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In this paper, we present two quite general approximation theorems for the propagators of higher order (in time) abstract Cauchy problems, which extend largely the classical Trotter-Kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions. Then, we apply the approximation theorems to deal with the second… (More)

- Britta Dorn, Eberhard Karls, Rainer Nagel, Abdelaziz Rhandi
- 2005

Hiermit erkläre ich, dass ich die Arbeit selbständig und nur mit den angegebenen Hilfsmitteln angefertigt habe und dass alle Stellen, die dem Wortlaut oder dem Sinne nach anderen Werken entnommen sind, durch Angabe der Quellen als Entlehnung kenntlich gemacht worden sind.

- RAINER NAGEL
- 1998

For a strongly continuous semigroup (T (t)) t0 with generator A we introduce its critical spectrum crit (T (t)). This yields in an optimal way the spectral mapping theorem (T(t)) = e tt(A) crit (T (t)) and improves classical stability results.

- Tanja Eisner, Rainer Nagel, Andr´as Serény
- 2008

In this article we survey results concerning asymptotic properties of C0-semigroups on Banach spaces with respect to the weak operator topology. The property " no eigenvalues of the generator on the imaginary axis " is equivalent to weak stability for most time values; a phenomenon called " almost weak stability ". Further, sufficient conditions implying… (More)

- Rainer Nagel
- 1997

We propose an abstract framework for the computation of the spectrum (A) of a linear operator A : D(A) X ! X on a Banach space X through a condition in a smaller Banach space X 1. If this space is nite dimensional this yields a characteristic equation for (A). The method is tested for delay, integro-diierential and population equations and is applicable to… (More)

- SIMON BRENDLE, RAINER NAGEL
- 2002

We consider a strongly continuous semigroup (T(t)) t0 with generator A on a Banach space X, an Abounded perturbation B, and the semigroup (S(t)) t0 generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t)) t0. The results are applied to a cell equation with age… (More)

For a generator A of a C 0-semigroup T (·) on a Banach space X we consider the semi-norm M k x := lim sup t→0+ t −1 (T (t) − I)A k−1 x on the Favard space F k of order k associated with A. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities… (More)

We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0-semigroup on C 0 (R − ,X) yielding the solutions of the equation.

Introduction Partial differential equations on bounded domains of R n have traditionally been equipped with homogeneous boundary conditions (usually Dirichlet, Neumann, or Robin). However, other kinds of boundary conditions can also be considered, and for a number of concrete application it seems that dynamic (i.e., time-dependent) boundary conditions are… (More)