Hindawi Publishing Corporation
Journal of Function Spaces and Applications
Volume 2013, Article ID 827458, 7 pages
http://dx.doi.org/10.1155/2013/827458
Research Article
Multivalued Pseudo-Picard Operators and Fixed Point Results
Gülhan MJnak, Özlem Acar, and Ishak Altun
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey
Correspondence should be addressed to Ishak Altun; ishakaltun@yahoo.com
Received 24 May 2013; Accepted 25 July 2013
Academic Editor: Josip E. Peˇ cari´ c
Copyright © 2013 G¨ ulhan Mınak et al. 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 introduce the concept of multivalued pseudo-Picard (MPP) operator on a metric space. Tis concept is weaker than multivalued
weakly Picard (MWP) operator, which is given by M. Berinde and V. Berinde (2007). Ten, we give both fxed point results and
examples for MPP operators. Also, we obtain some ordered fxed 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 :()×()→ R defned by
(,)= max {sup
∈
(,), sup
∈
(,)} (1)
is a metric on (), which is called Hausdorf metric, where
(,)= inf {(,):∈}. (2)
Let : → () be a map; then, is called multivalued
contraction if for all ,∈ there exists ∈[0,1) such that
(,)≤(,). (3)
In 1969, Nadler [1] proved a fundamental fxed point theo-
rem for multivalued maps: every multivalued contraction on
complete metric space has a fxed point.
Ten, a lot of generalizations of the result of Nadler were
given (see, e.g., [2–4]). 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 [7–9]).
Defnition 1. Let (,) be a metric space, and let :
→ P() (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)
0
=,
1
=,
(ii)
+1
∈
,
(iii) the sequence {
} is convergent and its limit is a fxed
point of .
Ten M. Berinde and V. Berinde [5] show that every
Nadler [1], Reich [10], Rus [11] and Petrus ¸el [12] type multival-
ued contractions on complete metric space are MWP opera-
tors. Mizoguchi and Takahashi [6], proved the following fxed
point theorem. Tis is also an example of MWP operator.
Teorem 2. Let (,) be a complete metric space, and let :
→ () be a multivalued map. Assume that
(,)≤((,))(,) (4)
for all ,∈, where is an MT-function (i.e., it satisfes
lim sup
→
+
()<1
(5)
for all ∈[0,∞)). Ten is an MWP operator.
In the same paper, M. Berinde and V. Berinde [5] intro-
duced the concepts of multivalued (,)-weak contraction
and multivalued (,)-weak contraction and proved the
following nice fxed point theorems.