Oscillatory property for second order linear delay differential equations (Q580572)

This interesting paper deals with the equation (1) \(x''(t)+a(t)x(g(t))=0,\) where \(a\in C[0,\infty)\to [0,\infty)\), a(t)\(\not\equiv 0\) on \([t_ 0,\infty)\) \((t_ 0\geq 0)\); \(g\in C[0,\infty)\to [0,\infty)\); \(0\leq g(t)\leq t\), \(t\geq 0\), \(\lim_{t\to \infty}g(t)=\infty\). The function sequence \(\{\alpha_ n(t)\}\) for \(n=1,2,..\). and \(t\geq t_ 0\), where \(\alpha_ 0(t)=\epsilon \int^{\infty}_{t}\frac{g(s)}{s}a(s)ds,\alpha_ n(t)=\int^{\infty}_{t}\alpha^ 2_{n-1}(s)ds+\alpha_ 0(t),\) \(n=1,2,..\). and \(0<\epsilon <1\) is introduced here. Sufficient conditions for (1) to be oscillatory are formulated by the functions \(\alpha_ n(t)\). The main result of the paper extends the well-known oscillation criteria of Hille, Kneser and Opial in the ordinary differential equation case and Erbe in the delay differential equation case.