# On the Unstable Solutions to Functional Vector Differential Equations of the Seventh Order

Authors: Cemil Tunç
Journal: International Journal of Analysis
 (2013)

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## Abstract

This paper studies the instability of the zero solution for a certain nonlinear functional vector differential equation of the seventh order with multiple deviating arguments. Under sufficient conditions, we prove a result on the instability of the zero solution. This work contributes and complements to previously known results in the literature.


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## Body

## 1. Introduction

More than 100 years ago, the world famous mathematician Lyapunov established the Lyapunov direct method to study stability problems. From then on, the Lyapunov’s direct method was also widely used to study the instability of solutions of ordinary differential equations and functional differential equations, see for example, Bereketoğlu [1], Sadek [2], Tejumola [3], Tunç [4–7], C. Tunç and E. Tunç [8], and the references therein. However, a review to date of the literature indicates that the instability of the solutions to the nonlinear functional vector differential equation of the seventh order has not been investigated. This paper is the first work on the subject. It is also known that the expressions of Lyapunov-Krasovskii functional are very complicated and hard to construct. In this paper, we define a Lyapunov-Krasovskii functional [9] and base on the Krasovskii criteria to prove a new theorem on the topic for the nonlinear functional vector differential equation of the seventh order. In this paper, intend to make a contribution to the subject since the functional differential equations have an important place in various fields of science and engineering.Meanwhile, some respective contributions on the topic can be summarized as follows.In 2000, Tejumola [3] discussed the instability of the zero solution of the seventh order scalar nonlinear differential equation without a deviating argument
(1)x(7)+a1x(6)+a2x(5)+a3x(4)+ψ4(x,.x′,…,x(6))x′′′+ψ5(x′)x′′+ψ6(x,x′,…,x(6))+ψ7(x)=0.Later, in 2003, Sadek [2] proved an instability theorem for the seventh order scalar nonlinear differential equation without a deviating argument
(2)x(7)+a1x(6)+a2x(5)+a3x(4)+a4x′′′+f(x′)x′′+g(x)x'+h(x)=0.Recently, Tunç [6, 7] discussed the same problem for the seventh order scalar nonlinear delay differential equations of the form:
(3)x(7)+a1x(6)+a2x(5)+a3x(4)+a4x′′′+f(x′)x′′+g(x)x′+h(x,x(t-r),x′,…,x(6)(t-r))=0,(4)x(7)+a1x(6)+a2x(5)+a3x(4)+ψ4(x,x(t-r),x′,…,x(6)(t-r))x′′′+ψ5(x′)x′′+ψ6(x,x(t-r),x′,…,x(6)(t-r))+ψ7(x(t-r))=0.In this paper, instead of (4), we consider the seventh order nonlinear functional vector differential equation with n-deviating arguments, τi, (i=1,2,…,n):
(5)X(7)+A1X(6)+A2X(5)+A3X(4)+Ψ4(X,X(t-τ1),…,X(t-τn),…,X(6)(t-τn))×X′′′+Ψ5(X′)X′′+Ψ6×(X,X(t-τ1),…,X(t-τn),…,X(6)(t-τn))X'+∑i=1nHi(X(t-τi))=0.
Setting X=X1, X'=X2, X′′=X3, X′′′=X4, X(4)=X5, X(5)=X6, X(6)=X7, we can write (5) in system form as follows
(6)X1′=X2,X2′=X3,X3′=X4,X4′=X5,X5′=X6,X6′=X7,X7′=-A1X7-A2X6-A3X5-Ψ4(X1,…,X1(t-τn),…,X7(t-τn))X4-Ψ5(X2)X3-Ψ6(X1,…,X1(t-τn),…,X7(t-τn))X2-∑i=1nHi(X1)+∑i=1n∫t-τitJHi(X1(s))X2(s)ds,
where τi are certain positive constants, the fixed delays, t-τi>0, A1, A2,  and A3 are constant n×n-symmetric matrices, the primes in (5) denote differentiation with respect to t, t∈ℜ+, ℜ+=[0,∞); Ψ4, Ψ5, and Ψ6 are continuous n×n-symmetric matrix functions for the arguments displayed explicitly, Hi:ℜn→ℜn, Hi(0)=0, Hi(X1)≠0, (X1≠0), and Hi are continuous for all the respective arguments. The Jacobian matrices of Hi(X) are given by
(7)JH1(X)=(∂h1i∂xj),…,JHn(X)=(∂hni∂xj)(i,j=1,2,…,n),
where (x1,…,xn) and (h1i),…,(hni) are the components of X and Hi, respectively. It is also assumed that the Jacobian matrices JHi(X) exist and are continuous. The existence and uniqueness of the solutions of (5) are assumed (see Èl’sgol’ts [10, pages 14, 15]). Throughout what follows that X1(t),…,X7(t) are abbreviated as X1,…,X7, respectively.Letr≥0 be given, and let C=C([-r,0],ℜn) with
(8)∥ϕ∥=max-r≤s≤0|ϕ(s)|,ϕ∈C.ForH>0 define CH⊂C by
(9)CH={ϕ∈C:∥ϕ∥<H}.Ifx:[-r,A)→ℜn is continuous, 0<A≤∞, then, for each t in [0,A),xt in C is defined by
(10)xt(s)=x(t+s),-r≤s≤0,t≥0.LetG be an open subset of C and consider the general autonomous delay differential system
(11)x˙=F(xt),xt=x(t+θ),-r≤θ≤0,t≥0,
where F:G→ℜn is continuous and maps closed and bounded sets into bounded sets. It follows from the conditions on F that each initial value problem
(12)x˙=F(xt),x0=ϕ∈G
has a unique solution defined on some interval [0,A), 0<A≤∞. This solution will be denoted by x(ϕ)(·) so that x0(ϕ)=ϕ.Definition 1.
The zero solution,x=0, of x˙=F(xt) is stable if for each ε>0 there exists δ=δ(ε)>0 such that ∥ϕ∥<δ implies that |x(ϕ)(t)|<ε for all t≥0. The zero solution is said to be unstable if it is not stable.Consider the linear constant coefficient differential equation of the seventh order:(13)x(7)+a1x(6)+a2x(5)+a3x(4)+a4x⃛+a5x¨+a6x˙+a7x=0.
It is known from the qualitative behavior of solutions of linear differential equations that the zero solution of (13) is unstable if and only if, the associated auxiliary equation:
(14)ψ(λ)≡λ7+a1λ6+a2λ5+a3λ4+a4λ3+a5λ2+a6λ+a7=0
has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1,a2,…,a7 in (14). For example, if
(15)a1<0
then it is clear from a consideration of the fact that the sum of the roots of (14) equals a1 and that at the least one root of (14) has a positive real part for arbitrary values of a2, a3, a4, a5, a6, and a7. An analogue consideration, combined with the fact that the product of the roots (14) equals (-a7) will verify that at least one root of (14) have a positive real part if
(16)a1=0,a7≠0
for arbitrary a2, a3, a4, a5 and a6. The condition a1=0 here in (16) is, however, superfluous when
(17)a7<0;
then for ψ(0)=a7<0 and ψ(R)>0 if R>0 is sufficiently large, thus showing that there is a positive real root of (14) subject to (17) and for arbitrary a1, a2, a3, a4, a5 and a6. Moreover, a necessary and sufficient condition for (14) to has a purely imaginary root λ=iβ (β real) is that the two equations
(18)-a1β6+a3β4-a5β2+a7=0,-β6+a2β4-a4β2+a6=0
are satisfied at the same time. If
(19)a1≤0,a3≥0,a5≤0,a7>0
or
(20)a1≥0,a3≤0,a5≥0,a7<0,
then (14) cannot have any purely imaginary root whatever.It should be noted that there are no restrictions on the constantsa2, a4, and a6 in (13).


## 2. Main Result

First, we give the following lemma.Lemma 2.
LetA be a real symmetric n×n-matrix and
(21)a'≥λi(A)≥a>0,(i=1,2,…,n),
where a'and a are constants, and λi(A) are the eigenvalues of the matrix A.
Then(22)a′〈X,X〉≥〈AX,X〉≥a〈X,X〉,a'2〈X,X〉≥〈AX,AX〉≥a2〈X,X〉
(Bellman [11]).Letτ=maxτi,(i=1,2,…,n).The following theorem is our main result.Theorem 3.
In addition to all the assumptions imposed toA1, A2, A3, Ψ4, Ψ5, Ψ6, and Hi that appear in (5), we assume that there exist constants a2<0, αi>0, βi>0, and δ>0 such that the conditions
(23)λi(A2)≤a2,β1≤λi(JH1(X1))≤α1,…,βn≤λi(JHn(X1))≤αn,14a2[λi(Ψ4(·))]2-λi(Ψ6(·))≥δ>0
hold. If τ<δ/n(α1+⋯+αn), then the zero solution of (5) is unstable.Remark 4.
For the proof of the theorem, under the conditions sated in the theorem, it suffices to find that there exists a continuous Lyapunov-Krasovskii functionalV(·)=V(X1t,…,X7t), which has the Krasovskii properties [9], say (K1), (K2), and (K3). (K1)
In every neighborhood of(0,0,0,0,0,0,0) there exists a point (ξ1,…,ξ7) such that V(ξ1,…,ξ7)>0.(K2)
The time derivativeV˙(·)=(d/dt)V(X1t,…,X7t) along solution paths of (6) is positive semidefinite.(K3)
The only solution(X1,…,X7)=(X1(t),…,X7(t)) of (6) which satisfies (d/dt)V(X1t,…,X7t)=0,(t≥0), is the trivial solution (0,0,0,0,0,0,0).Proof.
Consider the Lyapunov-Krasovskii functionalV(·)=V(X1t,…,X7t) defined by
(24)V(·)=〈X7,X2〉+〈A1X2,X6〉+〈A2X2,X5〉+〈A3X2,X4〉-〈X3,X6〉-〈A1X3,X5〉-〈A2X3,X4〉+〈X4,X5〉-12〈A3X3,X3〉+12〈A1X4,X4〉+∫01〈H1(σX1),X1〉dσ+⋯+∫01〈Hn(σX1),X1〉dσ+∫01〈Ψ5(σX2)X2,X2〉dσ-∑i=1nλi∫-τi0∫t+st∥X2(θ)∥2dθds,
where s is a real variable such that the integrals ∫-τi0∫t+st∥X2(θ)∥2dθds are nonnegative, and λi are certain positive constants to be determined later in the proof.
We see that(25)V(0,0,0,0,0,0,0)=0.
Since(∂/∂σ)Hi(σX1)=JHi(σX1)X1, Hi(0)=0, then
(26)∫01〈H1(σX1),X1〉dσ=∬01〈σ1JH1(σ1σ2X1)X1,X1〉dσ2dσ1≥β1∥X1∥2,∫01〈H2(σX1),X1〉dσ=∬01〈σ1JH2(σ1σ2X1)X1,X1〉dσ2dσ1≥β2∥X1∥2,⋮∫01〈Hn(σX1),X1〉dσ=∬01〈σ1JHn(σ1σ2X1)X1,X1〉dσ2dσ1≥βn∥X1∥2.
Let
(27)ε-=(ε11,…,ε1n).
Then, we get
(28)V(ε-,0,0,0,0,0,0)=(∑i=1nβi)∥ε-∥2>0
for all ε-,ε-∈ℜ, so that every neighborhood of the origin in the (X1,…,X7)-space contains points (ξ1,…,ξ7) such that V(ξ1,…,ξ7)>0.
Let(29)(X1,…,X7)=(X1(t),…,X7(t))
be an arbitrary solution of (6).
Calculating the time derivative of the Lyapunov-Krasovskii functionalV along this solution, we obtain
(30)V˙(·)=〈X5,X5〉-〈A2X4,X4〉-〈Ψ4(·)X2,X4〉-〈Ψ6(·)X2,X2〉+〈X2,∫t-τ1tJH1(X1(s))X2(s)ds〉+⋯+〈X2,∫t-τntJHn(X1(s))X2(s)ds〉-〈λ1τ1X2,X2〉-⋯-〈λnτnX2,X2〉+λ1∫t-τ1t∥X2(θ)∥2dθ+⋯+λn∫t-τnt∥X2(θ)∥2dθ.
Using the assumption 0<λi(JHi(X1))≤αi and the Schwarz inequality, we have
(31)〈X2,∫t-τ1tJH1(X1(s))X2(s)ds〉≥-∥X2∥∥∫t-τ1tJH1(X1(s))X2(s)ds∥≥-nα1∥X2∥∥∫t-τ1tX2(s)∥ds≥-nα1∥X2∥∫t-τ1t∥X2(s)∥ds≥-12nα1τ1∥X2∥2-12nα1∫t-τ1t∥X2(s)∥2ds,〈X2,∫t-τ2tJH2(X1(s))X2(s)ds〉≥-∥X2∥∥∫t-τ2tJH2(X1(s))X2(s)ds∥≥-nα2∥X2∥∥∫t-τ2tX2(s)∥ds≥-nα2∥X2∥∫t-τ2t∥X2(s)∥ds≥-12nα2τ2∥X2∥2-12nα2∫t-τ2t∥X2(s)∥2ds,⋮〈X2,∫t-τntJHn(X1(s))X2(s)ds〉≥-∥X2∥∥∫t-τntJH2(X1(s))X2(s)ds∥≥-nαn∥X2∥∥∫t-τntX2(s)∥ds≥-nαn∥X2∥∫t-τnt∥X2(s)∥ds≥-12nαnτn∥X2∥2-12nαn∫t-τnt∥X2(s)∥2ds.
Hence
(32)V˙(·)≥∥X5∥2-a2∥X4+2-1a2-1Ψ4(·)X2∥2+14a2〈Ψ4(·)X2,Ψ4(·)X2〉-〈Ψ6(·)X2,X2〉-(λ1+12nα1)τ1∥X2∥2-⋯-(λn+12nαn)τn∥X2∥2+(λ1-12nα1)∫t-τ1t∥X2(s)∥2ds+⋯+(λn-12nαn)∫t-τnt∥X2(s)∥2ds.
Letλi=(1/2)nαi. Then, using the assumptions of the theorem and the estimate τ=maxτi,we have
(33)V˙(·)≥∥X5∥2-a2∥X4+2-1a2-1Ψ4(·)X2∥2+14a2〈Ψ4(·)X2,Ψ4(·)X2〉-〈Ψ6(·)X2,X2〉-nα1τ1∥X2∥2-⋯-nαnτn∥X2∥2≥{δ-n(α1+⋯+αn)τ}∥X2∥2.
Ifτ<δ/n(α1+⋯+αn), then, for some positive constant k, we have
(34)V˙(·)≥k∥X2∥2>0.
Finally,V˙(·)=0 for all t≥0 necessarily implies that X2=0. Hence, it follows that
(35)X1=ξ1(constantvector),X2=X′=0,X3=X′′=0,X4=X′′′=0,X5=X(4)=0,X6=X(5)=0,X7=X(6)=0,
for all t≥0 so that
(36)X1=ξ1(constantvector),X2=X3=X4=X5=X6=X7=0,(t≥0).
Therefore, the estimates (d/dt)V(·)=0 and (6) imply X1=X2=X3=X4=X5=X6=X7=0 since Hi(ξ1)=0 if and only if ξ1=0. It now follows that functional V thus has all the requisite Krasovskii [9] properties subject to the conditions of the theorem. Hence, we can conclude that the zero solution of (5) is unstable.
The proof of the theorem is completed.



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