Semicontinuity and closedness of parametric generalized lexicographic quasiequilibrium problems

Abstract

This paper is mainly concerned with the upper semicontinuity, closedness, and the lower semicontinuity of the set-valued solution mapping for a parametric lexicographic equilibrium problem where both two constraint maps and the objective bifunction depend on both the decision variable and the parameters. The sufficient conditions for the upper semicontinuity, closedness, and the lower semicontinuity of the solution map are established. Many examples are provided to ensure the essentialness of the imposed assumptions.

Remarks on Pseudo-continuity

“…However, in [8] the authors showed that the same does not hold for transfer upper continuity. In a similar way, it does not hold for upper pseudo-continuity, a contra-example can be found in [29].…”

“…The notion of pseudo-continuity has gained more attention, for instance Anh, Khanh and Van [3] studied the well-posedness for quasi-equilibrium problems and quasioptimization problems. Wangkeeree, Bantaojai and Yimmuang [29] dealt with continuity properties of the set-valued solution map for a special kind of quasi-equilibrium problem. Al-Homidan, Hadjisavvas and Shaalan [1] implicitly used the concept of pseudo-continuity in order to characterize quasi-convex functions as a composition of two functions where one of them has the property that every local minimum is a global minimum.…”