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
and the Cauchy problem
has more than one solution.
Similar content being viewed by others
References
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Leningrad (1949).
I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1973).
P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).
A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations [in Ukrainian], Lybid’, Kyiv (2003).
A. N. Godunov, “On the Peano theorem in Banach spaces,” Funkts. Anal. Prilozhen., 9, Issue 1, 59–60 (1975).
J. Dieudonné, “Deux exemples singuliers d’équations différentielles,” Acta. Sci. Math., 12, Pt. B, 38–40 (1950).
S. G. Lobanov, “Peano theorem is not true for any infinite-dimensional Fréchet space,” Mat. Sb., 184, No. 2, 83–86 (1993).
S. A. Shkarin, “On one Smolyanov’s problem related to the infinite-dimensional Peano theorem,” Differents. Uravn., 28, No. 6, 1092 (1992).
S. G. Lobanov and O. G. Smolyanov, “Ordinary differential equations in locally convex spaces,” Usp. Mat. Nauk, 49, Issue 3 (297), 93–168 (1994).
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).
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).
A. Beck, “Uniqueness of flow solutions of differential equations,” Lect. Notes Math., 318, 30–50 (1973).
P. Hartman, “A differential equation with nonunique solutions,” Amer. Math. Monthly, 70, 255–259 (1963).
M. A. Lavrentjev, “Sur une équation différentielle du premier ordre,” Math. Z., 23, 197–209 (1925).
S. G. Krein, Linear Equations in a Banach Space [in Russian], Nauka, Moscow (1971).
T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).
V. Yu. Slyusarchuk, “Cauchy problem with nonunique solutions,” Nauk. Visn. Cherniv. Univ., Ser. Mat., 1, No. 4, 117–118 (2011).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 7, pp. 1001–1006, July, 2012.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0705-2