Skip to main content
SpringerLink
Log in

Existence and Stability Results for Generalized Fractional Differential Equations

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this paper, a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of order α (1 < α < 2) are given. The main results are obtained by using Krasnoselskii’s fixed point theorem in a weighted Banach space. Two examples are given to demonstrate the validity of the proposed 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. Almeida R, Malinowska A B, Odzijewicz T. Fractional differential equations with dependence on the Caputo-Katugampola derivative. J Comput Nonlinear Dynam, 2016, 11: 061017

    Article  Google Scholar 

  2. Baleanu D, Wu G C, Zeng S D. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos, Solitons and Fractals, 2017, 102: 99–105

    Article  MathSciNet  Google Scholar 

  3. Ben Makhlouf A. Stability with respect to part of the variables of nonlinear Caputo fractional differential equations. Math Commun, 2018, 23: 119–126

    MathSciNet  MATH  Google Scholar 

  4. Ben Makhlouf A, Nagy A M. Finite-time stability of linear Caputo-Katugampola fractional-order time delay systems. Asian Journal of Control, 2018, https://doi.org/10.1002/asjc.1880

    Google Scholar 

  5. Boroomand A, Menhaj M B. Fractional-order Hopfeld neural networks. Lecture Notes in Computer Science, 2009, 5509: 883–890

    Article  Google Scholar 

  6. Boucenna D, Ben Makhlouf A, Naifar O, Guezane-Lakoud A, Hammami M A. Linearized stability analysis of Caputo-Katugampola fractional-order nonlinear systems. J Nonlinear Funct Anal, 2018, 2018: Article 27

  7. Burov S, Barkai E. Fractional Langevin equation: overdamped, underdamped, and critical behaviors. Phys Rev E, 2008, 78(3): 031112

    Article  MathSciNet  Google Scholar 

  8. Corduneanu C. Integral Equations and Stability of Feedback Systems. New York, London: Academic Press, 1973

    MATH  Google Scholar 

  9. Debnath L. Fractional integrals and fractional differential equations in fluid mechanics. Frac Calc Appl Anal, 2003, 6: 119155

    MathSciNet  Google Scholar 

  10. Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, Castro-Linares R. Using general quadratic Lya- punov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simul, 2015, 22(1/3): 650–659

    Article  MathSciNet  Google Scholar 

  11. Ge F, Kou C. Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations. Appl Math Comput, 2015, 257: 308316

    MathSciNet  MATH  Google Scholar 

  12. Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Science Publishing, 2000

    Book  Google Scholar 

  13. Katugampola U N. Existence and uniqueness results for a class of generalized fractional differential equations. arXiv:1411.5229, 2014

    MATH  Google Scholar 

  14. Kilbas A A, Srivastava H M, Trujillo J J. Theory and Application of Fractional Differential Equations. New York: Elsevier, 2006

    MATH  Google Scholar 

  15. Kou C, Zhou H, Yan Y. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal, 2011, 74: 5975–5986

    Article  MathSciNet  Google Scholar 

  16. Krasnoselskii M A. Some problems of nonlinear analysis. Amer Math Soc Transl, 1958, 10: 345–409

    MathSciNet  Google Scholar 

  17. Laskin N. Fractional market dynamics. Phys A, 2000, 287(3/4): 482–492

    Article  MathSciNet  Google Scholar 

  18. Li Y, Chen Y Q, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8): 1965–1969

    Article  MathSciNet  Google Scholar 

  19. Magin R. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 2004, 32: 195–377

    Article  Google Scholar 

  20. Matignon D. Stability result on fractional differential equations with applications to control processing//IMACS SMC Proc, Lille, France, 1996: 963–968

    Google Scholar 

  21. Naifar O, Ben Makhlouf A, Hammami M A. Comments on “Mittag-Leffler stability of fractional order nonlinear dynamic systems [Automatica, 2009, 45(8): 1965–1969]”. Automatica, 2017, 75: 329

    Article  Google Scholar 

  22. Naifar O, Ben Makhlouf A, Hammami M A. Comments on Lyapunov stability theorem about fractional system without and with delay. Commun Nonlinear Sci Numer Simul, 2016, 30: 360–361

    Article  MathSciNet  Google Scholar 

  23. Naifar O, Ben Makhlouf A, Hammami M A, Chen L. Global practical mittag leffler stabilization by output feedback for a class of nonlinear fractional-order systems. Asian Journal of Control, 2018, 20: 599–607

    Article  MathSciNet  Google Scholar 

  24. Podlubny I. Fractional Differential Equations. New York: Academic Press, 1999

    MATH  Google Scholar 

  25. Soczkiewicz E. Application of fractional calculus in the theory of viscoelasticity. Molecular and Quantum Acoustics, 2002 23: 397–404

    Google Scholar 

  26. Sun H, Abdelwahad A, Onaral B. Linear approximation of transfer function with a pole of fractional order. IEEE Trans Automat Contr, 1984, 29(5): 441–444

    Article  Google Scholar 

  27. Tripathil D, Pandey S, Das S. Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Applied Mathematics and Computation, 2010, 215: 3645–3654

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Jouf University, Aljouf, Saudi Arabia

    A. Ben Makhlouf

  2. Department of Mathematics, Faculty of Sciences of Sfax, Route Soukra, BP 1171, 3000, Sfax, Tunisia

    A. Ben Makhlouf & M. A. Hammami

  3. Higher School for Professors of Technological Education, Skikda, Algeria

    D. Boucenna

Authors
  1. A. Ben Makhlouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Boucenna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Hammami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ben Makhlouf.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben Makhlouf, A., Boucenna, D. & Hammami, M.A. Existence and Stability Results for Generalized Fractional Differential Equations. Acta Math Sci 40, 141–154 (2020). https://doi.org/10.1007/s10473-020-0110-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0110-3

Key words

2010 MR Subject Classification

Search

Navigation