Semi-continuity of the solution maps of set-valued equilibrium problems with equilibrium constraints

Abstract

In this paper, we study the parametric set-valued equilibrium problems with equilibrium constraints based on the comparison of objective values of set-valued maps by set order relations. We introduce new notions of generalized concavity for set-valued maps and study their properties as well as their relationship with other existing well-known notions. By using the generalized concavity and semi-continuity of set-valued maps, we study the existence of solutions for set-valued equilibrium problems when the equilibrium condition is missing. We further establish sufficient conditions for lower/upper semi-continuity of the solution maps of the set-valued equilibrium problems involving set order relations. Several examples are provided to illustrate the derived results.

Keywords:

• Set order relations
• set-valued equilibrium problems
• generalized concavity
• upper/lower semi-continuity

AMS CLASSIFICATIONS:

• 90C31
• 90C33
• 90C46

Acknowledgments

The authors are grateful to the handling editor and two anonymous referees for their valuable comments and suggestions, which helped us to improve the previous version of this paper.

Disclosure statement

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

Additional information

Funding

In this research, first author was supported by DST-SERB Project No. MTR/2017/000135, second author was supported by UGC-Dr. D.S. Kothari Post Doctoral Fellowship No. F.4-2/2006 (BSR)/MA/19-20/0040, and third author was support by CSIR-UGC Junior Research Fellowship, India, Human Resource Development Group (Ref. No. 19/06/2016(i)EU-V).