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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 248717, 7 pages

http://dx.doi.org/10.1155/2013/248717

## Stability and Uniform Boundedness in Multidelay Functional Differential Equations of Third Order

Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey

Received 11 February 2013; Accepted 18 April 2013

Academic Editor: Tonghua Zhang

Copyright © 2013 Cemil Tunç and Melek Gözen. 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 consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiì functional approach, we give certain sufficient conditions to guarantee the asymptotic stability and uniform boundedness of the solutions.

#### 1. Introduction

Differential equations of third order are valuable tools in the modeling of many phenomena in various fields of science and engineering (Chlouverakis and Sprott [1], Cronin-Scanlon [2], Eichhorn et al. [3], Friedrichs [4], Linz [5], and Rauch [6]). In reality, the stability and boundedness of solutions of certain nonlinear differential equations of the third order have been received intensive attentions by authors (Ademola et al. [7], Afuwape and Castellanos [8], Chukwu [9], Ezeilo ([10, 11]), Hara [12], Mehri and Shadman [13], Ogundare and Okecha [14], Omeike [15], Reissig et al. [16], Swick [17], Tejumola ([18, 19]), Tunç [20–33], and Yoshizawa [34]).

In 2009, Omeike [15] considered the nonlinear differential equation of the third order with the constant delay : and he discussed the stability and boundedness of solutions of this equation.

In this paper, instead of the above equation, we consider the nonautonomous differential equation of the third order with multiple deviating arguments: where are certain positive constants, and and are real valued and continuous functions in their respective arguments with . The existence and uniqueness of the solutions of (2) are also assumed.

The motivation for this paper is a result of the researches mentioned regarding ordinary differential equations of the third order. It follows that the equation discussed in [15] is a special case of (2). Our aim is to improve the results established in [15] from one deviating argument to the multiple deviating arguments for the asymptotic stability and uniform boundedness of solutions. This work contributes to and complements previously known results on the topic in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions. It should be noted that in recent years scores of papers have been published on the qualitative behaviors of solutions (stability of solutions, boundedness of the solutions, existence of the periodic solutions, etc.) of the functional differential equations of the second order with multiple deviating arguments. However, very little attention was given to stability and boundedness of functional differential equations of the third order with multiple deviating arguments ([32]). Therefore, it is worth investigating the qualitative behaviors of solutions in multidelay functional differential equations of the third order. This case is the novelty of the present paper. It should also be noted that the results to be established here are different from those in Tunç [20–33] and the literature.

#### 2. Main Results

Let .

Theorem 1. *One assumes that there exist positive constants , and such that the following conditions hold:*(i)*,
* *,
*(ii)*. *

If

then every solution of (2) is uniform bounded and satisfies

*Remark 2. *It should be noted that it follows from (ii) that and are nonincreasing functions on . Therefore, since these functions are continuous on this interval and bounded below by , they are bounded on and the limit of each exists as . Since in (ii) is an arbitrary selected bound, we can also assume the following estimates:

*Proof. *We write (2) in the system form as follows:

Define a Lyapunov-Krasovskiì functional ([35]) by
where and are certain positive constants, which will be determined later in the proof.

This functional can be arranged as follows:
where

Using the assumptions of Theorem 1, it follows that since and .

Thus, there exist constants and such that
since . Further, using the assumptions of Theorem 1 and , it follows that
so that
where

In view of the previous discussion, we can get

Using a basic calculation, the time derivative of along solutions of (6) results in

Using , and the estimate , we have
where

Noting the previous discussion, it follows that

If , then

If , then it follows that

Since , and , then
where and .

Thus, we get

Let . Hence,

If , then since and . For those such that , we have

Thus,

Therefore, if
then we have

The proof for Theorem 1 is complete.

Let .

Theorem 3. *One assumes that all the assumptions of Theorem 1 and the assumption
**
hold. If
**
then all solutions of (2) are bounded. *

*Proof. *Equation (2) is equivalent to the system

Along any solution of (6), we have

Since , then it follows that
where . Noting that , we get
recalling that .

Let , then
or

Multiplying each side of this estimate by the integrating factor , we get

Integrating each side of this estimate from 0 to *t*, we obtain
or
where .

Since for all , this implies

Since the right-hand side of the last estimate is a constant and when , it follows that there exists a positive constant such that

From the system (30) this implies that

The proof for Theorem 3 is complete.

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