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

Let H:R^mxR^n->R^n be a locally Lipschitz function in a neighborhood of ([email protected]?,[email protected]?) and H([email protected]?,[email protected]?)=0 for some [email protected][email protected]?R^m and [email protected][email protected]?R^n. 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 @p"[email protected]?H([email protected]?,[email protected]?) is of maximal rank, then there exist a neighborhood Y of [email protected]? and a Lipschitz function G(.):Y->R^n such that G([email protected]?)[email protected]? 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 ([email protected]?,[email protected]?), then G has a superlinear (quadratic) approximate property at [email protected]?. This result is useful in designing Newton's methods for nonsmooth equations.