Research On Solutions Of Operator Equations And Fixed Points Problems For Operator In Probabilistic Metric Spaces

The theory of nonlinear operator equations and fixed point problems is an important part of the nonlinear function analysis theory, meanwhile, it is also a useful method to study a large number of differential-integral equations.This thesis mainly studies the problems for the solution of one class of nonlinear operators and a few classes of fixed point problems by using partial ordering methods and iterative methods in probabilistic metric spaces; moreover, this thesis is also study the existence of the solutions for one class of nonlinear differential equations by using the fixed point theorem for nonlinear operators. This thesis is divided into three chapters:In chapter one, a brief introduction of research are given. Besides, we introduced some elementary concepts which will be used in this thesis.In capter two, using the partial order method in probabilistic metric spaces, we discuss the existence of solutions for a mixed monotone operator equation Lx ?N?x,x?.Our results generalize and improve some important results in the corresponding literatures.In chapter three, in probabilistic metric spaces we establish the concept of a sequence of generalized ?-admissible mappings. Using the partial order method, we get the coincidence point of a sequence of mappings under different contractive conditions. Finally, we apply the new results presented to integral equations. Moreover,employing the tangential property, some coincidence and common fixed point theorems for two hybrid pairs of mappings via an implicit relation are proved.