Positivity of solutions of adapted generalized Riccati equation with consequences in oscillation theory
The authors present a new version of the adapted generalized Riccati equation. A simple condition is given which guarantees that any solution of this equation has to be positive in at least one point. Using this result, the authors establish oscillation criteria for the half-linear differential equation \[ \left( \left( \frac{t}{r(t)} \right)^{p-1}\Phi(x'(t)) \right)'+\frac{s(t)}{t\log^p{t}}\Phi(x(t))=0, \] where \(\Phi(x)=|x|^{p-1}\operatorname{sgn}x\) (\(p>1\)), \(\log\) denotes the natural logarithm, and \(r: [e,\infty)\to (0,\infty)\) and \(s: [e,\infty)\to\mathbb{R}\) are continuous functions such that \[ \int_e^\infty \frac{r(\tau)}{\tau} \, d\tau=\infty, \quad \int_e^\infty \frac{s(\tau)}{\tau\log^p{\tau}} \, d\tau\in\mathbb{R}. \]
scientific article; zbMATH DE number 7340780
scientific article