The Hölder continuity of solutions to generalized vector equilibrium problems

Abstract

In this paper, by using a weaker assumption, we discuss the Hölder continuity of solution maps for two cases of parametric generalized vector equilibrium problems under the case that the solution map is a general set-valued one, but not a single-valued one.These results extend the recent ones in the literature. Several examples are given for the illustration of our results.

Error analysis of approximate solutions to parametric vector quasiequilibrium problems

“…Motivated by the work on Hölder continuity and perturbation analysis reported in [10,11,22,26,39], this paper aims to establish some results of error analysis of approximate solutions to parametric vector quasiequilibrium problems in metric linear spaces, especially, the current paper may be viewed as a natural continuation and extension of our recent work [26]. In many practical problems, exact solutions do not exist because the perturbation of data of the problem.…”

“…Naturally, there is a need to study the Hölder continuous properties of the set-valued solution mappings. Very recently, Li et al [22] have first studied Hölder continuity of the solution mappings when they are set-valued ones for parametric generalized vector equilibrium problems. Furthermore, Li et al [26] have established the Hölder continuity of the solution mappings, which are not single-valued ones, for two parametric multivalued vector quasiequilibrium problems in metric spaces, and the results obtained improve the corresponding ones in [9][10][11]22] by weakening the Hölder-related assumptions.…”

“…Recently, the semicontinuity, especially the lower semicontinuity, of solution mappings to parametric vector variational inequalities and parametric vector equilibrium problems has been intensively studied in the literature, such as [6][7][8][15][16][17][18][19][20][21][23][24][25][27][28][29][30][31][32]. On the other hand, Hölder continuity of solutions to parametric vector equilibrium problems has also been discussed recently, see [9][10][11]14,22,26], although there may be less works in the literature devoted to this property than to semicontinuity. It is worth mentioning that Anh and Khanh [13] presented the main contributions of their recent works on various kinds of stability for weak and strong vector quasiequilibrium problems.…”

On Hölder continuity of approximate solution maps to vector equilibrium problems

“…The first is the semicontinuity, continuity (or Hausdorff continuity) of solution mappings [3,5,7,10,20,26,30,31] and references therein. The second is the Hölder/Lipschitz continuity of solution mappings [1,2,4,6,8,9,13,17,18,21,[32][33][34]. Observing that most of the works on the Hölder/Lispchitz continuity of the solutions maps imposed strong monotonicity/convexity properties in the data, the solution sets of the problems will be singleton in the neighborhood of the considered point.…”

“…There are some contributions to this field. In [33,34], the authors impose an assumption involving the solutions sets. This assumption is hard to verify since when the stability and sensitivity analyses of the problems are studied, it is assumed that the solution sets are unknown.…”