On the Dichotomy of the Evolution Families: A Discrete-Argument Approach

Canadian mathematical bulletin

Ciprian Sipos¹  Ciprian Preda² 

[1] Department of Economics, West University of Timişoara, Timişoara 300115, Romania;Department of Mathematics, University of California, Los Angeles, CA 90095, U.S.A.

关键词: evolution families;    exponential dichotomy;    Orlicz sequence spaces;    admissibility;   

DOI  :  10.4153/CMB-2010-105-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We establish a discrete-time criteria guaranteeing the existence of anexponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family ${cal U}={U(t,s)}_{tgeq sgeq 0}$ acting on a Banach space $X$ is uniformlyexponentially dichotomic (with respect to its continuous-timebehavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued 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.

【 授权许可】

Unknown