Asymptotic behavior of solutions of certain functional-differential equations (Q2386976)

The author studies the asymptotic behavior of solutions of the differential equation \[ \dot x= a(t) x(t-\omega), \quad t\in\mathbb{R}_+, \qquad x(\xi)= 0, \quad \xi< 0,\tag \(*\) \] where \(\omega> 0\), \(a: \mathbb{R}_+\to \mathbb{C}\) and \(a(t+\omega)= Ma(t)\) for some \(M\in \mathbb{C}\). The study of the asymptotic behavior of solutions of \((*)\) is reduced to the study of the asymptotic behavior of \(X(t)\), where \(X(t)\) is the solution of \[ \dot X= a(t)X (t-\omega),\quad t\in\mathbb{R}_+, \;X(\xi)= 0, \;\xi< 0; \;X(0)= 1. \]