ACTA issues

An elementary method for the study of Meissner's equation and its application to proving the Oscillation Theorem

László Hatvani

Acta Sci. Math. (Szeged) 79:1-2(2013), 87-105

Abstract. A geometric method is presented to describe the dynamics of the linear second order differential equation with step function coefficient $ x'' + a^2(t) x =0, a(t):= a_k \hbox{ if } t_{k-1} \le t< t_k (k\in{\msbm N}), $ where $a_k > 0$, $t_0 =0$, $t_k\nearrow\infty $ as $k\to\infty $. We rewrite this equation into a discrete dynamical system on the plane. The method is applied to the Meissner equation $ x'' + \lambda ^2 Q (t) x = 0, $ where $\lambda > 0$ is a real parameter; $Q$ is a $2L$-periodic real function which is $1$ on $[0,2)$ and $a^2$ on $[2,2L)$; $a$, $L$ ($0< a\not=1$, $L>1$) are given constants. We give a complete elementary proof for the classical oscillation theorem on the $2L$-periodic and $4L$-periodic solutions of this equation not using even Floquet's theorem from the theory of differential equations.

AMS Subject Classification (1991): 34B24, 34B30, 34C10; 74H45

Keyword(s): second order linear differential equations, non-autonomous equations, equations with periodic coefficients, Hill's equation, step function coefficients, eigenvalues of first and second type, oscillation

Received March 4, 2013, and in revised form April 23, 2013. (Registered under 17/2013.)