Mathematical Notes

, Volume 62, Issue 1, pp 108–115 | Cite as

Counterexample to Peano's theorem in infinite-dimensionalF′-spaces

  • S. A. Shkarin
Article

Abstract

LetE be a nonnormable Fréchet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt0∈ℝ,x0E′, the Cauchy problemx=f(x), x(t0)=x0 has no solutions is constructed.

Key words

Cauchy problem ordinary differential equation locally convex space Fréchet space F′-space 

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References

  1. 1.
    I. G. Petrovskii,A Course of Ordinary Differential Equations [in Russian], Nauka, Moscow (1967).Google Scholar
  2. 2.
    J. Dieudonné, “Deux exemples singuliers d'équations différentielles”,Acta Sci. Math. (Szeged),12, 38–40 (1950).MATHGoogle Scholar
  3. 3.
    J. A. Yorke, “A continuous differential equation in Hilbert space without existence”,Funkcial. Ekvac.,13, 19–21 (1970).MATHMathSciNetGoogle Scholar
  4. 4.
    A. N. Godunov, “Counterexample to Peano's theorem in infinite-dimensional Hilbert space”Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. [Moscow Univ. Math. Bull.], No 5, 19–21 (1972).Google Scholar
  5. 5.
    A. N. Godunov “Peano's theorem in infinite-dimensional Hilbert space is invalid even in the weakened formulation”,Mat. Zametki [Math. Notes],13, No. 3, 467–477 (1974).MathSciNetGoogle Scholar
  6. 6.
    A. Cellina “On the nonexistence of solution of differential equations in nonreflexive spaces”,Bull. Amer. Math. Soc.,78, No. 6, 1069–1072 (1972).MATHMathSciNetGoogle Scholar
  7. 7.
    A. N. Godunov, “On Peano's theorem in Banach spaces”,Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],9, No. 1, 59–60 (1975).MATHMathSciNetGoogle Scholar
  8. 8.
    K. Astala, “On Peano's theorem in locally convex spaces”,Studia Math.,73, 213–223 (1982).MATHMathSciNetGoogle Scholar
  9. 9.
    S. A. Shkarin, “Peano's theorem in infinite-dimensional Fréchet spaces is invalid”,Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.]27, No. 2, 90–92 (1993).MATHMathSciNetGoogle Scholar
  10. 10.
    S. G. Lobanov, “Ordinary differential equations with continuous right-hand side in Fréchet spaces”,Mat. Zametki [Math. Notes],53, No. 4, 77–91 (1993).MATHMathSciNetGoogle Scholar
  11. 11.
    S. G. Lobanov and O. G. Smolyanov, “Ordinary differential equations in locally convex spaces”,Uspekhi Mat. Nauk [Russian Math. Surveys],49, No. 3, 93–168 (1994).MathSciNetGoogle Scholar
  12. 12.
    S. A. Shkarin, “On Peano's theorem in locally convex spaces”,Differentsial'nye Uravneniya [Differential Equations],28, No. 6, 1064 (1992).Google Scholar
  13. 13.
    E. Mickael, “A selection theorem”,Proc. Amer. Math. Soc.,17, No. 6, 1404–1406 (1966).MathSciNetGoogle Scholar
  14. 14.
    A. P. Robertson and W. Robertson,Topological Vector Spaces, Cambridge Univ. Press, Cambridge (1964).Google Scholar
  15. 15.
    H. H. Schaefer,Topological Vector Spaces, The Macmillan Co., New York, Collier-Macmillan Ltd., London (1966).Google Scholar
  16. 16.
    R. Engelking,General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa (1977).Google Scholar
  17. 17.
    J. Bonet and P. Carreras,Barrelled Locally Convex Spaces, Vol. 131, Stud. Math. Appl, North Holland, Amsterdam (1987).Google Scholar
  18. 18.
    C. Bessaga and A. Pelczyński,Selected Topics in Infinite-Dimensional Topology, Polish scientific publ., Warszawa (1975).Google Scholar
  19. 19.
    C. J. R. Borges, “Continuous extensions”,Proc. Amer. Math. Soc.,18, No. 5, 874–878 (1967).MATHMathSciNetGoogle Scholar
  20. 20.
    J. Saint Raymond “Une équation différentielle sans solution”, in:Séminaire Initiation à l'Analyse (G. Choquet, M. Rogalski, and J. Saint Raymond, editors), Vol. 2, 20 année (1980/81), pp. 7–12.Google Scholar
  21. 21.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • S. A. Shkarin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowUSSR

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