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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 604578, 7 pages
http://dx.doi.org/10.1155/2013/604578
Research Article

Coupled Coincidence Point Theorems for New Types of Mixed Monotone Multivalued Mappings in Partially Ordered Metric Spaces

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 19 June 2013; Accepted 15 September 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Chalongchai Klanarong and Suthep Suantai. 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

We introduce and study new types of mixed monotone multivalued mappings in partially ordered complete metric spaces. We give relationships between those two types of mappings and prove their coupled fixed point and coupled common fixed point theorems in partially ordered complete metric spaces. Some examples of each type of mappings satisfying the conditions of the main theorems are also given. Our main result includes several recent developments in fixed point theory of mixed monotone multivalued mappings.

1. Introduction and Preliminaries

Let be a metric space, and let be the class of all nonempty bounded and closed subsets of . For and , we denote . For , define If , then we write . Also in addition, if , then . For all , the definition of gives the following formulas: (i);(ii);(iii) if and only if ;(iv).

It is easy to see that the following inequality holds, for , for all . By using the above inequality, the following lemma is obtained.

Lemma 1. Let be a metric space, and let be a nonempty subset of . If is defined by then is continuous.

Let be a nonempty set, (collection of all nonempty subsets of ), and . An element is called(i)coupled fixed point of if and ,(ii)coupled coincidence point of a hybrid pair if and ,(iii)coupled common fixed point of a hybrid pair if and .

We denote the set of coupled coincidence point of mappings and by . Note that if , then is also in .

The hybrid pair is called -compatible if whenever . The mapping is called -weakly commuting at some point if and .

Let be a partially ordered set, and suppose that there is a metric on such that is a metric space. For , we write if and . On the product space , we consider the following partial order:

The existence of a fixed point for contraction type of mappings in partially ordered metric spaces has been considered recently by Ran and Reurings [1], Gnana Bhaskar and Lakshmikantham [2], Nieto and Rodríguez-López [3, 4], Agarwal et al. [5], Lakshmikantham and Ciric [6], and Harjani and Sadarangani [7].

In [2], Gnana Bhaskar and Lakshmikantham introduced the notions of mixed monotone mappings and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings.

In [3], Nieto and Rodríguez-López studied some fixed point theorems for monotone nondecreasing mappings in partially ordered metric spaces. They proved the existence of a fixed point in partially ordered metric spaces and applied the obtained results to study a problem of ordinary differential equations.

In [6], Lakshmikantham and Ciric introduced notions of a mixed -monotone mapping and proved coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. They defined the mixed -monotone property as follows.

Let be a partially ordered set, and let and . We say that has the mixed -monotone property if is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument; that is, for any ,

They proved the following theorem.

Theorem 2 (see [6, Theorem 2.1]). Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Assume that there is a function with and for each , and also suppose that and are such that has the mixed -monotone property and for all for which and . Suppose , is continuous and commutes with , and also suppose that either (a) is continuous or(b) has the following properties:(i)if a nondecreasing sequence in , then for all ;(ii)if a nonincreasing sequence, in , then for all .If there exist such that and , then there exist such that and . That is, and have a coupled coincidence point.

Coupled coincidence and coupled common fixed point theorems for mixed -monotone single valued mappings can be applied for solving solutions of some ordinary differential equations, so there are many authors who pay their attention for the existence problem of coupled coincidence and coupled common fixed point theorems; see more detail in [6, 811].

In [8], Abbas et al. studied coupled coincidence point and common fixed point theorems for a pair of multivalued and single valued mappings which satisfy a generalized contractive condition in complete metric spaces. In [12], Huang and Fang introduced a class of multivalued mixed increasing operators in Banach spaces and proved some new fixed point and coupled fixed point theorems. In [13], Choudhury and Metiya proved some fixed point theorems for multivalued and single valued mappings in partially ordered metric spaces. In [14], Yin and Guo introduced the new notion of -monotone mapping and proved some fixed point theorems for multivalued and single valued -increasing mappings in partially ordered metric spaces. The mappings considered in [14] are assumed to satisfy certain metric inequalities which are established by an altering distance function. The main results of [14] extended and improved those of Choudhury and Metiya [13].

Motivated and inspired by these works, we introduce new classes of mixed monotone multivalued mapping in partially ordered metric spaces and prove some existence theorems for coupled fixed point, coupled coincidence point, and coupled common fixed point.

Next, we define two types of relations between the two sets and use these relations to introduce new types of multivalued mappings and show some relationships between these classes of mappings.

Definition 3. Let be a partially ordered set and let , be two nonempty subsets of . Then we denote(i) if for each there exists such that ;(ii) if for each there exists such that .

Definition 4. Let be a partially ordered set, and let and . We say that is mixed -monotone of type if for any

Definition 5. Let be a partially ordered set, let be a multivalued mapping, and let . We say that is mixed -monotone of type if, for any ,
The following proposition gives a relation between multivalued mappings of these two types.

Proposition 6. Let be a partially ordered set and a metric on , and . If is mixed -monotone of type , then it is of type .

Proof. Let such that and . Since is mixed -monotone of type , we have and . This implies that .

The converse of the above proposition is not true in general as seen in the following example.

Example 7. Let , and let be a usual partially ordered on ; that is, if and only if . Let and be defined by and let . It is easy to show that is mixed -monotone of type , but it is not of type . Consider , , and , we have and . It follows that . Hence, is not mixed -monotone of type .

2. Main Results

In this section, we prove the existence of a coupled fixed points of mixed -monotone multivalued mappings of types and .

Theorem 8. Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Suppose and are such that is mixed -monotone of type . Assume that there is a function with and for each and for all for which and . Suppose that , is a complete subset of , and suppose that(a)there exist such that (b) has the following properties:(i)if an increasing sequence , then for all ;(ii)if a decreasing sequence , then for all .Then there exist such that

Proof. Let be such that and . Since , it follows that there exist such that , and , . Since is mixed -monotone of type , we obtain Again from , we can choose such that , and , . By continuing this process, we obtain sequences and in such that
Put We show that Since and , from (14) and (10) we have Similarly, we have Adding (19) and (20), we obtain (18).
Since for , it follows From (18) that a sequence is monotone decreasing. Therefore, there is some such that We show that . Suppose, to the contrary, that . Then, by taking the limit as on both sides of (18), we have which is a contradiction. Thus, ; that is,
Now we prove that and are Cauchy sequences. Suppose, to the contrary, that at least one of or is not a Cauchy sequence. Then there exists an and two subsequences of integers , , with We may also assume that by choosing to be the smallest number exceeding for which (24) holds. From (24) and (25) and by the triangle inequality, we have By taking the limit as , we get By the triangle inequality, we have Hence, From (15) and (16), we have and . By (14) and (iii), we have Similarly, we have It follows from (30), (31), and (29) that By using (23) and (27) together with the property of , we have which is a contradiction. Thus, and are Cauchy sequences. Since is complete, there exist such that Finally, we claim and .
From (34) and (ii), we obtain , and , for all . By (iii), we have This implies by Lemma 1 and the property of that . Hence, . Similarly, we can show that . Thus, is a coupled coincidence point of and .

Example 9. Let and let be a usual partially ordered on ; that is, if and only if . Let and be defined by and let . It is easy to check that is a mixed -monotone of type . Now we prove that (10) holds. Let such that and . Then and . So and . And we have where . By Theorem 8, we obtain that has a coupled fixed point. Note that is a coupled fixed point of .

The following results are obtained directly from Theorem 8 and Proposition 6.

Corollary 10. Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Suppose that and are such that is a mixed -monotone of type . Assume that there is a function with and for each and for all for which and . Suppose that , that is a complete subset of ; suppose that(a)there exist such that (b) has the following properties:(i)if an increasing sequence , then for all ;(ii)if a decreasing sequence , then for all .Then there exist such that

Note that the contractive condition (38) is weaker than the condition (5) in [6].

Theorem 11. In addition to the hypotheses of Theorem 8 and Corollary 10, if one of the following conditions holds: (a) and are -compatible, is an increasing sequence which converges to , is a decreasing sequence which converges to for some , , and is continuous at and ,(b) is -weakly commuting for some , , and ,(c) is continuous at and for some ; and , for some ,(d) is singleton subset of .Then and have a coupled common fixed point.

Proof. By Theorem 8, and have a coupled coincidence point . Suppose that (a) holds. Then is an increasing sequence which converges to , and is a decreasing sequence which converges to for some , . Hence, and for all . Since is continuous at and , we have that and are fixed points of ; that is, and . As and are -compatible, for all and , . Using (10), we obtain By taking limit as and using the property of , we obtain , and hence . Similarly, we can show that . Consequently, and . Hence, is a coupled common fixed point of and . Suppose that (b) holds. That is is -weakly commuting, , and for some . Then , and . Hence, is a coupled common fixed point of and . Suppose now that (c) holds. Then there exists such that is continuous at and and , for some . By the continuity of at and , we get and . Hence, is coupled common fixed point of and . Finally, suppose that (d) holds. Let . Then . Hence, is a coupled common fixed point of and .

Acknowledgment

This work was supported by Chiang Mai University, Chiang Mai, Thailand.

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