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Lyapunov-type inequalities for a nonlinear fractional boundary value problem

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we obtain a Lyapunov-type and a Hartman–Wintner-type inequalities for a nonlinear fractional hybrid equation with left Riemann–Liouville and right Caputo fractional derivatives of order \(1/2<\alpha \le 1,\) subject to Dirichlet boundary conditions. It is also shown that failure of the Lyapunov-type and Hartman–Wintner-type inequalities, corresponding nonlinear boundary value problem has only trivial solutions. In the case \(\alpha =1\), our results coincide with the classical Lyapunov and Hartman–Wintner inequalities, respectively.

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References

  1. Agarwal, R.P., Özbekler, A.: Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term. J. Comput. Appl. Math. 314, 69–78 (2017)

    Article  MathSciNet  Google Scholar 

  2. Cabrera, I., Sadarangani, K., Samet, B.: Hartman–Wintner-type inequalities for a class of nonlocal fractional boundary value problems. Math. Meth. Appl. Sci. 40, 129–136 (2017)

    Article  MathSciNet  Google Scholar 

  3. Chidouh, A., Torres, D.F.M.: A generalized Lyapunov’s inequality for a fractional boundary value problem. J. Comput. Appl. Math. 312, 192–197 (2017)

    Article  MathSciNet  Google Scholar 

  4. Ferreira, R.A.C.: A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16(4), 978–984 (2013)

    Article  MathSciNet  Google Scholar 

  5. Ferreira, R.A.C.: On a Lyapunov-type inequality and the zeros of a certain Mittag–Leffler function. J. Math. Anal. Appl. 412(2), 1058–1063 (2014)

    Article  MathSciNet  Google Scholar 

  6. Ferreira, R.A.C.: Novel Lyapunov-type inequalities for sequential fractional boundary value problems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113(1), 171–179 (2019)

    Article  MathSciNet  Google Scholar 

  7. Guezane-Lakoud, A., Khaldi, R., Torres, D.F.M.: Lyapunov-type inequality for a fractional boundary value problem with natural conditions. SeMA J. 75, 157–162 (2018)

    Article  MathSciNet  Google Scholar 

  8. Hartman, P., Wintner, A.: On an oscillation criterion of Lyapunov. Am. J. Math. 73, 885–890 (1951)

    Article  Google Scholar 

  9. Jleli, M., Samet, B.: Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions. Math. Inequal. Appl. 18(2), 443–451 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Jleli, M., Kirane, M., Samet, B.: Lyapunov-type inequalities for fractional quasilinear problems via variational methods. J. Nonlinear Sci. Appl. 10, 2471–2486 (2017)

    Article  MathSciNet  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Mathematics studies. Elsevier, North-Holland (2006)

    MATH  Google Scholar 

  12. Liapounoff, A.M.: Probléme général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Toulouse 9, 203–474 (1907)

    Article  MathSciNet  Google Scholar 

  13. Lopez, B., Rocha, J., Sadarangani, K.: Lyapunov type inequality for a nonlinear fractional hybrid boundary value problem. Zeitschrift für Analysis und ihre Anwendungen. 38(1), 97–107 (2019)

    Article  MathSciNet  Google Scholar 

  14. Ma, Q., Ma, Ch., Wang, J.: A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative. J. Math. Inequal. 11(1), 135–141 (2017)

    Article  MathSciNet  Google Scholar 

  15. Meng, X., Stynes, M.: Green’s functions, positive solutions, and a Lyapunov inequality for a Caputo fractional-derivative boundary value problem. Fract. Calc. Appl. Anal. 22, 750–766 (2019)

    Article  MathSciNet  Google Scholar 

  16. Khaldi, R., Guezane-Lakoud, A.: On a generalized Lyapunov inequality for a mixed fractional boundary value problem. AIMS Math. 4(3), 506–515 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank to the reviewers for their valuable comments and remarks. This research is financially supported by the FWO Odysseus 1 grant G.0H94.18N:Analysis and Partial Differential Equations and by the “5–100” program of RUDN University.

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Authors and Affiliations

  1. Al–Farabi Kazakh National University, Al–Farabi Ave. 71, 050040, Almaty, Kazakhstan

    Aidyn Kassymov & Berikbol T. Torebek

  2. Institute of Mathematics and Mathematical Modeling, 125 Pushkin Street, 050010, Almaty, Kazakhstan

    Aidyn Kassymov

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Aidyn Kassymov & Berikbol T. Torebek

  4. RUDN University, 6 Miklukho-Maklay Street, 117198, Moscow, Russia

    Berikbol T. Torebek

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  1. Aidyn Kassymov
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  2. Berikbol T. Torebek
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Correspondence to Berikbol T. Torebek.

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Kassymov, A., Torebek, B.T. Lyapunov-type inequalities for a nonlinear fractional boundary value problem. RACSAM 115, 15 (2021). https://doi.org/10.1007/s13398-020-00954-9

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  • DOI: https://doi.org/10.1007/s13398-020-00954-9

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