## Advanced Studies in Pure Mathematics

### Dissipative delay endomorphisms and asymptotic equivalence

Christian Pötzsche

#### Abstract

Using an invariant manifold theorem we demonstrate that the dynamics of nonautonomous dissipative delayed difference equations (with delay $M$) is asymptotically equivalent to the long-term behavior of an $N$-dimensional first order difference equation (with $N \leq M)$ – assumed the nonlinearity is small Lipschitzian on the absorbing set. As consequence we obtain a result of Kirchgraber that multi-step methods for the numerical solution of ordinary differential equations are essentially one-step methods, and generalize it to varying step-sizes.

#### Article information

Dates
First available in Project Euclid: 28 November 2018

https://projecteuclid.org/ euclid.aspm/1543447658

Digital Object Identifier
doi:10.2969/aspm/05310237

Mathematical Reviews number (MathSciNet)
MR2582422

Zentralblatt MATH identifier
1182.39009

#### Citation

Pötzsche, Christian. Dissipative delay endomorphisms and asymptotic equivalence. Advances in Discrete Dynamical Systems, 237--259, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05310237. https://projecteuclid.org/euclid.aspm/1543447658