Abstract

We develop a theory of monotone operators in the framework of abstract convexity. We present a definition for abstract monotone operators and some examples of an abstract convex function such that its subdifferential is a maximal abstract monotone operator. Finally, we give an abstract convex representation for maximal abstract monotone operators.

Author information

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

H. Mohebi

Mahani Mathematical Research Center and Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

hmohebi@mail.uk.ac.ir

Andrew C. Eberhard

School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Victoria 3001, Australia

andy.eb@rmit.edu.au

Keywords

  • Monotone operator
  • abstract monotonicity
  • abstract convex function
  • abstract convexity
  • IPH function

Mathematics Subject Classification

47H05, 47H04, 52A01, 26A48, 26A51

References

  1. V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976).https://doi.org/10.1007/978-94-010-1537-0Google Scholar
  2. J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis, Springer, New York (2005).Google Scholar
  3. F. E. Browder: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Math. 18, Part 2, American Mathematical Society, Providence (1976).https://doi.org/10.1090/pspum/018.2Google Scholar
  4. R. S. Burachick, A. M. Rubinov: On abstract convexity and set valued analysis, J. Nonlinear Convex Anal. 9(1) (2008) 105–123.Google Scholar
  5. R. S. Burachick, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  6. R. S. Burachick, B. F. Svaiter: Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131 (2003) 2379–2383.https://doi.org/10.1090/S0002-9939-03-07053-9Google Scholar
  7. A. R. Doagooei, H. Mohebi: Monotonic analysis over ordered topological vector spaces: III, to appear.Google Scholar
  8. S. Fitzpatrick: Representing monotone operators by convex functions, in: Functional Analysis and Optimization, Workshop / Miniconference (Canberra, 1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  9. D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and their Applications, Classics in Appl. Math. 31, SIAM, Philadelphia (2000).https://doi.org/10.1137/1.9780898719451Google Scholar
  10. E. Krauss: A representation of maximal monotone operators by saddle functions, Rev. Roum. Math. Pures Appl. 30 (1985) 823–836.Google Scholar
  11. V. L. Levin: Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-Valued Anal. 1 (1999) 7–32.https://doi.org/10.1023/A:1008753021652Google Scholar
  12. J.-E. Martínez-Legaz, B. Svaiter: Monotone operators representable by l.s.c. functions, Set-Valued Anal. 13 (2005) 21–46.https://doi.org/10.1007/s11228-004-4170-4Google Scholar
  13. J.-E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001) 243–247.Google Scholar
  14. H. Mohebi, H. Sadeghi: Monotonic analysis over ordered topological vector spaces: I, Optimization 56 (2007) 305–321.https://doi.org/10.1080/02331930600819746Google Scholar
  15. J.-P. Penot: The relevance of convex analysis for the study of monotonicity, Nonlinear Anal. 58 (2004) 855–871.https://doi.org/10.1016/j.na.2004.05.018Google Scholar
  16. J.-P. Penot: Autoconjugate functions and representations of monotone operators, Bull. Austral. Math. Soc. 67 (2003) 277–284.https://doi.org/10.1017/S0004972700033748Google Scholar
  17. A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9Google Scholar
  18. A. M. Rubinov, M. Y. Andramonov: Minimizing increasing star-shaped functions based on abstract convexity, J. Glob. Optim. 15(1) (1999) 19–39.https://doi.org/10.1023/A:1008344317743Google Scholar
  19. A. M. Rubinov, B. M. Glover: Increasing convex-along-rays functions with applications to global optimization, J. Optimization Theory Appl. 102(3) (1999) 615–542.https://doi.org/10.1023/A:1022602223919Google Scholar
  20. A. M. Rubinov, A. Uderzo: On global optimality conditions via separation functions, J. Optimization Theory Appl. 109(2) (2001) 345–370.https://doi.org/10.1023/A:1017566406216Google Scholar
  21. S. Simons: Minimax and Monotonicity, Lectures Notes in Mathematics 1693, Springer, New York (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  22. S. Simons, C. Zălinescu: Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005) 1–22.Google Scholar
  23. E. Zeidler: Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer, New York (1990).https://doi.org/10.1007/978-1-4612-0981-2Google Scholar

Suggested citation

H. Mohebi, A. C. Eberhard. “An Abstract Convex Representation of Maximal Abstract Monotone Operators.” Journal of Convex Analysis 18 (2011), No. 1, 259–275. https://doi.org/10.68381/jca18014

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.