Discussion On Several Classes Of Set-valued Operators

The purpose of this thesis is to discuss the existence problems of the fixed points for set-valued operators and for single-valued operators in linear spaces.Since the 1940's,set-valued analysis has been developed rapidly,and then it has turned out to be one important branch of modern mathematics and one important component of nonlinear analysis. Since fixed point theorems of operators are the basis of nonlinear analysis,the study of fixed point theorems of set-valued operators and their applications are of important significance.In chapter I and II,we discuss the fixed point problems of set-valued operators.In chapter I,by means of partial ordering method and monotone iterative tech-niques,we discuss the set-valued operators in partially ordered sets and partially ordered topological spaces.In §l.l,we weaken the definition of close and upper semi-continuity for set-valued operators and get ordered close and ordered upper semi-continuity and give their examples respectively.In §1.2,\ve give some fixed point theorems of strongly increasing operators and mixed monotone operators in partially ordered sets, partially ordered topological spaces and topological linear spaces of which order is introduced by cone.The results obtained improve and extend the corresponding results.The main result is as follows:Theorem 1.2.3. Let X be a partially ordered topological space,M = [u0, v0) be a given ordered interval of X. Suppose A : M --> 2~M is an ordered close strongly increasing operator and any monotone sequence of M has limit.Then A must have minimal and maximal fixed points.In chapter II,we discuss the set-valued mappings in metric spaces and locally convex spaces and obtain some relevant fixed point theorems.S.Nadler set-valued contraction mapping and Ky Fan fixed point theorem extend the single-valued contraction mapping principle and Brouwer fixed point theorem respectively. In §2.1.we give a new definition of contraction for set-valued mapping and show this kind of set-valued contraction mapping and the other kind of set-valued contraction mapping,which is defined by Pompeiu-Hausdorff measure(by S.Nadler),neither contains the other by two examples.and we obtain that this kind of set-valued contraction mapping must have fixed points in complete metric spaces.In additioni,in ordered Banach space X, we extend Krasnoselskii fixed point theorem and obtain fixed point theorem of set-valued mapping T + S, where Tx = ax + b, -1 < a < 0,6 6 X is a special contraction mapping and 5 is a set-valued mapping. In §2.2,we eastablish a fixed point theorem of K set-valued mapping in locally convex spaces under certain boundary conditions.