Iterated nonlinear maps and Hilbert’s projective metric. II. (English) Zbl 0669.47031

Nussbaum, Roger D.

Mem. Am. Math. Soc. 401, 118 p. (1989).

From author’s abstract: If \(u\in\overset\circ K\) is an eigenvector of \(f\) of norm one and if \(g(x) = f(x)\| f(x)\|^{-1}\), is it true that \(\lim_{k\to\infty} g^ k (x) = u\) for every \(x \in \overset\circ K\)?
If f has no eigenvector in \(\overset\circ K\), what can be said about the behavior of \(f^ k(x)\) or \(g^ k(x)\) for \(k\geq1\) and \(x\in \overset\circ K\)?
After an introduction and preliminaries, the chapters deal with applications to some examples from mathematical biology, eigenvectors for \(f\in M_-\) and D-A-D theorems.
[For part I see ibid. 391, 137 p. (1988; Zbl 0666.47028); for a preliminary version see also NATO ASI Ser., Ser. F 37, 231-248 (1987; Zbl 0643.47050).]

Reviewer: D.Pascali

MSC:

47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10	Fixed-point theorems
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems
47B60	Linear operators on ordered spaces
92D25	Population dynamics (general)

Keywords:

iterated nonlinear maps; Hilbert’s projective metric; order preserving maps; population biology; eigenvector; mathematical biology; eigenvectors; D-A-D theorems

Citations:

Zbl 0666.47028; Zbl 0643.47050