Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

The well-known Favard's theorem states that the linear differential equation x′=A(t)x+f(t) Turn MathJax on with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded s...

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Detalles Bibliográficos
Autores: Caraballo Garrido, Tomás, Cheban, David
Tipo de recurso: artículo
Fecha de publicación:2009
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/23636
Acceso en línea:http://hdl.handle.net/11441/23636
https://doi.org/10.1016/j.jde.2008.04.001
Access Level:acceso abierto
Palabra clave:Almost periodic solution
Almost automorphic solutions
Non-autonomous dynamical systems
Favard's condition
Cocycle
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spelling Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. ICaraballo Garrido, TomásCheban, DavidAlmost periodic solutionAlmost automorphic solutionsNon-autonomous dynamical systemsFavard's conditionCocycleThe well-known Favard's theorem states that the linear differential equation x′=A(t)x+f(t) Turn MathJax on with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations x′=B(t)x, Turn MathJax on where B∈H(A):={B|B(t)=limn→+∞A(t+tn)}. If there are bounded solutions which are non-separated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan). In this paper we prove that linear differential equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equation x′=A(t)x Turn MathJax on are homoclinic to zero (i.e. lim|t|→+∞|φ(t)|=0 for all bounded solutions φ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution. The analogue of this result for difference equations is also given. We study the problem of existence of Bohr/Levitan almost periodic solutions of Eq. (1) in the framework of general non-autonomous dynamical systems (cocycles).Ecuaciones Diferenciales y Análisis Numérico2009info:eu-repo/semantics/articleapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/23636https://doi.org/10.1016/j.jde.2008.04.001reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of Differential Equations, 246(1), 108-128info:eu-repo/semantics/openAccessoai:idus.us.es:11441/236362026-06-17T12:51:07Z
dc.title.none.fl_str_mv Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
title Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
spellingShingle Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
Caraballo Garrido, Tomás
Almost periodic solution
Almost automorphic solutions
Non-autonomous dynamical systems
Favard's condition
Cocycle
title_short Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
title_full Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
title_fullStr Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
title_full_unstemmed Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
title_sort Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I
dc.creator.none.fl_str_mv Caraballo Garrido, Tomás
Cheban, David
author Caraballo Garrido, Tomás
author_facet Caraballo Garrido, Tomás
Cheban, David
author_role author
author2 Cheban, David
author2_role author
dc.contributor.none.fl_str_mv Ecuaciones Diferenciales y Análisis Numérico
dc.subject.none.fl_str_mv Almost periodic solution
Almost automorphic solutions
Non-autonomous dynamical systems
Favard's condition
Cocycle
topic Almost periodic solution
Almost automorphic solutions
Non-autonomous dynamical systems
Favard's condition
Cocycle
description The well-known Favard's theorem states that the linear differential equation x′=A(t)x+f(t) Turn MathJax on with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations x′=B(t)x, Turn MathJax on where B∈H(A):={B|B(t)=limn→+∞A(t+tn)}. If there are bounded solutions which are non-separated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan). In this paper we prove that linear differential equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equation x′=A(t)x Turn MathJax on are homoclinic to zero (i.e. lim|t|→+∞|φ(t)|=0 for all bounded solutions φ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution. The analogue of this result for difference equations is also given. We study the problem of existence of Bohr/Levitan almost periodic solutions of Eq. (1) in the framework of general non-autonomous dynamical systems (cocycles).
publishDate 2009
dc.date.none.fl_str_mv 2009
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/23636
https://doi.org/10.1016/j.jde.2008.04.001
url http://hdl.handle.net/11441/23636
https://doi.org/10.1016/j.jde.2008.04.001
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Differential Equations, 246(1), 108-128
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
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