Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 45, 1-23; http://www.math.u-szeged.hu/ejqtde/ Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation Takaˆ si Kusano a , Jelena Manojlovi´ c b∗ a Professor Emeritus at: Hiroshima University Department of Mathematics, Faculty of Science Higashi-Hiroshima 739-8526, Japan E-mail: kusanot@zj8.so-net.ne.jp b University of Niˇ s, Faculty of Science and Mathematics Department of Mathematics Viˇ segradska 33, 18000 Niˇ s, Serbia E-mail: jelenam@ pmf.ni.ac.rs Abstract. The asymptotic behavior of solutions of the odd-order differential equation of Emden-Fowler type x (2n+1) (t)+ q(t)|x(t)| γ sgn x(t)=0, is studied in the framework of regular variation, under the assumptions that 0 <γ< 1 and q(t):[a, ∞) → (0, ∞) is regularly varying function. It is shown that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity. Keywords: odd-order differential equation, intermediate solution, regularly varying function, slowly varying function, asymptotic behavior of solutions MSC 2010: 34C11, 26A12 1 Introduction The objective of this paper is to make a detailed study of the existence and the asymptotic behavior of positive solutions of the nonlinear odd-order differential equation (A) x (2n+1) (t)+ q(t)|x(t)| γ sgn x(t)=0, where γ is a constant such that 0 <γ< 1 and q :[a, ∞) → (0, ∞) is a continuous function. Equation (A) is often referred to as sublinear differential equation in this case, while equation (A) for which γ> 1 is called superlinear differential equation. * Corresponding author. EJQTDE, 2012 No. 45, p. 1