Stability And Iterative Approximation Of Positive Solutions For Neutral Functional Differential Equations

Neutral functional differential equation is saw more broadly than functional differential equation which can usually be transformed to neutral functional differ-ential equation. Recently, many research programs basing on such mathematical models have been involved, such as genetic problem, enzymatic reaction kinetics, population dynamics and so on. Due to functional expression taking the deriva-tive with respect to time in the neutral system, the solution behavior will be too complicated to tackle this kind of equation. Therefore, the research method and solution properties about neutral functional differential equation are quite different from non-neutral functional differential equation.At present, research results about the stability of neutral functional differential equation aren’t still perfect. Meanwhile, study of the existence for the functional differential equation only gives sufficient conditions for the solution existence, but hardly there is approximate representation for the solution which is more valuable in the applications. Basing on the above situation, stability of the zero solution and the iterative approximation of positive solution about neutral differential functional equation will be further discussed in this article.Firstly, Chapter 1 introduces the research background and research advance of properties and stability about neutral functional differential equation. Meanwhile, our research contents and methods are presented. Using krasnoselskii fixed point theorem, Chapter 2 shows sufficient condition of stability of the zero solution about first order nonlinear neutral functional differential equation and result of which provides an improvement to corresponding that in literature.In chapter 3, existence of the positive solution and the iterative approxima-tion of positive solution about below neutral differential functional equation are discussed, Basing on Banach contraction mapping principle, we give the sufficient conditions of uncountably many bounded positive solutions for different range of allowed values about function a. Specially, iterative approximate sequence and error estimate will be presented through establishing a sequence of appropriate iterative functions. So, the results of this chapter are more valuable for practical applications.