Positive solutions of fourth order Emden–Fowler type differential equations in the framework of regular variation Takaši Kusano, Jelena V. Manojlovic ´ ⇑ Hiroshima University, Department of Mathematics, Faculty of Science, Higashi-Hiroshima 739-8526, Japan University of Niš, Faculty of Science and Mathematics, Department of Mathematics and Computer Science, Višegradska 33, 18000 Niš, Serbia article info Keywords: Fourth order Nonlinear differential equations Positive solutions Intermediate solutions Regularly varying functions Asymptotic behavior abstract The fourth order nonlinear differential equations x ð4Þ þ qðtÞjxj c sgn x ¼ 0; 0 < c < 1; ðAÞ with regularly varying coefficient q(t) are studied in the framework of regular variation. It is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of (A) and their accurate asymptotic behavior at infinity. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction This paper is concerned with the fourth order Emden–Fowler type differential equations x ð4Þ þ qðtÞjxj c sgn x ¼ 0; ðAÞ where 0 < c < 1 and q(t) is a positive continuous function on [a, 1), a > 0. Eq. (A) are called sublinear or superlinear according as c < 1 or c > 1. A solution x(t) of (A) existing in an infinite interval of the form [T x , 1) is said to be proper if supfjxðtÞj : t P T g > 0 for any T P T x : A proper solution is called oscillatory if it has an infinite sequence of zeros clustering at infinity and nonoscillatory otherwise. Thus, a nonoscillatory solution is eventually positive or eventually negative. Beginning with the papers [6,7] of Kiguradze, oscillation theory of higher order nonlinear differential equations has been the subject of intensive investigations in recent years. Kiguradze’s oscillation theorem (see also the book of Kiguradze et al. [8, Theorems 15.1 and 15.3]) specialized to Eqs. (A) read as follows. Theorem A. Any proper solution of (A) is oscillatory if and only if Z 1 a t 3c qðtÞdt ¼1: ð1:1Þ 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.029 ⇑ Corresponding author at: University of Niš, Faculty of Science and Mathematics, Department of Mathematics and Computer Science, Višegradska 33, 18000 Niš, Serbia. E-mail addresses: kusanot@zj8.so-net.ne.jp (T. Kusano), jelenam@pmf.ni.ac.rs (J.V. Manojlovic ´). Applied Mathematics and Computation 218 (2012) 6684–6701 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc