Fixed Point Theorem of Half-Continuous Mappings on Topological Vector Spaces

Abstract

Some fixed point theorems of half-continuous mappings which are possibly discontinuous defined on topological vector spaces are presented. The results generalize the work of Philippe Bich (2006) and several well-known results.

1. Introduction

Almost a century ago, L. E. J. Brouwer proved a famous theorem in fixed point theory, that any continuous mapping from the closed unit ball of the Euclidean space to itself has a fixed point. Later in 1930, J. Schauder extended Brouwer's theorem to Banach spaces (see [1]).

In 2008, Herings et al. (see [2]) proposed a new type of mapping which is possibly discontinuous. They called such mappings locally gross direction preserving and proved that every locally gross direction preserving mapping defined on a nonempty polytope (the convex hull of a finite subset of ) has a fixed point. Their work both allows discontinuities of mappings and generalizes Brouwer's theorem.

Later, Bich (see [3]) extended the work of Herings et al. to an arbitrary nonempty compact convex subset of Moreover, in [4], Bich established a new class of mappings which contains the class of locally gross direction preserving mappings. He called the mappings in that class half-continuous and proved that if is a nonempty compact convex subset of a Banach space and is half-continuous, then has a fixed point. Furthermore, in the same work, Bich extended the notion of half-continuity to multivalued mappings and proved fixed point theorems which generalize several well-known results.

All vector spaces considered are real vector spaces. In this paper, we prove that some results of Bich (see [4]) are also valid in locally convex Hausdorff topological vector spaces and also show that several well-known theorems can be obtained from our results. The paper is organized as follows. In Section 2, some notations, terminologies, and fundamental facts are reviewed. Sections 3 and 4, the fixed point theorems are proved. Finally, in Section 5, we give some consequent results on inward and outward mappings.

2. Preliminaries

A mapping from a set into (the set of nonempty subsets of a set ) is called a multivalued mapping from into , and the fibers of at are the set A mapping is called a selection of if for all

Let be topological spaces. A mapping is called upper semicontinuous (u.s.c.) if for each and neighborhood of in , there exists a neighborhood of in such that for all . By a neighborhood of a point in , we mean any open subset of that contains

Let be a topological vector space (t.v.s.), not necessarily Hausdorff and the topological dual of . In this paper, we consider equipped with the topology of compact convergence. Then is a t.v.s. We say that separates points of , if whenever and are distinct points of then for some . If separates points of , then a topology on is Hausdorff. By Hahn-Banach theorem, if is locally convex Hausdorff, then separates points of , but the converse is not true, for an example, see [5, 6].

Let and . A mapping is called upper demicontinuous (u.d.c) if for each and any open half-space (the set of the form , where and ) in containing , there exists a neighborhood of in such that for all . It is clear that a u.s.c. multivalued mapping is u.d.c. but the converse is not true (see [7]). It is convenient to write instead of for and The reason for this is that often the vector and/or the continuous linear functional may be given in a notation already containing parentheses or other complicated form.

The following useful results are recalled to be referred.

Theorem 2.1 (Browder [8]).

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. If is a continuous mapping, then there exists such that for all .

Theorem 2.2 (Ben-El-Mechaiekh et al. [1]).

Let be a paracompact Hausdorff space and a convex subset of a t.v.s. Suppose is a multivalued mapping having nonempty convex values and open fibers, then has a continuous selection.

Theorem 2.3 (see [6]).

Let be disjoint nonempty convex subsets of a locally convex Hausdorff t.v.s. . If is compact and is closed, then there exists and such that for all and .

Theorem 2.4 (see [6]).

Let be a t.v.s. whose separates points. Suppose that and are disjoint nonempty compact convex sets in Then there exists such that .

Theorem 2.5 (see [9]).

Let be a topological space, a compact Hausdorff space, and a multivalued mapping with nonempty closed values. Then is u.s.c. if and only if the graph of is closed in .

3. Half-Continuous Mappings

Now, we introduce the notion of half-continuity on t.v.s., and investigate some of their properties.

Definition 3.1.

Let be a subset of a t.v.s. A mapping is said to be half-continuous if for each with there exist and a neighborhood of in such that

(3.1)

for all with .

By the name "half-continuous," it induces us to think that continuous mappings should be half-continuous. The following theorem tells us that if separates points of , then the statement is affirmative.

Proposition 3.2.

Let be a t.v.s. whose separates points and a nonempty subset of Then every continuous mapping is half-continuous.

Proof.

Let be such that Since separates points on , we may assume that, for some Since the mapping is continuous, there exists a neighborhood of in such that for all Therefore, is half-continuous.

The hypothesis that separates points of cannot be relaxed as will be shown in the following examples.

Example 3.3.

Let be a nontrivial vector space. Then the topology makes into a locally convex t.v.s. that is not Hausdorff and (see [10]). So does not separate points on . Consequently, every continuous self-mapping on which is not the identity, is not half-continuous.

Example 3.4.

For ,   is a Hausdorff t.v.s. with (see [6]).

Remark 3.5.

There are some half-continuous mappings which are not continuous. For example [4], let be defined by

(3.2)

It is clear that is half-continuous but not continuous.

Moreover, half-continuity is not closed under the composition, the addition, and the scalar multiplication. To see this consider a half-continuous mapping on defined by for and for . It is easy to see that and are not half-continuous. In fact, the composition of and a homeomorphism is not half-continuous yet.

Proposition 3.6.

Let be a nonempty subset of a t.v.s. and . Then is half-continuous if and only if for any , the mapping is half-continuous.

Proof.

The sufficiency is clear. To prove the necessity, let and let be defined by for all Let be such that Then and hence there exist and a neighborhood of in such that for all with . Then for each with ,

(3.3)

If , then done. Otherwise, consider instead of

Next, we give a sufficient condition for mappings on t.v.s. to be half-continuous.

Proposition 3.7.

Let be a nonempty subset of a t.v.s. and . Suppose that for each with , there exist such that and is lower semicontinuous at . Then is half-continuous.

Proof.

Let be such that Then there exists such that and is lower semicontinuous at Let be such that . Since is continuous at , there exists a neighborhood of in such that for all This implies that

(3.4)

By lower semicontinuity of , there exists a neighborhood of in such that

(3.5)

for all Then, for each with , we have from (3.4) and (3.5) that

(3.6)

Therefore, is half-continuous.

The latter case follows from the fact that is upper semicontinuous if and only if is lower semicontinuous.

Remark 3.8.

If is a Banach space, then Proposition 3.7 is Proposition in [4]. By considering the mapping in Remark 3.5, we note that the converse of Proposition 3.7 is not true (see [4]).

Let and be sets. Let and be mappings from to . The set is said to be the coincidence set of and . The next result is inspired by the idea of [4, Theorem ].

Theorem 3.9.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. and . Suppose that is bijective continuous and for each with there exist and a neighborhood of in such that for all with . Then is nonempty.

Proof.

Suppose that . Define by

(3.7)

for all Clearly, is nonempty for all Let and There are neighborhoods and of in such that

(3.8)

Clearly, and is a neighborhood of in . For each with

(3.9)

Hence, This implies that is convex.

Next, let and There exists a neighborhood of in such that for all with . Then Since is open, is open in From Theorems 2.1 and 2.2, there exists a continuous selection of and such that for every ,

(3.10)

Since is surjective, for some , and hence . Also, since , which is a contradiction.

If in Theorem 3.9 is the identity mapping, then the following result is immediate.

Corollary 3.10.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . If is half-continuous, then has a fixed point.

Remark 3.11.

If is a Banach space, then the previous corollary is the Theorem in [4].

The following result is obtained from Proposition 3.2 and Corollary 3.10.

Corollary 3.12 (Brouwer-Schauder-Tychonoff, see [1]).

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Then every continuous mapping has a fixed point.

4. Half-Continuous Multivalued Mappings

Now, we consider half-continuity of multivalued mappings and prove that under a certain assumption they have fixed point.

Definition 4.1.

Let be a subset of a t.v.s. A mapping is said to be half-continuous if for each with there exists and a neighborhood of in such that

(4.1)

The following proposition gives a sufficient condition for a multivalued mapping to be half-continuous.

Proposition 4.2.

Let be a nonempty subset of a locally convex Hausdorff t.v.s. . If is a u.d.c. mapping with nonempty closed convex values, then is half-continuous.

Proof.

Assume that is u.d.c. with nonempty closed convex values. Let be such that . Suppose that fails to be half-continuous. By Theorem 2.3, there exists and such that

(4.2)

for all . This implies that . Since is u.d.c., there exists a neighborhood of in such that for all . Set . Then is a neighborhood of in . Since is not half-continuous, there exists and such that

(4.3)

Since , , so . Then, by (4.3), . This means that , which is a contradiction. Therefore, is half-continuous.

Remark 4.3.

However, there are some half-continuous mappings which are not u.d.c.. To see this, consider the mapping defined by

(4.4)

Then is half-continuous but not u.d.c. at 0.

In case that is a t.v.s. whose separates points, we need more assumptions on the mapping as the following result. The proof is analogous to that of Proposition 4.2, by applying Theorem 2.4.

Proposition 4.4.

Let be a t.v.s. whose separates points and a nonempty subset of If is u.d.c. with nonempty compact convex values, then is half-continuous.

Next, we will prove the main result which guarantees the possessing of fixed points if the multivalued mapping is half-continuous. To do this, we need the following lemma.

Lemma 4.5.

Let be a nonempty subset of a t.v.s. and . If is half-continuous, then has a half-continuous selection.

Proof.

Assume that is half-continuous. Let be any selection of . Define by

(4.5)

Clearly, is a selection of . To show that is half-continuous, let be such that . Then and hence there exists and a neighborhood of in such that

(4.6)

It follows that for every with .

Corollary 3.10 and Lemma 4.5 yield the following main result.

Theorem 4.6.

Let be a nonempty compact subset of a locally convex Hausdorff t.v.s. . If is half-continuous, then has a fixed point.

The following result is immediately obtained from Theorem 4.6 and Proposition 4.2.

Corollary 4.7.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . If is u.d.c. with nonempty closed convex values, then has a fixed point.

It is well known that if is a subset of a topological space and has closed graph, then the set of fixed points of is closed in . From Corollary 4.7 and Theorem 2.5, we have the following corollary.

Corollary 4.8 (Kakutani-Fan-Glicksberg, see [11, 12]).

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . If is u.s.c. with nonempty closed convex values, then the set of fixed points of is nonempty and compact.

5. Inward and Outward Mappings

In case that the half-continuous mapping is a nonself-mapping on but has some nice property, then still possesses a fixed point in . We state the results in the following theorem.

Theorem 5.1.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Suppose that is half-continuous and for each with there exists such that then has a fixed point.

Proof.

Suppose that has no fixed point. For each let Define by

(5.1)

for all Then for every It is not difficult to see that is half-continuous. By Theorem 4.6, there exists and such that . It follows that , which is a contradiction.

Remark 5.2.

From Theorem 5.1, for with , if there is such that , then , in fact, is the element in . Indeed, by setting , then and so, by convexity of ,

Recall that the line segment joining vectors and in is the set . As a special case of Theorem 5.1 we obtain the following corollary.

Corollary 5.3 (Fan-Kaczynski, see [1]).

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Suppose that is continuous and for each with the line segment contains at least two points of then has a fixed point.

Next, we derive a generalization of a fixed point theorem due to F. E. Browder and B. R. Halpern. To do this, let us recall the definition of inward and outward mappings.

Definition 5.4 (see [1]).

Let be a subset of a vector space . A mapping is called inward (resp., outward) if for each there exists (resp., ) satisfying .

Theorem 5.5.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Then every half-continuous inward (or outward) mapping has a fixed point.

Proof.

Suppose that is a half-continuous inward mapping. Let be such that There exists such that . By letting and apply Theorem 5.1, has a fixed point.

Next, assume that is outward. Define by for all Then is inward and, by Proposition 3.6, is half-continuous. Hence, there is such that . That is

Remark 5.6.

In Theorem 5.5, if is a continuous inward (or outward) mapping, then Theorem 5.5 is the theorem proved by F. E. Browder (1967) and B. R. Halpern (1968) (see [1]).

In the final part, we prove the fixed points theorem for half-continuous inward and outward multivalued mappings.

Definition 5.7 (see [7]).

Let be a subset of a vector space . A mapping is called inward (resp., outward) if for each there exists and (resp., ) satisfying .

Theorem 5.8.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Then every half-continuous inward (or outward) mapping has a fixed point.

Proof.

Let be a half-continuous mapping. Suppose that is inward but it has no fixed point. Define by

(5.2)

for all We can see that is nonempty for all and is half-continuous. By Theorem 4.6, there exists and such that . That is , which is a contradiction.

Next, assume that is outward. Define by for all . It is easy to see that is half-continuous. Let be arbitrary. There exists and satisfying . Then . Since , is inward. Thus for some and . That is .

Any selection of half-continuous inward multivalued mappings may not be inward as shown in the following example. Let be defined by

(5.3)

Clearly, is inward half-continuous but a selection of defined by if and if is not inward.

Remark 5.9.

If the half-continuity of is replaced by upper semicontinuity, then Theorem 5.8 is the result of Halpern-Bergman (1968) (see [7]) and Fan (1969) (see [13]).

As an interesting special case of Theorem 5.8, we obtain the following corollary.

Corollary 5.10.

Let be a nonempty compact convex subset of a locally convex Hausdorff t.v.s. . Suppose that is half-continuous and for each , is nonempty, then has a fixed point.

6. Discussion

It is worth to notice that there exists a multivalued mapping which is not half-continuous but some of its selection is half-continuous. For example, let be defined by

(6.1)

Then is not half-continuous since (4.1) fails for . Nevertheless, a mapping defined by

(6.2)

is a half-continuous selection of .

From Theorem 4.6 we see that if a multivalued mapping has a half-continuous selection, then has a fixed point. It is interesting to investigate the condition(s) for a multivalued mapping to induce a half-continuous selection.