Two-Parametric Conditionally Oscillatory Half-Linear Differential Equations

Abstract

We study perturbations of the nonoscillatory half-linear differential equation (r(t)Φ(x'))'+c(t)Φ(x)=0, Φ(x):=|x|p-2x, p>1. We find explicit formulas for the functions r̂, ĉ such that the equation [(r(t)+λr̂(t))Φ(x')]'+[c(t)+μĉ(t)]Φ(x)=0 is conditionally oscillatory, that is, there exists a constant γ such that the previous equation is oscillatory if μ-λ>γ and nonoscillatory if μ-λ<γ. The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.