2 results
On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
-
- Journal:
- Canadian Mathematical Bulletin / Volume 54 / Issue 3 / 01 September 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 527-537
- Print publication:
- 01 September 2011
-
- Article
-
- You have access
- Export citation
-
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
$\mathcal{U}\,=\,{{\{U(t,\,s)\}}_{t\ge s\ge 0}}$ 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 (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.
Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line
-
- Journal:
- Canadian Mathematical Bulletin / Volume 54 / Issue 2 / 01 June 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 364-369
- Print publication:
- 01 June 2011
-
- Article
-
- You have access
- Export citation
