On Fredholm's Theorem of the Alternative and a Corollary of Rohn's Residual Existence Theorem

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F12%3AA130163Y" target="_blank" >RIV/61988987:17310/12:A130163Y - isvavai.cz</a>

Result on the web

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DOI - Digital Object Identifier

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Result language

angličtina

Original language name

On Fredholm's Theorem of the Alternative and a Corollary of Rohn's Residual Existence Theorem

Original language description

By Fredholm's Theorem of the alternative, the system ${mib A} {mib x} = {mib b}$ of linear equations has no solution if and only if ${mib u}^{sl T} ! {mib A} = {mib o}^{sl T}$ and ${mib u}^{sl T} {mib b} neq 0$ for some ${mib u} in {msbmR}^m$. Recently, Rohn proved as a corollary of the Residual Existence Theorem for linear equations [{it Optim. Lett./} 4 (2010), 287--292] that the system ${mib A} {mib x} = {mib b}$ has a solution if and only if the residual set ${, {mib A} {mib x} - {mib b} : {mib x} in {msbm R}^n ,}$ intersects all the orthants of~${msbm R}^m$. We study the relation between both the results in the more general setting of a vector space over a linearly ordered (possibly skew) field, obtain a new proof of the corollary, and give a generalisation of Fredholm's Theorem of the alternative.

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BB - Applied statistics, operational research

OECD FORD branch

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Project

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Continuities

S - Specificky vyzkum na vysokych skolach

Publication year

2012

Confidentiality

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

Name of the periodical

Mathematica Pannonica

ISSN

0865-2090

e-ISSN

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Volume of the periodical

23

Issue of the periodical within the volume

2

Country of publishing house

HU - HUNGARY

Number of pages

10

Pages from-to

311-320

UT code for WoS article

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EID of the result in the Scopus database

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