, Volume 120, Issue 1-2, pp 147-163
Date: 14 Mar 2008

Conditionally oscillatory half-linear differential equations

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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.

Research supported by the grant 201/07/0145 of the Grant Agency of the Czech Republic and by the Research Project MSM0021622409 of the Ministry of Education of the Czech Government.