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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/GA15-00735S" target="_blank" >GA15-00735S: Analýza stability optim a ekvilibrií v ekonomii</a><br>

Návaznosti

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

Rok uplatnění

2016

Kód důvěrnosti údajů

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

Název periodika

SIAM Journal on Control and Optimization

ISSN

0363-0129

e-ISSN

—

Svazek periodika

54

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

24

Strana od-do

3273-3296

Kód UT WoS článku

000391960900014

EID výsledku v databázi Scopus

2-s2.0-84959112113