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Maximal monotonicity criteria for the composition and the sum under weak interiority conditions

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Abstract

The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).

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References

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

    MATH  MathSciNet  Google Scholar 

  2. Borwein J.M.: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Amer. Math. Soc. 135, 3917–3924 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Edwards, R.E.: Functional Analysis. Theory and Applications. Corrected reprint of the 1965 original. Dover Publications, Inc., New York (1995)

  5. Fitzpatrick, S.: Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), In: Proceedings of the Centre for Mathematics Anal. Austral. Nat. Univ. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Gossez J.P.: On a convexity property of the range of a maximal monotone operator. Proc. Am. Math. Soc. 55, 359–360 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Holmes R.: Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics, No. 24. Springer, New York (1975)

    Google Scholar 

  9. Martínez-Legaz J.E., Théra M.: A convex representation of maximal monotone operators. J. Nonlinear Convex Anal. 2, 243–247 (2001)

    MATH  MathSciNet  Google Scholar 

  10. Penot J.-P.: Miscellaneous incidences of convergence theories in optimization and nonlinear analysis II: applications in nonsmooth analysis. In: Du, D.-Z. (eds) Recent Advances in Nonsmooth Optimization, pp. 289–321. World Scientific, Singapore (1995)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Penot J.-P., Zălinescu C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47, 1–20 (2005)

    Article  MATH  Google Scholar 

  13. Penot J.-P., Zălinescu C.: On the convergence of maximal monotone operators. Proc. Am. Math. Soc. 134, 1937–1946 (2006)

    Article  MATH  Google Scholar 

  14. Rockafellar R.T.: Local boundedness of nonlinear, monotone operators. Michigan Math. J. 16, 397–407 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  16. Simons S.: Minimax and monotonicity. Lecture Notes in Mathematics, 1693. Springer, Berlin (1998)

    Google Scholar 

  17. Simons S.: Dualized and scaled Fitzpatrick functions. Proc. Am. Math. Soc. 134, 2983–2987 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Simons S.: From Hahn–Banach to monotonicity. Springer, Berlin (2008)

    MATH  Google Scholar 

  19. Simons S., Zălinescu C.: A new proof for Rockafellar’s characterization of maximal monotone operators. Proc. Am. Math. Soc. 132, 2969–2972 (2004)

    Article  MATH  Google Scholar 

  20. Simons S., Zălinescu C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6, 1–22 (2005)

    MATH  MathSciNet  Google Scholar 

  21. 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)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Voisei, M.D., Zălinescu, C.: Strongly-representable monotone operators. J. Convex Anal. http://arxiv.org/abs/0802.3640 (2009, to appear)

  26. Zălinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    MATH  Google Scholar 

  27. Zălinescu C.: A new proof of the maximal monotonicity of the sum using the Fitzpatrick function. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications, pp. 1159–1172. Springer, New York (2005)

    Chapter  Google Scholar 

  28. Zălinescu C.: Hahn–Banach extension theorems for multifunctions revisited. Math. Meth. Oper. Res. 68, 493–508 (2008)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Towson University, Towson, MD, USA

    M. D. Voisei

  2. Faculty of Mathematics, University “Al. I. Cuza” Iaşi, 700506, Iasi, Romania

    C. Zălinescu

  3. Institute of Mathematics Octav Mayer, Iasi, Romania

    C. Zălinescu

Authors
  1. M. D. Voisei
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  2. C. Zălinescu
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Correspondence to M. D. Voisei.

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Voisei, M.D., Zălinescu, C. Maximal monotonicity criteria for the composition and the sum under weak interiority conditions. Math. Program. 123, 265–283 (2010). https://doi.org/10.1007/s10107-009-0314-5

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  • DOI: https://doi.org/10.1007/s10107-009-0314-5

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