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Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems

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Ukrainian Mathematical Journal Aims and scope

We prove the following theorem: Let E be an arbitrary Banach space, let G be an open set in the space \( \mathbb{R}\times E \), and let f: G → E be an arbitrary continuous mapping. Then, for an arbitrary point (t 0, x 0) ∈ G and an arbitrary number ε > 0, there exists a continuous mapping g: G → E such that

$$ \mathop{\sup}\limits_{{\left( {t,x} \right)\in G}}\left\| {g\left( {t,x} \right)-f\left( {t,x} \right)} \right\|\leqslant \varepsilon $$

and the Cauchy problem

$$ \frac{dz(t) }{dt }=g\left( {t,z(t)} \right),\quad z\left( {{t_0}} \right)={x_0} $$

has more than one solution.

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References

  1. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Leningrad (1949).

    Google Scholar 

  2. I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  3. P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  4. A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations [in Ukrainian], Lybid’, Kyiv (2003).

    Google Scholar 

  5. A. N. Godunov, “On the Peano theorem in Banach spaces,” Funkts. Anal. Prilozhen., 9, Issue 1, 59–60 (1975).

    MathSciNet  Google Scholar 

  6. J. Dieudonné, “Deux exemples singuliers d’équations différentielles,” Acta. Sci. Math., 12, Pt. B, 38–40 (1950).

    MATH  Google Scholar 

  7. S. G. Lobanov, “Peano theorem is not true for any infinite-dimensional Fréchet space,” Mat. Sb., 184, No. 2, 83–86 (1993).

    Google Scholar 

  8. S. A. Shkarin, “On one Smolyanov’s problem related to the infinite-dimensional Peano theorem,” Differents. Uravn., 28, No. 6, 1092 (1992).

    Google Scholar 

  9. S. G. Lobanov and O. G. Smolyanov, “Ordinary differential equations in locally convex spaces,” Usp. Mat. Nauk, 49, Issue 3 (297), 93–168 (1994).

    MathSciNet  Google Scholar 

  10. V. E. Slyusarchuk, “On the denseness of the set of unsolvable Cauchy problems in the set of all Cauchy problems in the case of an infinite-dimensional Banach space,” Nelin. Kolyvannya, 5, No. 1, 86–89 (2002); English translation: Nonlin. Oscillations, 5, No. 1, 79–82 (2002).

  11. H. Kneser, “Über die Lösungen eines Systems gewöhnlicher Differentialgleichungen das der Lipschitzschen Bedingung nicht genügt,” S.-B. Preuß. Akad. Wiss., Phys.-Math. Kl., 171–174 (1923).

  12. A. Beck, “Uniqueness of flow solutions of differential equations,” Lect. Notes Math., 318, 30–50 (1973).

    Article  Google Scholar 

  13. P. Hartman, “A differential equation with nonunique solutions,” Amer. Math. Monthly, 70, 255–259 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. A. Lavrentjev, “Sur une équation différentielle du premier ordre,” Math. Z., 23, 197–209 (1925).

    Article  MathSciNet  Google Scholar 

  15. S. G. Krein, Linear Equations in a Banach Space [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  16. T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  17. V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  18. V. Yu. Slyusarchuk, “Cauchy problem with nonunique solutions,” Nauk. Visn. Cherniv. Univ., Ser. Mat., 1, No. 4, 117–118 (2011).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. National University of Water Management and Nature Resources Use, Rivne, Ukraine

    V. Yu. Slyusarchuk

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  1. V. Yu. Slyusarchuk
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 7, pp. 1001–1006, July, 2012.

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Slyusarchuk, V.Y. Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems. Ukr Math J 64, 1144–1150 (2012). https://doi.org/10.1007/s11253-012-0705-2

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  • DOI: https://doi.org/10.1007/s11253-012-0705-2

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