# Instability of a Fifth-Order Nonlinear Vector Delay Differential

Authors: Cemil Tunç
Journal: Journal of Mathematics
 (2013)

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## Abstract

We study a fifth-order nonlinear vector delay differential equation with multiple deviating arguments. Some criteria for guaranteeing the instability of zero solution of the equation are given by using the Lyapunov-Krasovskii functional approach. Comparing with the previous literature, our result is new and complements some known results.


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## Body

## 1. Introduction

Lyapunov functions and functionals have been successfully used and are still being used to obtain stability, instability, boundedness, and the existence of periodic solutions of differential equations, differential equations with functional delays, and functional differential equations [1–18]).It should be noted that in 2000, using the Lyapunov function approach,  Li and Duan [2] discussed the instability of the solutions of the fifth-order nonlinear scalar differential equation without delay:(1)x(5)(t)+f5(x(t),x′(t),…,x(4)(t))x(4)(t)+f4(x(t),x′(t),...,x(4)(t))x(3)(t)+f3(x′′(t))+f2(x′(t))+a1x(t)=0.Later, in 2011, using the same technique,  Tunç [12] studied the instability of the solutions for the fifth-order nonlinear scalar delay differential equation:(2)x(5)(t)+f5(x(t-r),x′(t-r),…,x(4)(t-r))x(4)(t)+f4(x(t-r),x′(t-r),…,x(4)(t-r))x(3)(t)+f3(x′′(t-r))+f2(x′(t))+a1x(t)=0.In this paper, we consider the fifth-order nonlinear vector delay differential equation with multiple-deviating arguments:(3)X(5)(t)+F(X(4)(t-τn)X(t),X(t-τ1),…,X(t-τn),…,X(4)(t),…,X(4)(t-τn))X(4)(t)+G(X(4)(t),…,X(4)(t-τn)X(t),X(t-τ1),…,X(t-τn),…,X(4)(t),…,X(4)(t-τn))X(3)(t)+∑i=1nHi(X′′(t-τi))+Ψ(X′(t))+AX(t)=0,
where the primes in (3) denote differentiation with respect to  t, t∈ℜ+, ℜ+=[0,∞); X∈ℜn, τi  are certain positive constants, the fixed delays, t-τi>0,A is a constant n×n-symmetric matrix,  F and  G are continuous n×n-symmetric matrix functions for  the respective arguments,  Hi:ℜn→ℜn,Ψ:ℜn→ℜn with Hi(0)=Ψ(0)=0 are continuous for the respective arguments. Let   JHi(X′′)  and JΨ(X′) denote the linear operators from Hi(X′′) and  Ψ(X′) to
(4)JH1(X′′)=(∂h1i∂xj′′),…,JHn(X′′)=(∂hni∂xj′′),JΨ(X')=(∂ψi∂xj′),(i,j=1,2,…,n),
where (x1′,…,xn′), (x1′′,…,xn′′), (h1i),…,(hni), and (ψ1,…,ψn)   are the components of X′, X′′, Hi,  and Ψ, respectively. In what follows, it is assumed that  JHi(X′′) and JΨ(X′) exist and are symmetric and continuous.Equation (3) is the vector version for systems of real nonlinear differential equations of the fifth order:
(5)xi(5)+∑k=1nfik(x(4)(t-τ1),…,x(4)(t-τn)x,x(t-τ1),…,x(t-τn),…,x(4)(t-τ1),…,x(4)(t-τn))xk(4)+∑k=1ngik(x(4)(t-τ1),…,x(4)(t-τn)x,x(t-τ1),…,x(t-τn),…,x(4)(t-τ1),…,x(4)(t-τn))xk(3)+h1i(x1′′(t-τ1),…,xn′′(t-τn))+⋯+hni(x1′′(t-τ1),…,xn′′(t-τn))+ψi(x1′,…,xn′)+∑k=1naikxk=0,(i=1,2,…,n).Instead of (3), we consider the corresponding differential system
(6)X˙=Y,Y˙=Z,Z˙=W,W˙=U,U˙=-F(X,…,X(t-τn),…,U,…,U(t-τn))U-G(X,…,X(t-τn),…,U,…,U(t-τn))W-∑i=1nHi(Z)+∑i=1n∫t-τitJHi(Z(s))W(s)ds-Ψ(Y)-AX,
which was obtained  by setting X˙=Y,X¨=Z,X(3)=W, and X(4)=U from (3).However, a review to date of the literature indicates that the instability of solutions of vector differential equations of the fifth order with a deviating argument has not been investigated up to now. This paper is the first known work regarding the instability of solutions for the nonlinear vector delay differential equations of the fifth order with multiple-deviating arguments. The motivation of this paper comes from the above papers done on scalar nonlinear differential equations of the fifth order without and with delay and the vector differential equations of the fifth order without delay. By this work, we improve the results in [2, 12] to a vector delay differential equation of the fifth order with multiple-deviating arguments.  Defining the Lyapunov-Krasovskii functional and taking into account the Krasovskiĭ’s criteria [19],  we prove our main result on the subject.  The result to be obtained is new, has a contribution to the topic, and may be useful for the researchers working on the qualitative behaviors of solutions of the differential equations.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.
The symbol〈X,Y〉 corresponding to any pair X,Y in ℜn stands for the usual scalar product   ∑i=1nxiyi, that is, 〈X,Y〉=∑i=1nxiyi; thus 〈X,X〉=∥X∥2 and λi(Ω),(i=1,2,…,n) are the eigenvalues of the real symmetric n×n-matrix Ω. The matrix Ω is said to be negative-definite, when 〈ΩX,X〉≤0  for all nonzero X in ℜn.


## 2. Main Result

Before stating  the main result, we need the following result.Lemma 2.
LetA be a real symmetric n×n-matrix and
(7)a′≥λi(A)≥a>0,(i=1,2,...,n),
where a′ and  a are constants.
Then(8)a′〈X,X〉≥〈AX,X〉≥a〈X,X〉,a′2〈X,X〉≥〈AX,AX〉≥a2〈X,X〉. See [20].
Letr≥0 be given and let C=C([-r,0],ℜn) with(9)∥ϕ∥=max-r≤s≤0|ϕ(s)|,ϕ∈C.
ForH>0 define CH⊂C by
(10)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
(11)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 with finite delay:
(12)x˙=F(xt),F(0)=0,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 these conditions on F that each initial value problem,
(13)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.The main result of this paper is given by the following theorem.Theorem 3.
In addition to the basic assumptions imposed onF,G,Hi,Ψ, and  A   that appear in (3), we assume that there exist positive constants  a1,a4,βi, and δ such that the following conditions hold:
F
(
·
)
,
G
(
·
)
,
J
H
i
(
·
)
,
J
Ψ
(
·
),  and  A are symmetric,
(14)Hi(0)=0,Hi(Z)≠0,(Z≠0),Ψ(0)=0,Ψ(Y)≠0,(Y≠0),λi(A)≤-a1,λi(JΨ(Y))≥a4,|λi(JHi(Z))|≤βi,14λi(F(·))2+λi(G(·))≤-δ.
If
(15)τ<δn(β1+⋯+βn),
then  the zero solution  of (3) is unstable.Proof.
We define the Lyapunov-Krasovskii functionalV(·)=V(Xt,Yt,Zt,Wt,Ut):(16)V(·)=〈W,U〉+〈AX,Z〉-12〈AY,Y〉+〈Ψ(Y),Z〉+∫01〈H1(σZ),Z〉dσ+⋯+∫01〈Hn(σZ),Z〉dσ-∑i=1nμi∫-τi0∫t+st∥W(θ)∥2dθds,
where μi are certain positive constants and  will be determined later in the proof.
It is clear thatV(0,0,0,0,0)=0.
Let(17)ε-=(ε21,...,ε2n).
Using the assumptionλi(A)≤-a1, we have
(18)V(0,ε-,0,0,0)=-12〈Aε-,ε-〉≥12a1∥ε-∥2>0
for all arbitrary ε-≠0, ε-∈ℜn, which verifies the property (P1) of   Krasovskiĭ [19].
Using a basic calculation, the time derivative ofV  along solutions of  (6) results in
(19)ddtV(·)=〈U,U〉-〈F(·)W,U〉-〈G(·)W,W〉+〈JΨ(Y)Z,Z〉+〈W,∫t-τ1tJH1(Z(s))W(s)ds〉+⋯+〈W,∫t-τntJHn(Z(s))W(s)ds〉-〈μ1τ1W,W〉-⋯-〈μnτnW,W〉+μ1∫t-τ1t∥W(θ)∥2dθ+⋯+μn∫t-τnt∥W(θ)∥2dθ.
Under the assumptions of the theorem and Schwarz inequality, it can be easily seen that
(20)〈W,∫t-τ1tJH1(Z(s))W(s)ds〉≥-∥W∥∥∫t-τ1tJH1(Z(s))W(s)ds∥≥-nβ1∥W∥∥∫t-τ1tW(s)∥ds≥-nβ1∥W∥∫t-τ1t∥W(s)∥ds≥-12nβ1τ1∥W∥2-12nβ1∫t-τ1t∥W(s)∥2ds,〈W,∫t-τ2tJH2(Z(s))W(s)ds〉≥-∥W∥∥∫t-τ2tJH2(W(s))W(s)ds∥≥-nβ2∥W∥∥∫t-τ2tW(s)∥ds≥-nβ2∥W∥∫t-τ2t∥W(s)∥ds≥-12nβ2τ2∥W∥2-12nβ2∫t-τ2t∥W(s)∥2ds,⋮〈W,∫t-τntJHn(Z(s))W(s)ds〉≥-∥W∥∥∫t-τntJHn(Z(s))W(s)ds∥≥-nβn∥W∥∥∫t-τntW(s)∥ds≥-nβn∥W∥∫t-τnt∥W(s)∥ds≥-12nβnτn∥W∥2-12nβn∫t-τnt∥W(s)∥2ds.
Hence
(21)V˙(·)=∥U-2-1F(·)W∥2-14〈F(·)W,F(·)W〉-〈G(·)W,W〉+〈JΨ(Y)Z,Z〉-μ1τ1∥W∥2-⋯-μnτn∥W∥2-12nβ1τ1∥W∥2-⋯-12nβnτn∥W∥2+(μ1-12nβ1)∫t-τ1t∥W(s)∥2ds+⋯+(μn-12nβn)∫t-τnt∥W(s)∥2ds.
Letμi=(1/2)nβi.    Then, using the assumptions of the theorem and the estimate τ=maxτi,we get
(22)V˙(·)=∥U-2-1F(·)W∥2-14〈F(·)W,F(·)W〉-〈G(·)W,W〉+〈JΨ(Y)Z,Z〉-nβ1τ1∥W∥2-···-nβnτn∥W∥2≥-14〈F(·)W,F(·)W〉-〈G(·)W,W〉+〈JΨ(Y)Z,Z〉-nβ1τ1∥W∥2-···-nβnτn∥W∥2≥a4∥Z∥2+{δ-n(β1+···+βn)τ}∥W∥2>0.
Ifτ<δ/n(β1+⋯+βn),  then, for some positive constant  k1,we have
(23)V˙(·)≥a4∥Z∥2+k1∥W∥2>0,
which verifies that V(·)  has the property (P2) of   Krasovskiĭ [19].
Besides,(24)V˙(·)=0⟺Z=W=U=0.
The substitution of this estimate into system (6) results in
(25)Ψ(Y)+AX=0.
That is,
(26)Ψ(X′)+AX=0.
Because Z=X′′=0,  X′= constant vector, for all t>0.Hence, since A is not the zero matrix, we have  X=constant vector, for all t>0. But, in view of the assumptions of the theorem,  this implies that X′=0 and thus also, by  Ψ(X′)+AX=0, that  X=0,  forallt>0. These estimates result in   X=Y=Z=W=U=0. Hence,  the property (P3)of   Krasovskiĭ  [19] holds for the Lyapunov-Krasovskii functional V(·).
The proof of the theorem is completed.



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*Source: 809748-2012-12-20.xml*