On Fixed Point Existence For φ-Contractive Mappings In Probabilistic And Fuzzy Metric Spaces

The concept of the fixed point is proposed by a French mathematician called H.Poincaré in nineteenth Century. Many scholars found that we can use the fixed point theorems to solve important problems. So in 1910, L. E. J. Brouwer specifically focused on the concept of the fixed point for the first time, starting from the fixed point itself and proving that the fixed point satisfied the need of conditions required by existence uniqueness. They were the pioneers to open the study of the fixed point theorem.The representative study was that Polish mathematician Banach in 1922 used the Banach contraction mapping principle to solve the existence problems of some important questions. He also established the fixed point theorem for an important branch of applied mathematics, and made the fixed point theorem become a powerful mathematical tool in many important application problems.The contractive condition is the algebraic condition of the contractive mapping and the fixed point theorems. According to the different kinds of contractive conditions, the contractive fixed point theorems are divided into different classes. This thesis aims to discuss the fixed point theorem in probability metric spaces and fuzzy metric spaces. The main results are arranged as follows:(1) In the first chapter, it firstly introduces the basic concepts and the differences between probability metric spaces and fuzzy metric spaces, also including the concepts of convergence, Cauchy sequence and completeness. And then it introduces the current research of the fixed point theorems.(2) In the second chapter, it mainly introduces the fixed point theorems in probability metric spaces. On the basis of Jin-Xuan Fang’s studies [13], it reduces the conditions of φ- contractive mapping to a weaker shape and proves the fixed point theorems. It provides a theoretical basis for the research of φ- contractive mapping and the fixed point theorems in the future.(3) In the third chapter, it creates new fuzzy(j,y) - contractive mappings in the fuzzy metric spaces on the basis of the classical Banach contractive mappings and Mihet’s y - contractive mappings. And it also proves the fixed point theorems for(j,y) - contractive mappings.(4) The main task of the third chapter is that it creates weaker conditions on fuzzy coupled coincidence fixed point theorem on the basis of Choudhury’s studies. The research results shows that the mappings are not required to be compatible, continuous or commutable, and the t - norm is not required to be of Had?i?-type. Finally, two examples are presented to illustrate the main result of this chapter.