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.