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

• Main Authors: Mervan Pašić, Satoshi Tanaka
• Format: Article
• Language: English
• Published: University of Szeged 2016-10-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: non-monotonic behaviour; positive solutions; existence; nonexistence; criteria
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4940

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 $\theta (t)$ of the corresponding integrable linear equation: $(p(t)\theta')'=e(t)$. 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.