Set-Valued and Variational Analysis

, Volume 18, Issue 3–4, pp 327–335

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

  • Regina S. Burachik
  • Juan Enrique Martínez-Legaz
  • Marco Rocco
Article

Abstract

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever \(T(x)\cap S(x)\not=\emptyset \) for every x ∈ D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their ε-subdifferentials intersect at every point of that domain.

Keywords

Maximal monotone operators Subdifferential Convex functions Enlargements ε-subdifferential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bot, R.I., Grad, S.M., Wanka, G.: Fenchel’s duality theorem for nearly convex functions. J. Optim. Theory Appl. 132 (3), 509–515 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Burachik, R.S., Fitzpatrick, S.: On the Fitzpatrick family associated to some subdifferentials. J. Nonlinear Convex Anal. 6(1), 165–171 (2005)MATHMathSciNetGoogle Scholar
  3. 3.
    Burachik, R.S., Iusem, A.N.: Set Valued Mappings and Enlargements of Monotone Operators. Springer Optimization and Its Applications, vol. 8. Springer, New York (2008)Google Scholar
  4. 4.
    Burachik, R.S., Iusem, A.N., Svaiter, B.F.: Enlargements of maximal monotone operators with application to variational inequalities. Set-Valued Anal. 5, 159–180 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burachik, R.S., Sagastizábal, C.A., Svaiter, B.F.: ε-enlargements of maximal monotone operators: theory and applications. In: Fukushima, M., Qi, L. (eds.) Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 25–43. Kluwer, Dordrecht (1997)Google Scholar
  6. 6.
    Burachik, R.S., Sagastizábal, C.A., Svaiter, B.F.: Bundle methods for maximal monotone operators. In: Tichatschke, R., Théra, M. (eds.) Ill–posed Variational Problems and Regularization Techniques, pp. 49–64. Springer, Berlin (1999)Google Scholar
  7. 7.
    Burachik, R.S., Svaiter, B.F.: ε-enlargements of maximal monotone operators in Banach spaces. Set-Valued Anal. 7, 117–132 (1999)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Burachik, R.S., Svaiter, B.F.: Operating enlargements of monotone operators: new connection with convex functions. Pac. J. Optim. 2(3), 425–445 (2006)MATHMathSciNetGoogle Scholar
  10. 10.
    Fitzpatrick, S.: Representing monotone operators by convex functions. In: Functional Analysis and Optimization, Workshop and Miniconference, pp. 59–65. Canberra, Australia (1988) Proc. Center Math. Anal. Australian Nat. Univ., vol. 20 (1988)Google Scholar
  11. 11.
    Kocourek, P.: An elementary new proof of the determination of a convex function by its subdifferential. Optimization. doi:10.1080/02331930903086803
  12. 12.
    Martínez-Legaz, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8)(A), 855–871 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Phelps, R.R.: Convex functions, monotone operators and differentiability. Lecture Notes in Math., 2nd edn., vol. 1364. Springer (1993)Google Scholar
  15. 15.
    Simons, S.: From Hahn–Banach to monotonicity. Lecture Notes in Math., 2nd edn., vol. 1693. Springer (2008)Google Scholar
  16. 16.
    Svaiter, B.F.: A family of enlargements of maximal monotone operators. Set-Valued Anal. 8, 311–328 (2000)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Thibault, L., Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces. J. Math. Anal. Appl. 189(1), 33–58 (1995)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Regina S. Burachik
    • 1
  • Juan Enrique Martínez-Legaz
    • 2
  • Marco Rocco
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of South AustraliaAdelaideAustralia
  2. 2.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Dipartimento di Matematica, Statistica, Informatica e ApplicazioniUniversità degli Studi di BergamoBergamoItaly

Personalised recommendations