Second-order half-linear delay differential equations: Oscillation tests

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form $ \bigl(r(y^{\prime})^\gamma\bigr)^{\prime}(t)+ q(t)y^\alpha(\tau(t))=0\,.$ We study this equation under the assumption $\int^{\infty}\big(r(\eta)\big)^{-1/\gamma} d\eta=\infty$ and consider two cases when $\gamma > \alpha$ and $\gamma < \alpha$. We provide examples, illustrating the results and state an open problem.

Keywords:

Oscillation, nonoscillation, delay half-linear, Lebesgue's dominated convergence theorem,

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