Oscillation criteria of Kneser-Hille type for second-order differential equations with nonlinear perturbed terms

This paper is concerned with the oscillation problem for nonlinear differential equations of Euler type, which are denoted by (E-n) with n = 1, 2, 3,... Equation (E-n) consists of a linear main term and a nonlinear perturbed term. If the nonlinear perturbation vanishes, then all nontrivial solutions of (E-n) are nonoscillatory. A pair of sufficient and necessary conditions on the perturbed term for all nonlinear solutions of (E-n) to be oscillatory is given. It is also proved that all solutions of (E-n) tend to zero.