Lyapunov Functions and Stability of Caputo Fractional Differential Equations with Delays

  • Ravi AgarwalEmail author
  • Snezhana Hristova
  • Donal O’Regan
Original Research


The direct Lyapunov method is extended to nonlinear Caputo fractional differential equations with variable bounded delays. A brief overview of the literature on derivatives of Lyapunov functions is given and applications to fractional equations are discussed. Advantages and disadvantages are illustrated with examples. Sufficient conditions using three derivatives of Lyapunov functions are given and our results are compared with results in the literature. Also fractional order extensions of comparison principle are established.


Caputo fractional differential equations Lyapunov functions Stability Fractional derivative of Lyapunov functions 



Research was partially supported by Fund MU17-FMI-007, University of Plovdiv Paisii Hilendarski.


  1. 1.
    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional-order predatorprey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alsaedi, A., Ahmad, B., Kirane, M.: Maximum principle for certain generalized time and space fractional diffusion equations. Q. Appl. Math. (2015). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agarwal, R.P., Hristova, S., O’Regan, D.: Lyapunov functions and strict stability of Caputo fractional differential equations. Adv. Differ. Equ. 2015, 346 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Agarwal, R.P., Hristova, S., O’Regan, D.: A survey of Lyapunov functions and impulsive Caputo fractional differential equations. FCAA 19(2), 290–318 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Agarwal, R.P., O’Regan, D., Hristova, S.: Stability of Caputo fractional differential equations by Lyapunov functions. Appl. Math. 60(6), 653–676 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Agarwal, R., O’Regan, D., Hristova, S., Cicek, M.: Practical stability with respect to initial time difference for Caputo fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 42, 106–120 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baleanu, D., Sadati, S. J., Ghaderi, R., Ranjbar, A., Abdeljawad (Maraaba), T., Jarad, F.: Fractional systems with delay. Abstr. Appl. Anal. 2010(124812), 1–9 (2010). MathSciNetzbMATHGoogle Scholar
  8. 8.
    Baleanu, D., Ranjbar, N.A., Sadati, R.S.J., Delavari, H., Abdeljawad, T., Gejji, V.: Lyapunov–Krasovskii stability theorem for fractional order systems with delay. Rom. J. Phys. 56(56), 636–643 (2011)zbMATHGoogle Scholar
  9. 9.
    Chen, B., Chen, J.: Razumikhin-type stability theorems for functional fractional-order differential systems and applications, Appl. Math. Comput. 254, 63–69 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Das, Sh: Functional Fractional Calculus. Springer, Berlin (2011)CrossRefGoogle Scholar
  11. 11.
    Devi, J.V., Mc Rae, F.A., Drici, Z.: Variational Lyapunov method for fractional differential equations. Comput. Math. Appl. 64, 2982–2989 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)CrossRefGoogle Scholar
  13. 13.
    Gao, X., Yu, J.B.: Chaos in the fractional order periodically forced complex duffings oscillators. Chaos Solitons Fractals 24, 1097–1104 (2005)CrossRefGoogle Scholar
  14. 14.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New-York (1993)CrossRefGoogle Scholar
  15. 15.
    Hu, J.-B., Lu, G.-P., Zhang, S.-B., Zhao, L.-D.: Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 20(3), 905–913 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jalilian, Y., Jalilian, R.: Existence of solution for delay fractional differential equations. Mediterr. J. Math. 10(4), 1731–1747 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1999)zbMATHGoogle Scholar
  18. 18.
    Kucche, K.D., Sutar, S.T.: On existence and stability results for nonlinear fractional delay differential equations. Bol. Soc. Paran. Mat. 36(4), 55–75 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lakshmikantham, V.: Theory of fractional functional differential equations. Nonlinear Anal. Theory Methods Appl. 69(10), 3337–3343 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lakshmikantham, V., Leela, S., Sambandham, M.: Lyapunov theory for fractional differential equations. Commun. Appl. Anal. 12(4), 365–376 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. CSP, Cambridge (2009)zbMATHGoogle Scholar
  22. 22.
    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(5), 1810–1821 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  24. 24.
    Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 16671694 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sadati, S.J., Ghaderi, R., Ranjbar, A.: Some fractional comparison results and stability theorem for fractional time delay systems. Rom. Rep. Phys. 65(1), 94102 (2013)Google Scholar
  26. 26.
    Stamova, I., Stamov, G.: Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys. 54(4), 043502 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stamova, I., Stamov, G.: Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, p. 276. CRC Press, London (2016)CrossRefGoogle Scholar
  28. 28.
    Wang, Y., Li, T.: Stability analysis of fractional-order nonlinear systems with delay. Math. Probl. Eng. 2014(301235), 1–8 (2014)MathSciNetGoogle Scholar
  29. 29.
    Wen, Y., Zhou, X.-F., Zhang, Z., Liu, S.: Lyapunov method for nonlinear fractional differential systems with delay. Nonlinear Dyn. (2015). MathSciNetCrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  • Ravi Agarwal
    • 1
    • 2
    Email author
  • Snezhana Hristova
    • 3
  • Donal O’Regan
    • 4
  1. 1.Department of MathematicsTexas A&M University-KingsvilleKingsvilleUSA
  2. 2. Florida Institute of TechnologyMelbourneUSA
  3. 3.Department of Applied Mathematics and ModelingUniversity of Plovdiv Paisii HilendarskiPlovdivBulgaria
  4. 4.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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