Abstract and Applied Analysis

Volume 2014, Article ID 424512, 6 pages

http://dx.doi.org/10.1155/2014/424512

## Convergence of Solutions to a Certain Vector Differential Equation of Third Order

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

Received 12 December 2013; Accepted 18 January 2014; Published 26 February 2014

Academic Editor: Bingwen Liu

Copyright © 2014 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 give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature.

#### 1. Introduction

This paper is concerned with the following nonlinear vector differential equation of third order: where and and are continuous functions in their respective arguments.

It should be noted that, in 2005, Afuwape and Omeike [1] considered the following nonlinear vector differential equation of third order: where is real symmetric -matrix. The author established a new result on the convergence of solutions of (2) under different conditions on the function . For some related papers on the convergence of solutions to certain vector differential equations of third order, the readers can referee to the papers of Afuwape [2], Afuwape and Omeike [3], and Olutimo [4]. Further, it is worth mentioning that in a sequence of results Afuwape [2, 5, 6], Afuwape and Omeike [3], Afuwape and Ukpera [7], Ezeilo [8], Ezeilo and Tejumola [9, 10], Meng [11], Olutimo [4], Reissig et al. [12], Tiryaki [13], Tunç [14–16], Tunç and Ateş [17], C. Tunç and E. Tunç [18], and Tunç and Karakas [19] investigated the qualitative behaviors of solutions, stability, boundedness, uniform boundedness and existence of periodic solutions, and so on, except convergence of solutions, for some kind of vector differential equations of third order.

The Lyapunov direct method was used with the aid of suitable differentiable auxiliary functions throughout the mentioned papers. However, to the best of our knowledge, till now, the convergence of the solutions to (1) has not been discussed in the literature. Thus, it is worthwhile to study the topic for (1). It should be noted that the result to be established here is different from that in Afuwape [2], Afuwape and Omeike [1, 3], Olutimo [4], and the above mentioned papers. This paper is an extension and generalization of the result of Afuwape and Omeike [3]. It may be useful for the researchers working on the qualitative behaviors of solutions (see, also, Tunç and Gözen [20]).

It should be noted that throughout the paper will denote the real Euclidean space of -vectors and will denote the norm of the vector in .

*Definition 1. *Any two solutions , of (1) in will be said to converge to each other if
as .

#### 2. Main Result

The main result of this paper is the following theorem.

Theorem 2. *We assume that there are positive constants , , , , , , and such that the following conditions hold: *(i)*the Jacobian matrices , , and exist and are symmetric and their eigenvalues satisfy
* *for all , , in ;*(ii)* satisfies
* *for any , , , , in .**If**
then any two solutions , of (1) necessarily converge, where , , , are some positive constants with and ,
*

*Remark 3. *The mentioned theorem itself still holds valid with (5) replaced by the much weaker condition
for arbitrary any , , , , in , where it is assumed that for .

The following lemma is needed in our later analysis.

*Lemma 4. Let be a real symmetric -matrix and
where and are constants.*

Then

*Proof (see Afuwape [5]). *Our main tool in the proof of our result is the continuous function defined for any triple vectors , , in , by
This function can be rearranged as
where and

The following result is immediate from the estimate (11).

*Lemma 5. Assume that all the conditions on the vectors , , and in the theorem hold. Then, there exist positive constants and such that
for arbitrary in .*

*Proof. *Let
Then the proof can be easily completed by using Lemma 4. Therefore, we omit the details of the proof.

*Proof of the Theorem. *Let in be any solution of (1). For such a solution, let and be denoted, respectively, by and . Then, we can rewrite (1) in the following equivalent system form:

Let in be any solution of (1), define by
where is the function defined in (11) with , , replaced by , and , respectively.

By Lemma 5, it follows that there exist and such that

When we differentiate the function with respect to along the system (15), it follows, after simplification, that
where

Note that the existence of the following estimates is clear (see Afuwape and Omeike [1]):
where , , , , , .

Subject to the assumptions, it can be easily obtained that

In view of the assumptions of the theorem, it is also clear that
Hence,
Using the estimate , it follows that
Then
if with .

Further, since
then
where .

Moreover, it is obvious that
where .

Hence,
Using the assumption (5), we get
so that
There exists a constant such that
provided that , where is a sufficiently small positive constant.

In view of (17), the last estimate implies that
for some positive constant .

The conclusion of the theorem is immediate if, provided that , on integrating in (33) between and , we have
which implies that
By (17), this shows that
This completes the proof of the theorem.

*Example 6. *Let us consider (1),
with
where are bounded continuous functions on .

Then, it can be easily seen that
Thus, , , , , , and .

Let us choose
Then,
Since , then all the conditions of Theorem 2 hold. Therefore, all solutions of the equation considered converge (see, also, [1]).

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*References*

*References*

- A. U. Afuwape and M. O. Omeike, “Convergence of solutions of certain system of third order non-linear ordinary differential equations,”
*Annals of Differential Equations*, vol. 21, no. 4, pp. 533–540, 2005. View at Google Scholar · View at MathSciNet - A. U. Afuwape, “Convergence of solutions of some third order systems of non-linear ordinary differential equations,”
*Analele Ştiinţifice ale Universitatii ‘Alexandru Ioan Cuza’*, vol. 55, no. 1, pp. 11–20, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. U. Afuwape and M. O. Omeike, “Convergence of solutions of certain third order systems of non-linear ordinary differential equations,”
*Journal of the Nigerian Mathematical Society*, vol. 25, pp. 1–12, 2006. View at Google Scholar · View at MathSciNet - A. L. Olutimo, “Convergence results for solutions of certain third-order nonlinear vector differential equations,”
*Indian Journal of Mathematics*, vol. 54, no. 3, pp. 299–311, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. U. Afuwape, “Ultimate boundedness results for a certain system of third-order nonlinear differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 97, no. 1, pp. 140–150, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - A. U. Afuwape, “Further ultimate boundedness results for a third-order nonlinear system of differential equations,”
*Analisi Funzionale e Applicazioni*, vol. 6, no. 1, pp. 348–360, 1985. View at Google Scholar · View at MathSciNet - A. U. Afuwape and A. S. Ukpera, “Existence of solutions of periodic boundary value problems for some vector third order differential equations,”
*Journal of the Nigerian Mathematical Society*, vol. 20, pp. 1–17, 2001. View at Google Scholar · View at MathSciNet - J. O. C. Ezeilo, “$n$-dimensional extensions of boundedness and stability theorems for some third order differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 18, no. 3, pp. 395–416, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. O. C. Ezeilo and H. O. Tejumola, “Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations,”
*Annali di Matematica Pura ed Applicata*, vol. 74, no. 1, pp. 283–316, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. O. C. Ezeilo and H. O. Tejumola, “Further results for a system of third order differential equations,”
*Atti della Accademia Nazionale dei Lincei*, vol. 58, no. 2, pp. 143–151, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. W. Meng, “Ultimate boundedness results for a certain system of third order nonlinear differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 177, no. 2, pp. 496–509, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - R. Reissig, G. Sansone, and R. Conti,
*Non-Linear Differential Equations of Higher Order*, Noordhoff, Groningen, The Netherlands, 1974. View at MathSciNet - A. Tiryaki, “Boundedness and periodicity results for a certain system of third order nonlinear differential equations,”
*Indian Journal of Pure and Applied Mathematics*, vol. 30, no. 4, pp. 361–372, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order,”
*Applied Mathematics and Mechanics*, vol. 20, no. 2, pp. 163–170, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “On the boundedness of solutions of certain nonlinear vector differential equations of third order,”
*Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie*, vol. 49, no. 3, pp. 291–300, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç, “On the stability and boundedness of solutions of nonlinear vector differential equations of third order,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 6, pp. 2232–2236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç and M. Ateş, “Stability and boundedness results for solutions of certain third order nonlinear vector differential equations,”
*Nonlinear Dynamics*, vol. 45, no. 3-4, pp. 273–281, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç and E. Tunç, “New ultimate boundedness and periodicity results for certain third-order nonlinear vector differential equations,”
*Mathematical Journal of Okayama University*, vol. 48, pp. 159–172, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç and B. Karakas, “On boundedness of solutions to nonlinear vector differential equations of third-order,”
*Nonlinear Studies*, vol. 18, no. 1, pp. 63–73, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Tunç and M. Gözen, “Stability and uniform boundedness in multidelay functional differential equations of third order,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 248717, 7 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

*
*