# Conditional oscillation of half-linear differential equations with periodic coefficients

## Petr Hasil

Address: Department of Mathematics and Statistics, Masaryk University Janáčkovo nám. 2a, 602 00 Brno, Czech Republic

E-mail: hasil@math.muni.cz

Abstract: We show that the half-linear differential equation
\[ \big [r(t)\Phi (x^{\prime })\big ]^{\prime } + \frac{s(t)}{t^p} \Phi (x) = 0 \ast \]
with $\alpha $-periodic positive functions $r, s$ is conditionally oscillatory, i.e., there exists a constant $K>0$ such that () with $\frac{\gamma s(t)}{t^p}$ instead of $\frac{s(t)}{t^p}$ is oscillatory for $\gamma > K$ and nonoscillatory for $\gamma < K$.

AMSclassification: Primary: 34C10.

Keywords: oscillation theory, conditional oscillation, half-linear differential equations