On the Zeroes and the Critical Points of a Solution of a Second Order Half-Linear Differential Equation

Abstract

This paper presents two methods 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 $(p(x)\mathrm{\Phi }(y\text{'}))\text{'}+q(x)\mathrm{\Phi }(y)=0$, with $p(x)$ and $q(x)$ piecewise continuous and $p(x)>0$, $\mathrm{\Phi }(t)=|t{|}^{r-2}t$ and $r$ being real such that $r>1$. It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.