Menger Pm-space, Two Types Of Mapping Fixed Point Theorem

This pape deal with the fixed point theory in Menger probabilistic metric space. We mainly research fixed point theorems for the complex mappings and common fixed point theorem for two weakly compatible selfmappings on Menger probabilistic metric spaces. The main includes four sections as follows:1. We introduce a set of real functions A, and use it to establish a new fixed point theorem of the complex mappings on complete Menger probabilistic metric spaces, which improves and generalizes the fixed point theorems for complex mappings on Menger probabilistic metric spaces given by Wu Da-wei freshly. This theorem is also a generalization of the corresponding fixed point theorem in metric spaces given by Fisher, Telci.2. We introduce a set of real functions H, and use to it to give a fixed point theorem of tother type for complex mappings on complete metric spaces, which improves and generalizes the corresponding results on compact metric spaces or under the condition of weakening compactness obtained by Fisher, Telci and Xu Xiao-li, respectively. Also, by using a set of real functions H*, we research the generalization of the fixed point theorem of this type in Menger probabilistic metric spaces.As the consequences of this result, some new fixed point theorems for complex mappings or single mapping on Menger probabilistic metric spaces are obtained.3. We utilize the results in Chapter 2 and Chapter 3 to prove the fixed point theorems and several corollaries for complex mappings or single mapping on fuzzy metric spaces.4. We introduce the concept of weak-compatible maps. By using this concept, we establish a common fixed point theorem for two weak-compatible self-mappings on Menger probabilistic metric space, thus the fixed point theorem on metric spaces proved by Aamri has been extended to Menger probabilistic metric spaces.