Stability Conditions of Second Order Integrodifferential Equations with Variable Delay

Abstract

We investigate integrodifferential functional differential equations $\ddot{x}+f(t,x,\dot{x})\dot{x}+{\int }_{t-r(t)}^t\mathrm{‍}a(t,s)g(x(s))ds=0$ with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.