A counterexample in the theory of linear singularly perturbed systems

In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0.  Here A and  f  are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R...

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Autores:
Naulin, Raúl
Tipo de recurso:
Article of journal
Fecha de publicación:
1991
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/43433
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/43433
http://bdigital.unal.edu.co/33531/
Palabra clave:
51 Matemáticas / Mathematics
Bounded solutions
system linear algebraic system
bounded functions
Lipschitz function
Soluciones acotadas
sistema lineal
sistema algebráico
funciones acotadas
función de Lipschitz
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openAccess
License
Atribución-NoComercial 4.0 Internacional
id UNACIONAL2_a378e94f2e9953f00e7a1ea97b8fc3bf
oai_identifier_str oai:repositorio.unal.edu.co:unal/43433
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
spelling Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de Colombiahttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Naulin, Raúl25df13a8-27e4-46d5-b20f-0cea2b9db1743002019-06-28T11:58:22Z2019-06-28T11:58:22Z1991-01-01ISSN: 2357-4100https://repositorio.unal.edu.co/handle/unal/43433http://bdigital.unal.edu.co/33531/In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0.  Here A and  f  are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R e λ (t )| ≥ y and gt; 0. It is known that for small ε, the following estimate is valid hasta ||kε (f) + A-1 f || ≤  εL || f ||1, where kε(f) denotes the bounded solution of ε X'  = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ ||  and L is a constant. We prove that this estimate cannot be replaced by ||kε (f) + A-1 f || ≤  εL || f ||. Futhermore, if, instead of the condition that A be C1, we require that the function be bounded and Lipschitz continuous, we show that the same estimate, ||kε (f) + A-1 || ≤  εL || f ||1, can be obtained.En esta nota se comparan las soluciones acotadas del sistema lineal singularmente perturbado ε X' = A (t) x + f (t), con las soluciones del sistema algebráico A (t) x + f (t) = 0. Aquí  A y f  son funciones acotadas de clase C1, con derivadas acotadas. Suponemos además que los valores propios de A (t) satisfacen la condición | R e λ (t)| ≥ y and gt; 0. Es sabido que para ϵ C1 y ε suficientemente pequeños vale la siguiente estimación: ||kε (f) + A-1 || ≤  εL || f ||1, donde kε (f) denota la única solución acotada de ε X'  = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ || y L es una constante que no depende de f ni de ε.  Probaremos que esta estimación no puede ser extendida hasta ||kε (f) + A-1 f || ≤  εL || f ||. Además, si en lugar de exigir que A sea de clase C1 pedimos que A sea una función de Lipschitz acotada, entonces sigue siendo válida la estimación ||kε (f) + A-1 || ≤  εL || f ||1.application/pdfspaUniversidad Nacional de Colombiahttp://revistas.unal.edu.co/index.php/recolma/article/view/33395Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de MatemáticasRevista Colombiana de MatemáticasRevista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426Naulin, Raúl (1991) A counterexample in the theory of linear singularly perturbed systems. Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426, 25 (1-4). pp. 95-102. ISSN 2357-410051 Matemáticas / MathematicsBounded solutionssystem linear algebraic systembounded functionsLipschitz functionSoluciones acotadassistema linealsistema algebráicofunciones acotadasfunción de LipschitzA counterexample in the theory of linear singularly perturbed systemsArtículo de revistainfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/ARTORIGINAL33395-123875-1-PB.pdfapplication/pdf2112598https://repositorio.unal.edu.co/bitstream/unal/43433/1/33395-123875-1-PB.pdfac05c72c7d3a2de2296ba08dedc22a0aMD51THUMBNAIL33395-123875-1-PB.pdf.jpg33395-123875-1-PB.pdf.jpgGenerated Thumbnailimage/jpeg7359https://repositorio.unal.edu.co/bitstream/unal/43433/2/33395-123875-1-PB.pdf.jpg52f46a5b1e5baf65c776a7e9d45f5cddMD52unal/43433oai:repositorio.unal.edu.co:unal/434332024-02-10 23:06:13.617Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co
dc.title.spa.fl_str_mv A counterexample in the theory of linear singularly perturbed systems
title A counterexample in the theory of linear singularly perturbed systems
spellingShingle A counterexample in the theory of linear singularly perturbed systems
51 Matemáticas / Mathematics
Bounded solutions
system linear algebraic system
bounded functions
Lipschitz function
Soluciones acotadas
sistema lineal
sistema algebráico
funciones acotadas
función de Lipschitz
title_short A counterexample in the theory of linear singularly perturbed systems
title_full A counterexample in the theory of linear singularly perturbed systems
title_fullStr A counterexample in the theory of linear singularly perturbed systems
title_full_unstemmed A counterexample in the theory of linear singularly perturbed systems
title_sort A counterexample in the theory of linear singularly perturbed systems
dc.creator.fl_str_mv Naulin, Raúl
dc.contributor.author.spa.fl_str_mv Naulin, Raúl
dc.subject.ddc.spa.fl_str_mv 51 Matemáticas / Mathematics
topic 51 Matemáticas / Mathematics
Bounded solutions
system linear algebraic system
bounded functions
Lipschitz function
Soluciones acotadas
sistema lineal
sistema algebráico
funciones acotadas
función de Lipschitz
dc.subject.proposal.spa.fl_str_mv Bounded solutions
system linear algebraic system
bounded functions
Lipschitz function
Soluciones acotadas
sistema lineal
sistema algebráico
funciones acotadas
función de Lipschitz
description In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0.  Here A and  f  are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R e λ (t )| ≥ y and gt; 0. It is known that for small ε, the following estimate is valid hasta ||kε (f) + A-1 f || ≤  εL || f ||1, where kε(f) denotes the bounded solution of ε X'  = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ ||  and L is a constant. We prove that this estimate cannot be replaced by ||kε (f) + A-1 f || ≤  εL || f ||. Futhermore, if, instead of the condition that A be C1, we require that the function be bounded and Lipschitz continuous, we show that the same estimate, ||kε (f) + A-1 || ≤  εL || f ||1, can be obtained.
publishDate 1991
dc.date.issued.spa.fl_str_mv 1991-01-01
dc.date.accessioned.spa.fl_str_mv 2019-06-28T11:58:22Z
dc.date.available.spa.fl_str_mv 2019-06-28T11:58:22Z
dc.type.spa.fl_str_mv Artículo de revista
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dc.type.content.spa.fl_str_mv Text
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format http://purl.org/coar/resource_type/c_6501
status_str publishedVersion
dc.identifier.issn.spa.fl_str_mv ISSN: 2357-4100
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/43433
dc.identifier.eprints.spa.fl_str_mv http://bdigital.unal.edu.co/33531/
identifier_str_mv ISSN: 2357-4100
url https://repositorio.unal.edu.co/handle/unal/43433
http://bdigital.unal.edu.co/33531/
dc.language.iso.spa.fl_str_mv spa
language spa
dc.relation.spa.fl_str_mv http://revistas.unal.edu.co/index.php/recolma/article/view/33395
dc.relation.ispartof.spa.fl_str_mv Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de Matemáticas
Revista Colombiana de Matemáticas
dc.relation.ispartofseries.none.fl_str_mv Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426
dc.relation.references.spa.fl_str_mv Naulin, Raúl (1991) A counterexample in the theory of linear singularly perturbed systems. Revista Colombiana de Matemáticas; Vol. 25, núm. 1-4 (1991); 95-102 0034-7426, 25 (1-4). pp. 95-102. ISSN 2357-4100
dc.rights.spa.fl_str_mv Derechos reservados - Universidad Nacional de Colombia
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.license.spa.fl_str_mv Atribución-NoComercial 4.0 Internacional
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dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv Atribución-NoComercial 4.0 Internacional
Derechos reservados - Universidad Nacional de Colombia
http://creativecommons.org/licenses/by-nc/4.0/
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
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dc.publisher.spa.fl_str_mv Universidad Nacional de Colombia
institution Universidad Nacional de Colombia
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