Wilhelm Schappacher

Hypercyclic and chaotic semigroups of linear operatorsWork by G. W. supported by the National Science Foundation under Grant No. DMS-9202550. Work by W. D. done within the Spezialforschungsbereich F 003, Optimierung und Kontrolle. Work by W. D. and W. S. supported in part by a SCIENCE project of the European Community

We investigate hypercyclic and chaotic behavior of linear strongly continuous semigroups. We give necessary and sufficient conditions on the semigroup to be hypercyclic, and sufficient conditions on the spectrum of an operator to generate a hypercyclic semigroup. A variety of examples is provided.

Some considerations to the fundamental theory of infinite delay equations

In recent years we see an increasing interest in infinite delay equations. The main reason is that equations of this type become more and more important for different applications. Regardless of the specific problem one has to deal with it is in most cases necessary to establish some fundamental theory as, for instance, existence, uniqueness of solutions and continuous dependence on initial data. Also in the last few years we find an increasing number of papers where the general theory of linear and nonlinear semigroups or evolution operators is applied to functional differential equations. Of fundamental importance for all approaches is the right choice of the state space which in most cases is a Banach space of functions or of equivalence classes of functions. For equations with bounded delays this in general is not a difficult problem. But for infinite delay equations the choice of an appropriate state space is no more trivial. It was natural to investigate which properties of the state space are sufficient in order to establish the fundamental theory for infinite delay equations. Up to now the most thorough discussion of this problem is contained in [12]. The set of axioms given there seems to be in more or less final form. Other systems of axioms are just slight modifications of those given in [12] (cf., for instance, [18, 211) or consider more special cases (as in [6, 17). Section 1 of this paper can be considered as a discussion of the axioms given in [12] which results in a somewhat streamlined version of these axioms. It should be mentioned here that state spaces for equations with bounded retardation are included as special case, of course. In Section 2 we prove existence, uniqueness and continuous dependence of solutions concentrating on the nonstandard situation where the right-hand side of the equation is not defined for elements in the state space but only for representatives of elements in the state space. Such situations occur for instance if difference-differential equations are considered in a space like UP

Wellposedness and wave propagation for a class of integrodifferential equations in Banach space

In recent years there has been an increasing interest in the mathematical treatment of problems which originate in the theory of viscoelastic materials. In such problems the constitutive relations for these materials incorporate the history of the strain or the strain rate, and, in the linear theory, lead to linear Volterra integrodifferential equations with infinite memory in a Banach space X. The purpose of this paper is to examine the problems of wellposedness and wave propagation for one such equation. In particular, we are concerned with equations of the form x’(t)=/4 x(r)+J’ [ F(t-s)x(s)ds -al

Spectral Properties of Finite-Dimensional Perturbed Linear Semigroups

Abstract Many interesting Cauchy problems arising in applications can be treated as finite-dimensional perturbations of relatively simple problems. In most applications these perturbations take place via boundary conditions. It is the aim of this paper to investigate such Cauchy problems and especially the spectral properties of the corresponding generators. Some general results are obtained that unify several well-known results for functional-differential equations and age-dependent population dynamics. Applications to control problems as well as some nonlinear generalizations will appear in forthcoming papers.

Persistent age-distributions for a pair-formation model

Conditions on the vital rates of a two-sex population are presented which imply the existence or nonexistence of exponentially growing persistent age-distributions.

Nonlinear age-dependent population dynamics with random diffusion

Abstract A model of population growth is developed which allows for consideration of both age and spatial effects. The mortality and fertility processes of the population are allowed to depend nonlinearly upon the age structure, and the population is allowed to diffuse randomly in a spatial region. The problem is studied abstractly as an evolution equation in a Banach space of density functions. The problems investigated are existence and uniquencess of solutions, continious dependence upon variations in the mortality and fertility functions, and continous dependence upon the initial population distribution.

A note on the comparison of Co-semigroups

If S(.) and T(.) are two Co-semigroups of linear operators on a Banach space X, we derive a characterization of their generators so that ||S(t)−T(t)||=O(t) as t→O+.

Mild and strong solutions for partial differential equations with delay

SummaryWe study the nonlinear functional differential equation

Some considerations for Linear integrodifferential equations

\begin{gathered} \dot x(t) = f(x(t)) + g(x(t),x_t ) \hfill \\ x(0) = \eta ,x_0 = \varphi \hfill \\ \end{gathered}