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

该奖项支持阿肯色大学W.H.萨默斯教授与西德埃森大学数学系的沃尔夫冈·M·鲁斯教授合作进行研究。在之前的联合工作中，这些研究人员结合了互补的背景来开发标准和技术，事实证明，这些标准和技术通过为渐近接近几乎周期的动力学现象提供新的见解，在解决一些长期存在的问题方面是有用的。拟议的合作研究计划将继续这一调查路线。研究的目的是确定关于半群、它的生成元以及轨道在某种适当意义上渐近几乎周期的初值的条件。这一方向的预期结果和已有结果都将应用于遍历均值的强弱极限的存在性和刻画，无穷时滞柯西问题和泛函微分方程解的渐近行为，以及年龄相关人口模型的渐近行为。物理、化学和生物中许多依赖于时间的过程的数学模型采用柯西问题的形式，该问题与适当的Banach状态空间上的算子半群的生成元有关。然后，可以通过研究相应半群的运动和几乎轨道的渐近性质来研究该过程的未来时间行为。近年来，许多研究人员致力于对非线性半群的渐近行为进行分类。不同的方法得出了各种各样的结果。萨默斯博士和鲁斯博士为这一问题带来了新的技术和见解，最近他们对这一主题做出了一些非常有趣的贡献。他们现在开始寻求他们的结果的新应用，特别是在非线性泛函微分方程方面。

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.