Based On The Theory Of Fixed Point, The Stability Of Nonlinear Functional Differential Equations

As a new tool for studying the stability, the fixed points theory attracts many scholars with its great advantages. Burton, Furumouchi and other pioneers of this field pored over the work for years and got many creative results.Inspired by their work, a series of papers on studying stability via fixed points theory came out successively, but most of which are limited to the contraction fixed point theory and Schauder’s fixed point theoryDifferent fixed point theory has different advantages. Compared to the contraction fixed point theory and Schauder’s fixed point theory, Krasnoselskii fixed point theory is not simple and convenient, but it has unique advantage which bases on perturbed theory. So, if we choose a suitable perturbed part, we can also get a good or even better result. Motivated by the idea above and experts’work, this paper will study the stability of nonlinear functional differential equations by using Krasnoselskii fixed point theory.This paper can be divided into three chapters:In the first chapter, contents of four respects are introduced, the background, simple applications of fixed points theory in stability, the Preliminaries and the work being done in this paper.The second chapter will generalize the equation and its stable conditions studied by Burton in2005.Generally, this chapter will study the stability ofx"+f(t, x(t), x’(t))x’(t)+g(t, x(t-r(t)))=0via the contraction fixed point theory and Krasnoselskii fixed point theory respectively. More over, this chapter will improve the two conditions " f(t,x(t),x’(t)) is nonnegative" and "g(x)/x≥β＞0", which are referred to by Burton in2005.The third chapter will discuss abstract nonlinear functional differential equations and give the stability conditions of two classes of concrete equations. Papers on studying stability of differential equations via fixed points theory always requires that the integral of the linear part goes to infinite when t goes to infinite.but for nonlinear differential equations this requirement always doesn’t satisfied. To deal with the problem, this chapter will set a linear term...