### 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.

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