Oscillation and nonoscillation of higher order self-adjoint differential equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (*) (-1)(n)(t(alpha) y((n))) ((n)) + q(t)y = 0 are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)(n)(t(alpha) y((n)))((n)) -mun,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.