Various existence results of a mild solution of the integro-differential equation
with a nonlocal initial value condition
are obtained. Here, is the generator of a -semigroup. Recall that a mild solution is a function which formally satisfies the corresponding variation-of-constants formula. The main hypotheses are some growth estimates for , , and (leading to a-priori bounds), compactness of either or of the semigroup, and either compactness of or that is Lipschitz with a sufficiently small constant. The proofs use Schauder’s or Schaefer’s fixed point theorem.