Some Research On Algorithms Of Fixed Point Problems And Variational Inequality System Problems

In this paper,we mainly study the fixed point problem and the variational in-equality system problem in Banach spaces,and establish a new viscosity iterative algorithm for asymptotically non-expansive mappings to approximate the solutions of the fixed point problem and the variational inequality system problem.The strong convergence theorems of the sequence generated by the iterative algorithms is obtained under certain parameter conditions.The main theorems are used to solve the standard constrained convex optimization problem.The main research contents are divided into four chapters as follows:In chapter 1,based on the fixed point theory,the research background of vari-ational inequality problem and generalized variational inequality system problem is described.The basic concepts and important lemmas related to this paper are also introduced.In chapter 2,we introduce the generalized viscous implicit double midpoint rule for asymptotically non-expansive mappings in Banach spaces:v_n+1=?_nv_n++v_nf(t_nv_n+(1-t_n)v_n+1)+?-nTⁿ(t——nv-n+(1-t_n)v_n+1),n?0.The iterative algorithm of the rule solves the fixed point problem and the variational inequality problem.The strong convergence theorem of the sequence generated by the algorithm is proved under suitable parameter conditions.In chapter 3,a new algorithm for the common solutions of generalized variational inequality systems and nonlinear operator equations in Banach spaces is studied:The strong convergence results of the algorithm is obtained by demonstration.The main results are applied to the standard constrained convex optimization problem,and a numerical example is given to illustrate the validity and scientificity of the main results.In chapter 4,we study the common elements of the fixed points of a generalized variational inequality system and an asymptotically nonexpansive mapping in Banach spaces by using the viscosity algorithm of the generalized implicit double-midpoint rule:...