Convergence of solutions of some second-order nonlinear differential equations

Convergence of solutions of some second-order...

Abstract

This paper is concerned with the convergence of solutions of the second-order differential equation
$$\ddot{x}+f(x)\dot{x}+g(x)=e(t,x,\dot{x}),$$ where $f(x),g(x)$ and $e(t,x,\dot{x})$ are continuous real-valued functions in their arguments. By using the direct method of Lyapunov and constructing a complete Lyapunov function, sufficient conditions which guarantee the convergence of solutions are obtained.