Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations
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| Title: | Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations |
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| Authors: | Preda, C. preda@math.ucla.edu, Preda, P.1 preda@math.uvt.ro, Craciunescu, A.1 craciunescu@math.uvt.ro |
| Source: | Journal of Functional Analysis. Feb2010, Vol. 258 Issue 3, p729-757. 29p. |
| Subjects: | Existence theorems, Exponential functions, Asymptotic expansions, Variational principles, Vector-valued measures, Function spaces, Continuous functions |
| Abstract: | Abstract: We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in . Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) , J. Daleckij and M. Krein (1974) , J.L. Massera and J.J. Schäffer (1966) , N. van Minh, F. Räbiger and R. Schnaubelt (1998) . [Copyright &y& Elsevier] |
| Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Preda%2C+C%2E%22">Preda, C.</searchLink><i> preda@math.ucla.edu</i><br /><searchLink fieldCode="AR" term="%22Preda%2C+P%2E%22">Preda, P.</searchLink><relatesTo>1</relatesTo><i> preda@math.uvt.ro</i><br /><searchLink fieldCode="AR" term="%22Craciunescu%2C+A%2E%22">Craciunescu, A.</searchLink><relatesTo>1</relatesTo><i> craciunescu@math.uvt.ro</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Functional+Analysis%22">Journal of Functional Analysis</searchLink>. Feb2010, Vol. 258 Issue 3, p729-757. 29p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Existence+theorems%22">Existence theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Exponential+functions%22">Exponential functions</searchLink><br /><searchLink fieldCode="DE" term="%22Asymptotic+expansions%22">Asymptotic expansions</searchLink><br /><searchLink fieldCode="DE" term="%22Variational+principles%22">Variational principles</searchLink><br /><searchLink fieldCode="DE" term="%22Vector-valued+measures%22">Vector-valued measures</searchLink><br /><searchLink fieldCode="DE" term="%22Function+spaces%22">Function spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Continuous+functions%22">Continuous functions</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract: We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in . Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) , J. Daleckij and M. Krein (1974) , J.L. Massera and J.J. Schäffer (1966) , N. van Minh, F. Räbiger and R. Schnaubelt (1998) . [Copyright &y& Elsevier] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jfa.2009.09.002 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 29 StartPage: 729 Subjects: – SubjectFull: Existence theorems Type: general – SubjectFull: Exponential functions Type: general – SubjectFull: Asymptotic expansions Type: general – SubjectFull: Variational principles Type: general – SubjectFull: Vector-valued measures Type: general – SubjectFull: Function spaces Type: general – SubjectFull: Continuous functions Type: general Titles: – TitleFull: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Preda, C. – PersonEntity: Name: NameFull: Preda, P. – PersonEntity: Name: NameFull: Craciunescu, A. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2010 Type: published Y: 2010 Identifiers: – Type: issn-print Value: 00221236 Numbering: – Type: volume Value: 258 – Type: issue Value: 3 Titles: – TitleFull: Journal of Functional Analysis Type: main |
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