Differential Equations on Measure Chains and Applications

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

NSF Award Abstract - DMS-0072505 Mathematical Sciences: Differential Equations on Measure Chains and Applications Abstract 0072505 Peterson This project is concerned with the study of qualitative properties of solutions of differential systems and dynamical equations on time scales. The theory of calculus on measure chains treats continuous, discrete, and more general situations in a unified way. A central theme of the study is the unification of discrete and continuous formulations. The research investigates: (1) oscillation, nonoscillation, and disconjugacy properties of solutions of self adjoint systems and generalized Hamiltonian systems arising in a number of areas of applied mathematics, including calculus of variations, control theory, filtering theory, and continued fractions; (2) extension of existence, uniqueness, and multiplicity results for solutions to nonlinear boundary value problems via fixed point theorems of cone expansion/compression type and methods applicable to monotone operators in abstract spaces; (3) the structure of positive solutions of nonlinear boundary value problems; and (4) problems related to singular dynamical equations on a measure chain. In scientific investigation of natural phenomena, mathematical models are used to give quantitative descriptions and to furnish predictions. One form such models can take is a dynamic equation on a measure chain. This project provides new insight into the relationship between continuous and discrete descriptions of phenomena, and has application to many fields, including mathematical biology and economics. In one application, a study of population growth, we will give conditions that ensure survival of a population, in particular for the case in which parents die before offspring are hatched. Two well-known examples of this are the seventeen-year cicada and the common mayfly.