Precise asymptotic behavior of intermediate solutions of even order nonlinear differential equation in the framework of regular variation

The aim of this paper is to show that if the even-order differential equation of Emden–Fowler type
$$ x^{(2n)}(t)+q(t)|x(t)|^\gamma\operatorname{sgn}x(t)=0,\qquad0<\gamma<1, $$
with regularly varying coefficient $q(t)$ is studied in the framework of regular variation, not only necessary and sufficient conditions for the existence of intermediate regularly varying solutions can be established, but also precise information can be acquired about the asymptotic behavior at infinity of these solutions.