Almost periodic at infinity solutions to integro-differential equations with non-invertible operator at derivative

In the paper we consider an integro-differential equation with a non-invertible operator at a derivative in the space of uniformly continuous bounded functions. The integral part of the operator is a convolution of an operator-valued compactly supported Borel measure and a bounded continuous vector function. We obtain sufficient conditions (spectral conditions) of almost periodicity at infinity for bounded solutions of this equation.
The above results are based on the proven statement that if the right-hand side of the equation in question belongs to $C_0(\mathbb{J},X)$, which is the space of functions tending to zero at infinity, then the Beurling spectrum of each weak solution is contained in the singular set of a characteristic equation. In particular, for the equations of the form $ \mu \ast x = \psi, $ where the function $\psi\in C_{0} (\mathbb{J}, X) $ and the support $\mathrm{supp} \mu$ of a scalar measure $ \mu $ are compact, we establish that each classical solution is almost periodic at infinity. We show that if the singular set of the characteristic function of the considered equation has no accumulation points in $\mathbb{R}$, then each weak solution is almost periodic at infinity. We study the structure of bounded solutions in terms of slowly varying at infinity functions.
We provide applications of our results to nonlinear integro-differential equations. We establish that when the right hand side of a nonlinear integro-differential equation is a decaying at infinity mapping and a singular set of the characteristic function has no finite accumulation points on $\mathbb{R}$, a bounded solution of this equation is almost periodic at infinity.
The main results of the paper are obtained by means of the methods of abstract harmonic analysis. The spectral theory of Banach modules is essentially employed.