Growth of semigroups in discrete and continuous time

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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)
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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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