Mathematical Sciences: Monotonicity in Differential Equations

该项目支持对动力系统和微分方程式研究中出现的问题进行数学研究。该方程是滞后型泛函微分方程。在生物模型中出现的某些方程中，记忆效应的迟缓是一种自然特征。工作将集中在四个主要主题上。第一个是关于单调动力系统上的单调流。将努力修改标准系统理论以统一处理平衡点集合包含所有(相空间)常量函数的微分方程式。第二个主题涉及物理系统的不稳定性。应用有限时滞方程的不变性原理，得到有限时滞泛函微分方程和无限时滞泛函微分方程的不稳定性结果。不稳定性的特征在于，所有经过不稳定点附近的解都必须离开该点的每个邻域。中立型泛函微分方程是这个项目的第三个要素。将在建立近似解的比较收敛技术方面开展工作。本研究的最终目的是利用单调迭代技术研究泛函微分方程的周期边值问题。周期解的存在性和构造周期解的技术都将被考虑。

This project supports mathematical research on problems arising in the study of dynamical systems and differential equations. The equations are functional differential equations of retarded type. In certain equations occurring in biological models, the retardation of memory effect is a natural feature. Work will focus on four main topics. The first concerns monotone flows on monotone dynamical systems. Efforts to modify standard systems theory to present a unified treatment of differential equations for which the set of equilibrium points contains all (phase space) constant functions will be made. The second topic relates to instability of physical systems. Applications of invariance principles of finite delay equations will be made to obtain instability results for finite and infinite delay functional differential equations. Instability is characterized by the property that all solutions passing near an unstable point must leave every neighborhood of the point. Neutral functional differential equations make up the third element of this project. Work will be done in establishing comparison-convergence techniques for approximating solutions. The final goal of the research is that of using monotone iteration techniques to study periodic boundary value problems for functional differential equations. Both the existence of periodic solutions and techniques for constructing them will be considered.