The stability relation between ordinary and delay-integro-differential equations

This paper deals with the exponential stability of a class of nonlinear delay-integro-differential equations of the form ẋ(t)=f(t,x(t),x(t−τ1(t)),∫t−τ2(t)tg(t,s,x(s))ds),t≥t0, where τi(t)>0τi(t)>0 for i=1,2i=1,2 and t≥t0t≥t0. The stability relation between ordinary and delay-integro-differential equations is given. It is shown under some suitable conditions that a delay-integro-differential equation will remain exponential stability of the corresponding ordinary differential equation. Finally, some examples with numerical experiments illustrate the theoretical result.