Iterative Methods For Common Solutions Of A Split Equilibrium Problem And Related Problems In Hilbert Space

Equilibrium problems are widely used in many fields such as economy,mechanics and transportation network.The study of equilibrium problems greatly promotes the development of optimization and fixed point theory.The mathematical model of equilibrium problems not only provides a unified form for variational inequalities,fixed point theory and complementarity problems,but also includes some other important models,such as Nash equilibrium.With the in-depth study of equilibrium problems and related problems,some scholars began to study the iterative methods of common solutions of equilibrium problems and related problems,such as viscous approximation method,external gradient method,hybrid projection method and so on.Based on the fixed point theory,this paper presents an iterative method for solving the common solution of the split equilibrium problem and its related problems.Firstly,we consider an iterative algorithm for solving the common solution of split equilibrium problem and split common fixed point problem in Hilbert space.It is a general problem to solve the common solutions of these two kinds of problems.The split common fixed point problem involves nonexpansive mapping and firmly nonexpansive mapping.In this paper,we construct an iterative algorithm for solving the common solutions of these two kinds of problems by using the properties of equilibrium function,and prove the strong convergence of the iterative sequence generated by the algorithm under appropriate conditions.On the other hand,we consider the iterative algorithm for solving the common solution of the split equilibrium problem,the hierarchical fixed point problem and the system of general variational inequalities in Hilbert space.By using the equivalence between variational inequality and fixed point of projection operator and the properties of equilibrium function,an iterative method for solving the common solution of these three kinds of problems is constructed,and the strong convergence of the algorithm is proved under certain conditions.The results obtained generalize the corresponding results in related literatures.