Research On Split Pseudo-monotone Equilibrium Problem Algorithm

In recent years,the equilibrium problem has been widely used in mathematical programming,economics,physics,transportation,engineering and cybernetics.Therefore,it is of great significance to study how to solve the equilibrium problem.Up to now,the equilibrium problem has formed a relatively systematic theoretical system.Solving the equilibrium problem has always been a hot topic for scholars.In order to solve the monotone equilibrium problem in Hilbert space,some algorithms such as viscosity approximation and hybrid optimization algorithms have been proposed,and the strong convergence and weak convergence of these algorithms have been proved.Then the split equilibrium problem and its corresponding solving algorithm are proposed by the scholars.But in fact,these algorithms have certain difficulties in calculation,it’s difficult to find a solution of the equilibrium problem.Until some scholars proposed an iterative algorithm that based on convex optimization to solve the pseudo-monotone equalization problem,this algorithm can be easily calculated to obtain the final solution.Inspired by this algorithm,this paper proposes a convex optimization method to solve the common solution of the pseudo-monotone equilibrium problem and the nonexpansive mapping fixed point problem in the Hilbert space.The convergence of the algorithm is proved,and two numerical examples are given to illustrate the effectiveness of the algorithm.On the other hand,in the existing algorithms for solving the equilibrium problem,the norm of the operator A is inevitably used,which actually increases the difficulty of the algorithm implementation.It is inspired by Hieu’s research,in this paper,we introduce a novel non-convex combination iterative algorithm to solve a family of split equilibrium problems in Hilbert space and prove the strong convergence for the designed algorithm.Finally,a numerical example in infinite dimension space is given to illustrate the effectiveness of the algorithm and another example is used to compare the computed result with Hieu.