Bálint Farkas

Operator splitting for non-autonomous evolution equations

Abstract We establish general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems. The results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential. We also prove convergence rates for the sequential splitting applied to this problem.

Invariant decomposition of functions with respect to commuting invertible transformations

Consider a 1 ,..., , an ∈ R arbitrary elements. We characterize those functions f: R→ R that decompose into the sum of a j -periodic functions, i.e., f=f 1 +---+f n with Δ aJ f(x):= f(x+aj)-f(x) = 0. We showthat f has such a decomposition if and only if for all partitions Β 1 UB 2 U...US N = {a 1 ,..., a n ) } with Bj consisting of commensurable elements with least common multiples b j one has Δ b1 ... Δb N f = 0. Actually, we prove a more general result for periodic decompositions of functions f: A→ R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f: A → R with respect to commuting, invertible self-mappings of some abstract set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

POSITIVE FORMS ON BANACH SPACES

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case.

Operator Splitting with Spatial-temporal Discretization

Continuing earlier investigations, we analyze the convergence of operator splitting procedures combined with spatial discretization and rational approximations.

Operator splitting for nonautonomous delay equations

We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated by some typical numerical examples.

Weak Stability for Orbits of C 0-semigroups on Banach Spaces

A result of Huang and van Neerven [12] establishes weak individual stability for orbits of C 0-semigroups under boundedness assumptions on the local resolvent of the generator. We present an elementary proof for this using only the inverse Fourier-transform representation of the orbits of the semigroup in terms of the local resolvent.

On the Effect of Small Delays to the Stability of Feedback Systems

The problem considered in this paper is the following: Assume that some asymptotic properties are known for the solutions of the equation

Variations on Barbalat's lemma

\left( {{\rm{DE}}} \right)_{\rm{0}} \left\{ \begin{array}{l} u\prime {\rm{ = }}\left( {B + C} \right)u\left( t \right), t \ge {\rm{0,}} \\ + {\rm{initial conditions}}, \\ \end{array} \right.