Asymptotic equivalence relations for rapidly varying solutions of sublinear differential equations of Emden–Fowler type

Abstract

We discuss sublinear differential equations of the Emden–Fowler type $x''=q(t) x^{\gamma }$ x ″ = q ( t ) x γ under the assumption that the coefficient q is a rapidly varying function. We show that all of their strongly decreasing and strongly increasing solutions are rapidly varying functions and are in the asymptotic equivalence relation with a precisely defined function determined by the coefficient q.