Semicontinuity of the solution maps to vector equilibrium problems with equilibrium constraints

ABSTRACT

In this paper, we are concerned with parametric vector equilibrium problems with equilibrium constraints. We first introduce concepts of generalized semicontinuity and convexity of vector-valued maps in ordered spaces and discuss their properties, as well as their relationships with well-known concepts. Next, employing mentioned concepts of generalized semicontinuity of vector-valued maps, we study sufficient conditions of upper semicontinuity for solution maps to such problems. Then, combining these generalized semicontinuity along with generalized convexity assumptions, sufficient conditions for lower semicontinuity of solution maps to the reference problems are established. Several examples are provided to illustrate the relaxation and essentialness of assumptions. Our results are new, even for special cases of the underlying problems.

Keywords:

• Equilibrium problems with equilibrium constraints
• upper/lower semicontinuity
• level closedness
• generalized convexity

AMS CLASSIFICATIONS:

• 49J15
• 49K40
• 90C31

Acknowledgements

The author wishes to thank the anonymous referees for the careful reading of the paper and valuable comments and suggestions. 

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01-2020.11].