On the Rational Dynamic Equation on Discrete Time Scales

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger [16], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice-once for differential equations and once again for difference equations. 

The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have enormous potential for applications such as in population dynamics. For example, it can model insect population 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 (see Dosly et al. [10]). 

Several authors have expounded on various aspects of this new theory, see the survey paper by Agarwal et al. [3] and the references cited therein. The first course on dynamic equations on time scales as in Bohner et al. [7]. For advance of dynamic equations on time scales as in Bohner et al. [6]. For completeness, we give a short introduction to the time scale calculus. the introduction of the paper should explain the nature of the problem, previous work, purpose, and the contribution of the paper. The contents of each section may be provided to understand easily about the paper.