# General Decay Stability for Stochastic Functional Differential Equations with Infinite Delay

Authors: Yue Liu; Xuejing Meng; Fuke Wu
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

So far there are not many results on the stability for stochastic functional
differential equations with infinite delay. The main aim of this paper is to establish some new
criteria on the stability with general decay rate for stochastic functional differential equations
with infinite delay. To illustrate the applications of our theories clearly, this paper also
examines a scalar infinite delay stochastic functional differential equations with polynomial
coefficients.


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## Body

## 1. Introduction

Stability is one of the central problems for both deterministic and stochastic dynamic systems. Due to introduction of stochastic factors, stochastic stability mainly includes almost sure stability and the moment stability. In a series of papers (see [1–5]), Mao et al. examined the moment exponential stability and almost sure exponential stability for various stochastic systems.In many cases we may find that the Lyapunov exponent equals zero, namely, the equation is not exponentially stable, but the solution does tend to zero asymptotically. By this phenomenon, Mao [6] considered polynomial stability of stochastic system, which shows that solution tends to zero polynomially. Then in [7], he extended these two classes of stability into the general decay stability.In general, time delay and system uncertainty are commonly encountered and are often the source of instability (see [8]). Many studies focused on stochastic systems with delay. Especially, infinite delay systems have received the increasing attention in the recent years since they play important roles in many applied fields (cf. [7, 9–13]). Under the Lipschitz condition and the linear growth condition, Wei and Wang [14] built the existence-and-uniqueness theorem of global solutions to stochastic functional differential equations with infinite delay. There is also some other literature to consider stochastic functional differential equations with infinite delay and we here only mention [15–17].However, to the best knowledge of the authors, there are not many results on the stability with general decay rate for stochastic functional equations with infinite delay. It is therefore interesting to consider the stability of infinite delay stochastic systems. The main aim of this paper is to establish some new criteria forpth moment stability and almost surely asymptotic stability with general decay rate of the global solution to stochastic functional differential equations with infinite delay(1.1)dx(t)=f(t,x(t),xt)dt+g(t,x(t),xt)dw(t),
where f=(f1,…,fd)T:ℝ+×ℝd×Cb((-∞,0];ℝd)→ℝd, and g=[gij]d×r:ℝ+×ℝd×Cb((-∞,0];ℝd)→ℝd×r are Borel measurable functionals, and w(t) is an r-dimensional Brownian motion. Without the linear growth condition, we will show that (1.1) has the following properties.(i)
This equation almost surely admits a global solution on[0,∞).(ii)
There exists a pair of positive constantsp and q such that this global solution has properties
(1.2)lim supt→∞lnE|x(t,ξ)|plnψ(t)⩽-q,lim supt→∞ln|x(t,ξ)|lnψ(t)⩽-qp,a.s.,
where ψ(t) is a general decay function defined in the next section, namely, this solution is pth moment and almost surely asymptotically stable with general decay rate.In the next section, we introduce some necessary notation and definitions. Section3 gives the main result of this paper by establishing a new criteria for pth moment stability and almost surely asymptotic stability with general decay rate for the global solution of (1.1). To make our results more applicable, Section 4 gives the further result. To illustrate the application of our result, Section 5 considers a scalar stochastic functional differential equation with infinite delay in detail.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notation. Let(Ω,ℱ,{ℱt}t⩾0,ℙ) be a complete probability space with the filtration {ℱt}t⩾0 satisfying the usual conditions, that is, it is right continuous and increasing while ℱ0 contains all ℙ-null sets. w(t) is an r-dimensional Brownian motion defined on this probability space.Letℝ+=[0,+∞), ℝ++=(0,+∞), and ℝ-=(-∞,0]. Let |x| be the Euclidean norm of vector x∈ℝn. If A is a vector or matrix, its transpose is denoted by AT. For a matrix A, its trace norm is denoted by |A|=trace(ATA). Denote by Cb=Cb(ℝ-;ℝd) the family of all bounded continuous functions φ from ℝ- to ℝd with the norm ∥φ∥=sup-∞<θ⩽0|φ(θ)|, which forms a Banach space. In this paper, const always represents some positive constants whose precise value is not important. If x(t) is an ℝd-valued stochastic process on ℝ, for any t⩾0, define xt=xt(θ)={x(t+θ):θ∈ℝ-}. C2(ℝd,ℝ) denotes the family of continuously twice differentiable ℝ-valued functions defined on ℝd. For any V(x)∈C2(ℝd,ℝ+), define an operator ℒV:ℝ+×ℝd×Cb→ℝ by(2.1)ℒV(t,x,φ)=Vx(x)f(t,x,φ)+12trace[gT(t,x,φ)Vxx(x)g(t,x,φ)],
where(2.2)Vx(x)=(∂V(x)∂x1,∂V(x)∂x2,…,∂V(x)∂xd),Vxx(x)=[∂2V(x)∂xi∂xj]d×d.Ifx(t) is a solution of (1.1), for any V(x)∈C2(ℝd,ℝ), applying the Itô formula yields(2.3)dV(x(t))=LV(x(t))dt+Vx(x(t))g(t,x(t),xt)dw(t),whereLV(x(t))=ℒV(t,x(t),xt).Let us introduce the followingψ-type function, which will be used as the decay function.Definition 2.1.
The functionψ:ℝ→(0,∞) is said to be the ψ-type function if it satisfies the following conditions:(i)
it is continuous and nondecreasing inℝ and differentiable in ℝ+,(ii)
ψ(0)=1 and ψ(∞)=∞,(iii)
ϕ:=supt⩾0ψ1(t)<∞, where ψ1(t)=ψ′(t)/ψ(t),(iv)
for anyθ⩽0 and  t⩾0, ψ(t)⩽ψ(-θ)ψ(t+θ).It is is easy to find that functionsψ(t)=eγt and ψ(t)=(1+t+)γ̅ for any γ,γ̅>0 are ψ-type functions.For anyp,q⩾0 and φ∈Cb, define(2.4)𝒯p,q(φ)=∫-∞0ψq(θ)|φ(θ)|pdθ
and C(p,q)={φ∈Cb:𝒯p,q(φ)<∞}. Denote by M0 the family of all probability measures on ℝ-. For any μ∈M0 and ε⩾0, define(2.5)Mε={μ∈M0:με:=∫-∞0ψε(-θ)dμ(θ)<∞}.
We also impose the following standard assumption on coefficients f and g.Assumption 2.2.
Letf and g satisfy the Local Lipschitz condition. That is, for every integer n⩾1, there is kn>0 such that
(2.6)|f(t,x,φ)-f(t,x̅,φ̅)|∨|g(t,x,φ)-g(t,x̅,φ̅)|⩽kn(|x-x̅|+∥φ-φ̅∥),for all t⩾0 and those x,x̅∈ℝn, φ,φ̅∈Cb with |x|∨|x̅|∨∥φ∥∨∥φ̅∥⩽n.Let us present the continuous semimartingale convergence theory (cf. [18]).Lemma 2.3.
LetM(t) be a real-value 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
(2.7)X(t)≤ζ+M(t)fort≥0,then 𝔼X(t)≤ζ and X(t) is almost surely bounded, namely, limt→∞X(t)<∞, a.s.


## 3. Main Results

In this section, we establish the stability result with general decay rate for (1.1). This result includes the global existence and uniqueness of the solution, the pth moment stability, and almost surely asymptotic stability with general decay rate.In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [18, 19]) or a given non-Lipschitz condition and the linear growth condition (cf. [20, 21]). These show that the linear growth condition plays an important role to suppress the potential explosion of solutions and guarantee existence of global solutions. References [16, 22] extended these two classes conditions to infinite delay cases. However,  many well-known infinite delay systems such that the Lotka-Volterra (see [13]) do not satisfy the linear growth condition. It is therefore necessary to examine the global existence of the solution for (1.1).It is well known for stochastic differential equations that the linear growth condition for global solutions may be replaced by the use of the Lyapunov functions [23, 24]. By this idea, this paper establishes the existence-and-uniqueness theorem for (1.1).Fori=1,2,…,k, let ζi,αi∈ℝ+ and probability measures μi∈Mε. Define Γε:ℝn×Cb→ℝ as(3.1)Γε(x,φ)=∑i=0kζi(∫-∞0|φ(θ)|αidμi(θ)-μiε|x|αi),
where μiε is defined by (2.5). Then the following theorem follows.Theorem 3.1.
Assume that there exist positive constantsa, p, ε, ζi, αi and probability measures μi∈Mε, where i=1,2,…,k, such that for any x∈ℝd and φ∈Cb, the function V(x)=|x|p satisfies
(3.2)ℒV(t,x,φ)⩽Γε(x,φ)-a|x|p.
Under Assumption 2.2, there exists a constant q>0 such that for any ξ∈C(α̂,q), where α̂=min0⩽i⩽k{αi}, (1.1) almost surely admits a unique global solution x(t) on [0,∞) and this solution has the properties (1.2).Proof.
For sufficiently smallq∈(0,ε), fix the initial data ξ∈C(α̂,q). We divide this proof into the two steps.
Step 1 (existence and uniqueness of the global solution).
Under Assumption2.2, (1.1) has a unique maximal local solution x(t) on [0,ρe) (see [21]), where ρe is the explosion time. If we can show ρe=∞, a.s., then x(t) is actually a global solution. Let n0 be a positive integer such that supθ⩽0|ξ(θ)|<n0. For each integer n⩾n0, define the stopping time
(3.3)σn=inf{t∈[0,ρe):|x(t)|p⩾n}.Obviously, σn is increasing and σn→σ∞⩽ρe as n→∞. Thus, to prove ρe=∞ a.s., it is sufficient to show that σ∞=∞ a.s., which is equivalent to the statement that for any t>0, ℙ(σn⩽t)→0 as n→∞.
For anyt⩾0, define tn=t⋀σn. Applying the Itô formula to ψq(t)V(x(t)) yields
(3.4)nℙ(σn⩽t)=E[I{σn⩽t}V(x(tn))]⩽EV(x(tn))⩽E[ψq(tn)V(x(tn))]=const+E∫0tnL[ψq(s)V(x(s))]ds=const+E∫0tnψq(s)[LV(x(s))+qψ1(s)V(x(s))]ds⩽const+E∫0tnψq(s)[LV(x(s))+qϕV(x(s))]ds⩽const+E∫0tnψq(s)[Γε(x(s),xs)-a|x(s)|p+qϕV(x(s))]ds.
Note that by (2.5), μiε≥μiq for q≤ε. By the Fubini theorem and a substitution technique, we have
(3.5)∫0tnψq(s)Γε(x(s),xs)ds=∑i=0kζi[∫-∞0dμi(θ)∫0tnψq(s)|x(s+θ)|αids-μiε∫0tnψq(s)|x(s)|αids]⩽∑i=0kζi[∫-∞0dμi(θ)∫θtn+θψq(s-θ)|x(s)|αids-μiq∫0tnψq(s)|x(s)|αids]⩽∑i=0kζi[∫-∞0ψq(-θ)dμi(θ)∫θtn+θψq(s)|x(s)|αids-μiq∫0tnψq(s)|x(s)|αids]⩽∑i=0kζi[μiq∫-∞tnψq(s)|x(s)|αids-μiq∫0tnψq(s)|x(s)|αids]=∑i=0kζiμiq∫-∞0ψq(θ)|ξ(θ)|αidθ.Noting that ξ∈C(α̂,q), we have ξ∈C(αi,q), which implies that for all i=1,…,k,
(3.6)∫-∞0ψq(θ)|ξ(θ)|αidθ<∞.Hence, there exists
(3.7)∫0tnψq(s)Γε(x(s),xs)ds<∞.
By (3.4) and (3.7), we have
(3.8)nℙ(σn⩽t)⩽const+E∫0tnψq(s)[qϕV(x(s))-a|x(s)|p]ds.
Choosing q sufficiently small such that qϕ⩽a, by (3.8) we have nℙ(σn⩽t)⩽const, which implies that ℙ(σn⩽t)→0 as n→∞.
Step 2 (Proof of (1.2)).
Define(3.9)h(t)=ψq(t)V(x(t)).
By the Itô formula and (3.2),
(3.10)h(t)=h(0)+∫0tψq(s)[LV(x(s))+qψ1(s)V(x(s))]ds+M(t)⩽h(0)+∫0tψq(s)[Γε(x(s),xs)-a|x|p+qϕV(x(s))]ds+M(t),
where
(3.11)M(t)=∫0tψq(s)Vx(x(s))g(s,x(s),xs)dw(s)
is a continuous local martingale with M(0)=0. Similar to (3.7), there exists
(3.12)∫0tψq(s)Γε(x(s),xs)ds<∞.
By (3.10), (3.12), noting that qϕ⩽a,
(3.13)h(t)⩽const+∫0tψq(s)(qϕ-a)|x(s)|pds+M(t)⩽const+M(t).By Lemma 2.3, we have
(3.14)lim supt→∞Eh(t)<∞,lim supt→∞h(t)<∞,a.s.,which implies the required assertions.


## 4. Further Result

In Theorem3.1, it is not convenient to check condition (3.2) since it is not related to coefficients f and g explicitly. To make our theory more applicable, let us impose the following assumption on coefficients f and g.Assumption 4.1.
There exist positive constantsσ, σ̃, α, β, ε, and nonnegative constants σ̅, λ, λ̅, λ̃, σi, σ̅i, λj, λ̅j, αi, βj, such that for any x∈ℝn, φ∈Cb,
(4.1)xTf(t,x,φ)⩽-σ|x|α+2+σ̅∫-∞0|φ(θ)|α+2dμ(θ)-σ̃|x|2+∑i=0k(σi|x|αi+2+σ̅i∫-∞0|φ(θ)|αi+2dμi(θ)),(4.2)|g(t,x,φ)|⩽λ|x|β+1+λ̅∫-∞0|φ(θ)|β+1dν(θ)+λ̃|x|+∑j=0l(λj|x|βj+1+λ̅j∫-∞0|φ(θ)|βj+1dνj(θ)),
where 0⩽α0<α1<⋯<αk<α, 0⩽β0<β1<⋯<βl<β, 2β⩽α, and μ,μi,ν,νj∈Mε.We also need the following lemma.Lemma 4.2.
Letα,p>0. Assume that α0,α1,…,αk, c0,c1,…,ck are nonnegative constants such that 0⩽α0⩽α1⩽⋯⩽αk⩽α, b>c=:∑i=0kck and a>cρ, where
(4.3)ρ={1,ifα0=0,0,ifα0=α,(α-α0)(α0α0αα)1/(α-α0),ifα0∈(0,α),
then, there is a̅∈(0,a) such that for all t⩾0,
(4.4)atp+btα+p-∑i=0kcitαi+p⩾a̅tp.Proof.
Noting thata>cρ, choose the constant ã such that
(4.5)cρ<ã<a.
If we can show that for any t∈[0,∞), F(t)=:ã+btα-∑i=0kcitαi⩾0, then the inequality
(4.6)a+btα-∑i=0kcitαi⩾a-ãholds. Let a̅=a-ã. This is equivalent to prove that
(4.7)atp+btα+p-∑i=0kcitαi+p⩾a̅tp.
For allt∈(1,+∞), there exists F(t)⩾ã+btα-ctα. By ã>cρ⩾0 and b>c, we have F(t)⩾ã+btα-ctα>0.
For allt∈[0,1], there exists F(t)⩾F*(t)=:ã+btα-ctα0. To prove F(t)⩾0, we consider three cases of α0, respectively.
Case 1.
α0=0. By α0=0, we have F*(t)=ã+btα-c and ã∈(c,a). Then there exists F(t)⩾F*(t)>0.Case 2.
α0=α. By α0=α, we have F*(t)=ã+btα-ctα and ã∈(0,a). Noting b>c, we obtain F(t)⩾F*(t)>0.Case 3.
α0∈(0,α). Without the loss of generality, we assume that c>0. Obviously, on (0,+∞) the derivative function F*′(t)=bαtα-1-cα0tα0-1 has a unique null point t0=:(α0c/αb)1/(α-α0)<1. We can compute that
(4.8)F*(t0)=ã+b(α0cαb)α/(α-α0)-c(α0cαb)α0/(α-α0)=ã-c(cb)α0/(α-α0)(α-α0)(α0α0αα)1/(α-α0).
Since 0<α0<α and b>c, we know that
(4.9)0<(cb)α0/(α-α0)<1.
By (4.8) and (4.9), we obtain that F*(t0)>ã-cρ>0. Then we have that for any t∈[0,1], F(t)⩾F*(t)⩾F*(t0)>0. The proof is completed.For the purpose of simplicity, we introduce the following notations:(4.10)σ·=∑i=0kσi,σ̅·=∑i=0kσ̅i,λ·=∑j=0lλj,λ̅·=∑j=0lλ̅j,Q=σ-σ·-σ̅·-σ̅,S=λ+λ̅+λ̃+λ·+λ̅·Then the following theorem follows.Theorem 4.3.
Let Assumptions2.2 and 4.1 hold. Assume that
(4.11)2Q>S(S-λ̃),(4.12)2σ̃-2ρ(σ·+σ̅·)>S[λ̃+ρ(S-λ̃)],
where ρ is defined by Lemma 4.2 except that α0 is replaced by α0⋀2β. For any
(4.13)p∈(2,p1⋀p2),
where
(4.14)p1=1+2QS(S-λ̃),p2=1+2σ̃-2ρ(σ·+σ̅·)S[λ̃+ρ(S-λ̃)],
there exists a positive constant q such that for any initial data ξ∈C((α0⋀2β0)+p,q), (1.1) admits a unique global solution x(t) on [0,∞) and this solution has the properties (1.2).Proof.
DefineV(x)=|x|p for p>2. Applying (2.1) gives
(4.15)ℒV(t,x,φ)=p|x|p-2xTf+p2(p-2)|x|p-4|gTx|2+p2|x|p-2|g|2⩽p|x|p-2xTf+p2(p-1)|x|p-2|g|2=:I1+I2.
By (4.1) and the Young inequality,
(4.16)I1=p|x|p-2xTf⩽p|x|p-2[-σ|x|α+2+σ̅∫-∞0|φ(θ)|α+2dμ(θ)-σ̃|x|2+∑i=0k(σi|x|αi+2+σ̅i∫-∞0|φ(θ)|αi+2dμi(θ))]⩽-p(σ-σ̅p-2α+p)|x|α+p-pσ̃|x|p+pσ̅α+2α+p∫-∞0|φ(θ)|α+pdμ(θ)+p∑i=0k(σi+σ̅ip-2αi+p)|x|αi+p+p∑i=0kσ̅iαi+2αi+p∫-∞0|φ(θ)|αi+pdμi(θ).
Recall the following elementary inequality: for any λj⩾0 and xj∈ℝ, j=0,1,…,n, applying the Hölder inequality yields
(4.17)(∑j=0nλjxj)2⩽∑j=0nλj∑j=0nλjxj2.
By (4.2) and (4.17), applying the Young inequality and the Hölder inequality, we have
(4.18)I2=p2(p-1)|x|p-2|g|2⩽p(p-1)2|x|p-2[λ|x|β+1+λ̅∫-∞0|φ(θ)|β+1dν(θ)+λ̃|x|+∑j=0l(λj|x|βj+1+λ̅j∫-∞0|φ(θ)|βj+1dνj(θ))]2⩽Sp(p-1)2[(λ+λ̅p-22β+p)|x|2β+p+λ̅2β+22β+p∫-∞0|φ(θ)|2β+pdν(θ)+λ̃|x|p+∑j=0l(λj+λ̅jp-22βj+p)|x|2βj+p+∑j=0lλ̅j2βj+22βj+p∫-∞0|φ(θ)|2βj+pdνj(θ)].
Substituting (4.16) and (4.18) into (4.15) yields
(4.19)ℒV(t,x,φ)⩽Γε(x,φ)-p2H(x),
where
(4.20)Γε(x,φ)=∑i=0kpσ̅iαi+2αi+p(∫-∞0|φ(θ)|αi+pdμi(θ)-μiε|x|αi+p)+pσ̅α+2α+p(∫-∞0|φ(θ)|α+pdμ(θ)-με|x|α+p)+∑j=0lSp(p-1)2λ̅j2βj+22βj+p(∫-∞0|φ(θ)|2βj+pdνj(θ)-νjε|x|2βj+p)+Sp(p-1)2λ̅2β+22β+p(∫-∞0|φ(θ)|2β+pdν(θ)-νε|x|2β+p),
whose expression is similar to (3.1) and
(4.21)H(x)=a|x|p+b(ε)|x|α+p-c̃(ε)|x|2β+p-∑i=0kci(ε)|x|αi+p-∑j=0lc̃j(ε)|x|2βj+p,
in which
(4.22)a=2σ̃-S(p-1)λ̃,b(ε)=2σ-2σ̅p-2α+p-2σ̅α+2α+pμε,c̃(ε)=S(p-1)(λ+λ̅p-22β+p+λ̅2β+22β+pμε),ci(ε)=2σi+2σ̅ip-2αi+p+2σ̅iαi+2αi+pμiε,c̃j(ε)=S(p-1)(λj+λ̅jp-22βj+p+λ̅j2βj+22βj+pνjε).
Let c(ε)=c̃(ε)+∑i=0kci(ε)+∑j=0lc̃j(ε). Note that b(0)=2(σ-σ̅), c̃(0)=S(p-1)(λ+λ̅), ci(0)=2(σi+σ̅i), c̃j(0)=S(p-1)(λj+λ̅j), c(0)=S(p-1)(S-λ̃)+2(σ.+σ̅.). By (4.13), we obtain that c̃(0)>0, c(0)>0, and c̃j(0)>0 for all 0⩽j⩽l. By (4.11) and (4.13), we have b(0)>c(0). By (4.12) and (4.13), we obtain a(0)>ρc(0). Choose sufficiently small ε such that
(4.23)a>ρc(ε),b(ε)>c(ε).
By (4.23) and Lemma 4.2, there exists a constant a̅∈(0,a) such that
(4.24)a̅|x|p⩽a|x|p+b(ε)|x|α+p-c̃(ε)|x|2β+p-∑i=0kci(ε)|x|αi+p-∑j=0lc̃j(ε)|x|2βj+p.
By (4.19), (4.21), and (4.24), we therefore have
(4.25)ℒV(t,x,φ)⩽Γε(x,φ)-p2a̅|x|p,
which implies that condition (3.2) is satisfied. By (4.20), (4.25), and the fact that 0⩽α0<α1<⋯<αk<α and 0⩽β0<β1<⋯<βl<β, applying Theorem 3.1 yields that there exists q>0, such that for any ξ∈C((α0⋀2β0)+p,q), the desired assertions hold. The proof is completed.


## 5. A Scalar Case

To illustrate the application of our result, this section considers a scalar stochastic functional differential equations(5.1)dx(t)=[∑r=1nxr(t)ur(t)+∑0⩽r<r+s⩽nxr(t)∫-∞0xs(t+θ)urs(t,θ)dθ]dt+[∑k=1mxk(t)vk(t)+∑0⩽k<k+l⩽mxk(t)∫-∞0xl(t+θ)vkl(t,θ)dθ]dw(t),
where for r=1,2,…,n and k=1,2,…,m, ur(t),vk(t)∈C(ℝ+), for 0⩽r<r+s⩽n and 0⩽k<k+l⩽m,urs(t,θ),vkl(t,θ)∈C(ℝ+×ℝ-),n⩾3 is an odd number, m⩾2, and 2m⩽n+1. In this section, ∑0≤r<r+s≤n:=∑r=0n∑s=0n with r+s≤n and ∑0≤k<r+l≤m has similar explanation. Assume(5.2)u1(t)⩽-a1<0,un(t)⩽-an<0,|ur(t)|⩽ar,where2⩽r⩽n-1,|urs(t,θ)|⩽ars2ε(1-θ)-1-2ε,where0⩽r<r+s⩽n,(5.3)|vk(t)|⩽bk,where1⩽k⩽m,|vkl(t,θ)|⩽bkl2ε(1-θ)-1-2ε,where0⩽k<k+l⩽m,
in which ar, ars, bk, bkl are nonnegative constants and ε>0. Define(5.4)f(t,x,φ)=∑r=1nxr(t)ur(t)+∑0⩽r<r+s⩽nxr(t)∫-∞0φs(θ)urs(t,θ)dθ,g(t,x,φ)=∑k=1mxk(t)vk(t)+∑0⩽k<k+l⩽mxk(t)∫-∞0φl(θ)vkl(t,θ)dθ.
It is obvious that f(t,x,φ) and g(t,x,φ) satisfy the local Lipschtiz condition. By (5.4), (5.1) can be rewritten as (1.1).Choose theΨ-type function ψ(t)=1+t+. Let dμ(θ)=2ε(1-θ)-1-2εdθ. It is obvious that ∫-∞0dμ(θ)=1 and(5.5)∫-∞0ψε(-θ)dμ(θ)=∫-∞0[1+(-θ)+]ε2ε(1-θ)-1-2εdθ=∫-∞02ε(1-θ)-1-εdθ=2<∞,which shows thatμ∈Mε.By (5.2) and the Young inequality, we have that(5.6)xTf(t,x,φ)⩽-a1|x|2+∑i=1n-2ai+1|x|i+2-an|x|n+1+∑0⩽r<r+s⩽nars|x|r+1∫-∞0|φ(θ)|sdθ⩽-a1|x|2+∑i=1n-2ai+1|x|i+2-an|x|n+1+∑0⩽r<r+s⩽nars(r+1r+s+1|x|r+s+1+sr+s+1∫-∞0|φ(θ)|r+s+1dμ(θ))=-(an-∑0⩽r<r+s=n(r+1)arsn+1)|x|n-1+2+∑0⩽r<r+s=nsarsn+1∫-∞0|φ(θ)|n-1+2dμ(θ)-a1|x|2+12a01|x|2+∑i=1n-2(ai+1+∑0⩽r<r+s=i+1(r+1)arsi+2)|x|i+2+∑i=0n-2∑0⩽r<r+s=i+1sarsi+2∫-∞0|φ(θ)|i+2dμ(θ)=:-σ|x|α+2+σ̅∫-∞0|φ(θ)|α+2dμ(θ)-σ̃|x|2+∑i=0n-2(σi|x|αi+2+σ̅i∫-∞0|φ(θ)|αi+2dμ(θ)),which shows that condition (4.1) holds with(5.7)σ=an-∑0⩽r<r+s=n(r+1)arsn+1,σ̅=∑0⩽r<r+s=nsarsn+1,σ̃=a1,σi=ai+1+∑1⩽r<r+s=i+1(r+1)arsi+2,σ̅i=∑1⩽r<r+s=i+1sarsi+2,α=n-1,αi=i.By (5.3) and the Young inequality, we get that(5.8)|g(t,x,φ)|⩽b1|x|+∑j=1m-1bj|x|j+bm|x|m+∑0⩽k<k+l⩽mbkl|x|k∫-∞0|φ(θ)|ldθ⩽b1|x|+∑j=1m-1bj|x|j+bm|x|m+∑0⩽k<k+l⩽mbkl(kk+l|x|k+l+lk+l∫-∞0|φ(θ)|k+ldμ(θ))=(bm+∑0⩽k<k+l=mkbklm)|x|m-1+1+∑0⩽k<k+l=mlbklm∫-∞0|φ(θ)|m-1+1dμ(θ)+b1|x|+∑j=1m-2(bj+1+∑0⩽k<k+l=j+1kbklj+1)|x|j+1+∑j=0m-2∑0⩽k<k+l=j+1lbklj+1∫-∞0|φ(θ)|j+1dμ(θ)=:λ|x|β+1+λ̅∫-∞0|φ(θ)|β+1dμ(θ)-λ̃|x|+∑j=0m-2(λj|x|βj+1+λ̅j∫-∞0|φ(θ)|βj+1dμ(θ)),which shows that condition (4.2) holds with(5.9)λ=bm+∑0⩽k<k+l=mkbklm,λ̅=∑0⩽k<k+l=mlbklm,λ̃=b1,λ̅j=∑0⩽k<k+l=j+1lbklj+1,λj={0,ifj=0,bj+1+∑0⩽k<k+l=j+1kbklj+1,if1⩽j⩽m-2,β=m-1,βj=j.By2m⩽n+1, we have 2(m-1)⩽n-1, which implies 2β⩽α. It is easy to see that σ̅, σ̃, λ, λ̅, and λ̃ are positive, and σi, σ̅i, λj, λ̅j, are nonnegative, where 0⩽i⩽n-2,0⩽j⩽m-2.By the parameters in Theorem4.3, we can compute(5.10)ρ=1,σ·+σ̅·=∑i=2n-1ai+∑0⩽r<r+s⩽n-1ars,S=b·+∑0⩽k<k+l⩽mbkl,Q=an-(∑i=2n-1ai+∑0⩽r<r+s⩽nars),S-λ0=∑j=2mbj+∑0⩽k<k+l⩽mbkl.In Assumption4.1, the parameter σ is positive, so it is required that(5.11)an>∑0⩽r<r+s=n(r+1)arsn+1.
Let(5.12)W1=∑i=2n-1ai+∑0⩽r<r+s⩽n-1ars,W2=b·+∑0⩽k<k+l⩽mbkl,W3=∑i=2n-1ai+∑0⩽r<r+s⩽nars.
To apply Theorem 4.3, it is necessary to test that (4.11)–(4.13) are satisfied. This requires that(5.13)a1>W1+12W22,(5.14)an>W3+12W2(W2-b1).
Obviously, (5.11) can be obtained from (5.14). By (4.14),(5.15)p1=1+2(an-W3)W2(W2-b1),p2=1+2(a1-W1)W22.
Thus, we have the following corollary from Theorem 4.3.Corollary.
Let conditions (5.2), (5.3), (5.13), and (5.14) be satisfied, where W1, W2, and W3 are given in (5.12). For any p∈(2,p1⋀p2), where p1 and p2 are given in (5.15), there exist q>0, for any ξ∈C(p,q), (5.1) has a unique global solution x(t)=x(t,ξ), and this solution has properties
(5.16)lim supt→∞lnE|x(t,ξ)|pln(1+t+)⩽-q,lim supt→∞ln|x(t,ξ)|ln(1+t+)⩽-qp,a.s.



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*Source: 875908-2010-02-09.xml*