Fixed points in contractible spaces and convex subsets of topological vector spaces.

*(English)*Zbl 1101.54039This paper studies fixed point theorems for multi-valued maps. The results are for two different situations.

In one case, let \(X\) be a contractible topological space, \(D\) a compact subspace and \(T:X \to 2^D\) a semicontinuous map. Let us suppose that for every open cover \(W\) of \(D\), there exists an open finite refinement \(U\) such that, for every collection \(U_i\), \(i\in J\), of elements of \(U\) with nonempty intersection, the union of the \(U_i\) is contractible. Under this hypothesis it is proved that every upper semicontinuous map \(f:X \to 2^D\) with closed acyclic values has a fixed point.

In the second case, a main result is obtained under the following hypothesis. Let \(X\) be a compact convex subset of \(E\) (a topological vector space) and \(f: X \to 2^X\) an upper semicontinuous map with closed values. If a certain hypothesis (which is a little too technical to be stated here) holds and \(X\) has the simplicial approximation property, then \(f\) has a fixed point. This result is a generalization of fixed point results concerning Kakutani maps. The simplicial approximation property is the main issue of this result.

In the final section the authors study in more detail the simplicial approximation property. For example, they show that the existence of a topological vector space which does not satisfy the simplicial approximation property is equivalent to the existence of a metrizable topological vector space which does not satisfy the simplicial approximation property.

In one case, let \(X\) be a contractible topological space, \(D\) a compact subspace and \(T:X \to 2^D\) a semicontinuous map. Let us suppose that for every open cover \(W\) of \(D\), there exists an open finite refinement \(U\) such that, for every collection \(U_i\), \(i\in J\), of elements of \(U\) with nonempty intersection, the union of the \(U_i\) is contractible. Under this hypothesis it is proved that every upper semicontinuous map \(f:X \to 2^D\) with closed acyclic values has a fixed point.

In the second case, a main result is obtained under the following hypothesis. Let \(X\) be a compact convex subset of \(E\) (a topological vector space) and \(f: X \to 2^X\) an upper semicontinuous map with closed values. If a certain hypothesis (which is a little too technical to be stated here) holds and \(X\) has the simplicial approximation property, then \(f\) has a fixed point. This result is a generalization of fixed point results concerning Kakutani maps. The simplicial approximation property is the main issue of this result.

In the final section the authors study in more detail the simplicial approximation property. For example, they show that the existence of a topological vector space which does not satisfy the simplicial approximation property is equivalent to the existence of a metrizable topological vector space which does not satisfy the simplicial approximation property.

Reviewer: Daciberg Gonçalves (São Paulo)

##### MSC:

54H25 | Fixed-point and coincidence theorems (topological aspects) |

47H10 | Fixed-point theorems |

55M20 | Fixed points and coincidences in algebraic topology |

54C60 | Set-valued maps in general topology |

47H04 | Set-valued operators |