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References
H. Attouch, Convergences de fonctions convexes, des sous-différentiels et semigroupes associés. C. R. Acad. Sci. Paris Sér. I Math.284, 539–542 (1977).
H.Attouch, Variational convergence for functions and operators. New York, 1984.
H.Attouch, J.Ndoutoume and M.Thera, Epigraphical convergence of functions and convergence of their derivatives in Banach spaces. Séminaire d'Analyse Convexe Montpellier, Exposé N° 9, 1990.
H. Attouch andA. Damlamian, Strong solutions for parabolic variationai inequalities. Nonlinear Anal.2, 329–353 (1978).
H. Attouch andR. Wets, Quantitative Stability of Variationai Systems I. The Epigraphical Distance. Trans. Amer. Math. Soc.328, 695–730 (1991).
J.-P.Aubin and I.Ekeland, Applied nonlinear analysis. New York 1984.
J.-P.Aubin and H.Frankowska, Set valued analysis. Basel-Boston 1990.
D. Azé andJ.-P. Penot, Qualitative results about the convergence of convex sets and convex functions. Pitman Res. Notes Math. Ser.244, 1–25 (1992).
G. Beer, On Mosco convergence of convex sets. Bull. Austral. Math. Soc.38, 239–253 (1988).
G. Beer, On the Young-Fenchel transform for convex functions. Proc. Amer. Math. Soc.104, 1115–1123 (1988).
G.Beer, Topologies on closed and closed convex sets and the Effros measurability of set valued functions. Séminaire d'Analyse Convexe Montpellier, Exposé N° 2, 1991.
G.Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. J. Nonlinear Anal., to appear.
G.Beer, Wijsman convergence of convex sets under renorming. Preprint.
G. Beer andJ. Borwein, Mosco convergence and reflexivity. Proc. Amer. Math. Soc.109, 427–436 (1990).
G.Beer, Mosco convergence of level sets and graphs of linear functionals. J. Math. Anal. Appl., to appear.
G.Beer, A.Lechicki, S.Levi and S.Naimpally, Distance functionals and the suprema of hyperspace topologies. Ann. Mat. Pura Appl., to appear.
G. Beer andD. Pai, On convergence of convex sets and relative Chebyshev centers. J. Approx. Theory62, 147–169 (1990).
G. Beer andD. Pai, The Prox map. J. Math. Anal. Appl.156, 428–443 (1991).
E. Bishop andR. Phelps, The support functionals of a convex set. In: Proc. Sympos. Pure Math. Vol. VII, 27–35, Amer. Math. Soc., Providence, R. I. 1963.
J. Borwein, A note onε-subgradients and maximal monotonicity. Pacific J. Math.103, 307–314 (1982).
J. Borwein andS. Fitzpatrick, Mosco convergence and the Kadec property. Proc. Amer. Math. Soc.106, 843–849 (1989).
B.Cornet, Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la décision #7309, Université de Paris Dauphine 1973.
I. Ekeland andR. Temam, Convex analysis and variational problems. North Holland, Amsterdam 1978.
S. Francaviglia, A. Lechicki andS. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions. J. Math. Anal. Appl.112, 347–370 (1985).
J.Giles, Convex analysis with application in differentiation of convex functions. Melbourne 1982.
C.Hess, Contributions à l'étude de la mesurabilité, de la loi de probabilité, et de la convergence des multifonctions. Thèse d'état, Montpellier 1986.
R.Holmes, A course in optimization and best approximation. LNM257, Berlin-Heidelberg-New York 1972.
J. Joly, Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarité est bicontinue. J. Math. Pures Appl.52, 421–441 (1973).
K.Kuratowski, Topology, vol 1. New York 1966.
A. Lechicki andS. Levi, Wijsman convergence in the hyperspace of a metric space. Boll. Un. Mat. Ital. B1, 435–452 (1987).
B.Lemaire, Coupling optimization methods and variational convergence. In: Trends in mathematical optimization, series of Num. Math. vol. 84, K. H. Hoffmann, J.-B. Hiriart-Urruty, C. Lemarechal and J. Zowe, eds., 163–179, Basel-Boston 1988.
L. McLinden, Successive approximation and linear stability involving convergent sequences of optimization problems. J. Approx. Theory35, 311–354 (1982).
U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math.3, 510–585 (1969).
U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl.35. 518–535 (1971).
R.Phelps, Convex functions, monotone operators, and differentiability. LNM1364, Berlin-Heidelberg-New York 1989.
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings. Pacific J. Math.33, 209–216 (1970).
R. T.Rockafellar, Generalized second derivatives of convex functions and saddle functions. Trans. Amer. Math. Soc., to appear.
S.Simons, Subtangents with controlled slope. Preprint.
Y. Sonntag, Convergence au sens de Mosco; théorie et applications à l'approximation des solutions d'inéquations. Thèse d'Etat. Université de Provence, Marseille 1982.
Y.Sonntag and C.Zalinescu, Set convergences: An attempt of classification. In: Proceedings of Intl. Conf. on Diff. Equations and Control Theory, Iasi, Romania, August, 1990. Expanded version to appear in Trans. Amer. Math. Soc.
M. Tsukada, Convergence of best approximations in a smooth Banach space. J. Approx. Theory40, 301–309 (1984).
R. Wijsman, Convergence of sequences of convex sets, cones, and functions II. Trans. Amer. Math. Soc.123, 32–45 (1966).
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Research partially supported by NSF grant DMS-9001096. This author would like to thank l'équipe d'analyse convexe of U.S.T.L. Montpellier for its hospitality.
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Attouch, H., Beer, G. On the convergence of subdifferentials of convex functions. Arch. Math 60, 389–400 (1993). https://doi.org/10.1007/BF01207197
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DOI: https://doi.org/10.1007/BF01207197