LOCAL K-CONVOLUTED C-SEMIGROUPS AND COMPLETE SECOND ORDER ABSTRACT CAUCHY PROBLEMS

Abstract

Let C  be a bounded linear operator on a Banach space X over the
field F(=R or C), and K  a locally integrable function from  [0, T_0) into F for some 0 <T_0≦∞.
Under some suitable assumptions, we deduce some relationship between the generation of
a local (or an exponentially bounded) K-convoluted
„
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 wellposedness
of a complete second-order abstract Cauchy problem ACP(A,B, f, x, y): w''(t) =
Aw'(t) + Bw(t) + f(t) for a.e. t in (0, T_0) with w(0) = x and w'(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 α-times integrated C-cosine function on X for which A may
not be bounded; (iv) A is a subgenerator (resp., the generator) of a local α-times integrated
C-semigroup on X for which B may not be bounded.