Lower Semicontinuity of the Solution Mappings to a Parametric Generalized Ky Fan Inequality

Semicontinuity of approximate solution mappings for parametric generalized weak vector equilibrium problems

“…Under the assumptions which do not contain any information about solution mappings, Wang and Li [26] established the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem by using a scalarization method. The obtained results improve the corresponding ones in [14,15,7,21,28]. Wang et al [27] establish the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result.…”

“…Chen and Li [6] discussed the lower semicontinuity and continuity results of the solution sets to a parametric strong vector equilibrium problem and a parametric weak vector equilibrium problem without the uniform compactness assumption. By virtue of a key assumption that includes the information about the solution set, Li and Fang [21] established the lower semicontinuity of the solution mappings to a parametric weak vector equilibrium problem with vector-valued mappings. Under the assumption of the f-property, Xu and Li [28] obtained the lower semicontinuity of the solution mapping to a parametric generalized strong vector equilibrium problem by using a scalarization method.…”

Semicontinuity of the Solution Mapping to a Parametric Generalized Weak Ky Fan Inequality

“…The stability analysis of solution mappings for parametric Ky Fan inequalities is another important topic in the vector optimization theories and applications. Recently, there have been many results on the continuity, especially the lower semicontinuity, of the solution mappings for parametric (generalized) Ky Fan inequalities in the literature; for example, see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], where the (generalized) Ky Fan inequalities are called vector equilibrium problems or generalized systems.…”

“…By following the ideas in [8] and based on a theorem of Berge [23] saying that the union of a family of lower semicontinuous set-valued mappings is lower semicontinuous, Gong [9], Chen et al [13], and Li et al [15] have extended the lower semicontinuity results of Cheng and Zhu [8] to parametric generalized weak Ky Fan inequalities under a suitable strict cone-monotonicity assumption. Li and Fang [16], Peng et al [18], Chen and Huang [19], and Wang and Li [17] have used the linear scalarization approach to improve the lower semicontinuity results in [9,13,15] by weakening the strict cone-monotonicity assumption. However, the key assumption in [16,18,19] involves the information on the solution set.…”

“…Li and Fang [16], Peng et al [18], Chen and Huang [19], and Wang and Li [17] have used the linear scalarization approach to improve the lower semicontinuity results in [9,13,15] by weakening the strict cone-monotonicity assumption. However, the key assumption in [16,18,19] involves the information on the solution set.…”