Electron. J. Diff. Equ., Vol. 2013 (2013), No. 89, pp. 1-7.

Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems

Dhaou Lassoued

Abstract:
We prove that a family of $q$-periodic continuous matrix valued function $\{A(t)\}_{t\in \mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $\int_0^t e^{i\mu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $\mu$ and the solution of the Cauchy operator Problem
$$\displaylines{ \dot{Y}(t)=-Y(t)A(t)+ e^{i \mu t}(I-P) ,\quad t\geq s \cr Y(s)=0, }$$
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, $\{ U(t,s): t, s\in\mathbb{R}\}$ is the evolution family generated by $\{A(t)\}_{t\in \mathbb{R}}$, $\mu$ is a real number and q is a fixed positive number.

Submitted February 1, 2013. Published April 5, 2013.
Math Subject Classifications: 47A05, 34D09, 35B35.
Key Words: Periodic evolution families; exponential dichotomy; boundedness.

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Dhaou Lassoued
Laboratoire SAMM EA4543
Université Paris 1 Panthéon-Sorbonne
centre P.M.F., 90 rue de Tolbiac
75634 Paris cedex 13, France
email: Dhaou.Lassoued@univ-paris1.fr, dhaou06@gmail.com

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