Skip to main content

Advertisement

SpringerLink
Log in

Lyapunov-type inequalities for a fractional p-Laplacian system

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this paper, we establish Lyapunov-type inequalities for a fractional p-Laplacian system in an open bounded subset Ω⊂ℝN RN under Dirichlet boundary conditions. As an application of the obtained inequalities, we establish some geometric properties of the generalized spectrum associated to the considered system.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.A. Adams, J. JF Fournier, Sobolev Spaces. Academic Press (2003).

    MATH  Google Scholar 

  2. G. Borg, On a Liapounoff criterion of stability. Amer. J. Math. 71, No 1 (1949), 67–70.

    MathSciNet  MATH  Google Scholar 

  3. R.C. Brown, D.B. Hinton, Opial’s inequality and oscillation of 2nd order equations. Proc. Amer. Math. Soc. 125, No 4 (1997), 1123–1129.

    MathSciNet  MATH  Google Scholar 

  4. D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, No 2 (2010), 368–373.

    MathSciNet  MATH  Google Scholar 

  5. D. Çakmak, A. Tiryaki, Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p1p2 …,pnLaplacian. J. Math. Anal. Appl. 369, No 1 (2010), 76–81.

    MathSciNet  MATH  Google Scholar 

  6. A. Cañada, J. A. Montero, S. Villegas, Lyapunov inequalities for partial differential equations. J. Funct. Anal. 237, No 1 (2006), 176–193.

    MathSciNet  MATH  Google Scholar 

  7. A. Cañada, S. Villegas, Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues. J. Eur. Math. Soc. 12 (2010), 163–178.

    MathSciNet  MATH  Google Scholar 

  8. R. Cantrell, On coupled multiparameter nonlinear elliptic system. Trans. Amer. Math. Soc. 294, No 1 (1986), 263–285.

    MathSciNet  MATH  Google Scholar 

  9. R. Cantrell, C. Cosner, On the generalized spectrum for second order elliptic systems. Trans. Amer. Math. Soc. 303, No 1 (1987), 345–363.

    MathSciNet  MATH  Google Scholar 

  10. S.S. Cheng, A discrete analogue of the inequality of Lyapunov. Hokkaido Math. J. 12, No 1 (1983), 105–112.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. S. Dhar, Q. Kong, Lyapunov-type inequalities for higher order half-linear differential equations. Appl. Math. Comput. 273 (2016), 114–124.

    MathSciNet  MATH  Google Scholar 

  13. B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15, No 4 (2012), 536–555; 10.2478/s13540-012-0038-8. https://www.degruyter.eom/view/j/fca.2012.15.issue-4/s13540-012-0038-8/s13540-012-0038-8.xml

    MathSciNet  MATH  Google Scholar 

  14. Y.V. Egorov, V.A. Kondriatev, On Spectral Theory of Elliptic Operators (Operator Theory: Advances and Applications). Birkhauser (1996).

    Google Scholar 

  15. A. Elbert, A half-linear second order differential equation. Colloq. Math. Soc. Janos Bolyai. 30 (1979), 158–180.

    Google Scholar 

  16. Rui A.C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16, No 4 (2013), 978–984; 10.2478/s13540-013-0060-5. https://www.degruyter.com/view/j/fca.2013.16.issue-4/s13540-013-0060-5/s13540-013-0060-5.xml

    MathSciNet  MATH  Google Scholar 

  17. Rui 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.

    MathSciNet  MATH  Google Scholar 

  18. Rui A.C. Ferreira, Lyapunov-type inequality for an anti- periodic fractional boundary value problem. Fract. Calc. Appl. Anal. 20, No 1 (2017), 284–291; 10.1515/fca-2017-0015. https://www.degruyter.com/view/j/fca.2017.20.issue-1/fca-2017-0015/fca-2017-0015.xml

    MathSciNet  MATH  Google Scholar 

  19. G. Franzina, G. Palatucci, Fractional p-eigenvalues. Riv. Mat. Univ. Parma. 8, No 5 (2014), 315–328.

    MathSciNet  MATH  Google Scholar 

  20. G. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems. Comput. Math. Appl. 45, No 6–9 (2003), 1399–1416.

    MathSciNet  MATH  Google Scholar 

  21. P. Hartman, Ordinary Differential Equations. Wiley, New York (1964) Birkhauser, Boston 1982).

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. A. Iannizzotto, M. Squassina, Weyl-type laws for fractional p-eigenvalue problems. Asymptot. Anal. 88, No 4 (2014), 233–245.

    MathSciNet  MATH  Google Scholar 

  24. L. Ihnatsyeva, J. Lehrback, H. Tuominen, A.V. Vaahakangas, Fractional Hardy inequalities and visibility of the boundary. Available at: http://arxiv.org/abs/1305.4616

  25. M. Jleli, M. Kirane, B. Samet, Lyapunov-type inequalities for fractional partial differential equations. Appl. Math. Lett. 66 (2017), 30–39.

    MathSciNet  MATH  Google Scholar 

  26. 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 

  27. M. Jleli, B. Samet, Lyapunov-type inequalities for fractional boundary-value problems. Electron. J. Differential Equations. 88 (2015), 1–11.

    MathSciNet  MATH  Google Scholar 

  28. C. Lee, C. Yeh, C. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17, No 7 (2004), 847–853.

    MathSciNet  MATH  Google Scholar 

  29. E. Lindgren, P. Lindqvist, Fractional eigenvalues. Calc. Var. Partial Differential Equations. 49, No 1–2 (2014), 795–826.

    MathSciNet  MATH  Google Scholar 

  30. A.M. Lyapunov, Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Toulouse. 2 (1907), 203–407.

    Google Scholar 

  31. T.M. Michelitsch, G.A. Maugin, A.F. Nowakowski, F.C.G.A. Nicolleau, M. Rahman, The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion. Fract. Calc. Appl. Anal. 16, No 4 (2013), 827–859; 10.2478/s13540-013-0052-5. https://www.degruyter.com/view/j/fca.2013.16.issue-4/s13540-013-0052-5/s13540-013-0052-5.xml

    MathSciNet  MATH  Google Scholar 

  32. PL De Nápoli, JP Pinasco, Lyapunov-type inequalities for partial differential equations. J. Funct. Anal. 15, No 6 (2016), 1995–2018.

    MathSciNet  MATH  Google Scholar 

  33. P.L. De Napoli, J.P. Pinasco, A Lyapunov Inequality for monotone quasilinear operators. Differential Integral Equations. 18, No 10 (2005), 1193–1200.

    MathSciNet  MATH  Google Scholar 

  34. P.L. De Napoli, J.P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems. J. Differential Equations. 227, No 1 (2006), 102–115.

    MathSciNet  MATH  Google Scholar 

  35. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, No 5 (2012), 521–573.

    MathSciNet  MATH  Google Scholar 

  36. D. O’Regan, B. Samet, Lyapunov-type inequalities for a class of frac-tional differential equations. J. Inequal. Appl. 2015, No 247 (2015), 1–10.

    Google Scholar 

  37. B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195, No 2 (1995), 527–536.

    MathSciNet  MATH  Google Scholar 

  38. N. Parhi, S. Panigrahi, On Liapunov-type inequality for third-order differential equations. J. Math. Anal. Appl. 233, No 2 (1999), 445–460.

    MathSciNet  MATH  Google Scholar 

  39. J.P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian. Abstr. Appl. Anal. 2004 No 2 (2004), 147–153.

    MathSciNet  MATH  Google Scholar 

  40. M. Protter, The generalized spectrum of second order elliptic systems. Rocky Mountain J. Math. 9, No 3 (1979), 503–518.

    MathSciNet  MATH  Google Scholar 

  41. T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter (1996).

    MATH  Google Scholar 

  42. J. Sánchez, V. Vergara, A Lyapunov-type inequality for a p-Laplacian operator. Nonlinear Anal. 74, No 18 (2011), 7071–7077.

    MathSciNet  MATH  Google Scholar 

  43. I. Sim, Y.H. Lee, Lyapunov inequalities for one-dimensional p-Laplacian problems with a singular weight function. J. Inequal. Appl. 2010, No 1 (2010), 1–9.

    MathSciNet  MATH  Google Scholar 

  44. X. Tang, M. Zhang, Lyapunov inequalities and stability for linear Hamiltonian systems. J. Differential Equations. 252, No 1 (2012), 358–381.

    MathSciNet  MATH  Google Scholar 

  45. Y. Wang, Lyapunov-type inequalities for certain higher order differential equations with anti-periodic boundary conditions. Appl. Math. Lett. 25, No 12 (2012), 2375–2380.

    MathSciNet  MATH  Google Scholar 

  46. X. Yang, On inequalities of Lyapunov type. Appl. Math. Comput. 134, No 2 (2003), 293–300.

    MathSciNet  MATH  Google Scholar 

  47. X. Yang, K. Lo, Lyapunov-type inequality for a class of even-order differential equations. Appl. Math. Comput. 215, No 11 (2010), 3884–3890.

    MathSciNet  MATH  Google Scholar 

  48. Q-M. Zhou, Ke-Qi Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian. Fract. Calc. Appl. Anal. 18, No 1 (2015), 133–145; 10.1515/fca-2015-0009. https://www.degruyter.com/view/j/fca.2015.18.issue-1/fca-2015-0009/fca-2015-0009.xml

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Mohamed Jleli

  2. LaSIE, Pôle Sciences et Technologies, Université de La Rochelle Avenue M. Crépeau, 17042, La Rochelle, France

    Mokhtar Kirane

  3. NAAM Research Group, Department of Mathematics, Faculty of Science King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Mokhtar Kirane

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

    Bessem Samet

Authors
  1. Mohamed Jleli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mokhtar Kirane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bessem Samet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mokhtar Kirane.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jleli, M., Kirane, M. & Samet, B. Lyapunov-type inequalities for a fractional p-Laplacian system. FCAA 20, 1485–1506 (2017). https://doi.org/10.1515/fca-2017-0078

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2017-0078

MSC 2010

Key Words and Phrases

Search

Navigation