The authors study the linear equation
in a Banach space X. They prove that there exists a reasonable resolvent operator if and only if the autonomous equation (where
is well-posed (i.e., it has a unique solution that depends continuously on
. Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that
for all Re
that is a result of Hille-Yosida type. The authors also give an example showing that this condition can be satisfied although A does not generate a semigroup.