Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria - Mervan Pašić; Satoshi Tanaka (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
We study non-monotone positive solutions of the second-order linear differential equations: $(p(t)x')' + q(t) x = e(t)$, with positive $p(t)$ and $q(t)$. For the first time, some criteria as well as the existence and nonexistence of non-monotone positive solutions are proved in the framework of some properties of solutions $θ (t)$ of the corresponding integrable linear equation: $(p(t)θ')'=e(t)$. The main results are illustrated by many examples dealing with equations which allow exact non-monotone positive solutions not necessarily periodic. Finally, we pose some open questions.