# Article

Full entry | PDF   (0.2 MB)
Keywords:
difference equations; asymptotic behaviour; Lyapunov functions
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.
References:
[1] Bohner M., Došlý O., Kratz W.: Inequalities and asymptotics for Riccati matrix difference operators. J. Math. Anal. Appl. 221 (1998), 262–286. MR 1619144 | Zbl 0914.39012
[2] Hooker J. W., Patula W. T.: Riccati type transformations for second-order linear difference equations. J. Math. Anal. Appl. 82 (1981), 451–462. MR 0629769 | Zbl 0471.39007
[3] Kalas J.: Asymptotic behaviour of the system of two differential equations. Arch. Math. (Brno) 11 (1975), 175–186. MR 0412530
[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
[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. MR 0672484 | Zbl 0475.34028
[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
[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
[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
[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. MR 1313803 | Zbl 0814.34029
[10] Kalas J., Ráb M.: Asymptotic properties of dynamical systems in the plane. Demonstratio Math. 25 (1992), 169–185. MR 1170680 | Zbl 0757.34030
[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. MR 0724402 | Zbl 0541.65091
[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
[13] Lakshmikantham V., Matrosov V. M., Sivasundaram: Vector Lyapunov functions and stability analysis of nonlinear systems. Kluver Academic Publishers, 1991. MR 1206904 | Zbl 0721.34054
[14] Lakshmikantham V., Trigiante D.: Theory of difference equations. Academic Press, New York, 1987. MR 0939611
[15] Ráb M.: The Riccati differential equation with complex-valued coefficients. Czechoslovak Math. J. 20 (95) (1970), 491–503. MR 0268452
[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
[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
[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
[19] Ráb M., Kalas J.: Stability of dynamical systems in the plane. Differential Integral Equations 3 (1990), no. 1, 127–144. MR 1014730 | Zbl 0724.34060
[20] Řehák P.: Generalized discrete Riccati equation and oscillation of half-linear difference equations. Math. Comput. Modelling 34 (2001), 257–269. MR 1835825 | Zbl 1038.39002

Partner of