# Article

 Title: Asymptotic behaviour of a difference equation with complex-valued coefficients (English) Author: Kalas, Josef Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 41 Issue: 3 Year: 2005 Pages: 311-323 Summary lang: English . Category: math . Summary: 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. (English) Keyword: difference equations Keyword: asymptotic behaviour Keyword: Lyapunov functions MSC: 39A11 idZBL: Zbl 1122.39006 idMR: MR2188386 . Date available: 2008-06-06T22:46:22Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107961 . Reference: [1] Bohner M., Došlý O., Kratz W.: Inequalities and asymptotics for Riccati matrix difference operators.J. Math. Anal. Appl. 221 (1998), 262–286. Zbl 0914.39012, MR 1619144 Reference: [2] Hooker J. W., Patula W. T.: Riccati type transformations for second-order linear difference equations.J. Math. Anal. Appl. 82 (1981), 451–462. Zbl 0471.39007, MR 0629769 Reference: [3] Kalas J.: Asymptotic behaviour of the system of two differential equations.Arch. Math. (Brno) 11 (1975), 175–186. MR 0412530 Reference: [4] Kalas J.: Asymptotic behaviour of the solutions of the equation $dz/dt=f(t,z)$ 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. MR 0680606 Reference: [5] Kalas J.: On the asymptotic behaviour of the equation $dz/dt =f(t,z)$ with a complex-valued function $f$.Arch. Math. (Brno) 17 (1981), 11–22. Zbl 0475.34028, MR 0672484 Reference: [6] Kalas J.: Asymptotic properties of the solutions of the equation $\dot{z}~= f(t,z)$ with a complex-valued function $f$.Arch. Math. (Brno) 17 (1981), 113–123. MR 0672315 Reference: [7] Kalas J.: Asymptotic behaviour of equations $\dot{z}~\!=\!q(t,z)-p(t)z^2$ and $\ddot{x}\!=\!x\varphi (t,\dot{x}x^{-1})$.Arch. Math. (Brno) 17 (1981), 191–206. MR 0672659 Reference: [8] Kalas J.: On certain asymptotic properties of the solutions of the equation $\dot{z}=f(t,z)$ with a complex-valued function $f$.Czechoslovak Math. J. 33 (108) (1983), 390–407. MR 0718923 Reference: [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. Zbl 0814.34029, MR 1313803 Reference: [10] Kalas J., Ráb M.: Asymptotic properties of dynamical systems in the plane.Demonstratio Math. 25 (1992), 169–185. Zbl 0757.34030, MR 1170680 Reference: [11] Keckic J. D.: Riccati’s difference equation and a solution of the linear homogeneous second order difference equation.Math. Balkanica 8 (1978),145–146. Zbl 0541.65091, MR 0724402 Reference: [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. MR 0786022 Reference: [13] Lakshmikantham V., Matrosov V. M., Sivasundaram: Vector Lyapunov functions and stability analysis of nonlinear systems.Kluver Academic Publishers, 1991. Zbl 0721.34054, MR 1206904 Reference: [14] Lakshmikantham V., Trigiante D.: Theory of difference equations.Academic Press, New York, 1987. MR 0939611 Reference: [15] Ráb M.: The Riccati differential equation with complex-valued coefficients.Czechoslovak Math. J. 20 (95) (1970), 491–503. MR 0268452 Reference: [16] Ráb M.: Equation $Z^{\prime }=A(t)-Z^2$ coefficient of which has a small modulus.Czechoslovak Math. J. 21 (96) (1971), 311–317. MR 0287096 Reference: [17] Ráb M.: Global properties of a Riccati differential equation.University Annual Applied Mathematics 11 (1975), 165–175 (Državno izdatelstvo Technika, Sofia, 1976). MR 0501159 Reference: [18] Ráb M.: Geometrical approach to the study of the Riccati differential equation with complex-valued coefficients.J. Differential Equations 25 (1977), 108–114. MR 0492454 Reference: [19] Ráb M., Kalas J.: Stability of dynamical systems in the plane.Differential Integral Equations 3 (1990), no. 1, 127–144. Zbl 0724.34060, MR 1014730 Reference: [20] Řehák P.: Generalized discrete Riccati equation and oscillation of half-linear difference equations.Math. Comput. Modelling 34 (2001), 257–269. Zbl 1038.39002, MR 1835825 .

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