Spectral Problems Arising in the Theory of Differential Equations with Delay

In this article we present a review of results on asymptotic behavior and stability of strong solutions for functional differential equations (FDE). We also formulate several results about spectral properties (completeness and basisness) of exponential solutions of the above-mentioned equations. It is relevant to emphasize that our approach for the research of FDE is based on the spectral analysis of operator pencils that are symbols (characteristic quasi-polynomials) with operator coefficients. The article is divided into two parts. The first part is devoted to the research on FDE in a Hilbert space; the second part is devoted to the research on FDE in a finite-dimensional space.