Exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay

We show that if U ( t , s ) is an exponentially dichotomic evolution operator, then the unique solution of the Volterra equation V ( t , s ) = U ( t , s ) + ∫ s t U ( t , τ ) B ( τ ) V ( τ , s ) d τ is also an exponentially dichotomic evolution operator, for small B ( t ) . As a consequence, we prov...

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Bibliographic Details
Published inNonlinear analysis Vol. 71; no. 3; pp. 935 - 947
Main Author Popescu, Liviu Horia
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Ltd 01.08.2009
Elsevier
Subjects
Banach space
Evolution families
Exact sciences and technology
Exponential dichotomy
Functional analysis
Mathematical analysis
Mathematics
Ordinary differential equations
Sciences and techniques of general use
Structural stability
34D09
Evolution families
Structural stability
Banach space
Exponential dichotomy
34D30
Nonlinear analysis
Mathematical model
Numerical stability
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Summary:We show that if U ( t , s ) is an exponentially dichotomic evolution operator, then the unique solution of the Volterra equation V ( t , s ) = U ( t , s ) + ∫ s t U ( t , τ ) B ( τ ) V ( τ , s ) d τ is also an exponentially dichotomic evolution operator, for small B ( t ) . As a consequence, we prove that any exponentially dichotomic evolution family is structurally stable. We improve the previous results, showing that the condition of bounded growth and decay, present everywhere in the existing literature, can be completely removed.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2008.11.009