Research On Random Nonlinear Operators In Banach Space

Random nonlinear operator theory is becoming more and more important intergradient of random nonlinear functional analysis. It has much closed relation to the other mathematical branches. Especially, it plays an essential role in establishing the existence and uniqueness of the solutions of several kinds of random equations. In this thesis, random nonlinear operators and random operator equations in Banach Space are studied by applying the random topological degree, random fixed point index method, partial order and iterative methods. The material in this book is presented in three chapters.In chapter one, the backgrounds and current situation of random fixed-point theories of nonlinear operators are introduced, and the preliminaries in Banach spaces is given, which is needed to study the random fixed points of nonlinear operators.In chapter two, utilizing the method of random topological degree and random fixed-index, we discuss the existence of random fixed points and random solutions on random semi-closed 1-set-contractive operators, random constant 1-set-contractive operators and random operator equations such as A(ω, x)=μx(μ≥1) under different boundary conditions. We also obtain several new results and extend a serial of important theorems.In chapter three, by applying partial order and iterative methods, we first study the existence, uniqueness and convergence of iterative series of two point extend mixed random monotone operators and random fixed points of random nonmixed monotone operators in the partial order spaces induced by cones. Secondly, we discuss the problem on random common fixed points of random nonlinear operator pairs. Finally, we study the uniqueness of random fixed points of random monotone and that of mixed monotone operator in the partial order spaces induced by functions.