Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On a generalized Lyapunov inequality for a mixed fractional boundary value problem

Laboratory of Advanced Materials, Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this paper, we establish a new Lyapunov-type inequality for a differential equation involving left Riemann-Liouville and right Caputo fractional derivatives subject to Dirichlet-type boundary conditions.
  Figure/Table
  Supplementary
  Article Metrics

References

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

2. R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, London: Imperial College Press, 2015.

3. A. Chidouh and D. F. M. Torres, A generalized Lyapunovs inequality for a fractional boundary value problem, J. Comput. Appl. Math., 312 (2017), 192-197.    

4. D. Ma, A generalized Lyapunov inequality for a higher-order fractional boundary value problem, J. Inequal. Appl., 2016 (2016), 1-11.    

5. S. Dhar, Q. Kong and M. McCabe, Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions, Electron. J. Qual. Theory Differ. Equations, 2016 (2016), 1-16.

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

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

8. R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43.

9. R. A. C. Ferreira, Novel Lyapunov-type inequalities for sequential fractional boundary value problems, RACSAM Rev. R. Acad. A, 113 (2019), 171-179.

10. M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary value problems, Electron. J. Differ. Equations, 2015 (2015), 1-11.

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

12. P. Hartman and A. Wintner, On an oscillation criterion of Lyapunov, Amer. J. Math., 73 (1951), 885-890.    

13. R. Khaldi and A. Guezane-Lakoud, Lyapunov inequality for a boundary value problem involving conformable derivative, Prog. Frac. Diff. Appl., 3 (2017), 323-329.    

14. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Amsterdam: Elsevier Science, 2006.

15. A. M. Lyapunov, Problème général de la stabilité du mouvement, (French translation of a Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse, 2 (1907), 27-247, Reprinted as Ann. Math. Studies, No. 17, Princeton, 1947.

16. A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, In series of Springer Briefs in Applied Sciences and Technology, Springer Cham Heidelberg, 2015.

17. A. B. Malinowska and D. F. M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl., 59 (2010), 3110-3116.    

18. S. K. Ntouyas, B. Ahmad and T. P. Horikis, Recent developments of Lyapunov-type inequalities for fractional differential equations, in press. Available from: https://arxiv.org/pdf/1804.10760.pdf.

19. I. Podlubny, Fractional Differential Equation, Sain Diego: Academic Press, 1999.

20. Q. Ma, Ch. Ma and J. Wang, A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11 (2011), 135-141.

21. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Yverdon, Switzerland: Gordon and Breach, 1993.

22. D. O'Regan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 1-10.    

23. J. Rong and C. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Differ. Equations, 2015 (2015), 1-10.

24. A. Wintner, On the non-existence of conjugate points, Amer. J. Math., 73 (1951), 368-380.    

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved