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

Let H : ℜ^m × ℜⁿ → ℜⁿ be a locally Lipschitz function in a neighborhood of (&ymacr;,&xmacr;) and H(&ymacr;,&xmacr;) = 0 for some &ymacr; ∈ ℜ^m and &xmacr; ∈ ℜⁿ. The implicit function theorem in the sense of Clarke says that if πx∂H(&ymacr;,&xmacr;) is of maximal rank, then there exists a neighborhood Y of &ymacr; and a Lipschitz function G(·) : Y → ℜⁿ such that G(&ymacr;) = &xmacr; 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 (&ymacr;,&xmacr;), then G has a superlinear (quadratic) approximate property at &ymacr;. This result is useful in designing Newton's methods for nonsmooth equations.