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This paper is concerned with
characterizations of those linear, closed, but not necessarily densely defined operators
A on a Banach space E with nonempty resolvent set for which the abstract Cauchy
problem u′(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for
every initial value x ∈ D(An).
Investigating these operators we are led to the class of “integrated semigroups”.
Among others, this class contains the classes of strongly continuous semigroups and
cosine families and the class of exponentially bounded distribution semigroups.
The given characterizations of the generators of these integrated semigroups
unify and generalize the classical characterizations of generators of strongly
continuous semigroups, cosine families or exponentially bounded distribution
semigroups.
We indicate how integrated semigroups can be used studying second
order Cauchy problems u′′(t) − A1u′(t) − A2u(t) = 0, operator valued
equations U′(t) = A1U(t) + U(t)A2 and nonautonomous equations
u′(t) = A(t)u(t).
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