Convergence of the solutions for the equationx(iv)+ax⃛+bx¨+g(x˙)+h(x)=p(t,x,x˙,x¨,x⃛)

Abstract

This paper is concerned with differential equations of the formx(iv)+ax⃛+bx¨+g(x˙)+h(x)=p(t,x,x˙,x¨,x⃛)wherea,bare positive constants and the functionsg,handpare continuous in their respective arguments, with the functionhnot necessarily differentiable. By introducing a Lyapunov function, as well as restricting the incrementary ratioη−1{h(ζ+η)−h(ζ)},(η≠0), ofhto a closed sub-interval of the Routh-Hurwitz interval, we prove the convergence of solutions for this equation. This generalizes earlier results.