Asymptotic behavior of intermediate solutions of fourth-order nonlinear differential equations with regularly varying coefficients

Abstract

We study the fourth-order nonlinear differential equation $$ \big(p(t)|x''(t)|^{\alpha-1} x''(t)\big)''+q(t)|x(t)|^{\beta-1}x(t)=0,\quad \alpha>\beta, $$ with regularly varying coefficient $p,q$ satisfying $$ \int_a^\infty t\Big(\frac{t}{p(t)}\Big)^{1/\alpha}\,dt<\infty. $$ in the framework of regular variation. It is shown that complete information can be acquired about the existence of all possible intermediate regularly varying solutions and their accurate asymptotic behavior at infinity.

Keywords

• Fourth order differential equation
• asymptotic behavior of solutions
• positive solution, regularly varying solution, slowly varying solution