# A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation

Authors: A. M. A. Abou-El-Ela; A. I. Sadek; A. M. Mahmoud; R. O. A. Taie
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form:X(4)+F(X˙,X¨)X⃛+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0. By constructing a Lyapunov functional, we obtained the result of stability.


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## Body

## 1. Introduction

As is well known, the stability is a very important problem in the theory and applications of delay differential equations. Therefore, in the literature, some methods have been developed to obtain information on the stability behaviour of the delay differential equations when there is no analytical expression for the solutions. One of these methods is known as Lyapunov’s second method; since Lyapunov [1] proposed his famous second method on the stability of motion, the problems related to the investigation of stability of solutions of certain second-, third-, and fourth-order linear and nonlinear, scalar, and vector differential equations have been given great attention in the past five decades due to the importance of the subject.During this period, stability of solutions for various higher-order linear and nonlinear differential equations has been extensively studied and many results have been obtained in the literature (see, e.g., Krasovskii [2], Yoshizawa [3], Reissig et al. [4], Abou-El-Ela and Sadek [5–7], Bereketoglu and Kart [8], Sadek [9], Tunç [10–13], Abou-El-Ela et al. [14], and the references cited in those works), among which the results performed on asymptotic stability properties of linear and nonlinear scalar and vector differential equations of fourth-order can briefly be summarized as follows.First in 1990 Abou-El-Ela and Sadek [5] found sufficient conditions for the asymptotic stability of the zero solution of the scalar nonlinear differential equation of the form(1)x(4)+f1(x˙,x¨)x⃛+f2(x˙,x¨)+f3(x,x˙)+f4(x)=0.Later in 2004 Sadek [9] determined sufficient conditions, under which all solutions of the nonhomogeneous vector differential equation(2)X4+FX˙,X¨X⃛+ΦX¨+GX˙+A4X=P(t,X,X˙,X¨,X⃛)tend to zero as t→∞.Recently in 2012 Abou-El-Ela et al. [14] investigated sufficient conditions for the uniform stability of the zero solution of the real fourth-order vector delay differential equation(3)X(4)+AX⃛+Φ(X¨)+G(X˙)+H(X(t-r))=0.In the present paper, we are concerned with the uniform stability of the zero solutionX=0 of real nonlinear autonomous vector delay differential equation of the fourth-order(4)X4+FX˙,X¨X⃛+ΦX¨+GX˙t-r+H(X(t-r))=0,where X∈Rn; F is an n×n-symmetric matrix; Φ, G, and H are n-vector continuous functions; Φ(0)=G(0)=H(0)=0; and r is a bounded delay and positive constant.Equation (4) represents a system of real fourth-order differential equation with delay(5)xi4+∑k=1n‍fikx˙1,…,x˙n;x¨1,…,x¨nx⃛k+ϕix¨1,…,x¨n+gix˙1t-r,…,x˙nt-r+hix1t-r,…,xnt-r=0,(i=1,2,…,n).The Jacobian matricesJFY,ZY∣Z, JF(Y,Z)Z∣Z, JF(Y,Z)Y∣Y, JF(Y,Z)Z∣Y,JΦ(Z), JG(Y), and JH(X) are given by(6)JFY,ZY∣Z=∂∂zj∑k=1n‍fikyk=∑k=1n‍∂fik∂zjyk,JFY,ZZ∣Z=∂∂zj∑k=1n‍fikzk=F(Y,Z)+∑k=1n‍∂fik∂zjzk,JFY,ZY∣Y=∂∂yj∑k=1n‍fikyk=F(Y,Z)+∑k=1n‍∂fik∂yjyk,JFY,ZZ∣Y=∂∂yj∑k=1n‍fikzk=∑k=1n‍∂fik∂yjzk,JΦ(Z)=∂ϕi∂zj,JG(Y)=∂gi∂yj,JH(X)=∂hi∂xj,where (i,j=1,2,…,n), (x1,…,xn), (y1,…,yn), (z1,…,zn), (fik), (ϕ1,…,ϕn), (g1,…,gn), and (h1,…,hn) represent X, Y, Z, F, Φ, G, and H, respectively. It will also be assumed as basic throughout the paper that the Jacobian matrices JF(Y,Z)Y∣Z, JF(Y,Z)Z∣Z, JF(Y,Z)Y∣Y, JF(Y,Z)Z∣Y, JΦ(Z), JG(Y), and JH(X) exist and are continuous. The symbol X,Y will be used to denote the usual scalar product in Rn for any X,Y in Rn; that is, X,Y=∑i=1n‍xiyi; thus X,X=X2. It is well known that the real symmetric matrix A=(aij), (i,j=1,2,…,n) 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.


## 2. Main Result

In order to reach the main result of this paper, we will give some basic information to the stability criteria for a general autonomous delay differential system. We consider(7)x-˙=f-(x-t),x-t(s)=x-(t+s),-h≤s≤0,t≥0,where f-:CH→Rn is a continuous mapping, f-(0)=0, CH:={ϕ∈C([-h,0],Rn):ϕ≤H}, and for H1<H, there exists an L(H1)>0, with f-ϕ≤L(H1) when ϕ<H1.Theorem 1 (see [15]).
LetV(ϕ):CH→R be a continuous functional satisfying a local Lipschitz condition, V(0)=0, such that(i)
W1(|ϕ(0)|)≤V(ϕ)≤W2(ϕ), where W1, W2 are wedges;(ii)
V˙(7)(ϕ)≤0, for ϕ∈CH.Then the zero solution of (7) isuniformly stable.The following theorem will be our main stability result for (4).Theorem 2.
In addition to the essential assumptions imposed on the functionsF, Φ, G, and H, suppose the existence of arbitrary positive constants α1, α2, α3, α4, α1′, and α4′. Suppose also for i=1,2,…,n the following conditions are satisfied.(i)
F(Y,Z), JFY,ZY∣Z, and JFY,ZZ∣Z are symmetric; α1′≥λi(F(Y,Z))≥α1>0, for all Y,Z∈Rn.(ii)
G(0)=0,JG(Y) is symmetric and λi∫01‍JGσYdσ≥α3α42/α4′2, for all Y∈Rn.(iii)
There is a finite constantΔ>0 such that(8)α1α2-JGYα3α4-α1α4′2∫01‍FY,σZdσ≥Δ,for allY,Z∈Rn.(iv)
One has0≤λi(JG(Y)-∫01‍JG(σY)dσ)≤δ1<2Δ/α1α32, for all Y∈Rn.(v)
One hasλi(∫01‍F(Y,σZ)dσ-F(Y,Z))≤δ2<2Δ/α12α3α4, for all Y,Z∈Rn.(vi)
JFY,ZY∣Y-F(Y,Z) and JFY,ZZ∣Y are negative-definite.(vii)
AlsoH(0)=0, JH(X) is symmetric, and λi(∫01‍JH(σX)dσ)≥α4′, for all X∈Rn.(viii)
JH(X) commutes with JH(X′), for all X,X′∈Rn and 0≤λi(α4I-JH(X))≤εD0α12, for all X∈Rn, and D0:=α1α2+α2α3α4α4′-2.(ix)
AlsoΦ(0)=0, JΦ(Z) is symmetric, and 0≤λi(∫01‍JΦ(σZ)dσ-α2I)≤ε0α33α42/α4′4, for all Z∈Rn, where ε0 is a positive constant such that(9)ε0<ε=min⁡⁡1α1,α4′2α3α4,Δ4α1α3α4D0,α3α44α4′2D02α4′2Δα1α32α42-δ1,1α1,α4′2α3α4,Δ4α1α3α4D0α14D02Δα12α3α4-δ2.Then the zero solution of (4) is uniformly stable, provided that(10)r<min⁡εd1nα4+α1α2,α42/α4′2ε-ε0α3α4nd1+2d2+1+α1α2d2nΔ2α1α3α4nα4+α1α2d1+d2+2,α42/α4′2ε-ε0α3α4nd1+2d2+1+α1α2d2n,where(11)d1=ε+1α1,d2=ε+α4′2α3α4.The following two lemmas are important for proving Theorem2.Lemma 3.
LetA be a real symmetric n×n-matrix and(12)a′≥λi(A)≥a>0(i=1,2,…,n),where a′, a are constants. Then(13)a′X,X≥AX,X≥aX,X,a′2X,X≥AX,AX≥a2X,X.For a proof of the above lemma, see Bellman [16].Lemma 4.
Assume thatX˙=Y, Y˙=Z, and Z˙=W. Then(1)
d/dt∫01‍HσX,Xdσ=HX,Y;(2)
d/dt∫01‍GσY,Ydσ=GY,Z;(3)
d/dt∫01‍ΦσZ,Zdσ=ΦZ,W;(4)
d/dt∫01‍σFY,σZZ,Zdσ≦FY,ZZ,W;(5)
d/dt∫01‍FY,σZZ,Ydσ≦FY,ZY,W+∫01‍FY,σZdσZ,Z.Proof.
The proof is as follows:(14)1ddt∫01‍HσX,Xdσ=∫01‍σJHσXY,Xdσ+∫01‍HσX,Ydσ=∫01‍σJHσXX,Ydσ+∫01‍HσX,Ydσ=∫01‍σ∂∂σHσX,Ydσ+∫01‍HσX,Ydσ=σHσX,Y01=HX,Y.
The proofs of (2) and (3) are similar to that of (1):(15)(4)ddt∫01‍σFY,σZZ,Zdσ=∫01‍σFY,σZZ,Wdσ+∫01‍JFY,σZσZ∣YZ,Zdσ+∫01‍σJFY,σZσZ∣σZW,Zdσ≦∫01‍σFY,σZZ,Wdσ+∫01‍σJFY,σZσZ∣σZZ,Wdσ,since J(FZ∣Y) is negative-definite from assumption (vi) and J(FZ∣Z) is symmetric from assumption (i). Then(16)ddt∫01‍σFY,σZZ,Zdσ≦∫01‍σFY,σZZ,Wdσ+∫01‍σ∂∂σσFY,σZZ,Wdσ=σσFY,σZZ,W01=FY,ZZ,W,5ddt∫01‍FY,σZZ,Ydσ=ddt∫01‍FY,σZY,Zdσ=ddt∫01‍FY,σZY,Wdσ+∫01‍JFY,σZY∣YZ,Zdσ+∫01‍σJFY,σZY∣σZW,Zdσ.
SinceJFY∣Z is negative-definite from assumption (i), we have(17)∫01‍σJFY,σZY∣σZW,Zdσ=∫01‍σJFY,σZY∣σZZ,Wdσ=∫01‍σ∂∂σFY,σZY,Wdσ=FY,ZY,W-∫01‍FY,σZY,Wdσ,and then(18)ddt∫01‍FY,σZZ,Ydσ=FY,ZY,W+∫01‍JFY,σZY∣YZ,Zdσ=FY,ZY,W+∫01‍FY,σZZ,Zdσ+∫01‍JFY,σZY∣Y-FY,σZZ,Zdσ≦FY,ZY,W+∫01‍F(Y,σZ)dσZ,Z,since JFY∣Y-F is negative-definite from assumption (vi).


## 3. Proof of Theorem2

For the proof of the main stability theorem, it will be convenient to consider instead of (4) the equivalent system(19)X˙=Y,Y˙=Z,Z˙=W,W˙=-FY,ZW-ΦZ-GY-HX+∫t-rt‍JG(Y(s))Z(s)ds+∫t-rt‍JH(X(s))Y(s)ds.The proof of Theorem2 depends on a scalar differentiable function V(Xt,Yt,Zt,Wt); now we define the Lyapunov functional V as(20)2VXt,Yt,Zt,Wt=2d2∫01‍HσX,Xdσ+d2α2Y,Y-d1α4Y,Y+2∫01‍GσY,Ydσ+2d1∫01‍ΦσZ,Zdσ-d2Z,Z+2∫01‍σFY,σZZ,Zdσ+d1W,W+2HX,Y+2d1HX,Z+2d2∫01‍FY,σZZ,Ydσ+2d1GY,Z+2d2Y,W+2Z,W+2μ∫-r0‍∫t+st‍Yθ2dθds+2λ∫-r0‍∫t+st‍Z(θ)2dθds,where μ and λ are positive constants, which will be determined later. Let(21)F1(Y,Z)=∫01‍F(Y,σZ)dσ.Since λi(F(Y,Z))≥α1>0, for all Y,Z∈Rn, it follows that(22)λi(F1(Y,Z))≥α1>0,∀Y,Z∈Rn.Further we define(23)Γ(Y)=∫01‍JG(σY)dσ,and then it follows from (ii) and (iv) that(24)λi(Γ(Y))≥α3α42α4′2>0,for all Y∈Rn, and(25)0≤λiJGY-ΓY≤δ1,∀Y∈Rn.Since(26)∂∂σΦσZ=JΦσZZ,Φ(0)=0,then(27)Φ(Z)=∫01‍JΦ(σZ)Zdσ.Therefore(28)2d1∫01‍ΦσZ,Zdσ=2d1∫01‍∫01‍JΦσ1σ2Zσ2Z,Zdσ1dσ2=2d1∫01‍∫01‍JΦσ1Z~Z~,Zdσ1dσ2≥2d1∫01‍α2Z~,Zdσ2,byix=2d1∫01‍α2Z,Zσ2dσ2=d1α2Z,Z.Also since(29)2μ∫-r0‍∫t+st‍Y(θ)2dθds,2λ∫-r0‍∫t+st‍Z(θ)2dθdsare nonnegative, consequently we obtain(30)2VXt,Yt,Zt,Wt≥2d2∫01‍HσX,Xdσ+d2α2Y,Y-d1α4Y,Y+2∫01‍GσY,Ydσ+α2d1-d2Z,Z+2∫01‍σFY,σZZ,Zdσ+d1W,W+2HX,Y+2d1HX,Z+2d2∫01‍FY,σZZ,Ydσ+2d1GY,Z+2d2Y,W+2Z,W.Then we can find(31)2V≥2d2∫01‍HσX,Xdσ-Γ-1/2HX2+d2α2Y,Y-d1α4Y,Y-d22F11/2Y2+2∫01‍GσY,Ydσ-Γ1/2Y2+α2d1-d2Z2-d12Γ1/2Z2+2∫01‍σFY,σZZ,Zdσ-F11/2Z2+d1W2-F1-1/2W2+F1-1/2W+F11/2Z+d2F11/2Y2+Γ-1/2H(X)+Γ1/2Y+d1Γ1/2Z2.The matricesF1 and Γ are symmetric because F and JG are symmetric. The eigenvalues of F1 and Γ are positive because of (22) and (24).Consequently the square rootsF11/2 and Γ1/2 exist; these are again symmetric and nonsingular for all Y,Z∈Rn.Therefore we get(32)2V≥2d2∫01‍HσX,Xdσ-Γ-1HX,HX+d2α2Y,Y-d1α4Y,Y-d22F1Y,Y+2∫01‍GσY,Ydσ-ΓY,Y+α2d1-d2Z2-d12ΓZ,Z+2∫01‍σFY,σZZ,Zdσ-F1Z,Z+d1W2-F1-1W,W.Fromλi(F1-1)≤1/α1 and λi(Γ-1)≤α4′2/α3α42, because of (22) and (24), we get from Lemma 3 and Cauchy-Schwartz inequality that(33)2V≥2d2∫01‍HσX,Xdσ-Γ-1HX,HX+2∫01‍GσY,Ydσ-ΓY,Y+α2d2-α4d1-d22F1Y2+α2d1-d2-d12ΓZ2+2∫01‍σFY,σZZ,Zdσ-F1Z,Z+d1-1α1W2.From the definitions ofd1, d2 in (11), it follows that(34)2V(Xt,Yt,Zt,Wt)≥V1+V2+V3+εW2,where(35)V1:=2d2∫01‍HσX,Xdσ-Γ-1HX,HX,V2≔α2d2-α4d1-d22F1Y2V2+2∫01‍GσY,Ydσ-ΓY,Y,V3≔α2d1-d2-d12ΓZ2V3+2∫01‍σFY,σZZ,Zdσ-F1Z,Z.Since(36)∂∂σ1Hσ1X,Hσ1X=2JHσ1XX,Hσ1X,by integrating both sides from σ1=0 to σ1=1 and because of H(0)=0, we obtain(37)HX,HX=2∫01‍JHσ1XX,Hσ1Xdσ1.Thus(38)V1=2d2∫01‍HσX,Xdσ-2Γ-1∫01‍JHσ1XX,Hσ1Xdσ1=2∫01‍Hσ1X,d2I-Γ-1JHσ1XXdσ1.But from(39)∂∂σ2Hσ1σ2X,d2I-Γ-1JHσ1XX=σ1JHσ1σ2XX,d2I-Γ-1JHσ1XX,by integrating both sides from σ2=0 to σ2=1 and because of H(0)=0, we find(40)Hσ1σ2X,d2I-Γ-1JHσ1XX=∫01‍σ1JHσ1σ2XX,d2I-Γ-1JHσ1XXdσ2.Therefore by using (11), (24), (vii), (viii), and Lemma 3, we have(41)V1=2∫01‍∫01‍σ1JHσ1σ2XX,d2I-Γ-1JHσ1Xd2I-Γ-1JHσ1XXdσ2dσ1=2∫01‍∫01‍σ1JHσ1σ2Xd2I-Γ-1JHσ1XX,d2I-Γ-1JHσ1XXdσ2dσ1≥2ε∫01‍∫01‍JHσ1σ2Xσ1X,Xdσ2dσ1+2α4′2α3α42∫01‍∫01‍σ1JHσ1σ2XX,α4I-JHσ1XXdσ2dσ1≥2ε∫01‍∫01‍JHσ2X~X~,Xdσ2dσ1≥2ε∫01‍α4′X~,Xdσ1=2ε∫01‍α4′X,Xσ1dσ1=εα4′X,X=εα4′X2.To estimateV2 we need(42)α2d2-α4d1-d22F1=d2α2-d1JGY-d2F1+d1d2JGY-α4≥d2α2-d1JG(Y)-d2F1,since from (11) and (ii) we find that(43)d2JG(Y)-α4>ε+α4′2α3α4α3α42α4′2-α4=εα3α42α4′2>0.Now(44)α2-d1JGY-d2F1=α2-1α1JGY-α4′2α3α4F1-εJGY+F1=1α1α3α4α3α4α1α2-JGY-α1α4′2F1-εJGY+F1≥Δα1α3α4-εα1α2+α2α3α4α4′-2,from(iii).Thus we obtain from (viii)(45)α2-d1JG(Y)-d2F1≥Δα1α3α4-εD0.From the identity(46)∫01‍σJGσYY,Ydσ≡GY,Y-∫01‍GσY,Ydσ,we get from (25) and by Lemma 3(47)2∫01‍GσY,Ydσ-GY,Y=∫01‍GσY,Ydσ-∫01‍σJGσYY,Ydσ=∫01‍σΓσYY,Ydσ-∫01‍σJGσYY,Ydσ=-∫01‍σJGσY-ΓσYY,Ydσ≥-12δ1Y2.So we have from (9) and (11)(48)V2≥d2Δα1α3α4-εD0Y2-12δ1Y2≥α4′2α3α4Δα1α3α4-εD0-12δ1Y2≥142α4′2Δα1α32α42-δ1Y2,since ε<(α3α4/4α4′2D0)2α4′2Δ/α1α32α42-δ1.To estimateV3 we need(49)α2d1-d2-d12Γ=d1α2-d1Γ-d2F1+d2d1F1-1≥d1α2-d1Γ-d2F1≥d1α2-d1JGY-d2F1≥1α1Δα1α3α4-εD0,by (11), (25), and (45). So from the identity(50)∫01‍σFY,σZZ,Zdσ≡∫01‍FY,σZZ,Zdσ-∫01‍σF1Y,σZZ,Zdσ,we find(51)2∫01‍σFY,σZZ,Zdσ-F1Z,Z=∫01‍σFY,σZZ,Zdσ-∫01‍σF1Y,σZZ,Zdσ=-∫01‍σFY,σZ-F1Y,σZZ,Zdσ≥-12δ2Z2,by(v).Thus from (9), we obtain(52)V3≥1α1Δα1α3α4-εD0-12δ2Z2≥142Δα12α3α4-δ2Z2,since ε<(α1/4D0)2Δ/α12α3α4-δ2. Then it follows that(53)2VXt,Yt,Zt,Wt≥εα4′X2+142α4′2Δα1α32α42-δ1Y2+142Δα12α3α4-δ2Z2+εW2.Since the coefficients are positive constants from the definitions ofδ1, δ2, and ε in (iv), (v), and (9), then there exists a positive constant D1 such that(54)V(Xt,Yt,Zt,Wt)≥D1X2+Y2+Z2+W2.To prove that(55)VXt,Yt,Zt,Wt≤D2X2+Y2+Z2+W2,by using the hypotheses of Theorem 2 we find(56)F1≤nα2α3α4α4′-2,by(iii).Since(57)∂Φ(σZ)∂σ=JΦσZZ,Φ(0)=0,then from (ix) we have(58)ΦZ=∫01‍JΦσZZdσ≤∫01‍JΦσZZdσ≤nα2+εα33α42α4′4Z,and also since(59)∂G(σY)∂σ=JGσYY,G(0)=0,then from (iv) we have(60)GY=∫01‍JGσYYdσ≤∫01‍JGσYYdσ≤α1α2nY.Since(61)∂H(σX)∂σ=JHσXX,H(0)=0,then from (viii) we get(62)HX=∫01‍JHσXXdσ≤∫01‍JHσXXdσ≤α4nX.By using Cauchy-Schwartz inequalityu,v≤(1/2)(u2+v2) and from(63)2μ∫-r0‍∫t+st‍Yθ2dθds=2μ∫t-rt‍θ-t+rYθ2dθ≤2μY2∫t-rt‍θ-t+rdθ=μr2Y2,2λ∫-r0‍∫t+st‍Zθ2dθds=2λ∫t-rt‍θ-t+rZθ2dθ≤2λZ2∫t-rt‍θ-t+rdθ=λr2Z2.Hence there exists a positive constantD2 satisfying(64)V(Xt,Yt,Zt,Wt)≤D2X2+Y2+Z2+W2.Now from (19), (20), and Lemma 4, we have(65)dVdt=d2HX,Y+d2α2Y,Z-d1α4Y,Z+GY,Z+d1ΦZ,W-d2Z,W+FY,ZZ,W+d1W,-FY,ZW-ΦZ-GY-HX∫t-rt‍+∫t-rt‍JG(Y(s))Z(s)ds+∫t-rt‍JH(X(s))Y(s)ds+JHXY,Y+HX,Z+d1JHXY,Z+d1HX,W+d2FY,ZY,W+d2F1Z,Z+d1W,GY+d1JGYZ,Z+d2Y,-F(Y,Z)W∫t-rt‍-ΦZ-GY-HX+∫t-rt‍JGYsZsds+∫t-rt‍JHXsYsds+W,W+Z,-F(Y,Z)W∫t-rt‍-ΦZ-GY-HX+∫t-rt‍JGYsZsds+∫t-rt‍JH(X(s))Y(s)ds+d2Z,W+μrY2-μ∫t-rt‍Y(θ)2dθ+λrZ2-λ∫t-rt‍Z(θ)2dθ.Then we get(66)dVdt=d2α2Y,Z-d1α4Y,Z-d1W,FY,ZW-d2Y,ΦZ+JHXY,Y+d2F1Z,Z+d1JGYZ,Z+W,W+d1JHXY,Z-Z,ΦZ-d2Y,GY+d1W+Z+d2Y,∫t-rt‍JHXsYsds+d1W+Z+d2Y,∫t-rt‍JGYsZsds+μrY2-μ∫t-rt‍Yθ2dθ+λrZ2-λ∫t-rt‍Z(θ)2dθ,and it follows that(67)dVdt=α4Y,Y-d2Y,GY-α2Z,Z+d1JGYZ,Z+d2F1Z,Z-d1W,FY,ZW+W,W+d1W+Z+d2Y,∫t-rt‍JHXsYsds+d1W+Z+d2Y,∫t-rt‍JGYsZsds+μrY2-μ∫t-rt‍Yθ2dθ+λrZ2-λ∫t-rt‍Z(θ)2dθ+V4+V5,where(68)V4≔d2α2Z,Y-d2Y,ΦZV4-Z,ΦZ+α2Z,Z,V5≔-d1α4Z,Y+d1JHXY,ZV5+JHXY,Y-α4Y,Y.But(69)V4=-∫01JΦσZZ,Z-α2Z,Z+d2JΦσZZ,Y-α2Z,Ydσ=-∫01‍JΦσZ-α2IZ,Zdσ-d2∫01‍JΦσZ-α2IZ,Ydσ.Sinceλi∫01‍JΦ(σZ)dσ-α2I is nonnegative by (ix), then from (11) we get(70)V4≤d224∫01‍JΦσZ-α2IY,Ydσ≤14ε+α4′2α3α42ε0α33α42α4′-4Y,Y=14εα3α4α4′-2+12ε0α3Y2≤ε0α3Y2,since ε<α4′2/α3α4 by (9). Also(71)V5=-α4Y,Y-JHXY,Y+d1α4Z,Y-d1JHXY,Z=-α4I-JHXY,Y-d1α4I-JHXY,Z.Butλi(α4I-JH(X)) is nonnegative by (viii) and from (11), we get(72)V4≤d124α4I-JHXZ,Z≤14ε+1α12εD0α12Z,Z=14εα1+12εD0Z2≤εD0Z2,sinceε<1α1.Therefore(73)V˙≤-d2Y,GY-α4Y,Y+ε0α3Y2-α2Z,Z-d2F1Z2-d1JGZ2-d1W,FY,ZW-W,W+εD0Z2+d1W+Z+d2Y,∫t-rt‍JHXsYsds+d1W+Z+d2Y,∫t-rt‍JGYsZsds+μrY2-μ∫t-rt‍Yθ2dθ+λrZ2-λ∫t-rt‍Z(θ)2dθ.We know thatY,GY=Y,ΓYY and by Lemma 3, we get(74)dVdt≤-d2α3α42α4′2-α4Y2+ε0α3Y2-α2-d1JG-d2F1Z2+εD0Z2-α1d1-1W2+d1W+Z+d2Y,∫t-rt‍JHXsYsds+d1W+Z+d2Y,∫t-rt‍JGYsZsds+μrY2-μ∫t-rt‍Yθ2dθ+λrZ2-λ∫t-rt‍Zθ2dθ.SinceJH(X)≤α4n by (viii) and by using Cauchy-Schwartz inequality, we obtain(75)d1W+Z+d2Y,∫t-rt‍JHXsYsds≤d1W+Z+d2Y∫t-rt‍JHXsYsds≤d1W+Z+d2Y∫t-rt‍α4nYsds≤d1α4n2W2r+∫t-rt‍Ys2ds+α4n2Z2r+∫t-rt‍Ys2ds+d2α4n2Y2r+∫t-rt‍Y(s)2ds.Also, sinceJG(Y)≤α1α2n by (iii) and by using Cauchy-Schwartz inequality, we find(76)d1W+Z+d2Y,∫t-rt‍JGYsZsds≤d1W+Z+d2Y∫t-rt‍JGYsZsds≤d1W+Z+d2Y∫t-rt‍α1α2nZsds≤d1α1α2n2W2r+∫t-rt‍Zs2ds+α1α2n2Z2r+∫t-rt‍Zs2ds+d2α1α2n2Y2r+∫t-rt‍Z(s)2ds.Therefore it follows from (11) and (45) that(77)dVdt≤-α42α4′2ε-ε0α3-d2α4n2rα42α4′2ε-ε0-d2α1α2n2r-μrY2-Δ2α1α3α4-α4n2r-α1α2n2r-λrZ2-ε-d1α4n2r-d1α1α2n2rW2+d1α4n2+d2α4n2+α4n2-μ∫t-rt‍Ys2ds+d1α1α2n2+d2α1α2n2+α1α2n2-λ∫t-rt‍Z(s)2ds,and if we take(78)μ=α4n2d1+d2+1,λ=α1α2n2d1+d2+1,then we have(79)dVdt≤-α42α4′2ε-ε0α3-d2α4n2r-d2α1α2n2rα42α4′2ε-ε0-α4n2d1+d2+1rY2-Δ2α1α3α4-α4n2r-α1α2n2r-α1α2n2d1+d2+1rZ2-ε-d1α4n2r-d1α1α2n2rW2.Therefore if(80)r<min⁡εd1nα4+α1α2,((α42/α4′2)ε-ε0)α3α4n(d1+2d2+1)+α1α2d2nΔ2α1α3α4nα4+α1α2d1+d2+2,((α42/α4′2)ε-ε0)α3α4n(d1+2d2+1)+α1α2d2n,we obtain(81)dVdt(Xt,Yt,Zt,Wt)≤-αY2+Z2+W2,for some α>0. Therefore from (54), (64), and (81) the functional V(Xt,Yt,Zt,Wt) satisfies all the conditions of Theorem 1, so that the zero solution of (4) is uniformly stable.Thus the proof of Theorem2 is now complete.



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*Source: 618359-2015-04-02.xml*