Necessary and sufficient conditions for oscillation of second-order half-linear delay differential equations

Abstract

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(x^{\prime})^\gamma\bigr)^{\prime}(t)+ q(t)x^\alpha(\tau(t))=0\,.

\]

Under the assumption $\int^{\infty}\big(r(\eta)\big)^{-1/\gamma} d\eta=\infty$,

we consider the two cases when $\gamma > \alpha$ and $\gamma < \alpha$. Further, some illustrative examples showing applicability of the new results are

included, and state an open problem.