/ / Article

Research Article | Open Access

Volume 2012 |Article ID 392386 | https://doi.org/10.1155/2012/392386

Cemil Tunç, "Qualitative Behaviors of Functional Differential Equations of Third Order with Multiple Deviating Arguments", Abstract and Applied Analysis, vol. 2012, Article ID 392386, 12 pages, 2012. https://doi.org/10.1155/2012/392386

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

Accepted24 Sep 2012
Published05 Nov 2012

#### 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.  as a survey and the papers of Ademola et al. , Afuwape et al. , Ezeilo , Ezeilo and Tejumola [14, 15], Mehri and Shadman , Ogundare , Ogundare and Okecha , Omeike [19, 20], Ponzo [21, 22], Swick , Tejumola [26, 27], Tunç , Tunç and Ateş , Tunç and Ayhan , and the references cited in these papers for some works on the topic.

Besides, first, in 1973, Sinha  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 , Omeike , Sadek [40, 41], Tejumola and Tchegnani , Tunç , Yao and Meng , Zhu , 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 , Hale , and Krasovskiĭ ).

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  and Mitrinović ), 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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