Asymptotic behaviour of a difference equation with complex-valued coefficients

Josef Kalas

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 3, page 311-323
  • ISSN: 0044-8753

Abstract

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The asymptotic behaviour for solutions of a difference equation z n = f ( n , z n ) , where the complex-valued function f ( n , z ) is in some meaning close to a holomorphic function h , and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.

How to cite

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Kalas, Josef. "Asymptotic behaviour of a difference equation with complex-valued coefficients." Archivum Mathematicum 041.3 (2005): 311-323. <http://eudml.org/doc/249482>.

@article{Kalas2005,
abstract = {The asymptotic behaviour for solutions of a difference equation $z_n = f(n,z_n)$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.},
author = {Kalas, Josef},
journal = {Archivum Mathematicum},
keywords = {difference equations; asymptotic behaviour; Lyapunov functions; first order difference equation; asymptotic behaviour; Lyapunov functions},
language = {eng},
number = {3},
pages = {311-323},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic behaviour of a difference equation with complex-valued coefficients},
url = {http://eudml.org/doc/249482},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Kalas, Josef
TI - Asymptotic behaviour of a difference equation with complex-valued coefficients
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 3
SP - 311
EP - 323
AB - The asymptotic behaviour for solutions of a difference equation $z_n = f(n,z_n)$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.
LA - eng
KW - difference equations; asymptotic behaviour; Lyapunov functions; first order difference equation; asymptotic behaviour; Lyapunov functions
UR - http://eudml.org/doc/249482
ER -

References

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  1. Bohner M., Došlý O., Kratz W., Inequalities and asymptotics for Riccati matrix difference operators, J. Math. Anal. Appl. 221 (1998), 262–286. (1998) Zbl0914.39012MR1619144
  2. Hooker J. W., Patula W. T., Riccati type transformations for second-order linear difference equations, J. Math. Anal. Appl. 82 (1981), 451–462. (1981) Zbl0471.39007MR0629769
  3. Kalas J., Asymptotic behaviour of the system of two differential equations, Arch. Math. (Brno) 11 (1975), 175–186. (1975) MR0412530
  4. Kalas J., Asymptotic behaviour of the solutions of the equation d z / d t = f ( t , z ) with a complex-valued function f , Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 431–462, Colloq. Math. Soc. János Bolyai, 30, North-Holland, Amsterdam-New York, 1981. (1979) MR0680606
  5. Kalas J., On the asymptotic behaviour of the equation d z / d t = f ( t , z ) with a complex-valued function f , Arch. Math. (Brno) 17 (1981), 11–22. (1981) Zbl0475.34028MR0672484
  6. Kalas J., Asymptotic properties of the solutions of the equation z ˙ = f ( t , z ) with a complex-valued function f , Arch. Math. (Brno) 17 (1981), 113–123. (1981) MR0672315
  7. Kalas J., Asymptotic behaviour of equations z ˙ = q ( t , z ) - p ( t ) z 2 and x ¨ = x ϕ ( t , x ˙ x - 1 ) , Arch. Math. (Brno) 17 (1981), 191–206. (1981) MR0672659
  8. Kalas J., On certain asymptotic properties of the solutions of the equation z ˙ = f ( t , z ) with a complex-valued function f , Czechoslovak Math. J. 33 (108) (1983), 390–407. (1983) MR0718923
  9. Kalas J., On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients, Ann. Mat. Pura Appl. (4), 166 (1994), 155–173. (1994) Zbl0814.34029MR1313803
  10. Kalas J., Ráb M., Asymptotic properties of dynamical systems in the plane, Demonstratio Math. 25 (1992), 169–185. (1992) Zbl0757.34030MR1170680
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  12. Kwong M. K., Hooker J. W., Patula W. T., Riccati type transformations for second-order linear difference equations II, J. Math. Anal. Appl. 107 (1985), 182–196. (1985) MR0786022
  13. Lakshmikantham V., Matrosov V. M., Sivasundaram, Vector Lyapunov functions and stability analysis of nonlinear systems, Kluver Academic Publishers, 1991. (1991) Zbl0721.34054MR1206904
  14. Lakshmikantham V., Trigiante D., Theory of difference equations, Academic Press, New York, 1987. (1987) MR0939611
  15. Ráb M., The Riccati differential equation with complex-valued coefficients, Czechoslovak Math. J. 20 (95) (1970), 491–503. (1970) MR0268452
  16. Ráb M., Equation Z ' = A ( t ) - Z 2 coefficient of which has a small modulus, Czechoslovak Math. J. 21 (96) (1971), 311–317. (1971) MR0287096
  17. Ráb M., Global properties of a Riccati differential equation, University Annual Applied Mathematics 11 (1975), 165–175 (Državno izdatelstvo Technika, Sofia, 1976). (1975) MR0501159
  18. Ráb M., Geometrical approach to the study of the Riccati differential equation with complex-valued coefficients, J. Differential Equations 25 (1977), 108–114. (1977) MR0492454
  19. Ráb M., Kalas J., Stability of dynamical systems in the plane, Differential Integral Equations 3 (1990), no. 1, 127–144. (1990) Zbl0724.34060MR1014730
  20. Řehák P., Generalized discrete Riccati equation and oscillation of half-linear difference equations, Math. Comput. Modelling 34 (2001), 257–269. Zbl1038.39002MR1835825

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