IMPLICIT FUNCTION THEOREMS FOR GENERALIZED EQUATIONS

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 is an element of f(x, y) + F(x), where f is a function and F is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution to z is an element of g(x) + F(x), where g strongly approximates f in the sense of Robinson. In particular, the inverse map (f + F)(-1) has a local selection which is Lipschitz continuous near xo and Frechet (Gateaux, Bouligand, directionally) differentiable at xo if and only if the linearization inverse (f(x(0)) + del f(x(0))(.-x(0)) + F(.))(-1) has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.