Lyapunov-Razumikhin techniques for state-dependent delay differential equations

We present Lyapunov stability and asymptotic theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete delays, which may be nonautonomous the result, but autonomous (or periodically forced) result. main technique is replace DDE by a ordinary equation (ODE) where delayed terms become source in ODE. The result based on contradiction argument together Arzelà-Ascoli theorem. This approach alleviates need construct auxiliary functions ensure contraction, feature other we are aware. our model includes Hayes as special case, directly establish parts domain lower bounds size basin attraction.