Ulrich Krause

Asymptotic Properties for Inhomogeneous Iterations of Nonlinear Operators

The theorems on weak and strong ergodicity for inhomogeneous products of nonnegative matrices are extended to inhomogeneous iterations of nonlinear positive operators on Euclidean space. In particular some concave version of the Coale–Lopez theorem is presented and applied to a density-dependent Leslie model. The results are obtained, via Hilbert’s projective pseudometric, from general theorems on inhomogeneous iterations of operators mapping a metric space into itself.

A nonlinear extension of the Birkhoff-Jentzsch theorem☆

Abstract The Birkhoff-Jentzsch theorem for linear positive operators is extended to a certain class of nonlinear positive operators. These so-called p-ascending operators include concave operators and suprema of such operators. For the underlying positive cone to be complete for Hilberts metric a necessary and sufficient condition is given from which various criteria for completeness are derived. The extension made yields in particular a concave version of Jentzschs theorem.

Strong ergodicity for strictly increasing nonlinear operators

Abstract As a further generalization of the Perron-Frobenius theorem from linear to nonlinear operators, we prove uniqueness of the solution as well as ergodicity for nonlinear operators which are strictly increasing and weakly homogeneous on a certain subset of the Euclidean space. We also discuss higher order difference equations which involve this type of operators.

Stable inhomogeneous iterations of nonlinear positive operators on Banach spaces

For a sequence

Positive Nonlinear Systems in Economics

(f_n )_n

Path stability of prices in a nonlinear Leontief model

of nonlinear positive operators on a Banach space which converges to some operator f, conditions are specified under which the inhomogeneous iterates

Ergodic Price Setting with Technical Progress

f_n \circ f_{n - 1} \circ \cdots \circ f_2 \circ f_1

Unique representation in convex sets by extraction of marked components

, after normalization, converge to the unique positive and normalized eigenvector of f. This stability result extends, for discrete dynamical systems, the property of strong ergodicity from finite to infinite dimensions.

A limit set trichotomy for monotone nonlinear dynamical systems

In many cases an economic system can be modelled by a mapping transforming a state of the economic system at a certain period of time into the state of the system at the next period. If the transformation under consideration can be assumed to be linear then the well-established theory of linear operators can be applied; thereby spectral theory, including Perron-Frobenius theory for positive matrices and positive linear operators, is of particular importance. Very often, however, linearity is not an appropriate idealization, in which case a rigorous analysis may become very difficult or even impossible. It is this state of affairs which brings positive nonlinear systems into play, this not only in economics. Positivity and related mathematical properties are quite natural assumptions in economics. The state space is often given, e.g., if states are described by quantities or prices, by the positive orthant (or some more general convex cone) in Euclidean space. The transformation of such a state space may possess additional properties related to positivity as various forms of monotonicity. This is the case for the two economic problems considered in this paper: Balanced growth in a nonlinear multisectoral framework and price setting among several production units which depend on each other by technology. Given the transformation T mapping the state space K, a convex cone, into itself, the following questions will be addressed: Does there exist a unique equilibrium, that is does the fixed point equation Tx ⋆ = x ⋆ possess a unique solution x ⋆ є K (up to a positive scalar)?

Relative Stability for Ascending and Positively Homogeneous Operators on Banach Spaces

In a nonlinear Leontief model where the input-output coefficients depend on the level of outputs, the dynamics of cost-determined prices is examined. It is shown that under mild assumptions path stability holds in the sense that every path when being disturbed comes finally arbitrarily close to its original form. Path stability may occur even if each single path shows chaotic behaviour. By sharpening the assumptions considerably, convergence of all paths may be achieved.