Existence and asymptotic behavior of intermediate type of positive solutions of fourth-order nonlinear differential equations

Abstract

Under the assumptions that p and q are regularly
varying functions satisfying condition

$$\int_a^\infty\frac{dt}{p(t)^{\frac{1}{\alpha}}}<\infty,$$

existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation

$$\left(p(t)|x''(t)|^{\alpha-1}\,x''(t)\right)^{\prime\prime}+q(t)|x(t)|^{\beta-1}\,x(t)=0,\quad
\alpha>\beta>0.$$

It is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law.