Delay Dynamic Equations with Stability

The unification and extension of continuous calculus, discrete calculus, q-calculus, and indeed arbitrary real-number calculus to time-scale calculus, where a time scale is simply any nonempty closed set of real numbers, were first accomplished by Hilger in [4]. Since then, time-scale calculus has made steady inroads in explaining the interconnections that exist among the various calculuses, and in extending our understanding to a new, more general and overarching theory. The purpose of this work is to illustrate this new understanding by extending some continuous and discrete delay equations to certain time scales. Examples will include specific cases in differential equations, difference equations, q-difference equations, and harmonic-number equations. The definitions that follow here will serve as a short primer on the time-scale calculus; they can be found in [1, 2] and the references therein.