Skip to main content

Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces

Abstract

We consider the difference equation , , in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of directionally positive monotone increasing solutions.

[12345678910111213141516]

References

  1. 1.

    Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.

    Google Scholar 

  2. 2.

    Agarwal RP, Bohner M, Grace SR, O'Regan D: Discrete Oscillation Theory. Hindawi, New York, NY, USA; 2005:xiv+961.

    Google Scholar 

  3. 3.

    Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:viii+507.

    Google Scholar 

  4. 4.

    Agarwal RP, O'Regan D: Difference equations in abstract spaces. Journal of the Australian Mathematical Society 1998,64(2):277–284. 10.1017/S1446788700001762

    MATH  Article  Google Scholar 

  5. 5.

    Franco D, O'Regan D, Peran J: The antipodal mapping theorem and difference equations in Banach spaces. Journal of Difference Equations and Applications 2005,11(12):1037–1047. 10.1080/10236190500331305

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    González C, Jiménez-Melado A: Set-contractive mappings and difference equations in Banach spaces. Computers & Mathematics with Applications 2003,45(6–9):1235–1243.

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    González C, Jiménez-Melado A, Lorente M: Existence and estimate of solutions of some nonlinear Volterra difference equations in Hilbert spaces. Journal of Mathematical Analysis and Applications 2005,305(1):63–71. 10.1016/j.jmaa.2004.10.015

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Medina R: Delay difference equations in infinite-dimensional spaces. Journal of Difference Equations and Applications 2006,12(8):799–809. 10.1080/10236190600734192

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Medina R, Gil' MI: Delay difference equations in Banach spaces. Journal of Difference Equations and Applications 2005,11(10):889–895. 10.1080/10236190512331333860

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Medina R, Gil' MI: The freezing method for abstract nonlinear difference equations. Journal of Mathematical Analysis and Applications 2007,330(1):195–206. 10.1016/j.jmaa.2006.07.074

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Jiang J, Li X: Oscillation and nonoscillation of two-dimensional difference systems. Journal of Computational and Applied Mathematics 2006,188(1):77–88. 10.1016/j.cam.2005.03.054

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Dubé SG, Mingarelli AB: Note on a non-oscillation theorem of Atkinson. Electronic Journal of Differential Equations 2004,2004(22):1–6.

    Google Scholar 

  13. 13.

    Ehrnström M: Positive solutions for second-order nonlinear differential equations. Nonlinear Analysis: Theory, Methods & Applications 2006,64(7):1608–1620. 10.1016/j.na.2005.07.010

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Wahlén E: Positive solutions of second-order differential equations. Nonlinear Analysis: Theory, Methods & Applications 2004,58(3–4):359–366. 10.1016/j.na.2004.05.008

    MATH  Article  Google Scholar 

  15. 15.

    Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.

    Google Scholar 

  16. 16.

    Grace SR, El-Morshedy HA: Oscillation criteria for certain second order nonlinear difference equations. Bulletin of the Australian Mathematical Society 1999,60(1):95–108. 10.1017/S0004972700033360

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Cristóbal González.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

González, C., Jiménez-Melado, A. Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces. Adv Differ Equ 2007, 086925 (2007). https://doi.org/10.1155/2007/86925

Download citation

Keywords

  • Differential Equation
  • Hilbert Space
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis