Asymptotics of families of solutions of nonlinear difference equations
This article deals with first-order scalar difference equations. In a conventional notation they read as \[ y_{k+1}=f(k,y_k);\tag{\(\ast\)} \] note that these problems can be nonautonomous and nonlinear, where the right-hand side \(f\) is supposed to be continuously partially differentiable in the second argument. In the framework of Nonstandard Analysis, the author provides conditions for the existence of solutions to \((\ast)\) which have an asymptotic behavior given by \(\hat y_k\). Here, the family \(\hat y_k\) of ``asymptotic fixed points'' satisfies the so-called 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. \tag{\(\ast\ast\)} \] Under additional regularity conditions on \(f\) such asymptotic fixed points are denoted as approximate solution by the author. Various kinds of local asymptotic behavior of true solutions are presented using the notions of a (strongly, moderately, weakly exponential) attractive/repulsive river; a precise definition involves the notion of telescopes and macroscopes. In Sections 3 and 6 attractive and repulsive rivers are characterized in terms of the right-hand side \(f\) and the sequence \(\hat y_k\). Section 4 contains a convenient lemma which states that although a nonlinear difference equation \((\ast)\) in general lacks the property of uniqueness of solutions, it is possible to move back in time for a subclass of its solutions. Additionally, it is shown that repulsive solutions are unique in their asymptotic direction. Effects of the rescalings on the behavior of the equation and its solutions are treated in Section 5. Proofs of the main results are postponed to Section 6. In the final Section 7, the author considers some examples of quadratic equations, a natural class of drains which admits a simple formula for deviations and finally the question whether the river or drain itself satisfies the asymptotic functional equation \((\ast\ast)\).
scientific article; zbMATH DE number 5506510