An Inverse Mapping Theorem for Set-valued Maps

We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict derivative", the following properties of the inverse map F~x are inherited by (f + F)~x : F~x has a closed-valued pseudo-Lipschitz selection, F~~x is locally closed-valued and pseudo-Lipschitz, F~x has a Lipschitz selection, and F~x is locally single-valued and Lipschitz. Let us recall that the distance from a point x to a set A in a metric space (X, p) is defined by dist(x, A) = inf{p(x, y) : y e A} and the excess e from the set A to the set B (also called the Hausdorff semidistance from B to A) is given by e(B, A) = sup{dist(x, A): x e B}. We denote by Ba(x) the closed ball centered at x with radius a. For notational convenience, t500(x) denotes X. Also recall that in a linear metric space (Y, d) the metric d is invariant if it satisfies d(x + z, y + z) = d(x, y) for every x,y, ze Y. Let F: X —> Y be a set-valued map. The inverse map F~x is defined as F-x(y) = {xeX:yeF(x)}, while graph F is the set {(x, y) e X x Y: y e F(x)}. The map F from a metric space (X, p) into subsets of a metric space Y is pseudo-Lipschitz Received by the editors September 16, 1992. 1991 Mathematics Subject Classification. Primary 49K40; Secondary 26B10, 47H04, 49J52, 90C31.