The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation

Abstract

This paper reuses an idea first devised by Kwong to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second order half-linear differential equation $(p(x)\mathrm{\Phi }(y\mathrm{\text{'}}))\mathrm{\text{'}}+q(x)\mathrm{\Phi }(y)=\mathrm{0}$, with $p(x),q(x)>\mathrm{0}$, $\mathrm{\Phi }(t)=|t{|}^{r-\mathrm{2}}t$, and $r$ real such that $r>\mathrm{1}$. It also compares it with other methods developed by the authors.