋^′,𝐺^′) be a function. Then 𝑓 is said to be 𝐺-continuous at a point 𝑎∈𝑋 if and only if for every 𝜖>0, there is 𝛿>0 such that 𝑥,𝑦∈𝑋, and 𝐺(𝑎,𝑥,𝑦)<𝛿 implies 𝐺^′(𝑓(𝑎),𝑓(𝑥),𝑓(𝑦))<𝜖. A function 𝑓 is 𝐺-continuous at 𝑋 if and only if it is 𝐺-continuous at all 𝑎∈𝑋.

Proposition 1.9 (see [1]). Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be 𝐺-metric spaces, then a function 𝑓∶𝑋→𝑋^′ is 𝐺-continuous at a point 𝑥∈𝑋 if and only if it is 𝐺-sequentially continuous at 𝑥, that is, whenever (𝑥_𝑛) is 𝐺-convergent to 𝑥, (𝑓(𝑥_𝑛)) is 𝐺-convergent to 𝑓(𝑥).

Definition 1.10 (see [4]). Two self-mappings 𝑓 and 𝑔 of a 𝐺-metric space (𝑋,𝐺) are said to be weakly commuting if 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)≤𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥) for all 𝑥 in 𝑋.

Definition 1.11 (see [4]). Let 𝑓 and 𝑔 be two self-mappings from a 𝐺-metric space (𝑋,𝐺) into itself. Then the mappings 𝑓 and 𝑔 are said to be weakly compatible if 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)=0 whenever 𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥)=0.

Proposition 1.12 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, for all 𝑥,𝑦,𝑧,𝑎 in 𝑋, it follows that(i)if 𝐺(𝑥,𝑥,𝑦)=0, then 𝑥=𝑦=𝑧;(ii)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑥,𝑦)+𝐺(𝑥,𝑥,𝑧);(iii)𝐺(𝑥,𝑦,𝑦)≤2𝐺(𝑦,𝑥,𝑥);(iv)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧);(v)𝐺(𝑥,𝑦,𝑧)≤(2/3)(𝐺(𝑥,𝑦,𝑎)+𝐺(𝑥,𝑎,𝑧)+𝐺(𝑎,𝑦,𝑧)); (vi)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑦,𝑎,𝑎)+𝐺(𝑧,𝑎,𝑎).

2. Common Fixed Point Theorems

Theorem 2.1. Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 be six mappings of 𝑋 into itself satisfying the following conditions:(i)𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), ℎ(𝑋)⊂𝐴(𝑋);(ii)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, 𝐺𝐺,𝐺,𝐺()𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑘max𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧 (2.1) or 𝐺𝐺,𝐺,𝐺()𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑘max𝐴𝑥,𝐴𝑥,𝑔𝑦+𝐺𝐵𝑦,𝐵𝑦,𝑓𝑥𝐵𝑦,𝐵𝑦,ℎ𝑧+𝐺𝐶𝑧,𝐶𝑧,𝑔𝑦𝐶𝑧,𝐶𝑧,𝑓𝑥+𝐺𝐴𝑥,𝐴𝑥,ℎ𝑧, (2.2) where 𝑘∈[0,1/3). Then one of the pairs (𝑓,𝐴), (𝑔,𝐵), and (ℎ,𝐶) has a coincidence point in 𝑋. Further, if one of the following conditions is satisfied, then the mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 have a unique common fixed point in 𝑋.(a) Either 𝑓 or 𝐴 is 𝐺-continuous, the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible;(b) Either 𝑔 or 𝐵 is 𝐺-continuous, the pair (𝑔,𝐵) is weakly commutative, the pairs (𝑓,𝐴) and (ℎ,𝐶) are weakly compatible;(c) Either ℎ or 𝐶 is 𝐺-continuous, the pair (ℎ,𝐶) is weakly commutative, the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible.

Proof. Suppose that mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 satisfy condition (2.1).
Let 𝑥₀ in 𝑋 be arbitrary point, since (𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), ℎ(𝑋)⊂𝐴(𝑋), there exist the sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑦_3𝑛=𝑓𝑥_3𝑛=𝐵𝑥_3𝑛+1,𝑦_3𝑛+1=𝑔𝑥_3𝑛+1=𝐶𝑥_3𝑛+2,𝑦_3𝑛+2=ℎ𝑥_3𝑛+2=𝐴𝑥_3𝑛+3, (2.3) for 𝑛=0,1,2,….
If 𝑦_𝑛=𝑦_𝑛+1 for some 𝑛, with 𝑛=3𝑚, then 𝑝=𝑥_3𝑚+1 is a coincidence point of the pair (𝑔,𝐵); if 𝑦_𝑛+1=𝑦_𝑛+2 for some 𝑛, with 𝑛=3𝑚, then 𝑝=𝑥_3𝑚+2 is a coincidence point of the pair (ℎ,𝐶); if 𝑦_𝑛+2=𝑦_𝑛+3 for some 𝑛, with 𝑛=3𝑚, then 𝑝=𝑥_3𝑚+3 is a coincidence point of the pair (𝑓,𝐴). Without loss of generality, we can assume that 𝑦_𝑛≠𝑦_𝑛+1 for all 𝑛=0,1,2….
Now we prove that {𝑦_𝑛} is a 𝐺-cauchy sequence in 𝑋.
Actually, using the condition (2.1) and (𝐺₃), we have 𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1=𝐺𝑓𝑥_3𝑛,𝑔𝑥_3𝑛+1,ℎ𝑥_3𝑛−1𝐺≤𝑘max𝐴𝑥_3𝑛,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1+𝐺𝐵𝑥_3𝑛+1,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛,𝐺𝐵𝑥_3𝑛+1,ℎ𝑥_3𝑛−1,ℎ𝑥_3𝑛−1+𝐺𝐶𝑥_3𝑛−1,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1,𝐺𝐶𝑥_3𝑛−1,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛+𝐺𝐴_3𝑛,ℎ𝑥_3𝑛−1,ℎ𝑥_3𝑛−1𝐺𝑦=𝑘max^3𝑛−1,𝑦_3𝑛+1,𝑦_3𝑛+1+𝐺𝑦_3𝑛,𝑦_3𝑛,𝑦_3𝑛,𝐺𝑦_3𝑛,𝑦_3𝑛−1,𝑦_3𝑛−1+𝐺𝑦_3𝑛−2,𝑦_3𝑛+1,𝑦_3𝑛+1,𝐺𝑦_3𝑛−2,𝑦_3𝑛,𝑦_3𝑛+𝐺𝑦_3𝑛−1,𝑦_3𝑛−1,𝑦_3𝑛−1𝐺𝑦≤𝑘max^3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1,𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛,𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1+𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛−1+𝐺𝑦_3𝑛−1,𝑦_3𝑛+1,𝑦_3𝑛+1𝐺𝑦≤𝑘max^3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1,𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛,2𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1+𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛=𝑘2𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1+𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛, (2.4) which further implies that (1−2𝑘)𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1≤𝑘𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛. (2.5) Thus 𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1≤𝜆𝐺𝑦_3𝑛−2,𝑦_3𝑛−1,𝑦_3𝑛, (2.6) where 𝜆=𝑘/(1−2𝑘). Obviously 0≤𝜆<1.
Similarly it can be shown that 𝐺𝑦_3𝑛,𝑦_3𝑛+1,𝑦_3𝑛+2≤𝜆𝐺𝑦_3𝑛−1,𝑦_3𝑛,𝑦_3𝑛+1,𝐺𝑦_3𝑛+1,𝑦_3𝑛+2,𝑦_3𝑛+3≤𝜆𝐺𝑦_3𝑛,𝑦_3𝑛+1,𝑦_3𝑛+2. (2.7) It follows from (2.6) and (2.7) that, for all 𝑛∈ℕ, 𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+2≤𝜆𝐺𝑦_𝑛−1,𝑦_𝑛,𝑦_𝑛+1≤𝜆²𝐺𝑦_𝑛−2,𝑦_𝑛−1,𝑦_𝑛≤⋯≤𝜆^𝑛𝐺𝑦₀,𝑦₁,𝑦₂. (2.8) Therefore, for all 𝑛,𝑚∈ℕ, 𝑛<𝑚, by (𝐺₃) and (𝐺₅) we have 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚≤𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+1+𝐺𝑦_𝑛+1,𝑦_𝑛+2,𝑦_𝑛+2+𝐺𝑦_𝑛+2,𝑦_𝑛+3,𝑦_𝑛+3+⋯+𝐺𝑦_𝑚−1,𝑦_𝑚,𝑦_𝑚≤𝐺𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+2+𝐺𝑦_𝑛+1,𝑦_𝑛+2,𝑦_𝑛+3+⋯+𝐺𝑦_𝑚−1,𝑦_𝑚,𝑦_𝑚+1≤𝜆^𝑛+𝜆^𝑛+1+𝜆^𝑛+2+⋯+𝜆^𝑚−1𝐺𝑦₀,𝑦₁,𝑦₂≤𝜆^𝑛𝐺𝑦1−𝜆⁰,𝑦₁,𝑦₂→0,as𝑛→∞. (2.9) Hence {𝑦_𝑛} is a 𝐺-cauchy sequence in 𝑋, since 𝑋 is complete 𝐺-metric space, there exists a point 𝑢∈𝑋 such that 𝑦_𝑛→𝑢(𝑛→∞).
Since the sequences {𝑓𝑥_3𝑛}={𝐵𝑥_3𝑛+1}, {𝑔𝑥_3𝑛+1}={𝐶𝑥_3𝑛+2}, and {ℎ𝑥_3𝑛−1}={𝐴𝑥_3𝑛} are all subsequences of {𝑦_𝑛}, then they all converge to 𝑢, that is, 𝑦_3𝑛=𝑓𝑥_3𝑛=𝐵𝑥_3𝑛+1→𝑢,𝑦_3𝑛+1=𝑔𝑥_3𝑛+1=𝐶𝑥_3𝑛+2𝑦→𝑢,_3𝑛−1=ℎ𝑥_3𝑛−1=𝐴𝑥_3𝑛→𝑢(𝑛→∞). (2.10)
Now we prove that 𝑢 is a common fixed point of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 under the condition (a).
First, we suppose that 𝐴 is continuous, the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible.
Step  1. We prove that 𝑢=𝑓𝑢=𝐴𝑢.
By (2.10) and weakly commutativity of mapping pair (𝑓,𝐴), we have 𝐺𝑓𝐴𝑥_3𝑛,𝐴𝑓𝑥_3𝑛,𝐴𝑓𝑥_3𝑛≤𝐺𝑓𝑥_3𝑛,𝐴𝑥_3𝑛,𝐴𝑥_3𝑛→0(𝑛→∞). (2.11) Since 𝐴 is continuous, then 𝐴²𝑥_3𝑛→𝐴𝑢(𝑛→∞), 𝐴𝑓𝑥_3𝑛→𝐴𝑢(𝑛→∞). By (2.11), we know that 𝑓𝐴𝑥_3𝑛→𝐴𝑢(𝑛→∞).
From the condition (2.1), we get 𝐺𝑓𝐴𝑥_3𝑛,𝑔𝑥_3𝑛+1,ℎ𝑥_3𝑛+2𝐺𝐴≤𝑘max²𝑥_3𝑛,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1+𝐺𝐵𝑥_3𝑛+1,𝑓𝐴𝑥_3𝑛,𝑓𝐴𝑥_3𝑛,𝐺𝐵𝑥_3𝑛+1,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1,𝐺𝐶𝑥_3𝑛+2,𝑓𝐴𝑥_3𝑛,𝑓𝐴𝑥_3𝑛+𝐺𝐴²𝑥_3𝑛,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.12) Letting 𝑛→∞ and using the Proposition 1.12(iii), we have 𝐺()𝐺()(),𝐺()(),𝐺()()[𝐺()(().𝐴𝑢,𝑢,𝑢≤𝑘max𝐴𝑢,𝑢,𝑢+𝐺𝑢,𝐴𝑢,𝐴𝑢𝑢,𝑢,𝑢+𝐺𝑢,𝑢,𝑢𝑢,𝐴𝑢,𝐴𝑢+𝐺𝐴𝑢,𝑢,𝑢=𝑘𝐴𝑢,𝑢,𝑢+𝐺𝑢,𝐴𝑢,𝐴𝑢)]≤3𝑘𝐺𝐴𝑢,𝑢,𝑢 (2.13) Hence, 𝐺(𝐴𝑢,𝑢,𝑢)=0 and 𝐴𝑢=𝑢, since 0≤𝑘<1/3.
Again by using condition (2.1), we have 𝐺𝑓𝑢,𝑔𝑥_3𝑛+1,ℎ𝑥_3𝑛+2𝐺≤𝑘max𝐴𝑢,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1+𝐺𝐵𝑥_3𝑛+1,𝐺,𝑓𝑢,𝑓𝑢𝐵𝑥_3𝑛+1,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1,𝐺𝐶𝑥_3𝑛+2,𝑓𝑢,𝑓𝑢+𝐺𝐴𝑢,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.14) Letting 𝑛→∞, we have 𝐺𝑓𝑢,𝑢,𝑢≤𝑘𝐺𝑢,𝑓𝑢,𝑓𝑢. (2.15) From the Proposition 1.12(iii), we get 𝐺𝑓𝑢,𝑢,𝑢≤𝑘𝐺𝑢,𝑓𝑢,𝑓𝑢≤2𝑘𝐺𝑓𝑢,𝑢,𝑢. (2.16)
Hence, 𝐺(𝑓𝑢,𝑢,𝑢)=0 and 𝑓𝑢=𝑢, since 0≤𝑘<1/3.
So we have 𝑢=𝐴𝑢=𝑓𝑢.
Step  2. We prove that 𝑢=𝑔𝑢=𝐵𝑢.
Since 𝑓(𝑋)⊂𝐵(𝑋) and 𝑢=𝑓𝑢∈𝑓(𝑋), there is a point 𝑣∈𝑋 such that 𝑢=𝑓𝑢=𝐵𝑣. Again by using condition (2.1), we have 𝐺𝑓𝑢,𝑔𝑣,ℎ𝑥_3𝑛+2𝐺,≤𝑘max𝐺𝐴𝑢,𝑔𝑣,𝑔𝑣+𝐺𝐵𝑣,𝑓𝑢,𝑓𝑢𝐵𝑣,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝐺,𝑔𝑣,𝑔𝑣𝐶𝑥_3𝑛+2,𝑓𝑢,𝑓𝑢+𝐺𝐴𝑢,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.17) Letting 𝑛→∞, using 𝑢=𝐴𝑢=𝑓𝑢 and the Proposition 1.12(iii), we obtain 𝐺𝑢,𝑔𝑣,𝑢≤𝑘𝐺𝑢,𝑔𝑣,𝑔𝑣≤2𝑘𝐺𝑢,𝑔𝑣,𝑢. (2.18) Hence, 𝐺(𝑢,𝑔𝑣,𝑢)=0and so 𝑔𝑣=𝑢=𝐵𝑣, since 0≤𝑘<1/3.
Since the pair (𝑔,𝐵) is weakly compatible, we have 𝑔𝑢=𝑔𝐵𝑣=𝐵𝑔𝑣=𝐵𝑢. (2.19) Again by using condition (2.1), we have 𝐺𝑓𝑢,𝑔𝑢,ℎ𝑥_3𝑛+2𝐺,≤𝑘max𝐺𝐴𝑢,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑢,𝑓𝑢𝐵𝑢,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝐺,𝑔𝑢,𝑔𝑢𝐶𝑥_3𝑛+2,𝑓𝑢,𝑓𝑢+𝐺𝐴𝑢,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.20) Letting 𝑛→∞, using 𝑢=𝐴𝑢=𝑓𝑢, 𝑔𝑢=𝐵𝑢 and the Proposition 1.12(iii), we have 𝐺𝑢,𝑔𝑢,𝑢≤𝑘𝐺𝑢,𝑔𝑢,𝑔𝑢+𝐺𝑢,𝑔𝑢,𝑢≤3𝑘𝐺𝑢,𝑔𝑢,𝑢. (2.21) Hence, 𝐺(𝑢,𝑔𝑢,𝑢)=0 and so 𝑢=𝑔𝑢=𝐵𝑢, since 0≤𝑘<1/3.
So we have 𝑢=𝑔𝑢=𝐵𝑢.
Step  3. We prove that 𝑢=ℎ𝑢=𝐶𝑢.
Since 𝑔(𝑋)⊂𝐶(𝑋) and 𝑢=𝑔𝑢∈𝑔(𝑋), there is a point 𝑤∈𝑋 such that 𝑢=𝑔𝑢=𝐶𝑤. Again by using condition (2.1), we have 𝐺𝐺,𝐺(),𝐺()𝑓𝑢,𝑔𝑢,ℎ𝑤≤𝑘max𝐴𝑢,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑢,𝑓𝑢𝐵𝑢,ℎ𝑤,ℎ𝑤+𝐺𝐶𝑤,𝑔𝑢,𝑔𝑢𝐶𝑤,𝑓𝑢,𝑓𝑢+𝐺𝐴𝑢,ℎ𝑤,ℎ𝑤. (2.22) Using 𝑢=𝐴𝑢=𝑓𝑢, 𝑢=𝑔𝑢=𝐵𝑢=𝐶𝑤 and the Proposition 1.12(iii), we obtain 𝐺(𝑢,𝑢,ℎ𝑤)≤𝑘𝐺(𝑢,ℎ𝑤,ℎ𝑤)≤2𝑘𝐺(𝑢,𝑢,ℎ𝑤). (2.23) Hence, 𝐺(𝑢,𝑢,ℎ𝑤)=  0 and so ℎ𝑤=𝑢=𝐶𝑤, since 0≤𝑘<1/3.
Since the pair (ℎ,𝐶) is weakly compatible, we have ℎ𝑢=ℎ𝐶𝑤=𝐶ℎ𝑤=𝐶𝑢. (2.24) Again by using condition (2.1), we have 𝐺𝐺,𝐺(),𝐺()𝑓𝑢,𝑔𝑢,ℎ𝑢≤𝑘max𝐴𝑢,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑢,𝑓𝑢𝐵𝑢,ℎ𝑢,ℎ𝑢+𝐺𝐶𝑢,𝑔𝑢,𝑔𝑢𝐶𝑢,𝑓𝑢,𝑓𝑢+𝐺𝐴𝑢,ℎ𝑢,ℎ𝑢. (2.25) Using 𝑢=𝐴𝑢=𝑓𝑢, 𝑢=𝑔𝑢=𝐵𝑢, 𝐶𝑢=ℎ𝑢 and the Proposition 1.12(iii), we have 𝐺(𝑢,𝑢,ℎ𝑢)≤𝑘[𝐺(𝑢,ℎ𝑢,ℎ𝑢)+𝐺(ℎ𝑢,𝑢,𝑢)]≤3𝑘𝐺(𝑢,𝑢,ℎ𝑢). (2.26) Hence, 𝐺(𝑢,𝑢,ℎ𝑢)=  0 and so 𝑢=ℎ𝑢=𝐶𝑢, since 0≤𝑘<1/3.
Therefore, 𝑢 is the common fixed point of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 when 𝐴 is continuous and the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible.
Next, we suppose that 𝑓 is continuous, the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible.
Step  1. We prove that 𝑢=𝑓𝑢.
By (2.10) and weak commutativity of mapping pair (𝑓,𝐴), we have 𝐺𝑓𝐴𝑥_3𝑛,𝐴𝑓𝑥_3𝑛,𝐴𝑓𝑥_3𝑛≤𝐺𝑓𝑥_3𝑛,𝐴𝑥_3𝑛,𝐴𝑥_3𝑛→0(𝑛→∞). (2.27) Since 𝑓 is continuous, then 𝑓²𝑥_3𝑛→𝑓𝑢(𝑛→∞), 𝑓𝐴𝑥_3𝑛→𝑓𝑢(𝑛→∞). By (2.10), we know that 𝐴𝑓𝑥_3𝑛→𝑓𝑢(𝑛→∞).
From the condition (2.1), we have 𝐺𝑓²𝑥_3𝑛,𝑔𝑥_3𝑛+1,ℎ𝑥_3𝑛+2𝐺≤𝑘max𝐴𝑓𝑥_3𝑛,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1+𝐺𝐵𝑥_3𝑛+1,𝑓²𝑥_3𝑛,𝑓²𝑥_3𝑛,𝐺𝐵𝑥_3𝑛+1,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝑔𝑥_3𝑛+1,𝑔𝑥_3𝑛+1,𝐺𝐶𝑥_3𝑛+2,𝑓²𝑥_3𝑛,𝑓²𝑥_3𝑛+𝐺𝐴𝑓𝑥_3𝑛,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.28) Letting 𝑛→∞ and noting the Proposition 1.12(iii), we have 𝐺𝐺,𝐺()(),𝐺𝐺.𝑓𝑢,𝑢,𝑢≤𝑘max𝑓𝑢,𝑢,𝑢+𝐺𝑢,𝑓𝑢,𝑓𝑢𝑢,𝑢,𝑢+𝐺𝑢,𝑢,𝑢𝑢,𝑓𝑢,𝑓𝑢+𝐺𝑓𝑢,𝑢,𝑢=𝑘𝑓𝑢,𝑢,𝑢+𝐺𝑢,𝑓𝑢,𝑓𝑢≤3𝑘𝐺𝑓𝑢,𝑢,𝑢 (2.29) Hence, 𝐺(𝑓𝑢,𝑢,𝑢)=  0 and so 𝑓𝑢=𝑢, since 0≤𝑘<1/3.
Step  2. We prove that 𝑢=𝑔𝑢=𝐵𝑢.
Since 𝑓(𝑋)⊂𝐵(𝑋) and 𝑢=𝑓𝑢∈𝑓(𝑋), there is a point 𝑧∈𝑋 such that 𝑢=𝑓𝑢=𝐵𝑧. Again by using condition (2.1), we have 𝐺𝑓²𝑥_3𝑛,𝑔𝑧,ℎ𝑥_3𝑛+2𝐺≤𝑘max𝐴𝑓𝑥_3𝑛,𝑔𝑧,𝑔𝑧+𝐺𝐵𝑧,𝑓²𝑥_3𝑛,𝑓²𝑥_3𝑛,𝐺𝐵𝑧,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝐺,𝑔𝑧,𝑔𝑧𝐶𝑥_3𝑛+2,𝑓²𝑥_3𝑛,𝑓²𝑥_3𝑛+𝐺𝐴𝑓𝑥_3𝑛,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.30) Letting 𝑛→∞and using 𝑢=𝑓𝑢 and the Proposition 1.12(iii), we have 𝐺𝑢,𝑔𝑧,𝑢≤𝑘𝐺𝑢,𝑔𝑧,𝑔𝑧≤2𝑘𝐺𝑢,𝑔𝑧,𝑢. (2.31) Hence 𝐺(𝑢,𝑔𝑧,𝑢)=  0 and so 𝑔𝑧=𝑢=𝐵𝑧, since 0≤𝑘<1/3.
Since the pair (𝑔,𝐵) is weakly compatible, we have 𝑔𝑢=𝑔𝐵𝑧=𝐵𝑔𝑧=𝐵𝑢. (2.32) Again by using condition (2.1), we have 𝐺𝑓𝑥_3𝑛,𝑔𝑢,ℎ𝑥_3𝑛+2𝐺≤𝑘max𝐴𝑥_3𝑛,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛,𝐺𝐵𝑢,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2+𝐺𝐶𝑥_3𝑛+2,𝐺,𝑔𝑢,𝑔𝑢𝐶𝑥_3𝑛+2,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛+𝐺𝐴𝑥_3𝑛,ℎ𝑥_3𝑛+2,ℎ𝑥_3𝑛+2. (2.33) Letting 𝑛→∞and using 𝑢=𝑓𝑢, 𝑔𝑢=𝐵𝑢 and the Proposition 1.12(iii), we have 𝐺𝑢,𝑔𝑢,𝑢≤𝑘𝐺𝑢,𝑔𝑢,𝑔𝑢+𝐺𝑔𝑢,𝑢,𝑢≤3𝑘𝐺𝑢,𝑔𝑢,𝑢. (2.34) Hence, 𝐺(𝑢,𝑔𝑢,𝑢)=0 and so 𝑔𝑢=𝑢=𝐵𝑢, since 0≤𝑘<1/3.
So we have 𝑢=𝑔𝑢=𝐵𝑢.
Step  3. We prove that 𝑢=ℎ𝑢=𝐶𝑢.
Since 𝑔(𝑋)⊂𝐶(𝑋) and 𝑢=𝑔𝑢∈𝑔(𝑋), there is a point 𝑡∈𝑋 such that 𝑢=𝑔𝑢=𝐶𝑡. Again by using condition (2.1), we have 𝐺𝑓𝑥_3𝑛𝐺,𝑔𝑢,ℎ𝑡≤𝑘max𝐴𝑥_3𝑛,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛,𝐺(),𝐺𝐵𝑢,ℎ𝑡,ℎ𝑡+𝐺𝐶𝑡,𝑔𝑢,𝑔𝑢𝐶𝑡,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛+𝐺𝐴𝑥_3𝑛,ℎ𝑡,ℎ𝑡. (2.35) Letting 𝑛→∞and using 𝑢=𝑔𝑢=𝐵𝑢 and the Proposition 1.12(iii), we obtain 𝐺(𝑢,𝑢,ℎ𝑡)≤𝑘𝐺(𝑢,ℎ𝑡,ℎ𝑡)≤2𝑘𝐺(𝑢,𝑢,ℎ𝑡). (2.36) Hence, 𝐺(𝑢,𝑢,ℎ𝑡)=0 and so ℎ𝑡=𝑢=𝐶𝑡, since 0≤𝑘<1/3.
Since the pair (ℎ,𝐶) is weakly compatible, we have ℎ𝑢=ℎ𝐶𝑡=𝐶ℎ𝑡=𝐶𝑢. (2.37) Again by using condition (2.1), we have 𝐺𝑓𝑥_3𝑛𝐺,𝑔𝑢,ℎ𝑢≤𝑘max𝐴𝑥_3𝑛,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛,𝐺(),𝐺𝐵𝑢,ℎ𝑢,ℎ𝑢+𝐺𝐶𝑢,𝑔𝑢,𝑔𝑢𝐶𝑢,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛+𝐺𝐴𝑥_3𝑛,ℎ𝑢,ℎ𝑢. (2.38) Letting 𝑛→∞and using 𝑢=𝑔𝑢=𝐵𝑢 and the Proposition 1.12(iii), we have 𝐺(𝑢,𝑢,ℎ𝑢)≤𝑘[𝐺(𝑢,ℎ𝑢,ℎ𝑢)+𝐺(𝑢,𝑢,ℎ𝑢)]≤3𝑘𝐺(𝑢,𝑢,ℎ𝑢). (2.39) Hence, 𝐺(𝑢,𝑢,ℎ𝑢)=0 and so ℎ𝑢=𝑢=𝐶𝑢, since 0≤𝑘<1/3.
Step  4. We prove that 𝑢=𝐴𝑢.
Since ℎ(𝑋)⊂𝐴(𝑋) and 𝑢=ℎ𝑢∈ℎ(𝑋), there is a point 𝑝∈𝑋 such that 𝑢=ℎ𝑢=𝐴𝑝. Again by using condition (2.1), we have 𝐺𝐺,𝐺(),𝐺𝑓𝑝,𝑔𝑢,ℎ𝑢≤𝑘max𝐴𝑝,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑝,𝑓𝑝𝐵𝑢,ℎ𝑢,ℎ𝑢+𝐺𝐶𝑢,𝑔𝑢,𝑔𝑢𝐶𝑢,𝑓𝑝,𝑓𝑝+𝐺𝐴𝑝,ℎ𝑢,ℎ𝑢. (2.40) Using 𝑢=𝑔𝑢=𝐵𝑢, 𝑢=ℎ𝑢=𝐶𝑢, and the Proposition 1.12(iii), we obtain 𝐺𝑓𝑝,𝑢,𝑢≤𝑘𝐺𝑢,𝑓𝑝,𝑓𝑝≤2𝑘𝐺𝑓𝑝,𝑢,𝑢. (2.41) Hence 𝐺(𝑓𝑝,𝑢,𝑢)=0 and 𝑓𝑝=𝑢=𝐴𝑝, since 0≤𝑘<1/3.
Since the pair (𝑓,𝐴) is weakly compatible, we have 𝑓𝑢=𝑓𝐴𝑝=𝐴𝑓𝑝=𝐴𝑢=𝑢. (2.42) Therefore, 𝑢 is the common fixed point of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 when 𝑓 is continuous and the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible.
Similarly, we can prove the result that 𝑢 is a common fixed point of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 when under the condition of (b) or (c).
Finally, we prove uniqueness of common fixed point 𝑢.
Let 𝑢 and 𝑞 be two common fixed points of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶, by using condition (2.1), we have 𝐺𝐺,𝐺(),𝐺𝐺.𝑞,𝑢,𝑢=𝐺𝑓𝑞,𝑔𝑢,ℎ𝑢≤𝑘max𝐴𝑞,𝑔𝑢,𝑔𝑢+𝐺𝐵𝑢,𝑓𝑞,𝑓𝑞𝐵𝑢,ℎ𝑢,ℎ𝑢+𝐺𝐶𝑢,𝑔𝑢,𝑔𝑢𝐶𝑢,𝑓𝑞,𝑓𝑞+𝐺𝐴𝑞,ℎ𝑢,ℎ𝑢=𝑘𝑞,𝑢,𝑢+𝐺𝑢,𝑞,𝑞≤3𝑘𝐺𝑞,𝑢,𝑢 (2.43) Hence, 𝐺(𝑞,𝑢,𝑢)=0 and so 𝑞=𝑢, since 0≤𝑘<1/3.Thus common fixed point is unique.
The proof using (2.2) is similar. This completes the proof.

Now we introduce an example to support Theorem 2.1.

Example 2.2. Let 𝑋=[0,1], and let (𝑋,𝐺) be a 𝐺-metric space defined by 𝐺(𝑥,𝑦,𝑧) = |𝑥−𝑦|+|𝑦−𝑧|+|𝑧−𝑥| for all 𝑥,𝑦,and𝑧 in 𝑋. Let 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 be self-mappings defined by 1𝑓𝑥=1,𝑥∈0,2,781,𝑥∈2,1,𝑔𝑥=10111,𝑥∈0,2,781,𝑥∈2,9,1ℎ𝑥=110,𝑥∈0,2,781,𝑥∈2,1,1𝐴𝑥=𝑥,𝐵𝑥=1,𝑥∈0,2,781,𝑥∈2,1,10,𝑥=1,𝐶𝑥=1,𝑥∈0,2,781,𝑥∈2,1,1011,𝑥=1. (2.44) Note that 𝐴 is 𝐺-continuous in 𝑋, and 𝑓, 𝑔, ℎ, 𝐵, and 𝐶 are not 𝐺-continuous in 𝑋.

Clearly we can get 𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋),  and ℎ(𝑋)⊂𝐴(𝑋).

Actually, since 𝑓𝑋={7/8,1}, 𝐵𝑋={0,7/8,1}, 𝑔𝑋={7/8,10/11}, 𝐶𝑋={7/8,10/11,1}, ℎ𝑋 = {7/8,9/10},  and 𝐴𝑋=𝑋=[0,1], so we know 𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), and ℎ(𝑋)⊂𝐴(𝑋).

By the definition of the mappings of 𝑓 and 𝐴, for all 𝑥∈[0,1], 𝐺(𝑓𝐴𝑥,𝐴𝑓𝑥,𝐴𝑓𝑥) = 𝐺(𝑓𝑥,𝑓𝑥,𝑓𝑥)=0≤𝐺(𝑓𝑥,𝐴𝑥,𝐴𝑥), so we can get the pair (𝑓,𝐴) is weakly commuting.

By the definition of the mappings of 𝑔 and 𝐵, only for 𝑥∈(1/2,1), 𝑔𝑥=𝐵𝑥=7/8, at this time 𝑔𝐵𝑥=𝑔(7/8) = 7/8 = 𝐵(7/8) = 𝐵𝑔𝑥, so 𝑔𝐵𝑥=𝐵𝑔𝑥, so we can obtain that the pair (𝑔,𝐵) is weakly compatible. Similarly, we can show that the pair (ℎ,𝐶) is also weakly compatible.

Now we proof that the mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 are satisfying the condition (2.1) of Theorem 2.1 with 𝑘=5/16∈[0,1/3). Let 𝑀𝐺,𝐺,𝐺()𝑥,𝑦,𝑧=max𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧. (2.45)

Case 1. If 𝑥,𝑦,𝑧∈[0,1/2], then 𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,10,911=1105,𝐺𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥=𝐺𝑥,10,1110()11+𝐺1,1,1=2𝑥−10≥911.11 (2.46) Thus, we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧5<5⋅916𝐺.11≤𝑘𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥≤𝑘𝑀𝑥,𝑦,𝑧 (2.47)

Case 2. If 𝑥,𝑦∈[0,1/2],𝑧∈(1/2,1], then 𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,10,7118=14,𝐺𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥=𝐺𝑥,10,1110()11+𝐺1,1,1=2𝑥−10≥911.11 (2.48) Therefore, we get 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧4<5⋅916𝐺.11≤𝑘𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥≤𝑘𝑀𝑥,𝑦,𝑧 (2.49)

Case 3. If 𝑥,𝑧∈[0,1/2],𝑦∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,8,9=1104,𝐺()()9𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧=𝐺1,1,1+𝐺𝑥,,910910=2𝑥−≥4105. (2.50) Hence, we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧4=5⋅4165≤𝑘𝐺𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺(𝐴𝑥,ℎ𝑧,ℎ𝑧)≤𝑘𝑀𝑥,𝑦,𝑧. (2.51)

Case 4. If 𝑦,𝑧∈[0,1/2],𝑥∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺8,10,911=310,𝐺944𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦=𝐺1,,91010+𝐺1,10,1110=1121.55 (2.52) So we get 𝐺=3𝑓𝑥,𝑔𝑦,ℎ𝑧<544⋅1621𝐺.55≤𝑘𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦≤𝑘𝑀𝑥,𝑦,𝑧 (2.53)

Case 5. If 𝑥∈[0,1/2],𝑦,𝑧∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺1,8,78=14. (2.54) If 𝑦∈(1/2,1), then 𝐺7𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥=𝐺𝑥,8,787+𝐺87,1,1=2𝑥−8+14≥34+14=1. (2.55) If 𝑦=1, then 𝐺7𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥=𝐺𝑥,8,78()7+𝐺0,1,1=2𝑥−83+2≥4+2=114. (2.56) And so we have 𝐺𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥≥1 (2.57) for all 𝑦∈(1/2,1]. Hence we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧4<5𝐺.16⋅1≤𝑘𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥≤𝑘𝑀𝑥,𝑦,𝑧 (2.58)

Case 6. If 𝑦∈[0,1/2],𝑥,𝑧∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺8,10,7118=3,𝐺44𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥=𝐺𝑥,10,1110711+𝐺1,8,78=2𝑥−10+1114≥14. (2.59) Thus, we have 𝐺=3𝑓𝑥,𝑔𝑦,ℎ𝑧<544⋅1164≤𝑘𝐺𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥≤𝑘𝑀𝑥,𝑦,𝑧. (2.60)

Case 7. If 𝑧∈[0,1/2],𝑥,𝑦∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺8,78,9=110.20 (2.61) If 𝑦∈(1/2,1), then 𝐺7𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦=𝐺8,9,910710+𝐺1,8,78=3.10 (2.62) If 𝑦=1, then 𝐺9𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦=𝐺0,,910710+𝐺1,8,78=41.20 (2.63) And so we have 𝐺≥3𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦10 (2.64) for all 𝑦∈(1/2,1]. Hence we have 𝐺=1𝑓𝑥,𝑔𝑦,ℎ𝑧<520⋅316𝐺.10≤𝑘𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦≤𝑘𝑀𝑥,𝑦,𝑧 (2.65)

Case 8. If 𝑥,𝑦,𝑧∈(1/2,1], then 𝐺7𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺8,78,785=0≤𝑀.16𝑥,𝑦,𝑧=𝑘𝑀𝑥,𝑦,𝑧 (2.66) Then in all the above cases, the mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 are satisfying the condition (2.1) of Theorem 2.1 with 𝑘=5/16 so that all the conditions of Theorem 2.1 are satisfied. Moreover, 7/8 is the unique common fixed point for all of the mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶.

In Theorem 2.1, if we take 𝐴=𝐵=𝐶=𝐼 (𝐼 is identity mapping, the same as below), then we have the following corollary.

Corollary 2.3 (see [2, Theorem 2.4]). Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, and ℎ be three mappings of 𝑋 into itself satisfying the following conditions: 𝐺𝐺,𝐺,𝐺()𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑘max𝑥,𝑔𝑦,𝑔𝑦+𝐺𝑦,𝑓𝑥,𝑓𝑥𝑦,ℎ𝑧,ℎ𝑧+𝐺𝑧,𝑔𝑦,𝑔𝑦𝑧,𝑓𝑥,𝑓𝑥+𝐺𝑥,ℎ𝑧,ℎ𝑧 (2.67) or 𝐺𝐺,𝐺,𝐺()𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑘max𝑥,𝑥,𝑔𝑦+𝐺𝑦,𝑦,𝑓𝑥𝑦,𝑦,ℎ𝑧+𝐺𝑧,𝑧,𝑔𝑦𝑧,𝑧,𝑓𝑥+𝐺𝑥,𝑥,ℎ𝑧 (2.68)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1/3).Then 𝑓, 𝑔, and ℎ have a unique common fixed point in 𝑋.

Also, if we take 𝑓=𝑔=ℎ and 𝐴=𝐵=𝐶=𝐼 in Theorem 2.1, then we get the following.

Corollary 2.4 (see [3, Theorem  2.4]). Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓 be a mapping of 𝑋 into itself satisfying the following conditions: 𝐺𝐺,𝐺,𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝑘max𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑦,𝑓𝑥,𝑓𝑥𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑧,𝑓𝑦,𝑓𝑦𝑧,𝑓𝑥,𝑓𝑥+𝐺𝑥,𝑓𝑧,𝑓𝑧 (2.69) or 𝐺𝐺,𝐺,𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝑘max𝑥,𝑥,𝑓𝑦+𝐺𝑦,𝑦,𝑓𝑥𝑦,𝑦,𝑓𝑧+𝐺𝑧,𝑧,𝑓𝑦𝑧,𝑧,𝑓𝑥+𝐺𝑥,𝑥,𝑓𝑧, (2.70)forall𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1/3). Then, 𝑓 has a unique fixed point in 𝑋.

Remark 2.5. Theorem 2.1 and Corollaries 2.3 and 2.4 generalize and extend the corresponding results of Abbas and Rhoades [5] and Mustafa et al. [6].

Remark 2.6. In Theorem 2.1, if we take: (1) 𝑓=𝑔=ℎ; (2) 𝐴=𝐵=𝐶; (3) 𝑔=ℎ and 𝐵=𝐶; (4) 𝑔=ℎ, 𝐵=𝐶=𝐼, several new results can be obtained.

Theorem 2.7. Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 be six mappings of 𝑋 into itself satisfying the following conditions:(i)𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), ℎ(𝑋)⊂𝐴(𝑋);(ii)the pairs (𝑓,𝐴), (𝑔,𝐵), and (ℎ,𝐶) are commutative mappings;(iii)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧𝐺≤𝑘max𝐴𝑥,𝑔^𝑚𝑦,𝑔^𝑚𝑦+𝐺𝐵𝑦,𝑓^𝑚𝑥,𝑓^𝑚𝑥,𝐺𝐵𝑦,ℎ^𝑚𝑧,ℎ^𝑚𝑧+𝐺𝐶𝑧,𝑔^𝑚𝑦,𝑔^𝑚𝑦,𝐺𝐶𝑧,𝑓^𝑚𝑥,𝑓^𝑚𝑥+𝐺𝐴𝑥,ℎ^𝑚𝑧,ℎ^𝑚𝑧 (2.71)or 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧𝐺≤𝑘max𝐴𝑥,𝐴𝑥,𝑔^𝑚𝑦+𝐺𝐵𝑦,𝐵𝑦,𝑓^𝑚𝑥,𝐺𝐵𝑦,𝐵𝑦,ℎ^𝑚𝑧+𝐺𝐶𝑧,𝐶𝑧,𝑔^𝑚𝑦,𝐺𝐶𝑧,𝐶𝑧,𝑓^𝑚𝑥+𝐺𝐴𝑥,𝐴𝑥,ℎ^𝑚𝑧, (2.72) where 𝑘∈[0,1/2),𝑚∈ℕ, then 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 have a unique common fixed point in 𝑋.

Proof. Suppose that mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 satisfy the condition (2.71). Since 𝑓^𝑚𝑋⊂𝑓^𝑚−1𝑋⊂⋯⊂𝑓𝑋, 𝑓𝑋⊂𝐵𝑋 so that 𝑓^𝑚𝑋⊂𝐵𝑋. Similarly, we can show that 𝑔^𝑚𝑋⊂𝐶𝑋 and ℎ^𝑚𝑋⊂𝐴𝑋. From the the Theorem 2.1, we see that 𝑓^𝑚, 𝑔^𝑚, ℎ^𝑚, 𝐴, 𝐵, and 𝐶 have a unique common fixed point 𝑢.
Since 𝑓𝑢=𝑓(𝑓^𝑚𝑢)=𝑓^𝑚+1𝑢=𝑓^𝑚(𝑓𝑢), so that 𝐺𝑓^𝑚𝑓𝑢,𝑔^𝑚𝑢,ℎ^𝑚𝑢𝐺≤𝑘max𝐴𝑓𝑢,𝑔^𝑚𝑢,𝑔^𝑚𝑢+𝐺𝐵𝑢,𝑓^𝑚𝑓𝑢,𝑓^𝑚,𝐺𝑓𝑢𝐵𝑢,ℎ^𝑚𝑢,ℎ^𝑚𝑢+𝐺𝐶𝑢,𝑔^𝑚𝑢,𝑔^𝑚𝑢,𝐺𝐶𝑢,𝑓^𝑚𝑓𝑢,𝑓^𝑚𝑓𝑢+𝐺𝐴𝑓𝑢,ℎ^𝑚𝑢,ℎ^𝑚𝑢, (2.73) note that 𝐴𝑓𝑢=𝑓𝐴𝑢=𝑓𝑢 and the Proposition 1.12(iii), we obtain 𝐺𝐺,𝐺()(),𝐺𝐺.𝑓𝑢,𝑢,𝑢≤𝑘max𝑓𝑢,𝑢,𝑢+𝐺𝑢,𝑓𝑢,𝑓𝑢𝑢,𝑢,𝑢+𝐺𝑢,𝑢,𝑢𝑢,𝑓𝑢,𝑓𝑢+𝐺𝑓𝑢,𝑢,𝑢=𝑘𝑢,𝑓𝑢,𝑓𝑢+𝐺𝑓𝑢,𝑢,𝑢≤3𝑘𝐺𝑓𝑢,𝑢,𝑢 (2.74) Since 𝑘∈[0,1/3), hence 𝐺(𝑓𝑢,𝑢,𝑢)=0 and so 𝑓𝑢=𝑢.
By the same argument, we can prove that 𝑔𝑢=𝑢 and ℎ𝑢=𝑢. Thus, we have 𝑢 = 𝑓𝑢 = 𝑔𝑢 = ℎ𝑢 = 𝐴𝑢 = 𝐵𝑢 = 𝐶𝑢 so that 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 have a common fixed point 𝑢 in 𝑋. Let 𝑣 be any other common fixed point of 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶, then by using condition (2.71), we have 𝐺(𝑢,𝑢,𝑣)=𝐺𝑓^𝑚𝑢,𝑔^𝑚𝑢,ℎ^𝑚𝑣𝐺≤𝑘max𝐴𝑢,𝑔^𝑚𝑢,𝑔^𝑚𝑢+𝐺𝐵𝑢,𝑓^𝑚𝑢,𝑓^𝑚𝑢,𝐺𝐵𝑢,ℎ^𝑚𝑣,ℎ^𝑚𝑣+𝐺𝐶𝑣,𝑔^𝑚𝑢,𝑔^𝑚𝑢,𝐺𝐶𝑣,𝑓^𝑚𝑢,𝑓^𝑚𝑢+𝐺𝐴𝑢,ℎ^𝑚𝑣,ℎ^𝑚𝑣𝐺()(),𝐺()(),𝐺()()[𝐺()(().=𝑘max𝑢,𝑢,𝑢+𝐺𝑢,𝑢,𝑢𝑢,𝑣,𝑣+𝐺𝑣,𝑢,𝑢𝑣,𝑢,𝑢+𝐺𝑢,𝑣,𝑣=𝑘𝑢,𝑣,𝑣+𝐺𝑣,𝑢,𝑢)]≤3𝑘𝐺𝑢,𝑢,𝑣 (2.75) Hence, 𝐺(𝑢,𝑢,𝑣)=0 and so 𝑢=𝑣, since 0≤𝑘<1/3. Thus, common fixed point is unique.
The proof using (2.72) is similar. This completes the proof.

In Theorem 2.7, if we take 𝐴=𝐵=𝐶=𝐼, then we have the following corollary.

Corollary 2.8 (see [2, Corollary  2.5]). Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, and ℎ be three mappings of 𝑋 into itself satisfying the following conditions: 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧𝐺≤𝑘max𝑥,𝑔^𝑚𝑦,𝑔^𝑚𝑦+𝐺𝑦,𝑓^𝑚𝑥,𝑓^𝑚𝑥,𝐺𝑦,ℎ^𝑚𝑧,ℎ^𝑚𝑧+𝐺𝑧,𝑔^𝑚𝑦,𝑔^𝑚𝑦,𝐺𝑧,𝑓^𝑚𝑥,𝑓^𝑚𝑥+𝐺𝑥,ℎ^𝑚𝑧,ℎ^𝑚𝑧 (2.76) or 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧𝐺≤𝑘max𝑥,𝑥,𝑔^𝑚𝑦+𝐺𝑦,𝑦,𝑓^𝑚𝑥,𝐺𝑦,𝑦,ℎ^𝑚𝑧+𝐺𝑧,𝑧,𝑔^𝑚𝑦,𝐺𝑧,𝑧,𝑓^𝑚𝑥+𝐺𝑥,𝑥,ℎ^𝑚𝑧 (2.77)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1/3), 𝑚∈ℕ; then 𝑓, 𝑔, and ℎ have a unique common fixed point in 𝑋.

Also, if we take 𝑓=𝑔=ℎ and 𝐴=𝐵=𝐶=𝐼 in Theorem 2.7, then we get the following.

Corollary 2.9 (see [3, Corollary  2.5]). Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓 be a mapping of 𝑋 into itself satisfying the following conditions: 𝐺𝑓^𝑚𝑥,𝑓^𝑚𝑦,𝑓^𝑚𝑧𝐺≤𝑘max𝑥,𝑓^𝑚𝑦,𝑓^𝑚𝑦+𝐺𝑦,𝑓^𝑚𝑥,𝑓^𝑚𝑥,𝐺𝑦,𝑓^𝑚𝑧,𝑓^𝑚𝑧+𝐺𝑧,𝑓^𝑚𝑦,𝑓^𝑚𝑦,𝐺𝑧,𝑓^𝑚𝑥,𝑓^𝑚𝑥+𝐺𝑥,𝑓^𝑚𝑧,𝑓^𝑚𝑧 (2.78) or 𝐺𝑓^𝑚𝑥,𝑓^𝑚𝑦,𝑓^𝑚𝑧𝐺≤𝑘max𝑥,𝑥,𝑓^𝑚𝑦+𝐺𝑦,𝑦,𝑓^𝑚𝑥,𝐺𝑦,𝑦,𝑓^𝑚𝑧+𝐺𝑧,𝑧,𝑓^𝑚𝑦,𝐺𝑧,𝑧,𝑓^𝑚𝑥+𝐺𝑥,𝑥,𝑓^𝑚𝑧 (2.79)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, where 𝑘∈[0,1/3),𝑚∈ℕ; then 𝑓 has a unique fixed point in 𝑋.

Remark 2.10. In Theorem 2.7, if we take: (1) 𝑓=𝑔=ℎ; (2) 𝑔=ℎ and 𝐵=𝐶; (3) 𝑔=ℎ, 𝐵=𝐶=𝐼, several new results can be obtained.

Corollary 2.11. Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 be six mappings of 𝑋 into itself satisfying the following conditions:(i)𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), ℎ(𝑋)⊂𝐴(𝑋);(ii)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, 𝐺𝐺𝐺𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑎𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥+𝑏𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦+𝑐𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧) (2.80) or 𝐺𝐺𝐺𝐺(,𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑎𝐴𝑥,𝐴𝑥,𝑔𝑦+𝐺𝐵𝑦,𝐵𝑦,𝑓𝑥+𝑏𝐵𝑦,𝐵𝑦,ℎ𝑧+𝐺𝐶𝑧,𝐶𝑧,𝑔𝑦+𝑐𝐶𝑧,𝐶𝑧,𝑓𝑥+𝐺𝐴𝑥,𝐴𝑥,ℎ𝑧) (2.81) where 0≤𝑎+𝑏+𝑐<1/3.Then one of the pairs (𝑓,𝐴), (𝑔,𝐵), and (ℎ,𝐶) has a coincidence point in 𝑋. Further, if one of the following conditions is satisfied, then the mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 have a unique common fixed point in 𝑋. (a) Either 𝑓 or 𝐴 is 𝐺-continuous, the pair (𝑓,𝐴) is weakly commutative, the pairs (𝑔,𝐵) and (ℎ,𝐶) are weakly compatible; (b) Either 𝑔 or 𝐵 is 𝐺-continuous, the pair (𝑔,𝐵) is weakly commutative, the pairs (𝑓,𝐴) and (ℎ,𝐶) are weakly compatible; (c) Either ℎ or 𝐶 is 𝐺-continuous, the pair (ℎ,𝐶) is weakly commutative, the pairs (𝑓,𝐴) and (𝑔,𝐵) are weakly compatible.

Proof. Suppose that mappings 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 satisfy the condition (2.80). For 𝑥,𝑦,𝑧∈𝑋, let 𝑀𝐺,𝐺,𝐺()𝑥,𝑦,𝑧=max𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧. (2.82) Then 𝑎𝐺𝐺𝐺(≤()𝑀.𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥+𝑏𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦+𝑐𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧)𝑎+𝑏+𝑐𝑥,𝑦,𝑧 (2.83) So, if 𝐺𝐺𝐺𝐺(,𝑓𝑥,𝑔𝑦,ℎ𝑧≤𝑎𝐴𝑥,𝑔𝑦,𝑔𝑦+𝐺𝐵𝑦,𝑓𝑥,𝑓𝑥+𝑏𝐵𝑦,ℎ𝑧,ℎ𝑧+𝐺𝐶𝑧,𝑔𝑦,𝑔𝑦+𝑐𝐶𝑧,𝑓𝑥,𝑓𝑥+𝐺𝐴𝑥,ℎ𝑧,ℎ𝑧) (2.84) then 𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧)≤(𝑎+𝑏+𝑐)𝑀(𝑥,𝑦,𝑧). Taking 𝑘=𝑎+𝑏+𝑐 in Theorem 2.1, the conclusion of Corollary 2.11 can be obtained from Theorem 2.1 immediately.
The proof using (2.81) is similar. This completes the proof.

Corollary 2.12. Let (𝑋,𝐺) be a complete 𝐺-metric space, and let 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 be six mappings of 𝑋 into itself satisfying the following conditions:(i)𝑓(𝑋)⊂𝐵(𝑋), 𝑔(𝑋)⊂𝐶(𝑋), ℎ(𝑋)⊂𝐴(𝑋);(ii)the pairs (𝑓,𝐴), (𝑔,𝐵) and (ℎ,𝐶) are commutative mappings;(iii)𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦,𝑧∈𝑋, 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧≤𝑎𝐺𝐴𝑥,𝑔^𝑚𝑦,𝑔^𝑚𝑦+𝐺𝐵𝑦,𝑓^𝑚𝑥,𝑓^𝑚𝑥+𝑏𝐺𝐵𝑦,ℎ^𝑚𝑧,ℎ^𝑚𝑧+𝐺𝐶𝑧,𝑔^𝑚𝑦,𝑔^𝑚𝑦+𝑐𝐺𝐶𝑧,𝑓^𝑚𝑥,𝑓^𝑚𝑥+𝐺𝐴𝑥,ℎ^𝑚𝑧,ℎ^𝑚𝑧 (2.85) or 𝐺𝑓^𝑚𝑥,𝑔^𝑚𝑦,ℎ^𝑚𝑧≤𝑎𝐺𝐴𝑥,𝐴𝑥,𝑔^𝑚𝑦+𝐺𝐵𝑦,𝐵𝑦,𝑓^𝑚𝑥+𝑏𝐺𝐵𝑦,𝐵𝑦,ℎ^𝑚𝑧+𝐺𝐶𝑧,𝐶𝑧,𝑔^𝑚𝑦+𝑐𝐺𝐶𝑧,𝐶𝑧,𝑓^𝑚𝑥+𝐺𝐴𝑥,𝐴𝑥,ℎ^𝑚𝑧, (2.86) where 0≤𝑎+𝑏+𝑐<1/3, 𝑚∈ℕ; then 𝑓, 𝑔, ℎ, 𝐴, 𝐵, and 𝐶 have a unique common fixed point in 𝑋.

Proof. The proof follows from Corollary 2.11, and from an argument similar to that used in Theorem 2.7.

Remark 2.13. In Corollaries 2.11 and 2.12, if we take: (1) 𝐴=𝐵=𝐶=𝐼; (2) 𝑓=𝑔=ℎ; (3) 𝐴=𝐵=𝐶; (4) 𝑔=ℎ and 𝐵=𝐶; (5) 𝑔=ℎ, 𝐵=𝐶=𝐼, several new results can be obtained.

Acknowledgment

This work was supported by the National Natural Science Foundation of China Grant (nos. 11071169, 11271105) and the Natural Science Foundation of Zhejiang Province (nos. Y6110287, LY12A01030).