On The Stability And Existence Of Vector Optimization Problems

In this thesis, we study two problems: the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems with set-valued mappings and the existence of solution for the system of vector equilibrium problem.On the one hand, we study the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems (PGVEP and PGWVEP) with set-valued mappings in Banach spaces, which are generalizations of the parametric set-valued vector equilibrium problems. We firstly define an assumption which is weaker than C-strictly monotonicity. Under this assumption, the f -efficients solution may be a general set, but not a singleton. Then we obtain that f -efficient solutions set of PGVEP and PGWVEP are lower semicontinuous, respectively. By the Theorem 2 of [41, p.114], i.e., the union of a family of lower semicontinuous set-valued mappings is also lower semicontinuous, we establish the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems with set-valued mappings. Finally, some examples are given to illustrate our results. Our results improve and generalize the corresponding ones in [S.J. Li, H.M. Li, C.R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems[J], Bulletin of the Australian Mathematical Society, 2010 , 81: 85-95.] and [Z.M. Fang, S.J. Li and X.B. Li, Lower semicontinuity of solution mappings to parametric generalized vector equilibrium problems(submitted) ].On the other hand, in locally convex Banach space, we define two mappings with the help of the functionΔ_A and prove that they are the gap functions for system of vector equilibrium problems (SVEP and VEP), respectively. Simultaneously, some existence results for SVEP and VEP are also proved, respectively.