Mathematical Sciences: Nonlinear Functional Analysis

本课题涉及由算术-几何平均不等式的经典结果衍生出的数学问题，它与数学的各个部分有许多联系。我们可以将两个数的算术和几何平均值看作有序对，然后迭代这对。在这样做的过程中，一个人总是到达相等数量的极限。这一原理将在泛函分析的背景下进行研究，在泛函分析中，迭代的对象是非交换的线性算子，迭代的过程不是计算均值，而是将锥的内部映射到自身的更一般的映射。目标是找到唯一的固定射线（特征向量）。预计将应用于种群生物学中出现的问题。另外，我们将继续研究奇异摄动下的时滞微分方程。这里的目标是确定是否可以预测非恒定周期解-特别是第四个周期解，这是在边界层现象研究中出现的一个问题。

This project concerns mathematical questions deriving from classical results on the arithmetic-geometric mean inequalities which have many connections with diverse parts of mathematics. One can view the arithmetic and geometric mean of two numbers as an ordered pair and then iterate this pairing. In doing so, one always arrives at a limit of equal numbers. This principle will be studied in the context of functional analysis in which the objects of the iteration are noncommuting linear operators and the process, instead of computing means, is a more general one of mappings of the interior of a cone into itself. The objective will be to find unique fixed rays (eigenvectors). Applications are expected to be made to questions arising in population biology. Other, continuing, work will be done on delay- differential equations in the context of a singular perturbations. The objectives here are to determine whether or not nonconstant periodic solutions can be predicted - especially of period four, a problem which occurs in the study of boundary layer phenomena.