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Growth of semigroups in discrete and continuous time

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Gomilko A., Zwart H., Besseling N.

Studia Mathematica

2011

tom 206

nr 3

273-292

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ(t) = A^{-1} x(t)$, 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 $(e^{A^{-1}t})_{t≥0}$ is O(∜t), and for $((A+I)(A-I)^{-1})ⁿ$ it is O(∜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.

Ograniczanie wyników

1 Studia Mathematica

1 Besseling N.

1 Gomilko A.

1 Zwart H.

1 2011

Wyniki wyszukiwania

Growth of semigroups in discrete and continuous time