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

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

“…In [16,18,19], the authors used some assumptions, which also include information on the solution set. The main advantage of assumption (ii) in Theorem 16 is that it does not require any information on solution set ( ) for each ∈ .…”

“…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.…”

“…It is worth noting that the linear scalarization approach to the semicontinuity of solution mappings in [8,9,13,[15][16][17][18][19] requires (generalized) cone-convexity of the objective mappings or strict cone-monotonicity. To avoid using these assumptions, nonlinear scalarization approaches have been applied for discussing the stability analysis in parametric generalized Ky Fan inequalities.…”