A note on a theorem of Pucci and Serrin (Q1336326)

This note is connected with the paper of \textit{P. Pucci} and \textit{J. Serrin} ``Precise damping conditions for global asymptotic stability for nonlinear second order systems. II'' published in the same issue of the journal [ibid. 113, No. 2, 505-534 (1994; Zbl 0814.34033), see the preceding review)]. Pucci and Serrin required a condition of the form \[ \liminf_{t\to\infty} \int^ t_ T \delta(s) k(s)\exp\left(- \int^ t_ s k(r) dr\right) ds< \infty \] with appropriate nonnegative continuous scalar functions \(\delta\), \(k\). The author proves that this condition can be replaced by \[ \liminf_{t\to\infty} \int^ t_ T \delta(s) k(s) ds\Bigl/ \int^ t_ T k(s) ds< \infty, \] which is easier to use in applications.