Discrete Dynamics in Nature and Society

Volume 2008 (2008), Article ID 971534, 6 pages

http://dx.doi.org/10.1155/2008/971534

## A New Instability Result to Nonlinear Vector Differential Equations of Fifth Order

^{1}Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yıl University, 65080 Van, Turkey^{2}Institute of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey

Received 20 May 2008; Accepted 9 July 2008

Academic Editor: Binggen Zhang

Copyright © 2008 Cemil Tunç and Melike Karta. 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

By constructing a Lyapunov function, a new instability result is established, which guarantees that the trivial solution of a certain nonlinear vector differential equation of the fifth order is unstable. An example is also given to illustrate the importance of the result obtained. By this way, our findings improve an instability result related to a scalar differential equation in the literature to instability of the trivial solution to the afore-mentioned differential equation.

#### 1. Introduction

In 1990, Li and Yu [1] investigated the instability of trivial solution to fifth-order nonlinear scalar differential equation by introducting a Lyapunov function, where and are some positive constants.

In this paper, based on the result of Li and Yu [1], we are concerned with the instability of trivial solution to fifth-order nonlinear vector differential equation described by in the real Euclidean space (with the usual norm denoted in what follows by ), where and are constant -symmetric matrices; and are -symmetric continuous matrix functions depending, in each case, on the arguments shown. Throughout this paper, we consider, instead of (1.2), the equivalent differential system which was obtained as usual by setting from (1.2). For the sake of brevity, we assume that the symbol denotes the Jacobian matrix where and are components of and , respectively. In addition, it is assumed, as basic throughout the paper, that the Jacobian matrix exists and is continuous and symmetric. The symbol corresponding to any pair in stands for the usual scalar product , and are the eigenvalues of the -symmetric matrix , and the matrix is said to be positive definite if and only if the quadratic form is positive definite, where and denotes the transpose of .

At the same time, up to now, we should also recognize that some significant theoretical results related to instability of trivial solution of some nonlinear scalar and vector differential equations of fifth order have been achieved in the literature, see, for example, the papers of Ezeilo [2, 3], Tunç [4, 5], and the references registered in these papers. However, it should be noted that nearly all of the papers have been published on the subject without including any example related to the topic. The equation considered that the assumptions and Lyapunov [6] function that will be established here are completely different than those mentioned in the literature.

#### 2. Main Result

Our main result is the following theorem.

Theorem 2.1. *In addition to the
basic assumptions imposed on and appearing in (1.2), we assume there are constants and a positive constant such
that the following conditions hold:*

(i)*;*(ii)*.** Then the
trivial solution of (1.2) is unstable.*

Now, in order to prove our main result, we give a well-known lemma which plays an essential role throughout the proof of theorem.

*Remark 2.2. *It should be noted that there
is no restriction on symmetric matrix appearing in (1.2).

Lemma 2.3. *Let be a real symmetric -matrix and ** where are constants.**Then *

*Proof. *See [7].

*Proof of Theorem 2.1. *To achieve
the proof of theorem, we define the following Lyapunov function : where

Now, under the assumptions of the theorem, it will be shown that the Lyapunov
function satisfies the entire Krasovski*ĭ*
[8] criteria.

In every neighborhood of ,
there exists a point such that .

The time derivative along solution paths of system (1.3) is positive
semidefinite.

The only solution of system (1.3) which satisfies is the trivial solution .
These properties guarantee that the trivial solution of (1.2) is
unstable.

First, it is easy to see from (2.3) and (2.4) that Next, evidently,
one can easily get for all
arbitrary .

Finally, let be an arbitrary solution of system (1.3). Differentiating (2.4)
with respect
to , along this
solution, calculations give that Now, recall that Substituting (2.8) into (2.7), we
obtain Now, it follows from (2.3) and (2.9) that by (i) and (ii).

Thus, the assumptions of theorem show that for all ,
that is, is positive semidefinite. Furthermore, necessarily implies that for all ,
and for all .
Thus, it follows the estimates Therefore, the
function has the entire criteria of Krasovski*ĭ* [8] if
the assumptions of theorem hold. Thus, the basic properties of the
function ,
which are proved above, verify that the trivial solution of system (1.3) is unstable. The system of (1.3) is equivalent to the differential equation (1.2).

Hence,
this fact completes the proof of theorem.

*Example 2.4. *As a special case of system
(1.3), let us choose for , and :

Then,
respectively, we get Now, taking into account this
facts, it can be easily seen that Clearly, these last four expressions imply that

Thus, it is shown that all the assumptions of the theorem hold.

#### References

- W. J. Li and Y. H. Yu, “Instability theorems for some fourth-order and fifth-order differential equations,”
*Journal of Xinjiang University. Natural Science*, vol. 7, no. 2, pp. 7–10, 1990 (Chinese). View at Google Scholar · View at MathSciNet - J. O. C. Ezeilo, “Instability theorems for certain fifth-order differential equations,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 84, no. 2, pp. 343–350, 1978. View at Google Scholar · View at MathSciNet - J. O. C. Ezeilo, “A further instability theorem for a certain fifth-order differential equation,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 86, no. 3, pp. 491–493, 1979. View at Google Scholar · View at MathSciNet - C. Tunç, “On the instability of solutions of certain nonlinear vector differential equations of fifth order,”
*PanAmerican Mathematical Journal*, vol. 14, no. 4, pp. 25–30, 2004. View at Google Scholar · View at MathSciNet - C. Tunç, “Further results on the instability of solutions of certain nonlinear vector
differential equations of fifth order,”
*Applied Mathematics & Information Sciences*, vol. 2, no. 1, pp. 51–60, 2008. View at Google Scholar · View at MathSciNet - A. M. Lyapunov,
*Stability of Motion*, Academic Press, New York, NY, USA, 1966. View at MathSciNet - R. A. Horn and C. R. Johnson,
*Topics in Matrix Analysis*, Cambridge University Press, Cambridge, UK, 1994. View at MathSciNet - N. N. Krasovskiĭ, “On conditions of inversion of A. M. Lyapunov's theorems on instability for
stationary systems of differential equations,”
*Doklady Akademii Nauk SSSR*, vol. 101, pp. 17–20, 1955 (Russian). View at Google Scholar · View at MathSciNet