Eixstence And H(?)lder Continuous For Generalized Vector Equilibrium Prolems

In this thesis, we study two problems: the existence results of solutions for generalized vector quasi-equilibrium problems and H(o|¨)lder continuity of solutions to generalized vector equilibrium problems. The detailed contents are listed as following:In Hausdorff topological vector space, we discuss the existence results of solutions for three types of generalized vector quasi-equilibrium problem ( (GVQEP1), (GVQEP2) and (GVQEP3), for short) respectively. We firstly define the four kinds of the relatively weak generalized convexity, and discuss the relations among the four types of generalized convexity. And in virtue of the four types of generalized convexity definitions, we use Fan-Ky KKM theorem to obtain separately the existence theorems of solutions for the three typs of generalized vector quasi-equilibrium problem. Simultaneously, we give an example to expalin that the existence result for the generalized vector quasi-equilibrium problem (GVQEP1) is different from the one in [5], the existence result for generalized vector quasi-equilibrium problem (GVQEP2) is different from the one in [36]. In addition, we obtain an existence result of solutions for vector quasi optimization problems by using the existence result for (GVQEP1).In the metric spaces, we discuss the H(o|¨)lder continuities of solutions for two types of perturbed generalized vector equilibrium problems. In general, it is well-known that the solution set of vector equilibrium problems is not a single value. Until now there is no paper to discuss the H(o|¨)lder continuities of solution maps for perturbed vector equilibrium problems when their solution maps are not single-valued ones, but general set-valued ones. Thus, in this paper, we discuss the the H(o|¨)lder continuities of solution maps for the two classes of perturbed generalized vector equilibrium problems (PGVEP1) and (PGVEP2). We obtain a sufficient condition which guarantees the H(o|¨)lder continuities of the solution maps for (PGVEP1) and (PGVEP2), respectively。Simultaneously, we also use an examples to illustrate that theorems of H(o|¨)lder continuities of the solution maps for (PGVEP1) and (PGVEP2) are different from the corresponding ones in [7], [14] , respectively.