Non-monotone positive solutions of second-order linear differential equations: Existence, nonexistence and criteria

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.