Complete asymptotic analysis of positive solutions of odd-order nonlinear differential equation

We study the asymptotic behavior of solutions of the odd-order differential equation of Emden–Fowler type $$ {x^{{\left( {2n+1} \right)}}}(t)=q(t){{\left| {x(t)} \right|}^{\gamma }}\operatorname{sgn}x(t) $$ in the framework of regular variation under the assumptions that 0 < γ < 1 and q(t) : a, ∞) → (0, ∞) is regularly varying function. We show that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity.