The Study Of Problems Without Essential Components

The aim of this paper is to study problems which have not essential points and essential components by set-valued mapping . It consists of three chapters.Chapter 1 is the preliminaries. Firstly, we recall some notions and results about space theory, including Hausdorff space, Hausdorff distance between sets , Baire's category about sets and convex set. Secondly, the semi-continuity, closure, compactness of set-valued maps are introduced in set-valued analysis. Finally, essential point, essential set and essential component are introduced. Chapter 2 , we study the stability of approximate solutions of optimization problems(the continuous function is defined in the compact distance space),we prove that most of approximate solutions of optimization problems(under Baire's category) are stable, especially all accurate solutions are the essential points of the set of approximate solution. This chapter is the promotion of the result of Luo[34]. Chapter 3, we study the stability of approximate solutions of nonlinear equation problems(the continuous function is defined in the linear normed space).We prove that most of approximate solutions of nonlinear equation problems(under Baire's category) are stable; all accurate solutions are the essential points of the set of approximate solution; the connected component which contains the accurate solution is the essential component of the set of approximate solution; especially the number of essential components is finite.