On nonoscillatory solutions of third order nonlinear differential equations. (English) Zbl 1016.34036

The authors consider the third-order nonlinear differential equation \[ x'''+q(t)x'+r(t)f(x)=0,\tag{1} \] where \(q,r\in C([t_0,\infty), \mathbb{R})\), \(f\in C(\mathbb{R},\mathbb{R})\), \(r(t)>0\) and the nonlinearity \(f\) satisfies the sign condition \(f(u)u>0\) for \(u\neq 0\). The nonoscillatory solutions to equation (1) are classified into several subclasses according to the asymptotic behavior. Relations between the existence of all these types of nonoscillatory solutions and the role of the nonlinearity, especially with regard to its growth, are examined. In contrast to many of the papers concerning the asymptotic properties of equation (1), the authors do not suppose the nonoscillation of the second-order equation \(h''+q(t)h=0\).


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations