Stability Analysis For Nonlinear Functional Differential And Functional Equations With Variable Delay

Functional differential equations (FDEs) arise widely in physics, biology, engineering, medical science, economics and so on. It is meaningful to study the theory and application of numerical methods for FDEs. In the last several decades, many important results on the theory of computational methods for FDEs have been reported by a large number of researchers. Especially, the general stability theory and B-theory of numerical methods including Runge-kutta methods and general linear methods for nonlinear stiff Volterra functional equations (VFDEs) in Banach spaces has been established by Li Shoufo in recent years, which provides a unified theoretical foundation for the stability of the theoretical solution and B-stability and B-convergence of numerical methods for nonlinear stiff problems in delay differential equations (DDEs), integro-differential equations (IDEs), delay-integro-differential equations (DIDEs) and VFDEs of other types that appear in practice. However, that work isn't adapted for nonlinear neutral FDEs. FDEs, a class of hybrid systems, which consist of functional differential equation and functional equation and are more extended than neutral functional differential equation. Discussing the theoretical solution and numerical methods are more complex than other functional differential equations. At present, several authors only have investigated the asymptotic stability of numerical methods for linear FDFEs. In view of this, the thesis investigates the stability of the theoretical solution and B-stability and B-convergence of Runge-Kutta methods for a class of nonlinear FDFEs with variable delay The main results obtained are as followed:(1) We obtain the stability, general contractivity and asymptotic results for a class of nonlinear FDFEs with variable delay(2) Applying Runge-Kutta methods to nonlinear FDFEs with variable delay, we get B-stability, B-consistency and B-convergence results of numerical methods which are more general and deeper than the existing related results in literature.