Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 89, pp. 1-7.
Title: Exponential dichotomy of nonautonomous periodic systems in terms of
the boundedness of certain periodic Cauchy problems
Author: Dhaou Lassoued (Univ. Paris 1, Paris cedex 13, France)
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 01, 2013. Published April 05, 2013.
Math Subject Classifications: 47A05, 34D09, 35B35.
Key Words: Periodic evolution families; exponential dichotomy;
boundedness.