Boundedness and Vanishing of Solutions for a Forced Delay Dynamic Equation

Following Hilger’s landmark paper [8], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time-scale calculus, where a time scale is simply any nonempty closed set of real numbers. This paper illustrates this new understanding by extending some continuous results from differential equations to dynamic equations on time scales, thus including as corollaries difference equations and q-difference equations. Throughout this work, we consider the nonlinear forced delay dynamic equation