Study On Some Properties Of Solution Sets For Vector Equilibrium Problems

In this thesis, we discuss the connectedness on efficient solutions for vector equi-librium problems, the upper, lower semicontinuity and continuity on the solution map-pings to parametric (generalized) vector equilibrium problems, the Holder continuity on the (approximate) solution mappings to parametric (generalized) vector (quasi-)equilibrium problems, and Painleve-Kuratowski convergence of the solution sets for perturbed vector equilibrium problems with a sequence of mappings converging. This thesis is divided into nine chapters. It is organized as follows:1. In Chapter1, we describe the development and current researches on the topic of vector equilibrium problems, including the connectedness on efficient solutions to vector equilibrium problems, the upper, lower semicontinuity, continuity and Holder continuityon of the solution mappings to parametric (generalized) vector equilibrium problems, and Painleve-Kuratowski convergence of the solution sets for perturbed vector equilibrium problems with a sequence of mappings converging. We also give the motivation and the main research work.2. In Chapter2, some notions and propositions, which will be frequently used, are shown.3. In Chapter3, we give a density theorem of positive proper efficient solution to vector equilibrium problem. By using the density result, we provide a sufficient condition for the connectedness of efficient solutions to the vector equilibrium prob-lem. These results extend and improve the corresponding ones obtained in [42,43].4. In Chapter4, the lower semicontinuity of solution mappings for parametric (weak) generalized set-valued equilibrium problems is discussed. Under weaker conditions, we provide sufficient conditions for the lower semicontinuity of the solution map-pings to parametric (weak) generalized set-valued equilibrium problems in Haus-dorff topological vector spaces, where the f-solution set be a general set-valued one, but not a singleton. 5. In Chapter5, without the assumption of strict monotonicity, we discuss the lower semicontinuity of solution maps for parametric weak generalized vector equilibrium problems, and get the upper semicontinuity of solution maps to a parametric gen-eralized vector equilibrium problem by using a new proof method. In degenerate case, the continuity of solution maps for parametric generalized vector equilibrium problems are also given. These results extend and improve the corresponding ones obtained in [63,65,66].6. In Chapter6, by using a linear scalarization method, we establish sufficient condi-tions for the Holder continuity of the solution mappings to a parametric generalized vector quasi-equilibrium problem with set-valued mappings, where the considered operator is not necessary a bounded operator. The results extend the recent ones in the recent literature (eg.[87]).7. In Chapter7, without any information of solution set and the assumption of Holder strong monotonicity, we obtain sufficient conditions for Holder continuity of ap-proximate solution mappings to parametric generalized vector equilibrium prob-lems with set-valued mappings in normed spaces. These results are different from the recent ones in the literature([85,111]), and also extend and improve some known results in the literature([84,85,116,117]).8. In Chapter8, without the assumption of strict monotonicity, we obtain the Painleve-Kuratowski Convergence of the weak efficient solution sets, efficient solution sets and global efficient sets for the perturbed vector equilibrium problems (generalized systems) with a sequence of mappings converging under two different kinds of as-sumptions, respectively. Because our condition is weaker than the assumption in [101], the results extend and improve the recent ones in the literature [101].9. In Chapter9, we summarize the results of this thesis and make some discussions.