Coupled Coincidence Points in Partially Ordered Cone Metric Spaces with a c-Distance

Abstract

Cho et al. (2012) proved some coupled fixed point theorems in partially ordered cone metric spaces by using the concept of a c-distance in cone metric spaces. In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of a c-distance. Our results generalize several well-known comparable results in the literature. Also, we introduce an example to support the usability of our results.

1. Introduction and Preliminaries

Fixed point theory is an essential tool in functional nonlinear analysis. Consequently, fixed point theory has wide applications areas not only in the various branches of mathematics (see, e.g., [1, 2]) but also in many fields, such as, chemistry, biology, statistics, economics, computer science, and engineering (see, e.g., [3–11]). For example, fixed point results are incredibly useful when it comes to proving the existence of various types of Nash equilibria (see, e.g., [7]) in economics. On the other hand, fixed point theorems are vital for the existence and uniqueness of differential equations, matrix equations, integral equations (see, e.g., [1, 2]). Banach contraction mapping principle (Banach fixed point theorem) is one of the most powerful theorems of mathematics and hence fixed point theory. Huang and Zhang [12] generalized the Banach contraction principle by replacing the notion of usual metric spaces by the notion of cone metric spaces. Then many authors obtained many fixed and common fixed point theorems in cone metric spaces. For some works in cone metric spaces, we may refer the reader (as examples) to [13–24]. The concept of a coupled fixed point of a mapping 𝐹∶𝑋×𝑋→𝑋 was initiated by Bhaskar and Lakshmikantham [25], while Lakshmikantham and Ćirić [26] initiated the notion of coupled coincidence point of mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 and studied some coupled coincidence point theorems in partially ordered metric spaces. For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [27–34].

In the present paper, 𝐍^∗ is the set of positive integers and 𝐸 stands for a real Banach space. Let 𝑃 be a subset of 𝐸. We will always assume that the cone 𝑃 has a nonempty interior Int(𝑃) (such cones are called solid). Then 𝑃 is called a cone if the following conditions are satisfied: (1)𝑃is 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 a partial ordering ⪯ with respect to 𝑃 by 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝑃. We will write 𝑥≺𝑦 to indicate that 𝑥⪯𝑦 but 𝑥≠𝑦, while 𝑥≪𝑦 will stand for 𝑦−𝑥∈Int(𝑃). It can be easily shown that 𝜆Int(𝑃)⊆Int(𝑃) for all positive scalar 𝜆.

Definition 1.1 (see [12]). Let 𝑋 be a nonempty set. Suppose the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies (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.

Bhaskar and Lakshmikantham [25] introduced the notion of mixed monotone property of the mapping 𝐹∶𝑋×𝑋→𝑋.

Definition 1.2 (see [25]). Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋 be a mapping. Then the mapping 𝐹 is said to have mixed monotone property if 𝐹(𝑥,𝑦) is monotone nondecreasing in 𝑥 and is monotone nonincreasing in 𝑦; that is, for any 𝑥,𝑦∈𝑋, 𝑥₁≤𝑥₂implies𝐹𝑥₁,𝑦≤𝐹𝑥₂𝑦,𝑦,∀𝑦∈𝑋,₁≤𝑦₂implies𝐹𝑥,𝑦₂≤𝐹𝑥,𝑦₁,∀𝑥∈𝑋. (1.1)
Inspired by Definition 1.2, Lakshmikantham and Ćirić in [26] introduced the concept of a 𝑔-mixed monotone mapping.

Definition 1.3 (see [26]). Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋. Then the mapping 𝐹 is said to have mixed 𝑔𝑔-monotone property if 𝐹(𝑥,𝑦) is monotone 𝑔-nondecreasing in 𝑥 and is monotone 𝑔-nonincreasing in 𝑦; that is: 𝑔𝑥₁≤𝑔𝑥₂implies𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦,∀𝑦∈𝑋,𝑔𝑦₁≤𝑔𝑦₂implies𝐹𝑥,𝑦₂≤𝐹𝑥,𝑦₁,∀𝑥∈𝑋. (1.2)

Definition 1.4 (see [25]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of a mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (1.3)

Definition 1.5 (see [26]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled coincidence point of the mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝐹𝑥,𝑦=𝑔𝑥,𝐹𝑦,𝑥=𝑔𝑦. (1.4)
Recently, Cho et al. [35] introduced the concept of 𝑐-distance on cone metric space (𝑋,𝑑) which is a generalization of 𝑤-distance of Kada et al. [36] (see also [37, 38]).

Definition 1.6 (see [35]). 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 is 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 each 𝑥∈𝑋 and 𝑛≥1, if 𝑞(𝑥,𝑦_𝑛)⪯𝑢 for some 𝑢=𝑢_𝑥∈𝑃, then 𝑞(𝑥,𝑦)⪯𝑢 whenever (𝑦_𝑛) is a sequence in 𝑋 converging to a point 𝑦∈𝑋, (q4) for all 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑒∈𝐸 with 0≤𝑒 such that 𝑞(𝑧,𝑥)≪𝑒, and 𝑞(𝑧,𝑦)≪𝑒 implies 𝑑(𝑥,𝑦)≪𝑐.
Cho et al. [35] noticed the following important remark in the concept of 𝑐-distance on cone metric spaces.

Remark 1.7 (see [35]). 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. Very recently, Cho et al. [39] proved the following existence theorems.

Theorem 1.8 (see [39]). 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 𝑞𝐹⪯𝑘𝑥,𝑦,𝐹𝑥∗,𝑦∗2𝑞(𝑥,𝑥∗)+𝑞𝑦,𝑦∗, (1.5) 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 𝑦₀⊒𝐹(𝑦₀,𝑥₀), then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, one has 𝑞(𝑣,𝑣)=𝑞(𝑢,𝑢)=𝜃.

Theorem 1.9 (see [39]). 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 𝑞𝐹⪯𝑘𝑥,𝑦,𝐹𝑥∗,𝑦∗4𝑞(𝑥,𝑥∗)+𝑞𝑦,𝑦∗, (1.6) 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: (1)if (𝑥_𝑛) is a nondecreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥_𝑛⊑𝑥 for all 𝑛≥1, (2)if (𝑥_𝑛) is a nonincreasing sequence in 𝑋 with 𝑥_𝑛→𝑥, then 𝑥⊑𝑥_𝑛 for all 𝑛≥1.
If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⊑𝐹(𝑥₀,𝑦₀) and 𝑦₀⊒𝐹(𝑦₀,𝑥₀), then 𝐹 has a coupled fixed point (𝑢,𝑣). Moreover, one has q(𝑣,𝑣)=𝑞(𝑢,𝑢)=𝜃.

For other fixed point results using a 𝑐-distance, see [40].

In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of 𝑐-distance. Our results generalize Theorems 1.8 and 1.9. We consider an application to illustrate our result is useful (see Section 3).

2. Main Results

The following lemma is essential in proving our results.

Lemma 2.1 (see [35]). Let (𝑋,𝑑) be a cone metric space, and let 𝑞 be a cone 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 holds.(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 𝑧. (3)If 𝑞(𝑥_𝑛,𝑥_𝑚)⪯𝑢_𝑛 for 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.

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

Theorem 2.2. Let (𝑋,≤) be a partially ordered set and suppose that (𝑋,𝑑) is a cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗+𝑞𝐹𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯𝑘𝑞𝑔𝑥,𝑔𝑥∗+𝑞𝑔𝑦,𝑔𝑦∗, (2.1) for some 𝑘∈[0,1) and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑔𝑥≤𝑔𝑥∗)∧(𝑔𝑦≥𝑔𝑦∗) or (𝑔𝑥≥𝑔𝑥∗)∧(𝑔𝑦≤𝑔𝑦∗). Assume that 𝐹 and 𝑔 satisfy the following conditions:(1)𝐹 is continuous, (2)𝑔 is continuous and commutes with 𝐹, (3)𝐹(𝑋×𝑋)⊆𝑔𝑋, (4)(𝑋,𝑑) is complete, (5)𝐹 has the mixed 𝑔-monotone property. If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑔𝑦₀, then 𝐹 and 𝑔 have a coupled coincidence point (𝑢,𝑣). Moreover, one has 𝑞(𝑔𝑢,𝑔𝑢)=𝜃 and 𝑞(𝑔𝑣,𝑔𝑣)=𝜃.

Proof. Let 𝑥₀,𝑦₀∈𝑋 be such that 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑔𝑦₀. Since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₁,𝑦₁∈𝑋 such that 𝑔𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₁=𝐹(𝑦₀,𝑥₀). Again since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₂,𝑦₂∈𝑋 such that 𝑔𝑥₂=𝐹(𝑥₁,𝑦₁) and 𝑔𝑦₂=𝐹(𝑦₁,𝑥₁). Since 𝐹 has the mixed 𝑔-monotone property, we have 𝑔𝑥₀≤𝑔𝑥₁≤𝑔𝑥₂ and 𝑔y₂≤𝑔𝑦₁≤𝑔𝑦₀. Continuing this process, we can construct two sequences (𝑥_𝑛) and (𝑦_𝑛) in 𝑋 such that 𝑔𝑥_𝑛=𝐹𝑥_𝑛−1,𝑦_𝑛−1≤𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛≤𝑔𝑦_𝑛=𝐹𝑦_𝑛−1,𝑥_𝑛−1. (2.2) Let 𝑛∈𝐍^∗. Then by (2.1), we have 𝑞𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑞𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛⪯𝑘𝑞𝑔𝑥_𝑛−1,𝑔𝑥_𝑛+𝑞𝑔𝑦_𝑛−1,𝑔𝑦_𝑛. (2.3)
Repeating (2.3) 𝑛-times, we get 𝑞𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑞𝑔𝑦_𝑛,𝑔𝑦_𝑛+1⪯𝑘^𝑛𝑞𝑔𝑥₀,g𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.4) Thus, we have 𝑞𝑔𝑥_𝑛,𝑔𝑥_𝑛+1⪯𝑘^𝑛𝑞𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁,𝑞𝑔𝑦_𝑛,𝑔𝑦_𝑛+1⪯𝑘^𝑛𝑞𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.5)(2.6) Let 𝑚,𝑛∈𝐍^∗ with 𝑚>𝑛. Then by (𝑞₂) and (2.5), we have 𝑞𝑔𝑥_𝑛,𝑔𝑥_𝑚⪯^𝑚−1_𝑖=𝑛𝑞𝑔𝑥_𝑖,𝑔𝑥_𝑖+1⪯^𝑚−1_𝑖=𝑛𝑘^𝑖𝑞𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁⪯𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.7) Similarly, we have 𝑞𝑔𝑦_𝑛,𝑔𝑦_𝑚⪯𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.8) From part (3) of Lemma 2.1, we conclude that (𝑔𝑥_𝑛) and (𝑔𝑦_𝑛) are Cauchy sequences in in (𝑋,𝑑). Since 𝑋 is complete, there are 𝑢,𝑣∈𝑋 such that 𝑔𝑥_𝑛→𝑢 and 𝑔𝑦_𝑛→𝑣. Using the continuity of 𝑔, we get 𝑔(𝑔𝑥_𝑛)→𝑔𝑢 and 𝑔(𝑔𝑦_𝑛)→𝑔𝑣. Also, by continuity of 𝐹 and commutativity of 𝐹 and 𝑔, we have 𝑔𝑢=lim_𝑛→∞𝑔𝑔𝑥_𝑛+1=lim_𝑛→∞𝑔𝐹𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛=𝐹(𝑢,𝑣),𝑔𝑣=lim_𝑛→∞𝑔𝑔𝑦_𝑛+1=lim_𝑛→∞𝑔𝐹𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛=𝐹(𝑣,𝑢). (2.9) Hence, (𝑢,𝑣) is a coupled coincidence point of 𝐹 and 𝑔. Moreover, by (2.1) we have 𝑞(𝐹()((𝐹()(𝑞.𝑔𝑢,𝑔𝑢+𝑞𝑔𝑣,𝑔𝑣=𝑞𝑢,𝑣,𝐹𝑢,𝑣))+𝑞𝑣,𝑢,𝐹𝑣,𝑢))⪯𝑘𝑔𝑢,𝑔𝑢+𝑞𝑔𝑣,𝑔𝑣 (2.10) Since 𝑘<1, we conclude that 𝑞(𝑔𝑢,𝑔𝑢)+𝑞(𝑔𝑣,𝑔𝑣)=𝜃, and hence 𝑞(𝑔𝑢,𝑔𝑢)=𝜃 and 𝑞(𝑔𝑣,𝑔𝑣)=𝜃.

The continuity of 𝐹 in Theorem 2.2 can be dropped. For this, we present the following useful lemma which is a variant of Lemma 2.1, (1).

Lemma 2.3. Let (𝑋,𝑑) be a cone metric space, and let 𝑞 be a 𝑐-distance on 𝑋. Let (𝑥_𝑛) be a sequence in 𝑋. Suppose that (𝛼_𝑛) and (𝛽_𝑛) are sequences in 𝑃 converging to 𝜃. If 𝑞(𝑥_𝑛,𝑦)⪯𝛼_𝑛 and 𝑞(𝑥_𝑛,𝑧)⪯𝛽_𝑛, then 𝑦=𝑧.

Proof. Let 𝑐≫𝜃 be arbitrary. Since 𝛼_𝑛→𝜃, so there exists 𝑁₁∈ℕ such that 𝛼_𝑛≪𝑐/2 for all 𝑛≥𝑁₁. Similarly, there exists 𝑁₂∈ℕ such that 𝛽_𝑛≪𝑐/2 for all 𝑛≥𝑁₂. Thus, for all 𝑁≥max{𝑁₁,𝑁₂}, we have 𝑞𝑥_𝑛≪𝑐,𝑦2,𝑞𝑥_𝑛≪𝑐,𝑧2. (2.11) Take 𝑒=𝑐/2, so by (𝑞4), we get that 𝑑(𝑦,𝑧)≪𝑐 for each 𝑐≫𝜃; hence 𝑦=𝑧.

Theorem 2.4. Let (𝑋,≤) be a partially ordered set and suppose that (𝑋,𝑑) is a cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and let 𝑔∶𝑋→𝑋 be two mappings such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗+𝑞𝐹𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯𝑘𝑞𝑔𝑥,𝑔𝑥∗+𝑞𝑔𝑦,𝑔𝑦∗, (2.12) for some 𝑘∈[0,1) and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑔𝑥≤𝑔𝑥∗)∧(𝑔𝑦≥𝑔𝑦∗) or (𝑔𝑥≥𝑔𝑥∗)∧(𝑔𝑦≤𝑔𝑦∗). Assume that 𝐹 and 𝑔 satisfy the following conditions: (1)𝐹(𝑋×𝑋)⊆𝑔𝑋, (2)𝑔𝑋 is a complete subspace of 𝑋, (3)𝐹 has the mixed 𝑔-monotone property.

Suppose that 𝑋 has the following properties: (i)if a nondecreasing sequence 𝑥_𝑛→𝑥, then 𝑥_𝑛≤𝑥 for all 𝑛, (ii)if a nonincreasing sequence 𝑥_𝑛→𝑥, then 𝑥≤𝑥_𝑛 for all 𝑛. Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑔𝑦₀. Then 𝐹 and 𝑔 have a coupled coincidence point, say (𝑢,𝑣)∈𝑋×𝑋. Also, 𝑞(𝑔𝑢,𝑔𝑢)=𝑞(𝑔𝑣,𝑔𝑣)=𝜃.

Proof. As in the proof of Theorem 2.2, we can construct two Cauchy sequences (𝑔𝑥_𝑛) and (𝑔𝑦_𝑛) in the complete cone metric space (𝑔𝑋,𝑑). Then, there exist 𝑢,𝑣∈𝑋 such that 𝑔𝑥_𝑛→𝑔𝑢 and 𝑔𝑦_𝑛→𝑔𝑣. Similarly we have for all 𝑚>𝑛≥1𝑞𝑔𝑥_𝑛,𝑔𝑥_𝑚⪯𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁,𝑞𝑔𝑦_𝑛,𝑔𝑦_𝑚⪯𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.13) By (q3), we get that 𝑞𝑔𝑥_𝑛⪯𝑘,𝑔𝑢^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁,𝑞𝑔𝑦_𝑛⪯𝑘,𝑔𝑣^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.14)(2.15) By summation, we get that 𝑞𝑔𝑥_𝑛,𝑔𝑢+𝑞𝑔𝑦_𝑛𝑘,𝑔𝑣⪯2^𝑛𝑞1−𝑘𝑔x₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.16) Since (𝑔𝑥_𝑛) is nondecreasing and (𝑔𝑦_𝑛) is nonincreasing, using the properties (i), (ii) of 𝑋, we have 𝑔𝑥_𝑛≤𝑔𝑢,𝑔𝑣≤𝑔𝑦_𝑛,∀𝑛≥0. (2.17) From this and (2.14), we have 𝑞𝑔𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑔𝑦_𝑛,𝐹(𝑣,𝑢)=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹(𝑢,𝑣)+𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1()𝑞,𝐹𝑣,𝑢⪯𝑘𝑔𝑥_𝑛−1,𝑔𝑢+𝑞𝑔𝑦_𝑛−1,𝑔𝑣. (2.18) Therefore 𝑞𝑔𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑔𝑦_𝑛,𝐹(𝑣,𝑢)⪯𝑘𝑞𝑔𝑥_𝑛−1,𝑔𝑢+𝑞𝑔𝑦_𝑛−1,𝑔𝑣. (2.19) By (2.16), we have 𝑞𝑔𝑥_𝑛,𝐹(𝑢,𝑣)+𝑞𝑔𝑦_𝑛,𝐹(𝑣,𝑢)⪯𝑘𝑞𝑔𝑥_𝑛−1,𝑔𝑢+𝑞𝑔𝑦_𝑛−1,𝑔𝑣⪯𝑘2𝑘^𝑛−1𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁=2𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.20) This implies that 𝑞𝑔𝑥_𝑛,𝐹(𝑢,𝑣)⪯2𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞g𝑦₀,𝑔𝑦₁,𝑞𝑔𝑦_𝑛,𝐹(𝑣,𝑢)⪯2𝑘^𝑛𝑞1−𝑘𝑔𝑥₀,𝑔𝑥₁+𝑞𝑔𝑦₀,𝑔𝑦₁. (2.21)(2.22) By (2.14), (2.21) and Lemma 2.3, we obtain 𝑔𝑢=𝐹(𝑢,𝑣). Similarly, by (2.15), (2.22), and Lemma 2.3, we obtain 𝑔𝑣=𝐹(𝑣,𝑢). Also, adjusting as the proof of Theorem 2.2, we get that 𝑞𝑔𝑢,𝑔𝑢=𝑞𝑔𝑣,𝑔𝑣=𝜃. (2.23)

Corollary 2.5. Let (𝑋,≤) be a partially ordered set and suppose that (𝑋,𝑑) is a cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋, and let 𝑔∶𝑋→𝑋 be two mappings such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗⪯𝑎𝑞𝑔𝑥,𝑔𝑥∗+𝑏𝑞𝑔𝑦,𝑔𝑦∗, (2.24) for some 𝑎,𝑏∈[0,1) with 𝑎+𝑏<1 and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑔𝑥≤𝑔𝑥∗)∧(𝑔𝑦≥𝑔𝑦∗) or (𝑔𝑥≥𝑔𝑥∗)∧(𝑔𝑦≤𝑔𝑦∗). Assume that 𝐹 and 𝑔 satisfy the following conditions: (1)𝐹 is continuous, (2)𝑔 is continuous and commutes with 𝐹, (3)𝐹(𝑋×𝑋)⊆𝑔𝑋, (4)(𝑋,𝑑) is complete, (5)𝐹 has the mixed 𝑔-monotone property. If there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑔𝑦₀, then 𝐹 and 𝑔 have a coupled coincidence point (𝑢,𝑣). Moreover, one has 𝑞(𝑔𝑢,𝑔𝑢)=𝜃 and 𝑞(𝑔𝑣,𝑔𝑣)=𝜃.

Proof. Given 𝑥,𝑥∗,𝑦,𝑦∗∈𝑋 such that (𝑔𝑥≤𝑔𝑥∗)∧(𝑔𝑦≥𝑔𝑦∗). By (2.24), we have 𝑞𝐹,𝑞𝐹.𝑥,𝑦,𝐹𝑥∗,𝑦∗⪯𝑎𝑞𝑔𝑥,𝑔𝑥∗+𝑏𝑞𝑔𝑦,𝑔𝑦∗𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯𝑎𝑞𝑔𝑦,𝑔𝑦∗+𝑏𝑞𝑔𝑥,𝑔𝑥∗ (2.25) Thus 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗+𝑞𝐹𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯(𝑎+𝑏)𝑞𝑔𝑥,𝑔𝑥∗+𝑞𝑔𝑦,𝑔𝑦∗. (2.26) Since 𝑎+𝑏<1, the result follows from Theorem 2.2.

Corollary 2.6. 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. Let 𝐹∶𝑋×𝑋→𝑋 be a continuous mapping having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗⪯𝑎𝑞(𝑥,𝑥∗)+𝑏𝑞𝑦,𝑦∗, (2.27) for some 𝑎,𝑏∈[0,1) with 𝑎+𝑏<1 and for 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 (𝑥,𝑦). Moreover, one has 𝑞(𝑥,𝑥)=𝜃 and q(𝑦,𝑦)=𝜃.

Proof. It follows from Corollary 2.5 by taking 𝑔=𝐼_𝑋 (the identity map).

Corollary 2.7. Let (𝑋,≤) be a partially ordered set and suppose that (𝑋,𝑑) is a cone metric space. Let 𝑞 be a 𝑐-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two mappings such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗⪯𝑎𝑞𝑔𝑥,𝑔𝑥∗+𝑏𝑞𝑔𝑦,𝑔𝑦∗, (2.28) for some 𝑎,𝑏∈[0,1) with 𝑎+𝑏<1 and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑔𝑥≤𝑔𝑥∗)∧(𝑔𝑦≥𝑔𝑦∗) or (𝑔𝑥≥𝑔𝑥∗)∧(𝑔𝑦≤𝑔𝑦∗). Assume that 𝐹 and 𝑔 satisfy the following conditions: (1)𝐹(𝑋×𝑋)⊆𝑔𝑋, (2)𝑔𝑋 is a complete subspace of 𝑋, (3)𝐹 has the mixed 𝑔-monotone property.

Suppose that 𝑋 has the following properties: (i)if a nondecreasing sequence 𝑥_𝑛→𝑥, then 𝑥_𝑛≤𝑥 for all 𝑛, (ii)if a nonincreasing sequence 𝑥_𝑛→𝑥, then 𝑥≤𝑥_𝑛 for all 𝑛. Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑔𝑦₀. Then 𝐹 and 𝑔 have a coupled coincidence point.

Proof. It follows from Theorem 2.4 by similar arguments to those given in proof of Corollary 2.5.

Corollary 2.8. 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. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗⪯𝑎𝑞(𝑥,𝑥∗)+𝑏𝑞𝑦,𝑦∗, (2.29) for some 𝑎,𝑏∈[0,1) with 𝑎+𝑏<1 and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑥≤𝑥∗)∧(𝑦≥𝑦∗) or (𝑥≥𝑥∗)∧(𝑦≤𝑦∗). Suppose that 𝑋 has the following properties: (i)if a nondecreasing sequence 𝑥_𝑛→𝑥, then 𝑥_𝑛≤𝑥 for all 𝑛, (ii)if a nonincreasing sequence 𝑥_𝑛→𝑥, then 𝑥≤𝑥_𝑛 for all 𝑛. Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑦₀. Then 𝐹 has a coupled fixed point.

Proof. It follows from Corollary 2.7 by taking 𝑔=𝐼_𝑋 (the identity map).

Corollary 2.9. 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 mapping having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗+𝑞𝐹𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯𝑘𝑞(𝑥,𝑥∗)+𝑞𝑦,𝑦∗, (2.30) for some 𝑘∈[0,1) and for 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 (𝑥,𝑦). Moreover, we have 𝑞(𝑥,𝑥)=𝜃 and 𝑞(𝑦,𝑦)=𝜃.

Proof. It follows from Theorem 2.2 by taking 𝑔=𝐼_𝑋.

Corollary 2.10. 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. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having the mixed monotone property such that 𝑞𝐹𝑥,𝑦,𝐹𝑥∗,𝑦∗+𝑞𝐹𝑦,𝑥,𝐹𝑦∗,𝑥∗⪯𝑘𝑞(𝑥,𝑥∗)+𝑞𝑦,𝑦∗, (2.31) for some 𝑘∈[0,1) and for all 𝑥,𝑦,𝑥∗,𝑦∗∈𝑋 with (𝑥≤𝑥∗)∧(𝑦≥𝑦∗) or (𝑥≥𝑥∗)∧(𝑦≤𝑦∗). Suppose that 𝑋 has the following properties: (i)if a nondecreasing sequence 𝑥_𝑛→𝑥, then 𝑥_𝑛≤𝑥 for all 𝑛, (ii)if a nonincreasing sequence 𝑥_𝑛→𝑥, then 𝑥≤𝑥_𝑛 for all 𝑛. Assume there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑦₀. Then 𝐹 has a coupled fixed point.

Proof. It follows from Theorem 2.4 by taking 𝑔=𝐼_𝑋.

Example 2.11. Let 𝐸=𝐶¹_𝑅([0,1]) with ||𝑥||=||𝑥||_∞+||𝑥′||_∞ and 𝑃={𝑥∈𝐸∶𝑥(𝑡)≥0,𝑡∈[0,1]}. Let 𝑋=[0,1] with usual order ≤. Define 𝑑∶𝑋×𝑋→𝑋 by 𝑑(𝑥,𝑦)(𝑡)=|𝑥−𝑦|𝑒^𝑡 for all 𝑥,𝑦∈𝑋. Then (𝑋,𝑑) is a partially ordered cone metric space. Define 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)(𝑡)=𝑦𝑒^𝑡 for all 𝑥,𝑦∈𝑋. Then 𝑞 is a 𝑐-distance. Define 𝐹∶𝑋×𝑋→𝑋 by 𝐹𝑥,𝑦=𝑥−𝑦2,𝑥≥𝑦,0,𝑥<𝑦. (2.32) Then, (1)𝑞(𝐹(𝑥,𝑦),𝐹(𝑥∗,𝑦∗))+𝑞(𝐹(𝑦,𝑥),𝐹(𝑦∗,𝑥∗))≤1/2(𝑞(𝑥,𝑥∗)+𝑞(𝑦,𝑦∗)), for all 𝑥≤𝑥∗ and 𝑦≥𝑦∗, (2)there is no 𝑘∈[0,1) such that 𝑞(𝐹(𝑥,𝑦),𝐹(𝑥∗,𝑦∗))≤(𝑘/2)(𝑞(𝑥,𝑥∗)+𝑞(𝑦,𝑦∗)) for all 𝑥≤𝑥∗ and 𝑦≥𝑦∗, (3)there is no 𝑘∈[0,1) such that 𝑞(𝐹(𝑥,𝑦),𝐹(𝑥∗,𝑦∗))≤(𝑘/4)(𝑞(𝑥,𝑥∗)+𝑞(𝑦,𝑦∗)) for all 𝑥≤𝑥∗ and 𝑦≥𝑦∗. Note that 0≤𝐹(0,0) and 0≥𝐹(0,0). Thus by Corollary 2.10, we have 𝐹 which has a coupled fixed point. Here (0,0) is a coupled fixed point of 𝐹.

Proof. The proof of (2.1) is easy. To prove (2.3), suppose the contrary; that is, there is 𝑘∈[0,1) such that 𝑞(𝐹(𝑥,𝑦),𝐹(𝑥∗,𝑦∗))≤𝑘/2(𝑞(𝑥,𝑥∗)+𝑞(𝑦,𝑦∗)) for all 𝑥≤𝑥∗ and 𝑦≥𝑦∗. Take 𝑥=0,𝑦=1,𝑥∗=1 and 𝑦∗=0. Then 𝑞(𝐹()(𝑡)≤𝑘0,1,𝐹1,0))(2𝑞(0,1)+𝑞(1,0)(𝑡). (2.33) Thus 𝑞10,2(𝑡)=12𝑒^𝑡≤𝑘2𝑒^𝑡. (2.34) Hence 𝑘≥1 is a contradiction. The proof of (2.5) is similar to proof of (2.3).

Remark 2.12. Note that Theorems 3.1 and 3.2 of [39] are not applicable to Example 2.11.

Remark 2.13. Theorem 3.1 of [39] is a special case of Corollary 2.6 and Corollary 2.9.

Remark 2.14. Theorem 3.3 of [39] is a special case of Corollary 2.8 and Corollary 2.10.

3. Application

Consider the integral equations 𝑥(𝑡)=^𝑇₀𝑓(𝑠)(𝑠)[],𝑦(𝑡)=𝑡,𝑥,𝑦𝑑𝑠,𝑡∈0,𝑇_𝑇0𝑓𝑡,𝑦(𝑠),𝑥(𝑠)𝑑𝑠,𝑡∈[0,𝑇], (3.1) where 𝑇>0 and 𝑓∶[0,𝑇]×ℝ×ℝ→ℝ. Let 𝑋=𝐶([0,𝑇],ℝ) denote the space of ℝ-valued continuous functions on 𝐼=[0,𝑇]. The purpose of this section is to give an existence theorem for a solution (𝑥,𝑦) to (3.1) that belongs to 𝑋, by using the obtained result given by Corollary 2.10. Let 𝐸=ℝ², and let 𝑃⊂𝐸 be the cone defined by 𝑃=𝑥,𝑦∈ℝ²∣𝑥≥0,𝑦≥0. (3.2) We endow 𝑋 with the cone metric 𝑑∶𝑋×𝑋→𝐸 defined by 𝑑(𝑢,𝑣)=sup_𝑡∈𝐼|𝑢(𝑡)−𝑣(𝑡)|,sup_𝑡∈𝐼|𝑢(𝑡)−𝑣(𝑡)|,∀𝑢,𝑣∈𝑋. (3.3) It is clear that (𝑋,𝑑) is a complete cone metric space. Let 𝑞(𝑥,𝑦)=𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then, 𝑞 is a 𝑐-distance.

Now, we endow 𝑋 with the partial order ≤ given by 𝑢,𝑣∈𝑋,𝑢≤𝑣⟺𝑢(𝑥)≤𝑣(𝑥),∀𝑥∈𝐼. (3.4) Also, the product space 𝑋×𝑋 can be equipped with the partial order (still denoted ≤) given as follows: 𝑥,𝑦≤(𝑢,𝑣)⟺𝑥≤𝑢,𝑦≥𝑣. (3.5) It is easy that (i) and (ii) given in Corollary 2.10 are satisfied.

Now, we consider the following assumptions: (a)𝑓∶[0,𝑇]×ℝ×ℝ→ℝ is continuous, (b)for all 𝑡∈[0,𝑇], the function 𝑓(𝑡,⋅,⋅)∶ℝ→ℝ has the mixed monotone property, (c)for all 𝑡∈[0,𝑇], for all 𝑝,𝑞,𝑝′,𝑞′∈ℝ with 𝑝≤𝑞 and 𝑝′≥𝑞′, we have 𝑓𝑡,𝑞,𝑞^′−𝑓𝑡,𝑝,𝑝^′≤1𝑇𝜑𝑞−𝑝+𝑝^′−𝑞^′2, (3.6) where 𝜑∶[0,∞)→[0,∞) is continuous nondecreasing an satisfies the following condition: There exists 0<𝑘<1 such that 𝜑(𝑟)≤𝑘𝑟∀𝑟≥0, (3.7)(d)there exists 𝑥₀,𝑦₀∈𝐶([0,𝑇],ℝ) such that 𝑥₀(𝑡)≤^𝑇₀𝑓𝑡,𝑥₀(𝑠),𝑦₀(𝑠)𝑑𝑠,^𝑇₀𝑓𝑡,𝑦₀(𝑠),𝑥₀(𝑠)𝑑𝑠≤𝑦₀(𝑡),∀𝑡∈[0,𝑇]. (3.8)We have the following result.

Theorem 3.1. Suppose that (𝑎)-(𝑑) hold. Then, (3.1) has at least one solution (𝑥^∗,𝑦^∗)∈𝐶([0,𝑇],ℝ)×𝐶([0,𝑇],ℝ).

Proof. Define the mapping 𝐴∶𝐶([0,𝑇],ℝ)×𝐶([0,𝑇],ℝ)→𝐶([0,𝑇],ℝ) by 𝐴𝑥,𝑦(𝑡)=^𝑇₀𝑓𝑡,𝑥(𝑠),𝑦(𝑠)𝑑𝑠,𝑥,𝑦∈𝐶([0,𝑇],ℝ),𝑡∈[0,𝑇]. (3.9) We have to prove that 𝐴 has at least one coupled fixed point (𝑥^∗,𝑦^∗)∈𝐶([0,𝑇],ℝ)×𝐶([0,𝑇],ℝ).

From (b), it is clear that 𝐴 has the mixed monotone property.

Now, let 𝑥,𝑦,𝑢,𝑣∈𝐶([0,𝑇],ℝ) such that (𝑥≤𝑢 and 𝑦≥𝑣) or (𝑥≥𝑢 and 𝑦≤𝑣). Using (c), for all 𝑡∈[0,𝑇], we have 𝐴(𝑢,𝑣)(𝑡)−𝐴𝑥,𝑦(𝑡)≤^𝑇₀𝑓((𝑠)(𝑠(𝑠)(𝑠)≤1𝑡,𝑢,𝑣))−𝑓𝑡,𝑥,𝑦𝑑𝑠𝑇^𝑇₀𝜑𝑢(𝑠)−𝑥(𝑠)+𝑦(𝑠)−𝑣(𝑠)2≤1𝑑𝑠𝑇^𝑇₀𝜑sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−𝑣(𝑧)2𝑑𝑠=𝜑sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−𝑣(𝑧)2≤𝑘sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−𝑣(𝑧)2, (3.10) which implies that sup_{𝑧∈[0,𝑇]}𝐴(𝑢,𝑣)(𝑡)−𝐴𝑥,𝑦(𝑡)≤𝑘sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−𝑣(𝑧)2. (3.11) Similarly, one can get sup_{𝑧∈[0,𝑇]}𝐴(𝑣,𝑢)(𝑡)−𝐴𝑦,𝑥(𝑡)≤𝑘sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−𝑣(𝑧)2. (3.12) We deduce sup_{𝑧∈[0,𝑇]}𝐴(𝑢,𝑣)(𝑡)−𝐴𝑥,𝑦(𝑡)+sup_{𝑧∈[0,𝑇]}𝐴(𝑣,𝑢)(𝑡)−𝐴𝑦,𝑥(𝑡)≤𝑘sup_{𝑧∈[0,𝑇]}|𝑢(𝑧)−𝑥(𝑧)|+sup_{𝑧∈[0,𝑇]}𝑦(𝑧)−v(𝑧). (3.13) Thus 𝑑𝐴(𝑢,𝑣),𝐴𝑥,𝑦+𝑑𝐴(𝑣,𝑢),𝐴𝑦,𝑥⪯𝑘𝑑(𝑥,𝑢)+𝑑𝑦,𝑣. (3.14) Thus, we proved that condition (2.31) of Corollary 2.10 is satisfied. Moreover, from (d), we have 𝑥₀≤𝐴(𝑥₀,𝑦₀) and 𝐴(𝑦₀,𝑥₀)≤𝑦₀. Finally, applying our Corollary 2.10, we get the desired result.