Epi and coepi-analysis of one class of vector-valued mappings

Abstract

This article deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related to the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.