Approximation methods in equilibrium theory

We provide a sufficient condition for the existence elf (generalized) equil ibria, or fixed-points for correspondences F, defined on a compact set M su bset of R-n, with values in R-n, when the set M is neither assumed to be co nvex, nor smooth. We consider the class M of nonempty, compact subsets of R -n satisfying: 0 is not an element of partial derivative(+)d(M)((x) over ba r) <(def)double under bar> lim sup(x-->(x) over bar),x is not an element of M partial derivative d(M)(x), for every (x) over bar epsilon M. The corres pondence F is assumed to be upper semicontinuous with nonempty, convex, com pact values. If M has at least one connected component with a nonzero Euler characteristic, we prove that F admits a generalized equilibrium x* on M, i.e., x* epsilon M and 0 epsilon F(x*) - (N) over tilde(M)(x*), where (N) o ver tilde(M)(x*) is the cone spanned by partial derivative(+)d(M)(x*). Our approach extends previous results on the existence of generalized equilibri a: (i) by taking a (non-necessarily convex) cone (N) over tilde(M)(x*) smal ler than Clarke's normal cone, and (ii) by considering a larger class of se ts than the class of epi-Lipschitzian sets. (C) Academie des Sciences/Elsev ier, Paris.