US-Federal Republic of Germany Cooperative Research: Almost Periodicity and Asymptotics for Semigroups of Operators

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This award supports Professor W. H. Summers of the University of Arkansas to collaborate in research with Professor Wolfgang M. Ruess of the Mathematics Department, University of Essen, West Germany. In previous joint work, these investigators have combined complementary backgrounds to develop criteria and techniques which have proved useful in resolving some longstanding problems by providing new insight into dynamical phenomena that are asymptotically close to being almost periodic. The proposed cooperative research program will continue this line of investigation. The goal of the research is to identify conditions on the semigroup, its generator, and the initial values under which trajectories are asymptotically almost periodic in some appropriate sense. Both the expected and existing results in this direction will be applied to questions concerning the existence and description of strong or weak limits of ergodic means, asymptotic behavior of solutions to the Cauchy problem and functional differential equations with infinite delay, and asymptotic behavior of age-dependent population models. Mathematical models for many time-dependent processes in physics, chemistry and biology take the form of a Cauchy problem associated with the generator of a semigroup of operators on an appropriate Banach state space. The future-time behavior of the process can then be investigated by studying asymptotic properties of motions and almost-orbits of the corresponding semigroup. There has been considerable effort devoted by many researchers in recent years to classifying the asymptotic behavior of nonlinear semigroups. A great variety of results has been established by different methods. Drs. Summers and Ruess have brought new techniques and insights to the problem, and have recently produced some very interesting contributions to this subject. They are now beginning to seek new applications of their results, in particular to nonlinear functional differential equations.