Spectral analysis and representation of solutions of integro-differential equations with fractional exponential kernels

The aim of the present paper is to study the asymptotic behavior of solutions of integro-differential equations based on spectral analysis of their symbols. To this end, we obtain representations of strong solutions of these equations in the form of a sum of terms corresponding to the real and nonreal parts of the spectrum of operator functions that are the symbols of these equations. The resulting representations are new for this class of integro-differential equations.