On Some Discussion Of Set-valued Operaters

Since the 1940's, set-valued analysis has developed rapidly and become a branch of modern mathematics. As an important part of nonlinear analysis, it is extensively applied to many fields. The thoughts and methods of set-valued analysis has penetrated into the study of social science, natural science and technology. Owing to the importance of fixed points in theory and application, they have being paid much attention in mathematics. In the 1830's, Von Neumann discussed fixed point theories of set-valued mappings. Since then, Katutani, Brouwer, Bohnenblust, Karlin. Glicksberg and Ky Fan et al. discussed them in various spaces, respectively.In this thesis, we mainly discuss the properties of some set-valued mappings and the existence problem of their fixed points by means of partial ordering method and iterative technique and improve, extend and develop the corresponding results of single-valued and set-valued operators.In Chapter I, we give an analogue to Kuratowski noncompact measure and its properties in locally convex spaces with the order introduced by a cone defined in the thsis. And then we mainly give several fixed point theorems of set-valued mappings by using the cone theories and monotone iterative technique. And we improve ane extend the results in some references relative to them. The main theorems are as follows:Theorem 1.1 1) Let X be a sequentially complete locally convex space and P be a cone in A', u0, v0 ∈ X, u₀ ≤ v0, D = [u0, v0] C A' is bounded: 2) Let A : D -- 2D be a closed mapping, which satisfies that i) for any x.y ∈ D.x ≤ y, implies Ax ≤ Ay, ii) for any x ∈ D, Ax is a complete set and iii) for any p ∈ and B A (B) ≠ 0 implies ap(A(B)) < ap(B). Then A has a minimal fixed point and a maximal fixed point in D.Theorem 1.2 1) Let X be a sequentially complete locally convex space and P be a cone in X satisfying (H1.2). u0,v0 ∈ X,u0 ≤ v0,D = [u0, v0] C X; 2) Let A : D -- 2D be a closed mapping,which satisfies the condition i) and ii) in theorem 1.1. Then A has a minimal fixed point and a maximal fixde point in D.Theorem 1.3 Let E be a partial set. Any set with an upper (or a lower) bound has a supremum (or an infimum) in E. u0,v0 ∈ E,u0 ≤ v0,D = [u0,v0] C E. A : D -- 2^D is a set-valued mapping.1) Suppose that A satisfies that i) for any x ∈ D,x ≤ y implies Ax ≤ Ay.Then A has a maximal fixed point.2) Suppose that A satisfies that i) for any x ∈ D, x ≥ y implies Ax ≥ Ay.Then A has a minimal fixed point.