On connection between second-order delay differential equations and integrodifferential equations with delay
The second order delay differential equation \[ x''(t) + a(t)x'(g(t))+b(t)x(h(t)) = 0 \] appears in many applications. One of the methods used in the study of this equation is to transform it to a first order differential or integrodifferential equation with delays. As a result of such a transformation, the initial value problem of \[ x'(t) + \int^t_{t_0}K(t, s)x(h(s))ds = f(t),\quad t\geq t_0,\quad x\in\mathbb R^n,\;x(t) = \varphi(t),\;t < t_0 \] arises. In this paper, the existence and uniqueness of a solution to the above initial value problem is studied and a representation formula for the solution is provided. This work is a continuation of previous work involving the first author.
scientific article; zbMATH DE number 5690835