`Journal of Function Spaces and ApplicationsVolume 2013 (2013), Article ID 827458, 7 pageshttp://dx.doi.org/10.1155/2013/827458`
Research Article

Multivalued Pseudo-Picard Operators and Fixed Point Results

Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey

Received 24 May 2013; Accepted 25 July 2013

Copyright © 2013 Gülhan Mınak et al. 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 the concept of multivalued pseudo-Picard (MPP) operator on a metric space. This concept is weaker than multivalued weakly Picard (MWP) operator, which is given by M. Berinde and V. Berinde (2007). Then, we give both fixed point results and examples for MPP operators. Also, we obtain some ordered fixed point results for multivalued maps as application.

1. Introduction and Preliminaries

Let be a metric space, and let denote the class of all nonempty, closed, and bounded subsets of . It is well known that defined by is a metric on , which is called Hausdorff metric, where Let be a map; then, is called multivalued contraction if for all there exists such that

In 1969, Nadler [1] proved a fundamental fixed point theorem for multivalued maps: every multivalued contraction on complete metric space has a fixed point.

Then, a lot of generalizations of the result of Nadler were given (see, e.g., [24]). Two important generalizations of it were given by M. Berinde and V. Berinde [5] and Mizoguchi and Takahashi [6].

In [5], M. Berinde and V. Berinde introduced the concept of multivalued weakly Picard operator as follows (for single-valued Picard and weakly Picard operators we refer to [79]).

Definition 1. Let be a metric space, and let (the family of all nonempty subsets of ) be a multivalued operator. is said to be multivalued weakly Picard (MWP) operator if and only if for each and any , there exists a sequence in such that(i), (ii), (iii)the sequence is convergent and its limit is a fixed point of .

Then M. Berinde and V. Berinde [5] show that every Nadler [1], Reich [10], Rus [11] and Petruşel [12] type multivalued contractions on complete metric space are MWP operators. Mizoguchi and Takahashi [6], proved the following fixed point theorem. This is also an example of MWP operator.

Theorem 2. Let be a complete metric space, and let be a multivalued map. Assume that for all , where is an -function (i.e., it satisfies for all ). Then is an MWP operator.

In the same paper, M. Berinde and V. Berinde [5] introduced the concepts of multivalued ()-weak contraction and multivalued ()-weak contraction and proved the following nice fixed point theorems.

Theorem 3. Let be a complete metric space, and let be a multivalued ()-weak contraction; that is, there exist two constants and such that for all . Then is an MWP operator.

Theorem 4. Let be a complete metric space, and let be a multivalued ()-weak contraction; that is, there exist an -function and a constant such that for all . Then is an MWP operator.

We can find some detailed information about the single-valued case of ()-weak contraction and the nonlinear case of it in [1315].

Recently, Samet et al. [16] introduced the notion of --contractive mappings and gave some fixed point results for such mappings. Their results are closely related to some ordered fixed point results. Then, using their idea, some authors presented fixed point results for single and multivalued mappings (see, e.g., [1620]). First, we recall these results. Denote by the family of nondecreasing functions such that for all .

Definition 5 (see [16]). Let be a metric space, be a self map on , and be a function. Then is called --contractive whenever for all .

Note that every Banach contraction mapping is an --contractive mapping with and for some .

Definition 6 (see [16]). is called -admissible whenever implies that .

There exist some examples for -admissible mappings in [16]. For convenience, we mention in here one of them. Let . Define and by for all and for and for . Then is -admissible.

Definition 7 (see [17]). is said to have (B) property whenever is a sequence in such that for all and , then for all .

Theorem 8 (see [16]). Let be a complete metric space, and let be an -admissible and --contractive mapping. If there exists such that and is continuous, then has a fixed point.

Remark 9. If we assume that has (B) property instead of the continuity of , then again has a fixed point (Theorem 2.2 of [16]). If for each there exists such that and , then is said to have (H) property. Therefore, if has the (H) property in Theorems 2.1 and 2.2 in [16], then the fixed point of is unique (Theorem 2.3 of [16]).

Then, some generalizations of --contractive mappings are given as follows.

Definition 10 (see [17]). is called Ćirić type --generalized contractive mapping whenever for all , where

Note that every Ćirić type generalized contraction mapping is a Ćirić type --generalized contractive mapping with and for some .

Theorem 11 (see [17]). Let be a complete metric space, and let be an -admissible and Ćirić type --generalized contractive mapping. If there exists such that and is continuous or has (B) property, then has a fixed point. If has the (H) property, then the fixed point of is unique.

We can find some fixed point results for single-valued mappings in these directions in [18, 20]. Now we recall some multivalued cases.

Definition 12 (see [17, 19]). Let be a metric space, and let be a multivalued mapping. Then is called multivalued --contractive whenever for all , and is called multivalued --contractive whenever where . Similarly if we replace with we can obtain Ćirić type multivalued --generalized contractive and Ćirić type multivalued --generalized contractive mappings on .

Definition 13 (see [17, 19]). Let be a metric space, and let be a multivalued mapping.(a) is said to be -admissible whenever each and with imply that for all .(b) is said to be -admissible whenever each and with imply that .

Remark 14. It is clear that -admissible maps are also -admissible, but the converse may not be true as shown in the following example.

Example 15. Let , and is defined by and for . Define by Leting and , then , but . Thus, is not -admissible. Now we show that, is -admissible with the following cases.
Case 1. If , then and . Also, since .
Case 2. If , then and . Also, for all .
Case 3. If , then and . Also, since .

The multivalued version of the results for --contractive mappings is given [17, 19] as follows.

Theorem 16 (see [19]). Let be a complete metric space, let be a function, let be a strictly increasing map, and let , -admissible and --contractive multifunction on . Suppose that there exist and such that . Assume that is continuous or has (B) property; then has a fixed point.

Theorem 17 (see [17]). Let be a complete metric space, let be a function, let be a strictly increasing map, and let , -admissible and --contractive multifunction on . Suppose that there exist and such that . Assume that has (B) property. then has a fixed point.

The purpose of this paper is to introduce the concept of multivalued pseudo-Picard (MPP) operators and present fixed point results and examples.

2. Main Results

Definition 18. Let be a metric space, and let be a multivalued operator. is said to be multivalued pseudo-Picard (MPP) operator if and only if there exist , and a sequence in such that(i), (ii)the sequence is convergent and its limit is a fixed point of .

Remark 19. It is clear that the operators mentioned in Theorems 16 and 17 are MPP operators. Also, note that every MWP operator is an MPP operator, but the converse may not be true as shown in the following examples.

Example 20. Let and . Define by Then is not an MWP operator. Indeed, letting , then and so . Therefore, for , which is not convergent. But is MPP operator. To see this, let and . Continuing this way, we can construct a sequence in with such that it is convergent to , which is a fixed point of .

Example 21. Let and . Define by Then is an MPP but not MWP operator.

Before we give our main results, we recall the following. Let and be two topological spaces. Then a multivalued map is said to be upper semicontinuous (lower semicontinuous) if the inverse image of a closed set (open set) is closed (open). A multivalued map is continuous if it is upper as well as lower semicontinuous.

Lemma 22 (see [21]). Let be a metric space, and let be an upper semicontinuous map such that is closed for all . If and , then .

Theorem 23. Let be a complete metric space and let be an -admissible multivalued mapping such that for all , where is strictly increasing and . Suppose that there exist and such that . If is upper semicontinuous or has (B) property, then is an MPP operator.

Proof. Let and be as mentioned in the hypotheses. If or , then the proof is complete. Let and , then where is a constant. Therefore, there exists such that Also, since is -admissible, , and , then for all and so . Since is strictly increasing, we have Get . Then . If , then the proof is complete. Let . Then Therefore, there exists such that Since is -admissible, , and , then for all . Thus, since . Since is strictly increasing, we have Get . Then . If , then the proof is complete. By the way, we can construct a sequence in such that , and for all . Now, for each , we have Therefore, is a Cauchy sequence in . Since is complete, there exists such that . If is upper semicontinuous, then from Lemma 22, we have . Now assume that has (B) property. Then for all . Also, since and is continuous at , then Therefore, we have and so .

Although -admissibility implies -admissibility of , we will give the following theorem because the contractive condition is slightly different from (16).

Theorem 24. Let be a complete metric space, and let be an -admissible multivalued mapping such that for all , where is strictly increasing and . Suppose that there exist and such that . If is upper semicontinuous or has (B) property, then is an MPP operator.

Proof. Let and be as mentioned in the hypotheses, then since is -admissible. If or , then the proof is complete. Let and , then where is a constant. Therefore, there exists such that Since , then . Since is strictly increasing, Get . Then . If , then the proof is complete. Let . Then Therefore, there exists such that Since , then . By the way, we can construct a sequence in such that , and for all . Now, for each , we have Therefore, is a Cauchy sequence in . Since is complete, there exists such that . If is upper semicontinuous, then from Lemma 22, we have . Now assume that has (B) property. Then for all . Since is -admissible, . Therefore, and taking limit we have and so .

Remark 25. If we take by , then any multivalued mappings are -admissible as well as -admissible. Therefore, Theorem 3 is a special case of Theorems 23 and 24.

Remark 26. If we take in Theorems 23 and 24, then we have Theorems 16 and 17, respectively.

Now we give an example to illustrate our main results. Note that both Theorems 3 and 16 cannot be applied to this example.

Example 27. Let and . Define by and by Then is an -admissible, and condition (16) is satisfied for and . Indeed, first, we show that is an -admissible. Letting and with , then it should be . Thus, , and hence for all . Therefore, is an -admissible.
Now we consider the following cases.
Case 1. Letting with , then . Thus, (16) is satisfied. Also note that if , then and so (16) is satisfied. Therefore, we will consider in the following.
Case 2. Letting with , then . There are some subcases as follows.
Subcase 1. Consider that , and suppose that (without loss of generality) ; then and so again (16) is satisfied.
Subcase 2. Consider that and ; then
Subcase 3. If and , then Thus, (16) is satisfied.
Finally, has (B) property, then by Theorem 23, is an MPP operator.
Note that since , , and , then condition (6) is not satisfied. Therefore, Theorem 3 cannot be applied to this example.
Also, note that, since , , and , is not multivalued --contractive mapping. Therefore, Theorem 16 cannot be applied to this example.

3. Applications

Our results can be applied to some ordered fixed point results. First we recall some ordered notions. Let be a nonempty set and be a partial order on .

Definition 28 (see [22]). Let be two nonempty subsets of ; the relations between and are defined as follows. If for every , there exists such that , then . If for every , there exists such that , then . If and , then .

Remark 29 (see [22]). and are different relations between and . For example, let , , , and be a usual order on ; then , but ; if , , then , while .

Remark 30 (see [22]). , , and are reflexive and transitive but are not antisymmetric. For instance, let , , , and be a usual order on ; then and , but . Hence, they are not partial orders.

Theorem 31. Let be a partially ordered set, and suppose that there exists a metric in such that is complete metric space. Let be a multivalued mapping such that for all with , where is strictly increasing and . Suppose that there exists such that . Assume that for each and with , are has for all . If is upper semicontinuous or satisfies the following condition that then has a fixed point.

Proof. Define the mapping by Then we have for all . Also, since , then there exists such that and so . Now letting and with , then , and so by the hypotheses we have for all . Therefore, for all . This shows that is -admissible. Finally, if is upper semicontinuous or satisfies (41), then is upper semicontinuous or has (B) property. Therefore, from Theorem 23, has a fixed point.

Remark 32. We can give a similar result using instead of .

Acknowledgment

The authors are grateful to the referees because their suggestions contributed to the improvement of the paper.

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