Half-linear differential equations: Linearization technique and its application

Abstract

Oscillatory properties of the half-linear second-order differential equation(*)(r (t) Φ (x′))′ + c (t) Φ (x) = 0, Φ (x) = | x |p - 2 x, 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. © 2007 Elsevier Inc. All rights reserved. © 2007 Elsevier B.V., All rights reserved.