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

Intermediate solutions of fourth-order quasilinear differential equationp(t)|x″(t)|α-1x″(t)″+q(t)|x(t)|β-1x(t)=0,α>β>0are studied in the framework of regular variation. Under the assumptions that p(t),q(t) are regularly varying functions satisfying conditions∫a∞tp(t)1αdt=∞,∫a∞tp(t)1αdt=∞and∫a∞dtp(t)1α<∞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.