Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems
We prove that a family of $q$-periodic continuous matrix valued function ${A(t)}_{tin mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $int_0^t e^{imu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $mu$ and the solution of the Cauchy operator Problem...
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2013-04-01
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doaj-1bb1bd39109a4390a542801eeca7a91f2020-11-24T22:46:59ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912013-04-01201389,17Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problemsDhaou LassouedWe prove that a family of $q$-periodic continuous matrix valued function ${A(t)}_{tin mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $int_0^t e^{imu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $mu$ and the solution of the Cauchy operator Problem $$displaylines{ dot{Y}(t)=-Y(t)A(t)+ e^{i mu t}(I-P) ,quad tgeq s cr Y(s)=0, }$$ has a limit in $mathcal{L}(mathbb{C}^n)$ as s tends to $-infty$ which is bounded uniformly with respect to the parameter $mu$. Here, ${ U(t,s): t, sinmathbb{R}}$ is the evolution family generated by ${A(t)}_{tin mathbb{R}}$, $mu$ is a real number and q is a fixed positive number. http://ejde.math.txstate.edu/Volumes/2013/89/abstr.htmlPeriodic evolution familiesexponential dichotomyboundedness |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Dhaou Lassoued |
spellingShingle |
Dhaou Lassoued Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems Electronic Journal of Differential Equations Periodic evolution families exponential dichotomy boundedness |
author_facet |
Dhaou Lassoued |
author_sort |
Dhaou Lassoued |
title |
Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems |
title_short |
Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems |
title_full |
Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems |
title_fullStr |
Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems |
title_full_unstemmed |
Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems |
title_sort |
exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic cauchy problems |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2013-04-01 |
description |
We prove that a family of $q$-periodic continuous matrix valued function ${A(t)}_{tin mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $int_0^t e^{imu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $mu$ and the solution of the Cauchy operator Problem $$displaylines{ dot{Y}(t)=-Y(t)A(t)+ e^{i mu t}(I-P) ,quad tgeq s cr Y(s)=0, }$$ has a limit in $mathcal{L}(mathbb{C}^n)$ as s tends to $-infty$ which is bounded uniformly with respect to the parameter $mu$. Here, ${ U(t,s): t, sinmathbb{R}}$ is the evolution family generated by ${A(t)}_{tin mathbb{R}}$, $mu$ is a real number and q is a fixed positive number. |
topic |
Periodic evolution families exponential dichotomy boundedness |
url |
http://ejde.math.txstate.edu/Volumes/2013/89/abstr.html |
work_keys_str_mv |
AT dhaoulassoued exponentialdichotomyofnonautonomousperiodicsystemsintermsoftheboundednessofcertainperiodiccauchyproblems |
_version_ |
1725682885518688256 |