Fixed Point Theorem In Probabilistic Metric Space

This dissertation deals with some new fixed point theorems in Menger probabilistic metric spaces, which include common fixed point theorems of weakly compatible mappings under strict contractive conditions and common fixed point theorems of compatible and weakly compatible mappings underφ-contractive conditions. The main contents are as follows:In chapter 1, we introduce some basic concepts and lemmas that we will use in the sequel, such as Menger probabilistic metric space and its topological structure, the induced Menger-probabilistic metric space by metric space, compatible mappings and weakly compatible mappings and the property (E.A) in metric spaces and Menger probabilistic metric spaces.In chapter 2, we point out that two common fixed point theorems of noncompatible mappings under strict contractive conditions given by Pant etc are incorrect by a counterexample. At the same time, we correct the two theorems.In chapter 3, we introduce the property (E.A) of a pair of mappings in Menger probabilistic metric space. By using this concept, we establish some common fixed point theorems of weakly compatible mappings under strict contractive conditions in Menger probabilistic metric spaces. These results generalize the common fixed point theorems of non-compatible mappings in metric spaces obtained in chapter 2.In chapter 4, we introduce two classes of real functionsΦandΦ₁. By using the functionφinΦ₁ orΦ₁ as a contractive gauge function, we establish some new common fixed point theorems for compatible mappings and weakly compatible mappings in Menger probabilistic metric spaces. Our results generalize the fixed point theorem obtained by Singh and Jain recently.