Advances in Difference Equations

, 2007:086925 | Cite as

Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces

  • Cristóbal González
  • Antonio Jiménez-Melado
Open Access
Research Article


We consider the difference equation Open image in new window , Open image in new window , 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.


Differential Equation Hilbert Space Partial Differential Equation Ordinary Differential Equation Functional Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  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.MATHCrossRefGoogle 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.CrossRefGoogle 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/S1446788700001762MATHCrossRefGoogle 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. 10.1080/10236190500331305MATHMathSciNetCrossRefGoogle 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.MATHMathSciNetCrossRefGoogle 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.2004.10.015MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Medina R: Delay difference equations in infinite-dimensional spaces. Journal of Difference Equations and Applications 2006,12(8):799–809. 10.1080/10236190600734192MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Medina R, Gil' MI: Delay difference equations in Banach spaces. Journal of Difference Equations and Applications 2005,11(10):889–895. 10.1080/10236190512331333860MATHMathSciNetCrossRefGoogle 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.2006.07.074MATHMathSciNetCrossRefGoogle 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/ 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/ 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/ 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.CrossRefGoogle 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/S0004972700033360MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© C. González and A. Jiménez-Melado. 2007

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de MálagaMálagaSpain

Personalised recommendations