Skip to main content
Springer Nature Link
Log in

Robustness of Exponential Dichotomies for Generalized Ordinary Differential Equations

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

This paper deals with exponential dichotomy for generalized ODEs. We establish sufficient conditions to ensure that the exponential dichotomy of a linear generalized ODE is robust under certain perturbations. As a consequence, we obtain results on robustness of dichotomies for measure differential equations and, in particular, impulsive differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Afonso, S.M., Bonotto, E.M., Federson, M.: On exponential stability of functional differential equations with variable impulsive perturbations. Differ. Integral Equ. 27, 721–742 (2014)

    MATH  Google Scholar 

  2. Afonso, S.M., Bonotto, E.M., Federson, M., Gimenes, L.P.: Boundedness of solutions for functional differential equations with variable impulses via generalized ordinary differential equations. Mathematische Nachrichten 285, 545–561 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Afonso, S.M., Bonotto, E.M., Federson, M., Schwabik, Š.: Discontinuous local semiflows for Kurzweil equations leading to LaSalle’s invariance principle for differential systems with impulses at variable times. J. Differ. Equ. 250, 2936–3001 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Bainov, D.D., Kostadinov, S.I., van Minh, N.: Dichotomies and Integral Manifolds of Impulsive Differential Equations. Oxford Graphics Printers, Singapore (1994)

    Google Scholar 

  5. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monogr. Math. 96. Springer, Basel (2011)

    MATH  Google Scholar 

  6. Barreira, L., Valls, C.: On two notions of exponential dichotomy. Dyn. Syst. 33, 708–721 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Barreira, L., Valls, C.: Robustness for impulsive equations. Nonlinear Anal 72, 2542–2563 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Bartle, R.G.: A Modern Theory of Integration. Graduate Studies in Mathematics, vol. 32. AMS, New York (2000)

    MATH  Google Scholar 

  9. Bonotto, E.M., Federson, M., Santos, F.L.: Dichotomies for generalized ordinary differential equations and applications. J. Differ. Equ. 264(5), 3131–3173 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Coffman, C.V., Schäfer, J.J.: Dichotomies for linear difference equations. Math. Ann. 172, 139–166 (1967)

    MathSciNet  MATH  Google Scholar 

  11. Collegari, R., Federson, M., Frasson, M.: Linear FDEs in the frame of generalized ODEs: variation-of-constants formula. Czech. Math. J. 68(143), 889–920 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Congxin, W., Xiaobo, Y.: A Riemann-type definition of the Bochner integral. J. Math. Study 27, 32–36 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Coppel, W.A.: Dichotomies in Stability Theory, Lecture Notes in Mathematics. Springer, Berlin (1978)

  14. Daleckii, J.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 43. American Mathematical Society Providence, Rhode Island (1994)

    Google Scholar 

  15. Das, P.C., Sharma, R.R.: On optimal controls for measure delay-differential equations. J. SIAM Control 9, 43–61 (1971)

    MathSciNet  MATH  Google Scholar 

  16. Das, P.C., Sharma, R.R.: Existence and stability of measure differential equations. Czech. Math. J. 22(97), 145–158 (1972)

    MathSciNet  MATH  Google Scholar 

  17. Federson, M.: Substitution formulas for the Kurzweil and Henstock vector integrals. Math. Bohemica 127(1), 15–26 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Federson, M.: Some peculiarities of the Henstock and Kurzweil integrals of Banach space-valued functions. Real Anal. Exchange 29(1), 439–460 (2003/2004)

  19. Federson, M., Mesquisa, J.G., Slavík, A.: Basic results for functional differential and dynamic equations involving impulses. Mathematische Nachrichten 286, 181–204 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Federson, M., Mesquisa, J.G., Slavík, A.: Measure functional differential equations and functional dynamic equations on time scales. J. Differ. Equ. 252, 3816–3847 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Federson, M., Schwabik, Š.: Generalized ODE approach to impulsive retarded differential equations. Differ. Integral Equ. 19(11), 1201–1234 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Federson, M., Táboas, P.Z.: Topological dynamics of retarded functional differential equations. J. Differ. Equ. 195(2), 313–331 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Hale, J.K.: Ordinary Differential Equations. Wiley, Hoboken (1969)

    MATH  Google Scholar 

  24. Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1980)

    Google Scholar 

  25. Henstock, R.: Lectures on the Theory of Integration. World Scientific, Singapore (1988)

    MATH  Google Scholar 

  26. Hönig, C.S.: Volterra Stieltjes-Integral Equations. North-Holland Publ. Comp, Amsterdam (1975)

    MATH  Google Scholar 

  27. Imaz, C., Vorel, Z.: Generalized ordinary differential equations in Banach spaces and applications to functional equations. Bol. Soc. Mat. Mexicana 11, 47–59 (1966)

    MathSciNet  MATH  Google Scholar 

  28. Klyuchnyk, R., Kmit, I., Recke, L.: Exponential dichotomy for hyperbolic systems with periodic boundary conditions. J. Differ. Equ. 262, 2493–2520 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7(82), 418–448 (1957)

    MathSciNet  MATH  Google Scholar 

  30. Kurzweil, J.: Generalized ordinary differential equations. Czechoslovak Math. J. 8(83), 360–388 (1958)

    MathSciNet  MATH  Google Scholar 

  31. Lin, Z.S., Lin, Y.-X.: Linear Systems Exponential Dichotomy and Structure of Sets of Hyperbolic Points. Word Scientific Publishing Co. Pte. Ltd., Singapore (1999)

    Google Scholar 

  32. Meng, G., Zhang, M.: Measure Differential Equations I. Continuity of Solutions in Measures with Weak Topology. Tsinghua University (2009). http://faculty.math.tsinghua.edu.cn/%7Emzhang/publs/mde1.pdf

  33. Monteiro, G.A., Tvrdy, M.: On Kurzweil–Stieltjes integral in a Banach space. Math. Bohem. 137(4), 365–381 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Naralenkov, K.M.: On integration by parts for Stieltjes-type integrals of Banach space-valued functions. Real Anal Exchange 30(1), 235-260 (2004/2005)

  35. Naulin, R., Pinto, M.: Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559–571 (1998)

    MathSciNet  MATH  Google Scholar 

  36. Oliva, F., Vorel, Z.: Functional equations and generalized ordinary differential equations. Bol. Soc. Mat. Mexicana 11, 40–46 (1966)

    MathSciNet  MATH  Google Scholar 

  37. Palmer, K.J.: A generalization of Hartman’s linearization theorem. J. Math. Anal. Appl. 41, 753–758 (1973)

    MathSciNet  MATH  Google Scholar 

  38. Palmer, K.J.: Exponential dichotomies, the shadowing lemma and transversal homoclinic points. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported, vol. 1, pp. 265–306. Hohn Wiley and Sons, Chichester (1988)

    Google Scholar 

  39. Piccoli, B.: Measure Differential Equations. arXiv:1708.09738

  40. Popescu, L.H.: Exponential Dichotomy Roughness on Banach Spaces. J. Math. Anal. Appl. 314, 436–454 (2006)

    MathSciNet  MATH  Google Scholar 

  41. Sakamoto, K.: Estimates on the strength of exponential dichotomies and application to integral manifolds. J. Differ. Equ. 107, 259–279 (1994)

    MathSciNet  MATH  Google Scholar 

  42. Schmaedeke, W.W.: Optimal control theory for nonlinear vector differential equations containing measures. J. SIAM Control 3, 231–280 (1965)

    MathSciNet  MATH  Google Scholar 

  43. Schwabik, Š.: Abstract Perron–Stieltjes integral. Math. Bohem. 121, 425–447 (1996)

    MathSciNet  MATH  Google Scholar 

  44. Schwabik, Š.: Generalized Ordinary Differential Equations. Series in Real Anal., vol. 5. World Scientific, Singapore (1992)

    MATH  Google Scholar 

  45. Schwabik, Š.: Linear Stieltjes Integral Equations in Banach Spaces. Math. Bohem. 124, 433–457 (1999)

    MathSciNet  MATH  Google Scholar 

  46. Zhou, L., Kening, L., Zhang, W.: Equivalences between nonuniform exponential dichotomy and admissibility. J. Differ. Equ. 262, 682–747 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Ye, Y., Liang, H.: Asymptotic dichotomy in a class of higher order nonlinear delay differential equations. J. Inequal. Appl. (2019). https://doi.org/10.1186/s13660-018-1949-7

Download references

Acknowledgements

The authors thank the referee for the valuable comments and suggestions that improved the results of this article.

Author information

Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    E. M. Bonotto & M. Federson

  2. Departamento de Matemática, Universidade Federal Rural de Pernambuco, Rua Dom Manoel de Medeiros, s/n, Dois Irmãos, CEP 52171-900, Recife, PE, Brazil

    F. L. Santos

Authors
  1. E. M. Bonotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Federson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. L. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. M. Bonotto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

E.M. Bonotto is supported by FAPESP Grant 2016/24711-1 and by CNPq Grant 310497/2016-7. M. Federson is supported by FAPESP grant 2017/13795-2 and by CNPq grant 309344/2017-4. F. L. Santos: Supported by FAPESP Grant 2011/24027-0.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonotto, E.M., Federson, M. & Santos, F.L. Robustness of Exponential Dichotomies for Generalized Ordinary Differential Equations. J Dyn Diff Equat 32, 2021–2060 (2020). https://doi.org/10.1007/s10884-019-09801-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-019-09801-x

Keywords

Mathematics Subject Classification

Search

Navigation