Generalizations of sequential lower semicontinuity

W.A. Kirk, L.M. Saliga and Y. Chen, Y.J. Cho, 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 semicontinuity 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 semicontinuity; 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 Y. Chen, Y.J. Cho, L. Yang 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.