Stability Of Solutions For Vector Equilibrium Problems

In this thesis, we study two problems: the lower semicontinuity and continuity of the solution mapping to a parametric generalized weak vector equilibrium problem with set-valued mappings and the lower semicontinuity of the solution mapping to a parametric generalized strong vector equilibrium problem with set-valued mappings.In Hausdorff topological vector space, we discuss the lower semicontinuity and continuity of the solution mapping to a parametric generalized weak vector equilibrium problem with set-valued mappings. We firstly define the solution set and f -efficient solutions set to (PGWVEP), and give a kind of sufficient conditions which ensure that the f -efficient solutions set is not empty. Then we prove that the solution set of (PGWVEP) can be expressed as the union of its f -efficient solutions set, and obtain that f -efficient solutions set of (PGWVEP) is lower semicontinuous. By the Theorem 2 of [8, p.114], i.e., the union of a family of lower semicontinuous set-valued mappings is also lower semicontinuous, we establish the lower semicontinuity and continuity of the solution mapping to a parametric generalized weak vector equilibrium problem with set-valued mappings. Finally, some examples are given to illustrate our results. Our consequences improve and generalize the corresponding results in [Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, Journal of Global Optimization, 32 (2005) 543-550] and [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46].In Hausdorff topological vector space, we discuss the lower semicontinuity of the solution mapping to a parametric generalized strong vector equilibrium problem with set-valued mappings. Firstly, we introduce two notions of the efficient solution set and f -efficient solutions set to (PGSVEP), and obtain that the f -efficient solutions set to (PGSVEP) is lower semicontinuous. In particular, under our assumption which is weak than C -strictly monotone assumption, the f -efficients solution may be a general set, but not a singleton. Then, by Lemma 3.1 of [12], we prove that the efficient solution set of (PGSVEP) is dense in the union of its f -efficient solutions set. By the Theorem 2 of [8, p.114] and a conclusion in [21] (a conclusion about the sufficient and necessary condition of lower limits ), we get the lower semicontinuity of the solution mapping to a parametric generalized strong vector equilibrium problem with set-valued mappings. Finally, an example is given to illustrate our results. Our consequences extend the corresponding results in [X.H. Gong and J.C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, Journal of Optimization Theory and Applications, 138 (2008) 197-205.] to set-valued case.