An Instability Result to a Certain Vector Differential Equation of the Sixth Order

The nonlinear vector differential equation of the sixth order with constant delay is considered in this article. New criteria for instability of the zero solution are established using the Lyapunov-Krasovskii functional approach and the differential inequality techniques. The result of this article improves previously known results.


Introduction
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 , , , , , , 0 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, 0, by the Lyapunov-Krasovskii functional approach under assumptions A and B are constant ; , where   , , , n x x x  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 (5) (5) , .
We can write Equation (2) in the system form and . ( 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 (5) 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 1 2 6 in Equation ( 5).Basing on the relations between the roots and the coefficients of Equation ( 5) it can be said that if , , , a a a  then at the least one root of Equation ( 5) has a positive real part for arbitrary values of and or and respectively.
1 , a 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 , a , X Y corresponding to any pair X, Y in stands for the usual scalar prod- A 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 were discussed by Ezeilo [3], Tejumola [4], Tiryaki [5] and Tunç [6][7][8][9][10][11][12][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).

Main Result
First, we give an algebraic result.
Lemma.Let D be a real symmetric -matrix.Then for any Let G be an open subset of C and consider the general autonomous delay differential system with finite delay

    
, 0 0, 0, 0, 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   0, , A 0 A .
   This solution will be denoted by    .
x  so that  for all The zero solution is said to be unstable if it is not stable.

t 
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 1 and , a 2 , a 6 a  such that the following conditions hold: The matrices A, B, E, F, G and   H J X are symmetric and   . sgn ( .0, 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.
Using a basic calculation, the time derivative of along solutions of (3) results in The following estimates can be easily calculated: Using the assumptions of the theorem, we get Using these estimates in (3) and the assumptions of the theorem, we get for all So that the property 0. t   3 P of Krasovskii [2] holds.The proof of the theorem is complete.Example.For the particular case in Equation (2), we have x t H X t x t   then all the assumptions of the theorem hold.
E, F and G are continuous n n symmetric matrix functions depending, in each case, on the arguments shown; and H is continuous.Let : where d  and d are the least and greatest eigenval-