Hybrid Algorithms And Applications For Several Types Of Image Fixed Points And Related Problems

This paper gives a monotone hybrid algorithm to approximate the fixed points of these mappings and proves some strong convergence theorems for the fixed point of some mappings in Banach spaces.The results perfect and improve results of Mat-sushita,Takaha and others.It proves strong convergence theorems for the fixed point of a uniformly closed family of countable quasi-Bregman strictly pseudocon-tractive mappings and a family of countable Bregman quasi-Lipschitz mappings by accelerated hybrid algorithm in Banach spaces,of course,the results are ap-plied by a common element of solutions of a system of equilibrium problems,the problems of solution of a system of split variational inequality,the problems of solution of a system of split optimization,the common fixed point.The results improve the extension the latest research results of others.the paper useing a new multidirectional hybrid shrinking iterative algorithm for solving common problems which consist of a generalized split equilibrium problems and fixed point problems for a family of countable quasi-Lipschitz mappings in the framework of Hilbert spaces.The hybrid can accelerate the speed of convergence.The results improve the extension the latest research results of others.This paper includes five parts:The first part,we introduce the significance of fixed point theory and its appli-cation in nonlinear functional analysis,recall the history and present situation of fixed point theory of contraction mappings,iteration approximation methods.The second part,we research the existence theorems of fixed points of hemi-relatively nonexpansive mappings in Banach spaces,construct effective iterative schemes to approximate the set of fixed points of these mappings,obtain the corresponding convergence theorems and give the application.The third part,we deeply study a family of countable quasi-Bregman strictly pseudocontractive mappings and a family of countable Bregman quasi-Lipschitz mappings in Banach space,construct different effective iterative schemes,obtain the corresponding strong convergence theorems and give the application.The fourth part,we deeply study a family of countable quasi-Lipschitz mappings in Hilbert space,construct a new multi-directional hybrid shrinking iterative algorithm,obtain the corresponding strong convergence theorems and apply the result to solve common problems which con-sist of a generalized split equilibrium problems and fixed point problems.The last part draws a conclusion and future work.