Guangxi Normal University Exceptional Families Of Elements For Equilibrium Problems

Equilibrium problems are promotion of variational inequalities, convex optimization, fixed point theorem, complementarity problem and Nash Equilibrium problems. Further discussing equilibrium problems provide a research framework of system and extension for studying different fields such as finance, economics, network analysis, traffic and real estate. We study existence of equilibrium problems and dual equilibrium problems by employing the notion of exceptional family of elements.First of all, we present definition of exceptional family of equilibrium problems, where exceptional family is restrict Kr on two spaces for fixed r> dist(x, K). Next we restrict equilibrium problem on a bounded closed set Kr x Kr-a closed ballkrxKr. Considering the problem of(EP)r and (DEP)r, we discover that(EP)r has a solution on the boundary as EP has no solution. We develop a kind of way construction exceptional family as a radius of closed sphere tend to infinity. Using this way, we develop the relation between equilibrium problems and exceptional family. We has the main results as follows:Firstly, we consider dual equilibrium problems, if (DEP) has a solution, then there does not exist an exceptional family of elements for all x∈X.We add condition of-f to be pseudomonotone map, if(EP) has a solution, then there does not exist an exceptional family of elements for all x E X.We introduce condition of P4, if there does not exist an exceptional family of elements for all x E X, then imply (EP) has a solution.Under the conditions of pseudomonotone-f and P4, we prove (EP) problems have a solution if only if there not exist an exceptional family of elements for all x E X.We we develop the relation between equilibrium problems and exceptional family, and present two conditions of nonexist an exceptional family of elements.We define another conception of the exception family of elements, strengthened the fore condition of exceptional family, we develop corresponding existence results of equilib-rium problems, But, we have the nonexistence of exceptional family of elements is a weak condition. Finally, we apply the conclusions of(EP) problems to the convex optimization and variational inequality problems which are proved.