An Instability Result to a Certain Vector Differential Equation of the Sixth Order ()
1. 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
(1)
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, ![](https://www.scirp.org/html/6-7401050\6a0d506f-d381-44d7-ab3f-e94be999e377.jpg)
(2)
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; ![](https://www.scirp.org/html/6-7401050\1a59a57f-f590-496a-91a9-bf62364e2505.jpg)
and H is continuous. Let
denote the Jacobian matrix corresponding to
that is,
![](https://www.scirp.org/html/6-7401050\681170b3-4553-40d0-86b7-b320db333561.jpg)
![](https://www.scirp.org/html/6-7401050\c2dc52b1-577e-48fa-a475-ede12265292b.jpg)
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
![](https://www.scirp.org/html/6-7401050\8fb725ec-4961-4090-8007-e26db119f006.jpg)
We can write Equation (2) in the system form
(3)
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:
(4)
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
in Equation (5). Basing on the relations between the roots and the coefficients of Equation (5) it can be said that if
![](https://www.scirp.org/html/6-7401050\ead4ac2e-ce57-41fd-978b-1ba1ae62e950.jpg)
or
(6)
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 ![](https://www.scirp.org/html/6-7401050\63d2602c-ffc0-4c2a-93d2-5750acb9f6b3.jpg)
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).
2. Main Result
First, we give an algebraic result.
Lemma. Let D be a real symmetric
-matrix. Then for any ![](https://www.scirp.org/html/6-7401050\350b76f2-00ac-4f0e-8105-8e78c3e9b3b4.jpg)
![](https://www.scirp.org/html/6-7401050\b3bb5cfb-9185-484e-9b8e-ebd4e4160128.jpg)
where
and
are the least and greatest eigenvalues of
respectively (Bellman [14]).
Let
be given, and let
with
![](https://www.scirp.org/html/6-7401050\c752f205-5add-4183-9595-6e811d931fff.jpg)
For
define
by
![](https://www.scirp.org/html/6-7401050\3125945b-6fe7-4eb0-929f-368140348824.jpg)
If
is continuous,
then, for each t in
in C is defined by
![](https://www.scirp.org/html/6-7401050\a5fce1a2-b620-4db1-84b2-1cf0107626ea.jpg)
Let G be an open subset of C and consider the general autonomous delay differential system with finite delay
![](https://www.scirp.org/html/6-7401050\fd3604d5-4b80-43ed-a3d5-4147ccac906b.jpg)
where
is continuous and maps closed and bounded sets into bounded sets. It follows from these conditions on F that each initial value problem
![](https://www.scirp.org/html/6-7401050\bdffa095-b0ca-41e4-803f-4e2fe738994f.jpg)
![](https://www.scirp.org/html/6-7401050\d2436adf-5e77-42d2-9903-66d801f12b5a.jpg)
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
![](https://www.scirp.org/html/6-7401050\314165ac-5ca6-49f9-9a2e-61c1cafe4f7a.jpg)
![](https://www.scirp.org/html/6-7401050\27a981e9-0385-4f7f-843b-c14c6bb00cc5.jpg)
If
![](https://www.scirp.org/html/6-7401050\77157b11-f133-4248-b45d-0c0a8371d39b.jpg)
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
![](https://www.scirp.org/html/6-7401050\87588332-d623-4c10-8a02-96bd22da5b39.jpg)
![](https://www.scirp.org/html/6-7401050\54a0a274-fe98-4d76-b54d-4ac8c36534cf.jpg)
where
![](https://www.scirp.org/html/6-7401050\df7adf84-bca5-443a-9f0e-37930fe9c6f8.jpg)
where
is a certain positive constant and will be determined later in the proof.
It follows that
![](https://www.scirp.org/html/6-7401050\8ea97c96-b9f0-4455-98a2-8baeaca90fa0.jpg)
and
![](https://www.scirp.org/html/6-7401050\ac7abdca-f7ac-4098-8d03-8b540f8d2cb2.jpg)
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
![](https://www.scirp.org/html/6-7401050\3fcd1410-275e-4f0d-89f5-9024d0550152.jpg)
The following estimates can be easily calculated:
![](https://www.scirp.org/html/6-7401050\405886d8-8d5e-4025-a3b1-7738c833c2cc.jpg)
![](https://www.scirp.org/html/6-7401050\6c0ae7d8-c0fa-43ed-89a7-662da9d9737a.jpg)
![](https://www.scirp.org/html/6-7401050\17b02bfb-a0b1-4670-ac15-e0e298d943b0.jpg)
and
![](https://www.scirp.org/html/6-7401050\3bbc446d-1f7c-4512-89a4-caeb520f184c.jpg)
so that
![](https://www.scirp.org/html/6-7401050\d62ebe56-32f0-4887-8d28-3728b9136467.jpg)
Using the assumptions of the theorem, we get
![](https://www.scirp.org/html/6-7401050\b8772056-ecb5-4d00-83ac-9c8aad0dfb65.jpg)
Let
![](https://www.scirp.org/html/6-7401050\b7a42615-5c8c-488e-83af-eb4354093e63.jpg)
so that
![](https://www.scirp.org/html/6-7401050\5d7506be-e1e4-45a8-a88f-114ab88fd8d8.jpg)
If
then, for a positive constant
we have
![](https://www.scirp.org/html/6-7401050\96a6a07e-c694-4541-abd5-1c92d11284e8.jpg)
so that the property
of Krasovskii [2] holds.
It is seen that
![](https://www.scirp.org/html/6-7401050\feda76eb-13ce-45c6-931d-603c0fb6831f.jpg)
so that
![](https://www.scirp.org/html/6-7401050\104b2bf7-3a89-45fe-a8c7-052a5250c37a.jpg)
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
![](https://www.scirp.org/html/6-7401050\e075b296-14b4-4f5a-8b8b-e027c695125d.jpg)
![](https://www.scirp.org/html/6-7401050\7b74d32a-e307-4bc6-9532-6dccd52132ec.jpg)
![](https://www.scirp.org/html/6-7401050\d688415d-6ee1-405c-8639-dfe4142217a4.jpg)
![](https://www.scirp.org/html/6-7401050\e0246bdf-cff1-47c3-aa6c-d92b5919578a.jpg)
![](https://www.scirp.org/html/6-7401050\9b699d8a-b0bb-4df1-b610-468269167580.jpg)
![](https://www.scirp.org/html/6-7401050\f6b3c069-c82a-4299-b9cf-29f15fb31930.jpg)
![](https://www.scirp.org/html/6-7401050\f9691b63-75c5-49f1-bbb4-407d3fa2420d.jpg)
![](https://www.scirp.org/html/6-7401050\5d491f3e-fbc9-4e68-9d6b-b29447460c4e.jpg)
![](https://www.scirp.org/html/6-7401050\3d273aba-e100-40ea-a399-0097e4c1367d.jpg)
![](https://www.scirp.org/html/6-7401050\606225b4-3128-4d3d-ab1c-61e741021aff.jpg)
![](https://www.scirp.org/html/6-7401050\e7f615c4-e43d-427d-810a-9c3f441d2944.jpg)
![](https://www.scirp.org/html/6-7401050\6e5d1849-01cc-4d2f-a476-4c13233e6796.jpg)
![](https://www.scirp.org/html/6-7401050\99aee07f-45d0-4909-bc66-57f8d8235bfd.jpg)
![](https://www.scirp.org/html/6-7401050\cdafa401-2371-4924-8e2a-e8760cf2960c.jpg)
![](https://www.scirp.org/html/6-7401050\a91dea94-5e14-4f51-924d-82ca2e83a67c.jpg)
![](https://www.scirp.org/html/6-7401050\1b00ca5d-6867-43dc-905d-7110ffe9c16f.jpg)
![](https://www.scirp.org/html/6-7401050\3dd2d053-f5de-4024-b94a-f3b8c1984b10.jpg)
If
![](https://www.scirp.org/html/6-7401050\c614f44f-b674-448c-8745-ff421735d92e.jpg)
then all the assumptions of the theorem hold.