Stability analysis for nonlinear second order differential equations with impulses

Abstract

In this paper we investigate the impulsive equation \[\left\{\begin{array}{l} \left(r(t)x'\right)'+a(t)x+ f\left(t,x, x'\right)=p(t), t\geq t_{0},~t\neq t_{k}, \\ x( t_{k})=c_{k}x( t_{k}-0),~x'( t_{k})=d_{k}x'(t_{k}-0),~k=1,2,3,\ldots, \end{array}\right.\] and establish a couple of criteria to guarantee the equations of this type to possess the stability, including boundedness and asymptotic properties. Some examples are given to illustrate our results and the last one shows that, to some extent, our criteria have more comprehensive suitability than those given by G. Morosanu and C. Vladimirescu.

Keywords

• impulsive equations
• stability
• bernoulli type differential inequality
• bellman's inequality