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International Journal of Differential Equations
Volume 2013 (2013), Article ID 764389, 10 pages
http://dx.doi.org/10.1155/2013/764389
Research Article

Qualitative Analysis of Solutions of Nonlinear Delay Dynamic Equations

1Department of Mathematics, Canakkale Onsekiz Mart University, 17020 Canakkale, Turkey
2Department of Mathematics, University of Dayton, OH 45469-2316, USA

Received 21 October 2013; Revised 9 December 2013; Accepted 9 December 2013

Academic Editor: Tongxing Li

Copyright © 2013 Mehmet Ünal and Youssef N. Raffoul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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. Preliminaries on Time Scales

In this subsection, we recall some of the notations, definitions, and theorems on time scale calculus that we use throughout the paper. An excellent comprehensive treatment of calculus on time scales can be found in [8, 9]. Most of the material in this subsection can be found in [8, Chapter 1]. We should start mentioning that this theory was introduced in order to unify continuous and discrete analysis; however, it does not only unify the theories of differential equations and difference equations, but also it enable us to extend these classical cases to cases “in between,” for example, to so-called -difference equations.

A time scale is an arbitrary nonempty closed subset of the real numbers and is denoted by the symbol . The two most popular examples are and . Several other interesting time scales exist, and they give rise to plenty of applications such as the study of population dynamic models (see [8, pages 15 and 71]). Define the time scale interval by . Other time scale intervals are defined similarly. The forward jump operator at for is defined by with . The graininess function is defined by . The backward jump operator at for is defined by with . The point is called right scattered if , left scattered if , and dense if .

Fix and let . Define (called the delta derivatives of at ) to be the number (if exists) with the property that given there is a neighborhood of such that, for all ,

Some elementary facts concerning the delta derivative are as follows:(1)if is differentiable at , then (2)if and are differentiable at , then is differentiable at with (3)if and are differentiable at and , then is differentiable at with

We say is right-dense continuous provided is continuous at right-dense points in and its left-sided limit exists (finite) at left-dense points in . The importance of rd-continuous functions is that every rd-continuous function possesses an antiderivative. A function is called an antiderivative of provided holds for all .

Some elementary facts concerning the delta integral are as follows:

if , , , , and , , then(a);(b);(c)If on , then ;(d)If for all , then .

Now, we present a chain rule: Assume that is a strictly increasing and is a time scale. Let . If and exists for , then .

A function is called regressive if it is rd-continuous and satisfies The set of all regressive functions will be denoted by . Also (positively regressive) if and only if and , for all .

For , define the cylinder transformation by where is the principal logarithm function. For , we define for all .

If , then we define the generalized exponential function for all , . Note that one can also define the generalized exponential function to be the unique solution of the initial value problem Also, it is well know that, if , then for all . We will use many of the following properties of the generalized exponential function in our calculations.

If , and , , , , then the properties of generalized exponential function are:(1) and ;(2);(3), where ;(4);(5);(6), where ;(7);(8).

Another useful tool is a variation of parameters formula for first order linear nonhomogeneous dynamic equations which now we state next. Suppose that and is rd-continuous function. Let and . Then the unique solution of is given by

1.2. Solution

For each and for a given rd-continuous initial function , we say that is the solution of (1) if on and satisfies (1) for all . The zero solution of (1) is called stable if for any and , there exists a such that implies for all .

We need the following lemmas in proving our main theorem.

Lemma 1 (see [10, Lemma 2]). For nonnegative with one has the inequalities

Lemma 2 (see [3, Lemma 3], [4]). Suppose that is a time scale having a strictly increasing, and invertible delay function , such that and . Then for a given rd-continuous function , one has

Lemma 3. Assume is a time scale having a strictly increasing and invertible delay function , such that and . Then, the nonlinear delay equation (1) is equivalent to

Proof. Assume is a solution of (1). Then the proof immediately follows from Lemma 2 since

Lemma 4. Suppose that . If is a solution of (1) with initial function , then

Proof. We know from Lemma 3 that (1) is equivalent to (19). To create a linear term, we add and subtract in (19) to obtain Using the variation of constants formula page 77 [8] for (22) yields where denotes the delta derivative with respect to . The proof follows from using integration by parts formula on the last term of the right hand side of (23).

2. Main Results

Let be an arbitrary time scale which is unbounded above and consider the nonlinear delay dynamic equation with rd-continuous initial function .

In the sequel we assume the following:(a);(b)  rd-continuous and ;(c)the delay function is strictly increasing, invertible, and delta differentiable such that , for , and ;(d) is continuous, locally Lipschitz and odd while is rd-continuous, is nondecreasing and is increasing on an interval with , where is rd-continuous function.

We should remark here that condition (d) in our hypotheses ensures that the function and are locally Lipschitz with the same Lipschitz constant . Also it is clear that if , then the condition on given by (d) hold on . Moreover, for any given rd-continuous function with is bounded with , then for we have since is odd and nondecreasing on .

We need to construct a mapping which is suitable for fixed point theory. Instead of using the supremum norm, we will use nonconventional metric to define a new norm in order to overcome the difficulties that arise from the contraction constant which, in turn rely on the Lipschitz constant.

Lemma 5. Let (a)–(d) of our hypotheses hold. Let and be a fixed number for , and satisfy a Lipschitz condition with constant . Suppose that is rd-continuous and for define If , then for each there is a metric on such that is a contraction with constant and is a complete metric space.

Proof. Let be the Banach space of rd-continuous function for which exists. If , , then Since we obtain On the one hand, using Lemma 1 in the integral on the right hand side of (31), we have In addition, by setting and using the fact that yield where denotes the delta derivative with respect to variable . And hence, by fixing and using the integration by part formula by taking into account that we obtain Thus, we get Finally, substituting (38) into (31) we get Since is a subset of the Banach space and is closed, hence is complete. Thus, has a unique fixed point.

For any given rd-continuous initial function defined on with , we let where denotes the segment of on .

Theorem 6. Let (a)–(d) of our hypotheses hold, and suppose that for each the inequality and for the inequality hold. Then every solution of (24) is bounded. In addition if , then the zero solution of (24) is stable.

Proof. Define a mapping on using (21) in such a way that for we have By (41) there exists an such that if and for we have Since we obtain By choosing the initial function small enough we have where is the Lipschitz constant of on . Hence, we obtain Thus, we have showed that and any solution of (24) that is in is bounded.
Next we need to show that is a contraction. To do this, we proceed as in the proof of Lemma 5. First note that for , we have and hence we take the metric on which is induced by the norm Our aim is to simplify (48). It follows from Lemma 5 that the last term on the right hand side of (48) has a contraction constant since and both satisfy a Lipschitz condition with the same constant that is;
The first term satisfies Setting and using the fact that we obtain Thus, Now, we turn our attention to the second term of (48). Multiply this term by we get A substitution of (50), (55), and (57) into (48) gives Hence, we have showed that is a contraction for . Thus, taking into account Lemma 5, has a unique fixed point. This completes the proof.

When and , we have the following two corollaries which are immediate consequences of Theorem 6 and thus the proofs are omitted.

Corollary 7. Let (a)–(d) of our hypotheses hold in case of   and suppose that for each the inequality and for the inequality hold. Then every solution of (2) is bounded. In addition, if , then the zero solution of (2) is stable.

Corollary 8. Let (a)–(d) of our hypotheses hold in case of , and suppose that for each the inequality and for the inequality hold. Then every solution of (3) is bounded. In addition if , then the zero solution of (3) is stable.

Remark 9. We may deduce [11, Theorem 4.1] and [2, Theorem 2.2] as Corollaries 7 and 8, respectively. Thus, we have unified these results and moreover, we have extended them the general time scales. In particular, the next results concerning the -difference equation (4) are new.

Example 10. For any and fixed positive integer , define For any initial function , , Consider the following nonlinear delay initial value problem where . We show that the solution of (64) is bounded and zero solution is stable.

A simple comparison of (64) to (24) yield that , , , , and .

Notice that is increasing, continuous and odd and that and locally Lipschitz on . Thus, every condition of our hypotheses is satisfied.

In order to make use of Theorem 6, we must perform some calculations so that conditions (41) and (42) are satisfied.

For , we first calculate : and since we obtain

Next, we calculate . For , we have where . This implies that

Since we have Since , we have that where we have used the fact that

We know that , and hence, by taking , we obtain Thus As a consequence, we obtain

Left to calculate . Consider Thus, the above integral is independent of .

Left to compute

We have already shown which is independent of integration variable. This leads us to

As consequence of all of the above calculations, we arrive at, for that <