On nonoscillatory solutions of third order nonlinear differential equations (Q2761977)

The authors consider the third-order nonlinear differential equation NEWLINE\[NEWLINEx'''+q(t)x'+r(t)f(x)=0,\tag{1}NEWLINE\]NEWLINE 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 \(h''+q(t)h=0\).