Asymptotic stability of a second order integro-differential equation based on fixed point theory

In this paper, the asymptotic stability of a second order differential integral equation is studied without using lyapunov direct method. When differential integral equations have infinite terms or time delay is unbounded, it is difficult to use Lyapunov direct method to solve the asymptotic stability of zero solution of differential integral equations. In this paper, by using the fixed point theorem, the necessary and sufficient conditions for the asymptotic stability of the zero solution of a class of neutral second order differential integral equations with infinite delay is obtained. Then the fixed point theorem not only solves the asymptotic stability problem of the zero solution of the second order differential integral equation, but also relieves the previous strict restriction on infinite delay, and significantly reduces the restriction on function g.