Research On Some Problems For Several Classes Of Nonlinear Operators

A variety of nonlinear problems has drawn widespread attention, and nonlinearanalysis has become one of the most important branches of research in modernmathematics. Nonlinear operator theory is an indispensable part of nonlinear analysis,since the study on a variety of mathematical equations, such as differential equations,integral equations, and numerical theory can be converted to the study on some kindsof operator equations. In this thesis, utilizing partial ordering method as well astopological degree method, the existence and uniqueness of fixed points as well as thesolutions of operator equations for several classes of nonlinear operators areextensively studied. The thesis is divided into four chapters as follows:In chapter one, the historical background and current situation of the work relatedto this thesis are introduced and some preliminaries are given.In chapter two, by using the topological degree method, the existence for a classof semi-closed1-set-contractive operator equations are studied. Meanwhile, anexample is also given to illustrate the validity of our main results.In chapter three, by means of the techniques of partial order and cone theory, theexistence and uniqueness of solutions for tripled mixed g-monotone operatorequations Lx N(x,x,x)is discussed in complete metric spaces and real Banachspaces, respectively.In chapter four, the existence and uniqueness of coupled coincidence points andcoupled common fixed points for mixed g-monotone mappings in ordered conemetric spaces are extensively investigated by making use of a class of controlfunctions.