Stability of the set of solutions for generalized vector equilibrium problems with cone constraints

Abstract

In this paper, the Hausdorff quasi-upper and Hausdorff quasi-lower semicontinuity for set-valued mappings are introduced to derive the upper semicontinuity and lower semicontinuity of cone constraint mapping in normed vector spaces. We discuss existence of solutions for generalized vector equilibrium problems with cone constraints. Moreover, we establish the lower semicontinuity of (weakly) efficient approximate solution mappings, upper semicontinuity of weakly efficient approximate solution mappings, and Hausdorff upper semicontinuity of efficient approximate solution mappings to parametric generalized vector equilibrium problems with cone constraints under some suitable conditions. Some examples are also given to illustrate our main results.

Keywords:

• Generalized vector equilibrium problem
• cone constraint
• existence of solutions
• upper semicontinuity
• lower semicontinuity

AMS Subject Classifications:

• 49J40
• 49K40
• 90C31

Acknowledgments

The authors are grateful to the editors and reviewers for their constructive comments, which helps us to improve the paper.

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11671282, 11801257).