# Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Authors: Minggao Xue; Shaobo Zhou; Shigeng Hu
Journal: Journal of Applied Mathematics
 (2010)

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## Abstract

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.


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## Body

## 1. Introduction

Stochastic modelling has played an important role in many areas of science and engineering for a long time. Some of the most frequent and most important stochastic models used when dynamical systems not only depend on present and past states but also involve derivatives with functionals are described by the following neutral stochastic functional differential equation:(1.1)d[x(t)-u(xt)]=f(xt,t)dt+g(xt,t)dw(t).
The conditions imposed on their studies are the standard uniform Lipschitz condition and the linear growth condition. The classical result is described by the following well-known Mao's test see [1, page 202, Theorem 2.2].Theorem 1.1.
Assume that there exist positive constantsK, L, and κ∈(0,1) such that(1.2)|f(φ,t)-f(ψ,t)|2∨|g(φ,t)-g(ψ,t)|2≤K∥φ-φ̅∥2,|f(φ,t)|2∨|g(φ,t)|2≤L(1+∥φ∥2),|u(φ)-u(ψ)|≤κ∥φ-ψ∥
for all φ,ψ∈C([-τ,0];Rn). Then there exists a unique solution x(t) to (1.1) with initial data ξ∈Cℱ0b([-τ,0];Rn) (i.e., ξ is an ℱ0-measurable C([-τ,0];Rn)-valued random variable such that E∥ξ∥<∞).Theorem1.1 requires that the coefficients f and g satisfy the Lipschitz condition and the linear growth condition. However, there are many NSFDEs that do not satisfy the linear growth condition. For example, the following nonlinear NSFDE:(1.3)d[x(t)-u(xt)]=x(t)[a+bσ1(xt)-x(t)2]dt+cx(t)σ2(xt)dw(t),|σ1(φ)|∨|σ2(φ)|<κ∫-τ0|φ(θ)|dμ(θ),
where coefficients f(x,xt,t)=x(t)[a+bσ1(xt)-x(t)2] and g=cx(t)σ2(xt) do not obey the linear growth condition although they are Lipschitz continuous. To the authors' best knowledge, there is so far no result that shows that (1.3) has a unique global solution for any initial data.On the other hand, we still encounter a new problem when we attempt to deduce the exponential decay of the solution even if there is no problem with the existence of the solution. For example, Mao [2] initiated the following study of exponential stability for NSFDEs employing the Razumikhin technique.Theorem 1.2.
Letc1,c2,λ,p be all positive numbers and q>(c2/c1)(1-κ)-p,κ∈(0,1), for any φ∈LFt0p([-τ,0];Rn),(1.4)E|u(φ)|p≤κp∥φ∥0p,
and assume that there exists a function V(x,t)∈C2,1(Rn×[-τ,∞);R+) such that
(1.5)c1|x|p≤V(x,t)≤c2|x|p
for all (x,t)∈Rn×[-τ,∞) and also for all t≥0(1.6)ELV(φ,t)≤-λEV(φ̃(0),t)
provided φ={φ(θ):-τ≤θ≤0}∈LFtp([-τ,0];Rn) satisfying
(1.7)EV(φ(θ),t+θ)≤qEV(φ̃(0),t)
for all -τ≤θ≤0. Then for all ξ∈Cℱ0b([-τ,0];Rn),t≥0(1.8)E|x(t;ξ)|p≤c2c1(1+κ1-κ1)pe-γtE∥ξ∥0p,κ1=κeγτ/p,
where γ=μ∧τ-1ln[q1(1+κq11/p)-p],q1=c1q/c2.It is very difficult to verify the conditions of Theorem1.2, and it is clear that ELV(φ,t)≤-λEV(φ̃(0),t) does not hold for many NSFDEs. In fact, for (1.3), if one chooses V(x,t)=x2, then (1.9)LV=2x(x-u(xt))(a+bσ1(xt)-x2)+c2x2σ22(xt).
Here, the polynomial x4 appears on the right-hand side, and it has an order of 4 which is higher than the order of V(x)=x2. More recently, Mao [3–5], Zhou et al. [6, 7], Yue et al. [8] and Shen et al. [9] provided with some useful criteria on the exponential stability employing the Lyapunov function, but their tests encounters the same problem.Therefore, we see that there is a necessity to develop new criteria for NSFDEs where the linear growth condition may not hold while the bound on the operatorLV may take a much more general form. In the paper, we will establish a Khasminskii-type test for NSFDEs that cover a wide class of highly nonlinear NSFDEs referring to Khasminskii-type theorems [10] and Mao and Rassias [11] results of stochastic delay differential equations. To our best knowledge, there is no such result for NSFDEs and stochastic functional differential equations (SFDEs).In the next section, we will establish a general existence and uniqueness theorem of the global solution to (1.1) after giving some necessary notations. Boundedness and Moment stability are given under the Khasminskii-type condition in Section 3. Section 4 establishes asymptotic stability theorem by using semimartingale convergence theory. Section 5 gives corresponding criteria for stochastic functional differential equations. Finally, several examples are given to illustrate our results.


## 2. Global Solution of NSFDEs

Throughout this paper, unless otherwise specified, we let(Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0, satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let w(t)=(w1(t),…,wm(t))T be an m-dimensional continuous local martingale with w(0)=0 defined on the probability space. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA),  while its operator norm is denoted by ∥A∥=sup{|Ax|:|x|=1}(without any confusion with ∥φ∥). C([-τ,0];Rn) denote the family of all continuous functions φ from [-τ,0] to Rn with the norm ∥φ∥=sup-τ≤θ≤0|φ(θ)|, where |·| is the Euclidean norm in Rn. Denoted by Cℱ0b([-τ,0];Rn) the family of all bounded, ℱ0-measurable, C([-τ,0];Rn)-valued random variables.Consider ann-dimensional neutral stochastic functional differential equation(2.1)d[x(t)-u(xt,t)]=f(xt,t)dt+g(xt,t)dw(t)
on t≥0 with initial data x0=ξ∈Cℱ0b([-τ,0];Rn),u:C([-τ,0];Rn)→Rn, and(2.2)f:C([-τ,0];Rn)×R+→Rn,g:C([-τ,0];Rn)×R+→Rn×m
are Borel measurable. Let x(t;ξ) denote the solution of (2.1) while xt={x(t+θ):-τ≤θ≤0} which is regarded as a C([-τ,0];Rn)-valued stochastic process, denoted by x̃(t)=x(t)-u(xt).LetC2,1(Rn×R+;R) denote the family of all nonnegative functions V(x,t) on Rn×R+ which are continuously twice differentiable in x and once differentiable in t. If V(x,t)∈C2,1(Rn×R+;R), define an operator LV:C([-τ,0];Rn)×R+ to R by (2.3)LV(φ,t)=Vt(φ̃(0),t)+Vx(φ̃(0),t)f(φ,t)+12trace(gT(φ,t)Vxx(φ̃(0),t)g(φ,t)),
where Vt(x,t)=∂V(x,t)/∂t,Vx(x,t)=(∂V(x,t)/∂x1,∂V(x,t)/∂x2,…,∂V(x,t)/∂xn), Vxx(x,t)=(∂2V(x,t)/∂xixj)n×n.For the purpose of stability, assume thatf(0,t)=g(0,t)=u(0,t)=0. This implies that (2.1) admits a trivial solution, x(0,t)=0. Furthermore, we impose the following assumptions.(H1) (The local Lipschitz condition).  For each integerR≥1, there is a positive constant KR such that
(2.4)|f(φ,t)-f(ψ,t)|2∨|g(φ,t)-g(ψ,t)|2≤KR∥φ-ψ∥2
for those φ,ψ∈C([-τ,0];Rn) with ∥φ∥∨∥ψ∥≤R and t∈R+.(H2)  There exists a positive constantκ∈(0,1) and a probability measure ν such that
(2.5)|u(φ,t)-u(ψ,t)|≤κ∫-τ0|φ-ψ|dν(θ)
for any φ,ψ∈C([-τ,0];Rn).(H3) There are two functionsV∈C2,1(Rn×[-τ,+∞);R+) and U∈C(Rn×[-τ,+∞)];R+) as well as positive constants λ1,λ2,c1,c2 and a probability measure μ on [-τ,0] such that
(2.6)c1|x|2≤V(x,t)≤c2|x|2,(2.7)LV(φ,t)≤λ1[1+V(φ(0),t)+∫-τ0(V(φ(θ),t+θ)+U(φ(θ),t+θ))dμ(θ)]-λ2U(φ(0),t)
for all -τ≤θ≤0,(φ,t)∈C([-τ,0];Rn)×R+.Remark 2.1.
In condition (2.7), we see that the function U(x,t) plays a key role in allowing coefficients f and g to be nonlinear functions.Theorem 2.2.
Assume that (H1), (H2), and (H3) hold. Then for any initial conditionξ∈Cℱ0b([-τ,0];Rn), there exists a unique global solution x(t) to (2.1) on t∈[-τ,∞). Moreover, the solution has the properties that(2.8)EV(x(t),t)<∞,E∫0tU(x(s),s)ds<∞
for any t≥0.Proof.
It is clear that for any initial dataξ∈Cℱ0b([-τ,0];Rn), there exists a unique maximal local solution x(t) on t∈[-τ,τe), where τe is the explosion time [1], by applying the standing truncation technique (see Mao [12, 13]) to (2.1). According to (H2), we have(2.9)|x̃(0)|≤|x(0)|+|u(x0,0)|≤|x(0)|+κ∫-τ0|x(θ)|dν(θ)≤(1+κ)∥ξ∥.Letk0>(1+κ)∥ξ∥ be sufficiently large such that (2.10)1k0<min-τ≤t≤0|x̃(t)|<max-τ≤t≤0|x̃(t)|<k0.
Define the stopping time (2.11)τk=inf{t∈[0,τe]:|x̃(t)|∈̅Ik},Ik≡(1k,k),k≥k0,
where throughout this paper, we set inf∅=∞(∅ denotes the empty sets). Clearly, τk is increasing as k→∞. Denote τ∞=limk→∞τk,τ∞≤τe a.s. We will show that τe=∞ a.s., which implies that x(t) is global.Itô formula and condition (2.7) yield(2.12)dV(x̃(t),t)=LV(x,t)dt+Vx(x̃,t)g(xt,t)dw(t)≤λ1[1+V(x(t),t)+∫-τ0V(xt,t+θ)dμ(θ)+∫-τ0U(xt,t+θ)dμ(θ)]dt-λ2U(x(t),t)dt+Vx(x̃,t)g(xt,t)dw(t)
for t≥0. For any k≥k0 and t∈[0,τ], we integrate both sides of (2.12) from 0 to τk∧t and then take the expectations to get(2.13)EV(x̃(τk∧t),τk∧t)-V(x̃(0),0)≤E∫0τk∧tλ1[1+∫-τ0(V(xs,s+θ)+U(xs,s+θ))dμ(θ)]ds+E∫0τk∧t[λ1V(x(s),s)-λ2U(x(s),s)]ds.
According to the integral substitution technique, we estimate(2.14)∫0τk∧t∫-τ0V(xs,s+θ)dμ(θ)ds=∫0τk∧t∫-τ0V(x(s+θ),s+θ)dμ(θ)ds≤∫-τ0dμ(θ)∫θτk∧t+θV(x(s),s)ds≤∫-ττk∧tV(x(s),s)ds≤∫-ττV(x(s),s)ds<+∞,
Similiarly,(2.15)∫0τk∧t∫-τ0U(xs,s+θ)dμ(θ)ds≤∫-ττU(x(s),s)ds<+∞.
Substituting for (2.14) and (2.15) into (2.13), and by using the Fubini theorem, the result is(2.16)EV(x̃(τk∧t),τk∧t)≤V(x̃(0),0)+λ1τ+λ1E∫0τk∧tV(x(s),s)ds-λ2E∫0τk∧tU(x(s),s)ds+λ1E∫-ττ[V(x(s),s)+U(x(s),s)]ds≤C1+λ1E∫0τk∧tV(x(s),s)ds≤C1+λ1∫0tEV(x(τk∧s),τk∧s)ds,
where C1=V(x̃(0),0)+λ1τ+λ1E∫-ττ[V(x(s),s)+U(x(s),s)]ds. Equations (2.6) and (2.9) imply V(x̃(0),0)≤c2(1+κ)2∥ξ∥2; thus, C1 is a finite constant. By using inequality (a+b)2≤(1/(1-κ0))a2+(1/κ0)b2,a,b>0,κ0∈(0,1); thus,(2.17)E|x(τk∧t)|2≤(1-κ0)-1E|x̃(τk∧t)|2+κ0-1E|u(xτk∧t,τk∧t)|2,
condition (2.6) yields(2.18)c2-1V(x̃(τk∧t),τk∧t)≤|x̃(τk∧t)|2≤c1-1V(x̃(τk∧t),τk∧t).
(H2) and the Hölder inequality yield(2.19)E|u(xτk∧t,τk∧t)|2≤κ2E(∫-τ0|φ(θ)|dν(θ))2≤κ2∫-τ0E|x(τk∧t+θ)|2dν(θ).
Substituting for (2.16), (2.18), and (2.19) into (2.17), the result is(2.20)E|x(τk∧t)|2≤(1-κ0)-1c1-1(C1+λ1∫0tEV(x(τk∧s),τk∧s)ds)+κ0-1κ2∫-τ0E|x(τk∧t+θ)|2dν(θ).
For any t∈[-τ,τ], (2.20) implies (2.21)sup-τ≤s≤tE|x(τk∧s)|2≤(1-κ0)-1c1-1(C1+λ1∫0tEV(x(τk∧s),τk∧s)ds)+κ0-1κ2sup-τ≤s≤tE|x(τk∧s)|2.
Let κ0=κ, then (2.22)sup-τ≤s≤tE|x(τk∧s)|2≤(1-κ)-2c1-1(C1+λ1∫0tEV(x(τk∧s),τk∧s)ds).
Therefore, for any t∈[-τ,τ],(2.23)E|x(τk∧t)|2≤(1-κ)-2c1-1(C1+λ1∫0tEV(x(τk∧s),τk∧s)ds).
By (2.6), we may obtain (2.24)EV(x(τk∧t),τk∧t)≤c2E|x(τk∧t)|2≤(1-κ)-2c1-1c2(C1+λ1∫0tEV(x(τk∧s),τk∧s)ds).
For any k≥k0,t∈[0,τ], the Gronwall inequality implies (2.25)EV(x(τk∧τ),τk∧τ)≤c1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2t.
Thus, for all k≥k0,(2.26)EV(x(τk∧τ),τk∧τ)≤c1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2τ,
which implies(2.27)EV(x(t),t)≤c1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2τ,0≤t≤τ.
Since E[I{τk≤τ}V(x(τk∧τ),τk∧τ)]≤EV(x(τk∧τ),τk∧τ),and defining μk=inf|x|≥k,0≤t<∞V(x,t) for k≥k0, according to (2.26), then(2.28)μkP(τk≤τ)≤c1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2τ.
Clearly, condition (2.6) implies limk→∞μk=∞. Letting k→∞ in (2.28), then P(τ∞≤τ)=0, namely,(2.29)P(τ∞>τ)=1.
Moreover, setting t=τ in (2.16), we may obtain that (2.30)λ2E∫0τk∧τU(x(s),s)ds≤C1+λ1E∫0τk∧τV(x(s),s)ds≤C1+λ1τc1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2τ,
that is,(2.31)E∫0τU(x(s),s)ds≤C1λ2(1+λ1τc1-1c2C1(1-κ)-2ec1-1c2λ1(1-κ)-2τ)<∞.
Let us now proceed to prove τ∞>2τ a.s. given that we have shown (2.27)–(2.31). For any k≥k0 and t∈[0,2τ], we can integrate both sides of (2.12) from 0 to τk∧t and then take expectations to get(2.32)EV(x̃(τk∧t),τk∧t)≤C2+λ1E∫0τk∧tV(x(s),s)ds-λ2E∫0τk∧tU(x(s),s)ds,
where (2.33)C2=V(x̃(0))+2λ1τ+λ1E∫-τ2τ[V(x(s),s)+U(x(s),s)]ds<∞.
By the Gronwall inequality and (2.32), we have (2.34)EV(x̃(τk∧t),τk∧t)≤c1-1c2C2(1-κ)-2ec1-1c2λ1(1-κ)-2t,0≤t≤2τ,k≥k0.
In particular,(2.35)EV(x̃(τk∧2τ),τk∧2τ)≤c1-1c2C2(1-κ)-2ec1-1c2λ1(1-κ)-22τ,∀k≥k0.
This implies (2.36)μkP(τk≤2τ)≤c1-1c2C2(1-κ)-2ec1-1c2λ1(1-κ)-22τ.
Letting k→∞, by (2.6), then P(τ∞≤2τ)=0, that is, (2.37)P(τ∞>2τ)=1,EV(x(t),t)≤c1-1c2C2(1-κ)-2ec1-1c2λ1(1-κ)-22τ,0≤t≤2τ.
By (2.32), we may obtain that (2.38)λ2E∫0τk∧2τU(x,t)dt≤C2+λ1E∫0τk∧2τV(x(t),t)dt,
that is, (2.39)E∫02τU(x,t)dt≤C2λ2(1+2λ1τc1-1c2(1-κ)-2ec1-1c2λ1(1-κ)-22τ)<∞.
Repeating this procedure, we can show that, for any integer i≥1,τ∞>iτ a.s. and EV(x(t),t)≤c1-1c2Ci(1-κ)-2ec1-1c2(1-κ)-2λ1iτ,0≤t<iτ, and (2.40)E∫0iτU(x,t)dt≤Ciλ2(1+iλ1τc1-1c2(1-κ)-2ec1-1c2(1-κ)-2λ1iτ),
where (2.41)Ci=V(x̃(0),0)+λ1E∫-τiτ[1+V(x,t)+U(x,t)]dt<∞.
We must therefore have τ∞=∞ a.s. as well as the required assertion.Note that condition (2.6) may be replaced by more general condition c1|x|p≤V(x,t)≤c2|x|p,p≥2, which is suitable to the corresponding results below.


## 3. Boundedness and Moment Stability

In the previous section, we have shown that the solution of (2.1) has the properties that (3.1)EV(x(t),t)<∞,E∫0tU(x(s),s)ds<∞
for any t≥0. In the following, we will give more precise estimations under specified conditions; that is, we will establish the criteria of moment stability and asymptotic stability of the solution to (2.1) under specified conditions.Theorem 3.1.
Assume that (H1), (H2), and (H3) hold except (2.7) which is replaced by(3.2)LV(φ,t)≤μ1-μ2V(φ(0),t)+μ3∫-τ0V(φ,t+θ)dη1(θ)-μ4U(φ(0),t)+μ5∫-τ0U(φ,t+θ)dη2(θ)-μ6V(φ̃(0),t)
for all (φ,t)∈C([-τ,0];Rn)×R+,-τ≤θ≤0, where μ1≥0,μ2>μ3≥0,μ4>μ5>0,μ6>0 are constants and η1(θ) and η2(θ) are probability measures on [-τ,0]. Then for any initial data ξ, the global solution x(t) to (2.1) has the property that
(3.3)limsupt→∞EV(x(t),t)<c2μ1(1-κ0)2c1ε,
where ε=μ6∧ε1∧ε2∧τ-1logκ-2,κ0=κeετ, while ε1>0 and ε2>0 are the unique roots to the following equations:
(3.4)μ2=μ3eε1τ,μ4=μ5eε2τ,
respectively. If μ1=0, then
(3.5)limsupt→∞1tlnEV(x(t),t)<-ε,∫0∞EU(x(t),t)dt<∞.Proof.
We first observe that (3.2) implies (2.7) if we set λ1=μ1∨μ3∨μ5 and λ2=μ4. So, for any initial data, (2.1) has a unique global solution x(t) on t≥-τ, which has the properties (2.8). Based on these properties, we can apply the Itô formula and condition (3.2) to obtain that for any t≥0,(3.6)d[eεtV(x̃(t),t)]=eεt[εV(x̃(t),t)+LV(x(t),t)]dt+eεtVx(x̃(t),t)g(xt,t)dw(t)≤eεt[μ1-μ2V(x,t)+μ3∫-τ0V(xt,t+θ)dη1(θ)-μ4U(x,t)+μ5∫-τ0U(xt,t+θ)dη2(θ)]+eεtVx(x̃(t),t)g(xt,t)dw(t)-eεt(μ6-ε)V(x̃(t),t),
We integrate both sides of the above inequality from 0 to t and take expectations to get
(3.7)eεtEV(x̃(t),t)≤V(x̃(0),0)+μ1eεtε-μ2E∫0teεsV(x(s),s)ds+μ3E∫0t∫-τ0eεsV(xs,s+θ)dη1(θ)ds-μ4E∫0teεsU(x(s),s)ds+μ5E∫0t∫-τ0eεsU(xs,s+θ)dη2(θ)ds,
by using of ε=μ6∧ε1∧ε2<μ6. Compute
(3.8)E∫0t∫-τ0eεsV(xs,s+θ)dη1(θ)ds=E∫0t∫-τ0eεsV(x(s+θ),s+θ)dη1(θ)ds≤eετE∫0t∫-τ0eε(s+θ)V(x(s+θ),s+θ)dη1(s+θ)ds≤eετE∫-τ0dη1(θ)∫θt+θeεsV(x(s),s)dds≤eετE∫-τteεsV(x(s),s)ds≤eετE∫-τ0eεsV(x(s),s)ds+eετE∫0teεsV(x(s),s)ds,
Similiarly,
(3.9)E∫0t∫-τ0eεsU(xs,s+θ)dη2(θ)ds≤eετE∫-τ0eεsU(x(s),s)ds+eετE∫0teεsU(x(s),s)ds.
Substituting for (3.8) and (3.9) into (3.7), the result is
(3.10)eεtEV(x̃(t),t)≤V(x̃(0),0)+μ1eεtε-μ2E∫0teεsV(x(s),s)ds+μ3eετE∫-τ0eεsV(x(s),s)ds+μ3eετE∫0teεsV(x(s),s)ds-μ4E∫0teεsU(x,s)ds+μ5eετE∫-τ0eεsU(x(s),s)ds+μ5eετE∫0teε(s)U(x(s),s)ds=V(x̃(0),0)+μ1eεtε+eετE∫-τ0eεs(μ3V(x(s),s)+μ5U(x(s),s))ds-(μ2-μ3eετ)E∫0teεsV(x(s),s)ds-(μ4-μ5eετ)E∫0teεsU(x,s)ds=C+μ1eεtε-(μ2-μ3eετ)E∫0teεsV(x(s),s)ds-(μ4-μ5eετ)E∫0teεsU(x,s)ds,
where C=V(x̃(0),0)+eετE∫-τ0eεs(μ3V(x(s),s)+μ5U(x(s),s))ds. It is clear that, for ε≤ε1∧ε2, we have μ2-μ3eετ≥0,μ4-μ5eετ≥0, hence,
(3.11)eεtEV(x̃(t),t)≤C+μ1eεtε.
By (H2) and (H3) and inequality (a+b)2≤(1/(1-κ0))a2+(1/κ0)b2,a,b>0,κ0∈(0,1), we compute
(3.12)Eeεt|x(t)|2≤(1-κ0)-1Eeεt|x̃(t)|2+κ0-1Eeεt|u(xt,t)|2≤(1-κ0)-1c1-1eεtE|V(x̃(t),t)|2+κ0-1κ2∫-τ0eεtE|x(t+θ)|2dν(θ)≤(1-κ0)-1c1-1(C+μ1εeεt)+κ0-1κ2∫-τ0e-εθEeε(t+θ)|x(t+θ)|2dν(θ).
For any t≥0,(3.13)sup-τ≤s≤tEeεt|x(t)|2≤(1-κ0)-1c1-1(C+μ1εeεt)+κ0-1κ2sup-τ≤s≤tEeεt|x(t)|2∫-τ0e-εθdν(θ)≤(1-κ0)-1c1-1(C+μ1εeεt)+κ0-1κ2eετsup-τ≤s≤tEeεt|x(t)|2.
Leting κ0=κeετ<1, since ε<τ-1logκ-2, then κ0<1,(3.14)sup-τ≤s≤tEeεt|x(t)|2≤(1-κ0)-2c1-1(C+μ1εeεt),
and by (H3), EV(x,t)≤c2E|x(t)|2, then
(3.15)sup-τ≤s≤teεtEV(x,t)≤c2sup-τ≤s≤teεtE|x(t)|2≤(1-κ0)-2c1-1c2(C+μ1εeεt).
Therefore,
(3.16)limsupt→∞EV(x(t),t)≤c2μ1(1-κ0)2c1ε.
When μ1=0, then EV(x(t),t)≤c1-1c2C(1-κ0)-2e-εt,forallt≥0, that is,
(3.17)limsupt→∞1tlogEV(x(t),t)≤-ε.
On the other hand, when μ1=0, by (3.7) and the Itô formula, we may show easily that
(3.18)EV(x̃(t),t)=V(x̃(0),0)+E∫-τ0(μ3V(x(s),s)+μ5U(x(s),s))d(s)-(μ2-μ3)E∫0tV(x(s),s)ds-(μ4-μ5)E∫0tU(x(s),s)ds.
By μ2=μ3eε1τ>μ3,μ4=μ5eε2τ>μ5, and the Fubini theorem, we obtain
(3.19)∫0tEU(x,s)≤1μ4-μ5[V(x̃(0),0)+E∫-τ0(μ3V(x(s),s)+μ5U(x(s),s))d(s)]<∞.
The proof is complete.


## 4. Asymptotic Stability

In this section, we will establish asymptotic stability of (2.1) without the linear growth condition. It is well known that the linear growth condition is one of the most important conditions to guarantee asymptotic stability. Therefore we introduce the following semitingale convergence thoerem [14, 15], which will play a key role in dealing with nonlinear systems.Lemma 4.1.
LetM(t) be a real-valued local martingale with M(0)=0 a.s. Let ζ be a nonnegative ℱ0-measurable random variable. If X(t) is a nonnegative continuous ℱt-adapted process and satisfies X(t)≤ζ+M(t) for t≥0, then X(t) is almost surely bounded, namely, limt→∞X(t)<∞, a.s.Theorem 4.2.
Assume that (H1), (H2), and (H3) hold except (2.7) which is replaced by(4.1)LV(φ,t)≤-μ2V(φ(0),t)+μ3∫-τ0V(φ,t+θ)dη1(θ)-μ4U(φ(0),t)+μ5∫-τ0U(φ,t+θ)dη2(θ)-μ6V(φ̃(0),t)
for all (φ,t)∈Rn×R+,-τ≤θ≤0, where μ2>μ3≥0,μ4>μ5>0,μ6>0. Then, for any initial data, the unique global solution x(t) of (2.1) has the property that
(4.2)limsupt→∞1tlnV(x(t),t)≤-ε,∫0∞U(x(t),t)dt<∞,
where ε=μ6∧ε1∧ε2∧τ-1logκ-2, while ε1>0 and ε2>0 are the unique roots to the following equations:
(4.3)μ2=μ3eε1τ,μ4=μ5eε2τ,
respectively.Proof.
We first observe that (4.1) implies (2.7) if we set λ1=μ1∨μ3∨μ5 and λ2=μ4. So, for any initial data, (2.1) has a unique global solution x(t) on t≥-τ, which has the properties (2.8). Similar to the proof of Theorem 3.1, applying the Itô formula and condition (4.1), for any t≥0, we may obtain that(4.4)d[eεtV(x̃(t),t)]=eεt[εV(x̃(t),t)+LV(x(t),t)]dt+eεtVx(x̃(t),t)g(xt,t)dw(t)≤eεt[-μ2V(x(t),t)+μ3∫-τ0V(xt,t+θ)dη1(θ)-μ4U(x,t)+μ5∫-τ0U(xt,t+θ)dη2(θ)]+eεtVx(x̃(t),t)g(xt,t)dw(t)-(μ6-ε)V(x̃(t),t).
For t>0, we can integrate both sides of the above inequality from 0 to t and take expectations to get
(4.5)eεtEV(x̃(t),t)≤V(x̃(0),0)-μ2∫0teεsV(x(s),s)ds+μ3∫0t∫-τ0eεsV(xs,s+θ)dη1(θ)ds-μ4∫0teεsU(x,s)ds+μ5∫0t∫-τ0eεsU(xs,s+θ)dη2(θ)ds+M(t),
where M(t)=∫0teεsVx(x̃(s),s)g(xs,s)dw(s)ds is a real-valued continuous local martingale with M(0)=0. Similar to Theorem 3.1, we have
(4.6)eεtV(x̃(t),t)≤V(x̃(0),0)+eετ∫-τ0(μ3V(x(s),s)+μ5U(x(s),s))ds-(μ2-μ3eετ)∫0teεsV(x(s),s)ds-(μ4-μ5eετ)∫0teεsU(x,s)ds+M(t)≤const+M(t).
Lemma 4.1 implies
(4.7)limsupt→∞eεtV(x̃(t),t)<∞a.s.
Since c1|x|2≤V(x,t)≤c2|x|2, then
(4.8)limsupt→∞eεt|x̃(t)|2<∞a.s.
According to the definition of x̃(t), we compute
(4.9)eεt|x(t)|2=eεt|x̃(t)+u(xt,t)|2≤(1-κ0)-1eεt|x̃(t)|2+κ0-1eεt|u(xt,t)|2≤(1-κ0)-1eεt|x̃(t)|2+κ0-1κ2∫-τ0e-εθeε(t+θ)|x(t+θ)|2dν(θ).
Therefore, we may also compute
(4.10)sup-τ≤s≤teεs|x(s)|2≤∥ξ∥2+sup0≤s≤teεs|x(s)|2≤∥ξ∥2+(1-κ0)-1sup0≤s≤t[eεs|x̃(s)|2+κ0-1κ2∫-τ0e-εθeε(s+θ)|x(s+θ)|2dν(θ)]≤∥ξ∥2+(1-κ0)-1sup0≤s≤teεs|x̃(s)|2+κ0-1κ2sup-τ≤s≤teεs|x(s)|2∫-τ0e-εθdν(θ)≤∥ξ∥2+(1-κ0)-1sup0≤s≤teεs|x̃(s)|2+κ0-1κ2eετsup-τ≤s≤teεs|x(s)|2.
Noting that ε<τ-1logκ-2, choose κ0=κeετ. Then κ0<1, and we obtain
(4.11)sup-τ≤s≤teεs|x(s)|2≤(1-κ0)-1∥ξ∥2+(1-κ0)-2sup0≤s≤teεs|x̃(s)|2.
(4.8) and (4.11) yield
(4.12)sup-τ≤s≤teεs|x(s)|2<+∞a.s.
Recall the condition c1|x|2≤V(x,t)≤c2|x|2, which implies limsupt→∞eεtV(x(t),t)<Ca.s. The required result is obtained.Remark 4.3.
From the processes of the proof of Theorems3.1 and 4.2, we see that condition (2.6) plays an important role in dealing with the neutral term. Moreover, applying condition (2.6), we can also obtain more precise results(4.13)limsupt→∞1tlogE|x(t)|≤-ε2,limsupt→∞1tlog|x(t)|≤-ε2.
In the next section, condition (2.6) will be replaced by a more general condition for stochastic functional differential equation.


## 5. Stochastic Functional Differential Equation

Letu(xt)=0. Then (2.1) reduces to(5.1)dx(t)=f(xt,t)dt+g(xt,t)dw(t).
This is a stochastic functional differential equation. In this section, we will give the corresponding results for stochastic functional differential equation. We will also see that the conditions are more general.Define an operatorLV from C([-τ,0];Rn)×R+ to R by (5.2)LV(φ,t)=Vt(φ(0),t)+Vx(φ(0),t)f(φ,t)+12trace(gT(φ,t)Vxx(φ(0),t)g(φ,t)).
We impose the following assumption which is more general than (H3).(H3′) There are two functions V∈C2,1(Rn×[-τ,+∞);R+) and U∈C(Rn×[-τ,+∞);R+) as well as two positive constants λ1,λ2 and a probability measure μ on [-τ,0] such that
(5.3)lim|x|→∞inf0≤t<∞V(x,t)=∞,(5.4)LV(φ,t)≤λ1[1+V(φ(0),t)+∫-τ0(V(φ(θ),t+θ)+U(φ(θ),t+θ))dμ(θ)]-λ2U(φ(0),t)
for all -τ≤θ≤0,(φ,t)∈Rn×R+.Theorem 5.1.
Assume that (H1) and (H3′) hold. Then for any initial condition ξ∈Cℱ0b([-τ,0];Rn), there exists a unique global solution x(t) of (5.1) on t∈[-τ,∞). Moreover, the solution has the properties that(5.5)EV(x(t),t)<∞,E∫0tU(x(s),s)ds<∞
for any t≥0.Proof.
Since the proof is similar to Theorem2.2, we will only outline the proof. It is clear that for any initial data ξ∈Cℱ0b([-τ,0];Rn), there is a unique maximal local solution x(t) on t∈[-τ,τe), where τe is the explosion time [1]. Let k0>0 be sufficiently large for(5.6)1k0<min-τ≤t≤0|x(t)|<max-τ≤t≤0|x(t)|<k0.
Define the stopping time
(5.7)τk=inf{t∈[0,τe]:|x(t)|∈̅Ik},Ik≡(1k,k),k≥k0,
where throughout this paper, we set inf∅=∞(∅denotestheemptysets). Clearly,τk is increasing as k→∞. Denote τ∞=limk→∞τk,τ∞≤τe a.s. We will show that τ∞=∞ a.s., which implies τe=∞ a.s. By Itô formula and (5.4), for any k≥k0 and t∈[0,τ], we obtain
(5.8)EV(x(τk∧t),τk∧t)≤C1+λ1∫0tEV(x(τk∧s),τk∧s)ds-λ2∫0tEU(x(τk∧s),τk∧s)ds,
where C1=V(x(0),0)+λ1τ+λ1E∫-ττ[V(x(s),s)+U(x(s),s)]ds. For any k≥k0, the Gronwall inequality yields
(5.9)E[I{τk≤τ}V(x(τk∧τ),τk∧τ)]≤EV(x(τk∧τ),τk∧τ)≤C1eλ1τ,∀k≥k0,
which implies
(5.10)EV(x(t),t)≤C1eλ1τ,0≤t≤τ.
Defining μk=inf|x|≥k,0≤t<∞V(x,t) for k≥k0, according to (5.3), then
(5.11)μkP(τk≤τ)≤C1eλ1τ.
Condition (5.3) implies limk→∞μk=∞. Letting k→∞ in (5.11), then P(τ∞≤τ)=0, namely,
(5.12)P(τ∞>τ)=1.
Moreover, setting t=τ in (5.8), we may obtain that
(5.13)λ2E∫0τk∧τU(x,s)ds≤C1+λ1E∫0τk∧τV(x(s),s)ds≤C1+λ1C1τeλ1τ,
that is,
(5.14)E∫0τU(x,s)ds≤C1λ2(1+λ1τeλ1τ)<∞.
Let us now proceed to prove τ∞>2τ a.s. given that we have shown (5.10)–(5.14). For any k≥k0 and t∈[0,2τ], we get
(5.15)EV(x(τk∧t),τk∧t)≤C2+λ1E∫0τk∧tV(x(s),s)ds-λ2E∫0τk∧tU(x(s),s)ds,
where
(5.16)C2=V(x(0))+2λ1τ+λ1E∫-τ2τ[V(x(s),s)+U(x(s),s)]ds<∞.
By the Gronwall inequality and (5.8), we have
(5.17)EV(x(τk∧t),τk∧t)≤C2e2λ1τ,0≤t≤2τ,k≥k0.
In particular,
(5.18)EV(x(τk∧2τ),τk∧2τ)≤C2e2λ1τ,∀k≥k0.
This implies
(5.19)μkP(τk≤2τ)≤C2e2λ1τ.
Letting k→∞, by (5.3), then P(τ∞≤2τ)=0, that is,
(5.20)P(τ∞>2τ)=1.EV(x(t),t)≤C2e2λ1τ,0≤t≤2τ.
By (5.8), we may also obtain that
(5.21)λ2E∫0τk∧2τU(x,t)dt≤C2+λ1E∫0τk∧2τV(x(t),t)dt,
that is,
(5.22)E∫02τU(x,t)dt≤C2λ2(1+2λ1τe2λ1τ)<∞.
Repeating this procedure, we can show that, for any integer i≥1,τ∞>iτ a.s. and EV(x)≤Cieiλ1τ,0≤t<iτ, and
(5.23)E∫0iτU(x,t)dt≤Ciλ2(1+iλ1τeiλ1τ),
where
(5.24)Ci=V(x(0),0)+λ1E∫-τiτ[1+V(x,t)+U(x,t)]dt<∞.
We must therefore have τ∞=∞ a.s. as well as the required assertion (5.5).Theorem 5.2.
Assume that (H1) and (H3′) hold except (5.4) which is replaced by(5.25)LV(φ,t)≤μ1-μ2V(φ(0),t)+μ3∫-τ0V(φ,t+θ)dη1(θ)-μ4U(φ(0),t)+μ5∫-τ0U(φ,t+θ)dη2(θ)
for all (φ,t)∈Rn×R+,-τ≤θ≤0, where μ1≥0,μ2>μ3≥0,μ4>μ5>0. Then for any initial data, the global solution x(t) to (5.1) has the property that
(5.26)limsupt→∞EV(x(t),t)≤μ1ε,
where ε=ε1∧ε2, while ε1>0 and ε2>0 are the unique roots to the following equations:
(5.27)μ2=μ3eε1τ,μ4=μ5eε2τ,
respectively. If μ1=0, then
(5.28)limsupt→∞1tlnEV(x(t),t)≤-ε,∫0∞EU(x(t),t)dt<∞.Proof.
Since the proof is similar to Theorem3.1, we will only outline the proof. We first observe that (5.25) implies (5.4) if we set λ1=μ1∨μ3∨μ5 and λ2=μ4. So for any initial data, (5.1) has a unique global solution x(t) on t≥-τ, which has the properties (5.5). Based on these properties, we can apply the Itô formula and condition (5.4) to obtain that for any t≥0,(5.29)eεtEV(x(t),t)≤V(x(0),0)+μ1eεtε-μ2E∫0teεsV(x(s),s)ds+μ3E∫0t∫-τ0eεsV(xs,s+θ)dη1(θ)ds-μ4E∫0teεsU(x,s)ds+μ5E∫0t∫-τ0eεsU(xs,s+θ)dη2(θ)ds.
Applying for (3.8) and (3.9), similarly, we have
(5.30)eεtEV(x(t),t)≤V(x(0),0)+μ1eεtε-μ2E∫0teεsV(x(s),s)ds+μ3eετE∫-τ0eεsV(x(s),s)ds+μ3eετE∫0teεsV(x(s),s)ds-μ4E∫0teεsU(x,s)ds+μ5eετE∫-τ0eεsU(x(s),s)ds+μ5eετE∫0teε(s)U(x(s),s)ds=C¯+μ1eεtε-(μ2-μ3eετ)E∫0teεsV(x(s),s)ds-(μ4-μ5eετ)E∫0teεsU(x,s)ds,
where C¯=V(x(0),0)+eετE∫-τ0eεs(μ3V(x(s),s)+μ5U(x(s),s))ds. It is clear that, for ε≤ε1∧ε2, we have μ2-μ3eετ≥0,μ4-μ5eετ≥0; hence,
(5.31)eεtEV(x(t),t)≤C¯+μ1eεtε,
that is,
(5.32)EV(x(t),t)≤C¯e-εt+μ1ε,∀t≥0.
Therefore
(5.33)limsupt→∞EV(x(t),t)≤μ1ε.
When μ1=0, then EV(x(t),t)≤C¯e-εt,forallt≥0, that is,
(5.34)limsupt→∞1tlogEV(x(t),t)≤-ε.
On the other hand, when μ1=0, we may show easily that
(5.35)EV(x(t),t)=V(x(0),0)+E∫-τ0(μ3V(x(s),s)+μ5U(x(s),s))ds-(μ2-μ3)E∫0tV(x(s),s)ds-(μ4-μ5)E∫0tU(x(s),s)ds.
Recalling that μ2=μ3eε1τ>μ3,μ4=μ5eε2τ>μ5, the Fubini theorem yields
(5.36)∫0tEU(x,s)≤1μ4-μ5[V(x(0),0)+E∫-τ0(μ3V(x(s),s)+μ5U(x(s),s))ds]<∞.
The proof is complete.Theorem 5.3.
Assume that (H1) and (H3′) hold except (5.3) which is replaced by(5.37)LV(φ,t)≤-μ2V(φ(0),t)+μ3∫-τ0V(φ,t+θ)dη1(θ)-μ4U(φ(0),t)+μ5∫-τ0U(φ,t+θ)dη2(θ)
for all (φ,t)∈Rn×R+,-τ≤θ≤0, where μ2>μ3≥0,μ4>μ5>0. Then for any initial data, the unique global solution x(t) to (5.1) has the property that
(5.38)limsupt→∞1tlnV(x(t),t)≤-ε,∫0∞U(x(t),t)dt<∞,
where ε=ε1∧ε2, while ε1>0 and ε2>0 are the unique roots to the following equations
(5.39)μ2=μ3eε1τ,μ4=μ5eε2τ,
respectively.Proof.
It is clear that (5.1) has a unique global solution x(t) on t≥-τ, which has the properties (2.8). For any t≥0, we can obtain(5.40)eεtEV(x(t),t)≤V(x(0),0)-μ2∫0teεsV(x(s),s)ds+μ3∫0t∫-τ0eεsV(xs,s+θ)dη1(θ)ds-μ4∫0teεsU(x,s)ds+μ5∫0t∫-τ0eεsU(xs,s+θ)dη2(θ)ds+M(t),
where M(t)=∫0teεsVx(x̃(s),s)g(x,xs,s)dw(s)ds is a real-valued continuous local martingale with M(0)=0. Similar to Theorem 4.2,
(5.41)eεtV(x(t),t)≤C-(μ2-μ3eετ)∫0teεsV(x(s),s)ds-(μ4-μ5eετ)∫0teεsU(x,s)ds+M(t)≤const+M(t).
By Lemma 4.1, we have
(5.42)limsupt→∞eεtV(x(t),t)≤∞a.s.
The required result is obtained.


## 6. Example

In the following, we will consider several examples to illustrates our ideas.Example 6.1   6.1.
Consider a one-dimensional SFDE(6.1)dx(t)=x(t)[(a+bσ1(xt)-x2(t))dt+cσ2(xt)dw(t)],
where w(t) is a one-dimensional Brownian motion, a,b,c(b,c>0) are bounded real numbers, and the functions σ1,σ2∈C([-τ,0];R) having the property of
(6.2)|σ1(φ)|∨|σ2(φ)|≤κ∫-τ0|φ(θ)|dμ(θ)κ∈(0,1).
Let V(x)=x2. Then the corresponding operator LV:R×R×R+ has the form
(6.3)LV(xt,t)=2x2(t)(a+bσ1(xt)-x2(t))+c2x2(t)σ22(xt)=2ax2(t)+2bx2(t)σ1(xt)-2x4(t)+c2x2(t)σ22(xt)≤2ax2(t)+bx4(t)+bσ12(xt)-2x4(t)+0.5c2x4(t)+0.5c2σ24(xt)≤2ax2(t)+bσ12(xt)+0.5c2σ24(xt)-(2-b-0.5c2)x4(t)≤2ax2(t)+bκ2∫-τ0xt2(θ)dμ(θ)+0.5c2κ4∫-τ0xt4(θ)dμ(θ)-(2-b-0.5c2)x4(t),
where λ1=max{2a,bκ2,0.5c2κ4},λ2=2-b-0.5c2,U(x)=x4. If 2-b-0.5c2>0, then by Theorem 5.1, we can conclude that for any initial data {x(t):-τ≤t≤0}∈C([-τ,0];R), there is a unique global solution x(t) to (6.1) on t∈[-τ,∞). Moreover, the solution has the properties that
(6.4)E|x|2<∞,E∫0t|x(s)|4ds<∞
for any t≥0. If a<0,2-b-0.5c2>0.5c2κ4>0,-2a>bκ2≥0,U(x)=x4,ε1>0 and ε2>0 will be the unique roots to the following equations:
(6.5)-2a=bκ2eε1τ,2-b-0.5c2=0.5c2κ4eε2τ,
respectively. Set ε=ε1∧ε2, by Theorem 5.2, we can conclude that the unique global solution of (6.1) has the property that
(6.6)limsupt→∞1tlnE|x(t)|2≤-ε,limsupt→∞1tln|x(t)|2≤-ε,∫0∞Ex4(t)dt<+∞.
If we choose a=-2,b=1,c=1,κ=0.5,τ=8, then ε1=0.34657,ε2=0.34657, which implies
(6.7)limsupt→∞1tlnE|x(t)|2<-0.34657.Example 6.2   6.2.
Consider a one-dimensional NSFDE(6.8)d(x(t)-σ1(xt))=x(t)[(a-x2(t))dt+cσ2(xt)dw(t)],
where w(t) is a one-dimensional Brownian motion and both a,c are bounded positive real numbers, σ1,σ2∈C([-τ,0];R) having the property of
(6.9)|σ1(φ)|∨|σ2(φ)|≤κ∫-τ0|φ(θ)|dμ(θ)κ∈(0,1).
Let V(x)=x2. Then the corresponding operator LV:R×R×R+ has the form
(6.10)LV(x,xt,t)=2(x(t)-σ1(xt))(a-x2(t))x(t)+c2x2(t)σ22(xt)=2ax2(t)-2ax(t)σ1(xt)-2x4(t)+2x3(t)σ1(xt)+c2x2(t)σ22(xt)≤2ax2(t)+ax2(t)+aσ12(xt)-2x4(t)+2(x4(t))3/4(σ14(xt))1/4+0.5c2(x4(t)+σ24(xt))≤2ax2(t)+ax2(t)+aσ12(xt)-2x4(t)+1.5x4(t)+0.5σ14(xt)+0.5c2(x4(t)+σ24(xt))=3ax2(t)+aκ2∫-τ0‍xt2(θ)dμ(θ)+(0.5+0.5c2)κ4∫-τ0xt4(θ)dμ(θ)-0.5(1-c2)x4(t)=λ1(x2(t)+∫-τ0xt2(θ)dμ(θ)+∫-τ0xt4(θ)dμ(θ))-λ2x4(t),
where λ1=max{3a,aκ2,(0.5+0.5c2)κ4},λ2=0.5(1-c2),U(x)=x4. By Theorem 2.2, we can conclude that for any initial data, there is a unique global solution x(t) to (6.8) on t∈[-τ,∞). Moreover, the solution has the properties that for any t≥0(6.11)E|x(t)|2<∞,E∫0t|x(s)|4ds<∞.Example 6.3   6.3.
Consider a one-dimensional NSFDE(6.12)d[x(t)-σ1(xt)]=[(a-b)x(t)-aσ1(xt)-x(t)3]dt+cσ2(xt)dw(t),
where w(t) is a one-dimensional Brownian motion, a,b,c(b,c>0) are real numbers, σ1,σ2∈C([-τ,0];R) having the property of
(6.13)|σ1(φ)|∨|σ2(φ)|≤κ∫-τ0|φ(θ)|dμ(θ)κ∈(0,1).
Then, the corresponding operator LV has the form
(6.14)LV(x,xt,t)=2(x(t)-σ1(xt))[(a-b)x(t)-aσ1(xt)-x(t)3]+c2σ22(xt)=2a(x(t)-σ1(xt))2-2bx2(t)+2bx(t)σ1(xt)-2x4(t)+2x3(t)σ1(xt)+c2σ22(xt)≤2a(x(t)-σ1(xt))2-2bx2(t)+bx2(t)+bσ12(xt)-2x4(t)+2(x4(t))3/4(σ14(xt))1/4+c2σ22(xt)≤2a(x(t)-σ1(xt))2-bx2(t)+bσ12(xt)-2x4(t)+1.5x4(t)+0.5σ14(xt)+c2σ22(xt)≤-bx2(t)+(b+c2)κ2∫-τ0xt2(θ)dμ(θ)-0.5x4(t)+0.5κ4∫-τ0xt4(θ)dμ(θ)+2a(x(t)-σ1(xt))2,
where the first and second inequalities using the elementary inequality uαv1-α≤(αu+(1-α)v). If a<0,b>(b+c2)κ2≥0,ε1>0 and ε2>0 be the unique roots to the following equations,
(6.15)b=(b+c2)κ2eε1τ,0.5=0.5κ4eε2τ,
respectively. And set ε=-2a∧ε1∧ε2∧τ-1lnκ-2, by Theorem 3.1, we can conclude that the unique global solution of (6.12) has the property that
(6.16)limsupt→∞1tlnE|x(t)|2≤-ε,limsupt→∞1tln|x(t)|2≤-ε.
If we let a=-2,b=0.5=c,κ=0.5,τ=0.9, then
(6.17)0.5=2.6666e0.9ε1,0.5=0.55e0.9ε2,
which give their roots ε1=1.0898,ε2=3.08065, respectively, and τ-1lnκ-2=0.10168,(6.18)limsupt→∞1tln(E|x(t)|2)≤-0.10168,limsupt→∞1tln|x(t)|2≤-0.10168.



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*Source: 425762-2010-11-07.xml*