Skip to main content
SpringerLink
Log in

Location, Identification, and Representability of Monotone Operators in Locally Convex Spaces

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

Abstract

In this paper we study, in the relaxed context of locally convex spaces, intrinsic properties of monotone operators needed for the sum conjecture for maximal monotone operators to hold under classical interiority-type domain constraints.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauschke, H.H., McLaren, D.A., Sendov, H.S.: Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick. J. Convex Anal. 13(3–4), 499–523 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3–4), 561–586 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10(4), 297–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burachik, R.S., Svaiter, B.F.: Maximal monotonicity, conjugation and the duality product. Proc. Am. Math. Soc. 131(8), 2379–2383 (electronic) (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fitzpatrick, S., Phelps, R.R.: Bounded approximants to monotone operators on Banach spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(5), 573–595 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fitzpatrick, S., Phelps, R.R.: Some properties of maximal monotone operators on nonreflexive Banach spaces. Set-Valued Anal. 3(1), 51–69 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), volume 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., p 5965. Austral. Nat. Univ., Canberra (1988)

  8. Gossez, J.P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gossez, J.P.: On the range of a coercive maximal monotone operator in a nonreflexive Banach space. Proc. Am. Math. Soc. 35, 88–92 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Marques Alves, M., Svaiter, B.F.: On Gossez type (D) maximal monotone operators. J. Convex Anal. 17(3–4), 1077–1088 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Martínez-Legaz, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Penot, J.P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8), 855–871 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)

  15. Simons, S.: Minimax and Monotonicity, volume 1693 of Lecture Notes in Mathematics. Springer, Berlin (1998)

    Google Scholar 

  16. Verona, A., Verona, M.E.: Remarks on subgradients and 𝜖-subgradients. Set-Valued Anal. 1(3), 261–272 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Verona, A., Verona, M.E.: Regular maximal monotone multifunctions and enlargements. J. Convex Anal. 16(3–4), 1003–1009 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Voisei, M.D.: The sum and chain rules for maximal monotone operators. arXiv:math/0609296 (2006)

  19. Voisei, M.D.: A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces. Math. Sci. Res. J. 10(2), 36–41 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Voisei, M.D.: The sum theorem for linear maximal monotone operators. Math. Sci. Res. J. 10(4), 83–85 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Voisei, M.D.: Calculus rules for maximal monotone operators in general Banach spaces. J. Convex Anal. 15(1), 73–85 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Voisei, M.D.: The sum and chain rules for maximal monotone operators. Set-Valued Anal. 16(4), 461–476 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Voisei, M.D.: A sum theorem for (FPV) operators and normal cones. J. Math. Anal. Appl. 371, 661–664 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Voisei, M.D.: Characterizations and continuity properties for maximal monotone operators with non-empty domain interior. J. Math. Anal. Appl. 391, 119–138 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Voisei, M.D., Zălinescu, C.: Strongly-representable monotone operators. J. Convex Anal. 16(3–4), 1011–1033 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Voisei, M.D., Zălinescu, C.: Linear monotone subspaces of locally convex spaces. Set-Valued Var. Anal. 18(1), 29–55 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Voisei, M.D., Zălinescu, C.: Maximal monotonicity criteria for the composition and the sum under weak interiority conditions. Math. Program. 123(1, Ser. B), 265–283 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co. Inc., River Edge (2002)

    Book  MATH  Google Scholar 

  29. Zălinescu, C.: Letter to the editor: on J. M. Borwein’s paper: “Adjoint process duality” [Math. Oper. Res. 8 (1983), no. 3, 403–434; MR0716121 (85h:90092)]. Math. Oper. Res. 11(4), 692–698 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Towson University, Towson, MD, USA

    M. D. Voisei

Authors
  1. M. D. Voisei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. D. Voisei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Voisei, M.D. Location, Identification, and Representability of Monotone Operators in Locally Convex Spaces. Set-Valued Var. Anal 27, 151–168 (2019). https://doi.org/10.1007/s11228-017-0440-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0440-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation