Research On The Convergence Of Solutions For Equilibrium Problem

The equilibrium problem that has been widely used is a class of non-linear problems,and its exact solution is generally difficult to find.Therefore,finding its approximate solution has become the main content of mathematician-s' research on this problem,and iterative algorithm to approximate the solution of the equilibrium problem is the most common and most effective method.At present,many iterative algorithms have been proposed and many conclusions have been obtained.However,in practice,it is desired that the convergence rate of the algorithm will be as fast as possible,so it is a necessary and meaningful work for the continuous improvement of the algorithmThe purpose of this thesis is to study the equilibrium problem in Banach space and the J-equilibrium problem in the dual space of Banach space by using inertial algorithms and related techniques,it also removes a condition that was commonly used but not reasonable when solving the problem with inertial algorithms in the past,and speeds up the convergence rate by modifying the iterative parameters and obtains convergence.The results of this paper are new,and further promote and improve some existing results at home and abroad.in the textFirst,some basic concepts,research background and current status are in-troduced.The definition and application of the equilibrium problem and J-equilibrium problem are briefly reviewed.Some existing results of solving the equilibrium problem and the J-equilibrium problem at home and abroad are sum-marized.Based on this,the main research contents of this paper are proposedSecondly,the prerequisite knowledge needed in the subsequent chapters is given to provide a theoretical basis for the later proofsThirdly,in 2-uniformly convex and 2-uniformly smooth Banach spaces,com-bining inertia algorithm,proximity point algorithm and Krasnoselski-Mann itera-tion,an algorithm is constructed to solve the equilibrium problem,and the weak convergence theorem is obtained.Then,in the dual space of 2-uniformly convex and 2-uniformly smooth Ba-nach spaces,a new method with inertia term and the proximal regularity tech-nique is proposed to solve the J-equilibrium problem,and the strong convergence theorem is obtained.Finally,the conclusions and prospects are given,and some problems worth further investigation are put forward.