Growth of semigroups in discrete and continuous time

• Autorzy: Alexander Gomilko , Hans Zwart , Niels Besseling
• Języki publikacji: EN

Abstrakty

EN

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ(t) = A^{-1} x(t)$, and the difference equation $x_{d}(n+1) = (A+I)(A-I)^{-1} x_{d}(n)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup $(e^{A^{-1}t})_{t≥0}$ is O(∜t), and for $((A+I)(A-I)^{-1})ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of $((A+I)(A-I)^{-1})ⁿ$ is O(1), i.e., the operator is power bounded.

Kategorie tematyczne

• 47D06: One-parameter semigroups and linear evolution equations
• 34G10: Linear equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 206
• Numer: 3
• Strony: 273-292

Daty

wydano

2011

Twórcy

autor

Alexander Gomilko

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

autor

Hans Zwart

• Department of Applied Mathematics, University of Twente, P.O. 217, 7500 AE, Enschede, The Netherlands

autor

Niels Besseling

• Department of Applied Mathematics, University of Twente, P.O. 217 7500 AE, Enschede, The Netherlands

Identyfikatory

DOI

10.4064/sm206-3-3