Various existence results of a mild solution of the integro-differential equation $$u'(t)=Au(t)+f\Bigl(t,u(t),\int_0^tk(t,s,u(s))ds\Bigr)\quad(0<t<b)$$ with a nonlocal initial value condition $$u(0)=g(u)+u_0$$ are obtained. Here, $A$ is the generator of a $C_0$-semigroup. Recall that a mild solution is a function $u$ which formally satisfies the corresponding variation-of-constants formula. The main hypotheses are some growth estimates for $k$, $f$, and $g$ (leading to a-priori bounds), compactness of either $f$ or of the semigroup, and either compactness of $g$ or that $g$ is Lipschitz with a sufficiently small constant. The proofs use Schauder’s or Schaefer’s fixed point theorem.