A Fixed Point Theorem for Multivalued Mappings with -Distance

Abstract

We mainly study fixed point theorem for multivalued mappings with -distance using Wardowski’s technique on complete metric space. Let be a metric space and let be a family of all nonempty bounded subsets of . Define by Considering -distance, it is proved that if is a complete metric space and is a multivalued certain contraction, then has a fixed point.

1. Introduction

Fixed point theory concern itself with a very basic mathematical setting. It is also well known that one of the fundamental and most useful results in fixed point theory is Banach fixed point theorem. This result has been extended in many directions for single and multivalued cases on a metric space (see [1–9]). Fixed point theory for multivalued mappings is studied by both Pompeiu-Hausdorff metric [10, 11], which is defined on (the family of all nonempty, closed, and bounded subsets of ), and -distance, which is defined on (the family of all nonempty and bounded subsets of ). Using Pompeiu-Hausdorff metric, Nadler [12] introduced the concept of multivalued contraction mapping and show that such mapping has a fixed point on complete metric space. Then many authors focused on this direction [13–18]. On the other hand, Fisher [19] obtained different type of multivalued fixed point theorems defining -distance between two bounded subsets of a metric space . We can find some results about this way in [20–23].

In this paper, we give some new multivalued fixed point results by considering the -distance. For this we use the recent technique, which was given by Wardowski [24]. For the sake of completeness,we will discuss its basic lines. Let be the set of all functions satisfying the following conditions:(F1) is strictly increasing; that is, for all such that , .(F2)For each sequence of positive numbers if and only if .(F3)There exists such that .

Definition 1 (see [24]). Let be a metric space and let be a mapping. Given , we say that is -contraction, if there exists such that

Taking different functions in (1), one gets a variety of -contractions, some of them being already known in the literature. The following examples will certify this assertion.

Example 2 (see [24]). Let be given by the formulae . It is clear that . Then each self-mapping on a metric space satisfying (1) is an -contraction such that

It is clear that for , such that the inequality also holds. Therefore satisfies Banach contraction with ; thus is a contraction.

Example 3 (see [24]). Let be given by the formulae . It is clear that . Then each self-mapping on a metric space satisfying (1) is an -contraction such that

We can find some different examples for the function belonging to in [24]. In addition, Wardowski concluded that every -contraction is a contractive mapping, that is, Thus, every -contraction is a continuous mapping.

Also, Wardowski concluded that if , with for all and is nondecreasing, then every -contraction is an -contraction.

He noted that, for the mappings and , and a mapping is strictly increasing. Hence, it was obtained that every Banach contraction satisfies the contractive condition (3). On the other side, [24, Example 2.5] shows that the mapping is not an -contraction (Banach contraction) but still is an -contraction. Thus, the following theorem, which was given by Wardowski, is a proper generalization of Banach Contraction Principle.

Theorem 4 (see [24]). Let be a complete metric space and let be an -contraction. Then has a unique fixed point in .

Following Wardowski, Mınak et al. [25] introduced the concept of Ćirić type generalized -contraction. Let be a metric space and let be a mapping. Given , we say that is a Ćirić type generalized -contraction if there exists such that where Then the following theorem was given.

Theorem 5. Let be a complete metric space and let be a Ćirić type generalized -contraction. If or is continuous, then has a unique fixed point in .

Considering the Pompeiu-Hausdorff metric , both Theorems 4 and 5 were extended to multivalued cases in [26] and [27], respectively (see also [28, 29]). In this work, we give a fixed point result for multivalued mappings using the -distance. First recall some definitions and notations which are used in this paper.

Let be a metric space. For , we define If we write and also if , then . It is easy to prove that for If is a sequence in , we say that converges to and write if and only if(i) implies that for some sequence with for ,(ii)for any , such that for , where

Lemma 6 (see [20]). Suppose and are sequences in and is a complete metric space. If and then .

Lemma 7 (see [20]). If is a sequence of nonempty bounded subsets in the complete metric space and if for some , then .

2. Main Result

In this section, we prove a fixed point theorem for multivalued mappings with -distance and give an illustrative example.

Definition 8. Let be a metric space and let be a mapping. Then is said to be a generalized multivalued -contraction if and there exists such that for all with , where

Theorem 9. Let be a complete metric space and let be a multivalued -contraction. If is continuous and is closed for all , then has a fixed point in .

Proof. Let be an arbitrary point and define a sequence in as for all . If there exists for which , then is a fixed point of and so the proof is completed. Thus, suppose that, for every , . So and for all . Then, we have from (10) and so
Denote , for . Then, for all and, using (10), the following holds: From (14), we get . Thus, from (F2), we have From (F3) there exists such that By (14), the following holds for all : Letting in (17), we obtain that From (18), there exits such that for all . So we have for all . In order to show that is a Cauchy sequence consider such that . Using the triangular inequality for the metric and from (19), we have By the convergence of the series , we get as . This yields that is a Cauchy sequence in . Since is a complete metric space, the sequence converges to some point ; that is, . Now, suppose is continuous. In this case, we claim that . Assume the contrary; that is, . In this case, there exist an and a subsequence of such that for all . (Otherwise, there exists such that for all , which implies that . This is a contradiction, since .) Since for all , then we have Taking the limit and using the continuity of , we have , which is a contradiction. Thus, we get . This completes the proof.

Example 10. Let and . Then is a complete metric space. Define by We claim that is multivalued -contraction with respect to and . Because of the , , we can consider the following cases while and is empty or singleton.
Case  1. For and , we have
Case  2. For and , we have
Case  3. For , we have This shows that is multivalued -contraction; therefore, all conditions of theorem are satisfied and so has a fixed point in .
On the other hand, for and , since and , we get then does not satisfy for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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