A further result on an implicit function theorem for locally Lipschitz functions

Let H:Rm × Rn ? Rn be a locally Lipschitz function in a neighborhood of (?,x?) and H(?,x?) = 0 for some ? ? Rm and x? ? Rn. The implicit function theorem in the sense of Clarke (Pacific J. Math. 64 (1976) 97; Optimization and Nonsmooth Analysis, Wiley, New York, 1983) says that if ?x?H(?,x?) is of maximal rank, then there exist a neighborhood Y of ? and a Lipschitz function G(·):Y ? Rn such that G(?) = x? and for every y in Y, H(y,G(y)) = 0. In this paper, we shall further show that if H has a superlinear (quadratic) approximate property at (?,x?), then G has a superlinear (quadratic) approximate property at ?. This result is useful in designing Newton's methods for nonsmooth equations. © 2001 Elsevier Science B.V.