On the qualitative behavior of solutions of third order differential equations (Q1916748)

The paper presents an important contribution to the qualitative theory of third-order differential equations of the form \[ \Biggl[ {1\over p(t)} \Biggl({1\over r(t)} x'\Biggr)'\Biggr]'+ q(t) x= 0,\tag \(E_1\) \] where \(p\), \(r\), \(q\) are twice differentiable functions with \(p(t)\neq 0\), \(r(t)\neq 0\), \(q(t)\neq 0\) for \(t\in [0, \infty)\). Some criteria on the structure of the space of solutions of equation \((E_1)\) are derived and some conditions concerning oscillation and nonoscillation of solutions of \((E_1)\) and other results are given. All the conditions are presented as integral criteria and involve only the integral behavior of the functions \(p\), \(r\), \(q\) on \([0, \infty)\). The obtained results generalize many results of other papers.