Our aim in this paper is to obtain a new oscillation criterion for equation $$y^{\text{'}\text{'}\text{'}}+p\left(t\right)y^{\text{'}}+q\left(t\right)y=0$$ with a nonnegative coefficients which extends and improves some oscillation criteria for this equation. In the special case of equation (*), namely, for equation $y^{\text{'}\text{'}\text{'}}+q\left(t\right)y=0$, our results solve the open question of $Chanturiya$.

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.