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Lyapunov-Type Inequalities for Nonlinear Fractional Differential Equation with Hilfer Fractional Derivative Under Multi-Point Boundary Conditions

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Abstract

In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Hilfer fractional derivative under multi-point boundary conditions, the results are new and generalize and improve some early results in the literature.

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References

  1. R.C. Brown, D.B. Hinton, Lyapunov inequalities and their applications. In: Survey on Classical Inequalities (Ed. T.M. Rassias), Kluwer Academic Publishers, Dordrecht, 2000, 1–25.

  2. S. Cheng, Lyapunov inequalities for differential and difference equations. Fasc. Math. 23 (1991), 25–41.

    MathSciNet  MATH  Google Scholar 

  3. R.A.C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16, No 4 (2013), 978–984; 0.2478/s13540-013-0060-5; https://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.

    Article  MathSciNet  Google Scholar 

  4. R.A.C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function. J. Math. Anal. Appl. 412, No 2 (2014), 1058–1063.

    Article  MathSciNet  Google Scholar 

  5. R. Hilfer, Fractional calculus and regular variation in thermodynamics. In: Applications of Fractional Calculus in Physics (Ed. R. Hilfer), World Scientific, Singapore (2000).

    Chapter  Google Scholar 

  6. R. Hilfer, Y. Luchko and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, No 3 (2009), 299–318.

    MathSciNet  MATH  Google Scholar 

  7. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies # 204, Elsevier, Amsterdam, 2006.

    Google Scholar 

  8. A.M. Lyapunov, Problème général de la stabilité du mouvement (French Transl. of a Russian paper dated 1893). Ann. Fac. Sci. Univ. Toulouse 2 (1907), 27–247 (Reprinted as: Ann. Math. Studies, No 17, Princeton Univ. Press, Princeton, NJ, USA, 1947).

    Google Scholar 

  9. A. Tiryaki, Recent development of Lyapunov-type inequalities. Adv. Dyn. Syst. Appl. 5 No 2 (2010), 231–248.

    MathSciNet  Google Scholar 

  10. Z. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator. Nonlinear Analysis 75, No 7 (2012), 3364–3384.

    Article  MathSciNet  Google Scholar 

  11. M. Jleli, B. Samet, Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions. Math. Inequal. Appl. 18, No 2 (2015), 443–451.

    MathSciNet  MATH  Google Scholar 

  12. M. Jleli, B. Samet, Lyapunov-type inequalities for fractional boundary value problems. Electr. J. Differ. Equ. 88 (2015), 1–11.

    MathSciNet  MATH  Google Scholar 

  13. D. O’Regan, B. Samet, Lyapunov-type inequality for a class of fractional differential equations. J. Inequal. Appl. 247 (2015), 1–10.

    MATH  Google Scholar 

  14. J. Rong, C. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary condition. Adv. Difference Equ. 82 (2015), 1–10.

    MathSciNet  Google Scholar 

  15. M. Jleli, M. Kirane, B Samet, Lyapunov-type inequalities for a fractional p-Laplacian system. Fract. Calc. Appl. Anal. 20, No 6 (2017), 1485–1506.

    Article  MathSciNet  Google Scholar 

  16. A. Alsaedi, B. Ahmad, M. Kirane, A survey of useful inequalities in fractional calculus. Fract. Calc. Appl. Anal. 20, No 3 (2017), 574–594; 10.1515/fca-2017-0031; https://www.degruyter.com/view/j/fca.2017.20.issue-3/issue-files/fca.2017.20.issue-3.xml.

    Article  MathSciNet  Google Scholar 

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Correspondence to Youyu Wang.

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Wang, Y., Wang, Q. Lyapunov-Type Inequalities for Nonlinear Fractional Differential Equation with Hilfer Fractional Derivative Under Multi-Point Boundary Conditions. FCAA 21, 833–843 (2018). https://doi.org/10.1515/fca-2018-0044

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  • DOI: https://doi.org/10.1515/fca-2018-0044

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