Skip to main content
SpringerLink
Log in

Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations

  • Chapter
  • First Online:
Differential and Integral Inequalities

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 151))

Abstract

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm–Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal, and impulsive conditions. Furthermore, our study includes Riemann–Liouville, Caputo, Hadamard, Prabhakar, Hilfer, and conformable type fractional derivatives. Results for boundary value problems involving fractional p-Laplacian, fractional operators with nonsingular Mittag–Leffler kernels, q-difference, discrete, and impulsive equations are also taken into account.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Z. Nehari, On the zeros of solutions of second-order linear differential equations. Am. J. Math. 76, 689–697 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.M. Lyapunov, Probléme général de la stabilité du mouvement (French Transl. of a Russian paper dated 1893). Ann. Fac. Sci. Univ. Toulouse 2, 27–247 (1907). Reprinted in: Ann. Math. Stud. No. 17, Princeton (1947)

    Google Scholar 

  3. A. Wintner, On the nonexistence of conjugate points. Am. J. Math. 73, 368–380 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Hartman, Ordinary Differential Equations (Society for Industrial and Applied Mathematics, Philadelphia, 2002)

    Book  MATH  Google Scholar 

  5. D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, 368–373 (2010)

    MathSciNet  MATH  Google Scholar 

  6. S. Clark, D. Hinton, A Lyapunov inequality for linear Hamiltonian systems. Math. Inequal. Appl. 1, 201–209 (1998)

    MathSciNet  MATH  Google Scholar 

  7. A. Tiryaki, Recent developments of Lyapunov-type inequalities. Adv. Dyn. Syst. Appl. 5, 231–248 (2010)

    MathSciNet  Google Scholar 

  8. N. Parhi, S. Panigrahi, Lyapunov-type inequality for higher order differential equations. Math. Slovaca 52, 31–46 (2002)

    MathSciNet  MATH  Google Scholar 

  9. X. Yang, On Lyapunov-type inequality for certain higher-order differential equations. Appl. Math. Comput. 134, 307–317 (2003)

    MathSciNet  MATH  Google Scholar 

  10. X. Yang, K. Lo, Lyapunov-type inequality for a class of even-order differential equations. Appl. Math. Comput. 215, 3884–3890 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Q.-M. Zhang, X.H. Tang, Lyapunov inequalities and stability for discrete linear Hamiltonian systems. Appl. Math. Comput. 218, 574–582 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. R.C. Brown, D.B. Hinton, Lyapunov inequalities and their applications, in Survey on Classical Inequalities. Mathematics and Its Applications, vol. 517 (Springer, Dordrecht, 2000)

    Google Scholar 

  14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 (Elsevier Science, Amsterdam, 2006)

    Book  MATH  Google Scholar 

  15. I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Jleli, B. Samet, Lyapunov type inequalities for a fractional differential equations with mixed boundary value problems. Math. Inequal. Appl. 18, 443–451 (2015)

    MathSciNet  MATH  Google Scholar 

  19. M. Jleli, L. Ragoub, B. Samet, A Lyapunov-type inequality for a fractional differential equation under a Robin boundary condition. J. Function Spaces 2015, 468536 (5 pp.) (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Rong, C.Z. Bai, Lyapunov-type inequality for a fractional differential equations with fractional boundary value problems. Adv. Difference Equ. 2015, 82 (10 pp.) (2015)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. D. O’Regan, B. Samet, Lyapunov-type inequalities for a class of fractional differential equations. J. Inequal. Appl. 2015, 247 (10 pp.) (2015)

    Google Scholar 

  23. S. Sitho, S.K. Ntouyas, J. Tariboon, Existence results for hybrid fractional integro-differential equations. Bound. Value Probl. 2015, 113 (13 pp.) (2015)

    Google Scholar 

  24. M. Al-Qurashi, L. Ragoub, Lyapunov’s inequality for a fractional differential equation subject to a non-linear integral condition, in ADVCOMP 2016 : The Tenth International Conference on Advanced Engineering Computing and Applications in Sciences (2016), pp. 98–101

    Google Scholar 

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

    MathSciNet  Google Scholar 

  26. M. Al-Qurashia, L.Ragoub, Nonexistence of solutions to a fractional differential boundary value problem. J. Nonlinear Sci. Appl. 9, 2233–2243 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Dhar, Q. Kong, Lyapunov-type inequalities for α-th order fractional differential equations with 2 < α ≤ 3 and fractional boundary conditions. Electron. J. Differ. Equ. 2017 1–15 (2017)

    Article  MathSciNet  Google Scholar 

  29. M. Jleli, J. Nieto, B. Samet, Lyapunov-type inequalities for a higher order fractional differential equation with fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2017, 1–17 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Cabrera, B. Lopez, K. Sadarangani, Lyapunov type inequalities for a fractional two-point boundary value problem. Math. Method Appl. Sci. 40, 3409–3414 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. X. Wang, Y. Peng, W. Lu, Lyapunov-type inequalities for certain higher order fractional differential equations. J. Nonlinear Sci. Appl. 10, 5064–5071 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. R.P. Agarwal, A. Zbekler, 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 

  34. L. Zhang, Z. Zheng, Lyapunov type inequalities for the Riemann-Liouville fractional differential equations of higher order. Adv. Difference Equ. 2017, 270 (20 pp.) (2017)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Ma, A generalized Lyapunov inequality for a higher-order fractional boundary value problem. J. Inequal. Appl. 2016, 261 (11 pp.) (2016)

    Google Scholar 

  37. Y. Ru, F. Wang, T. An, Y. An, Some results on the fractional order Sturm-Liouville problems. Adv. Difference Equ. 2017, 320 (19 pp.) (2017)

    Google Scholar 

  38. I. Cabrera, J. Rocha, K.B. Sadarangani, Lyapunov type inequalities for a fractional thermostat model. RACSAM 112, 17–24 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Guidotti, S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation. Differ. Integr. Equ. 13, 1551–1568 (2000)

    MathSciNet  MATH  Google Scholar 

  40. I. Cabrera, K. Sadaragani, B. Samet, Hartman-Winter-type inequalities for a class of nonlocal fractional boundary value problems. Math. Methods Appl. Sci. 40, 129–136 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Y. Wang, Q. Wang, Lyapunov-type inequalities for fractional differential equations under multi-point boundary conditions. J. Comput. Anal. Appl. 26, 707–716 (2019)

    Google Scholar 

  42. N. Al Arifi, I. Altun, M. Jleli, A. Lashin, B. Samet, Lyapunov-type inequalities for a fractional p-Laplacian equation. J. Inequal. Appl. 2016, 189 (11 pp.) (2016)

    Google Scholar 

  43. Y. Liu, D. Xie, D. Yang, C. Bai, Two generalized Lyapunov-type inequalities for a fractional p-Laplacian equation with fractional boundary conditions. J. Inequal. Appl. 2017, 98 (11 pp.) (2017)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Q. Ma, C. Na, J. Wang, A Lyapunov-type inequality for fractional differential equation with Hadamard derivative. J. Math. Inequal. 11, 135–141 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities (Springer, Basel, 2017)

    Book  MATH  Google Scholar 

  47. N.G. Abuj, D.B. Pachpatte, Lyapunov-type inequality for fractional differential equation with k-Prabhakar derivative. Int. J. Pure Appl. Math. 113, 563–574 (2017)

    Google Scholar 

  48. S. Eshaghi, A. Ansari, Lyapunov inequality for fractional differential equations with Prabhakar derivative. Math. Inequal. Appl. 19, 349–358 (2016)

    MathSciNet  MATH  Google Scholar 

  49. D.B. Pachpatte, N.G. Abuj, A.D. Khandagale, Lyapunov-type inequality for hybrid fractional differential equation with Prabhakar derivative. Int. J. Pure Appl. Math. 113, 563–574 (2017)

    Google Scholar 

  50. M. Jleli, B. Samet, Lyapunov-type inequality for a fractional q-difference boundary value problem. J. Nonlinear Sci. Appl. 9, 1965–1976 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. B. Ahmad, S.K. Ntouyas, J. Tariboon, Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities (World Scientific, Singapore, 2016)

    Book  MATH  Google Scholar 

  52. M. Jleli, M. Kirane, B. Samet, Hartman-Wintner-type inequality for a fractional boundary value problem via a fractional derivative with respect to another function. Discret. Dyn. Nat. Soc. 2017, 5123240 (8 pp.) (2017)

    Google Scholar 

  53. Y. Chen, Y-C. Liou, C-H. Lo, Lyapunov inequality for a class of fractional differential equations with Dirichlet boundary conditions. J. Nonlinear Sci. Appl. 10, 3357–3363 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. J. Inequal. Appl. 2017, 130 (11 pp.) (2017)

    Google Scholar 

  56. Ch. Goodrich, A.C. Peterson, Discrete Fractional Calculus (Springer, Basel, 2015)

    Book  MATH  Google Scholar 

  57. R.A.C. Ferreira, Some discrete fractional Lyapunov-type inequalities. Frac. Differ. Calc. 5, 87–92 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  58. A. Chidouh, D.F.M. Torres, Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities. Opuscula Math. 38, 31–40 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. K. Ghanbari, Y. Gholami, New classes of Lyapunov-type inequalities of fractional Δ-difference Sturm-Liouville problems with applications. Bull. Iranian Math. Soc.43, 385–408 (2017)

    MathSciNet  MATH  Google Scholar 

  60. Z. Kayar, An existence and uniqueness result for linear fractional impulsive boundary value problems as an application of Lyapunov type inequality. Hacet. J. Math. Stat. 47, 287–297 (2018)

    MathSciNet  MATH  Google Scholar 

  61. R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)

    MATH  Google Scholar 

  62. R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials. J. Chem. Phys. 284, 399–408 (2002)

    Google Scholar 

  63. R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Frac. Calc. Appl. Anal. 12, 299–318 (2009)

    MathSciNet  MATH  Google Scholar 

  64. N.S. Pathak, Lyapunov-type inequality for fractional boundary value problems with Hilfer derivative. Math. Inequal. Appl. 21, 179–200 (2018)

    MathSciNet  MATH  Google Scholar 

  65. M. Kirane, B. Torebek, Lyapunov and Hartman-Wintner type inequalities for a nonlinear fractional boundary value problem with generalized Hilfer derivative, arXiv:1702.06073v1 [math.CA]

    Google Scholar 

  66. B. Lupinska, T. Odzijewicz, A Lyapunov-type inequality with the Katugampola fractional derivative. Math. Method Appl. Sci., 1–12 (2018)

    Google Scholar 

  67. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  68. T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  69. U.N. Katugampola, Correction to “What is a fractional derivative?” by Ortigueira and Machado [J. Comput. Phys. 293, 4–13 (2015)]. J. Comput. Phys. 321, 1255–1257 (2016)

    Google Scholar 

  70. M.D. Ortigueira, J.A. Tenreiro Machado. What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  71. R. Khaldi, A. Guezane-Lakoud, Lyapunov inequality for a boundary value problem involving conformable derivative. Progr. Fract. Differ. Appl. 3, 323–329 (2017)

    Article  MATH  Google Scholar 

  72. T. Abdeljawad, J. Alzabut, F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Difference Equ. 2017, 321 (10 pp.) (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Sotiris K. Ntouyas & Theodoros P. Horikis

  2. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Bashir Ahmad

Authors
  1. Sotiris K. Ntouyas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bashir Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Theodoros P. Horikis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Theodoros P. Horikis .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania

    Dorin Andrica

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ntouyas, S.K., Ahmad, B., Horikis, T.P. (2019). Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations. In: Andrica, D., Rassias, T. (eds) Differential and Integral Inequalities. Springer Optimization and Its Applications, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-27407-8_24

Download citation

Publish with us

Policies and ethics

Search

Navigation