Research On The Theory And Applications Of Nonlinear Operators In Generalized Metric Spaces

It is well known that the fixed point theory is one of the important foundations of management mathematics,decision engineering and economic equilibrium.In the fixed point theory the construction of new space has always been a hot topic.Once a new space is proposed,it will obtain some new results.In this thesis,(?)-metric spaces are generalized and G_?-metric spaces are introduced.The conditions of existence of fixed point of nonlinear operators on Modular space,G-metric space and S-metric space are analyzed.These results are applied to prove the existence of solutions of nonlinear operators equations.At the same time,through exploration some examples and counter examples are obtained.This thesis is divided into seven chapters:In the first chapter,the historical background and current situation of the theory and applications of nonlinear operators on generalized metric spaces are introduced.The problems and significance of the research are explianed.The related basic notions and some results are given.In the second chapter,some fixed point theorems by using contractive conditions with only one variable are expounded on G-metric space,to explain that some theorems on G-metric spaces cannot be transformed into theorems on quasi metric spaces or metric spaces.This conclusion contradicts the idea in a recent paper.The viewpoint of this thesis is of important significance to the study of G-metric space.In the third chapter,a generalized (?)-metric space is introduced.The existence of solutions of equations of nonlinear operator are thoroughly analyzed by using Geraghy type contractive conditions,JS type contractive conditions and the contractive condi-tions obtained by means of comparison functions.In the fourth chapter,a new space G_?-metric space is introduced.The topological structure and properties of G_?-metric space are analyzed.On G_?-metric spaces,the conditions for the existence of fixed points of nonlinear operators are analyzed and some application examples are given.The new space established in this thesis greatly enriches the theory of fixed point.In the fifth chapter,the S-metric space is studied and compared with G-metric space.On this basis,Meir-Keeler S type contractive conditions and contractive con-ditions with F control function are introduced on S metric spaces to obtain some new fixed point theorems.At the same time,an example is given to show that there exists Meir-Keeler S type contraction mapping,which has a fixed point,but it is discontinu-ous at the fixed point.In the sixth chapter,with the help of simulatioin functions and altering distance functions,firstly compatible contractive conditions on Modular spaces are established to get some common fixed point theorems of multiple pairs of mappings,secondly cyclic (?)_?-contractive conditions are constructed to get some fixed point theorems of a mapping.At the same time,some application examples are given.In the seventh chapter,a summary of the research work and the innovation of the work are made.A follow-up vision of the research in this thesis has been put forward.