Skip to main content
Log in

Separable Reduction in the Theory of Fréchet Subdifferentials

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

We prove a separable reduction theorem for the Fréchet subdifferential that contains all earlier results of that sort as particular cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bourgin, R.D.: Geometric Aspects of Convex Sets with the Radon–Nikodým Property. Lect. Notes, vol. 993. Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  2. Borwein, J.M., Moors, W.B.: Separable determination of integrability and minimality of the Clarke subdifferential mapping. Proc. Am. Math. Soc. 128, 215–221 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Books in Mathematics, Canadian Math. Soc. and Springer, New York (2005)

    Google Scholar 

  4. Cúth, M.: Separable reduction theorems of elementary submodels. Fundam. Math. 219, 191–222 (2012)

    Article  MATH  Google Scholar 

  5. Fabian, M.: On classes of subdifferentiability spaces of Ioffe. J. Nonlin. Anal. Theory Methods Appl. 12, 63–74 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carol. 30, 51–56 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Fabian, M., Mordukhovich, B.: Separable reduction and supporting properties of Fréchet–like normals in Banach spaces. Can. J. Math. 51, 26–48 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fabian, M., Mordukhovich, B.: Separable reduction and extremal principles in variational analysis. Nonlinear Analysis, Theory, Methods, Appl. 49, 265–292 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fabian, M., Zhivkov, N.V.: A characterization of Asplund spaces with help of local ε-supports of Ekeland and Lebourg. C. R. Acad. Bulg. Sci. 38, 671–674 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Ioffe, A.D.: On subdifferentiability spaces. Ann. N. Y. Acad. Sci. 410, 107–121 (1983)

    Article  MathSciNet  Google Scholar 

  11. Ioffe, A.D.: Separable reduction revisited. Optimization 60(1–2), 211–221 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ioffe, A.D.: On the general theory of subdifferentials. Adv. Nonlinear Anal. 1, 47–120 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Jules, F., Lassonde, M.: Subdifferential estimate of the directional derivative, optimality criterion and separation principles. Optimization doi:10.1080/02331934.2011.645034

  14. Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24, 684–687 (1980, in Russian)

    MathSciNet  MATH  Google Scholar 

  15. Lindenstrauss, J.: On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J. 10, 241–252 (1963)

    Article  MathSciNet  Google Scholar 

  16. Lindenstrauss, J., Preiss, D., Tišer, J.: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. Ann. Math. Studies, vol. 179. Princeton University Press (2012)

  17. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I. Basic Theory. Series of Comprehensive Studies in Mathematics, vol. 330. Springer-Verlag, Berlin (2006)

    Book  Google Scholar 

  18. Penot, J.P.: A short proof of the separable reduction theorem. Demonstratio Math. 4, 653–663 (2010)

    MathSciNet  Google Scholar 

  19. Penot, J.P.: Calculus Without Derivatives. Graduate Texts in Mathematics. Springer (2013)

  20. Phelps, R.R.: Convex Functions, Monotone Mappings and Differentiability, 2nd edn. Lect. Notes, vol. 1364. Springer-Verlag, Berlin (1993)

    Google Scholar 

  21. Preiss, D.: Gâteaux differentiable functions are somewhere Fréchet differentiable. Rend. Circ. Mat. Palermo 33, 122–133 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zajíček, L.: Fréchet differentiability, strict differentiability, and super-differentiability. Czechoslov. Math. J. 41, 471–489 (1991)

    Google Scholar 

  23. Zajíček, L.: Fréchet differentiability on Asplund spaces via almost everywhere differentiability on many lines. J. Convex Anal. 19, 23–48 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexander Ioffe.

Additional information

In honor of Petar S. Kenderov at the occasion of his 70th birthday.

Supported by grant P201/12/0290, and by RVO: 67985840.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fabian, M., Ioffe, A. Separable Reduction in the Theory of Fréchet Subdifferentials. Set-Valued Var. Anal 21, 661–671 (2013). https://doi.org/10.1007/s11228-013-0256-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-013-0256-1

Keywords

Mathematics Subject Classifications (2010)

Navigation