Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2017, Article ID 5123240, 8 pages
https://doi.org/10.1155/2017/5123240
Research Article

Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2LaSIE, Pôle Sciences et Technologies, Université de La Rochelle, avenue M. Crépeau, 17042 La Rochelle Cedex, France

Correspondence should be addressed to Bessem Samet; as.ude.usk@temasb

Received 2 January 2017; Revised 19 January 2017; Accepted 22 January 2017; Published 12 February 2017

Academic Editor: Thabet Abdeljawad

Copyright © 2017 Mohamed Jleli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Liapounoff, “Problème général de la stabilité du mouvement,” Annales de la faculté des sciences de Toulouse Mathématiques, vol. 9, pp. 203–474, 1907. View at Publisher · View at Google Scholar
  2. A. Beurling, “Un théorème sur les fonctions bornées et uniformément continues sur l'axe réel,” Acta Mathematica, vol. 77, pp. 127–136, 1945. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. G. Borg, “On a Liapounoff criterion of stability,” American Journal of Mathematics, vol. 71, pp. 67–70, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. C. Brown and D. B. Hinton, “Opial's inequality and oscillation of 2nd order equations,” Proceedings of the American Mathematical Society, vol. 125, no. 4, pp. 1123–1129, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R. S. Dahiya and B. Singh, “A Lyapunov inequality and nonoscillation theorem for a second order non-linear differential-difference equation,” Journal of Mathematical and Physical Sciences, vol. 7, pp. 163–170, 1973. View at Google Scholar · View at MathSciNet
  6. G. S. Guseinov and A. Zafer, “Stability criteria for linear periodic impulsive Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1195–1206, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964, Birkhuser, Boston, Mass, USA 1982.
  8. P. Hartman and A. Wintner, “On an oscillation criterion of Liapounoff,” American Journal of Mathematics, vol. 73, pp. 885–890, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. G. Pachpatte, “Inequalities related to the zeros of solutions of certain second order differential equations,” Facta Universitatis, Series: Mathematics and Informatics, vol. 16, pp. 35–44, 2001. View at Google Scholar
  10. W. T. Reid, “A matrix equation related to a non-oscillation criterion and Liapunov stability,” Quarterly of Applied Mathematics, vol. 23, pp. 83–87, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Singh, “Forced oscillation in general ordinary differential equations,” Tamkang Journal of Mathematics, vol. 6, pp. 5–11, 1975. View at Google Scholar
  12. A. Wintner, “On the non-existence of conjugate points,” American Journal of Mathematics, vol. 73, pp. 368–380, 1951. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Çakmak, “Lyapunov-type integral inequalities for certain higher order differential equations,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 368–373, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. S. B. Eliason, “Lyapunov type inequalities for certain second order functional differential equations,” SIAM Journal on Applied Mathematics, vol. 27, no. 1, pp. 180–199, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579–593, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. C. Lee, C. Yeh, C. Hong, and R. P. Agarwal, “Lyapunov and Wirtinger inequalities,” Applied Mathematics Letters, vol. 17, no. 7, pp. 847–853, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. B. G. Pachpatte, “Lyapunov type integral inequalities for certain differential equations,” Georgian Mathematical Journal, vol. 4, no. 2, pp. 139–148, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Panigrahi, “Liapunov-type integral inequalities for certain higher-order differential equations,” Electronic Journal of Differential Equations, vol. 28, pp. 1–14, 2009. View at Google Scholar · View at MathSciNet
  19. N. Parhi and S. Panigrahi, “Liapunov-type inequality for higher order differential equations,” Mathematica Slovaca, vol. 52, no. 1, pp. 31–46, 2002. View at Google Scholar · View at MathSciNet
  20. A. n. Tiryaki, “Recent developments of Lyapunov-type inequalities,” Advances in Dynamical Systems and Applications, vol. 5, no. 2, pp. 231–248, 2010. View at Google Scholar · View at MathSciNet
  21. X. Yang, “On Liapunov-type inequality for certain higher-order differential equations,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 307–317, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. X. Yang and K. Lo, “Lyapunov-type inequality for a class of even-order differential equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3884–3890, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. A. A. Kilbas, H. M. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  24. R. A. C. Ferreira, “A Lyapunov-type inequality for a fractional boundary value problem,” Fractional Calculus and Applied Analysis, vol. 16, no. 4, pp. 978–984, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. R. A. C. Ferreira, “On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function,” Journal of Mathematical Analysis and Applications, vol. 412, no. 2, pp. 1058–1063, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. R. A. C. Ferreira, “Lyapunov-type inequalities for some sequential fractional boundary value problems,” Advances in Dynamical Systems and Applications, vol. 11, no. 1, pp. 33–43, 2016. View at Google Scholar · View at MathSciNet
  27. M. Jleli and B. Samet, “Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions,” Mathematical Inequalities & Applications, vol. 18, no. 2, pp. 443–451, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. M. Jleli and B. Samet, “Lyapunov-type inequalities for fractional boundary value problems,” Electronic Journal of Differential Equations, vol. 88, pp. 1–11, 2015. View at Google Scholar
  29. M. Jleli, M. Kirane, and B. Samet, “Lyapunov-type inequalities for fractional partial differential equations,” Applied Mathematics Letters, vol. 66, pp. 30–39, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  30. M. Jleli, L. Ragoub, and B. Samet, “A Lyapunov-type inequality for a fractional differential equation under a Robin boundary condition,” Journal of Function Spaces, vol. 2015, Article ID 468536, 5 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. D. O'Regan and B. Samet, “Lyapunov-type inequalities for a class of fractional differential equations,” Journal of Inequalities and Applications, vol. 2015, article 247, 10 pages, 2015. View at Publisher · View at Google Scholar
  32. N. Al Arifi, I. Altun, M. Jleli, A. Lashin, and B. Samet, “Lyapunov-type inequalities for a fractional p-Laplacian equation,” Journal of Inequalities and Applications, vol. 2016, article 189, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. J. Rong and C. Bai, “Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions,” Advances in Difference Equations, vol. 2015, article 82, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. A. Chidouh and D. F. Torres, “A generalized Lyapunov's inequality for a fractional boundary value problem,” Journal of Computational and Applied Mathematics, vol. 312, pp. 192–197, 2017. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. R. P. Agarwal and A. Özbekler, “Lyapunov type inequalities for mixed nonlinear Riemann-Liouville fractional differential equations with a forcing term,” Journal of Computational and Applied Mathematics, vol. 314, pp. 69–78, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  36. D. Ma, “A generalized Lyapunov inequality for a higher-order fractional boundary value problem,” Journal of Inequalities and Applications, vol. 2016, article no. 261, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Q. Ma, C. Ma, and J. Wang, “A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative,” Journal of Mathematical Inequalities, vol. 11, no. 1, pp. 135–141, 2007. View at Publisher · View at Google Scholar
  38. D. Idczak and S. Walczak, “Fractional sobolev spaces via Riemann-Liouville derivatives,” Journal of Function Spaces and Applications, vol. 2013, Article ID 128043, 15 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet