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Representation of solutions of linear discrete systems with constant coefficients and pure delay

Abstract

The purpose of this contribution is to develop a method for construction of solutions of linear discrete systems with constant coefficients and with pure delay. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential having between every two adjoining knots the form of a polynomial. These polynomials have increasing degrees in the right direction. Such approach results in a possibility to express initial Cauchy problem in the closed form.

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References

  1. 1.

    Agarwal RP: Differential Equations and Inequalities. 2nd edition. Marcel Dekker, New York; 2000.

    Google Scholar 

  2. 2.

    Baštinec J, Diblík J: Asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations. Computers & Mathematics with Applications 2003,45(6–9):1163–1169.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Baštinec J, Diblík J: Subdominant positive solutions of the discrete equation Δ u ( k + n ) = - p ( k ) u ( k ). Abstract and Applied Analysis 2004,2004(6):461–470. 10.1155/S1085337504306056

    Article  MATH  Google Scholar 

  4. 4.

    Boichuk A, Růžičková M: Solutions of nonlinear difference equations bounded on the whole line. In Colloquium on Differential and Difference Equations, CDDE 2002 (Brno), Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math.. Volume 13. Masaryk University, Brno; 2003:45–60.

    Google Scholar 

  5. 5.

    Boichuk A, Růžičková M: Solutions bounded on the whole line for perturbed difference systems. In Proceedings of the Eighth International Conference on Difference Equations and Applications, 2005, Florida. Chapman & Hall/CRC; 51–59.

  6. 6.

    Čermák J: On the related asymptotics of delay differential and difference equations. Dynamic Systems and Applications 2005,14(3–4):419–429.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Diblík J: Discrete retract principle for systems of discrete equations. Computers & Mathematics with Applications 2001,42(3–5):515–528.

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Diblík J: Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique. Computers & Mathematics with Applications 2003,45(6–9):1041–1057.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Diblík J: Anti-Lyapunov method for systems of discrete equations. Nonlinear Analysis 2004,57(7–8):1043–1057. 10.1016/j.na.2004.03.030

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Khusainov DYa, Shuklin GV: Linear autonomous time-delay system with permutation matrices solving. Studies of the University of Žilina. Mathematical Series 2003,17(1):101–108.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Liz E, Pituk M: Asymptotic estimates and exponential stability for higher-order monotone difference equations. Advances in Difference Equations 2005,2005(1):41–55. 10.1155/ADE.2005.41

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Migda M, Musielak A, Schmeidel E: On a class of fourth-order nonlinear difference equations. Advances in Difference Equations 2004,2004(1):23–36. 10.1155/S1687183904308083

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Philos ChG, Purnaras IK: An asymptotic result for some delay difference equations with continuous variable. Advances in Difference Equations 2004,2004(1):1–10. 10.1155/S1687183904310058

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to J Diblík.

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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.

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Diblík, J., Khusainov, D. Representation of solutions of linear discrete systems with constant coefficients and pure delay. Adv Differ Equ 2006, 080825 (2006). https://doi.org/10.1155/ADE/2006/80825

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Cauchy Problem