Research On Fixed Point Problems About Nonlinear Operators In Probabilistic Metric Space

The study of nonlinear operators can solve many problems in some subjects, such as engineering, physics, biology, medicine and so on . In probabilistic metric space, the distance between two elements is measured by a distribution function, and an ordinary metric space can be viewed as a special case of a probabilistic metric space. So, the study of nonlinear operators in probablistic metric space is significant. In this paper, the problems of fixed point of nonlinear operators are studied. It is divided into the following four sections.In chapter one, the backgrounds and current situation of operator theory in PM-spaces are introduced and the preliminaries of PM-spaces are given.In chapter two, the problems of fixed point of nonlinear operators in probablistic metric space was discussed, and the existence and uniqueness theorems of fixed point for nonlinear multiple valued operators and compound nonlinear single valued operators are proved, and some corollaries are obtained.In chapter three, utilizing the concept of partial order in probabilistic metric space to discuss the exitence's problems of fixed point, theorems for fixed point and common fixed point of monotony increasing and monotony decreasing operators are obtained. At the same time, by the methods of partial order, some theorems for common fixed point of the squence of multiple valued operators are got.In chapter four, the types of contractive mappings in intuitionistic Menger PM-space and cone metric space are discussed, and some theorems of fixed point are obtained.