Abstract and Applied Analysis

Volume 2012 (2012), Article ID 392386, 12 pages

doi:10.1155/2012/392386

Research Article

Qualitative Behaviors of Functional Differential Equations of Third Order with Multiple Deviating Arguments

Cemil Tunç

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

Received 14 September 2012; Accepted 24 September 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Cemil Tunç. 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.

This paper considers nonautonomous functional differential equations of the third order with multiple constant deviating arguments. Using the Lyapunov-Krasovskii functional approach, we find certain sufficient conditions for the solutions to be stable and bounded. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.

1. Introduction

In this paper, we consider nonautonomous differential equation of the third order with constant multiple deviating arguments, , , as follows:

Writing (1.1) as a system of first order equations, we have

where and ,   , are positive constants, that is, fixed delays; the functions ,   , ,   , and are continuous for their all respective arguments and the primes in (1.1) denote differentiation with respect to ,  . It is also assumed that the derivatives ,   ,   , and exist and are continuous; throughout the paper ,   are abbreviated as , , and , respectively. Finally, the existence and uniqueness of solutions of (1.1) are assumed and all solutions considered are supposed to be real valued.

To the best of our knowledge from the literature, in the last five decades, there has been much attention paid to the discussion of stability and boundedness of solutions of nonlinear differential equations of the third order without a deviating argument. For a comprehensive treatment of the subject, we refer the readers to the book of Reissig et al. [1] as a survey and the papers of Ademola et al. [2], Afuwape et al. [3], Ezeilo [4–13], Ezeilo and Tejumola [14, 15], Mehri and Shadman [16], Ogundare [17], Ogundare and Okecha [18], Omeike [19, 20], Ponzo [21, 22], Swick [23–25], Tejumola [26, 27], Tunç [28–34], Tunç and Ateş [35], Tunç and Ayhan [36], and the references cited in these papers for some works on the topic.

Besides, first, in 1973, Sinha [37] studied the stability of solutions of a third order nonlinear differential equation with a deviating argument. Later, some authors dealt with the stability and boundedness of solutions for various third order nonlinear differential equations with a deviating argument. For some related works, one can refer to the papers of Afuwape and Omeike [38], Omeike [39], Sadek [40, 41], Tejumola and Tchegnani [42], Tunç [43–59], Yao and Meng [60], Zhu [61], and the references thereof.

It should be noted that throughout all the above mentioned papers, Lyapunov’s functions or the Lyapunov-Krasovskii functionals have been used as a basic tool to prove the results established there. It is also worth mentioning that the most effective method to study the stability and boundedness of solutions of nonlinear differential equations of higher orders without or with a deviating argument in the literature is still the Lyapunov’s direct method, despite its use for a past long period by now.

The motivation for this paper comes from the above mentioned papers and the books. Our results improve and include the results existing in the literature. This work makes also a contribution to the existing studies made in the literature.

2. Main Results

Let .

Our first result is given by the following theorem.

Theorem 2.1. In addition to the basic assumptions imposed to the functions , , , , and appearing in (1.1), we assume that there exist positive constants , , , , , , , , , and such that the following conditions hold:

(i), ,

(ii).

If

then the zero solution of (1.1) is stable.

Proof. Define the Lyapunov-Krasovskii functional as

where and are some positive constants to be chosen later.

Using the assumptions , , ,, and , we have

so that

On the other hand, it is obvious that

so that

Hence, we can obtain some positive constants , , such that

where , since the integrals and are nonnegative.

Let be a solution of (1.2). Differentiating the Lyapunov-Krasovskii functional along this solution, we find

Using the assumptions of Theorem 2.1 and the estimate , we get

so that

Let and Hence,

Let . Then

Thus, if

then

This completes the proof of Theorem 2.1 (Burton [62], Hale [63], and Krasovskiĭ [64]).

Let .

Our second main result is given by the following theorem.

Theorem 2.2. In addition to all the assumptions of Theorem 2.1, we assume that the condition

holds, where . If

then, there exists a finite positive constant such that the solution of (1.1) defined by the initial function

satisfies

for all , where .

Proof. Under the assumptions of Theorem 2.2, the time derivative of the Lyapunov-Krasovskii functional satisfies

Using the estimates and it follows that

where .

Integrating the above estimate from to , using the assumption and the Gronwall-Bellman inequality (see Gronwall [65] and Mitrinović [66]), we can conclude that all solutions of (1.1) are bounded.

Example 2.3. Consider the nonlinear differential equation of the third order with two deviating arguments as follows:

Writing (2.22) as a system of first order equations, we obtain

It follows that (2.22) is special case of (1.1), and when we compare (2.22) with (1.1) we obtain the following estimates:

In view of the above discussion, it follows that

that is, , and

Thus, all the assumptions of Theorems 2.1 and 2.2 hold. This shows that the zero solution of (2.22) is stable and all solutions of the same equation are bounded, when and , respectively.

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