Half-linear differential equations

Abstract

Oscillatory properties of the half-linear second-order differential equation (r (t) phi (x'))' + c(t) phi (x) = 0, phi(x) = vertical bar x vertical bar(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. Dosly, 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. (c) 2007 Elsevier Inc. All rights reserved.