Research On Some Problems In Menger PN-Space

In 1942, K. Menger firstly created probabilistic metric space (referred to as PM-space, formerly known as statistical metric space). He used a distribution function to describe the distance between two points in space, such a "distance" often referred to the probability of distance. According to his theory, the usual metric space is a special case of probabilistic metric spaces, therefore researching on probabilistic metric space is of great significance. In 1972, sehgal and Bharucha-Re id [5] the first successfully promoted the principle of compression to complete Menger PM-space, and created a research probability theory of fixed point in probability metric space. It is well known that in the normed space, we can use the A-proper topological degree theory to study the operator. So it is natural to consider the possibility of using A-proper mapping topological degree theory in probabilistic metric space to study the problems of operator theory? In the paper, based on the previous theory, using A-proper mapping of topological degree theory to study operator theory in probability metric space. There are three sections in this thesis.In chapter one, the background materials and recent developments of the nonlinear operator theory in PM-space is given. Moreover, we introduced some basic concepts and results which will be needed in this paper, and the results to be gotten and research significance of this paper.In chapter two, utilizing the properties of the topological degree for the A-proper mapping in the probabilistic metric spaces, the fixed point of nonlinear operator and the solutions for nonlinear operator equations in the projection complete Z-P-S space are studied, and some new results are obtained.In chapter three, a new concept of the W-M-PN space is introduced, and a new concept of the intrinsic value and the intrinsic element is introduced in the W-M-PN space. Utilizing the properties of the topological degree for the A-proper mapping in the probabilistic metric space, the solutions for operator equations and some problems of the intrinsic value and the intrinsic element in the projection complete W-M-PN space are studied, then we got some new theorems.