Research On Some Random Operator Problems In Banach Spaces

Random nonlinear operator theory is an important part of the random nonlinear function analysis theory. It has much closed relation to the neoteric mathematical branches. Especially, it plays an essential role in establishing the existence of the solutions of several kinds of random integral equations. In this thesis, random fixed points of random nonlinear operator, random operator equations and random ambiguous point in Banach Space are studied by applying the random topological degree theory, random fixed point index method, partial order and iterative methods. There are three chapters in this thesis.In chapter one, the backgrounds and current situation of random fixed point theories of random nonlinear operators are introduced and the knowledge about Banach Spaces are given.In chapter two, in Banach Space, utilizing the method of random topological degree and random random fixed point theory, we discuss the existence of random solutions on random semi-closed 1-set contractive operator equations, random fixed point on random semi-closed 1-set contractive operator and existence theorem of solution of random equation in the Z-C-X space. We also obtain several new results and extend a serial of important theorems.In chapter three, first of all, partial order space is refersed to in cone, we study the existence of coupled fixed point and convergence of iterative sequence of random mixed monotone operator by applying partial order and iterative methods; secondary, we discuss the problem on random ambiguous point of random condensing operator, a series of results are obtained.