Skip to main content
Springer Nature Link
Log in

E α -Ulam type stability of fractional order ordinary differential equations

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, the concepts of \(\mathbb{E}_{\alpha}\)-Ulam-Hyers stability, generalized \(\mathbb{E}_{\alpha}\)-Ulam-Hyers stability, \(\mathbb{E}_{\alpha}\)-Ulam-Hyers-Rassias stability and generalized \(\mathbb{E}_{\alpha}\)-Ulam-Hyers-Rassias stability for fractional order ordinary differential equations are raised. Without loss of generality, \(\mathbb{E}_{\alpha}\)-Ulam-Hyers-Rassias stability result is derived by using a singular integral inequality of Gronwall type. Two examples are also provided to illustrate our results.

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. András, S., Kolumbán, J.J.: On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions. Nonlinear Anal. 82, 1–11 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. András, S., Mészáros, A.R.: Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Appl. Math. Comput. 219, 4853–4864 (2013)

    Article  MathSciNet  Google Scholar 

  3. Burger, M., Ozawa, N., Thom, A.: On Ulam stability. Isr. J. Math. 193, 109–129 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cădariu, L.: Stabilitatea Ulam-Hyers-Bourgin Pentru Ecuatii Functionale. Univ. Vest Timişoara, Timişara (2007)

    Google Scholar 

  5. Cimpean, D.S., Popa, D.: Hyers-Ulam stability of Euler’s equation. Appl. Math. Lett. 24, 1539–1543 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Deng, W.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72, 1768–1777 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  8. Hegyi, B., Jung, S.M.: On the stability of Laplace’s equation. Appl. Math. Lett. 26, 549–552 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  10. Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  11. Jung, S.M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  12. Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  14. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    MATH  Google Scholar 

  15. Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lungu, N., Popa, D.: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385, 86–91 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  19. Miura, T., Miyajima, S., Takahasi, S.E.: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136–146 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Miura, T., Miyajima, S., Takahasi, S.E.: Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 258, 90–96 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Obłoza, M.: Hyers stability of the linear differential equation. Rocz. Nauk.-Dydakt. Pr. Mat. 13, 259–270 (1993)

    MATH  Google Scholar 

  22. Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocz. Nauk.-Dydakt. Pr. Mat. 14, 141–146 (1997)

    MATH  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  24. Rassias, T.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MATH  Google Scholar 

  25. Rassias, J.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rassias, J.M.: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rus, I.A.: Ulam stability of ordinary differential equations. Stud. Univ. Babes,-Bolyai, Math. 54, 125–133 (2009)

    MATH  MathSciNet  Google Scholar 

  28. Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math. 26, 103–107 (2010)

    MATH  MathSciNet  Google Scholar 

  29. Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers-Ulam stability of the Banach space-valued differential equation y′=λy. Bull. Korean Math. Soc. 39, 309–315 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer & HEP, Berlin (2011)

    Google Scholar 

  31. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)

    Google Scholar 

  32. Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)

    Article  MathSciNet  Google Scholar 

  33. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang, J., Fec̆kan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wang, J., Lv, L., Zhou, Y.: New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2530–2538 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Wang, J., Zhou, Y., Fec̆kan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wang, J., Fec̆kan, M., Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Spec. Top. 222, 1857–1874 (2013)

    Article  Google Scholar 

  38. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

    JinRong Wang

  2. Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

    JinRong Wang & Xuezhu Li

Authors
  1. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuezhu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Additional information

This work is supported by the National Natural Science Foundation of China (11201091), Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169), Key Support Subject (Applied Mathematics) and Key project on the reforms of teaching contents and course system of Guizhou Normal College.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Li, X. E α -Ulam type stability of fractional order ordinary differential equations. J. Appl. Math. Comput. 45, 449–459 (2014). https://doi.org/10.1007/s12190-013-0731-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-013-0731-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation