Abstract

Using the Fitzpatrick function, we characterize the solutions for different classes of deterministic and stochastic differential equations driven by maximal monotone operators (or in particular subdifferential operators) as the minimum point of a suitably chosen convex lower semicontinuous function. Such technique provides a new approach for the existence of the solutions for the considered equations.

Author information

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

Aurel Răşcanu

(1) Dept. of Mathematics, Al. I. Cuza University, Bd. Carol I 9-11, Iaşi, Romania
(2) Mathematics Institute, Romanian Academy of Sciences, Bd. Carol I 8, Iaşi, Romania

aurel.rascanu@uaic.ro

Keywords

  • Maximal monotone operators
  • Fitzpatrick function
  • Skorohod problem
  • stochastic differential equations

Mathematics Subject Classification

60H15; 65C30, 47H05, 47H15

References

  1. I. Asiminoaei, A. Răşcanu: Approximation and simulation of stochastic variational inequalities - splitting up method, Numer. Funct. Anal. Optimization 18 (3–4) (1997) 251–282.https://doi.org/10.1080/01630569708816759Google Scholar
  2. V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1975).https://doi.org/10.1007/978-94-010-1537-0Google Scholar
  3. V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces, Mathematics and its Applications (East European Series) 10, D. Reidel, Dordrecht (1986).Google Scholar
  4. A. Bensoussan, A. Răşcanu: Stochastic variational inequalities in infinite dimensional spaces, Numer. Funct. Anal. Optimization 18(1–2) (1997) 19–54.https://doi.org/10.1080/01630569708816745Google Scholar
  5. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam (1973).Google Scholar
  6. R. S. Burachik, S. Fitzpatrick: On a family of convex functions associated to subdifferentials, J. Nonlinear Convex Anal. 6(1) (2005) 165–171.Google Scholar
  7. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10(4) (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  8. E. Cépa: Multivalued Skorohod problem, Ann. Probab. 26(2) (1998) 500–532 (in French).https://doi.org/10.1214/aop/1022855642Google Scholar
  9. E. Cépa: Stochastic multivalued differential equations, in: Séminaire de Probabilités XXIX, Lecture Notes in Math. 1613, J. Azéma et al. (ed.), Springer, Berlin (1995) 86–107 (in French).https://doi.org/10.1007/BFb0094202Google Scholar
  10. 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
  11. I. Gyöngy, T. Martínez: Solutions of stochastic partial differential equations as extremals of convex functionals, Acta Math. Hung. 109(1–2) (2005) 127–145.https://doi.org/10.1007/s10474-005-0237-4Google Scholar
  12. J. E. Martinez-Legaz, M. Théera: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2(2) (2001) 243–247.Google Scholar
  13. E. Pardoux, A. Răşcanu: Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Processes Appl. 76(2) (1998) 191–215.https://doi.org/10.1016/S0304-4149(98)00030-1Google Scholar
  14. E. Pardoux, A. Răşcanu: Stochastic Differential Equations, in preparation.Google Scholar
  15. G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992).https://doi.org/10.1017/CBO9780511666223Google Scholar
  16. A. Răşcanu: Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operators, Panam. Math. J. 6(3) (1996) 83–119.Google Scholar
  17. S. Simons, C. Zălinescu: A new proof for Rockafellar’s characterization of maximal monotone operators, Proc. Amer. Math. Soc. 132 (2004) 2969–2972.https://doi.org/10.1090/S0002-9939-04-07462-3Google Scholar
  18. E. Zeidler: Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization, Springer, New York (1985).https://doi.org/10.1007/978-1-4612-5020-3Google Scholar

Suggested citation

A. Răşcanu, E. Rotenstein. “The Fitzpatrick Function - a Bridge between Convex Analysis and Multivalued Stochastic Differential Equations.” Journal of Convex Analysis 18 (2011), No. 1, 105–138. https://doi.org/10.68381/jca18006

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