Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations

Abstract

This paper discusses the behavior of real-valued solutions to the equation $$y''' + p(t)y'' + q(t)y' + r(t)y^\mu = 0$$ (1.1) where p, q, r are continuous and real-valued on some half-line (t0, + ∞) and μ is the quotient of odd positive integers. Oriteria are obtained for the existence of nonoscillatory solutions, several stability theorems are proved, and the existence of oscillatory solutions is shown. Of primary concern are the two cases q(t) ≤ 0, r(t) > 0, and q(t)>0, r(t)>0. Some of the main techniques used involve comparison theorems for linear equations and results in the theory of second order nonlinear oscillations.