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Fixed Point Properties Related to Multivalued Mappings
Fixed Point Theory and Applications volume 2010, Article number: 581728 (2010)
Abstract
We discuss fixed point properties of convex subsets of locally convex linear topological spaces. We derive equivalence among fixed point properties concerning several types of multivalued mappings.
1. Introduction
We present fundamental definitions related to multivalued mappings in order to fix our terminology. We assume Hausdorff separation axiom for all of the topological spaces which appear hereafter. Let and
be topological spaces. A multivalued mapping
from
to
is a function which attains a nonempty subset of
for each point
of
and the subset is denoted by
. For any subset
of
, the upper inverse
and the lower inverse
are defined by
and
, respectively. A multivalued mapping
is said to be upper semicontinuous (lower semicontinuous, resp.) if
(
, resp.) is open in
for any open subset
of
. Moreover,
is said to be upper demicontinuous if
is open in
for any open half-space
of
in case
is a linear topological space.
We are interested in fixed point properties of convex subsets of locally convex linear topological spaces. A topological space is said to have a fixed point property if every continuous functions from the topological space to itself has a fixed point. Following to this terminology, we define several fixed point properties depending on types of multivalued mappings we concern.
We always deal with convex-valued multivalued mappings defined on a convex subset of a locally convex topological linear space in this paper. Such situations appear often in arguments on fixed point theory for multivalued mappings, for example, Kakutani fixed point theorem [1], Browder fixed point theorem [2], and so forth. Let be a convex subset of a locally convex topological linear space and let
be a convex-valued multivalued mapping from
to
. We call
Kakutani-type if
is closed-valued and upper semicontinuous and weak Kakutani-type if
is closed-valued and demicontinuous. Similarly
is said to be Browder-type if
has open lower sections; that is,
is open for all
. We call
open graph-type if it has an open graph.
A convex subset of a locally convex linear topological space is said to have a Kakutani-type fixed point property if every Kakutani-type multivalued mapping from
to
has a fixed point. Similarly, we define weak Kakutani-type fixed point property, Browder-type fixed point property, and open graph-type fixed point property.
2. Result
Our main result is the following.
Theorem 2.1.
Let be a paracompact convex subset of a locally convex linear topological space
. Then each of the following statements is mutually equivalent.
(1) has a fixed point property.
(2) has a Browder-type fixed point property.
(3) has an open graph-type fixed point property.
(4) has a weak Kakutani-type fixed point property.
(5) has a Kakutani-type fixed point property.
Proof.
The proofs of and
are obvious.
(1) ⇒ (2). The method of the proof is similar to that of [2, Theorem ]. Let
be Browder-type. The family
is an open cover of
because any point
of
belongs to an open set
with
. Therefore, there is a partition of unity
subordinated to
. That is, each function
is continuous, the family
of open sets is a locally finite refinement of
, and
for all
. For each
, take
such that
, and we denote it by
. Then define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ1_HTML.gif)
Here the summation is well defined because there are only a finite number of indices
with
. The function
is continuous because the family
of open sets is locally finite. On the other hand, it follows that
since
is convex. Thus
has a fixed point
by the hypothesis. That is, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ2_HTML.gif)
It follows that for each
with
, and hence we have
. Since
is convex, we have
, and it is proved that
is a fixed point of
.
(3) ⇒ (4). The method of this proof is inspired by the discussions found in [3, 4]. Suppose that is weak Kakutani-type but it has no fixed point; that is,
for any
. Since
is closed and convex, there is a continuous linear functional
on
which separates
and
strictly. Thus there is a real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ3_HTML.gif)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ4_HTML.gif)
Then is a neighborhood of
in
, and we have
. Since
is an open cover of
, there is an open cover
of
such that
is locally finite and refines
because
is paracompact. For each
, take an
such that
and denote it by
. For each
, define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ5_HTML.gif)
Since for any
with
, we have
. Thus we have
. Therefore, we have
for all
, and the definition of
above defines a multivalued mapping
. It is easily seen that
is open and convex valued.
Next we show that has an open graph. Take any element
of the graph
of
and fix it. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ6_HTML.gif)
then is a neighborhood of
because
is locally finite. Thus
is a neighborhood of
. We show that
. Take any
. Since
, we have
for any
with
. Therefore, we have
. From this inclusion, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F581728/MediaObjects/13663_2010_Article_1308_Equ7_HTML.gif)
That is, . Therefore,
has an open graph.
On the other hand, take any . There is
such that
. Since
, we have
, and hence
. Thus
has no fixed point and this contradicts the assumption that
has open graph-type fixed point property.
Klee [5] proved that a convex subset of a locally convex metrizable linear topological space is compact if and only if it has a fixed point property. Since any metrizable topological space is paracompact, we have the following corollary of Theorem 2.1.
Corollary 2.2.
Let be a convex subset of a locally convex metrizable linear topological space. Then the following statements are mutually equivalent.
(1) is compact.
(2) has a fixed point property.
(3) has a Browder-type fixed point property.
(4) has an open graph-type fixed point property.
(5) has a weak Kakutani-type fixed point property.
(6) has a Kakutani-type fixed point property.
References
Kakutani S: A generalization of Brouwer's fixed point theorem. Duke Mathematical Journal 1941, 8: 457–459. 10.1215/S0012-7094-41-00838-4
Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721
Komiya H: Inverse of the Berge maximum theorem. Economic Theory 1997,9(2):371–375.
Yamauchi T: An inverse of the Berge maximum theorem for infinite dimensional spaces. Journal of Nonlinear and Convex Analysis 2008,9(2):161–167.
Klee VL Jr.: Some topological properties of convex sets. Transactions of the American Mathematical Society 1955, 78: 30–45. 10.1090/S0002-9947-1955-0069388-5
Acknowledgment
This paper is written with support from Research Center of Nonlinear Analysis and Discrete Mathematics, National Sun Yat-Sen University.
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Komiya, H. Fixed Point Properties Related to Multivalued Mappings. Fixed Point Theory Appl 2010, 581728 (2010). https://doi.org/10.1155/2010/581728
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DOI: https://doi.org/10.1155/2010/581728