Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}+q\left(t\right)y=0(*)$$ are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}-\frac{{\mu }_{n,\alpha }}{t^{2n-\alpha }}y=0,$$ where ${\mu }_{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.

In this paper we investigate oscillatory properties of the second order half-linear equation $${\left(r\left(t\right)\Phi \left(y^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(y\right)=0,\Phi \left(s\right):={\left|s\right|}^{p-2}s.(*)$$ Using the Riccati technique, the variational method and the reciprocity principle we establish new oscillation and nonoscillation criteria for (*). We also offer alternative methods of proofs of some recent oscillation results.