Advertisement

Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag-Leffler functions

  • Thabet Abdeljawad
  • Fahd JaradEmail author
  • Saed F. Mallak
  • Jehad Alzabut
Regular Article
  • 28 Downloads
Part of the following topical collections:
  1. Focus Point on Fractional Differential Equations in Physics: Recent Advantages and Future Direction

Abstract.

In this article, we prove Lyapunov type inequalities for a nonlocal fractional derivative, called fractional proportional derivative, generated by modified conformable or proportional derivatives in both Riemann-Liuoville and Caputo senses. Further, in the Riemann-Liuoville case we prove a Lyapunov inequality for a fractional proportional weighted boundary value problem and apply it on a weighted Sturm-Liouville problem to estimate an upper bound for the free zero disk of the Kilbas-Saigo Mittag-Leffler functions of three parameters. The proven results generalize and modify previously obtained results in the literature.

References

  1. 1.
    A.M. Lyapunov, Ann. Fac. Sci. Univ. Toulouse 2, 227 (1907)Google Scholar
  2. 2.
    J.P. Pinasco, Lyapunov-type Inequalities (Springer, New York, 2013)Google Scholar
  3. 3.
    D. Çakmak, Appl. Math. Comput. 216, 373 (2010)Google Scholar
  4. 4.
    S. Clark, D.B. Hinton, Math. Ineq. Appl. 1, 201 (2010)Google Scholar
  5. 5.
    N. Parhi, S. Panigrahi, Math. Slov. 52, 31 (2002)Google Scholar
  6. 6.
    X. Yang, Appl. Math. Comput. 134, 307 (2003)ADSMathSciNetGoogle Scholar
  7. 7.
    X. Yang, K. Lo, Appl. Math. Comput. 215, 3884 (2010)MathSciNetGoogle Scholar
  8. 8.
    R. Hilfer, Applications of Fractional Calculus in Physics (Word Scientific, Singapore 2000)Google Scholar
  9. 9.
    A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations, in Mathematics Studies, Vol. 204 (North Holland, 2006)Google Scholar
  10. 10.
    R.L. Magin, Fractional Calculus in Bioengineering (Begell House Publishers, 2006)Google Scholar
  11. 11.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego CA 1999)Google Scholar
  12. 12.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Yverdon 1993)Google Scholar
  13. 13.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  14. 14.
    A. Atangana, D. Baleanu, Therm. Sci. 20, 757 (2016)CrossRefGoogle Scholar
  15. 15.
    M. Hajipour, A. Jararmi, D. Baleanu, H. Sun, Commun. Nonlinear Sci. Numer. Simul. 69, 119 (2019)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    R. Meng, D. Yin, C.S. Drapaca, Comput. Mech. (2019)  https://doi.org/10.1007/s00466-018-1663-9
  17. 17.
    D. Baleanu, A. Jajarmi, M. Hajipour, Nonlinear Dyn. 94, 397 (2018)CrossRefGoogle Scholar
  18. 18.
    A. Jajarmi, D. Baleanu, Chaos, Solitons Fractals 113, 221 (2018)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    D. Baleanu, A. Jajarmi, E. Bonyah, M. Hajipour, Adv. Differ. Equ. 2018, 230 (2018)CrossRefGoogle Scholar
  20. 20.
    H. Khan, J.F. Gomez-Aguilar, A. Khan, T.S. Khan, Physica A 521, 737 (2019)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Khan, C. Tunç, D. Baleanu, A. Khan, A. Alkhazzan, Rev. R. Acad. Cienc. Exact. Fis. Nat. Ser. A Mat. (2019)  https://doi.org/10.1007/s13398-019-00624-5
  22. 22.
    H. Khan, A. Khan, T. Abdeljawad, A. Alkhazzan, Adv. Differ. Equ. 2019, 18 (2019)CrossRefGoogle Scholar
  23. 23.
    A. Alkhazzan, P. Jiang, D. Baleanu, H. Khan, A. Khan, Math. Methods Appl. Sci. 41, 9321 (2018)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    H. Khan, W. Chen, A. Khan, T.S. Khan, Q.M. Al-Madlal, Adv. Differ. Equ. 2018, 455 (2018)CrossRefGoogle Scholar
  25. 25.
    H. Khan, A. Khan, W. Chen, K. Shah, Math. Methods Appl. Sci. 42, 723 (2019)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    H. Khan, C. Tunç, W. Chen, A. Khan, J. Appl. Anal. Comput. 8, 1211 (2018)Google Scholar
  27. 27.
    A. Khan, M.I. Syam, A. Zada, H. Khan, Eur. Phys. J. Plus 133, 264 (2018)CrossRefGoogle Scholar
  28. 28.
    R.A.C. Ferreira, Fract. Calc. Appl. Anal. 16, 978 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    R.A.C. Ferreira, Adv. Dyn. Syst. Appl. 11, 33 (2016)MathSciNetGoogle Scholar
  30. 30.
    R.A.C. Ferreira, J. Math. Anal. Appl. 412, 1058 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Q. Ma, C. Ma, J. Wang, J. Math. Inequal. 11, 135 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    T. Abdeljawad, F. Madjidi, Eur. Phys. J. ST 226, 3355 (2017)CrossRefGoogle Scholar
  33. 33.
    T. Abdeljawad, R.P. Agarwal, J. Alzabut, F. Jarad, A. Özbekler, J. Inequal. Appl. 2018, 143 (2018)CrossRefGoogle Scholar
  34. 34.
    T. Abdeljawad, J. Alzabut, F. Jarad, Adv. Differ. Equ. 2017, 321 (2017)CrossRefGoogle Scholar
  35. 35.
    T. Abdeljawad, Adv. Differ. Equ. 2017, 313 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    T. Abdeljawad, J. Inequal. Appl. 2017, 130 (2017)CrossRefGoogle Scholar
  37. 37.
    T. Abdeljawad, Q.M. Al-Mdallal, M.A. Hajji, Discr. Dyn. Nat. Soc. 2017, 4149320 (2017)Google Scholar
  38. 38.
    F. Jarad, T. Abdeljawad, J. Alzabut, Eur. Phys. J. ST 226, 3457 (2017)CrossRefGoogle Scholar
  39. 39.
    D.R. Anderson, D.J. Ulness, Adv. Dyn. Sys. Appl. 10, 109 (2015)Google Scholar
  40. 40.
    D.R. Anderson, Commun. Appl. Nonlinear Anal. 24, 17 (2017)Google Scholar
  41. 41.
    T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2013)CrossRefGoogle Scholar
  42. 42.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math. 264, 65 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Kilbas, M. Saigo, Differ. Integr. Equ. 8, 993 (1995)Google Scholar
  44. 44.
    T. Abdeljawad, B. Benli, D. Baleanu, Abstr. Appl. Anal. 2012, 546062 (2012)Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Thabet Abdeljawad
    • 1
  • Fahd Jarad
    • 2
    Email author
  • Saed F. Mallak
    • 3
  • Jehad Alzabut
    • 1
  1. 1.Department of Mathematics and General SciencesPrince Sultan UniversityRiyadhSaudi Arabia
  2. 2.Department of MathematicsÇankaya UniversityAnkaraTurkey
  3. 3.Department of Applied mathematicsPalestine Technical University-KadoorieTulkarm, West BankPalestine

Personalised recommendations