THE LASALLE-TYPE THEOREM FOR NEUTRAL ...

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 32, Number 3, March 2012

doi:10.3934/dcds.2012.32.1065 pp. 1065–1094

THE LASALLE-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Fuke Wu and Shigeng Hu School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, Hubei 430074, China

Abstract. The main aim of this paper is to establish the LaSalle-type theorem to locate limit sets for neutral stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness and the almost sure stability with general decay rate and robustness are obtained. To make our theory more applicable, by the M -matrix theory, this paper also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. By specializing the general decay rate as the exponential decay rate and the polynomial decay rate, this paper examines two neutral stochastic integral-differential equations and shows that they are exponentially attractive and polynomially stable, respectively.

1. Introduction. Since Itˆ o established his stochastic calculus, the theory of stochastic differential equations has been developed very quickly. As one of the most important fields, stochastic stability has received increasing attention. A large number of stability methods in deterministic equations are introduced to examine stochastic stability, for example, the Lyapunov method has been developed to deal with stochastic stability by many authors. Here we only mention Arnold [2], Friedman [9], Khasminskii [15], Kolmanovskii and Nosov [16], Mohammed [35] and Mao [22, 27] among the others. As one of the most important developments of the Lyapunov method, the LaSalle theorem for deterministic systems may locate limit sets of nonautonomous systems (cf. [11, 19]), from which follow many classical Lyapunov results on stability. In his serial papers [28, 29, 30, 31, 36], Mao and his coauthors established various stochastic versions of the LaSalle-type theorems for stochastic differential equations, including stochastic functional differential equations with finite delay, by which many classical attraction and stability results for deterministic systems are extended to stochastic systems. However, these results only concern no delay as well as finite delay systems rather than the infinite delay cases. They also do not concern neutral systems. So far there seems to be no stochastic LaSalle theorem for neutral stochastic functional systems with infinite delay to locate limit sets although there exist some stability results (see, e.g. [13, 21, 42, 43]). 2000 Mathematics Subject Classification. Primary: 34K50, 34K40; Secondary: 34D45, 37C75. Key words and phrases. LaSalle-type theorem, attractor, neutral stochastic functional differential equations, infinite delay, the Itˆ o formula. Corresponding author: Fuke Wu. Tel: 0086-27-87557195. Fax: 0086-27-87543231.

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FUKE WU AND SHIGENG HU

Motivated by the chemical engineering systems as well as the theory of aeroelasticity, neutral systems have received the increasing attentions in the past decades, for example, [11, 16]. It is also well known that infinite delay systems (including deterministic and stochastic systems) have wide applications in many fields, for example, (cf. [1, 4, 10, 12, 17, 18, 34, 40, 41]). In general, time delay and system uncertainty are commonly encountered and are often sources of instability (see [16]). It is therefore interesting to locate limit sets of neutral stochastic systems infinite delay and further examine attraction, boundedness, stability and robustness of these systems. Consider the neutral stochastic differential equation with infinite delay d[x(t) − u(t, xt )] = f (t, xt )dt + g(t, xt )dw(t), (1.1) where xt = xt (θ) =: {x(t + θ) : θ ∈ (−∞, 0]}, u : R+ × BC((−∞, 0]; Rn ) → Rn is a continuous functional, f : R+ × BC((−∞, 0]; Rn ) → Rn and g : R+ × BC((−∞, 0]; Rn ) → Rn×m are Borel measurable, w(t) is an m-dimensional Brownian motion. Eq. (1.1) may be regarded as the stochastically perturbed equation of the deterministic neutral functional equation with infinite delay d[x(t) − u(t, xt )] = f (t, xt ). dt

(1.2)

If Eq. (1.2) is attractive and further stable, it is interesting to determine how much stochastic perturbation this equation can tolerate without losing these asymptotic properties, which is called as the robustness with respect to the stochastic perturbation. This paper will establish the LaSalle theorem for Eq. (1.1) to locate its limit set, by which some criteria on attraction, boundedness, almost sure stability with general decay rate and robustness will be obtained. To make our theory more applicable, by the M -matrix theory, this paper also examine some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. Compared with existing stochastic LaSalle theorems, the important contributions of this paper are that • the neutral term is considered, • infinite delay is considered, • the more general decay rate is considered, which may be specialized as the classically exponential decay rate and the polynomial decay rate. In the next section, we give some necessary notation and lemmas. Then we establish a LaSalle-type theorem for Eq. (1.1), which includes the existence and uniqueness of the global solution, attraction and boundedness in Section 3. In Section 4, we discuss the asymptotic attraction and stability with general decay rate. To make our theory more applicable, by the M -matrix theory, Section 5 also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. To illustrate the applications of our theory more clearly, Section 6 considers two neutral stochastic integral-differential equation with infinite delay and presents conditions under which one is exponentially attractive, and another is polynomial attractive by specializing the general decay rate as the exponential decay rate and the polynomial decay rate.

LASALLE-TYPE THEOREM FOR NSFDES WITH INFINITE DELAY

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2. Auxiliary definitions and statements. Throughout this paper, unless otherwise specified, we use the following notation. Let (Ω, F, P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions, that is, it is right continuous and increasing while F0 contains all P-null sets. Let w(t) be an m-dimensional Brownian motion defined on this probability space. If x(t) is an Rn -valued stochastic process, define xt = xt (θ) := {x(t + θ) : −∞ < θ ≤ 0} for t ≥ 0 and let x ˜(t) := x(t) − u(t, xt ). Let | · | be the Euclidean norm in Rn . If A is a vector or matrix,pits transpose is denoted by AT . If A is a matrix, denote its trace norm by |A| = trace(AT A). Let R+ = [0, ∞) and R++ = (0, ∞). Denote by C((−∞, 0]; Rn ) the family of continuous functions from (−∞, 0] to Rn . Similarly, denote by BC((−∞, 0]; Rn ) the family of bounded continuous functions from (−∞, 0] to Rn with the norm kϕk = supθ≤0 |ϕ(θ)| < ∞, which forms a Banach space. If a, b ∈ R, a ∨ b represents the maximum of a and b. Let a+ = a ∨ 0. For any x = (x1 , . . . , xn )T ∈ Rn , let diag(x1 , · · · , xn ) represent the n × n matrix with all elements zero except those on the diagonal which are x1 , . . . , xn . In this paper, const represents some positive constant whose value is not important. For a set D, denote by d(x, D) = inf{|x−y| : y ∈ D} the distance between x and D. Let Ker(G) = {x ∈ Rn : G(x) = 0} represent the kernel of the real-valued function G defined on Rn . Let C 1,2 (R+ × Rn ; R+ ) denote the family of all nonnegative functions V (t, x) from R+ × Rn to R+ which are continuously once differentiable in t and twice in x, and define an operator LV : R+ × BC((−∞, 0]; Rn ) → R by LV (t, ϕ)

=

Vt (t, ϕ(0) − u(t, ϕ)) + Vx (t, ϕ(0) − u(t, ϕ))f (t, ϕ) 1 + trace[g T (t, ϕ)Vxx (t, ϕ(0) − u(t, ϕ))g(t, ϕ)], 2

(2.1)

where Vt (t, x) =

∂V (t, x) , ∂t

Vx (t, x) =

∂V (t, x)

∂x1 h ∂ 2 V (t, x) i

,··· ,

∂V (t, x) , ∂xn

. ∂xi ∂xj n×n Let us emphasize that LV is a functional defined on R+ × BC((−∞, 0]; Rn ) while V is a function on R+ × Rn . R0 Let Lp ((−∞, 0]; Rn ) denote all functions h : (−∞, 0] → Rn with −∞ |h(s)|p ds < ∞. We give the following lemma. Vxx (t, x) =

Lemma 2.1. Let ϕ ∈ BC((−∞, 0]; Rn ) ∩ Lp ((−∞, 0]; Rn ) for p > 0. Then for any q > p, ϕ ∈ Lq ((−∞, 0]; Rn ). Proof. Clearly, by the definition of the norm in the space BC((−∞, 0]; Rn ), we have Z 0 Z 0 |ϕ(θ)|q dθ = |ϕ(θ)|p |ϕ(θ)|q−p dθ −∞

−∞

≤

q−p

Z

0

kϕk

|ϕ(θ)|p dθ < ∞,

−∞

which is the desired assertion. Let M0 denote all probability measures µ on (−∞, 0]. In this paper, all probability measures may be extended to any functions in (−∞, 0] with bounded variations. We also impose the following standard assumptions on coefficients u, f and g:

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Assumption 1. u(t, 0) ≡ 0 and there exists constant κ ∈ (0, 1) and probability measure η ∈ M0 such that Z 0 |ϕ(θ) − φ(θ)|dη(θ) |u(t, ϕ) − u(t, φ)| ≤ κ −∞ n

on t ≥ 0 for any ϕ, φ ∈ BC((−∞, 0]; R ). Assumption 2. Both f and g satisfy the local Lipschitz condition, that is, for any k > 0, there exists constant ck such that |f (t, ϕ) − f (t, φ)| ∨ |g(t, ϕ) − g(t, φ)| ≤ ck kϕ − φk on t ≥ 0 for those ϕ, φ ∈ BC((−∞, 0]; Rn ) with kϕk ∨ kφk ≤ k. It is obvious that Assumption 1 implies that Z 0 |u(t, ϕ)| ≤ κ |ϕ(θ)|dη(θ).

(2.2)

−∞

The following semimartingale convergence theorem plays an important role in this paper (see [20], [27]). We write it as a lemma. Lemma 2.2. Let M (t) be a real-valued continuous local martingale with M (0) = 0 a.s. Let ξ be a nonnegative F0 -measurable random variable with Eξ < ∞. If X(t) is nonnegative and satisfies X(t) ≤ ξ + M (t)

for t ≥ 0.

then lim sup EX(t) < ∞ and lim sup X(t) < ∞ t→∞

a.s.

t→∞

3. The LaSalle-type Theorem. This section will establish the LaSalle-type theorem which includes existence and uniqueness of the global solution for Eq. (1.1) as well as attraction and boundedness of this solution. Theorem 3.1. Let Assumptions 1 and 2 hold. If there exist functions V ∈ C 1,2 (R+ × Rn ; R+ ), G(x) ∈ C(Rn ; R+ ), nonnegative constants Lkj , αk , probability measures µk ∈ M0 for k = 1, · · · , K, j = 1, · · · , n such that for any x ∈ Rn and ϕ = (ϕ1 , · · · , ϕn )T ∈ BC((−∞, 0]; Rn ), h i lim inf V (t, x) = ∞, (3.1) |x|→∞

LV (t, ϕ) ≤

K X n X k=1 j=1

Lkj

Z

0

0≤t 2ε0 }. Ω0 = {ω ∈ Ω : lim sup h(t, t→∞

On the other hand, (3.7) implies that we can find a sufficiently large positive k, which may depend on ε0 , such that P(Ω00 ) ≥ 1 − ε0 ,

(3.14)

where Ω00 = {ω ∈ Ω : sup |˜ x(t)| < k}. t≥0

It is obvious that P(Ω0 ∩ Ω00 ) ≥ 2ε0 . Fix the above ε0 and k. Define the stopping times ρ = inf{t ≥ 0 : |˜ x(t; ξ)| ≥ k}, ˜ ≥ 2ε0 }, for i ≥ 1, %i = inf{t ≥ σi−1 : h(t) ˜ ≤ ε0 } for i ≥ 1. σ0 = 0 and σi = inf{t ≥ %i , h(t) It is obvious that ρ = ∞ implies that |˜ x(t, ω)| < k. Note that (3.11) implies that Z ∞ ˜ h(s)ds < ∞ a.s. 0

It then follows that ˜ = 0 a.s. lim inf h(t) t→∞

which implies that σi < ∞ if %i < ∞. Let Ai = {ω : %i < ∞, ρ = ∞} and li = E[IAi (σi − %i )]. By (3.11), we have ε0

∞ X

li

≤

i=1

Z ∞ h X E IAi

i ˜ h(t)dt

%i

i=1

Z ≤

σi

E

∞

˜ h(t)dt < ∞.

(3.15)

0

If we can prove that there exists a constant l such that li ≥ l > 0, this will produce a contradict. We may therefore prove that (3.8) holds for almost all ω ∈ Ω. To look

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for the contradict, noting that G(x) is continuous, there must exist a δ > 0 such that if |x − y| < δ, then |G(x) − G(y)| < ε0

(3.16)

for all x, y ∈ Rn with |x| ∨ |y| ≤ k. For the above δ > 0, let T > 0 be arbitrary and define n o ˜ i + t, ω) − h(% ˜ i , ω)| < ε0 , Ωi = ω ∈ Ω : sup |h(% 0≤t≤T

n o Si = ω ∈ Ω : sup |˜ x(%i + t, ω) − x ˜(%i , ω)| < δ . 0≤t≤T

By (3.16), we obtain Ai ∩ Si ⊂ Ωi . It is easy to test that Ai ∩ Ωi ⊂ {ω : σi − %i ≥ T }

(3.17)

Ω0 ∩ Ω00 ⊂ Ai .

(3.18)

and

Applying the Chebyshev inequality (see [27]) therefore yields li T −1

= E[IAi (σi − %i )]T −1 ≥ P(Ai ∩ Ωi ) ≥ P(Ai ∩ Si ) ≥ P(Ω0 ∩ Ω00 ) − P(Ai ∩ Sic ) ≥ 2ε0 − P(Ai ∩ Sic ).

(3.19)

Let βi := P(Ai ∩ Sic ). We claim that βi ≤ ε0 . By the definition of Si , applying the Chebyshev inequality and the elementary inequality (a + b)2 ≤ 2(a2 + b2 ) for any a, b ∈ R yield n o δ 2 βi = δ 2 P Ai ∩ ω ∈ Ω : sup |˜ x(%i + t, ω) − x ˜(%i , ω)|2 ≥ δ 2 0≤t≤T

≤

h i E IAi sup |˜ x(%i + t, ω) − x ˜(%i , ω)|2

≤

Z h E sup

≤

Z h E sup

≤

Z h 2E sup

0≤t≤T

0≤t≤T

0≤t≤T

0≤t≤T

%i +t

%i %i +t %i

0≤t≤T

2 i IAi (f (s, xs )ds + g(s, xs )dw(s))

%i +t

%i

Z h +2E sup =: I1 + I2 .

2 i IAi d[˜ x(s)]

2 i IAi f (s, xs )ds

%i +t

%i

2 i IAi g(s, xs )dw(s) (3.20)


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By the definition of the stopping time ρ, for any ω ∈ Ai , we have |˜ x(t)| ≤ k on all t ≥ 0. By the definition of x ˜(t) and Assumption 1, we have |x(t)|

≤

|˜ x(t) + u(t, xt )|

≤

|˜ x(t)| + |u(t, xt )| Z 0 |x(t + θ)|dη(θ) |˜ x(t)| + κ

≤

−∞

Z ≤

0

sup |x(s + θ)|dη(θ),

sup |˜ x(t)| + κ

−∞ 0≤s≤t

0≤s≤t

which implies that sup |x(t)| ≤ sup |˜ x(t)| + κ 0≤s≤t

|x(s)|.

sup −∞ 0 and given ε > 0, there exists constant ck and probability measure υ ∈ Mε such that coefficients f and g satisfies that Z 0 |f (t, ϕ) − f (t, φ)| ∨ |g(t, ϕ) − g(t, φ)| ≤ ck |ϕ(θ) − φ(θ)|dυ(θ) −∞ n

on t ≥ 0 for those ϕ, φ ∈ BC((−∞, 0]; R ) with kϕk ∨ kφk ≤ k. It is obvious that Assumption 3 is stronger than Assumption 1 and Assumption 4 also implies that Assumption 2 must hold since by the definition of the norm, Z 0 |f (t, ϕ) − f (t, φ)| ∨ |g(t, ϕ) − g(t, φ)| ≤ ck |ϕ(θ) − φ(θ)|dυ(θ) −∞

≤

ck kϕ − φk.

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Then we may establish the following result on the asymptotic ψ-type attraction and stability for Eq. (1.1). Theorem 4.4. Let Assumptions 3 and 4 hold for given ε > 0 and q ∈ (0, ε/2] be sufficiently small such that κq = κηq ∈ (0, 1), where ηq is defined by (4.2). If there exist c = (c1 , · · · , cn )T ∈ Rn++ , nonnegative constants Lkj , αk , probability measures µk ∈ Mε for k =P 1, · · · , K, j = 1, · · · , n and function G(x) ∈ C(Rn ; R+ ) such that n function V (x) = i=1 ci x2i satisfies

LV (t, ϕ) ≤

K X n hX Z ψ1 (t) Lkj

0

|ϕj (θ)|αk dµk (θ) − µkε |ϕj (0)|αk

−∞

k=1 j=1

i −2qV (ϕ(0) − u(t, ϕ)) − G(ϕ(0) − u(t, ϕ))

(4.5)

for any ϕ ∈ BC((−∞, 0]; Rn ), where µkε is defined by (4.2), for any initial data ξ ∈ BC((−∞, 0]; Rn ) ∩ Lαˆ ((−∞, 0]; Rn ), where α ˆ = min1≤k≤K {αk }, we then have the following assertions: (i) Eq. (1.1) almost surely admits a global solution x(t; ξ) on t ≥ 0. (ii) If G satisfies that G(x) ≤ 2 G(x) for any > 0 and f (t, 0) = g(t, 0) ≡ 0, then x ˜(t) = x(t) − u(t, xt ) is almost surely ψ-type attracted by Ker(G), namely, lim

t→∞

log d(˜ x(t), Ker(G)) ≤ −q, log ψ(t)

a.s.

(4.6)

(iii) Under the conditions in (ii), if Ker(G) is a bounded set, then the global solution x(t; ξ) is almost surely asymptotic ψ-type stable, namely, lim

t→∞

log |x(t; ξ)| ≤ −q, log ψ(t)

a.s.

(4.7)

Remark 3. Note that u(t, 0) = f (t, 0) = g(t, 0) ≡ 0 implies that Eq. (1.1) admits a trivial solution x(t) ≡ 0. Proof of Theorem 4.4. This proof is similar to Theorem 3.1, so we also divide it into three steps. Step 1. Existence and uniqueness of the global solution. Note that V (x) satisfies cˆ|x|2 ≤ V (x) ≤ cˆ|x|2 ,

(4.8)

where cˆ = min{c1 , · · · , cn } and cˆ = max{c1 , · · · , cn }. V (x) is therefore radial unbounded, namely, V (x) holds condition (3.1). By condition (4.5), we may obtain that

LV (t, ϕ) ≤ ψ1 (t)

K X n X k=1 j=1

Lkj

Z

0

−∞

|ϕ(θ)|αk dµk (θ) − µkε |ϕ(0)|αk .


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Noting that µkε ≥ 1 and ψ1 is decreasing in R+ , by the similar technique to (3.9), for any t ≥ 0, we have Z t hZ 0 i ψ1 (t) |xj (s + θ)|αk dµk (θ) − µkε |xj (s)|αk ds −∞ Z 0

0 0

Z ≤

dµk (θ) −∞ Z 0

ψ1 (s − θ)|xj (s)|αk ds

θ t+θ

Z

ψ1 (s − θ)|xj (s)|αk ds −

dµk (θ)

+ −∞ Z 0

≤ φ

0 Z 0

dµk (θ) −∞ t

Z −

t

Z

ψ1 (s)|xj (s)|αk ds

0

|xj (s)|αk ds +

Z

0

Z dµk (θ)

−∞

θ

t


0


0 Z 0

αk

≤ φ

|ξj (s)|

0

Z

|ξ(s)|αk ds < ∞

ds ≤ φ

−∞

(4.9)

−∞

by lemma 2.1 since ξ ∈ BC((−∞, 0]; Rn ) ∩ Lαˆ ((−∞, 0]; Rn ). Then repeating the proof of [42, Theorem 3.1] gives the assertion (i). Step 2. The proof of the assertion (ii) Similar to the proof of (ii) in Theorem ˜ = G(ψ q (t)˜ 3.1, to prove (4.6), redefine h(t) x(t)). We claim that for almost all ω ∈ Ω, lim sup |ψ q (t)˜ x(t, ω)| < ∞

(4.10)

t→∞

and ˜ ω) = 0. lim h(t,

t→∞

(4.11)

Choose a sequence {tk } such that tk → ∞ as k → ∞. By (4.10), there must exist x ˜∗ such that ψ q (tk )˜ x(tk , ω) → x ˜∗ (ω) ∈ Rn as tk → ∞, which also implies that ∗ ˜ h(tk , ω) → G(˜ x (ω)) = 0. This shows x ˜∗ (ω) ∈ Ker(G) 6= ∅ and d(ψ q (t)˜ x(t, ω), Ker(G)) → 0

(4.12)

2

as t → ∞. Note that G(x) ≤ G(x) for any > 0 shows that Ker(G) is a cone. (4.12) therefore implies ψ q (t)d(˜ x(t, ω), Ker(G)) → 0 as t → ∞, which is equivalent to the desired assertion (4.6). Hence, if we can prove (4.10) and (4.11) for almost all ω ∈ Ω, the desired assertion (ii) will follow. Noting that ψ 2q (t)V (˜ x(t)) = V (ψ q (t)˜ x(t)), by the Itô formula, we have Z t q ¯ (t), V (ψ (t)˜ x(t)) = V (˜ x(0)) + ψ 2q (s)[LV (s, xs ) + 2qψ1 (s)V (˜ x(s))]ds + M 0

¯ (t) is a real-valued continuous local martingale with M ¯ (0) = 0. By condiwhere M tion (4.5), we have V (ψ q (t)˜ x(t)) Z t ¯ (t) ≤ V (˜ x(0)) − ψ 2q (s)G(˜ x(s))ds + M 0

+

K X k=1

Z Lkj 0

t

Z ψ 2q (s)ψ1 (s)

0

−∞

|xj (s + θ)|αk dµk (θ) − µkε |xj (s)|αk ds.

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Similar to (4.9), by Definition 4.1, applying the Fubini theorem and a substitution technique gives that for any k = 1, · · · , K, Z t Z 0 ψ 2q (s)ψ1 (s) |xj (s + θ)|αk dµk (θ) − µkε |xj (s)|αk ds −∞

0 0

Z ≤

0

Z

ψ 2q (−θ)dµk (θ)

φ −∞ 0

ψ 2q (s)|xj (s)|αk ds

θ

Z

ψ 2q (−θ)dµk (θ)

+

t+θ

Z

−∞

ψ1 (s − θ)ψ 2q (s)|xj (s)|αk ds

0 t

Z

ψ1 (s)ψ 2q (s)|xj (s)|αk ds

−µkε 0

Z

0

ψ 2q (s)|xj (s)|αk ds + µkε

≤ φµk2q Z

−∞ t

−µkε

Z

t


0


0 Z 0

|ξj (s)|αk ds

≤ φµkε −∞ Z 0

≤ φµkε

|ξ(s)|αk ds < ∞.

−∞

Noting that G(x) ≤ 2 G(x) for any x ∈ Rn , we therefore have Z t ˜ ¯ (t). V (ψ q (t)˜ x(t)) ≤ const − h(s)ds +M 0

Now applying Lemma 2.2 yields Z

∞

E

˜ h(t)dt 0 sufficiently small and k > 0 sufficiently large such that P(Ω0 ) ≥ 3ε0 ˜ ω) > 2ε0 } and for the set Ω0 = {ω ∈ Ω : lim supt→∞ h(t, P(Ω00 ) ≥ 1 − ε0 00

q

(4.15)

(4.16)

for the set Ω = {ω ∈ Ω : supt≥0 ψ (t)|˜ x(t)| < k}. Similarly, we introduce the same notation of %i , σi , Ai , li , Ωi and βi as those in the proof of Theorem 3.1 for all i ≥ 1 and redefine ρ = inf{t ≥ 0 : ψ q (t)|˜ x(t; ξ)| ≥ k}, and n o Si = ω ∈ Ω : sup |ψ q (%i + t)˜ x(%i + t, ω) − ψ q (t)˜ x(%i , ω)| < δ . 0≤t≤T


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Repeating the proof of (ii) in Theorem 3.1 may show that (4.11) holds if we can prove that βi ≤ ε0 . By the definition of Si , applying the Chebyshev inequality and the elementary inequality (a + b + c)2 ≤ 3(a2 + b2 + c2 ) for any a, b, c ∈ R yields =

δ 2 βi o n δ 2 P Ai ∩ ω ∈ Ω : sup |ψ q (%i + t)˜ x(%i + t, ω) − ψ q (t)˜ x(%i , ω)|2 ≥ δ 2

≤

i h E IAi sup |ψ q (%i + t)˜ x(%i + t, ω) − ψ q (t)˜ x(%i , ω)|2

≤

Z h E sup

≤

Z h E sup

≤

Z h 3E sup

0≤t≤T

0≤t≤T

0≤t≤T

0≤t≤T

%i +t

%i %i +t %i

0≤t≤T

2 i IAi (ψ q (s)f (s, xs )ds + ψ q (s)g(s, xs )dw(s) + x ˜(s)dψ q (s))

%i +t

%i

Z h +3E sup 0≤t≤T

2 i IAi d[ψ q (s)˜ x(s)]

2 i IAi ψ q (s)f (s, xs )ds

%i +t

%i %i +t

Z h +3E sup 0≤t≤T

%i

2 i IAi ψ q (s)g(s, xs )dw(s) 2 i IAi x ˜(s)dψ q (s)

=: I1 + I2 + I3 .

(4.17)

Applying the H¨ older inequality gives that Z %i +T I1 ≤ 3T E IAi ψ 2q (s)|f (s, xs )|2 ds. %i

By the definition of the stopping time ρ, for any ω ∈ Ai , sup ψ q (s)|˜ x(t, ω)| ≤ k.

(4.18)

0≤s≤t

By Assumption 3 and the definition of x ˜(t), applying the property of the decay function ψ gives Z 0 h i ψ q (t)|x(t)| ≤ ψ q (t) |˜ x(t)| + κ |x(t + θ)|dη(θ) ≤ ψ q (t)|˜ x(t)| + κ

Z

−∞ 0

ψ q (−θ)ψ q (t + θ)|x(t + θ)|dη(θ)

−∞

≤ ψ q (t)|˜ x(t)| + κηq

sup [ψ q (s)|x(s)|], −∞ ¯b, the desired result follows. When α ¯ > 0, it is obvious that there ¯ exists a unique t0 = (¯ α¯b/αb)1/(α−α) such that F 0 (t) = 0 when t = t0 . By condition (5.1), ¯b(α − α ¯) α ¯¯b α−α¯ α¯ > 0. F (t0 ) = a − α αb Noting that F (0) = a > 0 and F (∞) = ∞, for any t ≥ 0, we have F (t) ≥ a∧F (t0 ) > 0, as required. If we can test condition (4.5), we will obtain existence and uniqueness of the global solution to Eq. (1.1), almost sure attraction and stability of this solution by Theorem 4.4. To specify condition (4.5), in this section, we impose some conditions on u, f and g to guarantee Theorem 4.4. These conditions clearly show coefficients u, f and g how to determine existence of the global solution and attraction. For any ϕ = (ϕ1 , · · · , ϕn )T ∈ BC((−∞, 0]; Rn ) and the given ε > 0, we list these conditions as follows. (H1) There exist constants κi ∈ (0, 1) and probability measures ηi ∈ Mε for i = 1, · · · , n such that Z 0 |ui (t, ϕ)| ≤ κi |ϕi (θ)|dηi (θ). −∞

(H2) Let Gi be a continuous function satisfying xi Gi (x) ∈ C(Rn ; R+ ) for all x = (x1 , · · · , xn )T ∈ Rn . Let f¯i (t, ϕ) = fi (t, ϕ) + Gi (ϕ(0) − u(t, ϕ)). There exist constants α, σi0 , σi > 0, σ ¯i0 , σij , σ ¯ij ≥ 0, probability measures µ, µ ¯ ∈ Mε for i, j = 1, · · · , n such that Z 0 h ϕi (0)f¯i (t, ϕ) ≤ ψ1 (t) − σi0 ϕ2i (0) − σi |ϕi (0)|α+2 + σ ¯i0 ϕ2 (θ)d¯ µ(θ) −∞

+

n X j=1

σij |ϕj (0)|α+2 + σ ¯ij

Z

0

−∞

i |ϕj (θ)|α+2 dµ(θ) .


1083

(H3) There exist nonnegative constants γi0 , γ¯i0 , γij and γ¯ij such that Z 0 h |f¯i (t, ϕ)| ≤ ψ1 (t) γi0 |ϕi (0)| + γ¯i0 |ϕi (θ)|d¯ µ(θ) −∞

+

n X

γij |ϕj (0)|α+1 + γ¯ij

0

Z

i |ϕj (θ)|α+1 dµ(θ) .

−∞

j=1

¯ i0 , λij λ ¯ ij , and probability (H4) There exist nonnegative constants β < α/2, λi0 , λ measures ν, µ ¯ ∈ Mε such that Z 0 h ¯ i0 |gi (t, ϕ)|2 ≤ ψ1 (t) λi0 |ϕi (0)|2 + λ |ϕi (θ)|2 d¯ ν (θ) −∞

Z n X 2β+2 ¯ + λij |ϕj (0)| + λij

0

|ϕ(θ)|2β+2 dν(θ)

i .

−∞

j=1

Note that these conditions still holds for any ε0 ∈ [0, ε], η, µ, µk , ν, νl ∈ Mε0 since Mε ⊂ Mε0 by Lemma 4.3. Remark 5. In some special cases, condition (H1) implies (2.2). Here we give three examples. (i) n = 1. It is obvious that condition (H1) implies (2.2). (ii) For any τ > 0, all η1 , · · · , ηn are Dirac measures in −τ . This shows that ui (t, ϕ) ≤ κi |ϕi (−τ )|. we have n hX i 21 |u(t, ϕ)| = κ2i |ϕi (−τ )|2 i=1

≤

κ ˇ |ϕ(−τ )| Z 0 =: κ ˇ |ϕ(θ)|dη(θ), −∞

where κ ˇ = max{κ1 , · · · , κn } ∈ (0, 1) and η is a Dirac measure in −τ . This shows that (2.2) holds. (iii) All κ1 , · · · , κn ∈ (0, n−1/2 ) and η1 = · · · ηn = η. This shows κ ˇ n1/2 ∈ (0, 1). |u(t, ϕ)|2

=

n X

|ui (t, ϕ)|2

i=1 n h_

i2 |ui (t, ϕ)|

≤

n

≤

Z n h_ n κi

i=1

nˇ κ2

i2 |ϕi (θ)|dη(θ)

−∞

i=1

≤

0

hZ

0

i2 |ϕ(θ)|dη(θ) ,

−∞

where

Wn

i=1

ui = u1 ∨ u2 ∨ · · · , ∨un . This implies that (2.2) holds.

For the purpose of simplicity, we introduce the following notations: γi =

n X j=1

γij ,

γ¯i =

n X j=1

γ¯ij ,

(5.3)

1084


α+1 1 ¯ ij ]n×n . [κi (γij + γ¯ij )]n×n , Σ = [σij + σ ¯ij ]n×n , Λ = [λij + λ α+2 2 Then by the M -matrix technique, the following theorem follows. Γ=

Theorem 5.4. Let Assumptions 3 and 4 hold for any given ε > 0. Define Q = diag(ω1 , · · · , ωn ) − Γ − Σ − Λ,

(5.4)

where h i κi (γi + γ¯i ) i 1 h 1 ¯ i0 ) ωi = σi − ∧ σi0 − σ ¯i0 − κi (γi0 + γ¯i0 ) − (λi0 + λ α+2 r 2 and  

1 (2β)2β α−2β (α − 2β) r= αα  1,

for β > 0,

(5.5)

for β = 0.

Under conditions (H1)–(H4), if Q is an M -matrix and Gi satisfies that for > 0 and x = (x1 , · · · , xn )T ∈ Rn , xi Gi (x) ≤ xi Gi (x),

(5.6)

then for any initial data ξ ∈ BC((−∞, 0]; Rn )∩L2 ((−∞, 0]; Rn ), there almost surely exists a global solution x(t; ξ) to Eq. (1.1) on t ≥ 0 and there exists q > 0 such that x ˜(t) = x(t) − u(t, xt ) is almost surely ψ-type attractive, namely, lim

t→∞

log d(˜ x(t), A) ≤ −q, log ψ(t)

(5.7)

where A = {x = (x1 , · · · , xn )T ∈ Rn : xi Gi (x) = 0 for all i = 1, · · · , n}. Moreover, if A is a bounded set, then x(t) has the property (4.7). Proof. By the properties of the M -matrix, there exists c = (c1 , · · · , cn )T ∈ Rn++ such that QT c ∈ Rn++ . Define G(x) = 2

n X

ci xi Gi (x).

(5.8)

i=1

It is obvious that G ∈ C(Rn ; R+ ), G(x) ≤ 2 G(x) and by the definition of A, G(x) = 0 if and only if xi Gi (x) = 0 for all i = 1, · · · , n, which implies A = Ker(G). To apply Theorem P 4.4, we only need to test condition (4.5). Applying (2.1) to n function V (˜ x(t)) = i=1 ci x ˜2i (t) yields LV (t, ϕ)

=

n X

ci [2(ϕi (0) − ui (t, ϕ))(f¯i (t, ϕ) − Gi (ϕ(0) − u(t, ϕ))) + |gi (t, ϕ)|2 ]

i=1

≤

n X

[2ci ϕi (0)f¯i (t, ϕ) + 2ci |ui (t, ϕ)||f¯i (t, ϕ)| + ci |gi (t, ϕ)|2 ]

i=1

−2

n X

ci x ˜i (t)Gi (˜ x(t))

i=1

=:

n X i=1

(Ii1 + Ii2 + Ii3 ) − G(˜ x(t)).

(5.9)


1085

By condition (H2), it directly follows that

≤

ψ1−1 (t)Ii1 h 2ci − σi |ϕi (0)|α+2 − (σi0 − σ ¯i0 µ ¯ε )|ϕi (0)|2 n Z 0 X α+2 + (σij + σ ¯ij µε )|ϕj (0)| +σ ¯i0 |ϕi (θ)|2 d¯ µ(θ) − µ ¯ε |ϕj (0)|2 +

j=1 n X

−∞

σ ¯ij

j=1

Z

0

|ϕj (θ)|α+2 dµ(θ) − µε |ϕj (0)|α+2

i

.

−∞

Recall the well-konwn Young inequality: for any α, β > 0, x, y ≥ 0, then xα y β ≤

1 (αxα+β + βy α+β ). α+β

By conditions (H1) and (H3), applying the Young inequality and the Hölder inequality yields ψ1−1 (t)Ii2 Z 0 Z h ≤ 2ci κi |ϕi (θ)|dηi (θ) γi0 |ϕi (0)| + γ¯i0 −∞

+

n X

|ϕi (θ)|d¯ µ(θ)

−∞

γij |ϕj (0)|α+1 + γ¯ij

Z

0

i |ϕj (θ)|α+1 dµ(θ)

−∞

j=1

n X γij (α + 1) |ϕj (0)|α+2 ϕ2i (θ)dηi (θ) + 2 α + 2 −∞ j=1 Z 0 Z (γij + γ¯ij ) γ¯ij (α + 1) 0 α+2 + |ϕi (θ)| dηi (θ) + |ϕj (θ)|α+2 dµ(θ) α+2 α+2 −∞ −∞ Z 0 Z 0 i γ¯i0 + |ϕi (θ)|2 dηi (θ) + |ϕi (θ)|2 d¯ µ(θ) 2 −∞ −∞ n h γ γ¯i0 α+1X i0 (1 + ηiε ) + (ηiε + µ ¯ε ) ϕ2i (0) + (γij + γ¯ij µε )|ϕj (0)|α+2 2ci κi 2 2 α + 2 j=1 Z n 1 X γi0 + γ¯i0 0 2 (γij + γ¯ij )ηiε |ϕi (0)|α+2 + ϕi (θ)dηi (θ) − ηiε ϕ2i (0) + α + 2 j=1 2 −∞ Z γ¯i0 0 + |ϕi (θ)|2 d¯ µ(θ) − µ ¯ε |ϕi (θ)|2 2 −∞ Z n X γij + γ¯ij 0 + |ϕi (θ)|α+2 dηi (θ) − ηiε |ϕi (0)|α+2 α+2 −∞ j=1 Z n i α+1X 0 + γ¯ij |ϕj (θ)|α+2 dµ(θ) − µε |ϕj (0)|α+2 . α + 2 j=1 −∞

≤ 2ci κi

=

0

hγ i0

ϕ2i (0) +

Z

0

1086


By (H4), it is obvious that Z h ¯ i0 ψ1−1 (t)Ii3 ≤ ci λi0 |ϕi (0)|2 + λ

0

|ϕi (θ)|2 d¯ ν (θ)

−∞

+

n X

¯ ij λij |ϕj (0)|2β+2 + λ

Z

0

i |ϕj (θ)|2β+2 dν(θ)

−∞

j=1

n h X ¯ i0 ν¯ε )|ϕi (0)|2 + ¯ ij νε )|ϕj (0)|2β+2 = ci (λi0 + λ (λij + λ j=1

¯ i0 +λ

Z

0

|ϕi (θ)|2 d¯ ν (θ) − ν¯ε |ϕi (0)|2

−∞

+

n X

¯ ij λ

Z

0

|ϕj (θ)|2β+2 dν(θ) − νε |ϕj (0)|2β+2

i .

−∞

j=1

By Condition (H1), applying the Hölder inequality gives that for any q¯ > 0,

≤

2¯ q ψ1 (t)V (ϕ(0) − u(t, ϕ)) n X 2¯ q ψ1 (t) ci (ϕi (0) − ui (t, ϕ))2

≤

Z n X 4¯ q ψ1 (t) ci ϕ2i (0) + κ2i

=

n hX Z 2 2 2 4¯ q ψ1 (t) ci (1 + ηiε κi )ϕi (0) + κ

i=1 0

ϕ2i (θ)|dηi (θ)

−∞

i=1

0

i ϕ2i (θ)dηi (θ) − ηiε ϕ2i (0) . (5.10)

−∞

i=1

Substituting Ii1 –Ii3 into (5.9) and applying (5.10) yield LV (t, ϕ)+2¯ q ψ1 (t)V (ϕ(0)−u(t, ϕ))+G(ϕ(0)−u(t, ϕ)) ≤ ψ1 (t)(Φε −I(ϕ(0))), (5.11) where Φε has the form similar to Z 0 K X n X Lkj ( |ϕj (θ)|αk dµk (θ) − µkε |ϕj (0)|αk ), −∞

k=1 j=1

so we omit their expression, and n X I(x) = 2 ai (ε) + bi (ε)|xi |α − ¯bi (ε)|xi |2β x2i , i=1

in which ai (ε)

= ci (σi0 − σ ¯i0 µ ¯ε ) −

ci κi [γi0 (ηiε + 1) + γ¯i0 (ηiε + µ ¯ε )] 2

ci ¯ i0 ν¯ε ) − 2¯ (λi0 + λ q ci (1 + κ2i ηiε ), 2 n ci κi (γi + γ¯i )ηiε X h (α + 1)κj (γji + γ¯ji µε ) i bi (ε) = ci σi − − cj σji + σ ¯ji µε + , α+2 α+2 j=1 −

n

X ¯bi (ε) = 1 ¯ ji νε ). cj (λji + λ 2 j=1


1087

Lemma 4.3 shows that functions ai , bi and ¯bi are continuous on ε. Noting that µ0 = ηi0 = ν0 = µ ¯0 = ν¯0 = 1, it is obvious that ai (0) = ci (σi0 − σ ¯i0 ) − ci κi (γi0 + γ¯i0 ) −

ci ¯ i0 ) − 2¯ (λi0 + λ q ci (1 + κ2i ), 2

n

bi (0) = ci σi −

ci κi (γi + γ¯i ) X h (α + 1)κj (γji + γ¯ji ) i , − cj σji + σ ¯ji + α+2 α+2 j=1 n

X ¯bi (0) = 1 ¯ ji ). cj (λji + λ 2 j=1 Note that QT c ∈ Rn++ implies that for all i = 1, · · · , n, bi (0) > ¯bi (0) ≥ 0

and

ai (0) − r¯bi (0) ≥ 0.

Choose ε sufficiently small and q¯ ∈ (0, ε/2] such that bi (ε) > ¯bi (ε) ≥ 0 and ai (ε) − r¯bi (ε) ≥ 0 and κq¯ ∈ (0, 1) for all i = 1, · · · , n. Applying Lemma 5.3 gives that I(x) ≥ 0, which implies that (5.9) satisfies condition (4.5). Letting q = q¯ and applying Theorem 4.4 give the desired assertions. ¯ i0 = λij = λ ¯ ij = 0 implies g ≡ 0. Choosing β = 0 (which Noting that λi0 = λ implies that r = 1), we may examine existence of the global solution to the deterministic neutral functional differential equation with infinite delay (1.2) and show that this solution is ψ-type attractive and stable. Theorem 5.5. Let Assumption 3 hold and f satisfy the local Lipschitz condition 4 for any given ε > 0. Define ¯ = diag(¯ Q ω1 , · · · , ω ¯ n ) − Γ − Σ,

(5.12)

where h κi (γi + γ¯i ) i ω ¯ i = σi − ∧ [σi0 − σ ¯i0 − κi (γi0 + γ¯i0 )]. α+2 ¯ is an M -matrix and Gi satisfies (5.6), then for Under conditions (H1)–(H3), if Q any initial data ξ ∈ BC((−∞, 0]; Rn )∩L2 ((−∞, 0]; Rn ), there exists a global solution x(t; ξ) to Eq. (1.2) on t ≥ 0 and x ˜(t) = x(t) − u(t, xt ) are ψ-type attractive, namely, there exists q > 0 such that log d(˜ x(t), A) ≤ −q, t→∞ log ψ(t) lim

where A is defined by Theorem 5.4. Moreover, if A is a bounded set, then x(t; ξ) is ψ-type stable, that is, log |x(t)| lim < −q. t→∞ log ψ(t) ¯ T c ∈ Rn++ . This shows that under the Remark 6. Clearly, QT c ∈ Rn++ implies Q conditions of Theorem 5.4, the ψ-type attraction and stability of the deterministic equation (1.2) is robust with respect to the stochastic perturbation satisfying condition (H4).

1088


6. Examples. In this section, by specializing the ψ-type attraction as the exponential attraction and the polynomial attraction, we consider two examples to illustrate the applications of our results. Firstly, we consider the following two-dimensional infinite delay neutral stochastic integro-differential equation with exponential kernel functions d˜ x1 (t)

=

h

− a1 x1 (t) − b1 x51 (t) + c1 x21 (t)

Z

0

i x32 (t + θ)eπ2 θ dθ − G1 (˜ x(t)) dt

−∞

Z h + ξ1 x1 (t)x2 (t) + ζ1 x1 (t)

0

i x2 (t + θ)eπ3 θ dθ dw1 (t),

−∞

d˜ x2 (t)

=

h

− a2 x2 (t) − b2 x52 (t) + c2 x22 (t)

Z

0

i x31 (t + θ)eπ2 θ dθ − G2 (˜ x(t)) dt

−∞

Z h + ξ2 x2 (t)x1 (t) + ζ2 x2 (t)

0

i x1 (t + θ)eπ3 θ dθ dw2 (t),

(6.1)

−∞

R0 where x ˜(t) = (˜ x1 (t), x ˜2 (t))T and x ˜i (t) = xi (t) − ρi −∞ xi (t + θ)eπ1 θ dθ, Gi (x) ∈ C(R2 ; R+ ) (i=1, 2) is local Lipschitz continuous and satisfies (5.6), all parameters are nonnegative and w(t) = (w1 (t), w2 (t))T is a two-dimensional Brownian motion. For any ϕ = (ϕ1 , ϕ2 )T ∈ BC((−∞, 0]; R2 ) and i, k = 1, 2 with i 6= k, define 0

Z

ϕi (θ)eπ1 θ dθ,

ui (t, ϕ) = ρi −∞

fi (t, ϕ) = −ai ϕi (0) − bi ϕ5i (0) + ci ϕ2i (0)

Z

0

ϕ3k (θ)eπ2 θ dθ − Gi (ϕ), ˜

−∞

Z

0

gi (t, ϕ) = ξi ϕi (0)ϕk (0) + ζi ϕi (0)

ϕk (θ)eπ3 θ dθ

−∞

and u = (u1 , u2 )T , f = (f1 , f2 )T and g = (g1 , g2 )T , where ϕ˜ = (ϕ˜1 , ϕ˜2 )T and R0 ϕ˜i = ϕi (0) − ρi −∞ ϕi (θ)eπ1 θ dθ. Then Eq. (6.1) may be rewritten as Eq. (1.1). For θ ∈ (−∞, 0], define probability measures η, µ and ν such that dη = π1 eπ1 θ dθ, dµ = π2 eπ2 θ dθ and dν = π3 eπ3 θ dθ. Then ui , fi and gi may be rewritten as ui (t, ϕ) =

ρi π1

Z

0

ϕi (θ)dη(θ), −∞

Z 0 ci 2 fi (t, ϕ) = −ai ϕi (0) − + ϕi (0) ϕ3k (θ)dµ(θ) − Gi (ϕ), ˜ π2 −∞ Z 0 ζi gi (t, ϕ) = ξi ϕi (0)ϕk (0) + ϕi (0) ϕk (θ)dν(θ). π3 −∞ bi ϕ5i (0)

For any ε ∈ (0, π1 ∧ π2 ∧ π3 ), choosing ψ(t) = et , it is easy to see that ψ1 (t) = 1 and (4.3) shows that η, µ, ν ∈ Mε . These shows Assumption 4 holds for any ε ∈ (0, π1 ∧ π2 ∧ π3 ).


1089

For any ϕ, φ ∈ BC((−∞, 0]; R2 ), |u(t, ϕ) − u(t, φ)|2 =

|u1 (t, ϕ) − u1 (t, φ)|2 + |u2 (t, ϕ) − u2 (t, φ)|2

≤

2[|u1 (t, ϕ) − u1 (t, φ)| ∨ |u2 (t, ϕ) − u2 (t, φ)|]2 h ρ Z 0 ρ Z 0 i2 1 2 2 |ϕ1 (θ) − φ1 (θ)|dη(θ) ∨ |ϕ2 (θ) − φ2 (θ)|dη(θ) π1 −∞ π1 −∞ i2 2 2 hZ 0 2(ρ1 ∨ ρ2 ) (|ϕ (θ) − φ (θ)| ∨ |ϕ (θ) − φ (θ)|)dη(θ) 1 1 2 2 π12 −∞ ρ ∨ ρ 2 h Z 0 i2 1 2 2 |ϕ(θ) − φ(θ)|dη(θ) , (6.2) π1 −∞

≤ ≤ ≤

it is therefore obvious that Assumption √ 3 and condition (H1) hold with κ = 2((ρ1 ∨ 2 ρ2 )/π1 ) , κi = ρi /π1 if ρ1 ∨ ρ2 ∈ (0, π1 / 2). Define f¯i (t, ϕ) = fi (t, ϕ) + Gi (ϕ) ˜ for i = 1, 2 and all ϕ ∈ BC((−∞, 0]; Rn ). Then Z 0 ci 3 2 6 ¯ ϕi (0)fi (t, ϕ) = −ai ϕi (0) − bi ϕi (0) + ϕi (0) ϕ3k (θ)dµ(θ) π2 −∞ Z 0 ci 6 ci ≤ −ai ϕ2i (0) − bi ϕ6i (0) + ϕi (0) + ϕ6 (θ)dµ(θ), 2π2 2π2 −∞ k which shows that condition (H2) holds with α = 4, σi0 = ai , σi = bi , σ ¯i0 = 0, σii = ci /(2π2 ), σik = 0, σ ¯ik = ci /(2π2 ) and σ ¯ii = 0. Applying the Young inequality yields Z 0 ci |f¯i (t, ϕ)| ≤ ai |ϕi (0)| + bi |ϕi (0)|5 + |ϕi (0)|2 |ϕk (θ)|3 dµ(θ) π2 −∞ Z 2ci 3ci 0 ≤ ai |ϕi (0)| + bi |ϕi (0)|5 + |ϕi (0)|5 + |ϕk (θ)|5 dµ(θ), 5π2 5π2 −∞ which implies that condition (H3) holds with γi0 = ai , γ¯i0 = 0, γii = 2ci /(5π2 ), γik = 0, γ¯ii = 0 and γ¯ik = 3ci /(5π2 ). Applying the Cauchy–Schwarz inequality and the Hölder inequality yields Z 0 h i2 ζi |gi (t, ϕ)|2 ≤ ξi ϕi (0)ϕk (0) + ϕi (0) ϕk (θ)dν(θ) π3 −∞ Z 0 h i ζi ζi 2 2 2 ≤ ξi + ξi ϕi (0)ϕk (0) + ϕi (0) ϕ2k (θ)dν(θ) π3 π3 −∞ Z 0 h i ζi ξi 4 ζ ζi i ≤ ξi + (ϕi (0) + ϕ4k (0)) + ϕ4i (0) + ϕ4k (θ)dµ(θ) , π3 2 2π3 2π3 −∞ ¯ i0 = 0, λii = which implies that condition (H4) holds with β = 1, λi0 = 0, λ ¯ ii = 0 and λ ¯ ik = ζi (ξi + ζi /π3 )/(2π3 ). (ξi + ζi /π3 )2 /2, λik = ξi (ξi + ζi /π3 )/2, λ By the definition of r in (5.5), we may compute r = 1/4. Substituting the above parameters into the definition of Q in Theorem 5.4 yields ρ i ci ρi 1 2ρ1 c1 3ρ1 c1 ωi = bi − ∧ 4ai 1 − , Γ= , 6π1 π2 π1 6π1 π2 3ρ2 c2 2ρ2 c2 1 (ξ1 + ζ1 /π3 )2 (ξ1 + ζ1 /π3 )2 1 c1 c1 , Λ= (6.3) Σ= c c 2π2 4 (ξ2 + ζ2 /π3 )2 (ξ2 + ζ2 /π3 )2 2 2

1090


and Q = [qij ]2×2 , where h ρ i ci ρi i ρi ci ci (ξi + ζi /π3 )2 qii = bi − ∧ 4ai 1 − − − − , 6π1 π2 π1 3π1 π2 2π2 4 ρi ci ci (ξi + ζi /π3 )2 qik = − − − . (6.4) 2π1 π2 2π2 4 By Lemma 5.2, Q is an M -matrix if and only if q11 > 0,

q11 q22 > q12 q21 .

(6.5)

Applying Theorem 5.4 therefore gives the following result. √ Corollary 1. Let ρ1 ∨ ρ2 ∈ (0, π1 / 2) and G1 (x), G2 (x) satisfy (5.6). For any initial data ξ ∈ BC((−∞]; Rn ) ∩ L2 ((−∞, 0]; Rn ), if condition (6.5) holds, there almost surely exists a unique global solution x(t; ξ) to Eq. (6.1). There exists q > 0 such that x ˜(t) = x(t) − u(t, xt ) is almost surely exponentially attractive, namely, log d(˜ x(t), A) < −q, a.s. t→∞ t where the limit set A = {x ∈ R2 : xi Gi (x) = 0, i = 1, 2}. Moreover, if A is a bounded set, then x(t) is almost surely exponentially stable, that is, lim

log |x(t)| < −q, a.s. t Remark 7. In this corollary, (6.5) is a key condition. By Lemma 5.2, sufficiently large ωi may guarantee condition (6.5), which implies that if ai and bi are sufficiently large, then (6.5) may be satisfied. lim

t→∞

Let π1 = π2 = π3 = 1, ρ1 = ρ2 = 0.5, c1 = c2 = 1, b1 = b2 = 2, a1 = a2 = 1. Condition (6.5) is specialized as (ξ1 + ζ1 )2 < 5

and

(ξ1 + ζ1 )2 + (ξ2 + ζ2 )2 < 8.

(6.6)

This shows that the deterministic neutral stochastic integro-differential equation with exponential kernel functions Z 0 d˜ x1 (t) 5 2 = −x1 (t) − 2x1 (t) + x1 (t) x32 (t + θ)eθ dθ − G1 (˜ x(t)) dt −∞ Z 0 d˜ x2 (t) = −x2 (t) − 2x52 (t) + x22 (t) x31 (t + θ)eθ dθ − G2 (˜ x(t)) (6.7) dt −∞ is exponentially attractive and may bear the environmental noise g under condition (6.6), which shows that this exponential attraction is robust, where x ˜i (t) = xi (t) − R0 0.5 −∞ xi (t + θ)eθ dθ for i = 1, 2. The different Gi (x) may produce the different results. Let G be defined by (5.8) for n = 2. We give the following several examples. (i) Choose Gi (x) = xi , which implies that xi Gi (x) = x2i and Ker(G) = {0}. By Corollary 1, there exists q > 0 such that x(t) is almost surely exponentially stable. (ii) Choose G1 (x) = x1 , G2 (x) = 0, which implies that x1 G1 (x) = x21 , x2 G2 (x) = 0 and Ker(G) = {0×R}. Applying the same way as Corollary 1 gives that there exists q > 0 such that log |x1 (t)| lim sup ≤ −q, a.s. t t→∞


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namely, Eq. (6.1) is partially almost surely exponentially stable. 2 2 (iii) Choose Gi (x) = x−1 i sin (xi ), which implies that xi Gi (x) = sin (xi ) and Ker(G) = {kπ, k = 0, 1, · · · }. Noting that G(x) does not satisfy G(x) ≤ 2 G(x), we cannot use Corollary 1. We may test that condition (3.2) is satisfied. Applying Theorem 3.1 therefore gives lim d(˜ x(t), Ker(G)) = 0, a.s.

t→∞

Secondly, we consider the following two-dimensional infinite delay neutral stochastic integro-differential equation with polynomial kernel functions Z 0 i h 1 x32 (t + θ) 5 2 − a1 x1 (t) − b1 x1 (t) + c1 x1 (t) dθ − G (˜ x (t)) dt d˜ x1 (t) = 1 π2 1+t −∞ (1 − θ) Z 0 1 h x2 (t + θ) i dθ dw1 (t), +√ ξ1 x1 (t)x2 (t) + ζ1 x1 (t) π3 1+t −∞ (1 − θ) Z 0 h 1 i x31 (t + θ) d˜ x2 (t) = dθ − G (˜ x (t)) dt − a2 x2 (t) − b2 x52 (t) + c2 x22 (t) 2 π 2 1+t −∞ (1 − θ) Z 0 1 h x1 (t + θ) i +√ dθ dw2 (t), (6.8) ξ2 x2 (t)x1 (t) + ζ2 x2 (t) π3 1+t −∞ (1 − θ) R0 where x ˜(t) = (˜ x1 (t), x ˜2 (t))T and x ˜i (t) = xi (t) − ρi −∞ xi (t + θ)/(1 − θ)π1 dθ, π1 , π2 , π3 > 1, the rest functions and parameters have the same definition as Eq. (6.1). For θ ∈ (−∞, 0], define probability measures η, µ and ν such that dη = (α1 −1)(1−θ)−α1 dθ, dµ = (α2 −1)(1−θ)−α2 dθ and dν = (α3 −1)(1−θ)−α3 dθ. Choose ψ(t) = 1 + t. Then ψ1 (t) = 1/(1 + t) and (4.4) shows that η, µ, ν ∈ Mε for any ε ∈ (0, π1 ∧ π2 ∧ π3 − 1). Eq. (6.8) may therefore be rewritten as Eq. (1.1) with Z 0 ρi ϕi (θ)dη(θ), ui (t, ϕ) = π1 − 1 −∞ Z 0 i 1 h ci fi (t, ϕ) = − ai ϕi (0) − bi ϕ5i (0) + ϕ2i (0) ϕ3k (θ)dµ(θ) − Gi (ϕ), ˜ 1+t π2 − 1 −∞ Z 0 i ζi 1 h ξi ϕi (0)ϕk (0) + gi (t, ϕ) = √ ϕi (0) ϕk (θ)dν(θ) . π3 − 1 1+t −∞ It is obvious that Assumption 4 holds. Repeating the same process as √ (6.2) gives that Assumption 3 and condition (H1) hold if ρ1 ∨ ρ2 ∈ (0, (π1 − 1)/ 2). By the similar computation process to the above example, we may test that conditions (H2), (H3) and (H4) hold with κ = 2((ρ1 ∨ρ2 )/(π1 −1))2 , κi = ρi /(π1 −1), α = 4, σi0 = ai , σi = bi , σ ¯i0 = 0, σii = ci /(2(π2 − 1)), σik = 0, σ ¯ik = ci /(2(π2 − 1)) and σ ¯ii = 0, γi0 = ai , γ¯i0 = 0, γii = 2ci /(5(π2 − 1)), γik = 0, γ¯ii = 0 and γ¯ik = 3ci /(5(π2 − 1)), ¯ i0 = 0, λii = (ξi + ζi /(π3 − 1))2 /2, λik = ξi (ξi + ζi /(π3 − 1))/2, β = 1, λi0 = 0, λ ¯ ii = 0 and λ ¯ ik = ζi (ξi + ζi /(π3 − 1))/(2(π3 − 1)). By the definition of Q = [qij ]2×2 , λ we have h ρi ci ρi i qii = bi − ∧ 4ai 1 − 6(π1 − 1)(π2 − 1) π1 − 1 ci (ξi + ζi /(π3 − 1))2 ρi ci − − − , 3(π1 − 1)(π2 − 1) 2(π2 − 1) 4 ρi ci ci (ξi + ζi /(π3 − 1))2 qik = − − − . (6.9) 2(π1 − 1)(π2 − 1) 2(π2 − 1) 4

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Applying Theorem 5.4 therefore gives the following result. √ Corollary 2. Let ρ1 ∨ ρ2 ∈ (0, (π1 − 1)/ 2) and G1 (x), G2 (x) satisfy (5.6). For any initial data ξ ∈ BC((−∞]; Rn )∩L2 ((−∞, 0]; Rn ), if condition (6.5) holds, there almost surely exists a unique global solution x(t; ξ) to Eq. (6.1). There exists q > 0 such that x ˜(t) = x(t) − u(t, xt ) is almost surely polynomially attractive, namely, lim

t→∞

log d(˜ x(t), A) < −q, a.s. log(1 + t)

where the limit set A is defined by Corollary 1. Moreover, if A is a bounded set, then x(t) is almost surely polynomial stable, that is, lim

t→∞

log |x(t)| < −q, a.s. log(1 + t)

Let π1 = π2 = π3 = 2, ρ1 = ρ2 = 0.5, c1 = c2 = 1, b1 = b2 = 2, a1 = a2 = 1. Condition (6.5) is specialized as (6.6) holds. This shows that the corresponding deterministic equation of Eq. (6.8) is polynomially attractive and this attraction is robust. Similarly, the different Gi (x) may produce the different results. (i) Choose Gi (x) = xi , which implies that xi Gi (x) = x2i and Ker(G) = {0}, where G is defined by (5.8). By Corollary 1, there exists q > 0, x(t) is almost surely polynomially stable. (ii) Choose G1 (x) = x1 , G2 (x) = 0, which implies that x1 G1 (x) = x21 , x2 G2 (x) = 0 and Ker(G) = {0 × R}, where G is defined by (5.8). Applying the same way as Corollary 1 gives that there exists q > 0 such that lim sup t→∞

log |x1 (t)| ≤ −q, a.s. log(1 + t)

namely, Eq. (6.8) is partially almost surely polynomially stable. Acknowledgments. The authors would like to thank the referees for their detailed comments and helpful suggestions. They also wish to thank the National Natural Science Foundation of China (Grant No. 11001091) and Chinese University Research Foundation (Grant No. 2010MS129) for their financial supports. REFERENCES [1] J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear Itˆ o-Volterra equations with damped stochastic perturbations, Electron. J. Probab., 8 (2003), 22 pp. [2] L. Arnold, “Stochastic Differential Equations: Theory and Applications,” Wiley, New York, 1972. [3] H. Bao and J. Cao, Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 215 (2009), 1732–1743 [4] H. Bereketoglu and I. Gy˝ ori, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210 (1997), 279–291. [5] A. Berman and R. J. Plemmons, “Nonnegative Matrices in the Mathematical Sciences,” SIAM, Philadelphia, PA, 1994. [6] T. Caraballo, M. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, System Control Lett., 48 (2003), 397–406. [7] F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, System Control Lett., 57 (2008), 262–270. [8] S. Fang and T. Zhang, A study of a class of stochastic differential equations with nonLipschitzian coefficients, Probab. Theory Related Fields, 132 (2005), 356–390. [9] A. Friedman, “Stochastic Differential Equations and their Applications,” Vol. 2, Academic Press, New York, 1976.


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Received September 2010; revised December 2010. E-mail address: [email protected] E-mail address: [email protected]