-type Maps and Birkhoff-kellogg Theorems

(a) Kakutani; (b) acyclic; (c) O’Neill; (d) approximable; (e) admissible in the sense of Górniewicz; or (f) in κ c . The maps considered will also satisfy various compactness criteria described in Section 2. Our analysis is elementary and combines properties of the Minkowski functional with fixed point theory for self-maps. Also using our new homotopy theorem, we will present an “invariant direction” result for particular classes of maps. The theory and results in this paper complement and extend previously known results in the literature (see [2, 7, 8, 10, 11, 12] and the references therein). 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 that 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 Čech 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