Fixed Point Theorem of Half-Continuous Mappings on Topological Vector Spaces
© I. Termwuttipong and T. Kaewtem. 2010
Received: 11 December 2009
Accepted: 10 January 2010
Published: 21 January 2010
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.
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 ).
In 2008, Herings et al. (see ) 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 ) extended the work of Herings et al. to an arbitrary nonempty compact convex subset of Moreover, in , 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 ) 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.
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 ). 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 ).
Theorem 2.2 (Ben-El-Mechaiekh et al. ).
Theorem 2.3 (see ).
Theorem 2.4 (see ).
Theorem 2.5 (see ).
3. Half-Continuous Mappings
Now, we introduce the notion of half-continuity on t.v.s., and investigate some of their properties.
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.
Let be a nontrivial vector space. Then the topology makes into a locally convex t.v.s. that is not Hausdorff and (see ). So does not separate points on . Consequently, every continuous self-mapping on which is not the identity, is not half-continuous.
For , is a Hausdorff t.v.s. with (see ).
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.
Next, we give a sufficient condition for mappings on t.v.s. to be half-continuous.
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 ].
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.
If is a Banach space, then the previous corollary is the Theorem in .
The following result is obtained from Proposition 3.2 and Corollary 3.10.
Corollary 3.12 (Brouwer-Schauder-Tychonoff, see ).
4. Half-Continuous Multivalued Mappings
Now, we consider half-continuity of multivalued mappings and prove that under a certain assumption they have fixed point.
The following proposition gives a sufficient condition for a multivalued mapping to be 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.
Corollary 3.10 and Lemma 4.5 yield the following main result.
The following result is immediately obtained from Theorem 4.6 and Proposition 4.2.
5. Inward and Outward Mappings
Corollary 5.3 (Fan-Kaczynski, see ).
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 ).
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 ).
In the final part, we prove the fixed points theorem for half-continuous inward and outward multivalued mappings.
Definition 5.7 (see ).
As an interesting special case of Theorem 5.8, we obtain the following corollary.
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.
The second author is financially supported by Mahidol Wittayanusorn School. This work is dedicated to Professor Wataru Takahashi on his retirement.
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