B-theory Of General Linear Methods For A Class Of Stif Nonlinear Functional Diferential And Functional Equations

Functional diferential equations are widely applied in various felds of na-ture sciences and engineering technology. Nearly recent half a century, peopleobtained a series of research results through extensive and profound research instability and convergence of the numerical methods for this kind of equations. Inparticular, Li obtained a general stability theory and B-theory of Runge-Kuttamethods and general linear methods in Banach spaces for stif Volterra func-tional diferential equations in recent years. Those research results laid unitedtheoretical foundation for the stability of the theoretical solutions and B-theoryof the numerical methods for the various functional diferential equations. How-ever, the theory doesn’t apply to the study of neutral functional diferentialequations.Functional diferential and functional equations are much wider appliedthan functional diferential equations and neutral delay diferential equations,so the research in stability of the theoretical solutions and the numerical meth-ods are more complicated and necessary. There are few research results aboutnonlinear stif functional diferential and functional equations at present. JiangChunhua made a further study of a class of nonlinear stif variable delay func-tional diferential and functional equations and fnally got the stability, generalcontractivity and asymptotic stability of the theoretical solutions and the B-theory of the Runge-Kutta methods in his master’s degree thesis in2010.Based on that, this paper will also make a study of a class of nonlinear stifvariable delay functional diferential and functional equation but I will use themore general methods here. I will obtain the B-theory results of the generallinear methods in the end.The B-theory results will be deeper and better thanthe existing results. I will verify the correctness of the theory through numericalexperiments at the end of the paper.