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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 971534, 6 pages
http://dx.doi.org/10.1155/2008/971534
Research Article

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

1Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yıl University, 65080 Van, Turkey
2Institute 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 Krasovskiĭ [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 Krasovskiĭ [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

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  2. 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
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