Asymptotic solutions of nonlinear difference equations (Q999705)

This paper is written in the framework of Nonstandard Analysis. It investigates the asymptotic behavior of explicitly time-dependent first-order nonlinear difference equations \[ y_{k+1}=f(k,y_k)\tag{\(*\)} \] in \({\mathbb R}\). The main result gives sufficient conditions to determine if \((\ast)\) possesses solutions with the asymptotic behavior \(\hat y_k\). Here, the so-called approximate solution \(\hat y_k\) should satisfy the asymptotic functional equation \[ \lim_{k\to\infty}\frac{f(k,\hat y_k)-\hat y_k}{\hat y_k(|D_2f(k,\hat y_k)|-1)}=0. \] Moreover, further conditions express that solutions of \((\ast)\) in a neighborhood of \(\hat y_k\) contract (attract or repel each other), faster than \(\hat y_k\) changes itself. Under additional regularity conditions there will be an actual solution \(\tilde y_k\) asymptotically equivalent to \(\hat y_k\).