Conditionally oscillatory half-linear differential equations

Abstract

We consider a nonoscillatory half-linear second order differential equation $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1, $$ (*) and suppose that we know its solution h. Using this solution we construct a function d such that the equation $$ (r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0 $$ (**) is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.