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On the convergence of subdifferentials of convex functions

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References

  1. H. Attouch, Convergences de fonctions convexes, des sous-différentiels et semigroupes associés. C. R. Acad. Sci. Paris Sér. I Math.284, 539–542 (1977).

    Google Scholar 

  2. H.Attouch, Variational convergence for functions and operators. New York, 1984.

  3. 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.

  4. H. Attouch andA. Damlamian, Strong solutions for parabolic variationai inequalities. Nonlinear Anal.2, 329–353 (1978).

    Google Scholar 

  5. H. Attouch andR. Wets, Quantitative Stability of Variationai Systems I. The Epigraphical Distance. Trans. Amer. Math. Soc.328, 695–730 (1991).

    Google Scholar 

  6. J.-P.Aubin and I.Ekeland, Applied nonlinear analysis. New York 1984.

  7. J.-P.Aubin and H.Frankowska, Set valued analysis. Basel-Boston 1990.

  8. 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).

    Google Scholar 

  9. G. Beer, On Mosco convergence of convex sets. Bull. Austral. Math. Soc.38, 239–253 (1988).

    Google Scholar 

  10. G. Beer, On the Young-Fenchel transform for convex functions. Proc. Amer. Math. Soc.104, 1115–1123 (1988).

    Google Scholar 

  11. 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.

  12. G.Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. J. Nonlinear Anal., to appear.

  13. G.Beer, Wijsman convergence of convex sets under renorming. Preprint.

  14. G. Beer andJ. Borwein, Mosco convergence and reflexivity. Proc. Amer. Math. Soc.109, 427–436 (1990).

    Google Scholar 

  15. G.Beer, Mosco convergence of level sets and graphs of linear functionals. J. Math. Anal. Appl., to appear.

  16. G.Beer, A.Lechicki, S.Levi and S.Naimpally, Distance functionals and the suprema of hyperspace topologies. Ann. Mat. Pura Appl., to appear.

  17. G. Beer andD. Pai, On convergence of convex sets and relative Chebyshev centers. J. Approx. Theory62, 147–169 (1990).

    Google Scholar 

  18. G. Beer andD. Pai, The Prox map. J. Math. Anal. Appl.156, 428–443 (1991).

    Google Scholar 

  19. 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.

    Google Scholar 

  20. J. Borwein, A note onε-subgradients and maximal monotonicity. Pacific J. Math.103, 307–314 (1982).

    Google Scholar 

  21. J. Borwein andS. Fitzpatrick, Mosco convergence and the Kadec property. Proc. Amer. Math. Soc.106, 843–849 (1989).

    Google Scholar 

  22. B.Cornet, Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la décision #7309, Université de Paris Dauphine 1973.

  23. I. Ekeland andR. Temam, Convex analysis and variational problems. North Holland, Amsterdam 1978.

    Google Scholar 

  24. 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).

    Google Scholar 

  25. J.Giles, Convex analysis with application in differentiation of convex functions. Melbourne 1982.

  26. 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.

  27. R.Holmes, A course in optimization and best approximation. LNM257, Berlin-Heidelberg-New York 1972.

  28. 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).

    Google Scholar 

  29. K.Kuratowski, Topology, vol 1. New York 1966.

  30. A. Lechicki andS. Levi, Wijsman convergence in the hyperspace of a metric space. Boll. Un. Mat. Ital. B1, 435–452 (1987).

    Google Scholar 

  31. 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.

  32. L. McLinden, Successive approximation and linear stability involving convergent sequences of optimization problems. J. Approx. Theory35, 311–354 (1982).

    Google Scholar 

  33. U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math.3, 510–585 (1969).

    Google Scholar 

  34. U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl.35. 518–535 (1971).

    Google Scholar 

  35. R.Phelps, Convex functions, monotone operators, and differentiability. LNM1364, Berlin-Heidelberg-New York 1989.

  36. R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings. Pacific J. Math.33, 209–216 (1970).

    Google Scholar 

  37. R. T.Rockafellar, Generalized second derivatives of convex functions and saddle functions. Trans. Amer. Math. Soc., to appear.

  38. S.Simons, Subtangents with controlled slope. Preprint.

  39. 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.

    Google Scholar 

  40. 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.

  41. M. Tsukada, Convergence of best approximations in a smooth Banach space. J. Approx. Theory40, 301–309 (1984).

    Google Scholar 

  42. R. Wijsman, Convergence of sequences of convex sets, cones, and functions II. Trans. Amer. Math. Soc.123, 32–45 (1966).

    Google Scholar 

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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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Keywords

  • Convex Function