Coupled Fixed Point Theorems under Weak Contractions

Abstract

Cho et al. [Comput. Math. Appl. 61(2011), 1254–1260] studied common fixed point theorems on cone metric spaces by using the concept of c-distance. In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance in cone metric spaces.

1. Introduction

Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang [1]. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to [2–15]. Recently, Bhaskar and Lakshmikantham [16] introduced the concept of a coupled coincidence point of a mapping 𝐹 from 𝑋×𝑋 into 𝑋 and a mapping 𝑔 from 𝑋 into 𝑋 and studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to [17–23].

Recently, Cho et al. [7] introduced a new concept of 𝑐-distance in cone metric spaces, which is a cone version of 𝑤-distance of Kada et al. [24] (see also [25]) and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the 𝑐-distance.

In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of 𝑐-distance.

2. Preliminaries

In this paper, assume that 𝐸 is a real Banach space. Let 𝑃 be 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 stand for 𝑦−𝑥∈int𝑃.

It can be easily shown that 𝜆int(𝑃)⊆int(𝑃) for all positive scalars 𝜆.

Definition 2.1 (see [1]). 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 2.2 (see [1]). 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). We denote 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 2.3 (see [7]). Let (𝑋,⊑) be a partially ordered set, and let 𝐹∶𝑋×𝑋→𝑋 be a function. Then the mapping 𝐹 is said to have the mixed monotone property if 𝐹(𝑥,𝑦) is monotone nondecreasing in 𝑥 and is monotone nonincreasing in 𝑦; that is, 𝑥₁⊑𝑥₂implies𝐹𝑥₁,𝑦⊑𝐹𝑥₂,𝑦 (2.1) for all 𝑦∈𝑋 and 𝑦₁⊑𝑦₂implies𝐹𝑥,𝑦₂⊑𝐹𝑥,𝑦₁ (2.2) for all 𝑥∈𝑋.

Definition 2.4 (see [7]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of a mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹(𝑥,𝑦)=𝑥 and 𝐹(𝑦,𝑥)=𝑦.

Recently, Cho et al. [7] introduced the concept of 𝑐-distance on cone metric space (𝑋,𝑑) which is a generalization of 𝑤-distance of Kada et al. [24].

Definition 2.5 (see [7]). 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≥1, 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 0≤𝑒 such that 𝑞(𝑧,𝑥)≪𝑒 and 𝑞(𝑧,𝑦)≪𝑐 imply 𝑑(𝑥,𝑦)≪𝑐.

Cho et al. [7] noticed the following important remark in the concept of 𝑐-distance on cone metric spaces.

Remark 2.6 (see [7]). Let 𝑞 be a 𝑐-distance on a cone metric space (𝑋,𝑑). Then(1)𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥) does not necessarily hold for all 𝑥,𝑦∈𝑋,(2)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 2.7 (see [7]). Let (𝑋,𝑑) be a cone metric space, and let 𝑞 be 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. Main Results

In this section, we prove some coupled fixed point theorems by using 𝑐-distance in partially ordered cone metric spaces.

Theorem 3.1. 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 let 𝐹∶𝑋×𝑋→𝑋 be a continuous function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘2𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.1) for some 𝑘∈[0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with (𝑥⊑𝑥^∗)∧(𝑦⊒𝑦^∗) or (𝑥⊒𝑥^∗)∧(𝑦⊑𝑦^∗). If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)⊑𝑦₀, then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, one has 𝑞(𝑣,𝑣)=𝜃 and 𝑞(𝑢,𝑢)=𝜃.

Proof. Let 𝑥₀,𝑦₀∈𝑋 be such that 𝑥₀⊑𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)⊑𝑦₀. Let 𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑦₁=𝐹(𝑦₀,𝑥₀). Since 𝐹 has the mixed monotone property, we have 𝑥₀⊑𝑥₁ and 𝑦₁⊑𝑦₀. Continuing this process, we can construct two sequences (𝑥_𝑛) and (𝑦_𝑛) in 𝑋 such that 𝑥_𝑛=𝐹𝑥_𝑛−1,𝑦_𝑛−1⊑𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛⊑𝑦_𝑛=𝐹𝑦_𝑛−1,𝑥_𝑛−1. (3.2) Let 𝑛∈𝐍. Now, by (3.1), 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. (3.3) From (3.3), it follows that 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. (3.4)
Similarly, 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. (3.5) Thus it follows from (3.4) and (3.5) that 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. (3.6) Repeating (3.6) 𝑛-times, 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. (3.7) Thus we have 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. (3.8) 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, (3.9) and 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. (3.10) From Lemma 2.7 (3), it follows that (𝑥_𝑛) and (𝑦_𝑛) are Cauchy sequences in (𝑋,𝑑). Since 𝑋 is complete, there exist 𝑢,𝑣∈𝑋 such that 𝑥_𝑛→𝑢 and 𝑦_𝑛→𝑣. Since 𝐹 is continuous, we have 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛(),𝑦⟶𝐹𝑢,𝑣_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛⟶𝐹(𝑣,𝑢). (3.11) By the uniqueness of the limits, we get 𝑢=𝑓(𝑢,𝑣) and 𝑣=𝐹(𝑣,𝑢). Thus (𝑢,𝑣) is a coupled fixed point of 𝐹.
Moreover, by (3.1), 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). (3.12) Therefore, we get 𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)⪯𝑘𝑞(𝑣,𝑣)+𝑞(𝑢,𝑢). (3.13) Since 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 3.2. In addition to the hypotheses of Theorem 3.1, 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.

Proof. As in the proof of Theorem 3.1, there exists a coupled fixed point (𝑢,𝑣)∈𝑋×𝑋. Here 𝑢=𝐹(𝑢,𝑣) and 𝑣=𝐹(𝑣,𝑢). By the additional assumption and (3.1), we have 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). (3.14) Thus we have 𝑞(𝑢,𝑣)+𝑞(𝑣,𝑢)⪯𝑘𝑞(𝑣,𝑢)+𝑞(𝑢,𝑣). (3.15) Since 𝑘<1, we get 𝑞(𝑢,𝑣)+𝑞(𝑣,𝑢)=𝜃. Hence 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. (3.16) From Lemma 2.7 (1), we have 𝑢=𝑣. Hence the coupled fixed point of 𝐹 has the form (𝑢,𝑢). This completes the proof.

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 let 𝐹∶𝑋×𝑋→𝑋 be a function having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥^∗,𝑦^∗⪯𝑘4𝑞𝑥,𝑥^∗+𝑞𝑦,𝑦^∗ (3.17) for some 𝑘∈(0,1) and all 𝑥,𝑦,𝑥^∗,𝑦^∗∈𝑋 with (𝑥⊑𝑥^∗)∧(𝑦⊒𝑦^∗) or (𝑥⊒𝑥^∗)∧(𝑦⊑𝑦^∗). 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≥1;(b)if (𝑥_𝑛) is a nonincreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥⊑𝑥_𝑛 for all 𝑛≥1.Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)⊑𝑦₀. If 𝑦₀⊑𝑥₀, then 𝐹 has a coupled fixed point.

Proof. As in the proof of Theorem 3.1, we can construct two Cauchy sequences (𝑥_𝑛) and (𝑦_𝑛) in 𝑋 such that 𝑥₀⊑𝑥₁⊑⋯⊑𝑥_𝑛𝑦⊑⋯,₀⊒𝑦₁⊒⋯𝑦_𝑛⊒⋯. (3.18)
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 (3.19) for each 𝑛>𝑚≥1. By (q3), we have 𝑞𝑥_𝑛⪯𝑘,𝑢^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁,𝑞𝑦_𝑛⪯𝑘,𝑣^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁, (3.20) and so 𝑞𝑥_𝑛,𝑢+𝑞𝑦_𝑛,𝑣⪯2𝑘^𝑛𝑞𝑥1−𝑘¹,𝑥₀+𝑞𝑦₁,𝑦₀+𝑞𝑥₀,𝑥₁+𝑞𝑦₀,𝑦₁. (3.21) By the properties (a) and (b), we have 𝑣⊑𝑦_𝑛⊑𝑦₀⊑𝑥₀⊑𝑥_𝑛⊑𝑢. (3.22) By (3.17), 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. (3.23) Thus we have 𝑞𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑦_𝑛()⪯𝑘,𝐹𝑣,𝑢2𝑞𝑥_𝑛−1,𝑢+𝑞𝑦_𝑛−1,𝑣. (3.24) By (3.21), 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. (3.25) 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. (3.26) By using (3.20) and (3.26), Lemma 2.7 (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.

Example 3.4. Let 𝐸=𝐶¹_𝐑[0,1] with ‖𝑥‖=‖𝑥‖_∞+‖𝑥′‖_∞ and 𝑃={𝑥∈𝐸∶𝑥(𝑡)≥0,𝑡∈[0,1]}. Let 𝑋=[0,+∞) (with usual order), and let 𝑑∶𝑋×𝑋→𝐸 be defined by 𝑑(𝑥,𝑦)(𝑡)=|𝑥−𝑦|𝑒^𝑡. Then (𝑋,𝑑) is an ordered cone metric space (see [7, Example 2.9]). Further, let 𝑞∶𝑋×𝑋→𝐸 be defined by 𝑞(𝑥,𝑦)(𝑡)=𝑦𝑒^𝑡. It is easy to check that 𝑞 is a 𝑐-distance. Consider now the function 𝐹∶𝑋×𝑋→𝑋 defined by 𝐹=1𝑥,𝑦8𝑥−𝑦,𝑥≥𝑦,0,𝑥<𝑦. (3.27) Then it is easy to see that 𝑞𝐹(⪯1𝑥,𝑦,𝐹𝑢,𝑣)6𝑞(𝑥,𝑢)+𝑞𝑦,𝑣 (3.28) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with (𝑥≤𝑢)∧(𝑦≥𝑣) or (𝑥≥𝑢)∧(𝑦≤𝑣). Note that 0≤𝐹(0,1) and 1≥𝐹(1,0). Thus, by Theorem 3.1, it follows that 𝐹 has a coupled fixed point in 𝐸. Here (0,0) is a coupled fixed point of 𝐹.

Acknowledgments

The first 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).