Existence of Oscillatory Solutions of Singular Nonlinear Differential Equations

Abstract

Asymptotic properties of solutions of the singular differential equation ${\left(p\right(t\left)u^{\prime }\right(t\left)\right)}^{\prime }=p\left(t\right)f\left(u\right(t\left)\right)$ are described. Here, $f$ is Lipschitz continuous on $ℝ$ and has at least two zeros 0 and . The function p is continuous on $[0,\infty )$ and has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Further conditions for f and p under which the equation has oscillatory solutions converging to 0 are given.