Stability Of The Solution Set For Vector Equilibrium Problems

Vector equilibrium problem is the generalization of variational inequality, it contains as special cases, for instance, optimization problem, Nash equilibrium problem, complementarity problem, fixed point problem and variational inequality. Its theories and methods are widely used in the areas of mathematical programming, theory of multi- objective programming, theory of management science, engineering techniques, mathematical and physical economic and the system of society economic, etc. Studying stability for vector equilibrium problems is an important aspect in vector equilibrium theory. In this thesis, we study stability of the solution set for vector equilibrium problems, such as symmetric vector quasi-equilibrium problem ,the system of vector quasi-equilibrium problems, andζ-efficiency optimization problem, and we study the existence result and stability of the solution set for symmetric vector quasi-equilibrium problem under the condition that its payoff mappings are cone-convex , and give some applications. The main research works are as follows:In Chapter 2, we obtain two stability theorems of the solution set for symmetric vector equilibrium problems. These theorems include a generic stability theorem and a existence result of essential component. We show that, under suitable assumptions, the solution set for symmetric vector quasi-equilibrium problems including many problems is stable and we also give some applications.In Chapter 3, we first derive a new existence theorem of the solutions for vector quasi- equilibrium problems. Under suitable assumptions, we prove that, for every vector quasi-equilibrium problem, there exists at least one essential component of the set of its solutions. As applications, we show that, for every system of vector quasi-equilibrium problems, there exists at least one essential component of its solution set in the uniform topological space of objective functions and constraint correspondences.In Chapter 4, in a normed space, the distance between two objective mappings is defined by the uniform topology. In this topology, using the famous Fort theorem, we show that, the solution set ofζ-efficiency in vector optimization problems is generically stable about its monotone continuous linear function, generically stable about its objective mapping, and generically stable about its monotone continuous linear function and objective mapping.In Chapter 5, applying the existence theorem of the solutions for vector quasiequilib- rium problems we obtain in Chapter 3, we obtain a new existence theorem of the solutions for symmetric vector quasi-equilibrium problem under the condition that its payoff mappings are cone-convex. The theorem, under weaker conditions, solves the second open problem proposed by Fu in [2], which is whether there is a solution for symmetric vector quasi- equilibrium problem when its payoff function are cone-convex. At last we discus generic stability of the solution set for symmetric vector quasi-equilibrium under the condition of cone-cone-convexity in normed linear spaces.Xu. Zhu and Lu introduced some relationships between Nash equilibrium , variational inequalities , and generalized equilibrium . They also investigated the relations among monot -onicities of their corresponding mappings and bifuctions . The results of their paper would be the theoretical basis for further studying in Nash equilibrium and generalized equilibrium problems. The stability of Nash equilibrium , variational inequalities , and generalized equili-brium is very important whether in real applications or in theory, but Xu, Zhu and Lu did not discussed their stability. In Chapter 6, applying the stability results, methods and techni- ques in Chapter 2, Chapter 3, Chapter 4 and Chapter 5, we study the stability of the Nash eq- uilibrium , variational inequalities , and generalized equilibrium in [3], and obtain some gene- ric stabilities and some essential component existence results for them.