Research On Some Problems Of Random Operator Equations And Random Fixed Points

Random functional analysis is an interdisciplinary branch of research which is originated from probability theory and functional analysis. Random nonlinear operator theory is an indispensable part of random nonlinear functional analysis, and plays an important role in studying various types of random differential equations and random integral equations. In this thesis, we use random topological degree theory, random fixed point index theory and iterative method to study the existence and uniqueness of the solutions for several classes of random nonlinear operator equations and random fixed points. The main content of the thesis contains the following three chapters:In chapter one, we briefly introduce the historical backgrounds and current situation of random functional analysis and fixed point theory for random nonlinear operators. Meanwhile, we introduce some preliminaries for the main results of this thesis.In chapter two, firstly, utilizing random topological degree theory, the random solutions of random semi-closed1-set contractive operator equations and the random fixed points of such operators are investigated in Z-C-X spaces and real separable Banach spaces. Secondly, by using random fixed point index theory, the random solutions of random operator equations under different boundary conditions in real separable Hilbert spaces are discussed and some new results are obtained, which turns out to be the generalizations of many famous theorems such as Petryshyn’s theorem、Krasnoselskii’s theorem and Altman’s theorem.In chapter three, using partial order method and iterative techniques, we first discussed the problems on random fixed points of random increasing or decreasing operators; then we studied the existence and uniqueness of random coupled fixed points of random mixed monotone operators.