Israel Journal of Mathematics

, Volume 54, Issue 1, pp 42–50 | Cite as

Partially monotone operators and the generic differentiability of convex-concave and biconvex mappings

  • J. M. Borwein
Article

Abstract

By studying partially monotone operators, we are able to show among other results that convex-concave and biconvex mappings defined on Asplund spaces or dually strictly convex spaces are respectively generically Fréchet or Gateaux differentiable.

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References

  1. 1.
    E. Asplund,Frechet-differentiability of convex functions, Acta Math.121 (1968), 31–47.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Asplund and R. T. Rockafellar,Gradients of convex functions, Trans. Amer. Math. Soc.139 (1969), 443–467.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. M. Borwein,Continuity and differentiability properties of convex operators, Proc. London Math. Soc.44 (1982), 420–444.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. M. Borwein,Subgradients of convex operators, Math. Operationforschung15 (1984), 179–191.MATHGoogle Scholar
  5. 5.
    J. P. R. Christensen,Topological and Borel Structure, North-Holland, New York, 1974.Google Scholar
  6. 6.
    J. P. Christensen and P. S. Kenderov,Dense strong continuity of mappings and the Radon-Nykodym property, preprint.Google Scholar
  7. 7.
    J. Diestel,Geometry of Banach Spaces, Lecture Notes in Mathematics485, Springer, New York, 1975.MATHGoogle Scholar
  8. 8.
    S. P. Fitzpatrick,Continuity of nonlinear monotone operators, Proc. Amer. Math. Soc.62 (1977), 111–116.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Jouak and L. Thibault,Equicontinuity of families of convex and concave-convex operators, Canad. Math. J.34 (1984), 883–893.MathSciNetGoogle Scholar
  10. 10.
    M. Jouak and L. Thibault,Directional derivatives and almost everywhere differentiability of biconvex and concave-convex operators, preprint.Google Scholar
  11. 11.
    P. Kenderov,Multivalued monotone mappings are almost everwhere single-valued, Studia Math.56 (1976), 199–203.MATHMathSciNetGoogle Scholar
  12. 12.
    P. Kenderov,Monotone operators in Asplund spaces, Compte Rendu Bulg. Sci.30 (1977), 936–964.MathSciNetGoogle Scholar
  13. 13.
    D. G. Larman and R. Phelps,Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc.20 (1979), 115–127.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Mazur,Über konvexe Mengen in linearen normierten Raumen, Studia Math.4 (1933), 70–84.MATHGoogle Scholar
  15. 15.
    G. J. Minty,Monotone (nonlinear) operators in Hilbert space, Duke Math. J.29 (1962), 341–346.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    I. Namioka and R. Phelps,Banach spaces which are Asplund spaces, Duke Math J.42 (1975), 735–750.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J.-P. Penot,Calcul sous-differential et optimisation, J. Funct. Anal.27 (1978), 248–276.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. T. Rockafellar,Convex Analysis, Princeton University Press, 1970.Google Scholar
  19. 19.
    R. T. Rockafellar,Local boundedness of nonlinear, monotone operators, Michigan Math. J.16 (1969), 397–407.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • J. M. Borwein
    • 1
  1. 1.Department of MathematicsDalhousie UniversityHalifaxCanada

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