Almost sure stability with general decay rate of neutral stochastic pantograph equations

Almost sure stability with general decay rate of neutral stochastic pantograph equations

with Markovian switching

Wei Mao

, Liangjian Hu

and Xuerong Mao

1School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China

2Department of Applied Mathematics, Donghua Univerisity, Shanghai, 201620, China

3Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, U.K.

Received 25 February 2019, appeared 5 August 2019 Communicated by Mihály Kovács

Abstract. This paper focuses on the general decay stability of nonlinear neutral stochas- tic pantograph equations with Markovian switching (NSPEwMSs). Under the local Lipschitz condition and non-linear growth condition, the existence and almost sure sta- bility with general decay of the solution for NSPEwMSs are investigated. By means of M-matrix theory, some sufficient conditions on the general decay stability are also established for NSPEwMSs.

Keywords: neutral stochastic pantograph equations, Markovian switching, existence and uniqueness results, general decay stability.

2010 Mathematics Subject Classification: 60H10, 93E15.

1 Introduction

In this paper, we are concerned with the asymptotic stability of neutral stochastic pantograph equations with Markovian switching (NSPEwMSs)

d[x(t)−D(x(qt),t,r(t))] = f(x(t),x(qt),t,r(t))dt+g(x(t),x(qt),t,r(t))dw(t), t≥t0, (1.1) where 0 < q< 1, the coefficients f : Rⁿ×Rⁿ×[t₀,∞)×S → Rⁿ andg : Rⁿ×Rⁿ×[t₀,∞)× S→R^n×m are Borel-measurable, D:Rⁿ×[t₀,∞)×S→Rⁿ is the neutral term andw(t)is an m-dimensional Brownian motion. Actually, Eq. (1.1) can be regarded as a perturbed system of the deterministic pantograph equations

d[x(t)−D(x(qt)_,t,r(t))]

dt = f(x(t),x(qt),t,r(t)). (1.2)

BCorresponding author. Email: ljhu@dhu.edu.cn

Since Eq. (1.2) possess a wide range of applications in applied mathematics and engineering, the asymptotic properties and numerical analysis of the solution have been widely studied, for example, Hale [11], Iserles et al. [5,15–17].

As a class of special stochastic delay systems, stochastic pantograph equations (SPEs) with unbounded delay have been investigated by many scholars, we can refer to Baker and Buck- war [3], Appleby and Buckwar [2], Fan et al. [8,9], Milosevic [20], Xiao et al. [35], Guo and Li [10]. In the study of stochastic delay systems, the stability is one of the important issues and has many important applications in practice. There are many results on the exponential sta- bility theorem for stochastic differential delay equations (SDDEs) and SDDEs with Markovian switching. We mention here [4,14,21–25,37] among others. On the other hand, some of the exponential stability criteria related to the moment exponential stability of solutions to neutral SDDEs and neutral SDDEs with Markovian switching were considered in [13,18,19,26,27,34]

and the references therein. Motivated by above mentioned works, some scholars began to study the exponential stability of SPEs with Markovian switching (SPEwMSs). For example, Zhou and Xue [38] investigated the exponential stability of a class of SPEwMSs, where the coefficients were dominated by polynomials with high orders. You et al. [36] discussed the robust exponential stability of highly nonlinear SPEwMSs, in virtue of M-matrix theory, they established exponential stability criterion for SPEwMSs. Similarly, Shen et al. [31] considered a class of nonlinear NSPEwMSs and established the exponential stability criteria for NSPEwMSs without the linear growth condition.

In fact, not all stochastic differential systems are exponentially stable, there are also a lot of stochastic systems which are stable but subject to a lower decay rate other than exponential decay. Therefore, much literatures focuses on the polynomial stability of stochastic differential systems. We mention here only [7,29]. These two kinds of stability show that the speed which the solution decays to zero is different. Then these stability concepts are extended to general decay stability (see [1,6,30,32,33]). To the best of our knowledge, there is no existing result on almost sure stability with general decay rate for NSPEwMSs (1.1). By applying the Itô formula and the non-negative semi-martingale convergence theorem, we study the almost sure decay stability of Eq.(1.1) and give the upper bound of general decay rate. Meanwhile, we impose some conditions on f,gand establish the sufficient criteria on general decay stability in terms of M-matrix.

The paper is organized as follows. In Section 2, we introduce some hypotheses concerning Eq. (1.1) and we establish the existence and uniqueness of solutions to NSPEwMSs under the local Lipschitz condition and nonlinear growth condition; In Section 3, by applying the Itô formula and stochastic inequality, we study the almost sure stability with general decay rate for NSPEwMSs (1.1); By means of M-matrix, we establish some sufficient criteria on general decay stability; Finally, we give two examples to illustrate our theory.

2 Preliminaries and the global solution

Let (_Ω,F,P) be a complete probability space with a filtration {F_t}_t≥t0 satisfying the usual conditions. Letw(t)be anm-dimensional Brownian motion defined on the probability space (_Ω,F,P). Let t ≥ t₀ > 0 and C([qt,t];Rⁿ) denote the family of the continuous functions ϕ from [qt,t] → Rⁿ with the norm kϕk = sup_qt≤θ≤t|ϕ(θ)|, where | · | is the Euclidean norm in Rⁿ. If A is a vector or matrix, its transpose is denoted by A^>. If A is a matrix, its norm kAk is defined by kAk = _sup{|Ax| _: |x| = ₁}_. L_F^p

t([qt,t]_;Rⁿ) denote the family of

all(F_t)-measurable,C([qt;t],Rⁿ)-valued random variables ϕ= {ϕ(θ):qt ≤θ ≤t}such that Ekϕk²<_∞.

Let r(t),t ≥ t₀ be a right-continuous Markov chain on the probability space (_Ω,F,P) taking values in a finite state space S={1, 2, . . . ,N}with generatorΓ= (γ_ij)_N×N given by:

P(r(t+_∆) =j|r(t) =i) = (

γ_ij∆+◦(_∆), ifi6= j, 1+γ_ij∆+◦(_∆), ifi=j.

where ∆ > 0. Here γ_ij ≥ 0 is the transition rate from i to j, i 6= j, While γ_ii = −_∑j6=iγ_ij. We assume that the Markov chain r(·) is independent of the Brownian motionw(·). Let us consider the nonlinear NSPEwMSs

d[x(t)−D(x(qt),t,r(t))] = f(x(t),x(qt),t,r(t))dt+g(x(t),x(qt),t,r(t))dw(t), t ≥t₀ (2.1) with initial data{x(t):qt₀≤ t≤t₀}= ξ ∈ L²_F

t0

([qt₀,t₀];Rⁿ).

In this paper, the following hypotheses are imposed on the coefficients f,g andD.

Assumption 2.1. For each integerd≥1, there exist a positive constantk_d such that

|f(x,y,t,i)− f(x, ¯¯ y,t,i)|²∨ |g(x,y,t,i)−g(x, ¯¯ y,t,i)|²≤k_d(|x−x¯|²+|y−y¯|²), (2.2) for those x,y, ¯x, ¯y∈Rⁿwith|x| ∨ |y| ∨ |x¯| ∨ |y¯| ≤dand(t,i)∈[t₀,∞)×S.

Assumption 2.2. For all u,v ∈ Rⁿ and (t,i) ∈ [t₀,∞)×S, there exists a constant k_i ∈ (0, 1) such that

|D(u,t,i)−D(v,t,i)|² ≤k_i|u−v|². (2.3) Letk₀ =max_i∈Sk_i andD(0,t,i) =0.

It is known that Assumptions2.1and2.2only guarantee that Eq. (2.1) has a unique maxi- mal solution, which may explode to infinity at a finite time. To avoid such a possible explosion, we need to impose an additional condition in terms of Lyapunov functions.

LetC(Rⁿ×[t0,∞)×S;R+)denote the family of continuous functions fromRⁿ×[t0,∞)× KSto R+. Also denote by C²(Rⁿ×[t₀,∞)×S;R+)the family of all continuous non-negative functionsV(x,t,i)defined on Rⁿ×[t₀,∞)×S such that for eachi∈ S, they are continuously twice differentiable in x. Given V ∈ C²(Rⁿ×[t0,∞)×S;R+), we define the function LV : Rⁿ×Rⁿ×[t₀,∞)×S→ Rby

LV(x,y,t,i) =Vt(x−D(y,t,i),t,i) +Vx(x−D(y,t,i),t,i)f(x,y,t,i) + ¹

2trace[g^>(x,y,t,i)Vxx(x−D(y,t,i),t,i)g(x,y,t,i)]

+

∑

γ_ijV(x−D(y,t,i)_,t,j)_{, (2.4)} where

V_t(x,t,i) = ^∂V(x,t,i)

∂t , V_x(x,t,i) =

∂V(x,t,i)

∂x₁ , . . . ,∂V(x,t,i)

∂xn

, Vxx(x,t,i) =

∂²V(x,t,i)

∂x_i∂x_j

n×n

.

Assumption 2.3. There exist a function V ∈ C^2,1(Rⁿ×[t₀,∞)×S;R+) and some positive constantsc₁,c2,α_i,(i=0, 1, 2, 3, 4),γ>2, such that for all(x,y,t,i)∈ Rⁿ×Rⁿ×[t0,∞)×S

c₁|x|²≤ V(x,t,i)≤c₂|x|², ∀(x,t,i)∈Rⁿ×[t₀,∞)×S (2.5) and

LV(x,y,t,i)≤α₀−α₁|x|²+α₂|y|²−α₃|x|^γ+α₄|y|^γ. (2.6) Lemma 2.4(see [28]). Let p≥1and a,b∈ Rⁿ. Then, for anyδ∈(0, 1),

|a+b|^p ≤ |a|^p

(1−δ)^p−1 + |b|^p δ^p−1.

Lemma 2.5(see [28]). Let A(t),U(t)be twoF_t-adapted increasing processes on t≥0with A(0) = U(0) = 0a.s. Let M(t)be a real-valued local martingale with M(0) = 0a.s. Let ζ be a nonnegative F₀-measurable random variable. Assume that x(t)is nonnegative and

x(t) =ζ+A(t)−U(t) +M(t) for t≥0.

Iflim_t→_∞A(t)<_∞a.s. then for almost allω ∈_Ω,lim_t→_∞x(t)< _∞ and lim_t→_∞U(t)< _∞,that is, both x(t)and U(t)converge to finite random variables.

Theorem 2.6. Let Assumptions2.1–2.3hold. Then for any given initial dataξ, there is a unique global solution x(t)to Eq.(2.1)on t∈[t₀,∞). Moreover, the solution has the properties that

E|x(t)|² ≤C (2.7)

for any t≥t0.

Proof. Since the coefficients of Eq. (2.1) are locally Lipschitz continuous, for any given initial dataξ, there is a maximal local solution x(t)on t ∈ [t₀,σ_∞), whereσ_∞ is the explosion time.

Let ¯k0 >0 be sufficiently large forkξk<k^¯0. For each integerk ≥k^¯0, define the stopping time τ_k =_inf{t ∈[t₀,σ_e)_:|x(t)| ≥k}_.

Clearly,τ_k is increasing ask → _{∞. Set}τ_∞ = lim_k→∞τ_k, whence τ_∞ ≤ σ_∞ a.s. Note if we can show thatτ_∞ =_∞a.s., thenσ_∞= _∞a.s. So we just need to show thatτ_∞ =_∞a.s.

We shall first show that τ_∞ > ^t_q⁰ a.s. By the generalised Itô formula (see e.g. [25]) and condition (2.6), we can show that, for anyk≥k^¯₀ andt₁≥ t₀,

EV(z(τ_k∧t₁),τ_k∧t₁,r(τ_k∧t₁))

≤ EV(z(t₀),t₀,r(t₀)) +E Z _τk∧t1

t0

α₀−α₁|x(t)|²+α₂|x(qt)|²−α₃|x(t)|^γ+α₄|x(qt)|^γdt, where z(t) = x(t)−D(x(qt),t,r(t)). Let us now restrict t₁ ∈ [t₀,^t_q⁰]. By condition (2.5), we then get

c₁E|z(τ_k∧t₁)|² ≤ H₁−α₁E Z _τk∧t1

t0

|x(t)|²dt−α₃E Z _τk∧t1

t0

|x(t)|^γ)dt, (2.8) where

H₁ =c₂E|z(t₀)|²+

Z ^t0

q

t0

α₀+α₂|x(qt)|²+α₄|x(qt)|^γdt

≤2c₂(1+k₀)Ekξk²+¹ qE

Z _t0

qt₀

α₀+α₂|x(t)|²+α₄|x(t)|^γdt<_∞.

It then follows that

E|z(τ_k∧t₁)|²≤ ^H1

c₁ , t₀≤t₁ ≤ ^t0

q (2.9)

for any k≥k^¯₀. By Lemma2.4, we get E|x(τ_k∧t₁)|²≤ ^E|z(τ_k∧t₁)|²

1−δ + ^E|D(x(q(τ_k∧t₁)),τ_k∧t₁,r(τ_k∧t₁))|² δ

≤ ^E|z(τ_k∧t₁)|²

1−δ + ^k0E|x(q(τ_k∧t₁))|²

δ .

This implies sup

t0≤t1≤^t_q⁰

E|x(τ_k∧t₁)|² ≤ sup_t

0≤t1≤^t_q⁰ E|z(τ_k∧t₁)|²

1−δ +

k₀sup_t

0≤t1≤^t_q⁰ E|x(q(τ_k∧t₁))|²

δ . (2.10)

Lettingδ =√

k0, it follows from (2.9) that sup

t0≤t1≤^t_q⁰

E|x(τ_k∧t₁)|²≤ ^H1 c₁(1−√

k0)+^pk₀Ekξk²+^pk₀ sup

t0≤t1≤^t_q⁰

E|x(τ_k∧t₁)|². (2.11)

Hence, we have

E|x(τ_k∧t₁)|² ≤ ^H1 c₁(₁−√

k₀)+

√k0

1−√

k₀Ekξk², t₀ ≤t₁≤ ^t0

q. (2.12)

In particular, E|x(τ_k∧ ^t_q⁰)|² ≤ ^H1

c1(1−√ k0)+

√k0

1−√

k0Ekξk², ∀k ≥ k^¯₀. This implies k²P(τ_k ≤ ^t_q⁰) ≤

H1

c1(1−√ k0) +

√k0

1−√

k0Ekξk². Lettingk→∞, we hence obtain that P τ_∞≤ ^t_q⁰=0, namely P

τ_∞> ^t0 q

=1. (2.13)

Lettingk→_∞in (2.13) yields

E|x(t₁)|² ≤ ^H1 c₁(1−√

k0)+

√k₀ 1−√

k0

Ekξk², t₀ ≤t₁≤ ^t0

q. (2.14)

Let us now proceed to proveτ_∞ > ^t0

q² a.s. given that we have shown (2.12)–(2.14). For any k≥k^¯₀andt₁∈[t₀,^t_q⁰₂], it follows from (2.6) that

c₁E|z(τ_k∧t₁)|² ≤H₂−α₁E Z _τk∧t1

t0

|x(t)|²dt−α₃E Z _τk∧t1

t0

|x(t)|^γ)dt, (2.15) where

H₂ =c₂E|z(t₀)|²+E Z ^t0

q2

t₀

α₀+α₂|x(qt)|²+α₄|x(qt)|^γdt

=H₁+E Z ^t0

q2 t0

q

α₀+α₂|x(qt)|²+α₄|x(qt)|^γdt

=H₁+¹ qE

Z ^t0

q

t0

α₀+α₂|x(t)|²) +α₄|x(t)|^γdt<_∞.

Consequently

E|z(τ_k∧t₁)|²≤ ^H2

c₁ , t₀≤ t₁ ≤ ^t0 q². Similar to (2.12), we can obtain that

E|x(τ_k∧t₁)|² ≤ ^H2 c₁(₁−√

k₀)+

√k0

1−√

k₀Ekξk², t₀≤t₁≤ ^t0

q². (2.16) In particular, E|x(τ_k∧ ^t0

q²)|² ≤ ^H2

c1(1−√ k0)+

√k0

1−√

k0Ekξk²_, ∀k ≥ k^¯₀. This implies k²P(τ_k ≤ ^t0

q²)≤

H2

c1(1−√ k0)+

√k0

1−√

k0Ekξk²_{. Letting}k → ∞, we then obtain that P(τ_∞ ≤ ^t0

q²) =_{0, namely} P(τ_∞ >

t₀

q²) =_{1. Letting}k→_∞in (2.16) yields E|x(t₁)|² ≤ ^H2

c₁(1−√ k₀)+

√k₀ 1−√

k₀Ekξk², t₀≤t₁≤ ^t0 q². Repeating this procedure, we can show that, for any integeri≥1,τ_∞ > ^t0

qⁱ a.s. and E|x(t)|² ≤ ^Hi

c₁(1−√ k₀)+

√k₀ 1−√

k₀Ekξk², t₀ ≤t₁≤ ^t0 qⁱ, where H_i = 2c2(1+k0)Ekξk²+ER

t0 qi

t₀ α₀+α₂|x(qt)|²+α₄|x(qt)|^γdt < ∞. We must there- fore have τ_∞ = _∞ a.s. and the required assertion (2.7) holds as well. The proof is therefore complete.

Remark 2.7. In [12,20,31,36,38], the authors proved that stochastic pantograph differential systems has a unique solution x(t) under the local Lipschitz condition and the generalized Khasminskii-type condition. In fact, the key of their proof is that the coefficientsα_i,i=1, 2, 3, 4 of (2.6) are required to satisfyα₁≥ α₂andα₃ ≥a₄. However, in our theorem, we remove this condition and prove that Eq. (2.1) has a unique global solution x(t). Hence, we improve and generalize the corresponding existence results of [12,20,31,36,38].

3 Stability of neutral stochastic pantograph systems

In this section, we shall study the almost sure stability with general decay rate of NSPEwMSs (2.1). Let us first introduce the following ψ-type function, which will be used as the decay function.

Definition 3.1. The function ψ : R+ → (_0,∞) is said to be ψ-type function if this function satisfies the following conditions:

(i) It is continuous and nondecreasing inRand continuously differentiable inR+. (ii) ψ(0) =1, ψ(_∞) =_∞andφ=sup_t>0|^ψ0(t)

ψ(t)|<_∞.

(iii) For anys,t≥_0, ψ(t)≤ψ(s)ψ(t−s)_.

Definition 3.2. The solution of Eq. 2.1is said to be almost surely ψ-type stable if there exists a constant ¯γsuch that

lim sup

t→_∞

log|x(t)|

logψ(t) <−γ¯ a.s.

Obviously, when ψ(t) =e^t andψ(t) = 1+t, this ψ-type stability implies the exponential stability and polynomial stability, respectively.

In order to obtain the almost sureψ-type stability of Eq. (2.1), we shall impose the following conditions on the neutral termD.

Assumption 3.3. For all u,v ∈ Rⁿ and (t,i) ∈ [t₀,∞)×S, there exists a constant k₀ ∈ (0, 1) andε≥0 such that

|D(u,t,i)−D(v,t,i)|²≤ k₀ψ^−ε((1−q)t)|u−v|² (3.1) andD(0,t,i) =0.

Theorem 3.4. Let Assumptions2.1,3.3and2.3hold except(2.6)which is replaced by

LV(x,y,t,i)≤ −α₁|x|²+α₂qψ^−ε((1−q)t)|y|²−α₃|x|^γ+α₄qψ^−ε((1−q)t)|y|^γ (3.2) for all (x,y,t,i)∈ Rⁿ×Rⁿ×[t₀,∞)×S, whereα₁ >α₂ ≥0andα₃> α₄ ≥0. Then for any given initial dataξ, the solution x(t)of Eq.(2.1)has the property that

lim sup

t→_∞

log|x(t)|

logψ(t) <−^η

2, a.s. (3.3)

whereη∈(0,ε∧η¯)whileη¯ is the unique root to the following equation α₁ =α₂+2

1+^k0

q

c₂C_ψη.¯

Proof. We first observe that (3.2) is stronger than (2.6). So, by Theorem2.6, for any given initial dataξ, Eq.(2.1) has a unique global solution x(t)on t ≥ t0. Letη ∈ (0,ε). For any t ≥ t0, by the generalized Itô formula toψ^η(t)V(z(t),t,r(t)), we obtain that

ψ^η(t)V(z(t),t,r(t)) =ψ^η(t₀)V(z(t₀),t₀,r(t₀)) +

Z _t

t0

ηψ⁰(s) ψ(s)^ψ

η(s)V(z(s),s,r(s))ds +

Z _t

t₀ ψ^η(s)LV(x(s),x(qs),s,r(s))ds+M_t, where Mt = Rt

t0ψ^η(s)Vx(z(s),s,r(s))g(x(s),x(qs),s,r(s))dw(s). By conditions (2.5) and (3.2), we then compute

c₁ψ^η(t)|z(t)|²≤c2ψ^η(t0)|z(t0)|²+c2ηCψ

Z _t

t0

ψ^η(s)|z(s)|²ds−α₁ Z _t

t0

ψ^η(s)|x(s)|²ds +α₂q

Z _t

t0

ψ^η(s)ψ^−ε((1−q)s)|x(qs)|²ds−α₃ Z _t

t0

ψ^η(s)|x(s)|^γds +α₄q

Z _t

t0

ψ^η(s)ψ^−ε((1−q)s)|x(qs)|^γds+Mt, (3.4) where C_ψ = sup_t

0≤t<_∞|^ψ0(t)

ψ(t)| < φ < ∞. By the basic inequality|a+b|² ≤ 2(|a|²+|b|²) and

the definition ofψfunction, we have Z _t

t0

ψ^η(s)|z(s)|²ds≤₂

Z _t

t0

ψ^η(s) |x(s)|²+k₀ψ^−ε((₁−q)s)|x(qs)|²ds

≤2 Z _t

t0

ψ^η(s)|x(s)|²ds+2k₀ Z _t

t0

ψ^η(s)ψ^−ε((1−q)s)|x(qs)|²ds

≤2 Z _t

t0

ψ^η(s)|x(s)|²ds+2k0

Z _t

t0

ψ^η(s)ψ^−η((1−q)s)|x(qs)|²ds

≤2 Z _t

t0

ψ^η(s)|x(s)|²ds+2k₀ Z _t

t0

ψ^η(qs)|x(qs)|²ds

≤2k₀ q

Z _t0

qt0

ψ^η(s)|x(s)|²ds+2

1+ ^k0 q

_{Z t}

t0

ψ^η(s)|x(s)|²ds. (3.5) Similarly, we get

α₂q Z _t

t0

ψ^η(s)ψ^−ε((₁−q)s)|x(qs)|²ds≤α₂ Z _t0

qt0

ψ^η(s)|x(s)|²ds+α₂ Z _t

t0

ψ^η(s)|x(s)|²ds (3.6) and

α₄q Z _t

t0

ψ^η(s)ψ^−ε((₁−q)s)|x(qs)|^γds≤α₄ Z _t0

qt0

ψ^η(s)|x(s)|^γds+α₄ Z _t

t0

ψ^η(s)|x(s)|^γds. (3.7) Hence,

c₁ψ^η(t)|z(t)|²≤C−

α₁−α₂−2

1+ ^k0 q

c₂Cψη

_{Z t}

t0

ψ^η(s)|x(s)|²ds

−(α₃−α₄)

Z _t

t0

ψ^η(s)|x(s)|^γds+M_t, (3.8) where

C= c₂ψ^η(t₀)|z(t₀)|²+

2c₂ηC_ψk₀ q +α₂

_{Z t}

0

qt₀ ψ^η(s)|x(s)|²ds+α₄ Z _t0

qt₀ψ^η(s)|x(s)|^γds

≤ c₂ψ^η(t₀)(|x(t₀)|²+k₀|x(qt₀)|²) +

2c2ηCψ

k₀ q +α₂

Z _t0

qt0

ψ^η(s)|x(s)|²ds+α₄ Z _t0

qt0

ψ^η(s)|x(s)|^γds<_∞.

Sinceη∈(0,ε∧η¯)andα₃ >α₄, then

c₁ψ^η(t)|z(t)|²≤C+Mt. (3.9) By Lemma2.5, we have that lim sup_t→∞ψ^η(t)|z(t)|² ≤ _∞a.s. Hence, there is a finite positive random variableζ such that

sup

t₀<t<_∞

ψ^η(t)|z(t)|²≤ζ a.s. (3.10) Similar to (2.12), it follows that for any t₁>t₀

sup

t0≤t≤t₁

ψ^η(t)|x(t)|² ≤ ^{supt0≤t≤t1ψ}

η(t)|z(t)|² (₁−√

k0)² +

√k₀ 1−√

k0

ψ^η(t₀)kξk².

≤ ^ζ

(1−√ k₀)² +

√k₀ 1−√

k₀ψ^η(t₀)kξk²_.

This implies

lim sup

t→_∞

log|x(t)|

logψ(t) <−^η 2 a.s.

as required. The proof is therefore complete.

Remark 3.5. In Theorem 3.4, if ψ(t) = e^t and ψ(t) = 1+t, then (3.3) implies that Eq. (2.1) is almost surely exponentially stable and polynomially stable. Hence, we obtain the general stability result as it contains both exponential and polynomial stability as special cases. In other words, we extend these two classes of stability into the general decay stability in this paper. And this will be fully illustrated by Examples3.11and3.12.

Remark 3.6. From Theorem3.4, the almost sure stability with general decay rate of Eq. (2.1) has been examined and the upper bound of the convergence rate has been estimated.

Obviously, it is not convenient to check condition (3.2) of Theorem 3.4, since it is not related to coefficients f and g explicitly. Now, we shall impose some conditions on f and g to guarantee Theorem3.4 and establish a sufficient criteria on almost sureψ-type stability in terms of M-matrix.

Let us now state our hypothesis in terms of an M-matrix, which will replace condition (3.2).

Assumption 3.7. Let γ > 2 and assume that for eachi ∈ S, there are nonnegative numbers α_2i,α_3i,α_4i,β_1i,β_2i,β_3i,β_4i and a real numberα_1i as well as bounded functions h_i(·)such that

(x−D(y,t,i))^>f(x,y,t,i)

≤ α_1i|x|²+α_2iqψ^−ε((1−q)t)|y|²−α_3i|x|^γ+α_4iqψ^−ε((1−q)t)|y|^γ (3.11) and

|g(x,y,t,i)|² ≤β_1i|x|²+β_2iqψ^−ε((1−q)t)|y|²+β_3i|x|^γ+β_4iqψ^−ε((1−q)t)|y|^γ (3.12) for any (x,y,t)∈ Rⁿ×Rⁿ×[t₀,∞).

Assumption 3.8. Assume that

A:=−diag(_2α11+β₁₁, . . . , 2α1N+β_1N)−(₁+^pk₀)_{Γ (3.13)} is a nonsingular M-matrix.

Lemma 3.9 (see [25]). If A ∈ Z^N×N = {A = (a_ij)_N×N : a_ij ≤ 0, i 6= j}, then the following statements are equivalent:

(1) A is a nonsingular M-matrix.

(2) A is semi-positive; that is, there exists x0in R^N such that Ax 0.

(3) A⁻¹exists and its elements are all nonnegative.

(4) All the leading principal minors of A are positive; that is

a₁₁ · · · a_1k ... ... a_k1 · · · a_kk

>0 for every k=1, 2, . . . ,N.

In fact, by Assumption3.8 and Lemma3.9, it follows that θ = (θ₁, . . . ,θN)^>:= A⁻¹−→

1 >0 (3.14)

for alli∈ S, where−→

1 = (1, . . . , 1)^>.

Theorem 3.10. Let Assumptions2.1,3.7and3.8hold. Assume that maxi∈S (_2α2i+β_2i)θ_i+^pk₀(₁+^pk₀)

∑

γ_ijθ_j

!

<_{1 (3.15)}

and

mini∈S(2α_3i−β_3i)θ_i >max

i∈S (2α_4i+β_4i)θ_i. (3.16) Then for any given initial dataξ, there is a unique global solution x(t)of Eq.(2.1) and the solution is almost surelyψ-type stable.

Proof. Let us define the function V(x−D(y,t,i),t,i) = θ_i|x−D(y,t,i)|². Clearly, V obeys conditions (2.5) withc₁ =min_i∈Sθ_i andc₂ =max_i∈Sθ_i. To verify condition (3.2), we compute the operator LVas follows

LV(x,y,t,i) =2θ_i(x−D(y,t,i))^>f(x,y,t,i) +θ_i|g(x,y,t,i)|²+

∑

γ_ijθ_j|x−D(y,t,i)|². (3.17) By the basic inequality

|a+b|²≤(1+ε)|a|²+

1+¹ ε

|b|², for anya,b≥0 andr∈ [0, 1] and Assumption3.3, we have

|x−D(y,t,i)|² ≤(1+^pk₀)|x|²+

1+√¹ k0

|D(y,t,i)|²

≤(1+^pk₀)|x|²+^pk₀(1+^pk₀)ψ^−ε((1−q)t)|y|². (3.18) By Assumption3.7, it follows from (3.17) that

LV(x,y,t,i)≤ (2α_1i+β_1i)θ_i+ (1+^pk₀)

∑

γ_ijθ_j

!

|x|²

+ (2α_2i+β_2i)θ_i+^pk₀(1+^pk₀)

∑

γ_ijθ_j

!

qψ^−ε((1−q)t)|y|²

−(2α_3i−β_3i)θ_i|x|^γ+ (2α_4i+β_4i)θ_iqψ^−ε((1−q)t)|y|^γ. (3.19) By the definition ofθ_i, we have

(_2α1i+β_1i)θ_i+ (₁+^pk₀)

∑

γ_ijθ_j =−1.

Hence,

LV(x,y,t,i)≤ −α₁|x|²+α₂qψ^−ε((₁−q)t)|y|²−α₃|x|^γ+α₄qψ^−ε((₁−q)t)|y|^γ_{, (3.20)}

where

α₁=1, α₂=max

i∈S (2α_2i+β_2i)θ_i+^pk₀(1+^pk₀)

∑

γ_ijθ_j

! , α₃=min

i∈S(2α_3i−β_3i)θ_i, α₄ =max

i∈S (2α_4i+β_4i)θ_i. (3.21) Recalling (3.15) and (3.16), condition (3.2) is fulfilled. By Theorem3.4, we can conclude that for any given initial dataξ, there is a unique global solution x(t)and the solution of Eq. (2.1) is almost surelyψ-type stable. The proof is therefore complete.

Finally, we shall give two examples to illustrate the applications of our results.

Example 3.11. Let w(t)be a scalar Brownian motion. Let r(t) be a right-continuous Markov chain taking values inS= {1, 2}with the generator

Γ=

−1 1 4 −4

.

Of course, w(t) and r(t) are assumed to be independent. Consider the following scalar NSPEwMSs

d[x(t)−D(x(0.75t),t,r(t))] = f(x(t),t,r(t))dt+g(x(0.75t),t,r(t))dw(t), t≥1, (3.22) with initial data ξ(t) = x₀ (0.75 ≤ t ≤ 1) and r(1) = 1. Moreover, for (x,y,t,i) ∈ R×R× [0.75,∞)×S,

D(y,t,i) =

(0.1(₁+0.25t)^−0.5y, ifi=_1, 0.2(1+0.25t)^−0.5y, ifi=2, f(x,t,i) =

(−x−2x³, ifi=1, x−x³, ifi=2, and

g(y,t,i) =

(0.1(1+0.25t)^−0.5y², ifi=1, 0.5(1+0.25t)^−0.5y², ifi=2.

We note that Eq. (3.22) can be regarded as the result of the two equations d[x(t)−0.1(1+0.25t)^−0.5x(0.25t)]

= (−x(t)−2x³(t))dt+0.1(1+0.25t)^−0.5x²(0.25t)dw(t) (3.23) and

d[x(t)−0.2(1+0.25t)^−0.5x(0.25t)]

= (x(t)−x³(t))dt+_0.05(₁+0.25t)^−0.5x²(0.25t)dw(t) _(3.24) switching among each other according to the movement of the Markov chainr(t). It is easy to see that Eq. (3.23) is polynomially stable but Eq. (3.24) is unstable. However, we shall see that due to the Markovian switching, the overall system (3.22) will be polynomially stable. Note

that the coefficients f and g satisfy the local Lipschitz condition but they do not satisfy the linear growth condition. Through a straight computation, we have

(x−D(y,t, 1))^>f(x,t, 1)≤ −0.8|x|²+0.05(1+0.25t)⁻¹|y|²

−1.85|x|⁴+0.05(1+0.25t)⁻¹|y|⁴, (3.25) (x−D(y,t, 2))^>f(x,t, 2)≤0.6|x|²+0.1(1+0.25t)⁻¹|y|²

−0.85|x|⁴+0.05(1+0.25t)⁻¹|y|⁴, (3.26)

|g(y,t, 1)|² ≤0.01(1+0.25t)⁻¹|y|⁴, (3.27)

|g(y,t, 2)|² ≤0.25(1+0.25t)⁻¹|y|⁴ (3.28) where ψ^−ε(0.25t) = (1+0.25t)⁻¹,(ε = 1) and α₁₁ = −0.8, α₂₁ = 0.2, α₃₁ = 1.85, α₄₁ = 0.2, α₁₂ =0.6, α22 =0.4, α32 =0.85, α₄₂ =0.2, β₁₁ =0, β₂₁ = 0, β₃₁ =0, β₄₁ = 0.04, β₁₂ = 0, β₂₂=0, β₃₂ =0, β₄₂=1, γ=4. So the inequalities (3.25)–(3.28) show that the Assumption 3.7holds. By (3.13), we get the matrixA

A=−diag(2α₁₁+β₁₁, 2α₁₂+β₁₂)−(1+^pk₀)_Γ

=

2.8 −1.2

−4.8 3.6

. It is easy to compute

A⁻¹ =

0.833 0.278 1.111 0.648

.

By Lemma3.9, we see thatAis a non-singular M-matrix. Compute (θ₁,θ₂)^T =A⁻¹~₁= (1.111, 1.759)^T, and by (3.21), we have

α₂=max

i=1,2 (2α_2i+β_2i)θ_i+^pk₀(1+^pk₀)

∑

γ_ijθ_j

!

=0.7851, α3=min

i=1,2(2α_3iθ_i−β_3iθ_i) =2.9903, α₄ =max

i=1,2(2α_4iθ_i+β_4iθ_i) =2.4626.

Hence, we conclude that the conditions (3.15), (3.16) hold. By Theorem3.4, we can obtain that lim sup

t→_∞

log|x(t)|

logt ≤ −^η 2 a.s.

where η ∈ (0, 0.1159). That is to say, the solution of Eq. (3.22) decays at the polynomial rate of at least 0.05795.

Example 3.12. Let w(t)is a scalar Brownian motion. Let r(t) be a right-continuous Markov chain taking values inS={1, 2}with the generator

Γ=

−1 1 2 −2

.

Of course, w(t) and r(t) are assumed to be independent. Consider the following scalar NSPEwMSs

d[x(t)−D(x(0.5t)_,t,r(t))]

= f(x(t),x(0.5t),t,r(t))dt+g(x(t),x(0.5t),t,r(t))dw(t), t≥1,

(3.29) with initial data ξ(t) = x₀ (0.5 ≤ t ≤ 1) and r(1) = 1. Moreover, for (x,y,t,i) ∈ R×R× [0.5,∞)×S,D(y,t,i) =0.25e^−0.5ty,i=1, 2.

f(x,y,t,i) =

(−3x−2.5x³+e^−0.5ty, ifi=1 2x−1.5x³+_0.8e^−0.5t_{y, ifi}=_2, and

g(x,y,t,i) = (

ρ₁e^−0.5ty², ifi=1 ρ₂e^−0.5ty², ifi=2,

butρ₁andρ₂are unknown parameters. Eq. (3.29) can be regarded as a stochastically perturbed system of the follwing neutral pantograph equations with Markovian switching

d[x(t)−D(x(0.5t),t,r(t))]

dt = f(x(t),x(0.5t),t,r(t)).

Our aim here is to get the bounds on the unknown parameters ρ₁ and ρ₂ so that Eq. (3.29) remain stable. To apply Theorem3.10, we let γ=4. Noting

(_x−D(_{y,t, 1}))^>_f(_{x,y,t, 1})≤ −1.469|x|²+_0.25e^−t|y|²−2.125|x|⁴+_0.125e^−t|y|⁴, (3.30) (x−D(y,t, 2))^>f(x,y,t, 2)≤2.045|x|²+0.3e^−t|y|²−1.219|x|⁴+0.094e^−t|y|⁴, (3.31)

|g(x,y,t, 1)|² ≤ρ²₁e^−t|y|⁴, |g(x,y,t, 2)|² ≤ρ²₂e^−t|y|⁴ (3.32) where ψ^−ε(0.5t) = e^−t,(ε = 2) andα₁₁ = −1.469, α₂₁ = 0.5, α₃₁ = 2.125, α₄₁ = 0.25, α₁₂ = 2.045, α₂₂ = 0.6, α₃₂ = 1.219, α₄₂ = 0.188, β₁₁ = 0, β₂₁ = 0, β₃₁ = 0, β₄₁ = 2ρ²₁, β₁₂ = 0, β₂₂ = _0, β₃₂= _0, β₄₂ =_2ρ²₂. Then, the inequalities (3.30)–(3.32) show that the Assumption 3.7 holds. By (3.13), we see that the matrixAis

A=−diag(2α₁₁+β₁₁, 2α₁₂+β₁₂)−(1+^pk0)_Γ

=

4.188 −1.25

−2.5 6.59

. It is easy to compute

A⁻¹ =

0.269 0.051 0.102 0.171

.

By Lemma 3.9, we see that Ais a non-singular M-matrix. By (3.14), we then haveθ₁ = 0.32 and θ2 = 0.273. Clearly, α2 = max_i=1,2

(2α_2i+β_2i)θ_i +√

k0(1+√

k0)_∑²_j=1γ_ijθ_j

= 0.3569, while condition (3.16) becomes

min{1.36, 0.665}>max{0.16+0.64ρ²₁, 0.103+0.546ρ²₂}. i.e.,

ρ²₁ <_0.789, ρ²₂<_{1.0293. (3.33)}

By Theorem3.10, we can conclude that if the parametersρ_i,i=1, 2 satisfy (3.33), then for any initial data x0, there is a unique global solution x(t)to Eq. (2.1) on t ∈ [1,∞). Moreover, the solution has the property that

lim sup

t→_∞

log|x(_t)|

t ≤ −^η

2 a.s.

whereη ∈ (_{0, 0.8932}). That is to say, the solution of Eq. (3.29) decays at the exponential rate of at least 0.4466.

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments and suggestions. The research of W. Mao was supported by the National Natural Science Foun- dation of China (11401261) and “333 High-level Project” of Jiangsu Province. The research of L. Hu was supported by the National Natural Science Foundation of China (11471071). The research of X. Mao was supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1).

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