Tripled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces

Abstract

Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example. The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

1. Introduction and Preliminaries

In recent times coupled fixed point theory has experienced a rapid growth in partially ordered metric spaces. The speciality of this line of research is that the problems herein utilize both order theoretic and analytic methods. References [1–19] are some instances of these works.

Definition 1.1 (see [14]). A function 𝑔∶𝑅→𝑅 is said to be monotone nondecreasing (or increasing) if 𝑥≥𝑦 implies 𝑔(𝑥)≥𝑔(𝑦).

Definition 1.2 (see [14]). Let 𝑋 be a nonempty set. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping. An element (𝑥,𝑦) is called a coupled fixed point of 𝐹 if 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (1.1)

Recently, Berinde and Borcut [20] extended the idea of coupled fixed points to tripled fixed points. The definition is as follows.

Definition 1.3 (see [20]). Let 𝑋 be a nonempty set. Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be a mapping. An element (𝑥,𝑦,𝑧) is called a tripled fixed point of 𝐹 if 𝐹𝑥,𝑦,𝑧=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑥=𝑧. (1.2)
They also extended the mixed monotone property to functions with three arguments.

Definition 1.4 (see [20]). Let (𝑋,⪯) be a partially ordered set and 𝐹∶𝑋×𝑋×𝑋→𝑋. The mapping 𝐹 is said to have the mixed monotone property if for any 𝑥,𝑦,𝑧∈𝑋𝑥₁,𝑥₂∈𝑋,𝑥₁⪯𝑥₂⟹𝐹𝑥₁,𝑦,𝑧⪯𝐹𝑥₂,𝑦,𝑦,𝑧₁,𝑦₂∈𝑋,𝑦₁⪯𝑦₂⟹𝐹𝑥,𝑦₁,𝑧⪰𝐹𝑥,𝑦₂,𝑧,𝑧₁,𝑧₂∈𝑋,𝑧₁⪯𝑧₂⟹𝐹𝑥,𝑦,𝑧₁⪯𝐹𝑥,𝑦,𝑧₂. (1.3)

Our purpose here is to establish tripled coincidence point results in metric spaces with partial ordering. For that purpose we define mixed 𝑔-monotone property in the following. Mixed g-monotone property was already defined in the context of coupled fixed points [14]. Here in the spirit of Definition 1.4 we have made an extension of that.

Definition 1.5. Let (𝑋,⪯) be a partially ordered set. Let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋×𝑋→𝑋. The mapping 𝐹 is said to have the mixed 𝑔-monotone property if for any 𝑥,𝑦,𝑧∈𝑋. 𝑥₁,𝑥₂∈𝑋,𝑔𝑥₁⪯𝑔𝑥₂⟹𝐹𝑥₁,𝑦,𝑧⪯𝐹𝑥₂,𝑦,𝑦,𝑧₁,𝑦₂∈𝑋,𝑔𝑦₁⪯𝑔𝑦₂⟹𝐹𝑥,𝑦₁,𝑧⪰𝐹𝑥,𝑦₂,𝑧,𝑧₁,𝑧₂∈𝑋,𝑔𝑧₁⪯𝑔𝑧₂⟹𝐹𝑥,𝑦,𝑧₁⪯𝐹𝑥,𝑦,𝑧₂. (1.4)

Coupled coincidence point was defined by Lakshmikantham and Ćirić [14]. We also extend the concept of coupled coincidence point to tripled coincidence point in the following.

Definition 1.6. Let 𝑋 be any nonempty set. Let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋×𝑋→𝑋. An element (𝑥,𝑦,𝑧) is called a tripled coincidence point of 𝑔 and 𝐹 if 𝐹𝑥,𝑦,𝑧=𝑔𝑥,𝐹𝑦,𝑥,𝑦=𝑔𝑦,𝐹𝑧,𝑦,𝑥=𝑔𝑧. (1.5)

We extend the concept of commuting mappings given by Lakshmikantham and Ćirić [14], in the following definition.

Definition 1.7. Let 𝑋 be a nonempty set. Then one says that the mappings 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋×𝑋→𝑋 are commuting if for all 𝑥,𝑦,𝑧∈𝑋𝑔𝐹𝑥,𝑦,𝑧=𝐹𝑔𝑥,𝑔𝑦,𝑔𝑧. (1.6)

The following is the definition of compatible mappings which is an extension of the compatibility defined by Choudhury and Kundu in [8].

Definition 1.8 (see [8]). Let (𝑋,𝑑) be a metric space. The mappings 𝑔 and 𝐹, where 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋×𝑋→𝑋 are said to be compatible if lim_𝑛→∞𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛=0,lim_𝑛→∞𝑑𝑔𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑔𝑦_𝑛=0,lim_𝑛→∞𝑑𝑔𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑔𝑧_𝑛,𝑔𝑦_𝑛,𝑔𝑥_𝑛=0, (1.7) whenever {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} are sequences in 𝑋 such that lim_𝑛→∞𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛=𝑔𝑦_𝑛=𝑦,lim_𝑛→∞𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛=𝑔𝑧_𝑛=𝑧. (1.8)

2. Main Results

Theorem 2.1. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose 𝐹∶𝑋×𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are such that, 𝑔 is monotone increasing, 𝐹 has the mixed 𝑔-monotone property and 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝜓max𝑑𝑔𝑥,𝑔𝑢,𝑑𝑔𝑦,𝑔𝑣,𝑑𝑔𝑧,𝑔𝑤 (2.1) for all 𝑥,𝑦,𝑧∈𝑋 for which 𝑔𝑥⪯𝑔𝑢, 𝑔𝑦≽𝑔𝑣 and 𝑔𝑧⪯𝑔𝑤, where 𝜓∶[0,+∞)→[0,+∞) is such that 𝜓(𝑡) is monotone, 𝜓(𝑡)<𝑡 and lim_𝑟→𝑡+𝜓(𝑟)<𝑡 for all 𝑡>0. Suppose 𝐹(𝑋×𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous, and {𝑔,𝐹} is a compatible pair. Suppose either (a)𝐹is continuous or (b)𝑋 has the following properties: (i)if a nondecreasing sequence {𝛼_𝑛}→𝛼, then 𝛼_𝑛⪯𝛼 for all 𝑛,(ii)if a nonincreasing sequence {𝛽_𝑛}→𝛽, then 𝛽_𝑛≽𝛽 for all 𝑛.
If there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑔𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑔𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑔𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), then there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝐹𝑥,𝑦,𝑧=𝑔𝑥,𝐹𝑦,𝑥,𝑦=𝑔𝑦,𝐹𝑧,𝑦,𝑥=𝑔𝑧, (2.2) that is, 𝑔 and 𝐹 have a tripled coincidence point.

Proof. By a condition of the theorem, there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑔𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑔𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑔𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀). Since 𝐹(𝑋×𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₁,𝑦₁,𝑧₁∈𝑋 such that 𝑔𝑥₁=𝐹𝑥₀,𝑦₀,𝑧₀,𝑔𝑦₁=𝐹𝑦₀,𝑥₀,𝑦₀,𝑔𝑧₁=𝐹𝑧₀,𝑦₀,𝑥₀. (2.3) Continuing this process, we can construct sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} in 𝑋 such that 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑔𝑧_𝑛+1=𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛. (2.4) Next we will show that, for 𝑛≥0, 𝑔𝑥_𝑛⪯𝑔𝑥_𝑛+1,𝑔𝑦_𝑛⪰𝑔𝑦_𝑛+1,𝑔𝑧_𝑛⪯𝑔𝑧_𝑛+1. (2.5) Since, 𝑔𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀),𝑔𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑔𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), by (2.3), we get 𝑔𝑥₀⪯𝑔𝑥₁,𝑔𝑦₀⪰𝑔𝑦₁,𝑔𝑧₀⪯𝑔𝑧₁, (2.6) that is, (2.5) holds for 𝑛=0.
We presume that (2.5) holds for some 𝑛=𝑚>0. As 𝐹 has the mixed 𝑔-monotone property and 𝑔𝑥_𝑚⪯𝑔𝑥_𝑚+1,𝑔𝑦_𝑚≽𝑔𝑦_𝑚+1 and 𝑔𝑧_𝑚⪯𝑔𝑧_𝑚+1, we obtain 𝑔𝑥_𝑚+1=𝐹𝑥_𝑚,𝑦_𝑚,𝑧_𝑚⪯𝐹𝑥_𝑚+1,𝑦_𝑚,𝑧_𝑚⪯𝐹𝑥_𝑚+1,𝑦_𝑚,𝑧_𝑚+1⪯𝐹𝑥_𝑚+1,𝑦_𝑚+1,𝑧_𝑚+1=𝑔𝑥_𝑚+2,𝑔𝑦_𝑚+1=𝐹𝑦_𝑚,𝑥_𝑚,𝑦_𝑚⪰𝐹𝑦_𝑚,𝑥_𝑚,𝑦_𝑚+1⪰𝐹𝑦_𝑚+1,𝑥_𝑚,𝑦_𝑚+1⪰𝐹𝑦_𝑚+1,𝑥_𝑚+1,𝑦_𝑚+1=𝑔𝑦_𝑚+2,𝑔𝑧_𝑚+1=𝐹𝑧_𝑚,𝑦_𝑚,𝑥_𝑚⪯𝐹𝑧_𝑚+1,𝑦_𝑚,𝑥_𝑚⪯𝐹𝑧_𝑚+1,𝑦_𝑚+1,𝑥_𝑚⪯𝐹𝑧_𝑚+1,𝑦_𝑚+1,𝑥_𝑚+1=𝑔𝑧_𝑚+2. (2.7)(2.8)(2.9) Thus, (2.5) holds for 𝑛=𝑚+1. Then, by induction, we conclude that (2.5) holds for 𝑛≥1.
If for some 𝑛∈ℕ, 𝑔𝑥_𝑛=𝑔𝑥_𝑛+1,𝑔𝑦_𝑛=𝑔𝑦_𝑛+1,𝑔𝑧_𝑛=𝑔𝑧_𝑛+1, (2.10) then, by (2.4), (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛) is a tripled coincidence point of 𝑔 and 𝐹. Therefore we assume, for any 𝑛∈ℕ, 𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛≠𝑔𝑥_𝑛+1,𝑔𝑦_𝑛+1,𝑔𝑧_𝑛+1. (2.11) Set 𝛿_𝑛=max{𝑑(𝑔𝑥_𝑛,𝑔𝑥_𝑛+1),𝑑(𝑔𝑦_𝑛,𝑔𝑦_𝑛+1),𝑑(𝑔𝑧_𝑛,𝑔𝑧_𝑛+1)}.
Then 𝛿_𝑛>0∀𝑛≥0. (2.12) Then, by (2.1), (2.4) and (2.5), we have 𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=𝑑𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑧_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛≤𝜓max𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛,𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑑𝑔𝑧_𝑛−1,𝑔𝑧_𝑛,𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=𝑑𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛≤𝜓max𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛,𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑑𝑔𝑧_𝑛,𝑔𝑧_𝑛+1=𝑑𝐹𝑧_𝑛−1,𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛≤𝜓max𝑑𝑔𝑧_𝑛−1,𝑔𝑧_𝑛,𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛. (2.13) Thus, from (2.13) we obtain that 𝛿_𝑛=max𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1,𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1,𝑑𝑔𝑧_𝑛,𝑔𝑧_𝑛+1≤𝜓max𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛,𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑑𝑔𝑧_𝑛−1,𝑔𝑧_𝑛. (2.14) It then follows from (2.12) and a property 𝜓, that for all 𝑛≥1, 𝛿_𝑛≤𝜓𝛿_𝑛−1<𝛿_𝑛−1. (2.15) Thus, {𝛿_𝑛} is a monotone decreasing sequence of nonnegative real numbers. So, there exist a 𝛿≥0 such that lim_𝑛→∞𝛿_𝑛=𝛿. (2.16) Suppose 𝛿>0. Letting 𝑛→∞ in (2.14), using (2.15), (2.16), and a property of 𝜓, we get 𝛿≤𝜓(𝛿)<𝛿, (2.17) which is a contradiction. Thus 𝛿=0, or lim_𝑛→∞𝛿_𝑛=0, (2.18) or lim_𝑛→∞𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛=0,lim_𝑛→∞𝑑𝑔𝑦_𝑛+1,𝑔𝑦_𝑛=0,lim_𝑛→∞𝑑𝑔𝑧_𝑛+1,𝑔𝑧_𝑛=0. (2.19) Now, we will prove that {𝑔𝑥_𝑛}, {𝑔𝑦_𝑛}, and {𝑔𝑧_𝑛} are Cauchy sequences. Suppose, to the contrary, that at least one of {𝑔𝑥_𝑛}, {𝑔𝑦_𝑛}, and {𝑔𝑧_𝑛} is not a Cauchy sequence. So, there exists an 𝜀>0 for which we can find subsequences {𝑔𝑥_𝑛(𝑘)} of {𝑔𝑥_𝑛}, {𝑔𝑦_𝑛(𝑘)} of {𝑔𝑦_𝑛}, and {𝑔𝑧_𝑛(𝑘)} of {𝑔𝑧_𝑛} with 𝑛(𝑘)>𝑚(𝑘)≥𝑘 such that 𝛼_𝑘=max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)≥𝜀. (2.20) Additionally, corresponding to 𝑚(𝑘), we may choose 𝑛(𝑘) such that it is the smallest integer satisfying (2.20). Then, for all 𝑘≥0, max𝑑𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_{𝑛(𝑘)−1},𝑔𝑧_𝑚(𝑘)<𝜀. (2.21) By using (2.20) and (2.21) we have for 𝑘≥0, 𝜀≤𝛼_𝑘=max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)≤max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_{𝑛(𝑘)−1}+𝑑𝑔𝑥_{𝑛(𝑘)−1},𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_{𝑛(𝑘)−1}+𝑑𝑔𝑦_{𝑛(𝑘)−1},𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_{𝑛(𝑘)−1}+𝑑𝑔𝑧_{𝑛(𝑘)−1},𝑔𝑧_𝑚(𝑘)≤max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_{𝑛(𝑘)−1},𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_{𝑛(𝑘)−1},𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_{𝑛(𝑘)−1}+𝜀≤𝛿_{𝑛(𝑘)−1}+𝜀. (2.22) Letting 𝑘→∞ in (2.22), and using (2.19), we get lim_𝑘→∞𝛼_𝑘=lim_𝑘→∞max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)=𝜀. (2.23) Let, for 𝑘≥0, 𝛽_𝑘=max𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1. (2.24)
Again, for all 𝑘≥0, 𝛼_𝑘=max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)≤max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘)+1+𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝑑𝑔𝑥_𝑚(𝑘)+1,𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘)+1+𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1+𝑑𝑔𝑦_𝑚(𝑘)+1,𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑛(𝑘)+1+𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1+𝑑𝑔𝑧_𝑚(𝑘)+1,𝑔𝑧_𝑚(𝑘)≤max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑛(𝑘)+1+max𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1+max𝑑𝑔𝑥_𝑚(𝑘),𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑚(𝑘),𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑚(𝑘),𝑔𝑧_𝑚(𝑘)+1≤𝛿_𝑛(𝑘)+1+𝛽_𝑘+𝛿_𝑚(𝑘)+1. (2.25) Analogously we have for 𝑘≥0,𝛽_𝑘=max𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘)+,𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1≤max𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑛(𝑘)+𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑥_𝑚(𝑘),𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑛(𝑘)+𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘)+𝑑𝑔𝑦_𝑚(𝑘),𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑛(𝑘)+𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)+𝑑𝑔𝑧_𝑚(𝑘),𝑔𝑧_𝑚(𝑘)+1≤max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑛(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑛(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑛(𝑘)+1+max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)+max𝑑𝑔𝑥_𝑚(𝑘),𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑚(𝑘),𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑚(𝑘),𝑔𝑧_𝑚(𝑘)+1≤𝛿_𝑛(𝑘)+1+𝛼_𝑘+𝛿_𝑚(𝑘)+1. (2.26) Letting 𝑘→∞ in (2.25) and (2.26), we get that lim_𝑘→∞max𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1,𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1,𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1=lim_𝑘→∞𝛽_𝑘=𝜀=lim_𝑘→∞𝛼_𝑘. (2.27) Since 𝑛(𝑘)>𝑚(𝑘), for 𝑘≥0, we have 𝑔𝑥_𝑛(𝑘)⪰𝑔𝑥_𝑚(𝑘),𝑔𝑦_𝑛(𝑘)⪯𝑔𝑦_𝑚(𝑘),𝑔𝑧_𝑛(𝑘)⪰𝑔𝑧_𝑚(𝑘). (2.28) Then from (2.1), (2.4), and (2.28), we have for 𝑘≥0, 𝑑𝑔𝑥_𝑛(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1=𝑑𝐹𝑥_𝑛(𝑘),𝑦_𝑛(𝑘),𝑧_𝑛(𝑘),𝐹𝑥_𝑚(𝑘),𝑦_𝑚(𝑘),𝑧_𝑚(𝑘)≤𝜓max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1=𝑑𝐹𝑦_𝑛(𝑘),𝑥_𝑛(𝑘),𝑦_𝑛(𝑘),𝐹𝑦_𝑚(𝑘),𝑥_𝑚(𝑘),𝑦_𝑚(𝑘)≤𝜓max𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘)+1,𝑔𝑧_𝑚(𝑘)+1=𝑑𝐹𝑧_𝑛(𝑘),𝑦_𝑛(𝑘),𝑥_𝑛(𝑘),𝐹𝑧_𝑚(𝑘),𝑦_𝑚(𝑘),𝑥_𝑚(𝑘)≤𝜓max𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘). (2.29) From (2.29) for 𝑘≥0, we get 𝛽_𝑘≤𝜓max𝑑𝑔𝑥_𝑛(𝑘),𝑔𝑥_𝑚(𝑘),𝑑𝑔𝑦_𝑛(𝑘),𝑔𝑦_𝑚(𝑘),𝑑𝑔𝑧_𝑛(𝑘),𝑔𝑧_𝑚(𝑘)=𝜓𝛼_𝑘. (2.30) Letting 𝑘→∞ in (2.30), using (2.20), (2.27), and a property of 𝜓, we get 𝜀≤𝜓(𝜀)<𝜀, (2.31) which is a contradiction. This shows that {𝑔𝑥_𝑛}, {𝑔𝑦_𝑛}, and {𝑔𝑧_𝑛} are Cauchy sequences.
Since 𝑋 is complete, there exist 𝑥,𝑦,𝑧∈𝑋 such that lim_𝑛→∞𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝑔𝑦_𝑛=𝑦,lim_𝑛→∞𝑔𝑧_𝑛=𝑧. (2.32) From (2.4) and (2.32), using the continuity of 𝑔, we have 𝑔𝑥=lim_𝑛→∞𝑔𝑔𝑥_𝑛+1=lim_𝑛→∞𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑔𝑦=lim_𝑛→∞𝑔𝑔𝑦_𝑛+1=lim_𝑛→∞𝑔𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑔𝑧=lim_𝑛→∞𝑔𝑔𝑧_𝑛+1=lim_𝑛→∞𝑔𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛. (2.33)(2.34)(2.35) Now we will show that 𝑔𝑥=𝐹(𝑥,𝑦,𝑧),𝑔𝑦=𝐹(𝑦,𝑥,𝑦), and 𝑔𝑧=𝐹(𝑧,𝑦,𝑥).
Since 𝑔 and 𝐹 are compatible, in addition with (2.33), (2.34), and (2.35), respectively imply lim_𝑛→∞𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛=0,lim_𝑛→∞𝑑𝑔𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑔𝑦_𝑛=0,lim_𝑛→∞𝑑𝑔𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑔𝑧_𝑛,𝑔𝑦_𝑛,𝑔𝑥_𝑛=0. (2.36)(2.37)(2.38) Suppose now the assumption (a) holds, that is, 𝐹 is continuous.
For all 𝑛≥0, we have 𝑑𝑔𝑥,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛≤𝑑𝑔𝑥,𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛. (2.39) Taking the limit as 𝑛→∞, using (2.32), (2.33), (2.36), and the facts that 𝑔 and 𝐹 are continuous, we have 𝑑(𝑔𝑥,𝐹(𝑥,𝑦,𝑧))=0.
Similarly, by using (2.32), (2.34), and (2.37) and (2.32), (2.35), and (2.38), respectively, and also the facts that 𝑔 and 𝐹 are continuous, we have 𝑑(𝑔𝑦,𝐹(𝑦,𝑥,𝑦))=0 and 𝑑(𝑔𝑧,𝐹(𝑧,𝑦,𝑥))=0.
Thus we have proved that 𝑔 and 𝐹 have a tripled coincidence point.
Suppose that the assumption (b) holds. Since {𝑔𝑥_𝑛},{𝑔𝑧_𝑛} are nondecreasing and 𝑔𝑥_𝑛→𝑥 with 𝑔𝑧_𝑛→𝑧 and also {𝑔𝑦_𝑛} is nonincreasing with 𝑔𝑦_𝑛→𝑦, by assumption (b) we have for all 𝑛𝑔𝑥_𝑛⪯𝑥,𝑔𝑦_𝑛⪰𝑦,𝑔𝑧_𝑛⪯𝑧. (2.40) By virtue of monotone increasing property of 𝑔 we have 𝑔𝑔𝑥_𝑛⪯𝑔𝑥,𝑔𝑔𝑦_𝑛⪰𝑔𝑦,𝑔𝑔𝑧_𝑛⪯𝑔𝑧. (2.41) Now using (2.4) we have 𝑑𝑔𝑥,𝐹𝑥,𝑦,𝑧≤𝑑𝑔𝑥,𝑔𝑔𝑥_𝑛+1+𝑑𝑔𝑔𝑥_𝑛+1,𝐹𝑥,𝑦,𝑧≤𝑑𝑔𝑥,𝑔𝑔𝑥_𝑛+1+𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛+𝑑𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛,𝐹𝑥,𝑦,𝑧≤𝑑𝑔𝑥,𝑔𝑔𝑥_𝑛+1+𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛+𝜓max𝑑𝑔𝑔𝑥_𝑛,𝑔𝑥,𝑑𝑔𝑔𝑦_𝑛,𝑔𝑦,𝑑𝑔𝑔𝑧_𝑛,𝑔𝑧,by(2.1),(2.41). (2.42) Taking the limit as 𝑛→∞ in the above inequality, using (2.33), (2.36), and (2.41) we have 𝑑𝑔𝑥,𝐹𝑥,𝑦,𝑧≤lim_𝑛→∞𝜓max𝑑𝑔𝑔𝑥_𝑛,𝑔𝑥,𝑑𝑔𝑔𝑦_𝑛,𝑔𝑦,𝑑𝑔𝑔𝑧_𝑛,𝑔𝑧. (2.43) By (2.33), (2.34), (2.35), and the property of 𝜓, we have 𝑑𝑔𝑥,𝐹𝑥,𝑦,𝑧≤𝜓(0)=0, (2.44) that is 𝑔𝑥=𝐹𝑥,𝑦,𝑧. (2.45) In a similar manner using (2.33), (2.34), (2.35), and (2.36), (2.37), (2.38), respectively, we obtain ,.𝑔𝑦=𝐹𝑦,𝑥,𝑦𝑔𝑧=𝐹𝑧,𝑦,𝑥 (2.46) Thus, we proved that 𝑔 and 𝐹 have a tripled coincidence point.
This completes the proof of the theorem.

Corollary 2.2. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose 𝐹∶𝑋×𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are such that 𝐹 has the mixed 𝑔-monotone property and 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝜓max𝑑𝑔𝑥,𝑔𝑢,𝑑𝑔𝑦,𝑔𝑣,𝑑𝑔𝑧,𝑔𝑤 (2.47) for any 𝑥,𝑦,𝑧∈𝑋 for which 𝑔𝑥⪯𝑔𝑢, 𝑔𝑦≽𝑔𝑣 and 𝑔𝑧⪯𝑔𝑤, where 𝜓∶[0,+∞)→[0,+∞) be such that 𝜓(𝑡) is monotone, 𝜓(𝑡)<𝑡 and lim_𝑟→𝑡+𝜓(𝑟)<𝑡 for all 𝑡>0. Suppose 𝐹(𝑋×𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous, and 𝐹 and 𝑔 are commuting. Suppose either (a)𝐹 is continuous, or   (b)𝑋 has the  following property: (i)if a nondecreasing sequence {𝑥_𝑛}→𝑥,  then 𝑥_𝑛⪯𝑥 for all 𝑛,(ii)if a nonincreasing sequence  {𝑦_𝑛}→𝑦, then 𝑦_𝑛≽𝑦 for all 𝑛.
If there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑔𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑔𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑔𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), then there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝐹𝑥,𝑦,𝑧=𝑔𝑥,𝐹𝑦,𝑥,𝑦=𝑔𝑦,𝐹𝑧,𝑦,𝑥=𝑔𝑧, (2.48) that is, 𝐹 and 𝑔 have a tripled coincidence point.

Proof. Since a commuting pair is also a compatible pair, the result of the Corollary 2.2 follows from Theorem 2.1.

Later, by an example, we will show that the Corollary 2.2 is properly contained in Theorem 2.1.

Corollary 2.3. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose 𝐹∶𝑋×𝑋×𝑋→𝑋 be such that 𝐹 has the mixed monotone property and 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝜓max𝑑(𝑥,𝑢),𝑑𝑦,𝑣,𝑑(𝑧,𝑤) (2.49) for any 𝑥,𝑦,𝑧∈𝑋 for which 𝑥⪯𝑢,𝑦≽𝑣 and 𝑧⪯𝑤, where 𝜓∶[0,+∞)→[0,+∞) be such that 𝜓(𝑡) is monotone, 𝜓(𝑡)<𝑡 and lim_𝑟→𝑡+𝜓(𝑟)<𝑡 for all 𝑡>0. Suppose(a)𝐹 is continuous, or(b)𝑋 has the following property: (i)if a nondecreasing sequence {𝑥_𝑛}→𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛,   (ii)if a nonincreasing sequence {𝑦_𝑛}→𝑦, then 𝑦_𝑛≽𝑦 for all 𝑛.   
If there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), then there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝐹𝑥,𝑦,𝑧=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑥=𝑧, (2.50) that is, 𝐹 has a tripled fixed point.

Proof. Taking 𝑔(𝑥)=𝑥 in Theorem 2.1 we obtain Corollary 2.3.

Corollary 2.4. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose 𝐹∶𝑋×𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are such that 𝐹 has the mixed monotone property and 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝑘max𝑑(𝑥,𝑢),𝑑𝑦,𝑣,𝑑(𝑧,𝑤) (2.51) for any 𝑥,𝑦,𝑧∈𝑋 for which 𝑥⪯𝑢,𝑦≽𝑣 and 𝑧⪯𝑤, where 0<𝑘<1. Suppose either(a)𝐹 is continuous,  or(b)𝑋 has the following  property:(i)if a nondecreasing sequence {𝑥_𝑛}→𝑥, then  𝑥_𝑛⪯𝑥 for all 𝑛,(ii)if a nonincreasing sequence {𝑦_𝑛}→𝑦, then  𝑦_𝑛≽𝑦 for all 𝑛.
If there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), then there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝐹𝑥,𝑦,𝑧=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑥=𝑧, (2.52) that is, 𝐹 has a tripled coincidence point.

Proof. Taking 𝜓(𝑡)=𝑘𝑡, 𝑡>0 where 0<𝑘<1, in Corollary 2.3 we obtain Corollary 2.4.

The following corollary is the result of Berinde and Borcut in [20].

Corollary 2.5. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose 𝐹∶𝑋×𝑋×𝑋→𝑋 be such that 𝐹 has the mixed monotone property and 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝑎₁𝑑(𝑥,𝑢)+𝑎₂𝑑𝑦,𝑣+𝑎₃𝑑(𝑧,𝑤) (2.53) for any 𝑥,𝑦,𝑧∈𝑋 for which 𝑥⪯𝑢,𝑦≽𝑣 and 𝑧⪯𝑤, where 𝑎₁+𝑎₂+𝑎₃<1. Suppose either(a)𝐹 is continuous, or(b)𝑋 has the following property:(i)if a nondecreasing sequence {𝑥_𝑛}→𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛(ii)if a nonincreasing sequence {𝑦_𝑛}→𝑦, then 𝑦_𝑛≽𝑦 for all 𝑛.
If there exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑧₀), 𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑥₀), then there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝐹𝑥,𝑦,𝑧=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑥=𝑧, (2.54) that is, 𝐹 has a tripled fixed point.

Proof. The proof follows from Corollary 2.4, since the inequality in Corollary 2.5 implies that Corollary 2.4.

Remark 2.6. The method used in the proof of Corollary 2.5 is different from that used by Berinde and Borcut [20].

Next we discuss an example.

Example 2.7. Let 𝑋=ℜ. Then (𝑋,⪯) is a partially ordered set with the partial ordering defined by 𝑥⪯𝑦 if and only if |𝑥|≤|𝑦| and 𝑥⋅𝑦≥0.

Let 𝑑(𝑥,𝑦)=|𝑥−𝑦| for 𝑥,𝑦∈ℜ. Then (𝑋,𝑑) is a complete metric space.

Let 𝑔∶𝑋→𝑋 be defined as 𝑔(𝑥)=𝑥²/10,for all 𝑥∈𝑋.

Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be defined as 𝐹=𝑥𝑥,𝑦,𝑧²−𝑦²+𝑧²9,∀𝑥,𝑦,𝑧∈𝑋. (2.55) Then F obeys the mixed 𝑔-monotone property.

Let 𝜓∶[0,∞)→[0,∞) be defined as 𝜓(𝑡)=(1/3)𝑡 for all 𝑡∈[0,∞).

Let, {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be three sequences in 𝑋 such that lim_𝑛→∞𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑎,lim_𝑛→∞𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑏,lim_𝑛→∞𝐹𝑧_𝑛,𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑧_𝑛=𝑐. (2.56) Then explicitly, lim_𝑛→∞𝑥²_𝑛−𝑦²_𝑛+𝑧²_𝑛9=lim_𝑛→∞𝑥²_𝑛10,∀𝑥,𝑦,𝑧∈𝑋,or,10𝑎−10𝑏+10𝑐9=𝑎imply𝑎−10𝑏+10𝑐=0. (2.57) Again, lim_𝑛→∞𝑦²_𝑛−𝑥²_𝑛+𝑦²_𝑛9=lim_𝑛→∞𝑦²_𝑛10,∀𝑥,𝑦,𝑧∈𝑋,or,10𝑏−10𝑎+10𝑏9=𝑏imply11𝑏−10𝑎=0. (2.58) And lim_𝑛→∞𝑧²_𝑛−𝑦²_𝑛+𝑥²_𝑛9=lim_𝑛→∞𝑧²_𝑛10,∀𝑥,𝑦,𝑧∈𝑋,or,10𝑐−10b+10𝑎9=𝑐imply𝑐−10𝑏+10𝑎=0. (2.59) Then from the above relations we have, 𝑎=0,𝑏=0, and 𝑐=0.

Therefore, 𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑧_𝑛𝑑𝑔𝐹𝑦⟶0as𝑛⟶∞,_𝑛,𝑥_𝑛,𝑦_𝑛,𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑔𝑦_𝑛𝑑𝑔𝐹𝑧⟶0as𝑛⟶∞,_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑔𝑧_𝑛,𝑔𝑦_𝑛,𝑔𝑥_𝑛⟶0as𝑛⟶∞. (2.60) Hence, the pair (𝑔,𝐹) is compatible in 𝑋.

Also, 𝑥₀=0,𝑧₀=𝑐(>0), and 𝑦₀=0 are three points in 𝑋 such that 𝑔(𝑥₀)=𝑔(0)=0<𝑐²/9=𝐹(0,0,𝑐)=𝐹(𝑥₀,𝑦₀,𝑧₀),𝑔(𝑦₀)=𝑔(0)=0=𝐹(0,0,0)=𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑔(𝑧₀)=𝑔(𝑐)=𝑐²/10<𝑐²/9=𝐹(𝑐,0,0)=𝐹(𝑧₀,𝑦₀,𝑥₀).

We next verify inequality (2.1) of Theorem 2.1. We take 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝑋, such that 𝑔𝑥⪯𝑔𝑢,𝑔𝑧⪯𝑔𝑤 and 𝑔𝑦≽𝑔𝑣, that is, 𝑥²≤𝑢²,𝑧²≤𝑤², and 𝑦²≥𝑣².

Let 𝐴=max{𝑑(𝑔𝑥,𝑔𝑢),𝑑(𝑔𝑦,𝑔𝑣),𝑑(𝑔𝑧,𝑔𝑤)}=max{|(𝑥²−𝑢²)|,|(𝑦²−𝑣²)|,|(𝑧²−𝑤²)|}.

Then 𝑑(𝐹(𝑥,𝑦,𝑧), 𝐹(𝑢,𝑣,𝑤))=𝑑((𝑥²−𝑦²+𝑧²)/9,(𝑢²−𝑣²+𝑤²)/9)=(|(𝑥²−𝑢²)−(𝑦²−𝑣²)+(𝑧²−𝑤²))/3|≤(|(𝑥²−𝑢²)|+|(𝑦²−𝑣²)|+|(𝑧²−𝑤²)|)/9≤3𝐴/9=𝐴/3=𝜓(𝐴)=𝜓(max{𝑑(𝑔𝑥,𝑔𝑢),𝑑(𝑔𝑦,𝑔𝑣),𝑑(𝑔𝑧,𝑔𝑤)}).

Thus it is verified that the functions 𝑔,𝐹, and 𝜓 satisfy all the conditions of Theorem 2.1. Here (0,0,0) is the tripled coincidence point of 𝑔 and 𝐹 in 𝑋.

Remark 2.8. It is observed that in Example 2.7 the function 𝐹 and 𝑔 do not commute, but they are compatible. Hence Corollary 2.2 cannot be applied to this example. This shows that Theorem 2.1 properly contains Corollary 2.2. Also 𝑔≠𝐼, so the results of Berinde and Borcut [20] cannot be applied to this example. This shows that result in [20] is effectively generalised.