Abstract

This article employs techniques from convex analysis to present characterizations of (maximal) n-monotonicity, similar to the well-known characterizations of (maximal) monotonicity found in the existing literature. These characterizations are further illustrated through examples

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Dual system
  • maximal cyclically monotone operator
  • Fitzpatrick function
  • Rockafellar's antiderivative
  • planar rotation
  • skew double-cone

Mathematics Subject Classification

47H05, 46N99, 47N10

References

  1. E. Asplund: A monotone convergence theorem for sequences of nonlinear mappings, in: Nonlinear Functional Analysis, Chicago 1968, Proc. Symp. Pure Math. XVIII, Part 1, American Mathematical Society, Providence (1970) 1–9.https://doi.org/10.1090/pspum/018.1/0275240Google Scholar
  2. S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich, X. Wang: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative, Nonlinear Anal., Theory Methods Appl. 66/5 (2007) 1198–1223.https://doi.org/10.1016/j.na.2006.01.013Google Scholar
  3. S. Bartz, S. Reich: Minimal antiderivatives and monotonicity, Nonlinear Anal., Theory Methods Appl. 74/1 (2011) 59–66.https://doi.org/10.1016/j.na.2010.08.015Google Scholar
  4. H. H. Bauschke, Y. Lucet, X. Wang: Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators, SIAM J. Control Optim. 46/6 (2007) 2031–2051.https://doi.org/10.1137/060675794Google Scholar
  5. H. H. Bauschke, D. A. McLaren, H. S. Sendov: Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick, J. Convex Analysis 13/3-4 (2006) 499–523.https://doi.org/10.68381/jca13043Google Scholar
  6. R. E. Edwards: Functional Analysis, corr. reprint of the 1965 original, Dover Publications, New York (1995).Google Scholar
  7. S. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization, Canberra 1988, Proc. Centre Math. Anal. Austral. Nat. Univ. Vol. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  8. R. T. Rockafellar: Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966) 497–510.https://doi.org/10.2140/pjm.1966.17.497Google Scholar
  9. M. D. Voisei: Extension theorems for k-monotone operators, Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacău 9 (1999) 235–242.Google Scholar
  10. M. D. Voisei: A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces, Math. Sci. Res. J. 10/2 (2006) 36–41.Google Scholar
  11. M. D. Voisei: Calculus rules for maximal monotone operators in general Banach spaces, J. Convex Analysis 15/1 (2008) 73–85.https://doi.org/10.68381/jca15006Google Scholar
  12. M. D. Voisei: The sum and chain rules for maximal monotone operators, Set-Valued Analysis 16/4 (2008) 461–476.https://doi.org/10.1007/s11228-007-0042-zGoogle Scholar
  13. M. D. Voisei: Characterizations and continuity properties for maximal monotone operators with non-empty domain interior, J. Math. Anal. Appl. 391 (2012) 119–138.https://doi.org/10.1016/j.jmaa.2012.02.025Google Scholar
  14. M. D. Voisei: Location, identification, and representability of monotone operators in locally convex spaces, Set-Valued Var. Analysis 27/1 (2019) 151–168.https://doi.org/10.1007/s11228-017-0440-9Google Scholar
  15. M. D. Voisei: Maximal monotone normal cones in locally convex spaces, J. Math. Anal. Appl. 476/2 (2019) 811–818.https://doi.org/10.1016/j.jmaa.2019.04.017Google Scholar
  16. M. D. Voisei, C. Zălinescu: Strongly-representable monotone operators, J. Convex Analysis 16/3-4 (2009) 1011–1033.https://doi.org/10.68381/jca16063Google Scholar
  17. M. D. Voisei, C. Zălinescu: Linear monotone subspaces of locally convex spaces, Set-Valued Var. Analysis 18/1 (2010) 29–55.https://doi.org/10.1007/s11228-009-0129-9Google Scholar
  18. M. D. Voisei, C. Zălinescu: Maximal monotonicity criteria for the composition and the sum under weak interiority conditions, Math. Programming, Ser. B 123/1 (2010) 265–283.https://doi.org/10.1007/s10107-009-0314-5Google Scholar

Suggested citation

M. D. Voisei. “General Monotonicity.” Journal of Convex Analysis 34 (2027), No. 1, 49–76. https://doi.org/10.68381/jca34005

Download BibTeXPublished by Banach Press LLC.

© 2026 Banach Press LLC. Published under CC BY 4.0.