Convergence Of The Common Solution Of Quasi-nonexpansive Mapping And Equilibrium Problem

The main purpose is to extend nonexpansive mappings to quasi-nonexpansive mappings, and study the common solution of an equilibrium problem, fixed points problem of a quasi-nonexpansive mapping and a variational inequality problem.This paper is organized as follows:The first part, the common solution problem of the fixed points problem of a quasi-nonexpansive mapping and an equilibrium problem is studied, it is proposed an iterative method for finding a common solution of them. It has proved that the iterative sequences converge weakly to a common solution of the problems mentioned above, and that the projections of the iterative sequences onto the set of common solutions of the above problems converge strongly to the common so-lution. By proving that nonexpansive mappings are quasi-nonexpansive mappings satisfying condition (B) of the theorem, it is shown that the iterative sequences converge weakly to a common solution of the equilibrium problem and the fixed points problem of the nonexpansive mapping. The results take a main result of Tada and Takahashi as special cases. In the finite space, a strong convergence theorem with a weaker condition was gotten. Furthermore, an example was given to show that there exists a quasi-nonexpansive mapping, which satisfies condi-tion (B) but is not a nonexpansive mapping.The rationality and convergence of the iterative algorithm about the equilibrium problem and the fixed points of quasi-nonexpansive mapping are verified by a numerical example.The second part, it is extended the first part to the case of solving the com-mon solution problem of an equilibrium problem and the fixed points problem of a quasi-nonexpansive mapping and a variational inequality problem, it is also proposed an iterative method for finding a common solution of them. It is proved that the iterative sequences converge weakly to a common solution of the research problems mentioned above under certain conditions, and that the projections of the iterative sequences onto the set of common solutions of the above problems converge strongly to the common solution. It is also shown that the iterative se-quences converge weakly to a common solution of the equilibrium problem and the fixed points problem of the nonexpansive mapping and the variational inequality problem. At last, in the finite space, a strong convergence theorem with a weaker condition was gotten.The third part, two iterative methods (the method about the fixed points pr- oblem of the quasi-nonexpansive mapping and the other method with compression mapping) are compared by numerical experiments. The experiments results show that the method in this paper is better than the method with compression mapping under certain conditions.