Research On Fixed Points Theory And Degree Theory In Probabilistic Metric Spaces

This dissertation is devoted to the fixed points and fixed degree problems in probabilistic metric space, the Leray-Schauder degree theory in probabilistic normed spaces, some fixed points theory based on it.The paper is organized in the following manner.In chapter one, the backgrounds materials and recent development of Menger spaces are given.In chapter two, some basic concepts and results which will be needed in this paper are introduced.In chapter three, we discuss the nonlinear operators in Menger probability metric spaces. The existence and uniqueness of the fixed point for nonlinear contraction and expansion are given. Further we extend the result for single operator to common fixed point for a family of exchangeable operators. The fixed degree under fuzzy mappings in Menger probability metric spaces are given. By weakening conditions, a new fixed degree theorem for fuzzy mapping is established.In chapter four, Leray-Schauder topological degree theory in probability norm space under the more general t-norm conditions are given. In the meantime, its unitarity, compact homotopy invariance, additivity, excision property, Kronecker poperty, Poincare-Bohl property,translatatability, connected region property, boundary value property and so on are given, then we get some fixed points theory based on it.