Abstract view
On the Dichotomy of the Evolution Families: A DiscreteArgument Approach


Published:20101231
Printed: Sep 2011
Ciprian Preda,
Department of Mathematics, University of California, Los Angeles, CA 90095, U.S.A.
Ciprian Sipos,
Department of Economics, West University of Timişoara, Timişoara 300115, Romania
Abstract
We establish a discretetime criteria guaranteeing the existence of an
exponential dichotomy in the continuoustime
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuoustime
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vectorvalued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.