Methods in half-linear asymptotic theory

Abstract

We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation $$ (r(t)|y'|^{\alpha-1}\hbox{sgn} y')'=p(t)|y|^{\alpha-1}\hbox{sgn} y, $$ where r(t) and p(t) are positive continuous functions on $[a,\infty)$, $\alpha\in(1,\infty)$. The aim of this article is twofold. On the one hand, we show applications of a wide variety of tools, like the Karamata theory of regular variation, the de Haan theory, the Riccati technique, comparison theorems, the reciprocity principle, a certain transformation of dependent variable, and principal solutions. On the other hand, we solve open problems posed in the literature and generalize existing results. Most of our observations are new also in the linear case.

Keywords

• Half-linear differential equation
• nonoscillatory solution
• regular variation
• asymptotic formula