Coupled Fixed Point Theorems for Weak Contraction Mappings under 𝐹-Invariant Set

Abstract

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems.

1. Introduction

Since Banach's fixed point theorem in 1922, because of its simplicity and usefullnes, it has become a very popular tool in solving the existence problems in many branches of nonlinear analysis. For some more results of the generalization of this principle, refer to [1–9] and references mentioned therein. Ran and Reurings [10] extended the Banach contraction principle to metric spaces endowed with a partial ordering, and they gave applications of their results to matrix equations. Afterward, Nieto and Rodriguez-López [11] extended Ran and Reurings’s theorems in [10] for nondecreasing mappings and obtained a unique solution for a first order-ordinary differential equation with periodic boundary conditions.

In 2006, Gnana Bhaskar and Lakshmikantham [12] introduced the concept of the mixed monotone property and a coupled fixed point. They also established some coupled fixed point theorems for mappings that satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. Because of their important role in the study of nonlinear differential equations, nonlinear integral equations and differential inclusions, a wide discussion on coupled fixed point theorems has been studied by many authors. A number of articles in this topic have been dedicated to the improvement and generalization; see [13–22] and references therein.

on the other hand, Huang and Zhang [23] introduced the notion of cone metric spaces and established the fixed point theorems for mappings on this space. In 2011, Cho et al. [24] introduced a new concept of the 𝑐-distance in cone metric spaces, which is a cone version of the 𝑤-distance of Kada et al. [25] and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the 𝑐-distance. Afterward, Sintunavarat et al. [26] studied and established the fixed point theorems for the generalized contraction mappings by using this concept. For some more results on the fixed point theory and applications in cone metric spaces by using the cone metric, 𝑤-cone-distance, and 𝑐-distance, we refer the readers to [27–35].

Recently, Cho et al. [36] established new coupled fixed point theorems under weak contraction mappings by using the concept of mixed monotone property and 𝑐-distance in partially cone metric spaces as follows.

Theorem 1.1 (see [36]). Let (𝑋,⊑) be a partially ordered set, and suppose that (𝑋,𝑑) is a complete cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋 and 𝐹∶𝑋×𝑋→𝑋 a continuous function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘2𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (1.1) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥⊑𝑥^∗∧𝑦⊒𝑦^∗𝑜𝑟𝑥⊒𝑥^∗∧𝑦⊑𝑦^∗. (1.2) If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹𝑥₀,𝑦₀,𝐹y₀,𝑥₀⊑𝑦₀, (1.3) then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, one has 𝑞(𝑣,𝑣)=𝜃 and 𝑞(𝑢,𝑢)=𝜃.

Theorem 1.2 (see [36]). In addition to the hypotheses of Theorem 1.1, suppose that any two elements 𝑥 and 𝑦 in 𝑋 are comparable. Then, the coupled fixed point has the form (𝑢,𝑢), where 𝑢∈𝑋.

Theorem 1.3 (see [36]). Let (𝑋,⊑) be a partially ordered set, and suppose that (𝑋,𝑑) is a complete cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋 and 𝐹∶𝑋×𝑋→𝑋 a function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘4𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (1.4) for some 𝑘∈(0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥⊑𝑥^∗∧𝑦⊒𝑦^∗𝑜𝑟𝑥⊒𝑥^∗∧𝑦⊑𝑦^∗. (1.5) Also, suppose that 𝑋 has the following properties:(a)if (𝑥_𝑛) is a nondecreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥_𝑛⊑𝑥 for all 𝑛∈ℕ;(b)if (𝑥_𝑛) is a nonincreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥⊑𝑥_𝑛 for all 𝑛∈ℕ.Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹𝑥₀,𝑦₀,𝐹𝑦₀,𝑥₀⊑𝑦₀. (1.6) If 𝑦₀⊑𝑥₀, then 𝐹 has a coupled fixed point.

In this paper, we weaken the condition of the mixed monotone property in results of Cho et al. [36] by using the concept of the 𝐹-invariant set due to Samet and Vetro [37] in the cone version. We also give the example of a nonlinear contraction mapping, which is not applied by the main results of Cho et al. but can be applied to our results. The presented results extend and complement some recent results of Cho et al. [36] and some classical coupled fixed point theorems and several results in the literature.

2. Preliminaries

Throughout this paper (𝑋,⊑) denotes a partially ordered set with the partial order ⊑. By 𝑥⊏𝑦, we mean 𝑥⊑𝑦 but 𝑥≠𝑦. A mapping 𝑓∶𝑋→𝑋 is said to be nondecreasing (non-increasing) if, for all 𝑥,𝑦∈𝑋, 𝑥⊑𝑦 implies 𝑓(𝑥)⊑𝑓(𝑦)(𝑓(𝑦)⊑𝑓(𝑥), resp.).

Definition 2.1 (see [12]). Let (𝑋,⊑) be a partial ordered set. A mapping 𝐹∶𝑋×𝑋→𝑋 is said to have a mixed monotone property if 𝐹 is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any 𝑥,𝑦∈𝑋𝑥₁,𝑥₂∈𝑋,𝑥₁⊑𝑥₂⟹𝐹𝑥₁,𝑦⊑𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑦₁⊑𝑦₂⟹𝐹𝑥,𝑦₁⊒𝐹𝑥,𝑦₂. (2.1)

Definition 2.2 (see [12]). Let 𝑋 be a nonempty set. An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of mapping 𝐹∶𝑋×𝑋→𝑋 if 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,𝑥. (2.2)
Next, we give some terminology of cone metric spaces and the concept of 𝑐-distance in cone metric spaces due to Cho et al. [24], which is a generalization of the 𝑤-distance of Kada et al. [25].
Let (𝐸,‖⋅‖) be a real Banach space, 𝜃 a zero element in 𝐸, and 𝑃 a subset of 𝐸 with int(𝑃)≠∅. Then, 𝑃 is called a cone if the following conditions are satisfied:(1)𝑃 is closed and 𝑃≠{𝜃};(2)𝑎,𝑏∈ℝ⁺, 𝑥,𝑦∈𝑃 implies 𝑎𝑥+𝑏𝑦∈𝑃;(3)𝑥∈𝑃∩−𝑃 implies 𝑥=𝜃.
For a cone 𝑃, define the partial ordering ⪯ with respect to 𝑃 by 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝑃. We write 𝑥≺𝑦 to indicate that 𝑥⪯𝑦 but 𝑥≠𝑦, while 𝑥≪𝑦 stands for 𝑦−𝑥∈int(𝑃).
It can be easily shown that 𝜆int(𝑃)⊆int(𝑃) for all positive scalars 𝜆.
The cone 𝑃 is called normal if there is a number 𝐾>0 such that, for all 𝑥,𝑦∈𝐸, 0≤𝑥≤𝑦⟹‖𝑥‖≤𝐾𝑦. (2.3) The least positive number satisfying the above is called the normal constant of 𝑃.

Definition 2.3 (see [23]). Let 𝑋 be a nonempty set. Suppose that the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies the following conditions:(1)𝜃⪯𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)𝑑(𝑥,𝑦)⪯𝑑(𝑥,𝑧)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋. Then, 𝑑 is called a cone metric on 𝑋 and (𝑋,𝑑) is called a cone metric space.

Definition 2.4 (see [23]). Let (𝑋,𝑑) be a cone metric space. Let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋.(1)If, for any 𝑐∈𝑋 with 𝜃≪𝑐, there exists 𝑁∈ℕ such that 𝑑(𝑥_𝑛,𝑥)≪𝑐 for all 𝑛≥𝑁, then {𝑥_𝑛} is said to be convergent to a point 𝑥∈𝑋 and 𝑥 is the limit of {𝑥_𝑛}. One denotes this by lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥_𝑛→𝑥 as 𝑛→∞.(2)If, for any 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑁∈ℕ such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐 for all 𝑛,𝑚≥𝑁, then {𝑥_𝑛} is called a Cauchy sequence in 𝑋.(3)The space (𝑋,𝑑) is called a complete cone metric space if every Cauchy sequence is convergent.

Definition 2.5 (see [24]). Let (𝑋,𝑑) be a cone metric space. Then a function 𝑞∶𝑋×𝑋→𝐸 is called a 𝑐-distance on 𝑋 if the following are satisfied:(q1)𝜃⪯𝑞(𝑥,𝑦) for all 𝑥,𝑦∈𝑋;(q2)𝑞(𝑥,𝑧)⪯𝑞(𝑥,𝑦)+𝑞(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋;(q3)for any 𝑥∈𝑋, if there exists 𝑢=𝑢_𝑥∈𝑃 such that 𝑞(𝑥,𝑦_𝑛)⪯𝑢 for each 𝑛∈ℕ, then 𝑞(𝑥,𝑦)⪯𝑢 whenever {𝑦_𝑛} is a sequence in 𝑋 converging to a point 𝑦∈𝑋;(q4)for any 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑒∈𝐸 with 𝜃≪𝑒 such that 𝑞(𝑧,𝑥)≪𝑒 and 𝑞(𝑧,𝑦)≪𝑐 imply 𝑑(𝑥,𝑦)≪𝑐.

Remark 2.6. The 𝑐-distance 𝑞 is a 𝑤-distance on 𝑋 if we take (𝑋,𝑑) is a metric space, 𝐸=ℝ⁺, 𝑃=[0,∞), and (q3) is replaced by the following condition:
for any 𝑥∈𝑋, 𝑞(𝑥,⋅)∶𝑋→ℝ⁺ is lower semicontinuous.
Moreover, (q3) holds whenever 𝑞(𝑥,⋅) is lower semi-continuous. Thus, if (𝑋,𝑑) is a metric space, 𝐸=ℝ⁺, and 𝑃=[0,∞), then every 𝑤-distance is a 𝑐-distance. But the converse is not true in the general case. Therefore, the 𝑐-distance is a generalization of the 𝑤-distance.

Example 2.7. Let (𝑋,𝑑) be a cone metric space and 𝑃 a normal cone. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then, 𝑞 is 𝑐-distance.

Example 2.8. Let 𝐸=𝐶¹_ℝ[0,1] with ‖𝑥‖=‖𝑥‖_∞+‖𝑥^′‖_∞ and 𝑃={𝑥∈𝐸∶𝑥(𝑡)≥0on[0,1]} (2.4) (this cone is not normal). Let 𝑋=[0,∞), and define a mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑𝑥,𝑦=𝑥−𝑦𝜑 (2.5) for all 𝑥,𝑦∈𝑋, where 𝜑∶[0,1]→ℝ such that 𝜑(𝑡)=𝑒^𝑡. Then, (𝑋,𝑑) is a cone metric space. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞𝑥,𝑦=𝑥+𝑦𝜑 (2.6) for all 𝑥,𝑦∈𝑋. Then, 𝑞 is a 𝑐-distance.

Example 2.9. Let (𝑋,𝑑) be a cone metric space and 𝑃 a normal cone. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞𝑥,𝑦=𝑑𝑢,𝑦 (2.7) for all 𝑥,𝑦∈𝑋, where 𝑢 is a fixed point in 𝑋. Then, 𝑞 is a 𝑐-distance.

Example 2.10. Let 𝐸=ℝ and 𝑃={𝑥∈𝐸∶𝑥≥0}. Let 𝑋=[0,∞), and define a mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑𝑥,𝑦=𝑥−𝑦 (2.8) for all 𝑥,𝑦∈𝑋. Then, (𝑋,𝑑) is a cone metric space. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞𝑥,𝑦=𝑦 (2.9) for all 𝑥,𝑦∈𝑋. Then, 𝑞 is a 𝑐-distance.

Remark 2.11. From Examples 2.9 and 2.10, we have two important results. For the 𝑐-distance, 𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥) does not necessarily hold and 𝑞(𝑥,𝑦)=𝜃 is not necessarily equivalent to 𝑥=𝑦 for all 𝑥,𝑦∈𝑋.

The following lemma is crucial in proving our results.

Lemma 2.12 (see [24]). Let (𝑋,𝑑) be a cone metric space and 𝑞 a 𝑐-distance on 𝑋. Let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝑋 and 𝑥,𝑦,𝑧∈𝑋. Suppose that {𝑢_𝑛} is a sequence in 𝑃 converging to 𝜃. Then, the following hold.(1)If 𝑞(𝑥_𝑛,𝑦)⪯𝑢_𝑛 and 𝑞(𝑥_𝑛,𝑧)⪯𝑢_𝑛, then 𝑦=𝑧.(2)If 𝑞(𝑥_𝑛,𝑦_𝑛)⪯𝑢_𝑛 and 𝑞(𝑥_𝑛,𝑧)⪯𝑢_𝑛, then {𝑦_𝑛} converges to a point 𝑧∈𝑋.(3)If 𝑞(𝑥_𝑛,𝑥_𝑚)⪯𝑢_𝑛 for each 𝑚>𝑛, then {𝑥_𝑛} is a Cauchy sequence in 𝑋.(4)If 𝑞(𝑦,𝑥_𝑛)⪯𝑢_𝑛, then {𝑥_𝑛} is a Cauchy sequence in 𝑋.

3. Coupled Fixed Point under 𝐹-Invariant Set

In this section, we prove some coupled fixed point theorems by using the 𝑐-distance under the concept of the 𝐹-invariant in cone metric spaces and apply our results in partially ordered cone metric spaces. First of all, we give the concept of the 𝐹-invariant set in the cone version.

Definition 3.1. Let (𝑋,𝑑) be a cone metric space and 𝐹∶𝑋×𝑋→𝑋 a given mapping. Let 𝑀 be a nonempty subset of 𝑋⁴. One says that 𝑀 is the 𝐹-invariant subset of 𝑋⁴ if and only if, for all 𝑥,𝑦,𝑧,𝑤∈𝑋, one has(a)(𝑥,𝑦,𝑧,𝑤)∈𝑀⇔(𝑤,𝑧,𝑦,𝑥)∈𝑀;(b)(𝑥,𝑦,𝑧,𝑤)∈𝑀⇒(𝐹(𝑥,𝑦),𝐹(𝑦,𝑥),𝐹(𝑧,𝑤),𝐹(𝑤,𝑧))∈𝑀.
We obtain that the set 𝑀=𝑋⁴ is trivially 𝐹-invariant.
The next example plays a key role in the proof of our main results in partially ordered set.

Example 3.2. Let (𝑋,𝑑) be a cone metric space endowed with a partial order ⊑. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping satisfying the mixed monotone property, that is, for all 𝑥,𝑦∈𝑋, we have 𝑥₁,𝑥₂∈𝑋,𝑥₁⊑𝑥₂⟹𝐹𝑥₁,𝑦⊑𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑦₁⊑𝑦₂⟹𝐹𝑥,𝑦₁⊒𝐹𝑥,𝑦₂. (3.1) Define the subset 𝑀⊆𝑋⁴ by 𝑀={(𝑎,𝑏,𝑐,𝑑)∶𝑐⊑𝑎,𝑏⊑𝑑}. (3.2) Then, 𝑀 is the 𝐹-invariant of 𝑋⁴.
Next, we prove the main results of this work.

Theorem 3.3. Let (𝑋,𝑑) be a complete cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋, 𝑀 a nonempty subset of 𝑋⁴, and 𝐹∶𝑋×𝑋→𝑋 a continuous function such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘2𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.3) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑀𝑜𝑟𝑥^∗,𝑦^∗,𝑥,𝑦∈𝑀. (3.4) If 𝑀 is an 𝐹-invariant and there exist 𝑥₀,𝑦₀∈𝑋 such that 𝐹𝑥₀,𝑦₀,𝐹𝑦₀,𝑥₀,𝑥₀,𝑦₀∈𝑀, (3.5) then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, if (𝑢,𝑣,𝑢,𝑣)∈𝑀, then 𝑞(𝑣,𝑣)=𝜃 and 𝑞(𝑢,𝑢)=𝜃.

Proof. As 𝐹(𝑋×𝑋)⊆𝑋, we can construct sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑥_𝑛=𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑦_𝑛=𝐹𝑦_𝑛−1,𝑥_𝑛−1∀𝑛∈ℕ. (3.6) Since (𝐹(𝑥₀,𝑦₀),𝐹(𝑦₀,𝑥₀),𝑥₀,𝑦₀)=(𝑥₁,𝑦₁,𝑥₀,𝑦₀)∈𝑀 and 𝑀 is an 𝐹-invariant set, we get 𝐹𝑥₁,𝑦₁,𝐹𝑦₁,𝑥₁,𝐹𝑥₀,𝑦₀,𝐹𝑦₀,𝑥₀=𝑥₂,𝑦₂,𝑥₁,𝑦₁∈𝑀. (3.7) Again, using the fact that 𝑀 is an 𝐹-invariant set, we have 𝐹𝑥₂,𝑦₂,𝐹𝑦₂,𝑥₂,𝐹𝑥₁,𝑦₁,𝐹𝑦₁,𝑥₁=𝑥₃,𝑦₃,𝑥₂,𝑦₂∈𝑀. (3.8) By repeating the argument to the above, we get 𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑥_𝑛−1,𝑦_𝑛−1=𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1,𝑦_𝑛−1∈𝑀 (3.9) for all 𝑛∈ℕ. From (3.3), we have 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛⪯𝑘2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑦_𝑛−1,𝑦_𝑛,𝑞𝑥_𝑛+1,𝑥_𝑛=𝑞𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥_𝑛−1,𝑦_𝑛−1⪯𝑘2𝑞𝑥_𝑛,𝑥_𝑛−1+𝑞𝑦_𝑛,𝑦_𝑛−1. (3.10)(3.11) Combining (3.10) and (3.11), we get 𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛⪯𝑘2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑦_𝑛−1,𝑦_𝑛+𝑞𝑥_𝑛,𝑥_𝑛−1+𝑞𝑦_𝑛,𝑦_𝑛−1. (3.12) Since (𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1,𝑦_𝑛−1)∈𝑀 for all 𝑛∈ℕ and 𝑀 is an 𝐹-invariant set, we get 𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛,𝑥_𝑛∈𝑀 (3.13) for all 𝑛∈ℕ. From (3.3), we have 𝑞𝑦_𝑛,𝑦_𝑛+1=𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛⪯𝑘2𝑞𝑦_𝑛−1,𝑦_𝑛+𝑞𝑥_𝑛−1,𝑥_𝑛,𝑞𝑦_𝑛+1,𝑦_𝑛=𝑞𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛−1,𝑥_𝑛−1⪯𝑘2𝑞𝑦_𝑛,𝑦_𝑛−1+𝑞𝑥_𝑛,𝑥_𝑛−1. (3.14) From (3.14), we have 𝑞𝑦_𝑛,𝑦_𝑛+1+𝑞𝑦_𝑛+1,𝑦_𝑛⪯𝑘2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑦_𝑛−1,𝑦_𝑛+𝑞𝑥_𝑛,𝑥_𝑛−1+𝑞𝑦_𝑛,𝑦_𝑛−1. (3.15) Adding (3.12) and (3.15), we get 𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+𝑞𝑦_𝑛,𝑦_𝑛+1+𝑞𝑦_𝑛+1,𝑦_𝑛⪯𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑦_𝑛−1,𝑦_𝑛+𝑞𝑥_𝑛,𝑥_𝑛−1+𝑞𝑦_𝑛,𝑦_𝑛−1. (3.16) We repeat the above process for 𝑛-times, and we get 𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+𝑞𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛⪯𝑘^𝑛𝑞𝑥₁,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.17) From (3.17), we can conclude that 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘^𝑛𝑞𝑥₁,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁,𝑞𝑦_𝑛,𝑦_𝑛+1⪯𝑘^𝑛𝑞𝑥₁,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.18) Let 𝑚,𝑛∈ℕ with 𝑚>𝑛. Since 𝑞𝑥_𝑛,𝑥_𝑚⪯^𝑚−1_𝑖=𝑛𝑞𝑥_𝑖,𝑥_𝑖+1,𝑞𝑦_𝑛,𝑦_𝑚⪯^𝑚−1_𝑖=𝑛𝑞𝑦_𝑖,𝑦_𝑖+1, (3.19) and 0≤𝑘<1, we have 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁,𝑞𝑦_𝑛,𝑦_𝑚⪯𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.20) Using Lemma 2.12(3), we have {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in (𝑋,𝑑). By the completeness of 𝑋, we get 𝑥_𝑛→𝑢 and 𝑦_𝑛→𝑣 for some 𝑢,𝑣∈𝑋.
Since 𝐹 is continuous, taking 𝑛→∞ in (3.6), we get lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞𝐹𝑥_𝑛,𝑦_𝑛=𝐹lim_𝑛→∞𝑥_𝑛,lim_𝑛→∞𝑦_𝑛=𝐹(𝑢,𝑣),lim_𝑛→∞𝑦_𝑛+1=lim_𝑛→∞𝐹𝑦_𝑛,𝑥_𝑛=𝐹lim_𝑛→∞𝑦_𝑛,lim_𝑛→∞𝑥_𝑛=𝐹(𝑣,𝑢). (3.21) By the uniqueness of the limits, we get 𝑢=𝐹(𝑢,𝑣) and 𝑣=𝐹(𝑣,𝑢). Therefore, (𝑢,𝑣) is a coupled fixed point of 𝐹.
Finally, we assume that (𝑢,𝑣,𝑢,𝑣)∈𝑀 and so (𝑣,𝑢,𝑣,𝑢)∈𝑀. By (3.3), we have 𝑞()(𝐹()(⪯𝑘𝑢,𝑢=𝑞𝑢,𝑣,𝐹𝑢,𝑣))2𝑞()(),𝑞()(𝐹()(⪯𝑘𝑢,𝑢+𝑞𝑣,𝑣𝑣,𝑣=𝑞𝑣,𝑢,𝐹𝑣,𝑢))2𝑞(𝑣,𝑣)+𝑞(𝑢,𝑢). (3.22) Therefore, we get 𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)⪯𝑘𝑞(𝑣,𝑣)+𝑞(𝑢,𝑢). (3.23) Since 0≤𝑘<1, we conclude that 𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)=𝜃 and hence 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. This completes the proof.

Theorem 3.4. In addition to the hypotheses of Theorem 3.3, suppose that any two elements 𝑥 and 𝑦 in 𝑋, we have 𝑦,𝑥,𝑥,𝑦∈𝑀𝑜𝑟𝑥,𝑦,𝑦,𝑥∈𝑀. (3.24) Then, the coupled fixed point has the form (𝑢,𝑢), where 𝑢∈𝑋.

Proof. As in the proof of Theorem 3.3, there exists a coupled fixed point (𝑢,𝑣)∈𝑋×𝑋. Hence, 𝑢=𝐹(𝑢,𝑣),𝑣=𝐹(𝑣,𝑢). (3.25) From the additional hypothesis and (3.3), we get 𝑞()(𝐹()(⪯𝑘𝑢,𝑣=𝑞𝑢,𝑣,𝐹𝑣,𝑢))2𝑞()(),𝑞()(𝐹()(⪯𝑘𝑢,𝑣+𝑞𝑣,𝑢𝑣,𝑢=𝑞𝑣,𝑢,𝐹𝑢,𝑣))2𝑞(𝑣,𝑢)+𝑞(𝑢,𝑣). (3.26) Therefore, we have 𝑞(𝑢,𝑣)+𝑞(𝑣,𝑢)⪯𝑘𝑞(𝑣,𝑢)+𝑞(𝑢,𝑣). (3.27) Since 0≤𝑘<1, we get 𝑞(𝑢,𝑣)+𝑞(𝑣,𝑢)=𝜃. Therefore, 𝑞(𝑢,𝑣)=𝜃 and 𝑞(𝑣,𝑢)=𝜃.
Let 𝑢_𝑛=𝜃 and 𝑥_𝑛=𝑢. Then, 𝑞𝑥_𝑛,𝑢⪯𝑢_𝑛,𝑞𝑥_𝑛,𝑣⪯𝑢_𝑛. (3.28) From Lemma 2.12(1), we have 𝑢=𝑣. Therefore, the coupled fixed point of 𝐹 has the form (𝑢,𝑢). This completes the proof.

Next, we apply Theorems 3.3 and 3.4 in partially ordered cone metric spaces. If we set 𝑀 as Example 3.2, then we get to Theorems 3.1 and 3.2 of Cho et al. [36].

Corollary 3.5 (see [36, Theorem 3.1]). Let (𝑋,⊑) be a partially ordered set, and suppose that (𝑋,𝑑) is a complete cone metric space. Let 𝑞 a 𝑐-distance on 𝑋 and 𝐹∶𝑋×𝑋→𝑋 be a continuous function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘2𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.29) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥⊑𝑥^∗∧𝑦⊒𝑦^∗𝑜𝑟𝑥⊒𝑥^∗∧𝑦⊑𝑦^∗. (3.30) If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹𝑥₀,𝑦₀,𝐹𝑦₀,𝑥₀⊑𝑦₀, (3.31) then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, one has 𝑞(𝑣,𝑣)=𝜃 and 𝑞(𝑢,𝑢)=𝜃.

Corollary 3.6 (see [36, Theorem 3.2]). In addition to the hypotheses of Corollary 3.5, suppose that any two elements 𝑥 and 𝑦 in 𝑋 are comparable. Then the coupled fixed point has the form (𝑢,𝑢), where 𝑢∈𝑋.

Theorem 3.7. Let (𝑋,𝑑) be a complete cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋, 𝑀 a subset of 𝑋⁴, and 𝐹∶𝑋×𝑋→𝑋 a function such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘4𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.32) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑀𝑜𝑟𝑥^∗,𝑦^∗,𝑥,𝑦∈𝑀. (3.33) Also, suppose that(i)there exist 𝑥₀,𝑦₀∈𝑋 such that (𝐹(𝑥₀,𝑦₀),𝐹(𝑦₀,𝑥₀),𝑥₀,𝑦₀)∈𝑀,(ii)if two sequences {𝑥_𝑛}, {𝑦_𝑛} are such that (𝑥_𝑛+1,𝑦_𝑛+1,𝑥_𝑛,𝑦_𝑛)∈𝑀 for all 𝑛∈ℕ and {𝑥_𝑛}→𝑥,{𝑦_𝑛}→𝑦, then (𝑥,𝑦,𝑥_𝑛,𝑦_𝑛)∈𝑀 for all 𝑛∈ℕ.If 𝑀 is an 𝐹-invariant set, then 𝐹 has a coupled fixed point.

Proof. As in the proof of Theorem 3.3, we can construct two Cauchy sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1,𝑦_𝑛−1∈𝑀 (3.34) for all 𝑛∈ℕ. Moreover, we have that {𝑥_𝑛} converges to a point 𝑢∈𝑋 and {𝑦_𝑛} converges to 𝑣∈𝑋: 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁,𝑞𝑦_𝑛,𝑦_𝑚⪯𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁ (3.35) for each 𝑚>𝑛≥1. By (q3), we have 𝑞𝑥_𝑛⪯𝑘,𝑢^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁𝑞𝑦_𝑛⪯𝑘,𝑣^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁ (3.36) and so 𝑞𝑥_𝑛,𝑢+𝑞𝑦_𝑛,𝑣⪯2𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.37) By assumption (ii), we have (𝑢,𝑣,𝑥_𝑛−1,𝑦_𝑛−1)∈𝑀 and (𝑦_𝑛−1,𝑥_𝑛−1,𝑣,𝑢)∈𝑀. From (3.32), we have 𝑞𝑥_𝑛,𝐹(𝑢,𝑣)=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1()⪯𝑘,𝐹𝑢,𝑣4𝑞𝑥_𝑛−1,𝑢+𝑞𝑦_𝑛−1,𝑞𝑦,𝑣_𝑛,𝐹(𝑣,𝑢)=𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1()⪯𝑘,𝐹𝑣,𝑢4𝑞𝑦_𝑛−1,𝑣+𝑞𝑥_𝑛−1,𝑢. (3.38) Thus, we have 𝑞𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑦_𝑛()⪯𝑘,𝐹𝑣,𝑢2𝑞𝑥_𝑛−1,𝑢+𝑞𝑦_𝑛−1,𝑣. (3.39) By (3.37), we get 𝑞𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑦_𝑛()⪯𝑘,𝐹𝑣,𝑢2⋅2𝑘^𝑛−1𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁=𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.40) Therefore, we have 𝑞𝑥_𝑛()⪯𝑘,𝐹𝑢,𝑣^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁,𝑞𝑦_𝑛(⪯𝑘,𝐹𝑣,𝑢)^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.41) By using (3.36) and (3.41), Lemma 2.12(1) shows that 𝑢=𝐹(𝑢,𝑣) and 𝑣=𝐹(𝑣,𝑢). Therefore, (𝑢,𝑣) is a coupled fixed point of 𝐹. This completes the proof.

Corollary 3.8 (see [36, Theorem 3.3]). Let (𝑋,⊑) be a partially ordered set, and suppose that (𝑋,𝑑) is a complete cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋 and 𝐹∶𝑋×𝑋→𝑋 a function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘4𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.42) for some 𝑘∈(0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with 𝑥⊑𝑥^∗∧𝑦⊒𝑦^∗𝑜𝑟𝑥⊒𝑥^∗∧𝑦⊑𝑦^∗. (3.43) Also, suppose that 𝑋 has the following properties:(a)if {𝑥_𝑛} is a non-decreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥_𝑛⊑𝑥 for all 𝑛∈ℕ;(b)if {𝑦_𝑛} is a non-increasing sequence in 𝑋 with 𝑦_𝑛→𝑦, then 𝑦⊑𝑦_𝑛 for all 𝑛∈ℕ. Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹𝑥₀,𝑦₀,𝐹𝑦₀,𝑥₀⊑𝑦₀. (3.44) If 𝑦₀⊑𝑥₀, then 𝐹 has a coupled fixed point.

Proof. Let 𝑀={(𝑎,𝑏,𝑐,𝑑)∶𝑐⊑𝑎,𝑏⊑𝑑}. We obtain that 𝑀 is an 𝐹-invariant set. By (3.42), we have 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘4𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.45) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with (𝑥,𝑦,𝑥^∗,𝑦^∗)∈𝑀 or (𝑥^∗,𝑦^∗,𝑥,𝑦)∈𝑀. From assumptions (a) and (b), we know that, for two sequences {𝑥_𝑛}, {𝑦_𝑛} such that {𝑥_𝑛} is a non-decreasing sequence in 𝑋 with 𝑥_𝑛→𝑥 and {𝑦_𝑛} is a non-increasing sequence in 𝑋 with 𝑦_𝑛→𝑦, 𝑥₀⊑𝑥₁⊑⋯⊑𝑥_𝑛𝑦⊑⋯⊑𝑥,₀⊒𝑦₁⊒⋯𝑦_𝑛⊒⋯⊒𝑦 (3.46) for all 𝑛∈ℕ. Therefore, we have (𝑥,𝑦,𝑥_𝑛,𝑦_𝑛)∈𝑀 for all 𝑛∈ℕ. Since 𝑦₀⊑𝑥₀, we have 𝑦⊑𝑦_𝑛⊑𝑦₀⊑𝑥₀⊑x_𝑛⊑𝑥 (3.47) for all 𝑛∈ℕ. Therefore, we can conclude that (𝑦_𝑛,𝑥_𝑛,𝑦,𝑥)∈𝑀 for all 𝑛∈ℕ. Now, all the hypotheses of Theorem 3.7 hold. Thus, 𝐹 has a coupled fixed point.

Example 3.9. Let 𝐸=𝐶¹_ℝ[0,1] with ‖𝑥‖=‖𝑥‖_∞+‖𝑥^′‖_∞ and 𝑃={𝑥∈𝐸∶𝑥(𝑡)≥0,𝑡∈[0,1]}. (3.48) Let 𝑋=[0,∞) (with usual order), and let 𝑑∶𝑋×𝑋→𝐸 be defined by 𝑑𝑥,𝑦(𝑡)=𝑥−𝑦𝑒^𝑡. (3.49) Then, (𝑋,𝑑) is an ordered cone metric space (see [24, Example 2.9]). Further, let 𝑞∶𝑋×𝑋→𝐸 be defined by 𝑞𝑥,𝑦(𝑡)=𝑦𝑒^𝑡. (3.50) It is easy to check that 𝑞 is a 𝑐-distance. Consider now the function 𝐹∶𝑋×𝑋→𝑋 defined by 𝐹=1𝑥,𝑦7𝑥+𝑦,𝑥≥𝑦,0,𝑥<𝑦. (3.51) For 𝑦₁=2 and 𝑦₂=3, we have 𝑦₁⊑𝑦₂ but 𝐹(𝑥,𝑦₁)⊑𝐹(𝑥,𝑦₂) for all 𝑥>3. So the mapping 𝐹 does not satisfy the mixed monotone property. Hence, the main results of Cho et al. [36] cannot be applied to this example. But it is easy to see that 𝑞𝐹(⪯1𝑥,𝑦,𝐹𝑢,𝑣)5𝑞(𝑥,𝑢)+𝑞𝑦,𝑣 (3.52) for all 𝑥,𝑦,𝑢,𝑣∈𝑋⁴. Note that there exists 0,1∈𝑋 such that (𝐹(0,1),𝐹(1,0),0,1)∈𝑋⁴. (3.53) Now, we can apply Theorem 3.3 with 𝑀=𝑋⁴. Therefore, 𝐹 has a unique coupled fixed point, that is, a point (0,0).

Remark 3.10. Although the main results of Cho et al. [36] are an essential tool in the partially ordered cone metric spaces to claim the existence of the coupled fixed points the mappings do not have the mixed monotone property in general case such as the mapping in the above example. Therefore, it is very interesting to consider our theorems as another auxiliary tool to claim the existence of a coupled fixed point.

Acknowledgments

The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821), and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund, and King Mongkut’s University of Technology Thonburi (KMUTT) (Grant no. MRG5380044) for financial support during the preparation of this paper.