Local C-semigroups and complete second order abstract Cauchy problems

Abstract

Let C : X ! X be an injective bounded linear operator on a Banach space X over the field
F(=R or C) and 0 < T0 1. Under some suitable assumptions, we deduce some relationship between the
generation of a local (or an exponentially bounded)

C 0
0 C

-semigroup on X×X with subgenerator (resp.,
the generator)

0 I
B A

and one of the following cases: (i) the well-posedness of a complete second-order
abstract Cauchy problem ACP(A,B, f, x, y): w00(t) = Aw0(t)+Bw(t)+f(t) for a.e. t 2 (0, T0) with w(0) = x
and w0(0) = y; (ii) a Miyadera-Feller-Phillips-Hille-Yosida type condition; (iii) B is a subgenerator (resp., the
generator) of a locally Lipschitz continuous local C-cosine function on X for which A may not be bounded;
(iv) A is a subgenerator (resp., the generator) of a local C-semigroup on X for which B may not be bounded.