, Volume 17, Issue 3, pp 601–620 | Cite as

Generalized positive sets and abstract monotonicity

  • H. Mohebi
  • A. R. Sattarzadeh


The theory of q-positive sets on SSD spaces has been introduced by Simons (J Convex Anal, 14:297–317, 2007; From Hahn–Banach to monotonicity, Springer, Berlin, 2008). Monotone sets can be considered as special case of q-positive sets. In this paper, we develop a theory of q-positive sets in the framework of abstract monotonicity. We use generalized Fenchel’s duality theorem and give some criteria for maximality of abstract q-positive sets. Finally, we investigate the relation between abstract q-positive sets and abstract convex functions.


Generalized Fenchel’s duality q-Positive set Abstract monotonicity Abstract convexity Abstract convex function 

Mathematics Subject Classification (2000)

47H05 47H04 52A01 26A51 26B25 


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  1. 1.
    Burachik R.S., Rubinov A.M.: On abstract convexity and set valued analysis. J. Nonlinear Convex Anal. 9(1), 105–123 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burachik R.S., Svaiter B.F.: Maximal monotonicity, conjugation and the duality product. Proc. Am. Math. Soc. 131, 2379–2383 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Borwein J.M., Zhu Q.J.: Techniques of Variational Analysis, Candian Mathematical Society. Springer, NewYork (2005)Google Scholar
  4. 4.
    Doagooei, A.R., Mohebi, H.: Sum formula for maximal abstract monotonicity and abstract Rockafellar’s surjectivity theorem (to appear) (2012)Google Scholar
  5. 5.
    Eberhard A.C., Mohebi H.: Maximal abstract monotonicity and generalized Fenchel’s conjugation formulas. Set Valued Variational Anal. 18, 79–108 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fitzpatrick, S: Representing monotone operators by convex functions. In: Workshop and Miniconference on Functional Analysis and Optimization, Canberra, 1988, Austral. Nat. Univ., Canberra, pp. 59–65 (1988)Google Scholar
  7. 7.
    Jeyakumar V., Rubinov A.M., Wu Z.Y.: Generalized Fenchel’s conjugation formulas and duality for abstract convex functions. J. Optim. Theory Appl. 132, 441–458 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Levin V.L.: Abstract convexity in measure theory and in convex analysis. J. Math. Sci. 4, 3432–3467 (2003)CrossRefGoogle Scholar
  9. 9.
    Levin V.L.: Abstract cyclical monotnoicity and Monge solutions for the general Monge–Kantorovich problem. Set Valued Anal. 1, 7–32 (1999)CrossRefGoogle Scholar
  10. 10.
    Martínez-Legaz J.-E.: Some generalizations of Rockafellar’s surjectivity theorem. Pacic J. Optim. 4(3), 527–535 (2008)zbMATHGoogle Scholar
  11. 11.
    Martínez-Legaz J.-E., Svaiter B.: Monotone operators representable by l.s.c. functions. Set Valued Anal. 13, 21–46 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Martínez-Legaz J.-E., Thera M.: A convex representation of maximal monotone operators. J. Nonlinear Convex Anal. 2, 243–247 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Martínez-Legaz J.E.: On maximally q-positive sets. J. Convex Anal. 16(3–4), 891–898 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mohebi H., Martínez-Legaz J.E., Rocco M.: Some criteria for maximal abstract monotonicity. J. Global Optim. 53, 137–163 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Mohebi H., Eberhard A.C.: An abstract convex representation of maximal abstract monotone operators. J Convex Anal. 18(1), 259–275 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rockafellar R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rubinov A.M.: Abstract Convexity and Global Optimization. Kluwer, Boston (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Simons, S.: Minimax and monotonicity. In: Lectures Notes in Mathematics, vol. 1963. Springer, New York (1998)Google Scholar
  19. 19.
    Simons S.: Positive sets and monotone sets. J. Convex Anal. 14, 297–317 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Simons S.: From Hahn–Banach to Monotonicity. Springer, Berlin (2008)zbMATHGoogle Scholar
  21. 21.
    Simons S., Zalinescu C.: Fenchel duality, Fitzpatrick functions and maximal mono-tonicity. J. Nonlinear Convex Anal. 6(1), 1–22 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsShahid Bahonar University of Kerman and Kerman Graduate University of TechnologyKermanIran

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