Lyusternik-Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping

In a paper of 1950 Graves proved that for a function f acting between Banach spaces and an interior point x̄ in its domain, if there exists a continuous linear mapping A which is surjective and the Lipschitz modulus of the difference f −A at x̄ is sufficiently small, then f is (linearly) open at x̄. 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 x̄ and F is a set-valued mapping. Roughly, we give conditions under which the mapping f + F is linearly open at x̄ for ȳ provided that for each element A of a certain set of continuous linear operators the mapping f(x̄) + A(· − x̄) + F is linearly open at x̄ for ȳ. 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 Páles, 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.