On asymptotic behaviour of oscillatory solutions for fourth order differential equations

Abstract

We establish sufficient conditions for the linear differential equations of fourth order $$ (r(t)y'''(t))' =a(t)y(t)+b(t)y'(t)+c(t)y''(t)+f(t) $$ so that all oscillatory solutions of the equation satisfy $$ lim_{toinfty}y(t)=lim_{toinfty}y'(t)=lim_{toinfty}y''(t)= lim_{toinfty}r(t)y'''(t)=0, $$ where $r:[0,infty)o(0,infty),a,b,c$ and $f:[0,infty)o R$ are continuous functions. A suitable Green's function and its estimates are used in this paper.

Keywords

• Oscillatory solution
• asymptotic behaviour