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Vladimir M Veliov - One of the best experts on this subject based on the ideXlab platform.
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lyusternik graves theorems for the sum of a lipschitz function and a set valued Mapping
Siam Journal on Control and Optimization, 2016Co-Authors: Radek Cibulka, Asen L Dontchev, Vladimir M VeliovAbstract: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...
Veliov, Vladimir M. - One of the best experts on this subject based on the ideXlab platform.
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Lyusternik-Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping
'Society for Industrial & Applied Mathematics (SIAM)', 2016Co-Authors: Cibulka Radek, Dontchev, Asen L., Veliov, Vladimir M.Abstract:Graves ve svém článku z roku 1950 dokázal, že zobrazení f mezi Banachovými prostory, které je definované v okolí referenčního bodu x a pro které existuje spojitý lineární operátor A takový, že lipschitzovský modulus rozdílu f-A v referenčním bodě x je dostatečně malý, je otevřené v bodě x s lineárním řádem. Jedná se o zobecnění Banachovy věty o otevřeném zobrazení pro spojitý lineární operátor na lipschitzovsky spojité funkce. Podobný výsledek, pro případ hladkého zobrazení, byl dokázan dříve Lyusternikem. V článku je dokázána Lyusternikova-Gravesova věta pro zobrazení f+F, kde f je lipschitzovsky spojitá funkce na okolí bodu x a F je mnohoznačné zobrazení. Jsou prezentovány podmínky zajišťující, že zobrazení f + F je otevřené s lineárním řádem za předpokladu, že pro každý prvek A z určité množiny spojitých lineárních operátorů je zobrazení f(x) +A(. - x) + F otevřené s lineárním řádem v bodě x pro y. Pokud F je identicky nulové, dostáváme jakožto důsledek Gravesovu větu, věty o otevřeném zobrazení Pourciaua a Pálese, a větu o otevřeném zobrazení pro zobrazení s omezující množinou Cibulky a Fabiana. Dále pak dostáváme také větu o inverzní funkci pro nehladká zobrazení dokázanou nedávno Cibulkou a Dontchevem. V závěru je prezentována aplikace na Nemytského operátory a jistá zobrazení z teorie řízení.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
Akhilesh Prasad - One of the best experts on this subject based on the ideXlab platform.
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hankel clifford transformations on some ultradifferentiable function spaces and pseudo differential operators
Journal of Pseudo-differential Operators and Applications, 2013Co-Authors: Akhilesh Prasad, Sumant KumarAbstract:Continuity of the first Hankel–Clifford transformation on the spaces of the type $$H_\mu $$ are investigated. Pseudo-differential operator $$h_{1,\mu ,a}$$ associated with Bessel type operator $$xD_{x}^{2}+(1-\mu )D_{x}$$ involving the symbol $$a(x,y)$$ whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain ultradifferentiable function spaces. It is shown that the operator $$h_{1,\mu ,a}$$ is a Continuous Linear Mapping of one ultradifferentiable function spaces into another spaces of same type.
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continuity of pseudo differential operator h μ a involving hankel translation and hankel convolution on some gevrey spaces
Integral Transforms and Special Functions, 2010Co-Authors: Akhilesh Prasad, Manish KumarAbstract:The pseudo-differential operator (p.d.o.) h μ, a associated with the Bessel operator involving the symbol a(x, y) whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. h μ, a on Hankel translation τ and Hankel convolution of Gevrey functions is a Continuous Linear Mapping into another Gevrey space.
Radek Cibulka - One of the best experts on this subject based on the ideXlab platform.
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lyusternik graves theorems for the sum of a lipschitz function and a set valued Mapping
Siam Journal on Control and Optimization, 2016Co-Authors: Radek Cibulka, Asen L Dontchev, Vladimir M VeliovAbstract: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...
Cibulka Radek - One of the best experts on this subject based on the ideXlab platform.
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Lyusternik-Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping
'Society for Industrial & Applied Mathematics (SIAM)', 2016Co-Authors: Cibulka Radek, Dontchev, Asen L., Veliov, Vladimir M.Abstract:Graves ve svém článku z roku 1950 dokázal, že zobrazení f mezi Banachovými prostory, které je definované v okolí referenčního bodu x a pro které existuje spojitý lineární operátor A takový, že lipschitzovský modulus rozdílu f-A v referenčním bodě x je dostatečně malý, je otevřené v bodě x s lineárním řádem. Jedná se o zobecnění Banachovy věty o otevřeném zobrazení pro spojitý lineární operátor na lipschitzovsky spojité funkce. Podobný výsledek, pro případ hladkého zobrazení, byl dokázan dříve Lyusternikem. V článku je dokázána Lyusternikova-Gravesova věta pro zobrazení f+F, kde f je lipschitzovsky spojitá funkce na okolí bodu x a F je mnohoznačné zobrazení. Jsou prezentovány podmínky zajišťující, že zobrazení f + F je otevřené s lineárním řádem za předpokladu, že pro každý prvek A z určité množiny spojitých lineárních operátorů je zobrazení f(x) +A(. - x) + F otevřené s lineárním řádem v bodě x pro y. Pokud F je identicky nulové, dostáváme jakožto důsledek Gravesovu větu, věty o otevřeném zobrazení Pourciaua a Pálese, a větu o otevřeném zobrazení pro zobrazení s omezující množinou Cibulky a Fabiana. Dále pak dostáváme také větu o inverzní funkci pro nehladká zobrazení dokázanou nedávno Cibulkou a Dontchevem. V závěru je prezentována aplikace na Nemytského operátory a jistá zobrazení z teorie řízení.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