ASYMPTOTIC ANALYSIS OF FOURTH ORDER QUASILINEAR DIFFERENTIAL EQUATIONS IN THE FRAMEWORK OF REGULAR VARIATION

Abstract

Under the assumptions that $p(t),\;q(t)$ 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 $$\left(p(t)|x''(t)|^{\alpha-1}\,x''(t)\right)^{\prime\prime}+q(t)|x(t)|^{\beta-1}\,x(t)=0,\quad \alpha \gt \beta\gt 0.$$ It is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law.