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Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2

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Abstract

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2+ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.

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References

  1. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: theory and applications (Gordon and Breach, Yverdon, 1993)

  2. I. Podlubny, Fractional differential equations (Academic Press, San Diego, CA, 1999)

  3. A. Kilbas, M.H. Srivastava, J.J. Trujillo, in Theory and application of fractional differential equations (North Holland Mathematics Studies, 2006), Vol. 204

  4. J.A. Tenreiro Machado, V. Kiryakova, F. Mainardi, FCAA J. Fract. Calc. Appl. Anal. 13, 329 (2010)

    Google Scholar 

  5. J.A. Tenreiro Machado, Commun. Nonlinear Sci. Numer. Simul. 16, 4596 (2011)

    Article  ADS  Google Scholar 

  6. M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)

    Google Scholar 

  7. J. Losada, J.J. Nieto, Progr. Fract. Differ. Appl. 1, 87 (2015)

    Google Scholar 

  8. A. Atagana, D. Baleanu, Therm. Sci. 20, 757 (2016)

    Article  Google Scholar 

  9. T. Abdeljawad, D. Baleanu, J. Nonlinear Sci. Appl. 10, 1098 (2017)

    Article  MathSciNet  Google Scholar 

  10. T. Abdeljawad, D. Baleanu, J. Rep. Math. Phys. 80, 11 (2017)

    Article  ADS  Google Scholar 

  11. T. Abdeljawad, D. Baleanu, Adv. Differ. Equ. 2016, 232 (2016)

    Article  Google Scholar 

  12. T. Abdeljawad, D. Baleanu, Adv. Differ. Equ. 2017, 78 (2017)

    Article  Google Scholar 

  13. O.J.J Algahtani, Choas Solitons Fractals 48, 552 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  14. T. Abdeljawad, Int. J. Math. Comput. 22, 144 (2014)

    Google Scholar 

  15. F. Jarad, T. Abdeljawad, D. Baleanu, Appl. Math. Comput. 218, 9234 (2012)

    MathSciNet  Google Scholar 

  16. O.P. Agrawal, D. Baleanu, J. Vib. Control 13, 1269 (2007)

    Article  MathSciNet  Google Scholar 

  17. C. Goodrich, A.C. Peterson, Discrete fractional calculus (Springer, 2015)

  18. A.M. Lyapunov, Ann. Fac. Sci. Univ. Toulouse 2, 27 (1907) [Reprinted in: Ann. Math. Studies, No. 17, Princeton 1947]

    Google Scholar 

  19. R.A.C. Ferreira, Fract. Calc. Appl. Anal. 6, 978 (2013)

    Google Scholar 

  20. A. Chdouh, D.F.M. Torres, J. Comput. Appl. Math. 312, 192 (2017)

    Article  MathSciNet  Google Scholar 

  21. M. Jleli, B. Samet, Electron J. Differ. Equ. 2015, 88 (2015)

    Article  Google Scholar 

  22. D. O’Regan, B. Samet, J. Inequal. Appl. 2015, 247 (2015)

    Article  Google Scholar 

  23. J. Rong, C. Bai, Adv. Differ. Equ. 2015, 82 (2015)

    Article  Google Scholar 

  24. M. Jleli, M. Kirane, B. Samet, Appl. Math. Lett. 66, 30 (2017)

    Article  MathSciNet  Google Scholar 

  25. R.A.C. Ferreira, Fract. Differ. Calc. 5, 87 (2015)

    Article  MathSciNet  Google Scholar 

  26. M. Jleli, B. Samet, J. Nonlinear Sci. Appl. 9, 1965 (2016)

    Article  MathSciNet  Google Scholar 

  27. T. Abdeljawad, Discret. Dyn. Nat. Soc. 2013, 406910 (2013)

    MathSciNet  Google Scholar 

  28. T. Abdeljawad, Adv. Differ. Equ. 2013, 36 (2013)

    Article  Google Scholar 

  29. T. Abdeljawad, F. Jarad, D. Baleanu, Adv. Differ. Equ. 1, 72 (2012)

    Article  Google Scholar 

  30. T. Abdeljawad, F. Atici, Abstr. Appl. Anal. 2012, 406757 (2012)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia

    Thabet Abdeljawad

  2. Department of Mathematics, University of Mohamed Boudiaf, M’sila, Algeria

    Fadila Madjidi

  3. Laboratoire des Mathématiques Appliquées, Université de Bejaia, Bejaia, 06000, Algeria

    Fadila Madjidi

Authors
  1. Thabet Abdeljawad
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  2. Fadila Madjidi
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Correspondence to Thabet Abdeljawad.

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Abdeljawad, T., Madjidi, F. Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2. Eur. Phys. J. Spec. Top. 226, 3355–3368 (2017). https://doi.org/10.1140/epjst/e2018-00004-2

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  • DOI: https://doi.org/10.1140/epjst/e2018-00004-2

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