Asymptotic behavior of linear dynamic equations on time scales

Until recently, mathematical models of natural occurrences were either exclusively continuous or discrete. These models worked well for continuous behavior, such as modeling plant growth, and for discrete behavior, such as the daily number of cranes in a field during migration. But such models are lacking when the behavior is sometimes continuous and sometimes discrete. The occurrences of both continuous and discrete behavior spark the need for a different kind of model. This is the idea behind dynamic equations on time scales. The theory seeks to unify and extend the continuous and discrete. Throughout this paper, we will be concerned with particular dynamic equations on time scales. We shall start with a brief overview of the times scale calculus and give some theory necessary for the new results. The main concern will then be the asymptotic behavior of solutions of a formally self-adjoint dynamic equation on a time scale. In particular one equation of interest is the perturbation of a nonoscillatory second-order formally self-adjoint equation. We will use the contraction mapping theorem to prove some asymptotic properties of our solutions and their derivatives. In Chapter 3, we again study the asymptotic behavior of solutions of a formally self-adjoint dynamic equation. However, in this chapter the equation has mixed derivatives. We will again use the contraction mapping theorem to prove some asymptotic properties of our solutions and their delta derivatives. Though the results are similar to those in Chapter 2, having both delta and nabla derivatives creates a unique challenge. Theorems from Chapter 14 of Hartman\u27s Ordinary Differential Equations are generalized to dynamic equations in Chapter 4. Instead of studying self-adjoint equations, here we address systems of linear equations. The common theme among the results is that they use monotonicity in the theorem statements or the proofs