Research Article
A Fixed Point Theorem for Multivalued
Mappings with -Distance
Özlem Acar and Ishak Altun
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey
Correspondence should be addressed to
¨
Ozlem Acar; acarozlem@ymail.com
Received 13 June 2014; Accepted 20 July 2014; Published 24 July 2014
Academic Editor: Poom Kumam
Copyright © 2014
¨
O. Acar and I. Altun. Tis 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.
We mainly study fxed 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 . Defne :()×()→ R
by (,)= sup {(,):∈,∈}. Considering -distance, it is proved that if (,) is a complete metric space and :→
() is a multivalued certain contraction, then has a fxed point.
1. Introduction
Fixed point theory concern itself with a very basic mathemat-
ical setting. It is also well known that one of the fundamental
and most useful results in fxed point theory is Banach
fxed point theorem. Tis 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-Hausdorf metric [10, 11],
which is defned on () (the family of all nonempty,
closed, and bounded subsets of ), and -distance, which is
defned on () (the family of all nonempty and bounded
subsets of ). Using Pompeiu-Hausdorf metric, Nadler [12]
introduced the concept of multivalued contraction mapping
and show that such mapping has a fxed point on complete
metric space. Ten many authors focused on this direction
[13–18]. On the other hand, Fisher [19] obtained diferent
type of multivalued fxed point theorems defning -distance
between two bounded subsets of a metric space . We can
fnd some results about this way in [20–23].
In this paper, we give some new multivalued fxed 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 F
be the set of all functions : (0,∞) → R satisfying the
following conditions:
(F1) is strictly increasing; that is, for all , ∈ (0,∞)
such that <, ()<().
(F2) For each sequence {
} of positive numbers
lim
→∞
=0 if and only if lim
→∞
(
)=−∞.
(F3) Tere exists ∈(0,1) such that lim
→0
+
()=0.
Defnition 1 (see [24]). Let (,) be a metric space and let
:→ be a mapping. Given ∈ F, we say that is
-contraction, if there exists >0 such that
,∈,
(,)>0⇒+((,))≤((,)).
(1)
Taking diferent functions ∈ F in (1), one gets a variety
of -contractions, some of them being already known in the
literature. Te following examples will certify this assertion.
Example 2 (see [24]). Let
1
: (0,∞) → R be given by the
formulae
1
() = ln . It is clear that
1
∈ F. Ten each
self-mapping on a metric space (,) satisfying (1) is an
1
-contraction such that
(,)≤
−
(,), ∀,∈, ̸ =. (2)
It is clear that for , ∈ such that =
the inequality (,) ≤
−
(,) also holds. Terefore
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 497092, 5 pages
http://dx.doi.org/10.1155/2014/497092