Some Variational Results Using Generalizations of Sequential Lower Semicontinuity

W.A. Kirk and L.M. Saliga and then 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 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 such our generalizations.