Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients

• Main Authors: László Székely, László Hatvani
• Format: Article
• Language: English
• Published: University of Szeged 2011-06-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: asymptotic stability; armellini-tonelli-sansone theorem; step function coefficients; half-linear differential equation
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=784

We give a sufficient condition guaranteeing asymptotic stability with respect to $x$ for the zero solution of the half-linear differential equation \[x''|x'|^{n-1} + q(t)|x|^{n-1}x=0, \qquad 1\le n \in \mathbb{R},\] with step function coefficient $q$. The geometric method of the proof can be applied also to two dimensional systems of linear non-autonomous difference equations. The application gives a new simple proof for a sharpened version of \'A. Elbert's asymptotic stability theorems for such difference equations and linear second order differential equations with step function coefficients.