Fixed Point Properties Related to Multivalued Mappings

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

(2.1)

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

(2.2)

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

(2.3)

Put

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

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.