Half-linear oscillation criteria: Perturbation in term involving derivative

We consider the nonoscillatory half-linear differential equation (r(t)Φ(x′))′+c(t)Φ(x)=0,Φ(x)≔|x|p−2x,p>1, and we study oscillatory properties of its perturbation equation(∗ )[(r(t)+r̃(t))Φ(x′)]′+(c(t)+c̃(t))Φ(x)=0. We use the Riccati technique and relationship between (∗) and a certain associated linear equation. The results are applied to the perturbed Euler-type equation.