A+ CATEGORY SCIENTIFIC UNIT

Growth of semigroups in discrete and continuous time

Volume 206 / 2011

Alexander Gomilko, Hans Zwart, Niels Besseling Studia Mathematica 206 (2011), 273-292 MSC: Primary 47D06; Secondary 34G10. DOI: 10.4064/sm206-3-3

Abstract

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.

Authors

  • Alexander GomilkoFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail
  • Hans ZwartDepartment of Applied Mathematics
    University of Twente
    P.O. 217
    7500 AE, Enschede,
    The Netherlands
    e-mail
  • Niels BesselingDepartment of Applied Mathematics
    University of Twente
    P.O. 217 7500 AE, Enschede
    The Netherlands
    e-mail

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