# OSCILLATION THEORY OF DYNAMIC EQUATIONS ON TIME SCALES

## Authors

Lynn H. Erbe

Allan C. Peterson

4-28-2008

## Document Type

Article

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors Lynn H. Erbe and Allan C. Peterson Lincoln, Nebraska May, 2008

Copyright (c) 2008 Raegan J. Higgins

## Abstract

In past years mathematical models of natural occurrences were either entirely continuous or discrete. These models worked well for continuous behavior such as population growth and biological phenomena, and for discrete behavior such as applications of Newton's method and discretization of partial differential equations. However, these models are deficient when the behavior is sometimes continuous and sometimes discrete. The existence of both continuous and discrete behavior created the need for a different type of model. This is the concept behind dynamic equations on time scales. For example, dynamic equations can model insect populations that are continuous while in season, die out in, say, winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. Throughout this work, we will be concerned with certain dynamic equations on time scales. We start with a brief introduction to the time scale calculus and some theory necessary for the new results. The main concern will then be the oscillatory behavior of solutions to certain second order dynamic equations. In Chapter 3, an equation of particular interest is one containing both advanced and delayed arguments. We will use the method of Riccati substitution to prove some oscillation results of the solutions. In Chapter 4 we again study the oscillatory behavior of a second dynamic equation. However, in this chapter, the equation only has delayed arguments. In addition to using Riccati substitution, we use the method of upper and lower solutions to develop necessary and sufficient conditions for oscillatory solutions. In the final chapter we are interested in the existence of nonoscillatory solutions of dynamic equations on time scales. The common theme among these results is the use of the Riccati substitution technique and the integration of dynamic inequalities.

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