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Israel Journal of Mathematics

, Volume 12, Issue 4, pp 373–390 | Cite as

On multivalued evolution equations in Hilbert spaces

  • H. Attouch
  • A. Damlamian
Article

Abstract

The Cauchy problemdu/dt+Au+B(t,u)∋0,u(0)=u 0 is studied in a separable Hilbert space setting, whenA is a multivalued maximal monotone operator, andB is a multivalued operator which is measurable with respect to the time variable and upper semi-continuous with respect to the space variable. Under some boundedness conditions onB, an existence theorem is proved, with the extra assumption, in the infinite dimensional case thatA is the subdifferential of a proper lower semi-continuous inf-compact convex function. A theorem of dependence upon the initial condition is also given.

Keywords

Multivalued Mapping Weak Topology Maximal Monotone Real Hilbert Space Maximal Monotone Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. Benilan andH. Brezis,Solutions faibles d’équations d’evolutions dans les espaces de 2 Hilbert, à paraître aux Ann. Inst. Fourier.Google Scholar
  2. 2.
    C. Berge,Espaces Topologiques, Fonctions Multivoques, Dunod, Paris, 2ème édition, 1959.zbMATHGoogle Scholar
  3. 3.
    H. Brezis,Problèmes unilateraux, J. Math. Pures Appl. (1972).Google Scholar
  4. 4.
    H. BrezisMonotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, ed., Zarantanello, Academic Press, 1971.Google Scholar
  5. 5.
    H. Brezis,Opérateurs maximaux monotones et semi-groupes non linéaires. Cours de 3ème cycle rédigé par P. Benilan, Paris, 1971.Google Scholar
  6. 6.
    H. Brezis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math.9 (1971), 513–534.zbMATHMathSciNetGoogle Scholar
  7. 7.
    H. Brezis,Semi-groupes non linéaires et applications Symposium sur les problèmes d’évolution. Symposia Mathematics VII, Academic Press, 1971.Google Scholar
  8. 8.
    H. Brezis andA. Pazy,Semi-groups of nonlinear contractions on convex sets, J. Functional Analysis6 (1970), 237–281.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    F. Browder,The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann.177 (1963), 283–301.CrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Castaing,Sur les multi-applications mesurables, R.I.R.O.1 (1967), 91–126.MathSciNetzbMATHGoogle Scholar
  11. 11.
    C. Castaing andM. Valadier,Equations différentielles multivoques dans les espaces vectoriels localement convexes, C. R. Acad. Sci.266 (1968), 985–987.zbMATHMathSciNetGoogle Scholar
  12. 12.
    N. Dunford andJ. Schwartz.Linear Operators Interscience.Google Scholar
  13. 13.
    C. Henry,An existence theorem for a class of differential equations with multivalued right-hand side, to appear in J. Math. Anal. and Appl.Google Scholar
  14. 14.
    C. Henry,Differential equations with discontinuous right-hand side for planning procedures, to appear in J. Economic Theory4 (1972).Google Scholar
  15. 15.
    T. Kato,Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan19, (1967), 508–520.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    T. Kato,Accretive operators and non linear evolution equations in Banach spaces; Nonlinear functional analysis, Proc. Symp. Pure Math., Amer. Math. Soc.,18 (1970), 138–161.Google Scholar
  17. 17.
    A. Lasota,Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Ann. Polon. Math.18 (1966), 65–77.zbMATHMathSciNetGoogle Scholar
  18. 18.
    A. Lasota andOpial,An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci.13 (1965), 781–786.zbMATHMathSciNetGoogle Scholar
  19. 19.
    J. L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969.Google Scholar
  20. 20.
    R. T. Rockafellar,On the maximality of the sums of nonlinear monotone operators, Trans. Amer. Math. Soc.,149 (1970), 75–88.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Seminaire, Semi-groupes et opérateurs non linéaires, Publications mathématiques d’Orsay, 1970–1971.Google Scholar
  22. 22.
    M. Valadier,Sur l’intégration d’ensembles convexes compacts en dimension infinie, C. R. Acad. Sci.266 (1968) 14–16.zbMATHMathSciNetGoogle Scholar
  23. 23.
    M. Valadier,Existence globale pour les équations différentielles multivoques, C. R. Acad. Sci.272 (1971), 474–477.zbMATHMathSciNetGoogle Scholar
  24. 24.
    K. Yosida,Functional Analysis, Springer Verlag, 3rd ed.Google Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

  • H. Attouch
    • 1
  • A. Damlamian
    • 1
  1. 1.Institute de MathématiquesUniversité Paris-Sud Centre d’OrsayOrsayFrance

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