Archiv der Mathematik

, Volume 60, Issue 4, pp 389–400 | Cite as

On the convergence of subdifferentials of convex functions

  • Hedy Attouch
  • Gerald Beer
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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. [2]
    H.Attouch, Variational convergence for functions and operators. New York, 1984.Google Scholar
  3. [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.Google Scholar
  4. [4]
    H. Attouch andA. Damlamian, Strong solutions for parabolic variationai inequalities. Nonlinear Anal.2, 329–353 (1978).Google Scholar
  5. [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. [6]
    J.-P.Aubin and I.Ekeland, Applied nonlinear analysis. New York 1984.Google Scholar
  7. [7]
    J.-P.Aubin and H.Frankowska, Set valued analysis. Basel-Boston 1990.Google Scholar
  8. [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. [9]
    G. Beer, On Mosco convergence of convex sets. Bull. Austral. Math. Soc.38, 239–253 (1988).Google Scholar
  10. [10]
    G. Beer, On the Young-Fenchel transform for convex functions. Proc. Amer. Math. Soc.104, 1115–1123 (1988).Google Scholar
  11. [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.Google Scholar
  12. [12]
    G.Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. J. Nonlinear Anal., to appear.Google Scholar
  13. [13]
    G.Beer, Wijsman convergence of convex sets under renorming. Preprint.Google Scholar
  14. [14]
    G. Beer andJ. Borwein, Mosco convergence and reflexivity. Proc. Amer. Math. Soc.109, 427–436 (1990).Google Scholar
  15. [15]
    G.Beer, Mosco convergence of level sets and graphs of linear functionals. J. Math. Anal. Appl., to appear.Google Scholar
  16. [16]
    G.Beer, A.Lechicki, S.Levi and S.Naimpally, Distance functionals and the suprema of hyperspace topologies. Ann. Mat. Pura Appl., to appear.Google Scholar
  17. [17]
    G. Beer andD. Pai, On convergence of convex sets and relative Chebyshev centers. J. Approx. Theory62, 147–169 (1990).Google Scholar
  18. [18]
    G. Beer andD. Pai, The Prox map. J. Math. Anal. Appl.156, 428–443 (1991).Google Scholar
  19. [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. [20]
    J. Borwein, A note onε-subgradients and maximal monotonicity. Pacific J. Math.103, 307–314 (1982).Google Scholar
  21. [21]
    J. Borwein andS. Fitzpatrick, Mosco convergence and the Kadec property. Proc. Amer. Math. Soc.106, 843–849 (1989).Google Scholar
  22. [22]
    B.Cornet, Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la décision #7309, Université de Paris Dauphine 1973.Google Scholar
  23. [23]
    I. Ekeland andR. Temam, Convex analysis and variational problems. North Holland, Amsterdam 1978.Google Scholar
  24. [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. [25]
    J.Giles, Convex analysis with application in differentiation of convex functions. Melbourne 1982.Google Scholar
  26. [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.Google Scholar
  27. [27]
    R.Holmes, A course in optimization and best approximation. LNM257, Berlin-Heidelberg-New York 1972.Google Scholar
  28. [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. [29]
    K.Kuratowski, Topology, vol 1. New York 1966.Google Scholar
  30. [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. [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.Google Scholar
  32. [32]
    L. McLinden, Successive approximation and linear stability involving convergent sequences of optimization problems. J. Approx. Theory35, 311–354 (1982).Google Scholar
  33. [33]
    U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math.3, 510–585 (1969).Google Scholar
  34. [34]
    U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl.35. 518–535 (1971).Google Scholar
  35. [35]
    R.Phelps, Convex functions, monotone operators, and differentiability. LNM1364, Berlin-Heidelberg-New York 1989.Google Scholar
  36. [36]
    R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings. Pacific J. Math.33, 209–216 (1970).Google Scholar
  37. [37]
    R. T.Rockafellar, Generalized second derivatives of convex functions and saddle functions. Trans. Amer. Math. Soc., to appear.Google Scholar
  38. [38]
    S.Simons, Subtangents with controlled slope. Preprint.Google Scholar
  39. [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. [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.Google Scholar
  41. [41]
    M. Tsukada, Convergence of best approximations in a smooth Banach space. J. Approx. Theory40, 301–309 (1984).Google Scholar
  42. [42]
    R. Wijsman, Convergence of sequences of convex sets, cones, and functions II. Trans. Amer. Math. Soc.123, 32–45 (1966).Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Hedy Attouch
    • 1
  • Gerald Beer
    • 1
  1. 1.Departement de MathematiquesUniversité Montpellier II, U. S. T. L.Montpellier Cedex 5France

Personalised recommendations