03-16-2009, 03:45 PM

Equilibrium problems which were introduced by Blum and Oettli [1] and Noor and Oettli [2] in 1994 have had a great impact and influence in the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization. It has been shown [1,2] that

equilibrium problems include variational inequalities, fixed point, Nash equilibrium and game theory as special cases. Hence collectively, equilibrium problems cover a vast range of applications. Due to the nature of the equilibrium problems, it is

not possible to extend the projection and its variant forms for solving equilibrium problems. To overcome this drawback, one usually uses the auxiliary principle technique. The main and basic idea in this technique is to consider an auxiliary equilibrium problem related to the original problem and then show that the solution of the auxiliary problems is a solution of the original problem. This technique has been used to suggest and analyze a number of iterative methods for solving various classes of equilibrium problems and variational inequalities, see [16,2529] and the references therein.

Related to the equilibrium problems, we also have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems.

In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of finitely many nonexpansive mappings, see [6,8,11, 17,30,3245] and the references therein. In this paper, we suggest and analyze a hybrid iterative method for finding a common element of a set of the solutions of mixed equilibrium problems and a set of fixed points of an infinite family.

equilibrium problems include variational inequalities, fixed point, Nash equilibrium and game theory as special cases. Hence collectively, equilibrium problems cover a vast range of applications. Due to the nature of the equilibrium problems, it is

not possible to extend the projection and its variant forms for solving equilibrium problems. To overcome this drawback, one usually uses the auxiliary principle technique. The main and basic idea in this technique is to consider an auxiliary equilibrium problem related to the original problem and then show that the solution of the auxiliary problems is a solution of the original problem. This technique has been used to suggest and analyze a number of iterative methods for solving various classes of equilibrium problems and variational inequalities, see [16,2529] and the references therein.

Related to the equilibrium problems, we also have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems.

In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of finitely many nonexpansive mappings, see [6,8,11, 17,30,3245] and the references therein. In this paper, we suggest and analyze a hybrid iterative method for finding a common element of a set of the solutions of mixed equilibrium problems and a set of fixed points of an infinite family.