Intermediate solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Intermediate solutions of Emden–Fowler type differential equationp(t)|x′(t)|α-1x′(t)′+q(t)|x(t)|β-1x(t)=0,α>β>0,are studied in the framework of regular variation. Under the assumptions that p(t),q(t)p(t),q(t) are generalized regularly varying functions necessary and sufficient conditions are established for the existence of three possible types of intermediate solutions witch are generalized regularly varying functions and it is shown that the asymptotic behavior of all such solutions of each type is governed by a unique explicit decay law.