An example in the theory of stability of equations with delay (Q1086422)

It is well known that the solutions of the ordinary differential equation \(\sum^{n}_{k=0}a_ ky^{(k)}(t)=0\) are (Lyapunov) exponentially stable if all roots of the characteristic polynomial \(\sum^{n}_{k=0}a_ k\lambda^ k=p_ n(\lambda)\) lie in the left half of the complex plane. A similar statement is true for a large class of differential equations with a delay. It is true for equations of the type \(\sum^{n}_{k=0}\int^{\infty}_{0}y^{(k)}(t-s)d\mu_ k=0,\) provided the measures \(\mu_ k\) do not contain singular (that is discrete point) measures, or such singular measures are sufficiently small. The author offers a counterexample to such equations with an exponentially growing solution.