FIXED-POINT THEOREMS AND MORSE LEMMA FOR LIPSCHITZIAN FUNCTIONS

We prove a fixed-point theorem for set-valued mappings defined on a nonempty compact subset X of Rn which can be represented by inequality constraints, i.e., X={x∈Rn| f(x)≤0}, f locally Lipschitzian and satisfying a nondegeneracy assumption outside of X. This class of sets extends significantly the class of convex, compact sets with a nonempty interior. Topological properties of such sets X are proved (continuous deformation retract of a ball, acyclicity) as a consequence of a generalization of Morse's lemma for Lipschitzian real-valued function defined on Rn a result also of interest for itself. © 1990.