Oscillation criteria for second-order retarded differential equations. (English) Zbl 0902.34061

The authors investigate oscillatory properties of second-order quasilinear equations \[ [r(t)| u'(t)|^{\alpha-1}u'(t)+p(t)| u(\tau(t))|^{\beta-1}u(\tau(t))]=0 \tag{*} \] with \(\alpha,\beta>0\), \(r(t)>0\), \(p(t)\geq 0\), \(\tau(t)\leq t\) and \(\lim_{t\to \infty}\tau(t)=\infty\). The results deal mostly with the case \(\alpha=\beta\) and extend some earlier criteria for linear equations \(u''+p(\tau(t))=0\) given by L. Erbe [Canadian Math. Bull. 16, 49-56 (1973; Zbl 0272.34095)] and J. Ohriska [Czech. Math. J. 34, 107-112 (1984; Zbl 0543.34054)]. A typical result is the following oscillation criterion:
Equation (*) with \(\alpha=\beta\) and \(r(t)\equiv 1\) is oscillatory provided one of the following conditions holds: \[ \lim_{t\to\infty}t^{\alpha}\int_t^\infty p(s) \left({\tau(s)\over s}\right)^{\alpha} ds>1,\quad\text{or}\quad \limsup_{t\to \infty}t^{\alpha}\int_{\gamma(t)}^\infty p(s) ds>1, \] with \(\gamma(t)=\sup\{s: \tau(s)\leq t\}\).
Reviewer: O.Došlý (Brno)


34K11 Oscillation theory of functional-differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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