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

Preda, Ciprian ; Sipos, Ciprian

Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 527-537

Résumé

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.

Détail

DOI : 10.4153/CMB-2010-105-1

Mots-clés : 34D05, 47D06, 93D20, evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility