In 2008, E. Tunç and C. Tunç [1] proved a theorem on the instability of the zero solution of the sixth order nonlinear vector differential equation

The objective of this article is to investigate the instability of the zero solution of the sixth order nonlinear vector differential equation with constant delay,

by the Lyapunov-Krasovskii functional approach under assumptions A and B are constant - symmetric matrices; E, F and G are continuous - symmetric matrix functions depending, in each case, on the arguments shown; and H is continuous. Let denote the Jacobian matrix corresponding to that is,

where and are the components of X and H, respectively. We also assume that the Jacobian matrix exists and is continuous.

It should be noted that Equation (2) is the vector version for systems of real nonlinear differential equations of the sixth order

We can write Equation (2) in the system form

which is obtained from (2) by setting and Throughout what follows are abbreviated as respectively. 

Consider, in the case the linear differential equation of the sixth order:

where are real constants.

It is known from the qualitative properties of solutions of Equation (4) that the zero solution of this equation is unstable if and only if the associated auxiliary equation

has at the least one root with a positive real part. The existence of such a root depends on (though not always all of) the coefficients in Equation (5). Basing on the relations between the roots and the coefficients of Equation (5) it can be said that if

or

then at the least one root of Equation (5) has a positive real part for arbitrary values of and or and respectively.

It should be noted that Equation (2) is an n-dimensional generalization of Equation (4), and when we establish our assumptions, we will take into consideration the estimates in (6). The symbol corresponding to any pair X, Y in stands for the usual scalar product and are the eigenvalues of the -matrix

It is worth mentioning that using the Lyapunov functions or Lyapunov-Krasovskii functionals and based on the Krasovskii properties [2], the instability of the solutions of the sixth order nonlinear scalar differential equations and the sixth order vector differential equations without delay were discussed by Ezeilo [3], Tejumola [4], Tiryaki [5] and Tunç [6-13]. The aim of this paper is to improve the results of ([1,3]) form the scalar and vector differential equations without delay to the sixth order nonlinear vector differential equation with delay, Equation (2).

First, we give an algebraic result.

Lemma. Let D be a real symmetric -matrix. Then for any

where and are the least and greatest eigenvalues of respectively (Bellman [14]).

Let be given, and let with

For define by

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

Let G be an open subset of C 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 F that each initial value problem

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

Definition. 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 result to be proved is the following theorem.

Theorem. In addition to the basic assumptions imposed on A, B, E, F, G and H that appear in Equation (2), we suppose that there are constants and such that the following conditions hold:

The matrices A, B, E, F, G and are symmetric and when and

If

then the zero solution of Equation (2) is unstable.

Remark. It is worth mentioning that there is no sign restriction on eigenvalues of F, and it is obvious that for the delay case our assumptions also have a very simple form and their applicability can be easily verified.

Proof. Define a Lyapunov-Krasovskii functional 

where

where is a certain positive constant and will be determined later in the proof. 

It follows that

and

for all arbitrary so that the property of Krasovskii [2] holds.

Using a basic calculation, the time derivative of along solutions of (3) results in

The following estimates can be easily calculated:

and

so that

Using the assumptions of the theorem, we get

Let

so that

If then, for a positive constant we have

so that the property of Krasovskii [2] holds.

It is seen that

so that

Using these estimates in (3) and the assumptions of the theorem, we get Thus, we have

for all So that the property of Krasovskii [2] holds.

The proof of the theorem is complete.

Example. For the particular case in Equation (2), we have

If

then all the assumptions of the theorem hold.