Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

Abstract

Abstract We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation ( p ⁢ ( t ) ⁢ | x ′ | α ⁢ sgn ⁡ x ′ ) ′ + q ⁢ ( t ) ⁢ | x | α ⁢ sgn ⁡ x = 0 , (p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0, where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions ∫ a ∞ d ⁢ s p ⁢ ( s ) 1 / α = ∞ or ∫ a ∞ d ⁢ s p ⁢ ( s ) 1 / α < ∞ . \int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty. The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.