Study On The Existence Of Solutions For Vector Equilibrium Problem

The vector equilibrium problem is a kind of universal mathematical model,which is widely used in the fields of economy,finance.transportation,resource allocation and project management.The existence of solution set for vector equilibrium problem is the basic problem of vector equilibrium problem.To ensure the existence of solutions to the vector equilibrium problem on unbounded set,generally need to impose coercivity conditions.For the case of the variational inequality problem,the exceptional family of elements method needs to be weaker than the coercivity conditions.There is no literature to use the excep-tional family of elements method to study the existence of solutions for vector equilibrium problem.In this paper,we mainly study the existence and boundedness of solution sets for vector equilibrium problem,equilibrium problem and vector optimization problem.We mainly use the exceptional family of elements to obtain the necessary and sufficient conditions for the solution of the corresponding problem.The details of this article are arranged as follows:In the first chapter,we briefly introduce the historical background and development of vector equilibrium problem and the exceptional family of elements.The basic concepts and lemmas are also given.In the second chapter,we mainly study the existence of weakly efficient solutions and the boundedness of the solution set for vector equilibrium problem in the self-reflex Banach space.by using the dual vector equilibrium problems in bounded set limit solution,we defined an exceptional family of elements for the vector equilibrium problems.We proved that the existence of weak efficient solutions for vector equilibrium problems is equivalent to the non-existence of the exceptional family of elements.We obtain the necessary and sufficient conditions for the existence of weakly efficient solutions of vector equilibrium problems.The necessary and sufficient conditions for the weakly efficient solutions of vector equilibrirum problems in reflex Banach spaces are given.At the same time,several equiva.lence conditions of the equilibrium problem in the literature[17]are extended to the vector equilibrium problem.