CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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

  Published:2010-12-31
 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
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time 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 continuous-time behavior) 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.
Keywords: evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility
MSC Classifications: 34D05, 47D06, 93D20 show english descriptions Asymptotic properties
One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
Asymptotic stability
34D05 - Asymptotic properties
47D06 - One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
93D20 - Asymptotic stability
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/