Stability Of Vector Equilibrium Problems

In this thesis, we study the upper, lower semicontinuity and continuity results on the solution mappings to parametric vector variational inequalities and vector equilibrium problems, minimax theorem in the sense of lexicographic order, and Painlevé-Kuratowski convergence of the solution sets for the perturbed vector equilibrium problems with a sequence of mappings converging. This thesis is divided into nine chapters. It is organized as follows:In Chapter 1, we describe the development and researches on the topic of vector optimization, including the semicontinuity of the solution mappings to parametric vector equilibrium problems and vector variational inequalities, minimax theorem in the sense of lexicographic order, and Painlevé-Kuratowski convergence of the solution sets for the perturbed vector equilibrium problems and vector variational inequalities with a sequence of mappings converging. Also, the motivation is given and main works are listed.In Chapter 2, the stability of the solution sets to parametric weak vector equilibrium problems (which are also called the weak efficient solution sets to parametric generalized systems), the solution sets to parametric vector equilibrium problems (which are also called the efficient solution sets to parametric generalized systems) and the global efficient solution sets to parametric vector equilibrium problems is established. Under a weaker assumption than the condition of C -strict monotonicity, the lower semicontinuity, upper semicontinuity and continuity are derived by a scalarization method and a property involving the union of a family of lower semicontinuous set-valued mappings. Our consequences extend the corresponding results in [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46] and [C.R. Chen and S.J. Li, On the solution continuity of parametric generalized systems, Pacific Journal of Optimization, 6 (2010) 141-151].In Chapter 3, by introducing the assumptions similar as H?lder-related, the lower semicontinuity of the solution mapping to two classes of parametric generalized vector equilibrium problems involving set-valued mappings is obtained in general metric spaces. Simultaneously, by using the same method, the lower semicontinuity of the solution mapping to two classes of parametric generalized strong vector equilibrium problems involving set-valued mappings is firstly investigated.In Chapter 4, the upper semicontinuity and continuity of solution mapping to a class of parametric weak vector variational inequality are investigated with the operator not continuity but v -hemicontinuity and C -pseudomonotonicity.In Chapter 5, lexicographic vector equilibrium problems, lexicographic weak vector equlibrium problems and lexicographic strong vector equlibrium problems are introduced and their relationships are studied. Particularly, by using the idea of sequential process, the upper semicontinuity and closedness of the solution set map are established for a parametric lexicographic strong vector equilibrium problem.In Chapter 6, based on the assumption of Cl ex-uniformly same order, a minimax theorem and a saddle point theorem are obtained for vector valued functions in the sense of lexicographic order, respectively. An equivalent relationship between the minimax inequality and the saddle point is established.In Chapter 7, the perturbed vector equilibrium problems with a sequence of mappings are studied in a real locally convex Hausdorff topological vector spaces. By using a scalarization method, Painlevé-Kuratowski convergences of the solution sets to perturbed weak vector equilibrium problems, the solution sets and various proper efficient solution sets to perturbed vector equilibrium problems are obtained.In Chapter 8, we summarize the results of this thesis and make some discussions.