Dynamic Equations on Time Scales

Abstract

The theory of time scales was introduced by Stefan Hilger in his 1988 PhD dissertation, [18]. The study of dynamic equations on time scales unifies and extends the fields of differential and difference equations, highlighting the similarities and providing insight into some of the differences. In his dissertation, [18], Hilger introduced the notion of the "delta-derivative" on a time scale. An analogous concept, the "nabla-derivative" was developed and explored by Ferhan Atici and Gusein Guseinov in [4]. It is interesting to look at what happens when these two kinds of derivatives are present in the same equation. The interaction between them yields some fascinating behavior, which in some cases is "cleaner" than the behavior found with only one type of derivative. In Chapter 2, we examine the second-order, self-adjoint dynamic equation which contains both delta- and nabla-derivatives. We develop a reduction of order theorem, explore oscillation and disconjugacy, and look at Riccati techniques. The material in Chapter 2 has been previously published in [23] and [22]. In Chapter 3, we look at solution techniques for linear dynamic equations which can be written in a factored form. We develop complete results in the case where the equation contains only one kind of derivative. We briefly discuss the mixed derivative case deferring further consideration to later work. The material in Chapter 3 has been previously published in [21]. In the final chapter, Chapter 4, we return to the self-adjoint equation. Here, we consider the matrix form of the equation. As in the scalar case, we develop a reduction of order theorem and explore Riccati techniques, culminating with the proof of Jacobi's Condition. Throughout much of this dissertation, the interaction between the delta- and nabla-derivatives plays a key role. In many cases, it is rather startling to see how all of the pieces fit together. It is our hope that this work will inspire further exploration in this area.