Advertisement

Set-Valued Analysis

, Volume 8, Issue 4, pp 375–403 | Cite as

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

  • Valery S. Melnik
  • José Valero
Article

Abstract

In this paper we define multivalued semiprocesses and give theorems providing the existence of global attractors for such systems. This theory generalizes the construction of nonautonomous dynamical systems given by V. V. Chepyzhov and M. I. Vishik to the case where the system is not supposed to have a unique solution for each initial state. Further, we apply these theorems to nonautonomous differential inclusions of reaction–diffusion type.

global attractor nonautonomous multivalued dynamical systems differential inclusions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, J. P. and Cellina, A.: Differential Inclusions, Springer-Verlag, Berlin, 1984.Google Scholar
  2. 2.
    Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.Google Scholar
  3. 3.
    Babin, A. V.: Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math. 44(2) (1995), 207–223.Google Scholar
  4. 4.
    Ball, J. M.: Continuity properties and global attractors of generalized semiflows and the Navier- Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502.Google Scholar
  5. 5.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.Google Scholar
  6. 6.
    Caraballo, T., Langa, J. A. and Valero J.: Global attractors for multivalued random dynamical systems, Nonlinear Anal. (to appear).Google Scholar
  7. 7.
    Chepyzhov, V. V. and Vishik, M. I.: Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. 73 (1994), 279–333.Google Scholar
  8. 8.
    Chepyzhov, V. V. and Vishik, M. I.: Evolution equations and their trajectory attractors, J. Math. Pures Appl. 76 (1997), 913–964.Google Scholar
  9. 9.
    Gutman, S.: Existence theorems for nonlinear evolution equations, Nonlinear Anal. 11(10) (1987), 1193–1206.Google Scholar
  10. 10.
    Hale, J. K.: Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence, 1988.Google Scholar
  11. 11.
    Kapustian, A. V. and Melnik, V. S.: On global attractors of multivalued semidynamical systems and their approximations, Kibernet. Sistem. Anal. 5 (1998), 102–111.Google Scholar
  12. 12.
    Kapustian, A. V. and Melnik, V. S.: Attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk Ukrainy 10 (1998), 21–25.Google Scholar
  13. 13.
    Kapustian, A. V. and Melnik, V. S.: On approximations and estimates of dimension of global compact attractors of multivalued semiflows, Nauchnye Vedomosti of National Technical University of Ukraine “KPI”, 1999 (to appear).Google Scholar
  14. 14.
    Kapustian, A. V. and Valero, J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal. (to appear).Google Scholar
  15. 15.
    Ladyzhenskaya, O. A.: Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.Google Scholar
  16. 16.
    Melnik, V. S.: Multivalued dynamics of nonlinear infinite-dimensional systems, Preprint No. 94-17, Academy of Science of Ukraine, Institute of Cybernetics, 1994.Google Scholar
  17. 17.
    Melnik, V. S.: Multivalued semiflows and their attractors, Dokl. Akad. Nauk 343(3) (1995), 302–305.Google Scholar
  18. 18.
    Melnik, V. S.: Families of m-semiflows and their attractors, Dokl. Akad. Nauk 353(2) (1997), 158–162.Google Scholar
  19. 19.
    Melnik, V. S., Slastikov, V. V. and Vasilkevich, S. I.: On global attractors of multivalued semiprocesses, Dokl. Akad. Nauk Ukrainy 7 (1999), 12–17.Google Scholar
  20. 20.
    Melnik, V. S., Slastikov, V. V. and Vasilkevich, S. I.: Multivalued semiprocesses in infinite-dimensional spaces and their attractors, submitted.Google Scholar
  21. 21.
    Melnik, V. S. and Valero, J.: On attractors of multivalued semidynamical systems in infinite-dimensional spaces, Preprint No. 5, Departamento de Matemáticas, Universidad de Murcia, 1997.Google Scholar
  22. 22.
    Melnik, V. S. and Valero, J.: On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111.Google Scholar
  23. 23.
    Papageorgiou, N. S.: Evolution inclusions involving a difference term of subdifferentials and applications, Indian J. Pure Appl. Math. 28(5) (1997), 575–610.Google Scholar
  24. 24.
    Tolstonogov, A. A.: On solutions of evolution inclusions. I, Sibirsk. Mat. Zh. 33(3) (1992), 161–174 (English translation in Siberian Math. J. 33(3) (1992)).Google Scholar
  25. 25.
    Valero, J.: Existence and dimension of attractors in dynamical systems generated by evolution inclusions, Ph.D. Thesis, Universidad de Murcia, 1997.Google Scholar
  26. 26.
    Valero, J.: Attractors of parabolic equations without uniqueness, Preprint No. 15, Departamento de Matemáticas, Universidad de Murcia, 1999.Google Scholar
  27. 27.
    Valero, J.: Finite and infinite-dimensional attractors for multivalued reaction- diffusion equations, Acta Math. Hungar. 88(3) (2000), 239–258.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Valery S. Melnik
    • 1
  • José Valero
    • 2
  1. 1.Institute of System Applied AnalysisKievUkraine
  2. 2.CEU San Pablo-ElcheAlicanteSpain

Personalised recommendations