Nonrectifiable oscillatory solutions of second order linear differential equations

Takanao Kanemitsu and Satoshi Tanaka

Address: Department of Applied Mathematics, Faculty of Science, Okayama University of Science, Ridaichou 1–1, Okayama 700–0005, Japan

E-mail: tanaka@xmath.ous.ac.jp

Abstract: The second order linear differential equation \begin{equation*} (p(x)y^{\prime })^{\prime }+q(x)y=0\,, \quad x \in (0,x_0] \end{equation*} is considered, where $p$, $q \in C^1(0,x_0]$, $p(x)>0$, $q(x)>0$ for $x \in (0,x_0]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.

AMSclassification: primary 34C10.

Keywords: oscillatory, nonrectifiable, second order linear differential equation.

DOI: 10.5817/AM2017-4-193