Ein Oszillationskriterium. (An oscillation criterion) (Q1100336)

For the equation \(x'(t)=-\int^{\infty}_{\tau =0}x(t-\tau)dr(t,\tau)\) \((-\infty <A\leq t<\infty)\), the following oscillation theorem is proved: If there is a continuous, positive, bounded function \(\psi\) on (0,\(\infty)\) which satisfies \[ \underline{\lim}_{t\to \infty}(\min_{\alpha \in W_ t}\frac{1}{\psi (\alpha)}\int^{t}_{t- \alpha}\int^{\infty}_{\tau =0}\psi (\tau)dr(s,\tau)ds)>(1/e) \] and \[ \underline{\lim}_{t\to \infty}(\int^{t}_{t-\delta (t)}\int^{\infty}_{\tau =0}\psi (\tau)dr(s,\tau)ds)>0, \] then every solution of the equation oscillates. \(W_ t\) denotes the set of the points of growth of the function r(t,\(\cdot)\). Using a special function \(\psi\) we get a generalization of a criterion of \textit{B. R. Hunt} and \textit{J. A. Yorke} [J. Differ. Equations 53, 139-145 (1984; Zbl 0571.34057)].