Journal of Mathematics

Volume 2013 (2013), Article ID 809748, 5 pages

http://dx.doi.org/10.1155/2013/809748

Research Article

Instability of a Fifth-Order Nonlinear Vector Delay Differential Equation with Multiple Deviating Arguments

Cemil Tunç

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

Received 30 August 2012; Accepted 25 September 2012

Academic Editor: Beny Neta

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.

We study a fifth-order nonlinear vector delay differential equation with multiple deviating arguments. Some criteria for guaranteeing the instability of zero solution of the equation are given by using the Lyapunov-Krasovskii functional approach. Comparing with the previous literature, our result is new and complements some known results.

1. Introduction

Lyapunov functions and functionals have been successfully used and are still being used to obtain stability, instability, boundedness, and the existence of periodic solutions of differential equations, differential equations with functional delays, and functional differential equations [1–18]).

It should be noted that in 2000, using the Lyapunov function approach, Li and Duan [2] discussed the instability of the solutions of the fifth-order nonlinear scalar differential equation without delay:

Later, in 2011, using the same technique, Tunç [12] studied the instability of the solutions for the fifth-order nonlinear scalar delay differential equation:

In this paper, we consider the fifth-order nonlinear vector delay differential equation with multiple-deviating arguments:

where the primes in (3) denote differentiation with respect to , , ; , are certain positive constants, the fixed delays, is a constant -symmetric matrix, and are continuous -symmetric matrix functions for the respective arguments, with are continuous for the respective arguments. Let and denote the linear operators from and to

where , , , and are the components of , , , and , respectively. In what follows, it is assumed that and exist and are symmetric and continuous.

Equation (3) is the vector version for systems of real nonlinear differential equations of the fifth order:

Instead of (3), we consider the corresponding differential system

which was obtained by setting , and from (3).

However, a review to date of the literature indicates that the instability of solutions of vector differential equations of the fifth order with a deviating argument has not been investigated up to now. This paper is the first known work regarding the instability of solutions for the nonlinear vector delay differential equations of the fifth order with multiple-deviating arguments. The motivation of this paper comes from the above papers done on scalar nonlinear differential equations of the fifth order without and with delay and the vector differential equations of the fifth order without delay. By this work, we improve the results in [2, 12] to a vector delay differential equation of the fifth order with multiple-deviating arguments. Defining the Lyapunov-Krasovskii functional and taking into account the Krasovskiĭ’s criteria [19], we prove our main result on the subject. The result to be obtained is new, has a contribution to the topic, and may be useful for the researchers working on the qualitative behaviors of solutions of the differential equations.

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.

The symbol corresponding to any pair in stands for the usual scalar product , that is, =; thus and are the eigenvalues of the real symmetric -matrix . The matrix is said to be negative-definite, when for all nonzero in .

2. Main Result

Before stating the main result, we need the following result.

Lemma 2. Let be a real symmetric -matrix and

where and are constants.

Then

See [20].

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 with finite delay:

where is continuous and maps closed and bounded sets into bounded sets. It follows from these conditions on that each initial value problem,

has a unique solution defined on some interval . This solution will be denoted by so that .

The main result of this paper is given by the following theorem.

Theorem 3. In addition to the basic assumptions imposed on , and that appear in (3), we assume that there exist positive constants , and such that the following conditions hold:

, and are symmetric,

If

then the zero solution of (3) is unstable.

Proof. We define the Lyapunov-Krasovskii functional

where are certain positive constants and will be determined later in the proof.

It is clear that .

Let

Using the assumption , we have

for all arbitrary , , which verifies the property of Krasovskiĭ [19].

Using a basic calculation, the time derivative of along solutions of (6) results in

Under the assumptions of the theorem and Schwarz inequality, it can be easily seen that

Hence

Let . Then, using the assumptions of the theorem and the estimate we get

If , then, for some positive constant we have

which verifies that has the property of Krasovskiĭ [19].

Besides,

The substitution of this estimate into system (6) results in

That is,

Because    constant vector, for all Hence, since is not the zero matrix, we have constant vector, for all . But, in view of the assumptions of the theorem, this implies that and thus also, by  , that ,  . These estimates result in . Hence, the property of Krasovskiĭ [19] holds for the Lyapunov-Krasovskii functional

The proof of the theorem is completed.

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