Birkhoff-kellogg Theorems on Invariant Directions for Multimaps

This paper presents Birkhoff-Kellogg type theorems on invariant directions for a large class of maps. A number of results which will enable to deduce results for upper semicontinuous maps which are either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) admissible (strongly) in the sense of Gorniewicz are given. The results in this paper, when the map is compact, complement and extend the previously known results in [8, 14, 16]. Also using the results in [7], we are able to present invariant direction results for countably condensing maps. For the remainder of this section, we present some definitions and some known facts. Let X and Y be subsets of Hausdorff topological vector spaces E1 and E2, respectively. We will look at maps F : X → K(Y), here K(Y) denotes the family of nonempty compact subsets of Y . We say F : X → K(Y) is Kakutani if F is upper semicontinuous with convex values. A nonempty topological space is said to be acyclic, if all its reduced C̆ech homology groups over the rationals are trivial. Now F : X → K(Y) is acyclic if F is upper semicontinuous with acyclic values. The map F : X → K(Y) is said to be an O’Neill map if F is continuous and if the values of F consist of one orm-acyclic components (herem is fixed). Given two open neighborhoods U and V of the origins in E1 and E2, respectively, a (U,V)-approximate continuous selection [6] of F : X → K(Y) is a continuous function s : X → Y satisfying