An Implicit Function Theorem for One-sided Lipschitz Mappings

Abstract

Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition. We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function. Applications are provided to the Crank-Nicolson scheme for differential inclusions and to the analysis of differential algebraic inclusions.

Semi-Implicit Euler Schemes for Ordinary Differential Inclusions

“…The construction of the spatial discretization of the implicit Euler scheme, however, requires explicit knowledge of the one-sided Lipschitz constant and the modulus of continuity of the right-hand side and is very sensitive to illestimated constants. This is the main motivation for the development of the semi-implicit Euler schemes (4) and (20a, 20b) analyzed in the present paper. In addition, their performance is considerably better than that of the implicit Euler scheme, because instead of implicit inclusions only implicit equations have to be solved for computing their images.…”

“…A set-valued implicit Euler scheme has been analyzed in [6]. It has very good analytical properties, and it is based on an implicit function theorem that is given in [4]. If applied to stiff differential inclusions, it is considerably more efficient than the explicit Euler scheme, because it senses the correct asymptotic behavior, while the reachable sets of the explicit Euler scheme do not only oscillate, but grow exponentially in diameter if the temporal step-size is not small enough.…”

On semiregularity of mappings

“…So the assumption in [5, Corollary 2 (ii)] that dist (ȳ, F (x)) is small enough holds trivially. Also note that "a slightly generalized definition of metric regularity" in [5] is nothing else but the usual definition of this property because F : R n ⇒ R n in [13] means neither that dom F = R n nor that x is an interior point of dom F .…”

“…The infimum of κ > 0 for which there exists a neighborhood U ofx in X such that (5) holds is called the subregularity modulus of F at (x,ȳ) and is denoted by subreg F (x,ȳ);…”