Order-preserving equations

A limit set trichotomy for order-preserving systems on time scales

(with S. Siegmund) Electronic Journal of Differential Equations 64, 2004, 1-18

In this paper we derive a limit set trichotomy for abstract order-preserving 2-parameter semiflows in normal cones of strongly ordered Banach spaces. Additionally, to provide an example, Müller’s theorem is generalized to dynamic equations on arbitrary time scales and applied to a model from population dynamics.

A limit set trichotomy for abstract order-preserving 2-parameter semiflows on time scales

Functional Differential Equations 11 (1-2), 2004, 133-140

Under certain contractivity conditions, we study the asymptotic behavior of abstract 2-parameter semiflows on normel cones in Banach spaces, and show that there are only three possible scenarios for their limit behavior.

Dedicated to Prof. István Győri on the occasion of his 60th birthday.

Order-preserving nonautonomous discrete dynamics:

Attractors and entire solutions

Positivity 19(3), 2015, 547-576

The concept of pullback convergence turned out to be a central idea to describe the long-term behavior of nonautonomous dynamical systems. This paper provides a general framework for the existence and structure of pullback attractors capturing the asymptotics of nonautonomous and order-preserving difference equations in Banach spaces. Furthermore we obtain criteria for the convergence to bounded entire solutions and additionally discuss various applications.

Monotonicity and discretization of Hammerstein integrodifference equations

(with Magdalena Nockowska-Rosiak), Journal of Computational Dynamics 10(1), 2023, 223-247

This paper provides sufficient conditions for monotonicity, subhomogeneity and order concavity of vector-valued Hammerstein integral operators over compact domains, as well as for the persistence of these properties under numerical discretization of degenerate kernel type. This has immediate consequences on the dynamics of Hammerstein integrodifference equations and allows to deduce a local-global stability principle.

Monotonicity and discretization of Urysohn integral operators

(with Magdalena Nockowska-Rosiak), Applied Mathematics and Computation 414, 1 February 2022, 126686

The property that a nonlinear operator on a Banach space preserves an order relation, is subhomogeneous or order concave w.r.t. an order cone has profound consequences. In Nonlinear Analysis it allows to solve related equations by means of suitable fixed point or monotone iteration techniques. In Dynamical Systems the possible long term behavior of associate integrodifference equations is drastically simplified. This paper contains sufficient conditions for vector-valued Urysohn integral operators to be monotone, subhomogeneous or concave. it also provides conditions guaranteeing that these properties are preserved under spatial discretization of particularly Nyström type. This fact is crucial for numerical schemes to converge, or for simulations to reproduce the actual behavior and asymptotics.

Positivity and discretization of Fredholm integral operators

(with Magdalena Nockowska-Rosiak), Journal of Mathematical Analysis and Applications, 524 (2023) 127137

We provide sufficient conditions for vector-valued Fredholm integral operators and their commonly used spatial discretizations to be positive in terms of an order relation induced by a corresponding order cone. It turns out that reasonable Nyström methods preserve positivity. Among the projection methods, persistence is obtained for the simplest ones based on polynomial, piecewise linear or specific cubic interpolation (collocation), as well as for piecewise constant basis functions in a Bubnov-Galerkin approach. However, for semi-discretizations using quadratic splines or sinc-collocation we demonstrate that positivity is violated. Our results are illustrated in terms of eigenpairs for Krein-Rutman operators and form the basis of corresponding investigations for nonlinear integral operators.