Smooth normal approximations of epi-Lipschitzian subsets of R-n

A sequence (M-k) of closed subsets of R-n converges normally to M subset of R-n if (sc) M = lim sup M-k = lim inf M-k in the sense of Painleve-Kuratow ski and (nc) lim sup G(N-Mk) subset of G(N-M), where G(N-M) (resp., G(N-Mk) ) denotes the graph of N-M (resp., N-Mk), Clarke's normal cone to M (resp., M-k). This paper studies the normal convergence of subsets of R-n and mainly show s two results. The first result states that every closed epi-Lipschitzian s ubset M of R-n, with a compact boundary, can be approximated by a sequence of smooth sets (M-k), which converges normally to M and such that the sets M-k and M are lipeomorphic for every k (i.e., the homeomorphism between M a nd M-k and its inverse are both Lipschitzian). The second result shows that , if a sequence (M-k) of closed subsets of R-n converges normally to an epi -Lipschitzian set M, and if we additionally assume that the boundary of M-k remains in a fixed compact set, then, for k large enough, the sets M-k and M are lipeomorphic. In Cornet and Czarnecki [Cahier Eco-Maths 95-55, 1995], direct applications of these results are given to the study (existence, stability, etc.) of th e generalized equation 0 is an element of f(x*) + N-M (x*) when M is a comp act epi-Lipschitzian subset of R-n and f : M --> R-n is a continuous map (o r more generally a correspondence).