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International Journal of Analysis
Volume 2013 (2013), Article ID 973917, 5 pages
http://dx.doi.org/10.1155/2013/973917
Research Article

On the Unstable Solutions to Functional Vector Differential Equations of the Seventh Order

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

Received 25 August 2012; Accepted 16 September 2012

Academic Editor: Yongsheng S. Han

Copyright © 2013 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 studies the instability of the zero solution for a certain nonlinear functional vector differential equation of the seventh order with multiple deviating arguments. Under sufficient conditions, we prove a result on the instability of the zero solution. This work contributes and complements to previously known results in the literature.

1. Introduction

More than 100 years ago, the world famous mathematician Lyapunov established the Lyapunov direct method to study stability problems. From then on, the Lyapunov’s direct method was also widely used to study the instability of solutions of ordinary differential equations and functional differential equations, see for example, Bereketoğlu [1], Sadek [2], Tejumola [3], Tunç [47], C. Tunç and E. Tunç [8], and the references therein. However, a review to date of the literature indicates that the instability of the solutions to the nonlinear functional vector differential equation of the seventh order has not been investigated. This paper is the first work on the subject. It is also known that the expressions of Lyapunov-Krasovskii functional are very complicated and hard to construct. In this paper, we define a Lyapunov-Krasovskii functional [9] and base on the Krasovskii criteria to prove a new theorem on the topic for the nonlinear functional vector differential equation of the seventh order. In this paper, intend to make a contribution to the subject since the functional differential equations have an important place in various fields of science and engineering.

Meanwhile, some respective contributions on the topic can be summarized as follows.

In 2000, Tejumola [3] discussed the instability of the zero solution of the seventh order scalar nonlinear differential equation without a deviating argument

Later, in 2003, Sadek [2] proved an instability theorem for the seventh order scalar nonlinear differential equation without a deviating argument

Recently, Tunç [6, 7] discussed the same problem for the seventh order scalar nonlinear delay differential equations of the form:

In this paper, instead of (4), we consider the seventh order nonlinear functional vector differential equation with -deviating arguments, , (): Setting , , , , , , , we can write (5) in system form as follows where are certain positive constants, the fixed delays, , , ,  and are constant -symmetric matrices, the primes in (5) denote differentiation with respect to , , ; , , and are continuous -symmetric matrix functions for the arguments displayed explicitly, , , , , and are continuous for all the respective arguments. The Jacobian matrices of are given by where and are the components of and , respectively. It is also assumed that the Jacobian matrices exist and are continuous. The existence and uniqueness of the solutions of (5) are assumed (see Èl’sgol’ts [10, pages 14, 15]). Throughout what follows that are abbreviated as , respectively.

Let be given, and let with

For define by

If is continuous, , then, for each in in is defined by

Let be an open subset of and consider the general autonomous delay differential system where is continuous and maps closed and bounded sets into bounded sets. It follows from the conditions on that each initial value problem has a unique solution defined on some interval , . This solution will be denoted by so that .

Definition 1. The zero solution, , of is stable if for each there exists such that implies that for all . The zero solution is said to be unstable if it is not stable.

Consider the linear constant coefficient differential equation of the seventh order: It is known from the qualitative behavior of solutions of linear differential equations that the zero solution of (13) is unstable if and only if, the associated auxiliary equation: has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients in (14). For example, if then it is clear from a consideration of the fact that the sum of the roots of (14) equals and that at the least one root of (14) has a positive real part for arbitrary values of , , , , , and . An analogue consideration, combined with the fact that the product of the roots (14) equals () will verify that at least one root of (14) have a positive real part if for arbitrary , , , and . The condition here in (16) is, however, superfluous when then for and if is sufficiently large, thus showing that there is a positive real root of (14) subject to (17) and for arbitrary , , , , and . Moreover, a necessary and sufficient condition for (14) to has a purely imaginary root ( real) is that the two equations are satisfied at the same time. If or then (14) cannot have any purely imaginary root whatever.

It should be noted that there are no restrictions on the constants , , and in (13).

2. Main Result

First, we give the following lemma.

Lemma 2. Let be a real symmetric -matrix and where and are constants, and are the eigenvalues of the matrix .
Then (Bellman [11]).

Let .

The following theorem is our main result.

Theorem 3. In addition to all the assumptions imposed to , , , , , , and that appear in (5), we assume that there exist constants , , , and such that the conditions hold. If , then the zero solution of (5) is unstable.

Remark 4. For the proof of the theorem, under the conditions sated in the theorem, it suffices to find that there exists a continuous Lyapunov-Krasovskii functional , which has the Krasovskii properties [9], say , , and . In every neighborhood of there exists a point such that . The time derivative along solution paths of (6) is positive semidefinite. The only solution of (6) which satisfies , is the trivial solution .

Proof. Consider the Lyapunov-Krasovskii functional defined by where is a real variable such that the integrals are nonnegative, and are certain positive constants to be determined later in the proof.
We see that
Since , , then Let Then, we get for all , so that every neighborhood of the origin in the -space contains points such that .
Let be an arbitrary solution of (6).
Calculating the time derivative of the Lyapunov-Krasovskii functional along this solution, we obtain Using the assumption and the Schwarz inequality, we have Hence
Let . Then, using the assumptions of the theorem and the estimate we have
If , then, for some positive constant , we have
Finally, for all necessarily implies that . Hence, it follows that for all so that Therefore, the estimates and (6) imply since if and only if . It now follows that functional thus has all the requisite Krasovskii [9] properties subject to the conditions of the theorem. Hence, we can conclude that the zero solution of (5) is unstable.
The proof of the theorem is completed.

References

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