Growth of semigroups in discrete and continuous time

Alexander Gomilko, Heiko J. Zwart, N.C. Besseling

    Research output: Contribution to journalArticleAcademicpeer-review

    2 Citations (Scopus)

    Abstract

    We show that the growth rates of solutions of the abstract differential equations $\dot{x}(t)=Ax(t), \dot{x}(t)=A^{-1}x(t)$, and the difference equation $x_d(n+1)=(A+I)(A-1)^{-1}x_d(n)$ 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 $\left(e^{A^{-1}t}\right)_{t\geq 0}$ is $O(\sqrt[4]{t})$, and for $\left((A+I)(A-I)^{-1}\right)^n$ 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 $\left((A+I)(A-I)^{-1}\right)^n$ is $O(1)$, i.e., the operator is power bounded.
    Original languageUndefined
    Pages (from-to)273-292
    Number of pages20
    JournalStudia mathematica
    Volume206
    Issue number3
    DOIs
    Publication statusPublished - 2011

    Keywords

    • EWI-20920
    • IR-78767
    • METIS-286269
    • MSC-47D06

    Cite this