Study On Fixed Points Of Nonlinear Operators In Several Spaces

As an important branch of modern functional analysis,the fixed point problem has received extensive attention from many domestic and foreign scholars.The multiple contractive forms of common fixed point and common coupled fixed point theorems in the metric space have achieved remarkable results.This paper is based on the existed results.The results of this study involve some spaces that are broader than the metric space,such as:fuzzy metric space,G-metric space,G_b-metric space and S-metric space.The problems of common fixed point and coupled fixed point are mainly studied in above four spaces.At the meantime,numerical and integral equation examples are given to illustrate the effectiveness of the results.The paper is divided into five chapters:The first chapter mainly describes the practical value of fixed point theory study,the current study status at home and abroad,as well as the main content and innovations of this paper;In the second chapter,in the fuzzy metric space,by using the incompatible R-weakly commutative mapping pairs of type(A_g)and the concept of(?)function,we establish several new common fixed point theorems for four mappings and mapping families;In the third chapter,in the G-metric space,the concepts of ?-coupled fixed point and quaternary modified F-control function are introduced.A coupled fixed point theorem for mixed monotone operators is established,and then a numerical example illustrates the validity of the theorem.As an application of the theorem,the results of the existence and uniqueness of the solutions of two types of nonlinear integral equations have been obtained at the same time.It is worth mentioning that the results of this chapter are brand new,and no scholars have been studied on the problem of ?-coupled fixed points in the G-metric space so far;In the fourth chapter,by defining two G_b-metrics on a set,a new type of rational fraction is discussed in this space.Two coupled fixed point theories in the G_b-metric space are established.We also give two numerical examples to prove the effectiveness of the new results;In the fifth chapter,for the first time,the concepts of sub-consistent and sub-sequential continuity of mapping pairs are introduced in the S-metric space,and some new fixed point theorems are proved by using the sub-consistent and relatively continuous mapping pairs.Practical example is given to verify the validity of the theorem.