Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Copyright © 2010 Ada Bottaro Aruffo and Gianfranco Bottaro. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kirk and Saliga and then Chen et al. introduced lower semicontinuity from above, a generalization of sequential
lower semicontinuity, and they showed that well-known results, such as
Ekeland's variational principle and Caristi's fixed point theorem, remain still
true under lower semicontinuity from above. In a previous paper we introduced
a new concept that generalizes lower semicontinuity from above. In the
present one we continue such study, also introducing other two new generalizations
of lower semicontinuity from above; we study such extensions, compare
each other five concepts (sequential lower semicontinuity, lower semicontinuity
from above, the one by us previously introduced, and the two here defined) and,
in particular, we show that the above quoted well-known results remain still
true under one of our such generalizations.