Mathematical Sciences: Nonlinear Functional Analysis

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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.