Some Nonlinear Weak Ergodic Theorems

Abstract

Let C be a cone with nonempty interior in a Banach space and, for $j \geqq 1$, let $f_j :\mathop C\limits^ \circ \to \mathop C\limits^ \circ $ be a sequence of maps. It is frequently assumed that each $f_j $ is homogeneous of degree 1 and order-preserving with respect to the partial ordering induced by C; but it is not assumed that $f_j (C - \{ 0\} ) \subset \mathop C\limits^ \circ $. If $F_m = f_m f_{m - 1} \cdots f_1 $, the composition of the first $mf_j $, and if d denotes Hilbert’s projective metric, then theorems (usually called weak ergodic theorems in the population biology literature) can be proved ensuring that, for all x and y in $\mathop C\limits^ \circ $, $\lim _{m \to \infty } d(F_m (x),F_m (y)) = 0$ and (if C is normal) $\lim _{m \to \infty } \| {({{F_m (x)} /{\| {F_m (x)} \|}}) - ({{F_m (y)} /{\| {F_m (y)} \|}})} \| = 0$. If $u \in \mathop C\limits^ \circ $ is fixed and assumptions on the $f_j $ are strengthened, it can be proved that for every $z \in \mathop C\limits^ \circ $ there exists $\lambda (x) > 0$ such that $\lim _{m \to \infty } \| {F_m (x) - \lambda (x)F_m (u)} \| = 0$. These theorems are applied to the case where $C = \{ x \in \mathbb{R}^n :x_i \geqq 0{\text{ for }}1 \leqq i \leqq n\} $ and where the maps $f_j $ belong to a class M arising in the theory of “means and their iterations” and in certain problems from population biology.