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

Abstract

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