Research On Some Nonlinear Problems In Menger Spaces

In this paper,some new concepts are introduced.Some problems,including solvability of nonlinear operator equation and the fixed point of nonlinear operator are investigated in Menger probabilistic metric spaces. Utilizing the topological degree method and the iterative method in probabilistic metric spaces, some important theorems are improved and generalized. Moreover, a series of new results are given. It is divided into the following three sections.In chapter one, the backgrounds, current situation and the preliminaries of Menger spaces are given.In chapter two, the existence of solution for nonlinear operator equations in Menger spaces is studied utilizing the topological degree method. Some new concepts are introduced and the fixed point of nonlinear operator are investigated in Menger probabilistic metric spaces.In chapter three,the concepts of Z-type contracting mapping and Menger half PM-space are defined. Some fixed point theorems of contracting mappings are proved utilizing the iterative method. These results generalize some important result on probabilistic metric spaces.