Skip to main content
Log in

Convergence of Solutions of Nonhomogeneous Linear Difference Systems with Delays

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Sufficient conditions are given for the asymptotic constancy of the solutions of a linear system of difference equations with delays. Moreover, it is shown that the limits of the solutions, as t→∞, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

  2. Arino, O., Pituk, M.: More on linear differential systems with small delays. J. Differ. Equ. 170, 381–407 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Atkinson, F.V., Haddock, J.B.: Criteria for asymptotic constancy of solutions of functional differential equations. J. Math. Anal. Appl. 91, 410–423 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bellman, R., Cooke, K.: Differential Difference Equations. Academic Press, Boston (1993)

    Google Scholar 

  5. Bereketoglu, H., Karakoc, F.: Asymptotic constancy for impulsive delay differential equations. Dyn. Syst. Appl. 17, 71–84 (2008)

    MATH  MathSciNet  Google Scholar 

  6. Bereketoglu, H., Pituk, M.: Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays. Discrete Contin. Dyn. Syst. (Supplement Volume) 100–107 (2003)

  7. Diblik, J.: Asymptotic representation of solutions of equation y (t)=β(t)[y(t)−y(tτ(t))]. J. Math. Anal. Appl. 217, 210–215 (1998)

    Article  MathSciNet  Google Scholar 

  8. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Wather, H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)

    MATH  Google Scholar 

  9. Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977)

    MATH  Google Scholar 

  10. Elaydi, S.: An Introduction to Difference Equations. Springer, New York (1996)

    MATH  Google Scholar 

  11. El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Argument. Holden Day, San Francisco (1966)

    Google Scholar 

  12. El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Argument. Academic Press, New York (1973)

    Google Scholar 

  13. Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992)

    Google Scholar 

  14. Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations. Clarendon, Oxford (1991)

    Google Scholar 

  15. Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977)

    MATH  Google Scholar 

  16. Jaros, J., Stavroulakis, I.P.: Necessary and sufficient conditions for oscillations of difference equations with several delays. Util. Math. 45, 187–195 (1994)

    MATH  MathSciNet  Google Scholar 

  17. Karakoc, F., Bereketoglu, H.: Some results for linear impulsive delay differential equations. Dyn. Contin. Discrete Impuls. Syst. (to appear)

  18. Kelley, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, New York (1991)

    MATH  Google Scholar 

  19. Kolmanovski, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, New York (1986)

    Google Scholar 

  20. Koplatadze, R., Kvinikadze, G., Stavroulakis, I.P.: Oscillation of second-order linear difference equations with deviating arguments. Adv. Math. Sci. Appl. 12, 217–226 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Krisztin, T.: A note on the convergence of the solutions of a linear functional differential equation. J. Math. Anal. Appl. 145, 17–25 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  23. Lakshmikhantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Academic Press, New York (1988)

    Google Scholar 

  24. MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  25. Murakami, K.: Asymptotic constancy for systems of delay differential equations. Nonlinear Anal. 30, 4595–4606 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian)

    MATH  Google Scholar 

  27. Shen, J., Stavroulakis, I.P.: Oscillation criteria for delay difference equations. Electron. J. Differ. Equ. 2001(10), 1–15 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Stavroulakis, I.P.: Oscillations of delay difference equations. Comput. Math. Appl. 29, 83–88 (1995)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Huseyin Bereketoglu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bereketoglu, H., Huseynov, A. Convergence of Solutions of Nonhomogeneous Linear Difference Systems with Delays. Acta Appl Math 110, 259–269 (2010). https://doi.org/10.1007/s10440-008-9404-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9404-2

Keywords

Mathematics Subject Classification (2000)

Navigation