Common Fixed Point Theorems of Altman Integral Type Mappings in 𝐺-Metric Spaces

Abstract

We introduce the concept of 𝜑-weakly commuting self-mapping pairs in 𝐺-metric space. Using this concept, we establish a new common fixed point theorem of Altman integral type for six self-mappings in the framework of complete 𝐺-metric space. An example is provided to support our result. The results obtained in this paper differ from the recent relative results in the literature.

1. Introduction and Preliminaries

Metric fixed point theory is an important mathematical discipline because of its applications in areas as variational and linear inequalities, optimization theory. Many results have been obtained by many authors considering different contractive conditions for self-mappings in metric space. In 1975, Altman [1] proved a fixed point theorem for a mapping which satisfies the condition 𝑑(𝑓𝑥,𝑓𝑦)≤𝑄(𝑑(𝑥,𝑦)), where 𝑄∶[0,+∞)→[0,+∞) is an increasing function satisfying the following conditions:(i)0<𝑄(𝑡)<𝑡,𝑡∈(0,∞);(ii)𝑝(𝑡)=𝑡/(𝑡−𝑄(𝑡)) is a decreasing function;(iii) for some positive number 𝑡₁, there holds ∫^𝑡1₀𝑝(𝑡)𝑑𝑡<+∞.

Remark 1.1. By condition (i) and that 𝑄 is increasing, we know that 𝑄(0)=0 and 𝑄(𝑡)=𝑡⇔𝑡=0.

Gu and Deng [2], Liu [3], Zhang [4], and Li and Gu [5] discussed common fixed point theorems for Altman type mappings in metric space. In 2006, a new structure of generalized metric space was introduced by Mustafa and Sims [6] as an appropriate notion of generalized metric space called 𝐺-metric space. Abbas and Rhoades [7] initiated the study of common fixed point in generalized metric space. Recently, many fixed point and common fixed point theorems for certain contractive conditions have been established in 𝐺-metric spaces, and for more details, one can refer to [8–42]. Coupled fixed point problems have also been considered in partially ordered 𝐺-metric spaces (see [43–56]). However, no one has discussed the common fixed point theorems for two or three pairs combining Altman type mappings recently.

Inspired by that, the purpose of this paper is to study common fixed point problem of Altman integral type for six self-mappings in 𝐺-metric space. We introduce a new concept of 𝜑-weakly commuting self-mapping pairs in 𝐺-metric space, and a new common fixed point theorem for six self-mappings has been established through this concept. The results obtained in this paper differ from the recent relative results in the literature.

Throughout the paper, we mean by ℕ the set of all natural numbers.

Definition 1.2 (see [6]). Let 𝑋 be a nonempty set, and let 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ be a function satisfying the following axioms: (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.

Definition 1.3 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space and let {𝑥_𝑛} be a sequence of points in 𝑋, a point 𝑥 in 𝑋 is said to be the limit of the sequence {𝑥_𝑛} if lim_{𝑚,𝑛→∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and one says that sequence {𝑥_𝑛} is 𝐺-convergent to 𝑥.

Thus, if 𝑥_𝑛→𝑥 in a 𝐺-metric space (𝑋,𝐺), then for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑁.

Proposition 1.4 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space, then the followings are equivalent:(1){𝑥_𝑛} is 𝐺-convergent to 𝑥, (2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0 as 𝑛→∞, (3)𝐺(𝑥_𝑛,𝑥,𝑥)→0 as 𝑛→∞, (4)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥)→0 as 𝑛,𝑚→∞.

Definition 1.5 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space. A sequence {𝑥_𝑛} is called 𝐺-Cauchy sequence if, for each 𝜖>0 there exists a positive integer 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜖 for all 𝑛,𝑚,𝑙≥𝑁; that is, 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→∞.

Definition 1.6 (see [6]). A 𝐺-metric space (𝑋,𝐺) is said to be 𝐺-complete, if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in 𝑋.

Proposition 1.7 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the followings are equivalent.(1) The sequence {𝑥_𝑛} is 𝐺-Cauchy;(2) For every 𝜖>0, there exists 𝑘∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜖, for all 𝑛,𝑚≥𝑘.

Proposition 1.8 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

Definition 1.9 (see [6]). Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be 𝐺-metric space, and 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 imply 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 1.10 (see [6]). Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be 𝐺-metric space. Then 𝑓∶𝑋→𝑋^′ is 𝐺-continuous at 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).

Proposition 1.11 (see [6]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, for any 𝑥,𝑦,𝑧,𝑎 in 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)≤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).

Definition 1.12 (see [8]). Self-mappings 𝑓 and 𝑔 of a 𝐺-metric space (𝑋,𝐺) are said to be compatible if lim_𝑛→∞𝐺(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛,𝑔𝑓𝑥_𝑛)=0 and lim_𝑛→∞𝐺(𝑔𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛)=0, whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑡, for some 𝑡∈𝑋.

In 2010, Manro et al. [9] introduced the concept of weakly commuting mappings, 𝑅-weakly commuting mappings into 𝐺-metric space as follows.

Definition 1.13 (see [9]). A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space is said to be weakly commuting if 𝐺𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥≤𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥,∀𝑥∈𝑋. (1.1)

Definition 1.14 (see [9]). A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space is said to be 𝑅-weakly commuting, if there exists some positive real number 𝑅 such that 𝐺𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥≤𝑅𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥,∀𝑥∈𝑋. (1.2)

Remark 1.15. If 𝑅≤1, then 𝑅-weakly commuting mappings are weakly commuting.

Now we introduce the new concept of 𝜑-weakly commuting mappings as follow.

Definition 1.16. A pair of self-mappings (𝑓,𝑔) of a 𝐺-metric space is said to be 𝜑-weakly commuting, if there exists a continuous function 𝜑∶[0,∞)→[0,∞),𝜑(0)=0, such that 𝐺𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥≤𝜑𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥,∀𝑥∈𝑋. (1.3)

Remark 1.17. Commuting mappings are weakly commuting mappings, but the reverse is not true. For example: let 𝑋=[0,1/2], 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑧−𝑥|, for all 𝑥,𝑦,𝑧∈𝑋, define 𝑓(𝑥)=𝑥/2, 𝑔(𝑥)=𝑥²/2, through a straightforward calculation, we have: 𝑓𝑔𝑥=𝑥²/4, 𝑔𝑓𝑥=𝑥²/8, 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)=𝐺(𝑥²/4,𝑥²/8,𝑥²/8)=𝑥²/4, but 𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥)=|𝑥−𝑥²|=𝑥−𝑥², hence, 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)≤𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥), but 𝑓𝑔𝑥≠𝑔𝑓𝑥.

Remark 1.18. Weakly commuting mappings are 𝑅-weakly commuting mappings, but the reverse is not true. For example: let 𝑋=[−1,1], define 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑧−𝑥|, for all 𝑥,𝑦,𝑧∈𝑋, 𝑓(𝑥)=|𝑥|, 𝑔(𝑥)=|𝑥|−1, then 𝑔𝑓𝑥=|𝑥|−1, 𝑓𝑔𝑥=1−|𝑥|, |𝑓𝑥−𝑔𝑥|=1, |𝑓𝑔𝑥−𝑔𝑓𝑥|=2(1−|𝑥|), 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)=2|𝑓𝑔𝑥−𝑔𝑓𝑥|=4(1−|𝑥|)≤4=4|𝑓𝑥−𝑔𝑥|=𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥), when 𝑅=2, we get that 𝑓 and 𝑔 are 𝑅-weakly commuting mappings, but not weakly commuting mappings.

Remark 1.19. 𝑅-weakly commuting mappings are 𝜑-weakly commuting mappings but the reverse is not true. For example: let 𝑋=[0,+∞), 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑧−𝑥|, for all 𝑥,𝑦,𝑧∈𝑋, 𝑓(𝑥)=𝑥²/4,𝑔(𝑥)=𝑥², thus, we have 𝑓𝑔𝑥=𝑥⁴/4, 𝑔𝑓𝑥=𝑥⁴/16, 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)=(3/8)𝑥⁴, 𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥)=(3/2)𝑥². Let 𝜑(𝑥)=(1/2)𝑥², then 𝐺=3𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥8𝑥⁴≤98𝑥⁴=1232𝑥²²3=𝜑2𝑥²=𝜑𝐺𝑓𝑥,𝑔𝑥,𝑔𝑥. (1.4) For any given 𝑅>0, since lim_𝑥→+∞(1/4)𝑥²=+∞, there exists 𝑥∈𝑋 such that (1/4)𝑥²>𝑅, so we get 𝐺(𝑓𝑔𝑥,𝑔𝑓𝑥,𝑔𝑓𝑥)=(1/4)𝑥²𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥)>𝑅𝐺(𝑓𝑥,𝑔𝑥,𝑔𝑥). Therefore, 𝑓 and 𝑔 are 𝜑-weakly commuting mappings, but not 𝑅-weakly commuting mappings.

Lemma 1.20. Let 𝛿(𝑡) be Lebesgue integrable, and 𝛿(𝑡)>0, for all 𝑡>0, let 𝐹(𝑥)=∫^𝑥₀𝛿(𝑡)𝑑𝑡, then 𝐹(𝑥) is an increasing function in [0,+∞).

Definition 1.21. Let 𝑓 and 𝑔 be self-mappings of a set 𝑋. If 𝑤=𝑓𝑥=𝑔𝑥 for some 𝑥 in 𝑋, then 𝑥 is called a coincidence point of 𝑓 and 𝑔, and 𝑤 is called s point of coincidence of 𝑓 and 𝑔.

2. Main Results

In this paper, we denote 𝜙∶[0,+∞)→[0,+∞) the function satisfying 0<𝜙(𝑡)<𝑡, for all 𝑡>0.

Theorem 2.1. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ be six mappings of 𝑋 into itself. If there exists an increasing function 𝑄∶[0,+∞)→[0,+∞) satisfying the conditions (i)~(iii) and the following conditions:(iv)𝑆(𝑋)⊆𝑔(𝑋),𝑇(𝑋)⊆ℎ(𝑋),𝑅(𝑋)⊆𝑓(𝑋), (v)∫₀^{𝐺(𝑆𝑥,𝑇𝑦,𝑅𝑧)}𝛿(𝑡)𝑑𝑡≤𝜙(∫₀^{𝑄(𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧))}𝛿(𝑡)𝑑𝑡), for all 𝑥,𝑦,𝑧∈𝑋,where 𝛿(𝑡) is a Lebesgue integrable function which is summable nonnegative such that ^𝜖₀𝛿(𝑡)𝑑𝑡>0,∀𝜖>0. (2.1) Then,(a) one of the pairs (𝑆,𝑓), (𝑇,𝑔), and (𝑅,ℎ) has a coincidence point in 𝑋,(b) if (𝑆,𝑓), (𝑇,𝑔), and (𝑅,ℎ) are three pairs of continuous 𝜑-weakly commuting mappings, then the mappings 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ have a unique common fixed point in 𝑋.

Proof. Let 𝑥₀ be an arbitrary point in 𝑋, from the condition (iv), there exist 𝑥₁,𝑥₂,𝑥₃∈𝑋 such that 𝑦₁=𝑆𝑥₀=𝑔𝑥₁,𝑦₂=𝑇𝑥₁=ℎ𝑥₂,𝑦₃=𝑅𝑥₂=𝑓𝑥₃. (2.2) By induction, there exist two sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋, such that 𝑦_3𝑛+1=𝑆𝑥_3𝑛=𝑔𝑥_3𝑛+1,𝑦_3𝑛+2=𝑇𝑥_3𝑛+1=ℎ𝑥_3𝑛+2,𝑦_3𝑛+3=𝑅𝑥_3𝑛+2=𝑓𝑥_3𝑛+3,𝑛∈ℕ. (2.3)
If 𝑦_𝑛=𝑦_𝑛+1 for some 𝑛, with 𝑛=3𝑚, then 𝑝=𝑥_3𝑚+1 is a coincidence point of the pair (𝑆,𝑓); 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 (𝑇,𝑔); 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 (𝑅,ℎ).
On the other hand, if there exists 𝑛₀∈ℕ such that 𝑦_𝑛0=𝑦_𝑛0+1=𝑦_𝑛0+2, then 𝑦_𝑛=𝑦_𝑛0 for any 𝑛≥𝑛₀. This implies that {𝑦_𝑛} is a 𝐺-Cauchy sequence.
In fact, if there exists 𝑝∈ℕ such that 𝑦_3𝑝=𝑦_3𝑝+1=𝑦_3𝑝+2, then applying the contractive condition (v) with 𝑥=𝑦_3𝑝, 𝑦=𝑦_3𝑝+1, and 𝑧=𝑦_3𝑝+2, and the property of 𝜙, we get ^{𝐺(𝑦3𝑝+1,𝑦3𝑝+2,𝑦3𝑝+3)}₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑆𝑥3𝑝,𝑇𝑥3𝑝+1,𝑅𝑥3𝑝+2)}₀𝛿(𝑡)𝑑𝑡≤𝜙^{𝑄(𝐺(𝑓𝑥3𝑝,𝑔𝑥3𝑝+1,ℎ𝑥3𝑝+2}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝐺(𝑦3𝑝,𝑦3𝑝+1,𝑦3𝑝+2}₀⁾⁾𝛿(𝑡)≤𝑑𝑡_{𝑄(𝐺(𝑦3𝑝,𝑦3𝑝+1,𝑦3𝑝+20))}𝛿(𝑡)𝑑𝑡. (2.4)
From Lemma 1.20 and the property of 𝑄, we have 𝐺𝑦_3𝑝+1,𝑦_3𝑝+2,𝑦_3𝑝+3≤𝑄𝐺𝑦_3𝑝,𝑦_3𝑝+1,𝑦_3𝑝+2=𝑄(0)=0. (2.5) Which implies that 𝑦_3𝑝+3=𝑦_3𝑝+1=𝑦_3𝑝. So we find 𝑦_𝑛=𝑦_3𝑝 for any 𝑛≥3𝑝. This implies that {𝑦_𝑛} is a 𝐺-Cauchy sequence. The same conclusion holds if 𝑦_3𝑝+1=𝑦_3𝑝+2=𝑦_3𝑝+3, or 𝑦_3𝑝+2=𝑦_3𝑝+3=𝑦_3𝑝+4 for some 𝑝∈ℕ.
Without loss of generality, we can assume that 𝑦_𝑛≠𝑦_𝑚 for all 𝑛,𝑚∈ℕ and 𝑛≠𝑚.
Now we prove that {𝑦_𝑛} is a 𝐺-Cauchy sequence in 𝑋.
Let 𝑡_𝑛=𝐺(𝑦_𝑛,𝑦_𝑛+1,𝑦_𝑛+2), then we have 𝑡_𝑛+1≤𝑄𝑡_𝑛<𝑡_𝑛 (2.6) for all 𝑛∈ℕ. Actually, from the condition (v), (2.3) and the property of 𝜙, we have ^𝑡3𝑛₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑦3𝑛,𝑦3𝑛+1,𝑦3𝑛+2)}₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑅𝑥3𝑛−1,𝑆𝑥3𝑛,𝑇𝑥3𝑛+1)}₀𝛿(𝑡)=𝑑𝑡_{𝐺(𝑆𝑥3𝑛,𝑇𝑥3𝑛+1,𝑅𝑥3𝑛−1)0}𝛿(𝑡)𝑑𝑡≤𝜙^{𝑄(𝐺(𝑓𝑥3𝑛,𝑔𝑥3𝑛+1,ℎ𝑥3𝑛−1}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝐺(𝑦3𝑛,𝑦3𝑛+1,𝑦3𝑛−1}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝑡3𝑛−1)}₀𝛿(𝑡)≤𝑑𝑡_{𝑄(𝑡3𝑛−1)0}𝛿(𝑡)𝑑𝑡. (2.7) By Lemma 1.20 and the property of 𝑄, we have 𝑡_3𝑛≤𝑄𝑡_3𝑛−1<𝑡_3𝑛−1. (2.8)
Again, using condition (v), (2.3) and the property of 𝜙, we get ^𝑡3𝑛+1₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑦3𝑛+1,𝑦3𝑛+2,𝑦3𝑛+3)}₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑆𝑥3𝑛,𝑇𝑥3𝑛+1,𝑅𝑥3𝑛+2)}₀𝛿(𝑡)𝑑𝑡≤𝜙^{𝑄(𝐺(𝑓𝑥3𝑛,𝑔𝑥3𝑛+1,ℎ𝑥3𝑛+2}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝐺(𝑦3𝑛,𝑦3𝑛+1,𝑦3𝑛+2}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝑡3𝑛)}₀𝛿(𝑡)𝑑𝑡≤^{𝑄(𝑡3𝑛)}₀𝛿(𝑡)𝑑𝑡. (2.9) From Lemma 1.20 and the property of 𝑄 we have 𝑡_3𝑛+1≤𝑄𝑡_3𝑛<𝑡_3𝑛. (2.10)
Similarly, we can get ^𝑡3𝑛+2₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑦3𝑛+2,𝑦3𝑛+3,𝑦3𝑛+4)}₀𝛿(𝑡)𝑑𝑡=^{𝐺(𝑆𝑥3𝑛+3,𝑇𝑥3𝑛+1,𝑅𝑥3𝑛+2)}₀𝛿(𝑡)𝑑𝑡≤𝜙^{𝑄(𝐺(𝑓𝑥3𝑛+3,𝑔𝑥3𝑛+1,ℎ𝑥3𝑛+2}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝐺(𝑦3𝑛+3,𝑦3𝑛+1,𝑦3𝑛+2}₀⁾⁾𝛿(𝑡)𝑑𝑡=𝜙^{𝑄(𝑡3𝑛+1)}₀𝛿(𝑡)𝑑𝑡≤^{𝑄(𝑡3𝑛+1)}₀𝛿(𝑡)𝑑𝑡. (2.11) From Lemma 1.20 and property of 𝑄, we have 𝑡_3𝑛+2≤𝑄𝑡_3𝑛+1<𝑡_3𝑛+1. (2.12) Combining (2.8), (2.10), and (2.12), we know that the (2.6) holds. This implies that {𝑡_𝑛} is a nonnegative sequence which is strictly decreasing, hence, {𝑡_𝑛} is convergent and 𝑡_𝑛+1≤𝑄(𝑡_𝑛)<𝑡_𝑛, for all 𝑛∈ℕ.
For any 𝑛,𝑚∈ℕ, 𝑚>𝑛, by combining (𝐺5), (𝐺3), and (2.6), we have 𝐺𝑦_𝑛,𝑦_𝑚,𝑦_𝑚≤^𝑚−1_𝑖=𝑛𝐺𝑦_𝑖,𝑦_𝑖+1,𝑦_𝑖+1≤^𝑚−1_𝑖=𝑛𝐺𝑦_𝑖,𝑦_𝑖+1,𝑦_𝑖+2=^𝑚−1_𝑖=𝑛𝑡_𝑖=^𝑚−1_𝑖=𝑛𝑡_𝑖𝑡_𝑖−𝑡_𝑖+1𝑡_𝑖−𝑡_𝑖+1≤^𝑚−1_𝑖=𝑛𝑡_𝑖𝑡_𝑖−𝑡_𝑖+1𝑡_𝑖−𝑄𝑡_𝑖≤^𝑚−1_𝑖=𝑛^𝑡𝑖_𝑡𝑖+1𝑡𝑡−𝑄(𝑡)𝑑𝑡=^𝑡𝑛_𝑡𝑚𝑝(𝑡)𝑑𝑡. (2.13) From the convergence of the sequence {𝑡_𝑛} and the condition (iii) we assure that lim_{𝑛,𝑚→∞}^𝑡𝑛_𝑡𝑚𝑝(𝑡)𝑑𝑡=0. (2.14) Thus, {𝑦_𝑛} is a 𝐺-Cauchy sequence in 𝑋, since (𝑋,𝐺) is a complete 𝐺-metric space, there exists 𝑢∈𝑋 such that lim_𝑛→∞𝑦_𝑛=𝑢, hence lim_𝑛→∞𝑦_3𝑛+1=lim_𝑛→∞𝑆𝑥_3𝑛=lim_𝑛→∞𝑔𝑥_3𝑛+1=𝑢,lim_𝑛→∞𝑦_3𝑛+2=lim_𝑛→∞𝑇𝑥_3𝑛+1=lim_𝑛→∞ℎ𝑥_3𝑛+2=𝑢,lim_𝑛→∞𝑦_3𝑛+3=lim_𝑛→∞𝑅𝑥_3𝑛+2=lim_𝑛→∞𝑓𝑥_3𝑛+3=𝑢. (2.15) Since (𝑆,𝑓) are 𝜑-weakly commuting mappings, thus we have 𝐺𝑆𝑓𝑥_3𝑛,𝑓𝑆𝑥_3𝑛,𝑓𝑆𝑥_3𝑛≤𝜑𝐺𝑆𝑥_3𝑛,𝑓𝑥_3𝑛,𝑓𝑥_3𝑛. (2.16) On taking 𝑛→∞ at both sides, noting that 𝑆 and 𝑓 are continuous mappings, we have 𝐺𝑆𝑢,𝑓𝑢,𝑓𝑢≤𝜑(𝐺(𝑢,𝑢,𝑢))=𝜑(0)=0. (2.17) Which gives that 𝑆𝑢=𝑓𝑢. Similarly, we can get 𝑇𝑢=𝑔𝑢,𝑅𝑢=ℎ𝑢.
By using condition (v) and the property of 𝜙, we get ₀^{𝐺(𝑆𝑢,𝑇𝑢,𝑅𝑢)}𝛿(𝑡)𝑑𝑡≤𝜙₀^{𝑄(𝐺(𝑓𝑢,𝑔𝑢,ℎ𝑢))}𝛿(𝑡)𝑑𝑡≤₀^{𝑄(𝐺(𝑓𝑢,𝑔𝑢,ℎ𝑢))}𝛿(𝑡)𝑑𝑡. (2.18) Thus, by Lemma 1.20, (𝐺5) and the property of 𝑄, noting that 𝑆𝑢=𝑓𝑢, 𝑇𝑢=𝑔𝑢, 𝑅𝑢=ℎ𝑢, we have 𝐺()𝐺()()()().𝑆𝑢,𝑇𝑢,𝑅𝑢≤𝑄𝑓𝑢,𝑔𝑢,ℎ𝑢≤𝐺𝑓𝑢,𝑔𝑢,ℎ𝑢≤𝐺𝑓𝑢,𝑆𝑢,𝑆𝑢+𝐺𝑆𝑢,𝑔𝑢,ℎ𝑢≤𝐺𝑓𝑢,𝑆𝑢,𝑆𝑢+𝐺ℎ𝑢,𝑅𝑢,𝑅𝑢+𝐺𝑅𝑢,𝑔𝑢,𝑆𝑢≤𝐺𝑓𝑢,𝑆𝑢,𝑆𝑢+𝐺ℎ𝑢,𝑅𝑢,𝑅𝑢+𝐺𝑅𝑢,𝑇𝑢,𝑆𝑢+𝐺𝑇𝑢,𝑇𝑢,𝑔𝑢=𝐺𝑆𝑢,𝑇𝑢,𝑅𝑢 (2.19) Which implies that 𝑄𝐺𝑓𝑢,𝑔𝑢,ℎ𝑢=𝐺(𝑆𝑢,𝑇𝑢,𝑅𝑢)=𝐺𝑓𝑢,𝑔𝑢,ℎ𝑢. (2.20) By Remark 1.1, we have 𝐺(𝑓𝑢,𝑔𝑢,ℎ𝑢)=0, therefore, 𝑓𝑢=𝑔𝑢=ℎ𝑢. So, immediately, we can have 𝑆𝑢=𝑇𝑢=𝑅𝑢=𝑓𝑢=𝑔𝑢=ℎ𝑢. Setting 𝑧=𝑆𝑢=𝑇𝑢=𝑅𝑢=𝑓𝑢=𝑔𝑢=ℎ𝑢. (2.21) Since (𝑆,𝑓) are 𝜑-weakly commuting mappings, we have 𝐺𝑆𝑧,𝑓𝑧,𝑓𝑧=𝐺𝑆𝑓𝑢,𝑓𝑆𝑢,𝑓𝑆𝑢≤𝜑𝐺𝑆𝑢,𝑓𝑢,𝑓𝑢=𝜑(0)=0. (2.22) Which gives that 𝑆𝑧=𝑓𝑧. By the same argument, we can get 𝑇𝑧=𝑔𝑧, 𝑅𝑧=ℎ𝑧, So we have 𝑆𝑓𝑢=𝑓𝑆𝑢, 𝑇𝑔𝑢=𝑔𝑇𝑢, 𝑅ℎ𝑢=ℎ𝑅𝑢. Again, by condition (v), we have ₀^{𝐺(𝑆𝑧,𝑧,𝑧)}𝛿(𝑡)𝑑𝑡=^𝐺(𝑆2₀^{𝑢,𝑇𝑢,𝑅𝑢)}𝛿(𝑡)𝑑𝑡≤𝜙₀^{𝑄(𝐺(𝑓𝑆𝑢,𝑔𝑢,ℎ𝑢))}𝛿(𝑡)𝑑𝑡≤₀^{𝑄(𝐺(𝑓𝑆𝑢,𝑔𝑢,ℎ𝑢))}𝛿(𝑡)𝑑𝑡. (2.23) By the Lemma 1.20 and the property of 𝑄, we have 𝐺(𝑆𝑧,𝑧,𝑧)≤𝑄𝐺𝑓𝑆𝑢,𝑔𝑢,ℎ𝑢≤𝐺𝑓𝑆𝑢,𝑔𝑢,ℎ𝑢=𝐺𝑆𝑓𝑢,𝑔𝑢,ℎ𝑢=𝐺(𝑆𝑧,𝑧,𝑧). (2.24) Which implies that 𝑄(𝐺(𝑆𝑓𝑢,𝑔𝑢,ℎ𝑢))=𝐺(𝑆𝑓𝑢,𝑔𝑢,ℎ𝑢). Thus, by the property of 𝑄, we have 𝑆𝑓𝑢=𝑔𝑢=ℎ𝑢, hence, 𝑧=𝑆𝑧=𝑓𝑧. Similarly, we can prove that 𝑧=𝑇𝑧=𝑔𝑧, 𝑧=𝑅𝑧=ℎ𝑧, so we get 𝑧=𝑆𝑧=𝑇𝑧=𝑅𝑧=𝑓𝑧=𝑔𝑧=ℎ𝑧, which means that 𝑧 is a common fixed point of 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ.
Now, we will show the common fixed point of 𝑆, 𝑇,𝑅, 𝑓, 𝑔, and ℎ is unique. Actually, assume 𝑤≠𝑧 is another common fixed point of 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ, then by condition (v), we have ₀^{𝐺(𝑧,𝑤,𝑤)}𝛿(𝑡)𝑑𝑡=₀^{𝐺(𝑆𝑧,𝑇𝑤,𝑅𝑤)}𝛿(𝑡)𝑑𝑡≤𝜙₀^{𝑄(𝐺(𝑓𝑧,𝑔𝑤,ℎ𝑤))}𝛿(𝑡)𝑑𝑡≤₀^{𝑄(𝐺(𝑓𝑧,𝑔𝑤,ℎ𝑤))}𝛿(𝑡)𝑑𝑡. (2.25) By Lemma 1.20 and the property of 𝑄, we have 𝐺(𝑧,𝑤,𝑤)≤𝑄𝐺𝑓𝑧,𝑔𝑤,ℎ𝑤=𝑄(𝐺(𝑧,𝑤,𝑤))<𝐺(𝑧,𝑤,𝑤). (2.26) It is a contradiction, unless 𝑧=𝑤, that is, 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ have a unique common fixed point in 𝑋. This completes the proof of Theorem 2.1.

Remark 2.2. If we take: (1) 𝑆=𝑇=𝑅; (2) 𝑓=𝑔=ℎ; (3) 𝑓=𝑔=ℎ=𝐼 (𝐼 is identity mapping); (4) 𝑇=𝑅 and 𝑔=ℎ; (5) 𝑇=𝑅, 𝑔=ℎ=𝐼; (6) 𝑦=𝑧 in Theorem 2.1, then several new results can be obtained.

Corollary 2.3. Let (𝑋,𝐺) be a complete 𝐺-metric space and let 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ be six mappings of 𝑋 into itself. If there exists an increasing function 𝑄∶[0,+∞)→[0,+∞) satisfying the conditions (i)~(iii) and the following conditions:(iv)𝑆(𝑋)⊆𝑔(𝑋), 𝑇(𝑋)⊆ℎ(𝑋), 𝑅(𝑋)⊆𝑓(𝑋),(v)𝐺(𝑆𝑥,𝑇𝑦,𝑅𝑧)≤𝜙(𝑄(𝐺(𝑓𝑥,𝑔𝑦,ℎ𝑧))), ∀𝑥,𝑦,𝑧∈𝑋.Then,(a) one of the pairs (𝑆,𝑓), (𝑇,𝑔), and (𝑅,ℎ) has a coincidence point in 𝑋.(b) if (𝑆,𝑓), (𝑇,𝑔), and (𝑅,ℎ) are three pairs of continuous 𝜑-weakly commuting mappings, then the mappings 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ have a unique common fixed point in 𝑋.

Proof. Taking 𝛿(𝑡)=1 in Theorem 2.1, the conclusion of Corollary 2.3 can be obtained from Theorem 2.1 immediately. This completes the proof of Corollary 2.3.

Now we give an example to support Corollary 2.3.

Example 2.4. Let 𝑋=[0,∞), 𝐺(𝑥,𝑦,𝑧)=|𝑥−𝑦|+|𝑦−𝑧|+|𝑧−𝑥|, for all 𝑥,𝑦,𝑧∈𝑋. Let 𝑆,𝑇,𝑅,𝑓,𝑔,ℎ∶𝑋→𝑋 be defined by 𝑆𝑥=𝑥/8, 𝑇𝑥=𝑥/16, 𝑅𝑥=𝑥/32, 𝑓𝑥=𝑥, 𝑔𝑥=𝑥/2, ℎ𝑥=𝑥/4. Clearly, we can get 𝑆(𝑋)⊆𝑔(𝑋), 𝑇(𝑋)⊆ℎ(𝑋), 𝑅(𝑋)⊆𝑓(𝑋). Through calculation, we have 𝐺𝑥𝑆𝑥,𝑇𝑦,𝑅𝑧=𝐺8,𝑦,𝑧16=𝑥328−𝑦+𝑦16−𝑧16+𝑥328−𝑧=1328𝑦𝑥−2+𝑦2−𝑧4+𝑧4,𝐺𝑦−𝑥𝑓𝑥,𝑔𝑦,ℎ𝑧=𝐺𝑥,2,𝑧4=𝑦𝑥−2+𝑦2−𝑧4+𝑧4−𝑥. (2.27) Thus, we have 𝐺=1𝑆𝑥,𝑇𝑦,𝑅𝑧8𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧. (2.28) Now we choose 𝜙(𝑡)=3𝑡/4, and 𝑄(𝑡)=𝑡/2, then we have 𝜙(𝑡)<𝑡 and 𝑄(𝑡) satisfies (i)~(iii). Thus, we have 𝐺=1𝑆𝑥,𝑇𝑦,𝑅𝑧8𝐺≤3𝑓𝑥,𝑔𝑦,ℎ𝑧4⋅12𝐺=3𝑓𝑥,𝑔𝑦,ℎ𝑧4𝑄𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧=𝜙𝑄𝐺𝑓𝑥,𝑔𝑦,ℎ𝑧. (2.29)

On the other hand, let 𝜑(𝑢)=𝑢/2 for all 𝑢∈[0,∞), we have 𝐺𝑥𝑆𝑓𝑥,𝑓𝑆𝑥,𝑓𝑆𝑥=𝐺8,𝑥8,𝑥8≤12⋅7𝑥4=12𝐺𝐺,𝐺𝑥𝑆𝑥,𝑓𝑥,𝑓𝑥=𝜑𝑆𝑥,𝑓𝑥,𝑓𝑥𝑇𝑔𝑥,𝑔𝑇𝑥,𝑔𝑇𝑥=𝐺,𝑥32,𝑥32≤1322⋅7𝑥8=12𝐺𝐺,𝐺()𝑥𝑇𝑥,𝑔𝑥,𝑔𝑥=𝜑𝑇𝑥,𝑔𝑥,𝑔𝑥𝑅ℎ𝑥,ℎ𝑅𝑥,ℎ𝑅𝑥=𝐺,𝑥128,𝑥128≤11282⋅7𝑥=1162𝐺(𝑅𝑥,ℎ𝑥,ℎ𝑥)=𝜑(𝐺(𝑅𝑥,ℎ𝑥,ℎ𝑥)), (2.30) for all 𝑥∈𝑋. Which means that (𝑆,𝑓), (𝑇,𝑔), and (𝑅,ℎ) are three pairs of continuous 𝜑-weakly commuting mappings in 𝑋. So that all the conditions of Corollary 2.3 are satisfied. Moreover, 0 is the unique common fixed point for all of the mappings 𝑆, 𝑇, 𝑅, 𝑓, 𝑔, and ℎ.

Corollary 2.5. Let (𝑋,𝐺) be a complete 𝐺-metric space, 𝑆,𝑇,𝑅∶𝑋→𝑋 are three self-mappings in 𝑋, and function 𝑄∶[0,+∞)→[0,+∞) satisfies conditions (i)~(iii) and the following condition: 𝐺𝑆𝑥,𝑇𝑦,𝑅𝑧≤𝜙𝑄𝐺𝑥,𝑦,𝑧,∀𝑥,𝑦,𝑧∈𝑋. (2.31) Then 𝑆, 𝑇, and 𝑅 have a unique common fixed point in 𝑋.

Proof. Taking ℎ=𝑔=𝑓=𝐼 in Corollary 2.3, where 𝐼 is an identity mapping. Then the conclusion of Corollary 2.5 can be obtained from Corollary 2.3 immediately. This completes the proof of Corollary 2.5.

Corollary 2.6. Let (𝑋,𝐺) be a complete 𝐺-metric space, 𝑆∶𝑋→𝑋 is a self-mapping in 𝑋, and function 𝑄∶[0,+∞)→[0,+∞) satisfies conditions (i)~(iii) and the following condition: 𝐺𝑆𝑥,𝑆𝑦,𝑆𝑧≤𝜙𝑄𝐺𝑥,𝑦,𝑧,∀𝑥,𝑦,𝑧∈𝑋. (2.32) Then 𝑆 has a unique fixed point in 𝑋.

Proof. Taking 𝑆=𝑇=𝑅 in Corollary 2.5, the conclusion of Corollary 2.6 can be obtained from Corollary 2.5 immediately. This completes the proof of Corollary 2.6.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), and the Physical Experiment Center in Hangzhou Normal University.