Intermediate solutions of fourth order quasilinear differential equations in the framework of regular variation Kusano Takas ˆi a , Jelena V. Manojlovic ´ b,1 , Jelena Miloševic ´ b,⇑ a Hiroshima University, Department of Mathematics, Faculty of Science, Higashi-Hiroshima 739-8526, Japan b University of Niš, Faculty of Science and Mathematics, Department of Mathematics, Višegradska 33, 18000 Niš, Serbia article info Keywords: Fourth order differential equation Regularly varying function Slowly varying function Asymptotic behavior of solutions Positive solutions abstract Intermediate solutions of fourth-order quasilinear differential equation pðtÞjx 00 ðtÞj a1 x 00 ðtÞ   00 þ qðtÞjxðtÞj b1 xðtÞ¼ 0; a > b > 0 are studied in the framework of regular variation. Under the assumptions that pðtÞ; qðtÞ are regularly varying functions satisfying conditions Z 1 a t pðtÞ 1 a dt ¼1; Z 1 a t pðtÞ  1 a dt ¼1 and Z 1 a dt pðtÞ 1 a < 1 necessary and sufficient conditions are established for the existence of regularly varying intermediate solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law. Ó 2014 Published by Elsevier Inc. 1. Introduction The equation to be studied in this paper is pðtÞjx 00 ðtÞj a1 x 00 ðtÞ   00 þ qðtÞjxðtÞj b1 xðtÞ¼ 0; t P a > 0; ðEÞ where (i) a and b are positive constants such that a > b, (ii) p; q : ½a; 1Þ ! ð0; 1Þ are continuous functions and pðtÞ satisfies conditions: Z 1 a t pðtÞ 1 a dt ¼1 ^ Z 1 a t pðtÞ  1 a dt ¼1 ð1:1Þ and http://dx.doi.org/10.1016/j.amc.2014.09.109 0096-3003/Ó 2014 Published by Elsevier Inc. ⇑ Corresponding author. E-mail addresses: kusanot@zj8.so-net.nejp (K. Takas ˆi), jelenam@pmf.ni.ac.rs (J.V. Manojlovic ´), jefimija@pmf.ni.ac.rs (J. Miloševic ´). 1 Author is supported by the Research project OI-174007 of the Ministry of Education and Science of Republic of Serbia. Applied Mathematics and Computation 248 (2014) 246–272 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc