Generalizations of Sequential Lower Semicontinuity

Aruffo, Ada, and Bottaro, Gianfranco. "Generalizations of Sequential Lower Semicontinuity." Bollettino dell'Unione Matematica Italiana 1.2 (2008): 293-318. <http://eudml.org/doc/290455>.

@article{Aruffo2008,
abstract = {In [7] W.A. Kirk and L.M. Saliga and in [3] Y. Chen, Y.J. Cho and L. Yang introduced lower semicontinuity from above, a generalization of sequential lower semicontinuity, and they showed that well-known results, such as some sufficient conditions for the existence of minima, Ekeland's variational principle and Caristi's fixed point theorem, remain still true under lower semicontinuity from above. In the second of the above papers the authors also conjectured that, for convex functions on normed spaces, lower semicontinuity from above is equivalent to weak lower semi-continuity from above. In the present paper we exhibit an example showing that such conjecture is false; moreover we introduce and study a new concept, that generalizes lower semicontinuity from above and consequently also sequential lower semi-continuity; moreover we show that: (1) such concept, for convex functions on normed spaces, is equivalent to its weak counterpart, (2) the above quoted results of [3] regarding sufficient conditions for minima remain still true for such a generalization,(3) the hypothesis of lower semicontinuity can be replaced by this generalization also in some results regarding well-posedness of minimum problems.},
author = {Aruffo, Ada, Bottaro, Gianfranco},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {293-318},
publisher = {Unione Matematica Italiana},
title = {Generalizations of Sequential Lower Semicontinuity},
url = {http://eudml.org/doc/290455},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Aruffo, Ada
AU - Bottaro, Gianfranco
TI - Generalizations of Sequential Lower Semicontinuity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/6//
PB - Unione Matematica Italiana
VL - 1
IS - 2
SP - 293
EP - 318
AB - In [7] W.A. Kirk and L.M. Saliga and in [3] Y. Chen, Y.J. Cho and L. Yang introduced lower semicontinuity from above, a generalization of sequential lower semicontinuity, and they showed that well-known results, such as some sufficient conditions for the existence of minima, Ekeland's variational principle and Caristi's fixed point theorem, remain still true under lower semicontinuity from above. In the second of the above papers the authors also conjectured that, for convex functions on normed spaces, lower semicontinuity from above is equivalent to weak lower semi-continuity from above. In the present paper we exhibit an example showing that such conjecture is false; moreover we introduce and study a new concept, that generalizes lower semicontinuity from above and consequently also sequential lower semi-continuity; moreover we show that: (1) such concept, for convex functions on normed spaces, is equivalent to its weak counterpart, (2) the above quoted results of [3] regarding sufficient conditions for minima remain still true for such a generalization,(3) the hypothesis of lower semicontinuity can be replaced by this generalization also in some results regarding well-posedness of minimum problems.
LA - eng
UR - http://eudml.org/doc/290455
ER -