Oscillation and nonoscillation of second order linear differential equations (Q1363603)

The paper deals with the oscillatory and nonoscillatory properties of the second-order linear differential equations \[ u''+ p(t)u=0,\tag{1} \] where \(p\in C[0,\infty)\) and \(p(t)\geq 0\). The following results are derived. Let \(\alpha_0= 3-2\sqrt 2\) and \(\mathbb{N}\) be the set of natural numbers. Theorem 1. If there exists a \(t_0>0\) such that \(\int^{2^{n+1}t_0}_{2^nt_0} p(t)dt\leq{\alpha_0\over 2^{n+1}t_0}\), for every \(n\in\mathbb{N}\), then equation (1) is nonoscillatory. Theorem 2. If there exist \(t_0>0\) and \(\alpha>\alpha_0\) such that \(\int^{2^{n+1}t_0}_{2^nt_0} p(t)dt\geq {\alpha\over 2^nt_0}\), for every \(n\in\mathbb{N}\), then equation (1) is oscillatory. By an example, the author illustrates that the number \(\alpha_0\) is the best constant for the validity of Theorem 1 and Theorem 2.