# Article

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Keywords:
oscillation theory; conditional oscillation; half-linear differential equations
Summary:
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 $\gamma < K$.
References:
[1] Došlý, O., Řehák, P.: Half-Linear Differential Equations. Elsevier, Mathematics Studies 202, 2005. MR 2158903 | Zbl 1090.34001
[2] Schmidt, K. M.: Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane. Proc. Amer. Math. Soc. 127 (1999), 2367–2374. DOI 10.1090/S0002-9939-99-05069-8 | MR 1626474 | Zbl 0918.34039

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