Research On Some Nonlinear Problems In Generalized Metric Spaces

The study on operator equations and fixed point problems play important roles in establishing the existence and uniqueness of solutions of the various types of equations. A large number of differential-integral equations in abstract spaces ultimately can be attributed for nonlinear operator equations or fixed point problems to be studied. As the promotion of the metric space theory, it is great significance to study the theory of nonlinear operators in probabilistic metric spaces and partial metric spaces. This thesis mainly studies the problems for the fixed points of nonlinear operators under certain conditions using partial ordering methods and iterative methods; moreover, this thesis is also to discuss the existence problem of solutions of operator equations, and we obtain the conclusion that such equation under certain conditions will have solutions. There are three chapters in this thesis.In chapter one, the historical backgrounds and current situation of probabilistic metric space theory and application are introduced and the preliminaries of PM-spaces are given.In chapter two, in probabilistic metric spaces, we introduce a specific partial ordering utilizing functional, and study the existence problem of fixed points of operator under certain conditions and then discuss the solvability of the form Ax=Lx (L not necessarily continuous) operator equations in PM-spaces.In chapter three, discuss the coupled fixed point theorems for mixed monotone operators under certain conditions in the probabilistic metric spaces defined on the abstract partially ordered set, and give an example to verify the above conclusions. Similarly, as the promotion of the metric space theory, we define new contractive operators and study the fixed point problems of these types of contractive operators.