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On connection between second-order delay differential equations and integrodifferential equations with delay. (English) Zbl 1195.34094

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.

MSC:

34K05 General theory of functional-differential equations
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