Inverse and implicit function theorems for H-differentiable and semismooth functions
چکیده
In this article, we present inverse and implicit function theorems forH -differentiable functions and semismooth functions, thereby generalizing the classical inverse and implicit function theorems to certain classes of nonsmooth functions. The classical inverse function theorem [47] asserts that a continuously differentiable function f : R → R is locally invertible at a point with a continuously differentiable inverse if the (Fréchet) derivative of the function at that point is nonsingular. The corresponding implicit function theorem says that when a continuously differentiable function f (x, y) vanishes at a point (x∗, y∗) with f ′ x(x∗, y∗) nonsingular, the equation f (x, y) = 0 can be solved for y in terms of x in a neighborhood of (x∗, y∗). There are numerous inverse and implicit function theorems in the literature. The classical inverse and implicit function theorems have been extended in various directions, e.g., to Banach spaces [1], to multivalued mappings [8,38], to nonsmooth functions [4,32,37,55], etc. Since our setting here is finite dimensional and our functions are single valued, we describe only those generalizations of the classical results that are relevant to our discussion. In 1976, Clarke extended the classical inverse and implicit function theorems to locally Lipschitzian functions by considering the so-called generalized Jacobian; see Refs. [37,55] for