An oscillation criterion for an \(n\)th order differential equation with damped term (Q909085)

Consider the equation (1) \(x^{(n)}(t)+p(t)x^{(n-1)}(t)+q(t)x(t)=0\), n even, where \(p,q: [t_ 0,\infty)\to [0,\infty)\) are continuous and q(t) is not identically zero on any ray of the form \([t^*,\infty)\) for some \(t^*\geq t_ 0\). We define the following conditions: (I) \(\lim_{t\to \infty}\int^{t}_{\bar t}\exp (-\int^{s}_{\bar t}p(\tau)d\tau)ds=\infty\) for every \(\bar t\geq t_ 0\). (II) \(\limsup_{t\to \infty}t^{-\alpha}\int^{t}_{t_ 0}(t- s)^{\alpha}s^{\beta}q(s)ds=\infty\) and \[ \limsup_{t\to \infty}t^{-\alpha}\int^{t}_{t_ 0}[(t-s)p(s)s+\alpha s-\beta (t- s)]^ 2(t-s)^{\alpha -2}s^{\beta -n}ds<\infty \] for some \(\alpha\in (1,\infty)\) and \(\beta\in [0,n-1).\) \[ (III)\quad \lim_{t\to \infty}\sup t^{-\alpha}\int^{t}_{t_ 0}(t-s)^{\alpha -2}s^{\beta}[(t-s)^ 2q(s)-2^{2n-5}(n-2)!\{(t- s)p(s)s+\alpha s-\beta (t-s)\}^ 2]ds=\infty \] for some \(\alpha\in (1,\infty)\) and \(\beta\in [0,n-1]\). The authors state and prove the following theorems: Theorem 1. Conditions (I) and (II) imply that every solution of equation (1) is oscillatory. Theorem 2. Conditions (I) and (III) imply that every solution of equation (1) is oscillatory.