Semicontinuity of the minimal solution mappings to parametric set optimization problems on Banach lattices

Abstract

This paper aims at investigating the stability of minimal solutions to set optimization problems on Banach lattices. We introduce the concepts of strictly quasi cone-convexlikeness and supremum mapping of the set-valued mappings and obtain an existence theorem of minimal solutions to set optimization problems by using Zorn's lemma. We first derive the continuity of supremum mapping of the set-valued mapping. Then, we establish the upper semicontinuity and lower semicontinuity of the minimal solution mapping and weak minimal solution mapping to parametric set optimization problems by utilizing strictly quasi cone-convexlikeness and the continuity of supremum mapping. Finally, our main results are applied to semicontinuity of the solution mappings to parametric vector optimization problems.

Keywords:

• Set optimization problem
• Banach lattice
• minimal solution
• upper semicontinuity
• lower semicontinuity

AMS Subject Classifications:

• 46B42
• 49J53
• 90C31

Acknowledgments

The authors are grateful to the editor and the anonymous referees for their valuable comments and suggestions that helped us to improve the quality of 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 [Grant No. 11801257,11961047], the Natural Science Foundation of Jiangxi Provincial Education Department [Grant No. GJJ190286].