Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem

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

“…The obtained results extend and generalize the corresponding ones in [14] and [6]. 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].…”

“…Since F(x, ·, µ) is C-subconvexlike on E, by Definition 2.5, F(x, ·, µ) + {tc 0 } is also C-subconvexlike on E. It follows from the proof similar to [26,Lemma 3.2] that…”

“…Motivated by the work reported in [17,23,25,26,27], the aim of this paper is to establish the lower semicontinuity and the upper semicontinuity of the approximate solution mapping to a parametric generalized weak vector equilibrium problem (in short, PGWVEP). By a scalarization method and introducing a new set-valued mapping, we establish the lower semicontinuity of the approximate solution mapping to PGWVEP without the assumptions about monotonicity and approximate solution mappings.…”

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.…”