Convergence of iterates in nonlinear Perron-Frobenius theory

Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x \in C^\circ$. This condition generalizes the notion of type K order-preserving for maps on $\mathbb{R}^n_{>0}$. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogeneous, analytic, multiplicatively convex functions on $\mathbb{R}^n_{>0}$. This later category includes the maps associated with the homogeneous eigenvalue problem for nonnegative tensors.