Abstract

We prove new coincidence point theorems for the -contractions and generalized Meir-Keeler-type --contractions in partially ordered metric spaces. Our results generalize many recent coincidence point theorems in the literature.

1. Introduction and Preliminaries

Throughout this paper, by , we denote the set of all nonnegative real numbers, while is the set of all natural numbers. Let be a metric space, a subset of , and a map. We say is contractive if there exists such that for all , The well-known Banach's fixed point theorem asserts that if , is contractive and is complete, then has a unique fixed point in . It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping is called a quasicontraction if there exists such that for any . In 1974, Ćirić [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

Recently, Eslamian and Abkar proved the following theorem.

Theorem 1 (see [3]). Let be a complete metric space and be such that where are as follows: is continuous and nondecreasing, is continuous, is lower semicontinuous, and Then has a fixed point in .

Recently, fixed point theory has developed rapidly in partially ordered metric spaces (e.g., [48]).

In 2012, Choudhury and Kundu [9] proved the following coincidence theorem as a generalization of Theorem 1.

Theorem 2 (see [9]). Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space and be such that , is -nondecreasing, is closed, and where are such that is continuous and nondecreasing, is continuous, is lower semicontinuous, and Also, if any nondecreasing sequence in converges to , then we assume that If there exists with , then and have a coincidence point in .

In this paper, we prove new coincidence point theorems for the -contractions and generalized Meir-Keeler-type --contractions in partially ordered metric spaces. Our results generalize many recent coincidence point theorems in the literature.

2. Main Results

We start with the following definition.

Definition 3 (-nondecreasing mapping [4]). Let be a partially ordered set and . Then is said to be -nondecreasing if, for ,

In the sequel, we denote by the class of functions satisfying the following conditions: ) is an increasing, continuous function in each coordinate, ()  for all , , , , and , if and only if .

Next, we denote by the class of functions satisfying the following conditions: () is a continuous function and monotone nondecreasing; () for and ; () is subadditive, that is, for all .

And, we denote the following sets of functions:

Let be a nonempty set and be a partially ordered set endowed with a metric . Then, the triple is called a partially ordered metric space.

We now state the -contraction and the main fixed point theorem for the -contraction in partially ordered metric spaces, as follows.

Definition 4. Let be a partially ordered complete metric space, and let . Then the pair is called a -contraction if the following inequality holds: for all with , where , , and .

We now state the main fixed point theorem for the -contraction in partially ordered metric spaces, as follows.

Theorem 5. Let be a partially ordered complete metric space, and let be such that , is -nondecreasing and is closed. Suppose the pair is a -contraction, and Also, if any nondecreasing sequence in converges to , then we assume that If there exists with , then and have a coincidence point in .

Proof. Since and there exists with , we can choose such that . Since is -nondecreasing, we have . In this process, we construct the sequence recursively as Thus, we also conclude that If any two consecutive terms in (14) are equal, then the conclusion of the theorem follows. So we may assume that Now, we claim that for all . If not, we assume that for some , substituting and in (10) and using the definition of the function , we have and hence Since for all , we have that , which contradicts to (15). Therefore, we conclude that From above argument, we also have that for each It follows (18) that the sequence is monotone decreasing, it must converge to some . Taking limit as in (19) and using the continuities of and and the lower semicontinuity of , we get which implies that . So we conclude that
We next claim that is a Cauchy sequence, that is, for every , there exists such that if , then .
Suppose the above statement is false. Then there exists such that for any , there are with satisfying Further, corresponding to , we can choose in such a way that it is the smallest integer with and . Therefore . Now we have that for all Letting , then we get On the other hand, we have Letting , then we get By (14), we have that the elements and are comparable. Substituting and in (10), we have that for all , By above argument and using inequality (10), we can conclude that which implies that , a contradiction. Therefore, the sequence is a Cauchy sequence.
Since is complete and is closed, there exists such that Later, we prove that is a coincidence point of and . From (14) and (29), we deduce that Substituting and in (10), we have that Taking in the above inequality, we have which implies that , that is, . So we complete the proof.

We give the following example to illustrate Theorem 5.

Example 6. Let . We define a partial order “” on as if and only if for all . We take the usual metric for all . Let be defined as Let be defined as and let denote Without loss of generality, we assume that and verity inequality (10).
For all with , we have Therefore, inequality (10) is satisfied and all the conditions of Theorem 5 are satisfied, and we obtained that is a coincidence point of and .

Applying Definition 4, Theorem 5, and Example 6, if we let we are easy to get the following theorem.

Theorem 7. Let be a partially ordered complete metric space, and let be such that , is -nondecreasing, is closed, and for all such that , where , , and , and Also, if any nondecreasing sequence in converges to , then one assumes that

If there exists with , then and have a coincidence point in .

In the other research of this paper, we recall the Meir-Keeler-type contraction [10] and -admissible mapping [11]. In 1969, Meir and Keeler [10] introduced the following notion of Meir-Keeler-type contraction in a metric space .

Definition 8. Let be a metric space, . Then is called a Meir-Keeler-type contraction whenever for each there exists such that

And, the following definition was introduced in [11].

Definition 9. Let be a self-mapping of a set and . Then is called a -admissible mapping if

We introduce the notion of --admissible mapping, as follows.

Definition 10. Let be a self-mapping of a set and . Then is called a --admissible mapping if

We give the following example to illustrate Definition 10.

Example 11. Let and we define Then is a --admissible mapping.

We now state the new notions of generalized Meir-Keeler-type -contractions and generalized Meir-Keeler-type --contractions in partially ordered complete metric spaces, as follows.

Definition 12. Let be a partially ordered complete metric space, and let . Then the pair is called a generalized Meir-Keeler-type -contraction whenever for each , there exists such that for all with , where .

Definition 13. Let be a partially ordered complete metric space, let , and . Then is called a generalized Meir-Keeler-type --contraction if the following conditions hold: (1) is --admissible; (2)for each there exists such that for all with , where .

Remark 14. Note that if is a generalized Meir-Keeler-type --contraction, then we have that for all Further, if then .
On the other hand, if then

We now state our main result for the generalized Meir-Keeler-type --contraction, as follows.

Theorem 15. Let be a partially ordered complete metric space, let be continuous in each coordinate, and let be such that , is -nondecreasing, and is closed. Suppose the pair is a generalized Meir-Keeler-type --contraction and the following conditions hold.(i)If any nondecreasing sequence in converges to , then we assume that (ii)There exists with and . (iii)If for all , then . Then and have a coincidence point in .

Proof. Since and by , there exists with and , we can choose such that . Since is -nondecreasing, we have . In this process, we construct the sequence recursively as Thus, we also conclude that If any two consecutive terms in (53) are equal, then the conclusion of the theorem follows. So we may assume that On the other hand, since is --admissible and , we have By continuing this process, we get By (53), (54), and (56), substituting and in (50), we have If , then the inequality (57) becomes which implies a contradiction, and we get that .
From the argument above, we have that the sequence is decreasing, and it must converge to some , that is, It follow from that (57) and (59), we have Notice that . We claim that . Suppose, to the contrary, that . Since is a generalized Meir-Keeler-type --contraction, corresponding to use, and taking into account the above inequality (60), there exist and a natural number such that which implies So we get a contradiction, since . Thus we have that
We next claim that is a Cauchy sequence, that is, for every , there exists such that if , then .
Suppose the above statement is false. Then there exists such that for any , there are with satisfying Further, corresponding to , we can choose in such a way that is it the smallest integer with and . Therefore . Now we have that for all Letting , then we get On the other hand, we have Letting , then we get By (53), we have that the elements and are comparable. Substituting and in (50), we have that for all , Letting in (69), then we get which implies a contradiction. Thus, is a Cauchy sequence.
Since is complete and is closed, there exists such that Since is continuous in each coordinate and by the condition , we have
Later, we prove that is a coincidence point of and . From (53) and (71), we deduce that By (72) and substituting and in (50), we have that Taking in the above inequality, we have This implies that . So we complete the proof.

Apply Theorem 15, we are easy to get the following theorem.

Theorem 16. Let be a partially ordered complete metric space, and let be such that , is -nondecreasing, and is closed. Suppose the pair is a generalized Meir-Keeler-type -contraction and the following conditions hold. (i)If any nondecreasing sequence in converges to , then we assume that (ii)There exists with . Then and have a coincidence point in .