The Research Of The Fixed Point And Iterative Algorithm For Some Nonlinear Maps

The fixed point theory and nonlinear operators theory is an important part of nonlinear functional analysis, it is applied to many fields such as differential equations, integral equations, control theory, optimization theory, operator spectral theory, mathematical programming, equilibrium problems of economic and traffic, modern mechanics and nonlinear evolution equations and so on, which is more applied to the modern mechanics and nonlinear evolution equations. In recent years, the research of iterative algorithm for nonlinear operators has become a hot issue.In recent years, the research is more and more attention about the common fixed point theory and application, and achieved many important research results. On the basis of these results,this article studied existence and uniqueness of the fixed point for commuting self-maping, weakly commuting self-maping, generalized quasi-weakly commuting self-maping, compatible self-mapping,weakly compatible self-mapping, (Ag) type weak compatible mappings,φ-contractive mapping and the power of five type contractive mapping, and also studied public solution set of a strict pseudo-contraction mapping fixed point set and the generalized variational inequality set. This article is divided into five parts.The first part is the preface, which introduces research background and current status of the fixed point in symmetric space and metric space and theory of nonlinear operator in Banach space and Hilbert space. The second part, which discusses the coincidence point and fixed point problems in symmetric space for the mappings satisfying the condition of properties (E-A),weak compatible,(Ag)type weak compatible, and gives examples of argumentative; The third part, which discussed common fixed point problem of the power of five type contractive mapping in the metric space for commuting self-maping, weakly commuting self-maping, generalized quasi-weakly commuting self-maping, compatible self-mapping,weakly compatible self-mapping, and in the part which is fixed point problem of generalized quasi-weakly commuting self-maping gives examples of argumentative. The forth part, which proved common fixed point theorem for the six compatible,weakly compatible mappings in metric space.The fifth part, which gives an new iterative algorithm in the Banach space and Hilbert space, and using the algorithm studid convergence problems of iterative for the problem of variational inequality and fixed points of strict pseudo-contraction mapping.In short,this article obtained some new fixed point theorem, common fixed point theorem and convergence theorem of iterative by introducing a new class of mappings, construct a new iterative scheme, using a new iterative technique. These results improved, extended, complemented and completed related results, furthermore enriched and developed the fixed point theory and the theory of nonlinear operator and iteration.