Fixed Point Theorems for Ćirić-Berinde Type Contractive Multivalued Mappings

Abstract

We give a Ćirić-Berinde type contractive condition for multivalued mappings and analyze the existence of fixed point for these mappings.

1. Introduction and Preliminaries

In 2012, Samet et al. [1] introduced the notions of --contractive mapping and -admissible mappings in metric spaces and obtained corresponding fixed point results, which are generalizations of ordered fixed point results (see [1]). Since then, by using their idea, some authors investigated fixed point results in the field. Asl et al. [2] extended some of results in [1] to multivalued mappings by introducing the notions of --contractive mapping and -admissible mapping.

Recently, Salimi et al. [3] modified the notions of --contractive mapping and -admissible mappings by introducing another function . And then, they gave generalizations of the results of Samet et al. [1] and Karapınar and Samet [4]. Hussain et al. [5] extended these modified notions to multivalued mappings. That is, they introduced the notion of --contractive multifunctions and gave fixed point results for these multifunctions.

Very recently, Ali et al. [6] generalized and extended the notion of --contractive mapping by introducing the notion of -contractive multivalued mappings and obtained fixed point theorems for these mappings in complete metric spaces.

The purpose of this paper is to introduce the notion of Ćirić-Berinde type contractive multivalued mappings and to generalize and extend the notion of --contractive multifunctions and to establish fixed point theorems for Ćirić-Berinde type contractive multivalued mappings.

Let be a metric space. We denote by the class of nonempty closed and bounded subsets of and by the class of nonempty closed subsets of . Let be the generalized Hausdorff distance on ; that is, for all , where is the distance from point to subset .

For , let .

Then, we have for all .

From now on, we denote by for a multivalued map and .

We denote by the class of all functions such that(1)is continuous;(2) is nondecreasing on ;(3) if and only if ;(4) is subadditive.

Also, we denote by the family of all nondecreasing functions such that for each , where is the th iterate of .

Note that if , then and for all .

Let be a metric space, and let be a function.

We consider the following conditions:(1)for any sequence in with for all and , we have (2)for any sequence in with for all and a cluster point of , we have (3)for any sequence in with for all and a cluster point of , there exists a subsequence of such that

Remark 1. (1) implies (2) and (2) implies (3).

Note that if is a metric space and , then is a metric space.

Let be a metric space, and let be a multivalued mapping. Then, we say that(1) is called  -admissible [2] if  where ;(2) is called -admissible [7] if, for each and with , we have for all .

Lemma 2. Let be a metric space, and let be a multivalued mapping. If is -admissible, then it is -admissible.

Proof. Suppose that is an -admissible mapping.
Let and be such that .
Let be given.
Since is -admissible, .

Lemma 3. Let be a metric space, and let and .
If and , then there exists such that .

Proof. Let .
Since and is metric on , there exists such that by definition of infimum. Hence, .

Let be a metric space.

A function is called upper semicontinuous if, for each and with , we have .

A function is called lower semicontinuous if, for each and with , we have .

For a multivalued map , let be a function defined by .

2. Fixed Point Theorems

In this section, we establish fixed point theorems for Ćirić-Berinde type contractive multivalued mappings.

Theorem 4. Let be a complete metric space, and let be a function. Suppose that a multivalued mapping is -admissible.
Assume that, for all implies where , , and is strictly increasing.
Also, suppose that the following are satisfied: (1)there exists and such that ;(2)either is continuous or is lower semicontinuous.
Then has a fixed point in .

Proof. Let and be such that . Let be a real number with .
If , then is a fixed point.
Let .
If , then is a fixed point. Let . Then .
From (7) we obtain If , then we have , which is a contradiction.
Thus, , and hence we have Hence, there exists such that Since is -admissible, from condition (1) and , we have If , then is a fixed point. Let .
Then , and so .
From (7) we obtain If , then we have , which is a contradiction.
Thus, , and hence we have Hence, there exists such that Since is -admissible, from and , we have By induction, we obtain a sequence such that, for all , Let be given.
Since , there exists such that For all , we have which implies for all . Hence, is a Cauchy sequence in .
It follows from the completeness of that there exists Suppose that is continuous.
We have By letting in the above inequality, we obtain , and so .
Assume that is lower semicontinuous.
Then, . Hence, . Thus, .

Corollary 5. Let be a complete metric space, and let be a function. Suppose that is an -admissible mapping.
Assume that, for all , where , , and is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem 4 are satisfied.
Then has a fixed point in .

Remark 6. If we have for all , , and is continuous, then Corollary 5 reduces to Theorem 3.4 of [7].

Let be an ordered set and . We say that whenever, for each , there exists such that .

Corollary 7. Let be a complete ordered metric space. Suppose that a multivalued mapping satisfies for all with (resp., ), where , , and is strictly increasing.
Assume that, for each and with (resp., ), we have (resp., ) for all .
Also, suppose that the following are satisfied:(1)there exists and such that (resp., );(2)either is continuous or is lower semicontinuous.
Then has a fixed point in .

Remark 8. If we have for all , , and is continuous, then Corollary 7 reduces to Corollary 3.6 of [7].

From Theorem 4 we obtain the following result.

Corollary 9. Let be a complete metric space, and let be a function. Suppose that is an -admissible mapping.
Assume that, for all implies where , , and is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem 4 are satisfied.
Then has a fixed point in .

Remark 10. If we have in Corollary 9, then Corollary 9 reduces to Theorem 2.5 of [6].

Corollary 11. Let be a complete metric space, and let be a function. Suppose that is an -admissible mapping.
Assume that, for all , where , , and is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem 4 are satisfied.
Then has a fixed point in .

Remark 12. In Corollary 11, let for all and for all and for all , where . If is single valued map, then Corollary 11 reduces to Theorem 2.2 of [8].

Theorem 13. Let be a complete metric space, and let be a function. Suppose that a multivalued mapping is -admissible.
Assume that, for all implies where , , and is strictly increasing and upper semicontinuous function. Also, suppose that the following are satisfied: (1)there exists and such that ;(2)for a sequence in with for all and a cluster point of , there exists a subsequence of such that, for all ,
Then has a fixed point in .

Proof. Following the proof of Theorem 4, we obtain a sequence with such that, for all , From (2) there exists a subsequence of such that Thus, we have where We have and so Suppose that .
Since is upper semicontinuous, Letting in inequality (29) and using continuity of , we obtain which is a contradiction. Hence, , and hence is a fixed point of .

The following example shows that upper semicontinuity of cannot be dropped in Theorem 13.

Example 14. Let , and let for all .
Define a mapping by Let for all , and let Then, , and and is a strictly increasing function.
Let be defined by Obviously, condition (2) of Theorem 13 is satisfied. Condition (1) of Theorem 13 is satisfied with .
We show that (7) is satisfied.
Let be such that .
Then, .
If , then obviously (7) is satisfied.
Let .
If and , then we obtain Let and .
Then, we have Thus, (7) is satisfied.
We now show that is -admissible.
Let be given, and let be such that .
Then, , .
Obviously, for all whenever .
If , then . Hence, for all , .
Hence, is -admissible. Thus, all hypotheses of Theorem 13 are satisfied. However, has no fixed points.

Note that is not upper semicontinuous.

Corollary 15. Let be a complete metric space, and let be a function. Suppose that is an -admissible mapping.
Assume that, for all , where , , and is strictly increasing and upper semicontinuous function. Also, suppose that conditions (1) and (2) of Theorem 13 are satisfied.
Then has a fixed point in .

Corollary 16. Let be a complete ordered metric space. Suppose that a multivalued mapping satisfies for all with (resp., ), where , , and is strictly increasing and upper semicontinuous function.
Assume that, for each and with (resp., ), we have (resp., ) for all .
Also, suppose that the following are satisfied:(1)there exists and such that (resp., );(2)for a sequence in with (resp., ) for all and a cluster point of , there exists a subsequence of such that, for all ,
Then has a fixed point in .

Remark 17. Corollary 16 is a generalization and extension of the result of [9] to multivalued mappings.

Corollary 18. Let be a complete metric space, and let be a function. Suppose that a multivalued mapping is -admissible.
Assume that, for all , implies where , , and is strictly increasing and upper semicontinuous function.
Also, suppose that conditions (1) and (2) of Theorem 13 are satisfied.
Then has a fixed point in .

Remark 19. By taking in Corollary 18 and by applying Remark 1, Corollary 18 reduces to Theorem 2.6 of [6].

Corollary 20. Let be a complete metric space, and let be a function. Suppose that is an -admissible mapping.
Assume that, for all , where , , and is strictly increasing and upper semicontinuous function.
Also, suppose that conditions (1) and (2) of Theorem 13 are satisfied.
Then has a fixed point in .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the anonymous reviewers for their valuable comments.