Serdica Math. J. 29 (2003), 301-314 APPLICATIONS OF THE FRÉCHET SUBDIFFERENTIAL

In this paper we prove two results of nonsmooth analysis involving the Fréchet subdifferential. One of these results provides a necessary optimality condition for an optimization problem which arise naturally from a class of wide studied problems. In the second result we establish a sufficient condition for the metric regularity of a set-valued map without continuity assumptions. 1. Preliminaries. Let X be a normed vector space and X its topological dual; we denote by BX , UX , SX the open unit ball, the closed unit ball and the unit sphere of X, respectively. By w and w we mean the weak topology on X and the weak star topology on X. If S is a subset of X we denote by clS the closure of S; if x ∈ X, we denote the distance from x to S by d(x, S) = infy∈S d(x, y) and by dS the distance function with respect to S, 2000 Mathematics Subject Classification: 46A30, 54C60, 90C26.