Lyusternik--Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping
Radek Cibulka, Asen L. Dontchev, Vladimir M. Veliov · SIAM Journal on Control and Optimization · 2016
In a paper of 1950 Graves proved that for a function $f$ acting between Banach spaces and an interior point $\bar x$ in its domain, if there exists a continuous linear mapping $A$ which is surjective and the Lipschitz modulus of the difference $f-A$ at $\bar x$ is sufficiently small, then $f$ is (linearly) open at $\bar x$. 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 $\bar x$ and $F$ is a set-valued mapping. Roughly, we give conditions under which the mapping $f+F$ is linearly open at $\bar x$ for $\bar y$ provided that for each element $A$ of a certain set of continuous linear operators the mapping $f(\bar{x}) +A(\cdot - \bar{x}) + F$ is linearly open at $\bar x$ for $\bar y$. 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 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.