Solutions of half-linear differential equations in the classes Gamma and Pi

Abstract

We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $$ (r(t)|y'|^ {\alpha-1}\text{sgn}\, y')'=p(t)|y|^{\alpha-1}\text{sgn}\, y , $$ where $\alpha\in(1,\infty)$ and $r,p$ are positive continuous functions on $[a,\infty)$, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class $\Gamma$ resp. $\Gamma_-$ under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case $\alpha=2$.