Differential Equations

, Volume 51, Issue 5, pp 592–604 | Cite as

Strength and stability of the Bohl index

  • V. S. KlimovEmail author
Ordinary Differential Equations


For a homogeneous multimapping A, we introduce the notion of Bohl index, which determines the asymptotics of solutions of the finite-dimensional differential inclusion 0 ∈ x′ + A(t, x). We focus our attention on lower bounds for the Bohl index. We study the dependence of solutions of the inclusion 0 ∈ x′ +A(t, x) on the initial value and the mapping A. We prove the stability of the Bohl index under small-in-mean perturbations.


Cauchy Problem Closed Interval Exponential Stability Superposition Operator Nonempty Compact 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gajewskii, H., Gröger, K., and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,Berlin: Akademie–Verlag, 1974. Translated under the title Nelineinye operatornye uravneniyai operatornye differentsial’nye uravneniya, Moscow: Mir, 1978.Google Scholar
  2. 2.
    Blagodatskikh, V.I. and Filippov, A.F., Differential Inclusions and Optimal Control, Tr. Mat. Inst.Steklova Akad. Nauk SSSR, 1985, vol. 169, pp. 194–252.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Borisovich, Yu.G., Gel’man, B.D., Myshkis, A.D., and Obukhovskii, V.V., Vvedenie v teoriyu mnogoznachnykhotobrazhenii i differentsial’nykh vklyuchenii (Introduction to the theory of Multivalued Mappingsand Differential Inclusions), Moscow: Librokom, 2011.Google Scholar
  4. 4.
    Ioffe, A.D. and Tikhomirov, V.M., Teoriya ekstremal’nykh zadach (Theory of Extremal Problems), Moscow: Nauka, 1974.Google Scholar
  5. 5.
    Aubin, J.-P. and Ekeland, I., Applied Nonlinear Analysis, New York: John Wiley & Sons, 1984. Translatedunder the title Prikladnoi nelineinyi analiz, Moscow: Mir, 1988.zbMATHGoogle Scholar
  6. 6.
    Tolstonogov, A.A., Differentsial’nye vklyucheniya v banakhovykh prostranstvakh (Differential Inclusionsin a Banach Space), Novosibirsk: Nauka, 1986.Google Scholar
  7. 7.
    Filippov, A.V., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations withDiscontinuous Right-Hand Side), Moscow: Nauka, 1985.Google Scholar
  8. 8.
    Bohl, P., Sobranie trudov (Collected Papers), Riga: Zinatne, 1974.Google Scholar
  9. 9.
    Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve(Stability of Solutions of Differential Equations in Banach Space), Moscow: Nauka, 1970.Google Scholar
  10. 10.
    Klimov, V.S., First Approximation Stability Theorems for Differential Inclusions, Mat. Zametki, 2004,vol. 76, no. 4, pp. 517–530.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Klimov, V.S., The Bohl Index of a Homogeneous Parabolic Inclusion, Izv. Ross. Akad. Nauk Ser. Mat.,2011, vol. 75, no. 2, pp. 99–122.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Donchev, T. and Farki, E., Stability and Euler Approximation of One-Sided Lipschitz Differential Inclusions, SIAM J. Control Optim., 1998, vol. 36, no. 2, pp. 780–796.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Sokolovskaya, E.V. and Filatov, O.P., Approximation from Above to Systems of Differential Equationswith Non-Lipschitz Right-Hand Side, Mat. Zametki, 2005, vol. 78, no. 5, pp. 763–772.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, Amsterdam: North-Holland,1976. Translated under the title Vypuklyi analiz i variatsionnye problemy, Moscow, 1979.zbMATHGoogle Scholar
  15. 15.
    Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Theoryof Stability), Moscow: Nauka, 1967.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.P.G. Demidov Yaroslavl State UniversityYaroslavlRussia

Personalised recommendations