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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 787342, 7 pages
http://dx.doi.org/10.1155/2013/787342
Research Article

Common Fixed Points of Generalized Cyclic Meir-Keeler-Type Contractions in Partially Ordered Metric Spaces

1Department of Applied Mathematics, National Hsinchu University of Education, Taiwan
2Department of Mathematics, National Kaohsiung Normal University, Taiwan

Received 19 June 2013; Accepted 2 August 2013

Academic Editor: E. Karapinar

Copyright © 2013 Chiming Chen and Ing-Jer Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type ()-contraction in partially ordered metric spaces. Our results generalize many recent common 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, let be a subset of , and let be a map. We say that is contractive if there exists such that for all , The well-known Banach 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.

The following definitions and results will be needed in the sequel. Let and be two nonempty subsets of a metric space . A mapping is called a cyclic map if and . In 2003, Kirk et al. [3, 4] proved the following fixed point theorem.

Theorem 1 (see [3, 4]). Let and be two nonempty closed subsets of a complete metric space , and suppose that satisfies(i) and , (ii) for all , , and .
Then is nonempty, and has a unique fixed point in .

Recently, many authors proved some fixed point theorems for cyclic maps satisfying various contractive conditions (see, [520]).

Let be a nonempty set, and let be a partially ordered set endowed with a metric . Then, the triple is called a partially ordered metric space. Two elements are said to be comparable if either or holds. Altun et al. [21] introduced the notion of weakly increasing mappings and proved some existing theorems.

Definition 2 (see [21]). Let be a partially ordered set and . Then are said to be weakly increasing if and for all .

And the following definition was introduced in [22].

Definition 3 (see [22]). Let be a partially ordered set, let be closed subsets of with , and let . Then the pair is said to be -weakly increasing if for all and for all .

In this paper, we introduce the new notion of generalized cyclic Meir-Keeler-type -contraction. The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type -contraction in partially ordered metric spaces. Our results generalize many recent common point theorems in the literature.

2. Main Results

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 .

We start with the following definition.

Definition 4 (see [23]). Let be a self-mapping of a set and . Then is called -admissible if

Definition 5. Let be two nonempty subsets of a set with , let , with and , and let . Then the pair is called -admissible if the following conditions hold: (1), , (2), .

In 1969, Meir and Keeler [24] introduced the following notion of Meir-Keeler-type contraction in a metric space .

Definition 6. Letting be a metric space, . Then is called a Meir-Keeler-type contraction whenever for each , there exists such that

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

Definition 7. Let be a partially ordered metric space, let be two nonempty subsets of with , and let , with and . Then the pair is called a generalized cyclic Meir-Keeler-type -contraction; if for any comparable elements with and , we have that for each there exists such that where .

Definition 8. Let be a partially ordered metric space, let be two nonempty subsets of with , and let , with and , and . Then is called a generalized cyclic Meir-Keeler-type -contraction if the following conditions hold: (1) the pair is -admissible; (2) for any comparable elements with and , we have that for each there exists such that where .

Remark 9. Note that if is a generalized cyclic Meir-Keeler-type -contraction, then we have that for any comparable elements with and , Further, if then .
On the other hand, if then

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

Theorem 10. Let be a partially ordered complete metric space, let be nonempty closed subsets of with , let , and let be two mappings such that the pair is a generalized cyclic Meir-Keeler-type ()-contraction and -weakly increasing. Suppose that the following conditions hold: (i) or is continuous; (ii)there exists with ; (iii)if for all and , then and .
Then and have a common fixed point in .

Proof. By (ii), there exists with . Since and the pair is -admissible, there exists such that Since and the pair is -admissible, there exists such that Continuing this process, we construct the sequence in such that and for all , Since the pair is -weakly increasing, we have that and so we conclude that for all ,
Step  1. We will show that is a Cauchy sequence in .
Case  1. Suppose that for some in the inequality (16). Since and are comparable in with and , by the Remark 9, we have If , then , . By Remark 9, we get a contradiction. So we conclude that ; that is, . Similarly, we may show that . Hence is a constant sequence, and so is a Cauchy sequence in .
Case  2.  Suppose that for all in the inequality (16).
Substep  1. We show that the sequence is decreasing.
Subcase  1. If is even, then we let for some . Since , , and , are comparable in , we have If , then the above inequality becomes which is a contradiction. So we have that
Subcase  2. If is odd, then we let for some . Since , and , are comparable in , we have If , then the above inequality becomes which is a contradiction. So we have that From (20) and (23), we conclude that From the above argument, we have that the sequence is decreasing, and it must converge to some ; that is,
Substep  2. We next claim that Notice that . We claim that . Suppose, to the contrary, that .
If is even, by the argument of Subcase 1 and the inequality (25), we have Since is a generalized cyclic Meir-Keeler-type -contraction, corresponding to use and taking into account the above (27), there exist and a natural number such that which implies So we get a contradiction, since . Thus we have that
If is odd, by the argument of Subcase 2 and the inequality (25), we have Similarly, we can prove that
Substep  3. We show that is a Cauchy sequence in . It is sufficient to show that is a Cauchy sequence in .
Suppose, to the contrary, that is not a Cauchy sequence in . Then there exist and two subsequences and of such that is the smallest integer for which , and we get Letting in the above inequality, we get On the other hand, we also obtain that Letting in the above inequality, we get Since letting in the above inequality, we have Since , , and , are comparable in , we have Letting in the above inequality and using (37) and (39), we get which implies a contradiction. So we get that is a Cauchy sequence in .
Step  2. Finally, we prove the existence of common fixed point of and .
Since is complete and is a Cauchy sequence in , there exists such that From (42) and since for all , we have and .
Since is a sequence in and is closed, by (42), we have that . Similarly, since is a sequence in and is closed, by (42), we have that . We now claim that is a common fixed point of and . Without loss of generality, we assume that is continuous, and by (42), we have By the uniqueness of the limit, we have that .
Since with and , we have This implies that . So we complete the proof.

Applying Theorem 10 and if we let , then we immediately get the following theorem.

Theorem 11. Let be a partially ordered complete metric space, let be nonempty closed subsets of with , and let be two mappings such that the pair is a generalized cyclic Meir-Keeler-type -contraction and -weakly increasing. If or is continuous, then and have a common fixed point in .

We next state our second main result for the generalized cyclic Meir-Keeler-type -contraction as follows.

Theorem 12. Let be a partially ordered complete metric space, let be nonempty closed subsets of with , let , and let be two mappings such that the pair is a generalized cyclic Meir-Keeler-type -contraction and -weakly increasing. Suppose that the following conditions hold: (i)if is a nondecreasing sequence in and , then ; (ii)there exists with ; (iii)if for all and , then and .
Then and have a common fixed point in .

Proof. From the same proof’s process of Theorem 10, we can construct a nondecreasing sequence in with , , and for some . Since and , are nonempty closed subsets of , we have , , and . By the condition (i), we get for all .
Since and , we have Letting in the above inequality, we get This implies that ; that is, . Similarly, we may show that . So is a common fixed point of and .

Applying Theorem 12, it is easy to get the following theorem.

Theorem 13. Let be a partially ordered complete metric space, let be nonempty closed subsets of with , and let be two mappings such that the pair is a generalized cyclic Meir-Keeler-type -contraction and -weakly increasing. Suppose that the following condition holds:if is a nondecreasing sequence in and , then .
Then and have a common fixed point in .

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