Oscillation and non-oscillation theorems for superlinear Emden–Fowler equations of the fourth order

摘要

We study the oscillatory behavior of solutions of the fourth-order Emden–Fowler equation: (E) y~((iv)) + q(t) y|~α sgny = 0, where α > 1 and q(t) is a positive continuous function on [t_0, ∞), t_0 > 0. Our main results Theorem 2–if (q(t)t~((α3+5)/2))' ≥ 0, then equation (E) has oscillatory solutions; Theorem 3 – if lim_(t→∞)q(t)t~(4+λ)(α-1) = 0,λ > 0, then every solution y(t) of equation (E) is either non-oscillatory or satisfies lim sup_(t→∞)t~(-μ+i)|y~((i))(t)| = ∞ for μ < λ and i = 0,1,2,3,4. These results complement those given by Kura for equation (E) when q(t) < 0 and provide analogues to the results of the second-order equation, y" + q(t)|y|~α sgny = 0, α > 1.