Necessary and sufficient conditions for oscillation to 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 $$\begin{aligned} (r(x^{\prime })^\gamma )^{\prime }(t)+ q(t)x^\alpha (\tau (t))=0. \end{aligned}$$ Under the assumption $$\int ^{\infty }(r(\eta ))^{-1/\gamma } \mathrm{d}\eta =\infty $$ , 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.