On Generalized Weakly 𝐺-Contractive Mappings in Partially Ordered 𝐺-Metric Spaces

Abstract

The aim of this paper is to present some coincidence and common fixed point results for generalized weakly 𝐺-contractive mappings in the setup of partially ordered 𝐺-metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weakly 𝐺-contractive mappings are obtained.

1. Introduction and Mathematical Preliminaries

The concept of a generalized metric space, or a 𝐺-metric space, was introduced by Mustafa et al. [1]. In recent years, many authors have obtained different fixed point theorems for mappings satisfying various contractive conditions on 𝐺-metric spaces. For a survey of fixed point theory, its applications, comparison of different contractive conditions, and related topics in 𝐺-metric spaces we refer the reader to [1–14] and the references mentioned therein.

Definition 1.1 (𝐺-metric space [1]). Let 𝑋 be a nonempty set and 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ be a function satisfying the following properties:(G1)𝐺(𝑥,𝑦,𝑧)=0 if and only 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 𝐺-metric on 𝑋 and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Definition 1.2 (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 one says that the 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 a positive integer 𝑁 such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑁.

Definition 1.3 (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 a positive integer 𝑁 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀, for all 𝑛,𝑚,𝑙≥𝑁, that is, if 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0, as 𝑛,𝑚,𝑙→∞.

Lemma 1.4 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then, the following 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_𝑚,𝑥_𝑛,𝑥)→0, as 𝑚,𝑛→∞.

Lemma 1.5 (see [15]). If (𝑋,𝐺) is a 𝐺-metric space, then {𝑥_𝑛} is a 𝐺-Cauchy sequence if and only if for every 𝜀>0, there exists a positive integer 𝑁 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all 𝑚>𝑛≥𝑁.

Definition 1.6 (see [1]). A 𝐺-metric space (𝑋,𝐺) is said to be 𝐺-complete (or complete 𝐺-metric space) if every 𝐺-Cauchy sequence in (𝑋,𝐺) is convergent in 𝑋.

Definition 1.7 (see [1]). Let (𝑋,𝐺) and (𝑋^′,𝐺^′) be two 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).

The concept of an altering distance function was introduced by Khan et al. [16] as follows.

Definition 1.8. The function 𝜓∶[0,∞)→[0,∞) is called an altering distance function, if the following properties are satisfied.(1)𝜓 is continuous and nondecreasing.(2)𝜓(𝑡)=0 if and only if 𝑡=0.

In [5], Aydi et al. established some common fixed point results for two self-mappings 𝑓 and 𝑔 on a generalized metric space 𝑋. They presented the following definitions.

Definition 1.9 (see [5]). Let (𝑋,𝐺) be a 𝐺-metric space and 𝑓,𝑔∶𝑋→𝑋 be two mappings. We say that 𝑓 is a generalized weakly 𝐺-contraction mapping of type 𝐴 with respect to 𝑔 if for all 𝑥,𝑦,𝑧∈𝑋, the following inequality holds: 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥, (1.1) where(1)𝜓 is an altering distance function;(2)𝜑∶[0,∞)³→[0,∞) is a continuous function with 𝜑(𝑡,𝑠,𝑢)=0 if and only if 𝑡=𝑠=𝑢=0.

Definition 1.10 (see [5]). Let (𝑋,𝐺) be a 𝐺-metric space and 𝑓,𝑔∶𝑋→𝑋 be given mappings. We say that 𝑓 is a generalized weakly 𝐺-contraction mapping of type 𝐵 with respect to 𝑔 if for all 𝑥,𝑦,𝑧∈𝑋, the following inequality holds: 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑔𝑥,𝑓𝑦+𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧+𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑔𝑥,𝑓𝑦,𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧,𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥, (1.2) where(1)𝜓 is an altering distance function;(2)𝜑∶[0,∞)³→[0,∞) is a continuous function with 𝜑(𝑡,𝑠,𝑢)=0 if and only if 𝑡=𝑠=𝑢=0.
Note that the concept of a generalized weakly 𝐺-contraction is the extension of the concept of weakly C-contraction which has been defined by Choudhury in [17]. For more details on weakly C-contractive mappings we refer the reader to [18, 19].

Definition 1.11 (see [20]). Let (𝑋,⪯) be a partially ordered set. A mapping 𝑓 is called a dominating map on 𝑋 if 𝑥⪯𝑓𝑥 for each 𝑥 in 𝑋.

Example 1.12 (see [20]). Let 𝑋=[0,1] be endowed with the usual ordering. Let 𝑓∶𝑋→𝑋 be defined by 𝑓𝑥=𝑥^1/3. Then, 𝑥≤𝑥^1/3=𝑓𝑥 for all 𝑥∈𝑋. Thus, 𝑓 is a dominating map.

Example 1.13 (see [20]). Let 𝑋=[0,∞) be endowed with the usual ordering. Let 𝑓∶𝑋→𝑋 be defined by 𝑓𝑥=√^𝑛𝑥 for 𝑥∈[0,1) and 𝑓𝑥=𝑥^𝑛 for 𝑥∈[1,∞), for any 𝑛∈ℕ. Then, for all 𝑥∈𝑋,𝑥≤𝑓𝑥; that is, 𝑓 is a dominating map.

A subset 𝑊 of a partially ordered set 𝑋 is said to be well ordered if every two elements of 𝑊 be comparable [20].

The following definition is Definition 2.5 of [21], but in the setup of partially ordered 𝐺-metric spaces.

Definition 1.14. Let (𝑋,⪯,𝐺) be a partially ordered 𝐺-metric space. We say that 𝑋 is regular if and only if the following hypothesis holds.
For any nondecreasing sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛→𝑧 as 𝑛→∞, it follows that 𝑥_𝑛⪯𝑧 for all 𝑛∈ℕ.

Jungck in [22] introduced the following definition.

Definition 1.15 (see [22]). Let (𝑋,𝑑) be a metric space and 𝑓,𝑔∶𝑋→𝑋. The pair (𝑓,𝑔) is said to be compatible if and only if lim_𝑛→∞𝑑(𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_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.

Let 𝑋 be a nonempty set and 𝑓∶𝑋→𝑋 be a given mapping. For every 𝑥∈𝑋, let 𝑓⁻¹(𝑥)={𝑢∈𝑋∣𝑓𝑢=𝑥}.

Definition 1.16 (see [21]). Let (𝑋,⪯) be a partially ordered set and 𝑓,𝑔,ℎ∶𝑋→𝑋 are given mappings such that 𝑓𝑋⊆ℎ𝑋 and 𝑔𝑋⊆ℎ𝑋. We say that 𝑓 and 𝑔 are weakly increasing with respect to ℎ if and only if for all 𝑥∈𝑋, we have 𝑓𝑥⪯𝑔𝑦,∀𝑦∈ℎ⁻¹𝑓𝑥,𝑔𝑥⪯𝑓𝑦,∀𝑦∈ℎ⁻¹𝑔𝑥. (1.3)
If 𝑓=𝑔, we say that 𝑓 is weakly increasing with respect to ℎ.

If ℎ=𝐼 (the identity mapping on 𝑋), then the above definition reduces to the weakly increasing mapping [23] (also see [21, 24]).

Definition 1.17. Let (𝑋,𝐺) be a 𝐺-metric space and 𝑓,𝑔∶𝑋→𝑋. The pair (𝑓,𝑔) is said to be compatible if and only if lim_𝑛→∞𝐺(𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_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.

Note that the concept of compatibility in a 𝐺-metric space has been defined by Kumar in [25] (Definition 2.1). In the above definition we only modify his definition, using the fact that 𝐺(𝑥,𝑦,𝑦)≤2𝐺(𝑥,𝑥,𝑦), for all 𝑥,𝑦∈𝑋.

The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly 𝐺-contractive mappings in partially ordered 𝐺-metric spaces.

2. Main Results

From now, we assume Φ=𝜑∣𝜑∶[0,∞)³→[0,∞)isacontinuousfunctionsuchthat𝜑𝑥,𝑦,𝑧=0⟺𝑥=𝑦=𝑧=0. (2.1)

Our first result is the following.

Theorem 2.1. Let (𝑋,⪯,𝐺) be a partially ordered complete 𝐺-metric space. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that 𝑓(𝑋)⊆𝑔(𝑋); 𝑓 is weakly increasing with respect to 𝑔 and 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥 (2.2) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑔𝑥⪯𝑔𝑦⪯𝑔𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then 𝑓 and 𝑔 have a coincidence point in 𝑋 provided that 𝑓 and 𝑔 are continuous and the pair (𝑓,𝑔) is compatible.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point. Since 𝑓(𝑋)⊆𝑔(𝑋), we can construct a sequence {𝑧_𝑛} defined by: 𝑧_𝑛=𝑔𝑥_𝑛=𝑓𝑥_𝑛−1, for all 𝑛≥0.
Now, since 𝑥₁∈𝑔⁻¹(𝑓𝑥₀) and 𝑥₂∈𝑔⁻¹(𝑓𝑥₁), as 𝑓 is weakly increasing with respect to 𝑔, we obtain 𝑔𝑥₁=𝑓𝑥₀⪯𝑓𝑥₁=𝑔𝑥₂⪯𝑓𝑥₂=𝑔𝑥₃. (2.3)
Continuing this process, we get: 𝑔𝑥₁⪯𝑔𝑥₂⪯𝑔𝑥₃⪯⋯⪯𝑔𝑥_𝑛⪯𝑔𝑥_𝑛+1⪯⋯. (2.4)
We complete the proof in three steps.
Step I. We will prove that lim_𝑛→∞𝐺(𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1)=0.
Since 𝑔𝑥_𝑛−1⪯𝑔𝑥_𝑛, using (2.2) we obtain that 𝜓𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1=𝜓𝐺𝑓𝑥_𝑛−1,𝑓𝑥_𝑛,𝑓𝑥_𝑛𝐺≤𝜓𝑔𝑥_𝑛−1,𝑓𝑥_𝑛,𝑓𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛−1,𝑓𝑥_𝑛−13−𝜑𝐺𝑔𝑥_𝑛−1,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛−1,𝑓𝑥_𝑛−1𝐺𝑧=𝜓^𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑧_𝑛,𝑧_𝑛3−𝜑𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛,𝑧_𝑛𝐺𝑧≤𝜓^𝑛−1,𝑧_𝑛,𝑧_𝑛+2𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+13−𝜑𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛,𝑧_𝑛𝐺𝑧≤𝜓^𝑛−1,𝑧_𝑛,𝑧_𝑛+2𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+13. (2.5)
Since 𝜓 is a nondecreasing function, from (2.5), we have 𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1≤𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+13≤𝐺𝑧_𝑛−1,𝑧_𝑛,𝑧_𝑛+2𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+13. (2.6)
Hence, we conclude that {𝐺(𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1)} is a nondecreasing sequence of nonnegative real numbers. Thus, there is an 𝑟≥0 such that lim_𝑛→∞𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1=𝑟. (2.7)
Letting 𝑛→∞ in (2.6), we get that 𝑟≤lim_𝑛→∞𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1+𝑟3≤𝑟, (2.8) that is, lim_𝑛→∞𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1=2𝑟. (2.9)
Again, from (2.5) we have 𝜓𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1𝐺𝑧=𝜓^𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑧_𝑛,𝑧_𝑛3−𝜑𝐺𝑧_𝑛−1,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1,𝐺𝑧_𝑛,𝑧_𝑛,𝑧_𝑛. (2.10)
Letting 𝑛→∞ and using (2.7), (2.9), and the continuities of 𝜓 and 𝜑, we get 𝜓(𝑟)≤𝜓((2𝑟+𝑟+0)/3)−𝜑(2𝑟,𝑟,0), and hence 𝜑(2𝑟,𝑟,0)=0. This gives us that lim_𝑛→∞𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1=0, (2.11) from our assumptions about 𝜑.
Step II. We will show that {𝑧_𝑛} is a 𝐺-Cauchy sequences in 𝑋. So, we will show that for every 𝜀>0, there exists 𝑘∈ℕ such that for all 𝑚,𝑛≥𝑘, 𝐺𝑧_𝑚,𝑧_𝑛,𝑧_𝑛<𝜀. (2.12)
Suppose the above statement is false. Then, there exists 𝜀>0 for which we can find subsequences {𝑧_𝑚(𝑘)} and {𝑧_𝑛(𝑘)} of {𝑧_𝑛} such that 𝑛(𝑘)>𝑚(𝑘)>𝑘 and 𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)≥𝜀, (2.13) where 𝑛(𝑘) is the smallest index with this property, that is, 𝐺𝑧_𝑚(𝑘),𝑧_{𝑛(𝑘)−1},𝑧_{𝑛(𝑘)−1}<𝜀. (2.14)
From rectangle inequality, 𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)≤𝐺𝑧_𝑚(𝑘),𝑧_{𝑛(𝑘)−1},𝑧_{𝑛(𝑘)−1}+𝐺𝑧_{𝑛(𝑘)−1},𝑧_𝑛(𝑘),𝑧_𝑛(𝑘). (2.15)
Making 𝑘→∞ in (2.15), from (2.11), (2.13), and (2.14) we conclude that lim_𝑘→∞𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)=𝜀. (2.16)
Again, from rectangle inequality, 𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+2𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1,𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)≤𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘),𝑧_𝑛(𝑘)+1. (2.17)
Hence in (2.17), if 𝑘→∞, using (2.11), and (2.16), we have lim_𝑘→∞𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1=𝜀. (2.18)
On the other hand, 𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1, (2.19) and 𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑚(𝑘)≤𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑚(𝑘). (2.20)
Hence in (2.19) and (2.20), if 𝑘→∞, from (2.11), (2.16) and (2.18) we have lim_𝑘→∞𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1=𝜀. (2.21)
In a similar way, we have 𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘),𝑧_𝑚(𝑘)+𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤2𝐺𝑧_𝑚(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1+𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1+𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1, (2.22) and therefore, from (2.22) by taking limit when 𝑘→∞, using (2.11) and (2.18), we get that lim_𝑘→∞𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1=𝜀. (2.23)
Also, 𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘),𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1≤𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘). (2.24)
So, from (2.11), (2.23), and (2.24), we have lim_𝑘→∞𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1=𝜀. (2.25)
Finally, 𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1≤𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘)+1,𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1,𝐺𝑧_𝑛(𝑘)+1,𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1≤𝐺𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1≤𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1. (2.26)
Hence in (2.26), if 𝑘→∞ and using (2.11) and (2.25), we have lim_𝑘→∞𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1=𝜀. (2.27)
Since 𝑔𝑥_𝑚(𝑘)⪯𝑔𝑥_𝑛(𝑘)⪯𝑔𝑥_𝑛(𝑘), putting 𝑥=𝑥_𝑚(𝑘), 𝑦=𝑥_𝑛(𝑘), and 𝑧=𝑥_𝑛(𝑘) in (2.2), for all 𝑘≥0, we have 𝜓𝐺𝑧_𝑚(𝑘)+1,𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1=𝜓𝐺𝑓𝑥_𝑚(𝑘),𝑓𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘)𝐺≤𝜓𝑔𝑥_𝑚(𝑘),𝑓𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘)+𝐺𝑔𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘)+𝐺𝑔𝑥_𝑛(𝑘),𝑓𝑥_𝑚(𝑘),𝑓𝑥_𝑚(𝑘)3−𝜑𝐺𝑔𝑥_𝑚(𝑘),𝑓𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘),𝐺𝑔𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘),𝑓𝑥_𝑛(𝑘),𝐺𝑔𝑥_𝑛(𝑘),𝑓𝑥_𝑚(𝑘),𝑓𝑥_𝑚(𝑘)𝐺𝑧≤𝜓^𝑚(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1+𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+13−𝜑𝐺𝑧_𝑚(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1,𝐺𝑧_𝑛(𝑘),𝑧_𝑛(𝑘)+1,𝑧_𝑛(𝑘)+1,𝐺𝑧_𝑛(𝑘),𝑧_𝑚(𝑘)+1,𝑧_𝑚(𝑘)+1. (2.28)
Now, if 𝑘→∞ in (2.28), from (2.11), (2.21), (2.25), and (2.27), we have 𝜓(𝜀)≤𝜓2𝜀3−𝜑(𝜀,0,𝜀). (2.29)
Hence, 𝜀=0 which is a contradiction. Consequently, {𝑧_𝑛} is 𝐺-Cauchy.
Step III. We will show that 𝑓 and 𝑔 have a coincidence point.
Since {𝑔𝑥_𝑛} is a 𝐺-Cauchy sequence in the complete 𝐺-metric space 𝑋, there exists 𝑧∈𝑋 such that lim_𝑛→∞𝐺𝑧_𝑛,𝑧_𝑛,𝑧=lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑧=lim_𝑛→∞𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑧=0. (2.30)
From (2.30) and the continuity of 𝑔, we get lim_𝑛→∞𝐺𝑔𝑧_𝑛,𝑔𝑧_𝑛,𝑔𝑧=lim_𝑛→∞𝐺𝑔𝑔𝑥_𝑛,𝑔𝑔𝑥_𝑛,𝑔𝑧=0. (2.31)
By the rectangle inequality, we have 𝐺𝑔𝑧,𝑓𝑧,𝑓𝑧≤𝐺𝑔𝑧,𝑔𝑔𝑥_𝑛+1,𝑔𝑔𝑥_𝑛+1+𝐺𝑔𝑓𝑥_𝑛,𝑓𝑧,𝑓𝑧≤𝐺𝑔𝑧,𝑔𝑔𝑥_𝑛+1,𝑔𝑔𝑥_𝑛+1+𝐺𝑔𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛+𝐺𝑓𝑔𝑥_𝑛,𝑓𝑧,𝑓𝑧. (2.32)
From (2.30), as 𝑛→∞, we have 𝑔𝑥_𝑛→𝑧,𝑓𝑥_𝑛→𝑧. (2.33)
Since the pair (𝑓,𝑔) is compatible, this implies that lim_𝑛→∞𝐺𝑔𝑓𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛=0. (2.34)
Now, from the continuity of 𝑓 and (2.30), we have lim_𝑛→∞𝐺𝑓𝑧_𝑛,𝑓𝑧,𝑓𝑧=0. (2.35)
Combining (2.31), (2.32), and (2.34) and letting 𝑛→∞ in (2.35), we obtain 𝐺𝑔𝑧,𝑓𝑧,𝑓𝑧≤0, (2.36) which implies that 𝑓𝑧=𝑔𝑧, that is, 𝑧 is a coincidence point of 𝑓 and 𝑔.

In the following theorem, we will omit the continuity of 𝑓 and 𝑔, and the compatibility of the pair (𝑓,𝑔).

Theorem 2.2. Let (𝑋,⪯,𝐺) be a partially ordered 𝐺-metric space. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that 𝑓(𝑋)⊆𝑔(𝑋); 𝑓 is weakly increasing with respect to 𝑔 and 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥, (2.37) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑔𝑥⪯𝑔𝑦⪯𝑔𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then, 𝑓 and 𝑔 have a coincidence point in 𝑋 if 𝑋 is regular and 𝑔(𝑋) is a 𝐺-complete subset of (𝑋,𝐺).

Proof. Following the proof of Theorem 2.1, there exists 𝑧∈𝑋 such that lim_𝑛→∞𝐺𝑧_𝑛,𝑧_𝑛,𝑧=lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑧=lim_𝑛→∞𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑧=0. (2.38)
Since 𝑔(𝑋) is 𝐺-complete and {𝑧_𝑛}⊆𝑔(𝑋), we have 𝑧∈𝑔(𝑋) and hence there exists 𝑢∈𝑋 such that 𝑧=𝑔𝑢 and lim_𝑛→∞𝐺𝑧_𝑛,𝑧_𝑛,𝑔𝑢=lim_𝑛→∞𝐺𝑔𝑥_𝑛,𝑔𝑥_𝑛,𝑔𝑢=lim_𝑛→∞𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝑔𝑢=0. (2.39)
Now, we will prove that 𝑢 is a coincidence point of 𝑓 and 𝑔.
We know that {𝑔𝑥_𝑛} is a nondecreasing sequence in 𝑋. Regularity of 𝑋 yields that 𝑔𝑥_𝑛⪯𝑧=𝑔𝑢. So, from (2.2) we have𝜓𝐺𝑧_𝑛+1,𝑧_𝑛+1,𝑓𝑢=𝜓𝐺𝑓𝑥_𝑛,𝑓𝑥_𝑛𝐺,𝑓𝑢≤𝜓𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛+𝐺g𝑥_𝑛,𝑓𝑢,𝑓𝑢+𝐺𝑔𝑢,𝑓𝑥_𝑛,𝑓𝑥_𝑛3−𝜑𝐺𝑔𝑥_𝑛,𝑓𝑥_𝑛,𝑓𝑥_𝑛,𝐺𝑔𝑥_𝑛,𝑓𝑢,𝑓𝑢,𝐺𝑔𝑢,𝑓𝑥_𝑛,𝑓𝑥_𝑛𝐺𝑧=𝜓^𝑛,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑓𝑢,𝑓𝑢+𝐺𝑔𝑢,𝑧_𝑛+1,𝑧_𝑛+13−𝜑𝐺𝑧_𝑛,𝑧_𝑛+1,𝑧_𝑛+1+𝐺𝑧_𝑛,𝑓𝑢,𝑓𝑢+𝐺𝑔𝑢,𝑧_𝑛+1,𝑧_𝑛+1. (2.40)
Letting 𝑛→∞ in (2.40), from the continuity of 𝜓 and 𝜑, we get 𝜓𝐺𝐺𝑧,𝑧,𝑓𝑢≤𝜓𝑧,𝑓𝑢,𝑓𝑢3−𝜑0,𝐺𝑧,𝑓𝑢,𝑓𝑢,0. (2.41)
As 𝐺(𝑧,𝑓𝑢,𝑓𝑢)≤2𝐺(𝑧,𝑧,𝑓𝑢), we have 𝜓𝐺𝑧,𝑧,𝑓𝑢≤𝜓2𝐺𝑧,𝑧,𝑓𝑢3−𝜑0,𝐺𝑧,𝑓𝑢,f𝑢,0. (2.42)
Hence, 𝜑(0,𝐺(𝑧,𝑓𝑢,𝑓𝑢),0)≤𝜓(2𝐺(𝑧,𝑧,𝑓𝑢)/3)−𝜓(𝐺(𝑧,𝑧,𝑓𝑢))≤0. So, 𝐺(𝑧,𝑓𝑢,𝑓𝑢)=0 and hence, 𝑔𝑢=𝑧=𝑓𝑢. This means that 𝑔 and 𝑓 have a coincidence point.

Taking 𝑔=𝐼_𝑋 (the identity mapping on 𝑋) and 𝜓=𝐼_[0,∞) in the above theorems, we obtain the following fixed point result.

Corollary 2.3. Let (𝑋,⪯,𝐺) be a partially ordered complete 𝐺-metric space. Let 𝑓∶𝑋→𝑋 be a mapping such that 𝑓𝑥⪯𝑓(𝑓𝑥), for all 𝑥∈𝑋 and 𝐺≤𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑧,𝑓𝑥,𝑓𝑥3−𝜑𝐺𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑧,𝑓𝑥,𝑓𝑥, (2.43) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑥⪯𝑦⪯𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then, 𝑓 has a fixed point in 𝑋 provided that one of the following two conditions is satisfied:(a)𝑓 is continuous, or,(b)𝑋 is regular.

Taking 𝜑(𝑥,𝑦,𝑧)=(1/3−𝛼)(𝑥+𝑦+𝑧), where 𝛼∈[0,1/3), in the above corollary, we obtain the following result.

Corollary 2.4. Let (𝑋,⪯,𝐺) be a partially ordered complete 𝐺-metric space. Let 𝑓∶𝑋→𝑋 be a mapping such that 𝑓𝑥⪯𝑓(𝑓𝑥), for all 𝑥∈𝑋 and 𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝛼𝐺𝑥,𝑓𝑥,𝑓𝑥+𝐺𝑦,𝑓𝑦,𝑓𝑦+𝐺𝑧,𝑓𝑧,𝑓𝑧, (2.44) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑥⪯𝑦⪯𝑧, where 𝛼∈[0,1/3). Then, 𝑓 has a fixed point in 𝑋 if one of the following two conditions is satisfied:(a)𝑓 is continuous, or,(b)𝑋 is regular.

Theorem 2.5. Under the hypotheses of Theorem 2.1, 𝑓 and 𝑔 have a common fixed point in 𝑋 if 𝑔 is a nondecreasing dominating map.
Moreover, the set of common fixed points of 𝑓 and 𝑔 is well ordered if and only if 𝑓 and 𝑔 have one and only one common fixed point.

Proof. Following the proof of the Theorem 2.1 we obtain that the sequence {𝑧_𝑛} is 𝐺-convergent to 𝑧 and 𝑓𝑧=𝑔𝑧. Since 𝑓 and 𝑔 are weakly compatible (since the pair (𝑓,𝑔) is compatible), we have 𝑓𝑔𝑧=𝑔𝑓𝑧. Let 𝑤=𝑔𝑧=𝑓𝑧. Therefore, we have 𝑓𝑤=𝑔𝑤. (2.45) As 𝑔 is a nondecreasing dominating map, 𝑧⪯𝑔𝑧⪯𝑔𝑔𝑧=𝑔𝑤. (2.46)
If 𝑧=𝑤, then 𝑧 is a common fixed point. If 𝑧≠𝑤, then, since from (2.46) 𝑔𝑧⪯𝑔𝑤, from (2.2) we have 𝜓𝐺𝐺𝑓𝑧,𝑓𝑧,𝑓𝑤≤𝜓𝑔𝑧,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑤,𝑓𝑤+𝐺𝑔𝑤,𝑓𝑧,𝑓𝑧3𝐺𝐺−𝜑𝑔𝑧,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑤,𝑓𝑤,𝐺𝑔𝑤,𝑓𝑧,𝑓𝑧≤𝜓𝑓𝑧,𝑓𝑧,𝑓𝑧+𝐺𝑓𝑧,𝑓𝑤,𝑓𝑤+𝐺𝑓𝑤,𝑓𝑧,𝑓𝑧3−𝜑𝐺𝑓𝑧,𝑓𝑧,𝑓𝑧,𝐺𝑓𝑧,𝑓𝑤,𝑓𝑤,𝐺𝑓𝑤,𝑓𝑧,𝑓𝑧≤𝜓2𝐺𝑓𝑧,𝑓𝑧,𝑓𝑤+𝐺𝑓𝑧,𝑓𝑧,𝑓𝑤3−𝜑0,𝐺𝑓𝑧,𝑓𝑤,𝑓𝑤,𝐺𝑓𝑤,𝑓𝑧,𝑓𝑧. (2.47)
Therefore, 𝜑(0,𝐺(𝑓𝑧,𝑓𝑤,𝑓𝑤),𝐺(𝑓𝑤,𝑓𝑧,𝑓𝑧))=0. So, 𝑓z=𝑓𝑤. Now, since 𝑤=𝑔𝑧=𝑓𝑧 and 𝑓𝑤=𝑔𝑤, we have 𝑤=𝑔𝑤=𝑓𝑤. This completes the proof.
Suppose that the set of common fixed points of 𝑓 and 𝑔 is well ordered. We claim that common fixed point of 𝑓 and 𝑔 is unique. Assume on contrary that, 𝑓𝑢=𝑔𝑢=𝑢 and 𝑓𝑣=𝑔𝑣=𝑣, and 𝑢≠𝑣. Without any loss of generality, we may assume that 𝑔𝑢=𝑢⪯𝑣=𝑔𝑣. Using (2.2), we obtain 𝜓(𝐺(𝐺𝐺𝑢,𝑢,𝑣))=𝜓𝑓𝑢,𝑓𝑢,𝑓𝑣≤𝜓𝑔𝑢,𝑓𝑢,𝑓𝑢+𝐺𝑔𝑢,𝑓𝑣,𝑓𝑣+𝐺𝑔𝑣,𝑓𝑢,𝑓𝑢3𝐺()()−𝜑𝑔𝑢,𝑓𝑢,𝑓𝑢,𝐺𝑔𝑢,𝑓𝑣,𝑓𝑣,𝐺𝑔𝑣,𝑓𝑢,𝑓𝑢≤𝜓2𝐺𝑣,𝑢,𝑢+𝐺𝑣,𝑢,𝑢3−𝜑(0,𝐺(𝑢,𝑣,𝑣),𝐺(𝑣,𝑢,𝑢)). (2.48)
Therefore, 𝑢=𝑣, a contradiction. Conversely, if 𝑓 and 𝑔 have only one common fixed point then, clearly, the set of common fixed points of 𝑓 and 𝑔 is well ordered.

Theorem 2.6. Under the hypotheses of Theorem 2.2, 𝑓 and 𝑔 have a common fixed point in 𝑋 provided that 𝑓 and 𝑔 are weakly compatible and 𝑔 is a nondecreasing dominating map.
Moreover, the set of common fixed points of 𝑓 and 𝑔 is well ordered if and only if 𝑓 and 𝑔 have one and only one common fixed point.

Proof. The proof is done as in Theorem 2.5.

Following arguments similar to those given in the proof of Theorems 2.1 and 2.2, we have the following results for a generalized weakly 𝐺-contractive mapping of type 𝐵.

Theorem 2.7. Let (𝑋,⪯,𝐺) be a partially ordered complete 𝐺-metric space. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that 𝑓(𝑋)⊆𝑔(𝑋)𝑓 is weakly increasing with respect to 𝑔 and 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑔𝑥,𝑓𝑦+𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧+𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑔𝑥,𝑓𝑦,𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧,𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥, (2.49) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑔𝑥⪯𝑔𝑦⪯𝑔𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then f and 𝑔 have a coincidence point in 𝑋 provided that 𝑓 and 𝑔 are continuous and the pair (𝑓,𝑔) is compatible.
Moreover, 𝑓 and 𝑔 have a common fixed point in 𝑋 if 𝑔 is a nondecreasing dominating map.
Also, the set of common fixed points of 𝑓 and 𝑔 is well ordered if and only if 𝑓 and 𝑔 have one and only one common fixed point.

Theorem 2.8. Let (𝑋,⪯,𝐺) be a partially ordered 𝐺-metric space. Let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that 𝑓(𝑋)⊆𝑔(𝑋)𝑓 is weakly increasing with respect to 𝑔 and 𝜓𝐺𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝜓𝑔𝑥,𝑔𝑥,𝑓𝑦+𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧+𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥3−𝜑𝐺𝑔𝑥,𝑔𝑥,𝑓𝑦,𝐺𝑔𝑦,𝑔𝑦,𝑓𝑧,𝐺𝑔𝑧,𝑔𝑧,𝑓𝑥, (2.50) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑔𝑥⪯𝑔𝑦⪯𝑔𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then 𝑓 and 𝑔 have a coincidence point in 𝑋 provided that 𝑋 is regular and 𝑔(𝑋) is a 𝐺-complete subset of (𝑋,𝐺).
Moreover, 𝑓 and 𝑔 have a common fixed point in 𝑋 if 𝑓 and 𝑔 are weakly compatible and 𝑔 is a nondecreasing dominating map.
Also, the set of common fixed points of 𝑓 and 𝑔 is well ordered if and only if 𝑓 and 𝑔 have one and only one common fixed point.

The following corollary is an immediate consequence of the above theorems.

Corollary 2.9. Let (𝑋,⪯,𝐺) be a partially ordered complete 𝐺-metric space. Let 𝑓∶𝑋→𝑋 be a mapping such that 𝑓𝑥⪯𝑓(𝑓𝑥), for all 𝑥∈𝑋 and 𝐺≤𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧𝑥,𝑥,𝑓𝑦+𝐺𝑦,𝑦,𝑓𝑧+𝐺𝑧,𝑧,𝑓𝑥3−𝜑𝐺𝑥,𝑥,𝑓𝑦,𝐺𝑦,𝑦,𝑓𝑧,𝐺𝑧,𝑧,𝑓𝑥, (2.51) for every 𝑥,𝑦,𝑧∈𝑋 such that 𝑥⪯𝑦⪯𝑧, where 𝜓∶[0,∞)→[0,∞) is an altering distance function and 𝜑∈Φ. Then 𝑓 has a fixed point in 𝑋 provided that one of the following two conditions is satisfied:(a)𝑓 is continuous, or,(b)𝑋 is regular.

Example 2.10. Let 𝑋=[0,∞) be endowed with the usual order in ℝ and 𝐺 on 𝑋 be given as 𝐺𝑥,𝑦,𝑧=max𝑥−𝑦,𝑦−𝑧,|𝑧−𝑥|. (2.52)
Define 𝑓,𝑔∶𝑋→𝑋 as 𝑓(𝑥)𝑔(𝑥)==1,2−𝑥²,if0≤𝑥≤√20,if𝑥>√2, (2.53) for all 𝑥∈𝑋.
Define 𝜓∶[0,∞)→[0,∞) by 𝜓(𝑡)=(1/4)𝑡² and 𝜑∶[0,∞)³→[0,∞) by 𝜑(𝑠,𝑡,𝑢)=(1/100)(𝑠+𝑡+𝑢)².
Let 0≤𝑥≤𝑦≤𝑧≤√2. Now, we have 𝜓𝐺1𝑓𝑥,𝑓𝑦,𝑓𝑧=0≤4𝑥²−1+𝑦²−1+𝑧²−13²−1𝑥100²−1+𝑦²−1+𝑧²−1²≤143−𝑥²−𝑦²−𝑧²3²−11003−𝑥²−𝑦²−𝑧²²1=𝜓3𝐺𝐺.𝑔𝑥,𝑓𝑥,𝑓𝑥+𝐺𝑔𝑦,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑧,𝑓𝑧,𝑓𝑧−𝜑𝑔𝑥,𝑓𝑥,𝑓𝑥,𝐺𝑔𝑦,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑧,𝑓𝑧,𝑓𝑧 (2.54)
There are other 3 cases as follows:(1)0≤𝑥≤𝑦≤1 and √2<𝑧.(2)0≤𝑥≤√2 and √2<𝑦≤𝑧.(3)√2<𝑥≤𝑦≤𝑧.
By a careful calculation for the remained cases above, we see that all the conditions of Theorems 2.1 and 2.5 are satisfied. Moreover, (1) is the unique common fixed point of 𝑓 and 𝑔.

Denote by Λ the set of all functions 𝜇∶[0,+∞)→[0,+∞) verifying the following conditions:(I)𝜇 is a positive Lebesgue integrable mapping on each compact subset of [0,+∞).(II)for all 𝜀>0, ∫^𝜀₀𝜇(𝑡)𝑑𝑡>0.

Other consequences of the main theorems are the following results for mappings satisfying a contraction of integral type.

Corollary 2.11. Replace the contractive condition (2.2) of Theorem 2.1 by the following condition.
There exists a 𝜇∈Λ such that ₀^{𝜓(𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑧))}𝜇(𝑡)𝑑𝑡≤₀^{𝜓((𝐺(𝑔𝑥,𝑓𝑦,𝑓𝑦)+𝐺(𝑔𝑦,𝑓𝑧,𝑓𝑧)+𝐺(𝑔𝑧,𝑓𝑥,𝑓𝑥))/3)}𝜇(𝑡)−𝑑𝑡_{0𝜑(𝐺(𝑔𝑥,𝑓𝑦,𝑓𝑦),𝐺(𝑔𝑦,𝑓𝑧,𝑓𝑧),𝐺(𝑔𝑧,𝑓𝑥,𝑓𝑥))}𝜇(𝑡)𝑑𝑡. (2.55)
Then, 𝑓 and 𝑔 have a coincidence point, if the other conditions of Theorem 2.1 are satisfied.

Proof. Consider the function Γ(𝑥)=∫^𝑥₀𝜇(𝑡)𝑑𝑡. Then, (2.55) becomes Γ𝜓𝐺𝜓𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤Γ𝑔𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥3−Γ𝜑𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥. (2.56)
Taking 𝜓₁=Γ∘𝜓 and 𝜑₁=Γ∘𝜑 and applying Theorem 2.1, we obtain the proof (it is easy to verify that 𝜓₁ is an altering distance function and 𝜑₁∈Φ).

Similar to [21], let 𝑁∈𝑁^∗ be fixed. Let {𝜇_𝑖}_{1≤𝑖≤𝑁} be a family of 𝑁 functions which belong to Λ. For all 𝑡≥0, we define 𝐼₁(𝑡)=^𝑡₀𝜇₁(𝑠)𝐼𝑑𝑠,₂(𝑡)=^𝐼1𝑡₀𝜇₂(𝑠)𝑑𝑠=^{∫𝑡0𝜇1}₀^{(𝑠)𝑑𝑠}𝜇₂(𝑠)𝐼𝑑𝑠,₃(𝑡)=^𝐼2𝑡₀𝜇₃(𝑠)𝑑𝑠=^{∫∫𝑡0𝜇10(𝑠)𝑑𝑠𝜇2}₀^{(𝑠)𝑑𝑠}𝜇₃(𝑠)⋮𝐼𝑑𝑠,_𝑁(𝑡)=^{𝐼(𝑁−1)𝑡}₀𝜇_𝑁(𝑠)𝑑𝑠. (2.57) We have the following result.

Corollary 2.12. Replace the inequality (2.2) of Theorem 2.1 by the following condition: 𝐼_𝑁𝜓𝐺𝑓𝑥,𝑓𝑦,𝑓𝑧≤𝐼_𝑁𝜓𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦+𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧+𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥3−𝐼_𝑁𝜑𝐺𝑔𝑥,𝑓𝑦,𝑓𝑦,𝐺𝑔𝑦,𝑓𝑧,𝑓𝑧,𝐺𝑔𝑧,𝑓𝑥,𝑓𝑥. (2.58)
Then, 𝑓 and 𝑔 have a coincidence point if the other conditions of Theorem 2.1 are satisfied.

Proof. Consider Ψ=𝐼_𝑁∘𝜓 and Φ=𝐼_𝑁∘𝜑.

3. Periodic Point Results

Let 𝐹(𝑓)={𝑥∈𝑋∶𝑓𝑥=𝑥}, the fixed point set of 𝑓.

Clearly, a fixed point of 𝑓 is also a fixed point of 𝑓^𝑛 for every 𝑛∈ℕ; that is, 𝐹(𝑓)⊂𝐹(𝑓^𝑛). However, the converse is false. For example, the mapping 𝑓∶ℕ→ℕ, defined by 𝑓𝑥=1/2−𝑥 has the unique fixed point 1/4, but every 𝑥∈ℕ is a fixed point of 𝑓². If 𝐹(𝑓)=𝐹(𝑓^𝑛) for every 𝑛∈ℕ, then 𝑓 is said to have property 𝑃. For more details, we refer the reader to [6, 26–28] and the references mentioned therein.

Theorem 3.1. Let 𝑋 and 𝑓 be as in Corollary 2.3. If 𝑓 is a dominating map on 𝑋, then 𝑓 has property 𝑃.

Proof. From Corollary 2.3, 𝐹(𝑓)≠∅. Let 𝑢∈𝐹(𝑓^𝑛) for some 𝑛>1. We will show that 𝑢=𝑓𝑢. Since 𝑓 is dominating on 𝑋, we have 𝑢⪯𝑓𝑢, which implies that 𝑓^𝑛−1𝑢⪯𝑓^𝑛𝑢, as 𝑓 is nondecreasing. Using (2.2), we obtain that 𝐺𝑢,𝑓𝑢,𝑓𝑢=𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢=𝐺𝑓𝑓^𝑛−1𝑢,𝑓𝑓^𝑛𝑢,𝑓𝑓^𝑛𝑢≤13𝐺𝑓^𝑛−1𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢+𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢+𝐺𝑓^𝑛𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢−𝜑𝐺𝑓^𝑛−1𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,𝐺𝑓^𝑛𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢≤13𝐺𝑓^𝑛−1𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢+2𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢𝐺𝑓+0−𝜑^𝑛−1𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,0, (3.1) that is, 𝐺𝑢,𝑓𝑢,𝑓𝑢=𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢≤𝐺𝑓^𝑛−1𝑢,𝑓^𝑛𝑢,𝑓^𝑛𝑢−3𝜑𝐺𝑓^𝑛−1𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,𝐺𝑓^𝑛𝑢,𝑓^𝑛+1𝑢,𝑓^𝑛+1𝑢,0. (3.2)
Repeating the above process, we get 𝐺𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢≤𝐺𝑓^{𝑛−(𝑖+1)}𝑢,𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖)}𝑢−3𝜑𝐺𝑓^{𝑛−(𝑖+1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢𝐺𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,0. (3.3)
From the above inequalities, we have 𝐺𝑢,𝑓𝑢,𝑓𝑢≤𝐺𝑢,𝑓𝑢,𝑓𝑢−3^𝑛−1_𝑖=0𝜑𝐺𝑓^{𝑛−(𝑖+1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝐺𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,0. (3.4)
Therefore, ^𝑛−1_𝑖=0𝜑𝐺𝑓^{𝑛−(𝑖+1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝐺𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,0=0, (3.5) which from our assumptions about 𝜑 implies that 𝐺𝑓^{𝑛−(𝑖+1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢=𝐺𝑓^{𝑛−(𝑖)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢,𝑓^{𝑛−(𝑖−1)}𝑢=0 (3.6) for all 0≤𝑖≤𝑛−1. Now, taking 𝑖=𝑛−1, we have 𝑢=𝑓𝑢.

Analogously, we have the following theorem.

Theorem 3.2. Let 𝑋 and 𝑓 be as in Corollary 2.12. If 𝑓 is a dominating map on 𝑋, then 𝑓 has property 𝑃.