LYUSTERNIK-GRAVES THEOREMS FOR THE SUM OF A LIPSCHITZ FUNCTION AND A SET-VALUED MAPPING

摘要

In a paper of 1950 Graves proved that for a function f acting between Banach spaces and an interior point (x) over bar in its domain, if there exists a continuous linear mapping A which is surjective and the Lipschitz modulus of the difference f - A at (x) over bar issufficiently small, then f is (linearly) open at (x) over bar. This is an extension of the Banach open mapping principle from continuous linear mappings to Lipschitz functions. A closely related result was obtained earlier by Lyusternik for smooth functions. In this paper, we obtain Lyusternik{Graves theorems for mappings of the form f + F, where f is a Lipschitz continuous function around (x) over bar and F is a set-valued mapping. Roughly, we give conditions under which the mapping f + F is linearly open at (x) over bar for (y) over bar provided that for each element A of a certain set of continuous linear operators the mapping f (x) + A (. - x) + F is linearly open at x for y. In the case when F is the zero mapping, as corollaries we obtain the theorem of Graves as well as open mapping theorems by Pourciau and Pales, and a constrained open mapping theorem by Cibulka and Fabian. From the general result we also obtain a nonsmooth inverse function theorem proved recently by Cibulka and Dontchev. Application to Nemytskii operators and a feasibility mapping in control are presented.