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

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F16%3A43931419" target="_blank" >RIV/49777513:23520/16:43931419 - isvavai.cz</a>

  • Result on the web

    <a href="http://dx.doi.org/10.1137/16M1063150" target="_blank" >http://dx.doi.org/10.1137/16M1063150</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1137/16M1063150" target="_blank" >10.1137/16M1063150</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

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

  • Original language description

    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 sufficiently 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 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.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

  • CEP classification

    BA - General mathematics

  • OECD FORD branch

Result continuities

  • Project

    <a href="/en/project/GA15-00735S" target="_blank" >GA15-00735S: Stability analysis of optima and equilibria in economics</a><br>

  • Continuities

    P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Others

  • Publication year

    2016

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    SIAM Journal on Control and Optimization

  • ISSN

    0363-0129

  • e-ISSN

  • Volume of the periodical

    54

  • Issue of the periodical within the volume

    6

  • Country of publishing house

    US - UNITED STATES

  • Number of pages

    24

  • Pages from-to

    3273-3296

  • UT code for WoS article

    000391960900014

  • EID of the result in the Scopus database

    2-s2.0-84959112113