On the concept of \(\Gamma \)-convergence for locally compact vector-valued mappings
The authors work in the following general framework: \(X,Y\) are real Banach spaces, \(Y\) being ordered by a pointed closed convex cone \(\Lambda\) with nonempty interior. One considers functions \(f:X\to Y^\bullet,\) where \(Y^\bullet=Y\cup\{\infty_\Lambda\},\) and one studies their properties, like sequential lower semi-continuity or local compactness. One defines the lower lsc locally compact regularization of a function \(f\), one proves its existence and one gives characterizations in terms of the sequential closure of the co-epigraph of the function \(f\). Also, the notion of \(\Gamma\)-convergence of sequences of functions is extended to this context and one establishes relations with the Kuratowski convergence of their co-epigraphs.NEWLINENEWLINESemicontinuity and regularization concepts for vector-valued functions were studied also by \textit{M. Ait Mansour, A. Metrane} and \textit{M. Théra}, J. Glob. Optim. 35, No. 2, 283--309 (2006; Zbl 1121.49013) and by \textit{M. Ait Mansour, C. Malivert} and \textit{M. Théra}, Optimization 56, No. 1--2, 241--252 (2007; Zbl 1120.26029).NEWLINENEWLINEThe paper contains some misprints (for instance, Def.\ 2.4 of the infimum is not correct; in Def. 2.5 there is an unexplained term \(L^1(D_T)\); in formula (4.1) \(\lim\) should be replaced with \(\liminf\)). Also, some of the results in the present paper overlap with results ones from another paper by the same authors [``Epi and coepi-analysis of one class of vector-valued mappings'', Optimization, \url{doi:10.1080/02331934.2012.676643}].
scientific article; zbMATH DE number 6049725