#### 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., [311]). 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 [1324]. The concept of a coupled fixed point of a mapping was initiated by Bhaskar and Lakshmikantham [25], while Lakshmikantham and Ćirić [26] initiated the notion of coupled coincidence point of mappings and 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 [2734].

In the present paper, is the set of positive integers and stands for a real Banach space. Let be a subset of . We will always assume that the cone has a nonempty interior (such cones are called solid). Then is called a cone if the following conditions are satisfied: (1)is closed and ,(2), implies , (3) implies . For a cone , define a partial ordering with respect to by if and only if . We will write to indicate that but , while will stand for . It can be easily shown that for all positive scalar .

Definition 1.1 (see [12]). Let be a nonempty set. Suppose the mapping satisfies (1) for all and if and only if ,(2) for all ,(3) for all . Then is called a cone metric on , and is called a cone metric space.

Bhaskar and Lakshmikantham [25] introduced the notion of mixed monotone property of the mapping .

Definition 1.2 (see [25]). Let be a partially ordered set and be a mapping. Then the mapping is said to have mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is, for any ,
Inspired by Definition 1.2, Lakshmikantham and Ćirić in [26] introduced the concept of a -mixed monotone mapping.

Definition 1.3 (see [26]). Let be a partially ordered set and . Then the mapping is said to have mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is:

Definition 1.4 (see [25]). An element is called a coupled fixed point of a mapping if

Definition 1.5 (see [26]). An element is called a coupled coincidence point of the mappings and if
Recently, Cho et al. [35] introduced the concept of -distance on cone metric space which is a generalization of -distance of Kada et al. [36] (see also [37, 38]).

Definition 1.6 (see [35]). Let be a cone metric space. Then a function is called a -distance on if the following is satisfied: (q1) for all , (q2) for all , (q3) for each and , if for some , then whenever is a sequence in converging to a point , (q4) for all with , there exists with such that , and implies .
Cho et al. [35] noticed the following important remark in the concept of -distance on cone metric spaces.

Remark 1.7 (see [35]). Let be a -distance on a cone metric space . Then (1) does not necessarily hold for all ,(2) is not necessarily equivalent to for all . Very recently, Cho et al. [39] proved the following existence theorems.

Theorem 1.8 (see [39]). Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on and let be a continuous function having the mixed monotone property such that for some and all with or . If there exist such that and , then has a coupled fixed point . Moreover, one has .

Theorem 1.9 (see [39]). Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on , and let be a function having the mixed monotone property such that for some and all with or . Also, suppose that has the following properties: (1)if is a nondecreasing sequence in with , then for all , (2)if is a nonincreasing sequence in with , then for all .
If there exist such that and , then has a coupled fixed point . Moreover, one has .

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

In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of -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 be a cone metric space, and let be a cone distance on . Let and be sequences in and . Suppose that is a sequence in converging to . Then the following holds.(1)If and , then . (2)If and , then converges to . (3)If for , then is a Cauchy sequence in . (4)If , then is a Cauchy sequence in .

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

Theorem 2.2. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and be two mappings such that for some and for all with or . Assume that and satisfy the following conditions:(1) is continuous, (2) is continuous and commutes with , (3), (4) is complete, (5) has the mixed -monotone property. If there exist such that and , then and have a coupled coincidence point . Moreover, one has and .

Proof. Let be such that and . Since , we can choose such that and . Again since , we can choose such that and . Since has the mixed -monotone property, we have and . Continuing this process, we can construct two sequences and in such that Let . Then by (2.1), we have
Repeating (2.3) -times, we get Thus, we have Let with . Then by and (2.5), we have Similarly, we have From part (3) of Lemma 2.1, we conclude that and are Cauchy sequences in in . Since is complete, there are such that and . Using the continuity of , we get and . Also, by continuity of and commutativity of and , we have Hence, is a coupled coincidence point of and . Moreover, by (2.1) we have Since , we conclude that , and hence and .

The continuity of 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 be a cone metric space, and let be a -distance on . Let be a sequence in . Suppose that and are sequences in converging to . If and , then .

Proof. Let be arbitrary. Since , so there exists such that for all . Similarly, there exists such that for all . Thus, for all , we have Take , so by , we get that for each ; hence .

Theorem 2.4. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and let be two mappings such that for some and for all with or . Assume that and satisfy the following conditions: (1), (2) is a complete subspace of , (3) has the mixed -monotone property.

Suppose that has the following properties: (i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all . Assume there exist such that and . Then and have a coupled coincidence point, say . Also, .

Proof. As in the proof of Theorem 2.2, we can construct two Cauchy sequences and in the complete cone metric space . Then, there exist such that and . Similarly we have for all By (q3), we get that By summation, we get that Since is nondecreasing and is nonincreasing, using the properties (i), (ii) of , we have From this and (2.14), we have Therefore By (2.16), we have This implies that By (2.14), (2.21) and Lemma 2.3, we obtain . Similarly, by (2.15), (2.22), and Lemma 2.3, we obtain . Also, adjusting as the proof of Theorem 2.2, we get that

Corollary 2.5. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let , and let be two mappings such that for some with and for all with or . Assume that and satisfy the following conditions: (1) is continuous, (2) is continuous and commutes with , (3), (4) is complete, (5) has the mixed -monotone property. If there exist such that and , then and have a coupled coincidence point . Moreover, one has and .

Proof. Given such that . By (2.24), we have Thus Since , the result follows from Theorem 2.2.

Corollary 2.6. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a continuous mapping having the mixed monotone property such that for some with and for all with or . If there exist such that and , then has a coupled fixed point . Moreover, one has and .

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

Corollary 2.7. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and be two mappings such that for some with and for all with or . Assume that and satisfy the following conditions: (1), (2) is a complete subspace of , (3) has the mixed -monotone property.

Suppose that has the following properties: (i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all . Assume there exist such that and . Then and 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 be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a mapping having the mixed monotone property such that for some with and for all with or . Suppose that has the following properties: (i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all . Assume there exist such that and . Then has a coupled fixed point.

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

Corollary 2.9. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on , and let be a continuous mapping having the mixed monotone property such that for some and for all with or .
If there exist such that and , then has a coupled fixed point . Moreover, we have and .

Proof. It follows from Theorem 2.2 by taking .

Corollary 2.10. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a mapping having the mixed monotone property such that for some and for all with or . Suppose that has the following properties: (i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all . Assume there exist such that and . Then has a coupled fixed point.

Proof. It follows from Theorem 2.4 by taking .

Example 2.11. Let with and . Let with usual order . Define by for all . Then is a partially ordered cone metric space. Define by for all . Then is a -distance. Define by Then, (1), for all and , (2)there is no such that for all and , (3)there is no such that for all and . Note that and . Thus by Corollary 2.10, we have which has a coupled fixed point. Here is a coupled fixed point of .

Proof. The proof of (2.1) is easy. To prove (2.3), suppose the contrary; that is, there is such that for all and . Take and . Then Thus Hence 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 where and . Let denote the space of -valued continuous functions on . The purpose of this section is to give an existence theorem for a solution to (3.1) that belongs to , by using the obtained result given by Corollary 2.10. Let , and let be the cone defined by We endow with the cone metric defined by It is clear that is a complete cone metric space. Let for all . Then, is a -distance.

Now, we endow with the partial order given by Also, the product space can be equipped with the partial order (still denoted ) given as follows: It is easy that and given in Corollary 2.10 are satisfied.

Now, we consider the following assumptions: (a) is continuous, (b)for all , the function has the mixed monotone property, (c)for all , for all with and , we have where is continuous nondecreasing an satisfies the following condition: There exists such that (d)there exists such that We have the following result.

Theorem 3.1. Suppose that hold. Then, (3.1) has at least one solution .

Proof. Define the mapping by We have to prove that has at least one coupled fixed point .

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

Now, let such that ( and ) or ( and ). Using (c), for all , we have which implies that Similarly, one can get We deduce Thus Thus, we proved that condition (2.31) of Corollary 2.10 is satisfied. Moreover, from (d), we have and . Finally, applying our Corollary 2.10, we get the desired result.