Geodesic
Generalizations of Sequential Lower Semicontinuity
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 293-318.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

In [7] W.A. Kirk and L.M. Saliga and in [3] 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 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 semi-continuity 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 semi-continuity; 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 [3] 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. 
@article{BUMI_2008_9_1_2_a0,
     author = {Aruffo, Ada and Bottaro, Gianfranco},
     title = {Generalizations of {Sequential} {Lower} {Semicontinuity}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {293--318},
     publisher = {mathdoc},
     volume = {Ser. 9, 1},
     number = {2},
     year = {2008},
     zbl = {1209.49010},
     mrnumber = {2424295},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a0/}
}
TY  - JOUR
AU  - Aruffo, Ada
AU  - Bottaro, Gianfranco
TI  - Generalizations of Sequential Lower Semicontinuity
JO  - Bollettino della Unione matematica italiana
PY  - 2008
SP  - 293
EP  - 318
VL  - 1
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a0/
LA  - en
ID  - BUMI_2008_9_1_2_a0
ER  - 
%0 Journal Article
%A Aruffo, Ada
%A Bottaro, Gianfranco
%T Generalizations of Sequential Lower Semicontinuity
%J Bollettino della Unione matematica italiana
%D 2008
%P 293-318
%V 1
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a0/
%G en
%F BUMI_2008_9_1_2_a0
Aruffo, Ada; Bottaro, Gianfranco. Generalizations of Sequential Lower Semicontinuity. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 293-318. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a0/

[1] J. M. Borwein - Q. J. Zhu, Techniques of Variational Analysis, CMS Books in Mathematics, 20, Springer (2005). | MR | Zbl

[2] A. Bottaro Aruffo - G. F. Bottaro, Some Variational Results Using Generalizations of Sequential Lower Semicontinuity, to appear. | MR | Zbl

[3] Y. Chen - Y. J. Cho - L. Yang, Note on the Results with Lower Semi-Continuity, Bull. Korean Math. Soc., 39 (2002), no. 4, 535-541. | DOI | MR | Zbl

[4] A. L. Dontchev - T. Zolezzi, Well-posed Optimization Problems, Lecture Notes in Mathematics, 1543, Springer (1993). | DOI | MR | Zbl

[5] I. Ekeland - R. Temam, Analyse convexe et probleÁmes variationnelles, Dunod, Gauthier-Villars (1974). | MR

[6] J. R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Pitman Res. Notes Math., 58, Pitman (1982). | MR | Zbl

[7] W. A. Kirk - L. M. Saliga, The Brézis-Browder Order Principle and Extensions of Caristi's Theorem, Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000), Nonlinear Anal., 47 (2001), no. 4, 2765-2778. | DOI | MR | Zbl

[8] J. Morgan - V. Scalzo, Pseudocontinuity in Optimization and Nonzero-Sum Games, J. Optim. Theory Appl., 120 (2004), no. 1, 181-197. | DOI | MR | Zbl

[9] J. Morgan - V. Scalzo, New Results on Value Functions and Applications to MaxSup and MaxInf Problems, J. Math. Anal. Appl., 300 (2004), no. 1, 68-78. | DOI | MR | Zbl

[10] J. Morgan - V. Scalzo, Discontinuous but Well-Posed Optimization Problems, SIAM J. Optim., 17 (2006), no. 3, 861-870 (electronic). | DOI | MR | Zbl

[11] A. E. Taylor - D. C. Lay, Introduction to Functional Analysis, second edition, Wiley (1980). | MR | Zbl