Half-linear differential equations: Linearization technique and its application

Oscillatory properties of the half-linear second-order differential equationequation(*)(r(t)Φ(x′))′+c(t)Φ(x)=0,Φ(x)=|x|p−2x,p>1, are investigated, where (*) is viewed as a perturbation of an equation of the same form, and these properties are compared with the oscillatory behavior of a certain associated linear second-order differential equation. As a corollary, a conjecture posed by the first author in the paper [O. Došlý, Perturbations of the half-linear Euler–Weber differential equations, J. Math. Anal. Appl. 323 (2006) 426–440], is proved and some new research problems are formulated.