On α-times integrated C-cosine functions and abstract Cauchy problem I

In this paper we first investigate some basic properties concerning nondegenerate α-times integrated C-cosine functions on a Banach space X, and then characterize their generator A in terms of the unique existence of strong solutions of the following abstract Cauchy problem: ACP(f, x, y) u (t) = Au(t) + f (t) for t > 0, u(0) = x, u (0) = y.  2005 Elsevier Inc. All rights reserved