Fixed Point Properties Related to Multivalued Mappings
© Hidetoshi Komiya. 2010
Received: 12 January 2010
Accepted: 2 April 2010
Published: 23 May 2010
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.
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 , Browder fixed point theorem , 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.
Our main result is the following.
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.
The proofs of and are obvious.
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 .
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.
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  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.
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.
This paper is written with support from Research Center of Nonlinear Analysis and Discrete Mathematics, National Sun Yat-Sen University.
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