Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 323487, 21 pages
Research Article

Some Variational Results Using Generalizations of Sequential Lower Semicontinuity

Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy

Received 1 October 2009; Accepted 14 February 2010

Academic Editor: Mohamed Amine Khamsi

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.