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

作者：Pasic Mervan; Tanaka Satoshi^*

来源：Electronic Journal of Qualitative Theory of Differential Equations, 2016, (93): 1-25.

DOI：10.14232/ejqtde.2016.1.93

• non-monotonic behaviour
• positive solutions
• existence
• nonexistence
• 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 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.

• 出版日期2016