Convergence of functions: equi-semicontinuity
HTML articles powered by AMS MathViewer
- by Szymon Dolecki, Gabriella Salinetti and Roger J.-B. Wets PDF
- Trans. Amer. Math. Soc. 276 (1983), 409-430 Request permission
Abstract:
We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem.References
- Hédy Attouch, Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35–111 (French, with English summary). MR 551062, DOI 10.1007/BF02411939 —, Sur la $\Gamma$-convergence, Seminaire Brézis-Lions, Collège de France.
- M. L. Bernard-Mazure, Équi-S.C.I., $\Gamma$-convergence et convergence simple, Travaux Sém. Anal. Convexe 11 (1981), no. 1, exp. no. 7, 16 (French). MR 636493
- Giuseppe Buttazzo, Su una definizione generale dei $\Gamma$-limiti, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 3, 722–744 (Italian, with English summary). MR 0500789
- G. Choquet, Convergences, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 23 (1948), 57–112. MR 0025716 S. Dolecki, Role of lower semicontinuity in optimality theory, Proc. Game Theory and Mathematical Economics (O. Moeschlin and D. Pallaschke, eds.), North-Holland, Amsterdam, pp. 265-274.
- Ennio De Giorgi and Tullio Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 6, 842–850 (Italian). MR 448194 C. Kuratowski, Topologie, PWN, Warsaw.
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
- Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, DOI 10.1016/0022-247X(77)90060-9
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 409-430
- MSC: Primary 58E30; Secondary 49D99, 54A20
- DOI: https://doi.org/10.1090/S0002-9947-1983-0684518-7
- MathSciNet review: 684518