Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. (German) Zbl 0695.34001

Hilger, Stefan

Würzburg: Univ. Würzburg, Diss. iv, 141 S. (1988).

The booklet is devoted to a uniform treatment of a theory of the most important equations of dynamical systems, i.e. differential and difference equations. The starting point is a linear ordered set T closed under the operations inf and sup, endowed with the order topology and considered together with a mapping \(\mu\) : \(T\times T\to {\mathbb{R}}\) having the following properties: (i) \(\mu (t_ 1,t_ 2)+\mu (t_ 2,t_ 3)=\mu (t_ 1,t_ 3)\) for \(t_ 1,t_ 2,t_ 3\in T\); (ii) if \(t_ 1,t_ 2\in T\) and \(t_ 1>t_ 2\) then \(\mu (t_ 1,t_ 2)>0\); (iii) \(\mu\) is continuous with respect to the first variable.
For functions mapping T into a Banach space the notion of a derivative is introduced which agrees with the classical derivative whenever T is a real interval and on the other hand generalizes the notion of the difference operator in the case \(T={\mathbb{Z}}\). Basing on this idea the author develops a theory of “differential” equations proving among others the existence and uniqueness of solutions of equations with Lipschitzian right-hand side, the Gronwall inequality, and a number of other theorems being analogous of classical results. A special emphasis is laid on quasilinear equations (stable and unstable manifolds, theorem of Hartman-Grobman). The appendix contains the proof of a product version of Banach’s contraction principle being a useful tool in the whole article.

Reviewer: W.Jarczyk

MSC:

34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C30	Manifolds of solutions of ODE (MSC2000)
39A70	Difference operators
37C10	Dynamics induced by flows and semiflows

Keywords:

stable manifold; dynamical systems; difference equations; Banach space; equations with Lipschitzian right-hand side; unstable manifolds