Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: $$\begin{cases} x(t+1) = X(x(t),y(t)), \\ y(t+1) = Y(x(t), y(t)), \end{cases}$$ where $X(x,y)= \lambda_1 x+ \mu y +\sum_{i+j \geq 2} c_{ij} x^i y^j$, $Y(x,y)= \lambda_2 y+\sum_{i+j \geq 2}$ $d_{ij} x^i y^j$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases “$|\lambda_1| \ne 1$ or $|\lambda_2| \ne 1$” or “$\mu=0$” in earlier studies. In the present paper, we will prove the existence of an analytic solution for the case $\lambda_1 = \lambda_2 = 1$ and $\mu=1$.