Skip to main content
SpringerLink
Log in

Existence Theory and Ulam’s Stabilities of Fractional Langevin Equation

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we consider fractional Langevin equation and derive a formula of solutions for fractional Langevin equation involving two Caputo fractional derivatives. Secondly, we implement the concept of Ulam–Hyers as well as Ulam–Hyers–Rassias stability. Then, we choose Generalized Diaz–Margolis’s fixed point approach to derive Ulam–Hyers as well as Ulam–Hyers–Rassias stability results for our proposed model, over generalized complete metric space. We give several examples which support our main results.

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

Access this article

Log in via an institution

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. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)

    Article  MathSciNet  Google Scholar 

  2. Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. Real World Appl. RWA 13(2), 599–602 (2012)

    Article  MathSciNet  Google Scholar 

  3. Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373–380 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Cadariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J Inequal. Pure Appl. Math. 4(1), 1–7 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Choi, G., Jung, S.M.: Invariance of Hyers-Ulam stability of linear differential equations and its applications. Adv. Differ. Equ. 14 (2015)

  6. Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete matric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    Article  Google Scholar 

  7. Fa, K.S.: Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E 73(6), 061104 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Jung, S.M.: A fixed point approach to the stability of differential equations \(y_0 = F(x, y)\). Bull. Malays. Math. Sci. Soc. 33, 47–56 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential equation. Elsevier Science B.V (2006)

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

  12. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), 1–19 (2019)

    Article  MathSciNet  Google Scholar 

  13. Li, T., Viglialoro, G.: Analysis and explicit solvability of degenerate tensorial problems. Bound. Value Probl. 1–13 (2018)

  14. Lim, S.C., Li, M., Teo, L.P.: Langevin equation with two fractional orders. Phys. Lett. A 372(42), 6309–6320 (2008)

    Article  MathSciNet  Google Scholar 

  15. Mainardi, F., Pironi, P.: The fractional Langevin equation: Brownian motion revisited. Extracta Math. 11(1), 140–154 (1996)

    MathSciNet  Google Scholar 

  16. Obloza, M.: Hyers stability of the linear differential equation. Rocznik Nauk. Dydakt. Prace Mat 13, 259–270 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Obloza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk. Dydakt. Prace Mat. 14, 3141–146 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations. Academic Press (1999)

  19. Popa, D., Rasa, I.: Hyers–Ulam stability of the linear differential operator with non-constant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Rizwan, R.: Existence theory and stability analysis of fractional Langevin equation. Int. J. Nonlinear Sci. Numer. Simul. 20(7–8) (2019)

  22. Rizwan, R., Zada, A., Ahmad, M., Shah, S.O., Waheed, H.: Existence theory and stability analysis of switched coupled system of nonlinear implicit impulsive Langevin equations with mixed derivatives. Math. Methods Appl. Sci. 1–23 (2021). https://doi.org/10.1002/mma.7324

  23. Rizwan, R., Zada, A., Wang, X.: Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses. Adv. Differ. Equ. 2019, 85 (2019)

    Article  Google Scholar 

  24. Rizwan, R., Zada, A.: Nonlinear impulsive Langevin equation with mixed derivatives. Math. Methods App. Sci. 43(1), 427–442 (2020)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  26. Shah, S.O., Zada, A., Hamza, A.E.: Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales. Qual. Theory Dyn. Syst. https://doi.org/10.1007/s12346-019-00315-x

  27. Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers–Ulam stability of the Banach space-valued differential equation \(f^{\prime }=\lambda f\). Bull. Korean Math. Soc. 39, 309–315 (2002)

    Article  MathSciNet  Google Scholar 

  28. Tang, S., Zada, A., Faisal, S., El-Sheikh, M.M.A., Li, T.: Stability of higher-order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 9, 4713–4721 (2016)

    Article  MathSciNet  Google Scholar 

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

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

    MATH  Google Scholar 

  31. Viglialoro, G., Murcia, J.: A singular elliptic problem related to the membrane equilibrium equations. Int. J. Comput. Math. 90(10), 2185–2196 (2013)

    Article  Google Scholar 

  32. Wang, J., Feckan, M., Zhou, Y.: Ulams type stability of impulsive ordinary differential equation. J. Math. Anal. Appl. 35, 258–264 (2012)

    Article  MathSciNet  Google Scholar 

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

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

  35. Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)

    Article  MathSciNet  Google Scholar 

  36. Wang, X., Rizwan, R., Lee, J.R., Zada, A., Shah, S.O.: Existence, uniqueness and Ulam’s stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives. AIMS Math. 6(5), 4915–4929 (2021)

    Article  MathSciNet  Google Scholar 

  37. Zada, A., Ali, S.: Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul. 19(7), 763–774 (2018)

    Article  MathSciNet  Google Scholar 

  38. Zada, A., Ali, S., Li, Y.: Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Differ. Equ. 2017, 317 (2017)

    Article  MathSciNet  Google Scholar 

  39. Zada, A., Ali, W., Farina, S.: Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. 40(15), 5502–5514 (2017)

    Article  MathSciNet  Google Scholar 

  40. Zada, A., Ali, W., Park, C.: Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall–Bellman–Bihari’s type. Appl. Math. Comput. 350, 60–65 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Zada, A., Rizwan, R., Xu, J., Fu, Z.: On implicit impulsive Langevin equation involving mixed order derivatives. Adv. Differ. Equ. (489) (2019)

  42. Zada, A., Shah, S.O.: Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses. Hacettepe J. Math. Stat. 47(5), 1196–1205 (2018)

    MathSciNet  Google Scholar 

  43. Zada, A., Shah, O., Shah, R.: Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Buner, Buner, Pakistan

    Rizwan Rizwan

  2. Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan

    Akbar Zada

Authors
  1. Rizwan Rizwan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akbar Zada
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

Corresponding author

Correspondence to Rizwan Rizwan.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interest regarding this research work.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rizwan, R., Zada, A. Existence Theory and Ulam’s Stabilities of Fractional Langevin Equation. Qual. Theory Dyn. Syst. 20, 57 (2021). https://doi.org/10.1007/s12346-021-00495-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00495-5

Keywords

Mathematics Subject Classification

Search

Navigation