ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 39 Migda, MałGorzata and Schmeidel, Ewa, Poznań University of Technology, Poland On the asymptotic behavior of solutions of nonlinear difference equation We consider the nonlinear difference equation

Δ^{2}x_{n} = ∑_{i=0}^{k} a_{n}^{i}f_{i}(x_{n+i}) + b_{n}

where a_{n}^{i}, b_{n} ∈ R, f_{i} : R → R, i = 0, 1, …, k. Using the Schauder fixed point theorem we prove that if the series ∑_{n=1}^{∞} na_{n}^{i}, ∑_{n=1}^{∞} b_{n} are absolutely convergent and the function f_{i} are continuous then for every constant c ∈ R there exists a solution (x_{n}) of the equation convergent to c. Moreover, if the functions f_{i} are bounded and uniformly continuous then for every c, d ∈ R there exists a solution (x_{n}) which possesses the asymptotic behavior x_{n} = cn + d + o(1). Reference: 1. M. Migda, J. Migda, Asymptotic properties of the solutions of the second order difference equation, (submitted). 2. J. Popenda, E. Schmeidel, On the asymptotic behavior of nonhomogeneous linear equation, Indian J. Math. 28(3), (1997), 319–327.