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 x ̇ ( t ) = f ( t , x ( t ) , x ( t − τ 1 ( t ) ) , ∫ t − τ 2 ( t ) t g ( t , s , x ( s ) ) d s ) , t ≥ t 0 , where τ i ( t ) > 0 for i = 1 , 2 and t ≥ t 0 . 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.