Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

The lower bounds of the spacings $b-a$ or $a'-a$ of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form $$y''' + q(t)y'+p(t)y=0\tag*{$(*)$}$$ are derived under very general assumptions on $p$ and $q$. These results are then used to show that $t_{n+1}-t_n\to \infty$ or $t_{n+2}-t_n\to \infty$ as $n\to \infty$ under suitable assumptions on $p$ and $q$, where $\langle t_n\rangle$ is a sequence of zeros of an oscillatory solution of $(*)$. The Opial-type inequalities are used to derive lower bounds of the spacings $d-a$ or $b-d$ for a solution $y(t)$ of $(*)$ with $y(a) = 0 = y'(a)$, $y'(c) = 0$ and $y''(d) = 0$ where $d\in (a, c)$ or $y'(c) = 0$, $y(b) = 0 = y'(b)$ and $y''(d) = 0$ where $d\in (c, b)$.