Exponential Stability Of A Differential Equation With State-Dependent Delays

This paper is concerned with the linearized stability of delay diferential equations. Under some assumptions, the trivial solution of the discrete and distributed state-dependent delay system is exponentially stable if and only if the zero solution of the associated linearized system is exponentially stable. Finally, a threshold-type delay equation is given as an illustration.The paper cinsist of three parts.In the first part , we introduce the development and significance of delay diferential equations.In the second part , we prove that the trivial solution of the discrete and distributed state-dependent delay system is exponentially stable if and only if the zero solution of the associated linearized system is exponentially stable.In the third part ,w e give an illustration of threshold-type delay equation.