Oscillation Criteria for Second Order Nonlinear Differential Equations of Euler Type

The purpose of this paper is to solve the oscillation problem for the nonlinear Euler differential equation t2x + g x = 0 and the extended equation x + a t g x = 0. Here g x satisfies the sign condition xg x > 0 if x = 0, but is not assumed to be monotone. We give necessary and sufficient conditions for all nontrivial solutions to be oscillatory. To this end, we use phase plane analysis of the Li´enard system and the oscillation result on the Riemann–Weber version of the linear Euler differential equation t2y + 1/4 + δ/ log t 2 y = 0. Our results are a negative answer to a conjecture which was given by Wong. Finally, we illustrate our results by two examples.