A new application method for nonoscillation criteria of Hille-Wintner type

The present paper deals with nonoscillation problem for the Sturm–Liouville half-linear differential equation

$$begin{aligned} big (r(t)phi _p(x')big )' + c(t)phi _p(x) = 0, end{aligned}$$

where r, (c!:[a,infty ) rightarrow mathbb {R}) are continuous functions, (r(t) > 0) for (t ge a), and (phi _p(z) = |z|^{p-2}z) with (p > 1). The purpose of this paper is to show that it is possible to broaden the application range of Hille-Wintner type nonoscillation criteria. To this end, we derive a comparison theorem by means of Riccati’s technique. Our result is new even in the linear case that (p = 2). By the obtained result, we can compare two differential equations having a different power p of the above-mentioned type. To illustrate our comparison theorem, we present two examples of which all non-trivial solutions of the Sturm-Liouville linear differential equation are nonoscillatory even if (int _a^t!frac{1}{r(s)}dsint _t^infty !!c(s)ds) or (int _t^infty !!frac{1}{r(s)}dsint _a^t!c(s)ds) is less than the lower bound (-3/4).