Existence Of Solutions Sets For Tensor Equilibrium Problems

The equilibrium problem is a kind of universal mathematical model,including fixed point problems,game problems,variational inequality problems and optimization problems,and is widely used in the fields of economy,finance,human resources,transportation and engineering management.We study the existence of solution sets for tensor equilibrium problems.The detailed arrangement of the dissertation is as follows:In chapter 1,we introduce the historical background and research status of equilibrium problems,tensor complementarity problems,tensor variational inequalities and the development of exceptional family of elements.Moreover,we introduce the common symbols,basic concepts and lemmas used in this dissertation.In chapter 2,by employing the notion of exceptional family of elements,we investigate the existence of solution sets for tensor equilibrium problems.Firstly,the definition of exceptional family of elements of tensor equilibrium problems is given,and it is proved that this definition is equivalent to the definition of exceptional family of elements of tensor mixed variational inequality problems.When the tensor is positive semidefinite on the set,it is proved that the existence of solutions of tensor equilibrium problems is equivalent to the nonexistence of exceptional family of elements.Secondly,some necessary and sufficient conditions for the existence of solutions of tensor equilibrium problems and the equivalent conditions for the solution sets to be nonempty and bounded are given.We present a sufficient condition for the solution sets of tensor equilibrium problems to be nonempty and compact.Finally,It is proved that,when the cost function of each player can be decomposed into the sum of a differentiable convex function and a convex function,and satisfies some conditions,a class of multi-person noncooperative games is equivalent to tensor equilibrium problems.In chapter 3,we investigate the existence of solutions of mixed tensor variational inequality problems by utilizing the existence of solutions for optimization problems.Firstly,when the tensor is not positive definite on the unbounded closed convex set,the existence of solutions of mixed tensor variational inequality problems is obtained by using the existence of solutions of coercive optimization problems on the unbounded closed convex set,the solution set of mixed tensor variational inequality problems is proved to be compact by using the properties of the solution set of variational inequality problems.Secondly,for the positive semidefinite tensor on the unbounded closed convex set,the mixed tensor variational inequality problems are transformed into a class of convex optimization problems,and several sufficient conditions are presented such that the solution sets of mixed tensor variational inequalities are nonempty and bounded.Finally,it is proved that,when the cost function of each player satisfies some conditions,a class of multi-person noncooperative games is equivalent to the mixed tensor variational inequality problems.