Research On Several Random Operator Problems

The fundamental theory of random operators is an important branch of stochastic analysis and its development is required for the study of several classes of random operator equations. The study of random fixed points which is considered as a central topic in this theory has received much attention since the publication of the survey article by Bharuch-Reid in 1976, in which the stochastic version of some well-known fixed point theorems were proved. In this thesis, random fixed points of random nonlinear operators, random multiple-solution of random operator equations and some new random contractions and iterative processes are studied by applying the random fixed point index method 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 is given.In chapter two, in Banach spaces, utilizing the random fixed point index and some important results, we discuss the existence of random solutions on random semi-closed 1-set contractive operator equations and, random fixed point and random multiple-solution on random semi-closed 1-set contractive operator. We also obtain several new results and extend a serial of important theorems, such as Krasnoselskii's theorem and Leggett-Williams'three-solution theorem.In chapter three, first of all, some new concepts such as random compatible operators, random sequentially commutative and so on are introduced in Polish spaces and, some common random fixed points for random compatible operators satisfying some new contractive conditions are investigated and the uniqueness is proved; secondary, some new random iterative processes for random strongly accretive and hemi-contractive operators are introduced in Banach spaces and the necessary conditions for the convergence of these processes are investigated.