# A New Instability Result to Nonlinear Vector Differential Equations of Fifth Order

Authors: Cemil Tunç; Melike Karta
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

By constructing a Lyapunov function, a new instability result is established,
which guarantees that the trivial solution of a certain nonlinear vector differential equation of
the fifth order is unstable. An example is also given to illustrate the importance of the result
obtained. By this way, our findings improve an instability result related to a scalar differential
equation in the literature to instability of the trivial solution to the afore-mentioned differential
equation.


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## Body

## 1. Introduction

In 1990, Li and Yu [1]
investigated the instability of trivial solution to fifth-order nonlinear scalar
differential equation (1.1)x(5)+ax(4)+bx⃛+ψ(x,x˙,x¨,x⃛,x(4))x¨+g(x)x˙+f(x)=0 by
introducting  a Lyapunov function, where a and b are some positive constants.In this paper, based on the result of Li
and Yu [1], we are concerned with the instability of trivial solution to
fifth-order nonlinear vector differential equation described by (1.2)X(5)+AX(4)+BX⃛+Ψ(X,X˙,X¨,X⃛,X(4))X¨+G(X)X˙+F(X)X=0 in the real
Euclidean space ℜn (with the usual norm denoted in what follows
by ∥⋅∥), where X∈ℜn;A and B are constant n×n-symmetric matrices; Ψ,G, and F are n×n-symmetric continuous matrix functions depending,
in each case, on the arguments shown. Throughout this paper, we consider,
instead of (1.2), the equivalent differential system (1.3)X˙=Y,Y˙=Z,Z˙=W,W˙=U,U˙=−AU−BW−Ψ(X,Y,Z,W,U)Z−G(X)Y−F(X)X, which was obtained
as usual by setting X˙=Y,X¨=Z,X⃛=W,X(4)=U from (1.2). For the sake of brevity, we assume that the symbol JG(X) denotes the Jacobian matrix (1.4)JG(X)=(∂gi∂xj)(i,j=1,2,…,n), where (x1,x2,…,xn) and (g1,g2,…,gn) are components
of X and G,
respectively. In addition, it is assumed, as basic throughout the paper, that
the Jacobian matrix JG(X) exists and is continuous and symmetric. The symbol 〈X,Y〉 corresponding to any pair X,Y in ℜn stands for the usual scalar product ∑i=1nxiyi,
and λi(A),(A=(aij))(i,j=1,2,…,n)
are the eigenvalues of the n×n-symmetric matrix A,
and the matrix A=(aij) is said to be positive definite if and only if
the quadratic form XTAX is positive definite, where X∈ℜn and XT denotes the transpose of X.At the same time, up to now, we should also recognize that some significant
theoretical results related to instability  of trivial solution of  some nonlinear scalar and vector differential
equations of fifth order have been achieved in the literature, see,  for example, the papers of  Ezeilo [2, 3], Tunç [4, 5], and the
references registered in these papers. However, it should be noted that nearly
all of the papers have been published on the subject without including any
example related to the topic. The equation considered that the assumptions and
Lyapunov [6] function that will be established here are completely different than
those mentioned in the literature.


## 2. Main Result

Our main result is the following theorem.Theorem 2.1.
In addition to the
basic assumptions imposed onA,B,Ψ,G, and F appearing  in (1.2), we assume there are constants a,b and a positive constant k1 such
that  the following conditions hold:
(i)
λi(A)≥a,λi(B)≥b,bsgna>0;(ii)
λi(F(X))sgna−(1/4|a|)(λi(Ψ(X,Y,Z,W,U))2>k1(i=1,2,…,n). Then the
trivial solution X=0 of (1.2) is unstable.Now, in order to prove our main result, we give
a well-known lemma which plays an essential role throughout the proof of
theorem.Remark 2.2.
It should be noted that there
is no restriction on symmetric matrixG appearing in (1.2).Lemma 2.3.
LetA be a real symmetric n×n-matrix and (2.1)a′≥λi(A)≥a>0(i=1,2,…,n), where a′,a are constants.
Then(2.2)a′〈X,X〉≥〈AX,X〉≥a〈X,X〉,a′2〈X,X〉≥〈AX,AX〉≥a2〈X,X〉.Proof.
See [7].Proof  of Theorem2.1.
To achieve
the proof of theorem, we define the following Lyapunov functionV:=V(X,Y,Z,W,U): (2.3)V=V0(X,Y,Z,W,U)sgna, where (2.4)V0=〈Y,W〉+〈Y,AZ〉−〈X,U〉−〈X,AW〉−〈X,BZ〉−12〈Z,Z〉+12〈BY,Y〉−∫01〈σG(σX)X,X〉dσ.
Now, under the assumptions of the theorem, it will be shown that the Lyapunov
functionV=V(X,Y,Z,W,U) satisfies the entire Krasovskiĭ
[8] criteria.
(K1) In every neighborhood of (0,0,0,0,0),
there exists a point (ξ,η,ζ,μ,ρ) such that V(ξ,η,ζ,μ,ρ)>0.
(K2) The time derivative V˙=(d/dt)V(X,Y,Z,W,U) along solution paths of system (1.3) is positive
semidefinite.
(K3) The only solution (X,Y,Z,W,U)=(X(t),Y(t),Z(t),W(t),U(t)) of system (1.3) which satisfies V˙=0(t≥0) is the trivial solution (0,0,0,0,0).
These properties guarantee that the trivial solution of (1.2) is
unstable.
First, it is easy to see from (2.3) and (2.4) that (2.5)V(0,0,0,0,0)=V0(0,0,0,0,0)sgna=0. Next, evidently,
one can easily get (2.6)V(0,εsgna,0,ε,0)=〈ε,ε〉+12bsgna〈ε,ε〉=∥ε∥2+12bsgna∥ε∥2>0 for all
arbitrary ε≠0,ε∈ℜn.
Finally, let(X,Y,Z,W,U)=(X(t),Y(t),Z(t),W(t),U(t)) be an arbitrary solution of system (1.3). Differentiating (2.4)
with respect
to t, along this
solution, calculations give that (2.7)V˙0=ddtV0(X,Y,Z,W,U)=〈AZ,Z〉+〈F(X)X,X〉+〈X,Ψ(X,Y,Z,W,U)Z〉+〈G(X)Y,X〉−ddt∫01〈σG(σX)X,X〉dσ. Now, recall that (2.8)ddt∫01〈σG(σX)X,X〉dσ=∫01〈σG(σX)Y,X〉dσ+∫01σ∂∂σ〈σG(σX)Y,X〉dσ=σ2〈G(σX)Y,X〉|01=〈G(X)Y,X〉. Substituting (2.8) into (2.7), we
obtain (2.9)V˙0=〈AZ,Z〉+〈F(X)X,X〉+〈X,Ψ(X,Y,Z,W,U)Z〉. Now, it follows from (2.3) and (2.9) that (2.10)V˙=sgna〈Z,AZ〉+sgna〈X,F(X)X〉+sgna〈X,Ψ(X,Y,Z,W,U)Z〉≥|a|〈Z,Z〉+sgna〈X,F(X)X〉+sgna〈X,Ψ(X,Y,Z,W,U)Z〉=|a|∥Z+12|a|Ψ(X,Y,Z,W,W,U)Xsgna∥2+[sgna〈X,F(X)X〉−14|a|〈Ψ(X,Y,Z,W,U)X,Ψ(X,Y,Z,W,U)X〉]≥[sgna〈X,F(X)X〉−14|a|〈Ψ(X,Y,Z,W,U)X,Ψ(X,Y,Z,W,U)X〉]≥k1∥X∥2≥0, by (i) and (ii).
Thus, the assumptions of theorem show thatV˙(t)≥0 for all t≥0,
that is, V˙ is positive semidefinite. Furthermore, V˙=0(t≥0) necessarily implies that X=0 for all t≥0,
and Z=Y˙=0,W=Y¨=0,W˙=Y⃛=0 for all t≥0.
Thus, it follows the estimates (2.11)X=Y=Z=W=U=0. Therefore, the
function V has the entire criteria of Krasovskiĭ [8] if
the assumptions of theorem hold. Thus, the basic properties of the
function V(X,Y,Z,W,U),
which are proved above, verify that the trivial solution of  system (1.3) is unstable. The system of (1.3) is equivalent to the differential equation (1.2).
Hence,
this fact completes the proof of theorem.Example 2.4.
As a special case of system
(1.3), let us choose for n=2,A,B,Ψ, and F: (2.12)A=[3113],B=[4224],Ψ(X,Y,Z,W,U)=[2+(1+x2+y2+z2+w2+u2)−1112+(1+x2+y2+z2+w2+u2)−1],F(X)=[20+11+x21120+11+x2].
Then,
respectively, we get(2.13)λ1(A)=2,λ2(A)=4,λ1(B)=2,λ2(B)=6,λ1(Ψ(X,Y,Z,W,U))=1+11+x2+y2+z2+w2+u2,λ2(Ψ(X,Y,Z,W,U))=3+11+x2+y2+z2+w2+u2,λi(Ψ(X,Y,Z,W,U))≥1,λ1(F(X))=19+11+x2,λ2(F(X))=21+11+x2. Now, taking into account this
facts, it can be easily seen that (2.14)λi(A)≥2=a,λi(B)≥2=b,bsgna>0,λi(F(X))sgn≥19,(i=1,2),(19+11+x2)sgn⁡2−18(1+11+x2+y2+z2+w2+u2)2>0,(19+11+x2)sgn⁡2−18(3+11+x2+y2+z2+w2+u2)2>0,(21+11+x2)sgn⁡2−18(1+11+x2+y2+z2+w2+u2)2>0,(21+11+x2)sgn⁡2−18(3+11+x2+y2+z2+w2+u2)2>0. Clearly, these last four expressions imply that (2.15)λi(F(X))sgna−14|a|(λiΨ(X,Y,Z,W,U))2>0(i=1,2).Thus, it is shown
that all the assumptions of the theorem hold.



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*Source: 971534-2008-09-01.xml*