Oscillation of solutions to second-order nonlinear differential equations of generalized Euler type

Abstract

We are concerned with the oscillatory behavior of the solutions of a generalized Euler differential equation where the nonlinearities satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. Some implicit necessary and sufficient conditions and some explicit sufficient conditions are given for all nontrivial solutions of this equation to be oscillatory or nonoscillatory. Also, it is proved that solutions of the equation are all oscillatory or all nonoscillatory and cannot be both.

Keywords

• Oscillation
• nonlinear differential equations
• Lienard system