Oscillation criteria for second-order delay differential equations.
The paper deals with the second-order nonlinear retarded differential equation \[ (r(t)| u'(t)|^{\alpha -1}u'(t))' +p(t)| u[\tau(t)]|^{\alpha -1}u[\tau(t)]=0,\tag{1} \] where \(\alpha\) is a positive number; \(r\in C^1(t_0,\infty)\), \(r(t)>0\), and \(R(t)=\int_{t_0}^tr^{-1/\alpha}(s)ds\to\infty\) as \(t\to\infty\); \(p\in C(t_0,\infty)\), \(p(t)>0\); \(\tau\in C^1(t_0,\infty)\), \(\tau(t)\leq t\), and \(\tau(t)\to\infty\) as \(t\to\infty\). The authors establish sufficient conditions for all solutions of (1) to be oscillatory in the case \(\alpha\geq 1\), and for \(0<\alpha <1\).
scientific article; zbMATH DE number 1978095
scientific article