On the Spectrum, Complete Trajectories, and Asymptotic Stability of Linear Semi-dynamical Systems

✍ Scribed by V.Q. Phong

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 594 KB
• Volume: 105
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1993.1081

✦ Synopsis

Let ({T(t)}{1 \geqslant 0}) be a (C{0})-semigroup in a Banach space (X) with generator (A). We prove that if ({T(t)}{t \geqslant 0}) is bounded and sun-reflexive, and the sun-dual semigroup is not asymptotically stable, then there exist bounded complete trajectories under ({T(t)}{t \geq 0}), provided: (i) there is (t_{0}>0) such that (\operatorname{ran}\left(T^{\odot}\left(t_{0}\right)\right)) is dense in (X \odot), or (ii) (\sigma(A) \nsupseteq i \mathbb{R}). Questions of almost periodicity of complete trajectories are also discussed and a new proof of our earlier theorem (jointly with Yu. I. Lyubich) on asymptotic stability is given. 1993 Academic Press, Inc.