Skip to main content

Compact perturbations of linear m-dissipative operators which lack Gihman's property

Part of the Lecture Notes in Mathematics book series (LNM,volume 1248)

Abstract

Some questions about abstract methods for initial value problems lead us to a study of the equation (*) u′(t)=(A+B)u(t), where A is m-dissipative and B is compact. Does a solution to (*) necessarily exist? Earlier studies of this question, reviewed and then continued here, depend on an analysis of the related quasiautonomous equation (**) u′(t)=Au(t)+f(t). We say A has Gihman's property if the mapping fu is continuous from L 1 w ([0,T],K) into C([0,T];X) for every compact KX; this condition is closely related to the Lie-Trotter-Kato product formula. If A has this property, then (*) is known to have a solution. In this paper, we consider linear, m-dissipative operators A which lack Gihman's property. We obtain partial results regarding the existence of solutions of (*); but in general, the existence question remains open. Our method applies the variation of parameters formula to (**), but this requires a weakened topology when Range (f) \(\nsubseteq \overline {D(A)}\). Two examples are studied: one in ℓ, the other in the space of bounded continuous functions.

Keywords

  • Banach Space
  • Weak Topology
  • Norm Topology
  • Semi Group
  • Bounded Continuous Function

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Arendt, P.R. Chernoff, and T. Kato, A generalization of dissipativity and positive semigroups, J. Oper. Th.8 (1982), 167–180.

    MathSciNet  MATH  Google Scholar 

  2. J. Banaś, A. Hajnosz, and S. Wedrychowicz, Some generalization of Szufla's theorem for ordinary differential equations in Banach space, Bull. Acad. Polon. Sci. Ser. Math.29 (1981), 459–464.

    MATH  Google Scholar 

  3. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

    CrossRef  MATH  Google Scholar 

  4. P. Bénilan, Équations d'évolution dans un espace de Banach quelconque et applications, thesis, Orsay, 1972.

    Google Scholar 

  5. P. Bénilan, M. G. Crandall, and A. Pazy, book in preparation.

    Google Scholar 

  6. A. Bressan, Solutions of lower semicontinuous differential inclusions on closed sets, Rend. Sem. Mat. Univ. Padova69 (1983), 99–107.

    MathSciNet  MATH  Google Scholar 

  7. A. J. Chorin, T. J. Hughes, M. F. Mc Cracken, and J. E. Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math.31 (1978), 205–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, M.R.C. Technical Summary Report2724. To appear in the proceedings of the Symposium on Nonlinear Functional Analysis and Applications, held in Berkeley in July, 1983.

    Google Scholar 

  9. M. G. Crandall and A. Pazy, An approximation of integrable functions by step functions with an application, Proc. Amer. Math. Soc.76 (1979), 74–80.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. M. Dawidowski, On some generalization of Bogoliubov averaging theorem, Functiones et Approximatio7 (1979), 55–70.

    MathSciNet  MATH  Google Scholar 

  11. B. Dembart, Perturbations of semigroups on locally convex spaces, Bull. Amer. Math. Soc.79 (1973), 986–991.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. M. A. Freedman, Product integrals of continuous resolvents: existence and nonexistence, Israel J. Math.46 (1983), 145–160.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. I. I. Gihman, Concerning a theorem of N. N. Bogolyubov, Ukr. Math. J.4 (1952), 215–218 (in Russian). (For English summary see Math. Reviews17, p. 738.)

    MathSciNet  Google Scholar 

  14. A.N. Godunov, Peano's theorem in Banach spaces, Funct. Anal. Appl.9 (1975), 53–55.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. J. A. Goldstein, Locally quasi-dissipative operators and the equation ∂u/∂t=ø(x, ∂u/∂x)∂2u/∂x2 + g(u), in: Evolution Equations and their Applications (proceedings of the Graz conference on nonlinear differential equations; F. Kappel and W. Schappacher, Ed.), Pitman Research Notes in Mathematics no. 68, Boston, 1982.

    Google Scholar 

  16. S. Gutman, Evolutions governed by m-accretive plus compact operators, Nonlin. Anal. Theory Methods Appl.7 (1983), 707–715.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. S. Gutman, Topological equivalence in the space of integrable vector-valued functions, Proc. Amer. Math. Soc.93 (1985), 40–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, AMS Colloq. Publns. 31, AMS, Providence R.I., Revised Edition, 1957.

    MATH  Google Scholar 

  19. J. L. Kelley, General Topology, Van Nostrand, N.Y., 1955; reprinted by Springer, N.Y., 1975.

    MATH  Google Scholar 

  20. Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan27 (1975), 640–665.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. M. A. Krasnosel'skii and S. G. Krein, On the principle of averaging in nonlinear mechanics, Usp. Mat. Nauk10 (1955), 147–152. Russian. (For English summary see Math. Reviews17 #152.)

    MathSciNet  Google Scholar 

  22. T. G. Kurtz, An abstract averaging theorem, J. Funct. Anal.23 (1976), 135–144.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. T. G. Kurtz and M. Pierre, A counterexample for the Trotter product formula, J. Diff. Eqns.52 (1984), 407–414.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J.7 (1957), 418–449.

    MathSciNet  MATH  Google Scholar 

  25. L. Lapidus, Generalization of the Trotter-Lie formula, Integral Equations and Operator Theory4 (1981), 366–415.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. R. H. Martin, Approximation and existence of solutions to ordinary differential equations in Banach spaces, Funk. Ekvac.16 (1973), 195–211.

    MathSciNet  MATH  Google Scholar 

  27. R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, N.Y., 1976.

    Google Scholar 

  28. J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eqns.26 (1977), 347–374.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. L. W. Neustadt; On the solutions of certain integral-like operator equations: existence, uniqueness, and dependence theorems; Arch. Rat. Mech. Anal.38 (1970), 131–160.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. C. Olech, An existence theorem for solutions of orientor fields, in: Dynamical Systems: An International Symposium, vol. 2, L. Cesari, J. K. Hale, and J. P. La Salle (eds.), Academic Press, N.Y., 1976; pp. 63–66.

    Google Scholar 

  31. E. Schechter, Evolution generated by continuous dissipative plus compact operators, Bull. London Math. Soc.13 (1981), 303–308.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. E. Schechter, Interpolation of nonlinear partial differential operators and generation of differentiable evolutions, J. Diff. Eqns.46 (1982), 78–102.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. E. Schechter, Perturbations of regularizing maximal monotone operators, Israel J. Math.43 (1982), 49–61.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. E. Schechter, Evolution generated by semilinear dissipative plus compact operators, Trans. Am. Math. Soc.275 (1983), 297–308.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. E. Schechter, Necessary and sufficient conditions for convergence of temporally irregular evolutions, Nonlin. Analysis Theory Methods Appl.8 (1984), 133–153.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. E. Schechter, Correction to “Perturbations of monotone operators” and a note on injectiveness, Israel J. Math.47 (1984), 236–240.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. S. Szufla, On the equation x′=f(t, x) in Banach spaces, Bull. Acad. Polon Ser. Sci. Math. Astron. Phys.26 (1978), 401–406.

    MathSciNet  MATH  Google Scholar 

  38. K. Yosida, Functional Analysis, Springer, New York, 1964.

    MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Schechter, E. (1987). Compact perturbations of linear m-dissipative operators which lack Gihman's property. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1248. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077423

Download citation

  • DOI: https://doi.org/10.1007/BFb0077423

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17741-8

  • Online ISBN: 978-3-540-47791-4

  • eBook Packages: Springer Book Archive