Solvability of an autonomous differential equation with aftereffect on the negative semi-axis

Abstract

We consider a linear autonomous homogeneous functional differential equation on the real negative semi-axis. We prove that if solutions belong to the special space of functions with integral limitations, then the space of solutions is finite-dimensional and its basis is formed by the solutions of the form (t m exp(pt)) generated by the roots of the characteristic equation. In contrast to the spaces used earlier, the pointwise estimation of solutions is replaced with the integral one. We adduce examples of differential equations with aftereffect and give the effective description of the space of solutions for these equations.