Oscillation and nonoscillation of solutions to even order self-adjoint differential equations

Abstract

We establish oscillation and nonoscilation criteria for the linear differential equation (-1)n (tαy(n)) (n) - γn, α/t2n-α y = q(t)y, α ∉ {1, 3, ..., 2n - 1}, where γn, α = 1/4n n∏k=1 (2k - 1 - α)2 and q is a real-valued continuous function. It is proved, using these criteria, that the equation (-1)n (tαy(n)) (n) - (γn,α/t2n-α + γ/t2n-α lg2t) y = 0 is nonoscillatory if and only if γ ≤ ~γn, α : = 1/4n n∏k=1 (2k - 1 - α)2 n∑k=1 1/(2k - 1 - α)2.