Growth of semigroups in discrete and continuous time

Alexander Gomilko¹ ; Hans Zwart² ; Niels Besseling³

¹Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland
²Department of Applied Mathematics University of Twente P.O. 217 7500 AE, Enschede, The Netherlands
³Department of Applied Mathematics University of Twente P.O. 217 7500 AE, Enschede The Netherlands

Studia Mathematica, Tome 206 (2011) no. 3, pp. 273-292

Résumé

We show that the growth rates of solutions of the abstract differential equations $\dot{x}(t) = A x(t)$, $\dot{x}(t)= A^{-1} x(t)$, 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 $(e^{A^{-1}t})_{t \geq 0}$ is $O(\sqrt[4]{t})$, and for $( (A+I)(A-I)^{-1})^n$ it is $O(\sqrt[4]{n})$. 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.

Détail

DOI : 10.4064/sm206-3-3

Keywords: growth rates solutions abstract differential equations dot dot difference equation a i closely related assuming generates exponentially stable semigroup general banach space lowest growth rate semigroup geq sqrt a i sqrt similarity growth holds banach spaces particular hilbert spaces best estimates log log respectively furthermore conditions growth rate a i operator power bounded