Asymptotic behavior of the solutions of a third-order nonlinear differential equation (Q1707071)

Consider the scalar differential equation \[ (x''+ p(t) x)'+ p(t) x'+ q(t)\,f(x)= 0,\tag{\(*\)} \] where \(p\), \(q\), \(f\) are continuous, \(q(t)>0\) and \(xf(x)>0\) for \(x\neq 0\). The authors study solutions of \((*)\) that are continuable and nontrivial. They derive conditions under which these solutions are oscillatory or nonoscillatory. Some necessary and sufficient relationships between the square integrability of the first and the second derivatives are also presented.