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

Abstract

Let H:Rm × Rn → Rn be a locally Lipschitz function in a neighborhood of (ȳ,x̄) and H(ȳ,x̄) = 0 for some ȳ ∈ Rm and x̄ ∈ 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 πx∂H(ȳ,x̄) is of maximal rank, then there exist a neighborhood Y of ȳ and a Lipschitz function G(·):Y → Rn such that G(ȳ) = x̄ 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 (ȳ,x̄), then G has a superlinear (quadratic) approximate property at ȳ. This result is useful in designing Newton's methods for nonsmooth equations. © 2001 Elsevier Science B.V.