Qualitative Analysis of Solutions of Nonlinear Delay Dynamic Equations

We use the fixed point theory to investigate the qualitative analysis of a nonlinear delay dynamic equation on an arbitrary time scales. We illustrate our results by applying them to various kind of time scales. 1. Introduction In this paper, we investigate the qualitative analysis of solutions of nonlinear delay dynamic equation of the form on an arbitrary time scale which is unbounded above, where the functions and are rd-continuous, the delay function is strictly increasing, invertible, and delta differentiable such that , for , and . Although it is assumed that the reader is already familiar with the time scale calculus, for completeness, we will provide some essential information about time scale calculus in the Section 1.1. We should only mention here that this theory was introduced in order to unify continuous and discrete analysis; however it is not only unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases to cases “in between,” for example, to so-called -difference equations. Also note that, when , (1) is reduced to the nonlinear delay differential equation and when , it becomes a nonlinear delay difference equation In the case of quantum calculus which defined as , is a real number, (1) leads to the nonlinear delay -difference equation where . Motivated by the papers [1, 2], in this paper we study the qualitative properties of solution of nonlinear delay dynamic equation (1) by means of fixed point theory. The results of this paper unify the results given by [1] for (2) and by [2] for (3). Moreover, we obtain new results for the -difference equation (4) and explicitly provide an example in which we show how our conditions can be applied. Our technique in proving the results naturally has some common features with the ones employed in both [1] and [2] but it is actually quite different due to difficulties that are peculiar to the time scale calculus. Also, our results may be considered as generalization of the ones obtained in [3, 4] and [5] in which the authors studied the stability of the delay dynamic equation In [6], the authors establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation One can easily see that the results of (6) cannot be applied to -difference equations. Moreover, it requires that be in the time scale. For resent results regarding existence, uniqueness and continuous dependence of the solution for nonlinear delay dynamic equations, we refer to [7]. 1.1.