Convergence of solutions of certain non-homogeneous third order ordinary differential equations

A. U. Afuwape, M.O. Omeike

This paper is concerned with differential equations of the form $$\st{...}{x}+a\ddot{x}+g(\dot{x})+h(x)=p(t,x,\dot{x},\ddot{x})$$ where $a$ is a positive constant and $g$,$h$ and $p$ are continuous in their respective arguments, with functions $g$ and $h$ not necessarily differentiable. By introducing a complete Lyapunov function, as well as restricting the incrementary ratio $\eta^{-1}\{h(\xi + \eta)-h(\xi)\},(\eta \neq 0),$ of $h$ to a closed sub-interval of the Routh-Hurwitz interval, we prove the convergence of solutions for this equation. This generalizes earlier results.