Integrated cosine functions

Quan Zheng

doi:10.1155/s0161171296000798

Abstract

In order to the second order Cauchy problem (CP2) : x″(t) = Ax(t), x(0) = x ∈ D(An), x″(0) = y ∈ D(Am) on a Banach space, Arendt and Kellermann recently introduced the integrated cosine function. This paper is concerned with its basic theory, which contain some properties, perturbation and approximation theorems, the relationship to analytic integrated semigroups, interpolation and extrapolation theorems.

Highlights

PERTURBATIONSWe first consider the perturbation problem of 2m-times integrated cosine functions. The following is a generalization of the Takenaka-Okazawa theorem (cf. [10, II])
It is known that a 0-times integrated cosine function consists with a cosine function, while the following relationship can be shown by the same method in [4, Th.3.5]
The basic properties of integrated cosine functions can be deduced from Prop.l.1, the properties of Laplace transforms and integrated semigroups

Summary

PERTURBATIONS

We first consider the perturbation problem of 2m-times integrated cosine functions. The following is a generalization of the Takenaka-Okazawa theorem (cf. [10, II]). The following is a generalization of the Takenaka-Okazawa theorem (cf [10, II]). (b) loo liin.\+oo/.\ < c, where Ix sup{f e-X’llBC(’-t)(t)xll,dt; x D(A"+’), Ilxll,. Combining (b]), Prop.l.1, [10, II, Th.4.2] with this yields that A + qB 6- G+(D(.A)) for Iql < i/t. In the case m 0, since loo 0 (see [9, Lemma]) and (b3) can be replaced by fg C(s)(X)ds 6D(B) for > 0, Theorem 3.2 (with m 0) is consistent with [9, Pop.l. We turn to the perturbation problem of (2rn + )-times integrated cosine fimctions. Let (A,C(t)) G+(X) and B be a closed linear operator on X satisfying (b) D(A)U C(t)(X)C D(B) for 0 and R(B) C D(A).

APPROXIMATIONS

RELATIONSttIP TO INTEGRATED SEMIGROUPS

INTEIIPOIATION AND EXTRAPOLATION