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R. Cibulka, A.L. Dontchev, V.M. Veliov:
"Lyusternik-Graves Theorems for the sum of a Lipschitz function and a set-valued mapping";
SIAM Journal on Control and Optimization, 6 (2016), 3273 - 3296.

In a paper of 1950 Graves proved that for a function f acting between Banach spaces and an interior point 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 x is suffciently small, then f is (linearly) open at 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 x and F is a set-valued mapping. Roughly, we give conditions under which the mapping f + F is linearly open at x for y 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.

open mapping theorem, inverse function theorem, linear openness, metric regularity, strict prederivative, feasibility in control

http://dx.doi.org/10.1137/16M1063150