We consider the integro-differential operator equations having the form
$$\sum _{i=1}^n{\alpha }_i{u}^{(i)}(t)-\alpha Au(t)+\beta \mu \ast u(t)=f(t),t\in \mathbb{R},$$
where the free term $f$ belongs to a closed subspace $M$ of $L^{\mathrm{\infty }}(\mathbb{R},X)$, $A$ is the generator of a $C_0$-semigroup of operators defined on a Banach space $X$, $\mu$ is a bounded Borel measure on $\mathbb{R}$ and $\alpha ,\beta ,{\alpha }_i\in \mathbb{C}$, $i=1,2,\dots ,n$. Certain conditions will be imposed to guarantee the existence of solutions in the class $M$.