Laplace transform and asymptotic behaviour of strongly continuous semigroups

ABSTRACT. In this note we present some results concerning the asymptotic behavior of the solutions of an abstract Cauchy problem it(t) = Au(t), u(0) = x where A is the generator of a strongly continuous emigroup on a Banach space. 0. Introduction. Let E be a Banach space and let A be a linear operator on E with dense domain D(A). If the resolvent set O(A) is nonempty, then the Cauchy problem (*) fi(t) = Au(t), u(0) = x has a unique solution u: [0,.o)- • D(A) in CI([0,*o),E) for every x C D(A) if and onl