# Growth of semigroups in discrete and continuous time

Gomilko, Alexander
and
Zwart, Hans
and
Besseling, Niels
(2011)
*Growth of semigroups in discrete and continuous time.*
Studia Mathematica, 206
(3).
pp. 273-292.
ISSN 0039-3223

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Abstract: | We show that the growth rates of solutions of the abstract differential equations x˙(t)=Ax(t), x˙(t)=A −1 x(t) and the difference equation xd(n+1)=(A+I)(A−I)−1 xd(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−1t) t≥0 is O(√4t) and for ((A+I)(A−I)−1)n it is O(√4n). 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. |

Item Type: | Article |

Copyright: | © 2011 Instytut Matematyczny PAN |

Faculty: | Electrical Engineering, Mathematics and Computer Science (EEMCS) |

Research Group: | |

Link to this item: | http://purl.utwente.nl/publications/78767 |

Official URL: | http://dx.doi.org/10.4064/sm206-3-3 |

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