Open Access

Some Variational Results Using Generalizations of Sequential Lower Semicontinuity

Fixed Point Theory and Applications20102010:323487

https://doi.org/10.1155/2010/323487

Received: 1 October 2009

Accepted: 14 February 2010

Published: 21 February 2010

Abstract

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.

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Authors’ Affiliations

(1)
Dipartimento di Matematica, Università di Genova

Copyright

© A. Bottaro Aruffo and G. Bottaro. 2010

This article is published under license to BioMed Central Ltd. 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.