Homeomorphisms and generalized derivatives

Abstract

When D is open in R”, f: D + Rk is said to be Lipschitzian provided each point in D has a neighborhood where f is Lipschitz continuous. For such maps the derivative f ‘(p) may not exist everywhere in D, but one can at least assign to each p a certain collection ~?f(p) of linear transformations from R” into R k called the generalized derivative (For the definition, see Section 4.) Reading Clarke [3] or Pourciau [20 J, one finds this notion of differentiability enjoys many nice properties, but only the following extension of the inverse function theorem needs to be singled out here: supposing f has values in R”, if af(p) is invertible (that is, if each linear transformation in af(p) is invertible) at every p, then f is a local homeomorphism. Of course local homeomorphisms are not generally homeomorphisms, and in this paper we seek additional conditions on the generalized derivative and the boundary behavior of the map, forcing f to be a homeomorphism. In mathematical economics, theorems of this type are crucial for establishing the uniqueness of competitive equilibrium points in abstract economies. For such applications, theorems that require the image off to satisfy a difficult to verify condition like simple connectedness are not very useful. Involving boundary restrictions and covering spaces, our results at first are topological, but then in Section 4 we blend these topological ideas with properties of the generalized derivative to obtain analytic conditions when the map is Lipschitzian. Throughout we suppose D is a domain with compact closure K in Euclidean n-space R”, B is the boundary K/D, F is a continuous map from K into R”, and f is the restriction F ( D. Our path will take us through covering spaces. We call f: D -m a covering if each point of fz, has an open neighborhood V whose preimage f -’ V is the pairwise disjoint union of open sets, each of which f maps homeomorphically onto V. When f is a covering, it is a local homeomorphism, yet not all local homeomorphisms are coverings.