[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Riccati technique and oscillation of linear second-order difference equations

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we analyse oscillatory properties of a general class of linear difference equations. Applying the modified Riccati technique, we prove an oscillation criterion for the studied equations and we formulate its consequences. In contrast to many known criteria, in the presented results, there are not considered any auxiliary sequences. The results are based directly on the coefficients of the treated equations, i.e., the obtained results are easy to use. In addition, recently, we have proved a non-oscillatory counterpart of the presented criterion. The combination implies that the studied equations are conditionally oscillatory.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abu-Risha, M.H.: Oscillation of second-order linear difference equations. Appl. Math. Lett. 13(1), 129–135 (2000)

    Article  MathSciNet  Google Scholar 

  2. Agarwal, R.P.: Difference Equations and Inequalities. Marcel Dekker, New York (2000)

    Book  Google Scholar 

  3. Agarwal, R.P., Bohner, M., Grace, S.R., O’Regan, D.: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York (2005)

    Book  Google Scholar 

  4. Chen, S.Z., Erbe, L.H.: Disconjugacy, disfocality, and oscillation of second order difference equations. J. Differential Equations 107, 383–394 (1994)

    Article  MathSciNet  Google Scholar 

  5. Chen, S.Z., Erbe, L.H.: Riccati techniques and discrete oscillations. J. Math. Anal. Appl. 142(2), 468–487 (1989)

    Article  MathSciNet  Google Scholar 

  6. Došlý, O., Řehák, P.: Half-Linear Differential Equations. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  7. Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2005)

    MATH  Google Scholar 

  8. Fišnarová, S.: Oscillatory properties of half-linear difference equations: two-term perturbations. Adv. Difference Equ. 2012(101), 1–16 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Fujimoto, K., Hasil, P., Veselý, M.: Riccati transformation and nonoscillation criterion for linear difference equations. Proc. Amer. Math. Soc. 148(10), 4319–4332 (2020)

    Article  MathSciNet  Google Scholar 

  10. Grace, S.R., Abadeer, A.A., El-Morshedy, H.A.: Summation averaging technique for the oscillation of second order linear difference equations. Publ. Math. 55, 333–347 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Hasil, P., Juránek, J., Veselý, M.: Non-oscillation of half-linear difference equations with asymptotically periodic coefficients. Acta Math. Hungar. 159(1), 323–348 (2019)

    Article  MathSciNet  Google Scholar 

  12. Hasil, P., Kisel’ák, J., Pospíšil, M., Veselý, M.: Non-oscillation of half-linear dynamic equations on time scales. Math. Methods Appl. Sci. 44(11), 8775–8797 (2021)

    Article  MathSciNet  Google Scholar 

  13. Hasil,P., Veselý,M.: Critical oscillation constant for difference equations with almost periodic coefficients. Abstract Appl. Anal. 2012, Art. ID 471435, 19 pp. (2012)

  14. Hasil, P., Veselý, M.: Oscillation and non-oscillation criteria for linear and half-linear difference equations. J. Math. Anal. Appl. 452(1), 401–428 (2017)

    Article  MathSciNet  Google Scholar 

  15. Hasil, P., Veselý, M.: Oscillation constants for half-linear difference equations with coefficients having mean values. Adv. Difference Equ. 2015(210), 1–18 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Hasil, P., Veselý, M.: Oscillation result for half-linear dynamic equations on timescales and its consequences. Math. Methods Appl. Sci. 42(6), 1921–1940 (2019)

    Article  MathSciNet  Google Scholar 

  17. Hasil, P., Veselý, M.: Oscillatory and non-oscillatory solutions of dynamic equations with bounded coefficients. Electron. J. Differential Equations 2018(24), 1–22 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Hinton, D.B., Lewis, R.T.: Spectral analysis of second order difference equations. J. Math. Anal. Appl. 63(2), 421–438 (1978)

    Article  MathSciNet  Google Scholar 

  19. Hongyo, A., Yamaoka, N.: General solutions for second-order linear difference equations of Euler type. Opuscula Math. 37(3), 389–402 (2017)

    Article  MathSciNet  Google Scholar 

  20. Hooker, J.W.: A Hille–Wintner type comparison theorem for second order difference equations. Int. J. Math. Sci. 6, 387–393 (1983)

    Article  MathSciNet  Google Scholar 

  21. Hooker, J.W., Kwong, M.K., Patula, W.T.: Oscillatory second order linear difference equations and Riccati equations. SIAM J. Math. Anal. 18(1), 54–63 (1987)

    Article  MathSciNet  Google Scholar 

  22. Hooker, J.W., Patula, W.T.: Riccati type transformations for second-order linear difference equations. J. Math. Anal. Appl. 82(2), 451–462 (1981)

    Article  MathSciNet  Google Scholar 

  23. Kalybay, A., Oinarov, R.: Weighted hardy inequalities with sharp constants. J. Korean Math. Soc. 57(3), 603–616 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Kelley, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (2001)

    MATH  Google Scholar 

  25. Kwong, M.K., Hooker, J.W., Patula, W.T.: Riccati type transformations for second-order linear difference equations II. J. Math. Anal. Appl. 107(1), 182–196 (1985)

    Article  MathSciNet  Google Scholar 

  26. Morshedy, H.A.: Oscillation and non-oscillation criteria for half-linear second order difference equations. Dyn. Syst. Appl. 15, 429–450 (2006)

    Google Scholar 

  27. Naĭman,P.B.: The set of isolated points of increase of the spectral function pertaining to a limit-constant Jacobi matrix (Russian). Izv. Vysš. Učebn. Zaved. Matematika 1959, 1(8), 129–135 (1959)

  28. Řehák, P.: Comparison theorems and strong oscillation in the half-linear discrete oscillation theory. Rocky Mountain J. Math. 33(1), 333–352 (2003)

    Article  MathSciNet  Google Scholar 

  29. Řehák, P.: Generalized discrete Riccati equation and oscillation of half-linear difference equations. Math. Comput. Mod. 34(3–4), 257–269 (2001)

    Article  MathSciNet  Google Scholar 

  30. Řehák, P.: Hartman–Wintner type lemma, oscillation, and conjugacy criteria for half-linear difference equations. J. Math. Anal. Appl. 252(2), 813–827 (2000)

    Article  MathSciNet  Google Scholar 

  31. Sugie, J.: Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation II. Appl. Math. Comput. 304, 142–152 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Sugie, J., Tanaka, M.: Nonoscillation of second-order linear equations involving a generalized difference operator. In: Advances in Difference Equations and Discrete Dynamical Systems, 241–258, Springer Proc. Math. Stat., 212. Springer, Singapore (2017)

  33. Sugie, J., Tanaka, M.: Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation. Proc. Amer. Math. Soc. 145(5), 2059–2073 (2017)

    Article  MathSciNet  Google Scholar 

  34. Veselý, M., Hasil, P.: Oscillation and non-oscillation of asymptotically almost periodic half-linear difference equations. Abstract Appl. Anal. 2013, Art. ID 432936, 12 pp. (2013)

  35. Vítovec, J.: Critical oscillation constant for Euler-type dynamic equations on time scales. Appl. Math. Comput. 243, 838–848 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Wu, F., She, L., Ishibashi, K.: More-type nonoscillation criteria for half-linear difference equations. Monatsh. Math. 194, 377–393 (2021)

    Article  MathSciNet  Google Scholar 

  37. Yamaoka, N.: Oscillation and nonoscillation criteria for second-order nonlinear difference equations of Euler type. Proc. Amer. Math. Soc. 146(5), 2069–2081 (2018)

    Article  MathSciNet  Google Scholar 

  38. Yamaoka, N.: Oscillation criteria for second-order nonlinear difference equations of Euler type. Adv. Difference Equ. 2012(218), 1–14 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Petr Hasil and Michal Veselý are supported by Grant GA20-11846S of the Czech Science Foundation. The authors would like to thank the reviewer for valuable comments that improved the presentation of the used method.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Petr Hasil & Michal Veselý

Authors
  1. Petr Hasil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Veselý
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michal Veselý.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hasil, P., Veselý, M. Riccati technique and oscillation of linear second-order difference equations. Arch. Math. 117, 657–669 (2021). https://doi.org/10.1007/s00013-021-01649-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-021-01649-2

Keywords

Mathematics Subject Classification