Almost sure stability with general decay rate of
neutral stochastic pantograph equations
with Markovian switching
Wei Mao
1,
Liangjian Hu
B2and
Xuerong Mao
31School 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→Rn×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)
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 {Ft}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 = supqt≤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}. LFp
t([qt,t];R
all(Ft)-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= (γij)N×N given by:
P(r(t+∆) =j|r(t) =i) =
(
γij∆+◦(∆), ifi6= j, 1+γij∆+◦(∆), ifi=j.
where ∆ > 0. Here γij ≥ 0 is the transition rate from i to j, i =6 j, While γii = −∑j6=iγij. 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 ∈ L2Ft
0([qt0,t0];R
n).
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 constantkd 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 ki ∈ (0, 1)
such that
|D(u,t,i)−D(v,t,i)|2 ≤ki|u−v|2. (2.3)
Letk0 =maxi∈Ski 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)]
+ N
∑
j=1
γijV(x−D(y,t,i),t,j), (2.4)
where
Vt(x,t,i) =
∂V(x,t,i)
∂t , Vx(x,t,i) =
∂V(x,t,i)
∂x1
, . . . ,∂V(x,t,i)
∂xn
,
Vxx(x,t,i) =
∂2V(x,t,i) ∂xi∂xj
Assumption 2.3. There exist a function V ∈ C2,1(Rn×[t
0,∞)×S;R+) and some positive
constantsc1,c2,αi,(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
δp−1.
Lemma 2.5(see [28]). Let A(t),U(t)be twoFt-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.
Iflimt→∞A(t)<∞a.s. then for almost allω ∈Ω,limt→∞x(t)< ∞ and limt→∞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
τk =inf{t ∈[t0,σe):|x(t)| ≥k}.
Clearly,τk is increasing ask → ∞. Setτ∞ = limk→∞τk, 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∞ > tq0 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(τk∧t1),τk∧t1,r(τk∧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)|γ
dt,
where z(t) = x(t)−D(x(qt),t,r(t)). Let us now restrict t1 ∈ [t0,tq0]. By condition (2.5), we
then get
c1E|z(τk∧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)|γ
dt
≤2c2(1+k0)Ekξk2+1
qE
Z t0
qt0
α0+α2|x(t)|2+α4|x(t)|γ
It then follows that
E|z(τk∧t1)|2≤
H1
c1
, t0≤t1 ≤
t0
q (2.9)
for any k≥k¯0. By Lemma2.4, we get
E|x(τk∧t1)|2≤
E|z(τk∧t1)|2
1−δ +
E|D(x(q(τk∧t1)),τk∧t1,r(τk∧t1))|2 δ
≤ E|z(τk∧t1)| 2
1−δ +
k0E|x(q(τk∧t1))|2
δ .
This implies
sup t0≤t1≤tq0
E|x(τk∧t1)|2 ≤
supt
0≤t1≤tq0
E|z(τk∧t1)|2
1−δ +
k0supt 0≤t1≤tq0
E|x(q(τk∧t1))|2
δ . (2.10)
Lettingδ = √
k0, it follows from (2.9) that
sup t0≤t1≤tq0
E|x(τk∧t1)|2≤
H1
c1(1− √
k0)
+pk0Ekξk2+ p
k0 sup
t0≤t1≤tq0
E|x(τk∧t1)|2. (2.11)
Hence, we have
E|x(τk∧t1)|2 ≤
H1
c1(1− √
k0)
+
√
k0
1−√k0
Ekξk2, t0 ≤t1≤
t0
q. (2.12)
In particular, E|x(τk∧ tq0)|2 ≤ c H1
1(1− √
k0)+ √
k0 1−√k0Ekξk
2, ∀k ≥ k¯
0. This implies k2P(τk ≤ tq0) ≤ H1
c1(1− √
k0) + √
k0 1−√k0Ekξk
2. Lettingk→∞, we hence obtain that P
τ∞≤ tq0=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∞ > qt02 a.s. given that we have shown (2.12)–(2.14). For any
k≥k¯0andt1∈[t0,tq02], it follows from (2.6) that
c1E|z(τk∧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)|γ
dt
=H1+E
Z t0
q2 t0
q
α0+α2|x(qt)|2+α4|x(qt)|γ
dt
=H1+
1 qE
Z t0
q
t0
α0+α2|x(t)|2) +α4|x(t)|γ
Consequently
E|z(τk∧t1)|2≤
H2
c1
, t0≤ t1 ≤
t0
q2.
Similar to (2.12), we can obtain that
E|x(τk∧t1)|2 ≤
H2
c1(1− √
k0)
+
√
k0
1−√k0
Ekξk2, t0≤t1≤
t0
q2. (2.16)
In particular, E|x(τk∧ tq02)|2 ≤
H2
c1(1− √
k0)+ √
k0 1−√k0Ekξk
2, ∀k ≥ k¯0. This implies k2P(τ
k ≤ tq02)≤
H2
c1(1− √
k0)+ √
k0 1−√k0Ekξk
2. Lettingk → ∞, we then obtain that P(τ
∞ ≤ t0
q2) =0, namely P(τ∞ >
t0
q2) =1. Lettingk→∞in (2.16) yields
E|x(t1)|2 ≤
H2
c1(1− √
k0)
+
√
k0
1−√k0
Ekξk2, t0≤t1≤
t0
q2.
Repeating this procedure, we can show that, for any integeri≥1,τ∞ > tq0i a.s. and
E|x(t)|2 ≤ Hi c1(1−
√
k0)
+
√
k0
1−√k0
Ekξk2, t0 ≤t1≤
t0
qi,
where Hi = 2c2(1+k0)Ekξk2+ER t0 qi
t0 α0+α2|x(qt)|
2+α
4|x(qt)|γ
dt < ∞. 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 coefficientsαi,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=supt>0|ψ
0(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)|
Obviously, when ψ(t) =et 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) ∈ [t
0,∞)×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
c2Cψη¯.
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 toψη(t)V(z(t),t,r(t)), we obtain that
ψη(t)V(z(t),t,r(t)) =ψη(t0)V(z(t0),t0,r(t0)) +
Z t
t0 ηψ
0(s)
ψ(s)ψ
η(s)V(z(s),s,r(s))ds
+
Z t
t0
ψη(s)LV(x(s),x(qs),s,r(s))ds+Mt,
where Mt = Rtt0ψη(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
c1ψη(t)|z(t)|2≤c2ψη(t0)|z(t0)|2+c2ηCψ
Z t
t0
ψη(s)|z(s)|2ds−α1
Z t
t0
ψη(s)|x(s)|2ds
+α2q
Z t
t0
ψη(s)ψ−ε((1−q)s)|x(qs)|2ds−α3
Z t
t0
ψη(s)|x(s)|γds
+α4q
Z t
t0
ψη(s)ψ−ε((1−q)s)|x(qs)|γds+Mt, (3.4)
where Cψ = supt0≤t<∞|
ψ0(t)
ψ(t)| < φ < ∞. By the basic inequality|a+b|
the definition ofψfunction, we have
Z t
t0
ψη(s)|z(s)|2ds≤2
Z t
t0
ψη(s) |x(s)|2+k0ψ−ε((1−q)s)|x(qs)|2ds
≤2
Z t
t0
ψη(s)|x(s)|2ds+2k0
Z t
t0
ψη(s)ψ−ε((1−q)s)|x(qs)|2ds
≤2
Z t
t0
ψη(s)|x(s)|2ds+2k0
Z t
t0
ψη(s)ψ−η((1−q)s)|x(qs)|2ds
≤2
Z t
t0
ψη(s)|x(s)|2ds+2k0
Z t
t0
ψη(qs)|x(qs)|2ds
≤2k0 q
Z t0
qt0
ψη(s)|x(s)|2ds+2
1+ k0 q
Z t
t0
ψη(s)|x(s)|2ds. (3.5)
Similarly, we get
α2q
Z t
t0
ψη(s)ψ−ε((1−q)s)|x(qs)|2ds≤α2
Z t0
qt0
ψη(s)|x(s)|2ds+α2
Z t
t0
ψη(s)|x(s)|2ds (3.6)
and
α4q
Z t
t0
ψη(s)ψ−ε((1−q)s)|x(qs)|γds≤α4
Z t0
qt0
ψη(s)|x(s)|γds+α4
Z t
t0
ψη(s)|x(s)|γds. (3.7)
Hence,
c1ψη(t)|z(t)|2≤C−
α1−α2−2
1+ k0 q
c2Cψη Z t
t0
ψη(s)|x(s)|2ds
−(α3−α4)
Z t
t0
ψη(s)|x(s)|γds+Mt, (3.8)
where
C= c2ψη(t0)|z(t0)|2+
2c2ηCψ k0
q +α2
Z t0
qt0
ψη(s)|x(s)|2ds+α4
Z t0
qt0
ψη(s)|x(s)|γds
≤ c2ψη(t0)(|x(t0)|2+k0|x(qt0)|2)
+
2c2ηCψ k0
q +α2
Z t0
qt0
ψη(s)|x(s)|2ds+α4
Z t0
qt0
ψη(s)|x(s)|γds<∞.
Sinceη∈(0,ε∧η¯)andα3 >α4, then
c1ψη(t)|z(t)|2≤C+Mt. (3.9)
By Lemma2.5, we have that lim supt→∞ψη(t)|z(t)|2 ≤ ∞a.s. Hence, there is a finite positive
random variableζ such that
sup t0<t<∞
ψη(t)|z(t)|2≤ζ a.s. (3.10)
Similar to (2.12), it follows that for any t1>t0
sup t0≤t≤t1
ψη(t)|x(t)|2 ≤
supt
0≤t≤t1ψ
η(t)|z(t)|2 (1−√k0)2
+
√
k0
1−√k0
ψη(t0)kξk2.
≤ ζ
(1−√k0)2
+
√
k0
1−√k0
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) = et 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
α2i,α3i,α4i,β1i,β2i,β3i,β4i and a real numberα1i as well as bounded functions hi(·)such that
(x−D(y,t,i))>f(x,y,t,i)
≤ α1i|x|2+α2iqψ−ε((1−q)t)|y|2−α3i|x|γ+α4iqψ−ε((1−q)t)|y|γ (3.11)
and
|g(x,y,t,i)|2 ≤β1i|x|2+β2iqψ−ε((1−q)t)|y|2+β3i|x|γ+β4iqψ−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+β1N)−(1+
p
k0)Γ (3.13)
is a nonsingular M-matrix.
Lemma 3.9 (see [25]). If A ∈ ZN×N = {A = (aij)N×N : aij ≤ 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 RN 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 · · · a1k ..
. ... ak1 · · · akk
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
max
i∈S (2α2i+β2i)θi+
p
k0(1+ p
k0)
N
∑
j=1 γijθj
!
<1 (3.15)
and
min
i∈S(2α3i−β3i)θi >maxi∈S (2α4i+β4i)θi. (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) = θi|x−D(y,t,i)|2. Clearly, V obeys conditions (2.5) withc1 =mini∈Sθi andc2 =maxi∈Sθi. To verify condition (3.2), we compute the operator LVas follows
LV(x,y,t,i) =2θi(x−D(y,t,i))>f(x,y,t,i) +θi|g(x,y,t,i)|2+ N
∑
j=1
γijθj|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+ p
k0(1+ p
k0)ψ−ε((1−q)t)|y|2. (3.18)
By Assumption3.7, it follows from (3.17) that
LV(x,y,t,i)≤ (2α1i+β1i)θi+ (1+
p
k0)
N
∑
j=1 γijθj
!
|x|2
+ (2α2i+β2i)θi+
p
k0(1+ p
k0)
N
∑
j=1 γijθj
!
qψ−ε((1−q)t)|y|2
−(2α3i−β3i)θi|x|γ+ (2α4i+β4i)θiqψ−ε((1−q)t)|y|γ. (3.19)
By the definition ofθi, we have
(2α1i+β1i)θi+ (1+ p
k0)
N
∑
j=1
γijθj =−1.
Hence,
where
α1=1, α2=max
i∈S (2α2i+β2i)θi+
p
k0(1+ p
k0)
N
∑
j=1 γijθj
!
,
α3=min
i∈S(2α3i−β3i)θi, α4 =maxi∈S (2α4i+β4i)θi. (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)
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+ p
k0)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 (2α2i+β2i)θi+ p
k0(1+ p
k0) 2
∑
j=1 γijθj
!
=0.7851,
α3=min
i=1,2(2α3iθi−β3iθi) =2.9903, α4 =maxi=1,2(2α4iθi+β4iθi) =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−0.5ty, 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 ≤ρ12e−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+ p
k0)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 = maxi=1,2
(2α2i+β2i)θi +
√
k0(1+ √
k0)∑2j=1γijθj
= 0.3569, while condition (3.16) becomes
min{1.36, 0.665}>max{0.16+0.64ρ21, 0.103+0.546ρ22}.
i.e.,
By Theorem3.10, we can conclude that if the parametersρi,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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