# The Global Solutions and Moment Boundedness of Stochastic Multipantograph Equations

Authors: Maosheng Tian; Xuejing Meng; Jihong Chen; Xiaoqi Tang
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

We consider the existence of global solutions and their moment boundedness for stochastic multipantograph equations. By the idea of Lyapunov function, we impose some polynomial growth conditions on the coefficients of the equation which enables us to study the boundedness more applicably. Methods and techniques developed here have the potential to be applied in other unbounded delay stochastic differential equations.


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## Body

## 1. Introduction

Delay differential equations (DDEs) play an important role in applied mathematics owing to providing a powerful model of many phenomena, such as some physical applications with noninstant transmission phenomena, neural networks, or other memory processes, and specially biological motivations (e.g., [1–3]) like species’ growth or incubating time on disease models among many others.An interesting case of DDEs which is the subject of a lot of papers is the pantograph equation:(1)x˙t=ft,xt,xqt,t≥0x0=x0,where 0<q≤1, x0∈Rn. The name pantograph originated from the work of Ockendon and Tayler [4] on the collection of current by the pantograph head of an electric locomotive. The pantograph equations appeared in modeling of various problems such as number theory, astrophysics, nonlinear dynamical systems, biology, economy, quantum mechanics, and electrodynamics. For some applications of this type of equations, we refer to [4–8].Since any realistic systems are inevitably subject to environmental noise, the stochastic pantograph equation(2)dxt=ft,xt,xqtdt+gt,xt,xqtdwt,t≥0,x0=x0therefore receives more and more attention. Fan et al. [9] have given the sufficient conditions of existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for (2). Appleby and Buckwar [10] have studied the asymptotic growth and decay properties of solutions of the linear stochastic pantograph equation with multiplicative noise. For more literatures we refer the interested reader to [11–13].Properties of the analytic solution of (1) and (2) as well as numerical methods have been studied by several authors, for example, Lü and Cui [14], Iserles [15, 16], Liu et al. [17], and Appleby and Buckwar [10]. A more general form than (1) is the multipantograph equation; it reads(3)x˙t=ft,xt,xq1t,xq2t,…,xqLt,t≥0,x0=x0,where 0<q1<q2<⋯<qL<1. Equation (3) was also studied by many authors numerically and analytically (see, e.g., [18–20] and the references cited therein).However, to the best of our knowledge, there are no corresponding numerical and analytical results on stochastic multipantograph equations which also have numerous applications as (2) in engineering and science. It has the form(4)dxt=ft,xt,xq1t,…,xqLtdt+gt,xt,xq1t,…,xqLtdwt,x0=x0,t≥0,where 0<q1<q2<⋯<qL<1, f:R+×Rn(L+1)→Rn, and g:R+×Rn(L+1)→Rn×m. In this paper, we mainly study the asymptotic properties of the analytic solution of (4). Owing to the fact that the delay is unbounded, many methods which are useful for the bounded delay systems are inefficient or impossible for these systems. For example, some classical techniques such as Lyapunov direct methods in [21–23] cannot be transferred directly to the study of boundedness properties for unbounded delay equation (4). By introducing a decay function to control the unbounded delay term, we develop the traditional techniques like Lyapunov direct methods to be applied in the pantograph equations’ cases.It is well known for stochastic differential equations that the linear growth condition plays an important role in suppressing the potential explosion of solutions and guarantees the existence of the global solutions (cf. [22–25]). This paper, without the linear growth condition, shows that (4) almost surely makes a global solution x(t,x0) and this solution is bounded in the sense(5)limsupt→∞⁡Ext,x0p≤Kp,(6)limsupt→∞⁡1t∫0tExs,x0qds≤Lq,where p, q, Kp, and Lq are positive constants independent of the initial data x0.The content of the paper is as follows. In Section2, we give some necessary notations and useful lemmas. Section 3 is devoted to presenting a general theorem for the existence and boundedness of the global solution. In Section 4, we apply Theorem 4 to obtain two useful criteria which can be easily verifiable in applications. Two examples are provided to show how our results will be applied in Section 5. Further remarks are made to conclude the paper in the final section.


## 2. Some Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. 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 the probability space. Let · be the Euclidean norm in Rn. If A is a vector or matrix, its transpose is denoted by AT; if A is a matrix, its trace norm is denoted by |A|=trace(ATA). Let R+=[0,∞); R×=R∖{0} and R×n=Rn∖{0}.Moreover, letC2(Rn;R+) denote the family of all functions V(x) from Rn to R+ which are continuously twice differentiable. For all t≥0, x∈Rn, and Y∈Rn×L and any V(x)∈C2(Rn;R+), define a function LV:R+×Rn×Rn×L→R by(7)LVt,x,Y=Vxxft,x,Y+12tracegTt,x,YVxxxgt,x,Y,where Y=(y1,y2,…,yL)∈Rn×L,(8)Vxx=∂Vx∂x1,∂Vx∂x2,…,∂Vx∂xn,Vxxx=∂2Vx∂xi∂xjn×n. Thus, if x(t) is a solution of (4), by Itô’s formula, we have (9)dVxt=LVxtdt+Vxxtgt,xt,Ytdwt, where(10)LVxt=LVt,xt,Yt,(11)Yt=y1t,y2t,…,yLt,yit=xqit,i=1,…,L.For coefficientsf and g, we will impose the following standing assumptions.Assumption 1.
Bothf and g satisfy the local Lipschitz condition. That is, for each j=1,2,…, there exists a constant cj>0 such that (12)ft,x,y1,…,yL-ft,x¯,y¯1,…,y¯L∨gt,x,y1,…,yL-gt,x¯,y¯1,…,y¯L≤cjx-x¯+∑i=1Lyi-y¯i for all t≥0 and those x,x¯,yi,y¯i∈Rn with x∨x¯∨yi∨y¯i≤j(1≤i≤L).The following lemma shows the bounded property of polynomial functions.Lemma 2.
For anyh(x)∈C(Rn;R+) and positive constants a and q, if h(x)=o(|x|q) as |x|→∞, then (13)supx∈Rn⁡-axq+oxq<∞.Proof.
Chooser>0 such that h(x)<a|x|q when x∈Rn,|x|>r, which implies that -a|x|q+o(|x|q)<0. We therefore have (14)supx∈Rn⁡-axq+oxq=supx∈Rn,x≤r⁡-axq+oxq<∞, as required.To proceed, we need a lemma which will play a crucial role in overcoming the difficulties for the existence of unbounded delays. For the sake of simplicity, we denote(15)Φɛ≔Φɛt,x,Y=∑k=1K∑l=1Lakylαke-ɛq¯lt-ql-1xαk,where ak, αk (1≤k≤K) are all positive constants and ɛ≥0, q¯l=1-ql. If ɛ=0, then we have(16)Φ≔Φx,Y=∑k=1K∑l=1Lakylαk-ql-1xαk.Lemma 3.
Assume that0≤γ≤ɛ. If x(t) is a solution to (4) with initial data x0∈Rn, then(17)∫0teγsΦɛs,xs,Ysds≤0.Proof.
LetI denote the left side of (17). We compute (18)I=∑k=1K∑l=1Lak∫0teγse-ɛq¯lsylsαk-ql-1xsαkds=∑k=1K∑l=1Lak∫0teγse-ɛq¯lsxqlsαkds-ql-1∫0teγsxsαkds≤∑k=1K∑l=1Lak∫0teγqlsxqlsαkds-ql-1∫0teγsxsαkds≤∑k=1K∑l=1Lakql-1∫0qlteγsxsαkds-ql-1∫0teγsxsαkds=∑k=1K∑l=1Lakql-1∫tqlteγsxsαkds≤0. The proof is complete.


## 3. A General Theorem

In this section, by Lyapunov function techniques, we establish a general theorem for the existence and boundedness of the global solution to (4).Theorem 4.
Assume that there exist positive constantsɛ,p,a, letting the function V(x)=|x|p, if(19)LVt,x,Y+ɛVx≤Φɛ-axq+oxq,t≥0,x∈Rn,Y∈Rn×L,where Φɛ is defined by (15) and LV(t,x,Y) is defined by (7). Then, for any given initial data x0∈Rn, there exists a unique global solution xt,x0(0≤t<∞) to (4) which obeys (5) and (6).Proof.
For initial datax0∈Rn, the proof will be divided into three steps.
Step
1 (existence of the global solution). By Assumption 1, there exists a unique maximal local solution x(t)=x(t,x0)(-τ0≤t<σ) to (4), where σ is the explosion time. Let k0 be a sufficiently large positive number such that |x0|≤k0. For each integer k≥k0, define the stopping time:(20)σk=inf⁡-τ0≤t<σ:Vxt≥k.Clearly, σk is increasing and σk→σ∞≤σ as k→∞. If we can show σ∞=∞, a.s., then σ=∞ a.s., which implies the desired result. This is also equivalent to proving that, for any t>0, P(σk≤t)→0 as k→∞. Letting tk=t∧σk, by the Itô formula (19), and noting that Φɛ is decreasing in ɛ, we have(21)kPσk≤t≤EVxtk=const+E∫0tkLVxsds≤const+E∫0tkΦɛs,xs,Ys-ɛVx-axsq+oxqds≤const+E∫0tkΦxs,Ys+Kds≤const+Kt,where we have used Lemmas 2 and 3. In this paper, const and K always represent some positive constants whose values are not important.
Therefore,(22)Pσk≤t⟶0ask⟶∞, as required.
Step
2 (moment boundedness). Applying the Itô formula and connecting (19) with Lemmas 2 and 3, we compute (23)eɛtEVxt=EVx0+E∫0teɛsLVxs+ɛVxsds≤const+E∫0teɛsΦɛs,xs,Ys+Kds≤const+ε-1Keɛt, which implies that (24)limsupt→∞⁡EVxt≤ε-1K, as desired.
Step
3 (moment boundedness average in time). By (19), applying the Itô formula to V(x) yields (25)a∫0tExsqds≤E∫0t-LVs,xs,Ys+Φxs,Ys+Kds≤-EVxt-EVx0+const+Kt≤-EVxt+const+Kt≤const+Kt,which implies the desired (6). The proof is complete.Remark 5.
In existence of the global solution, it is not necessary to specifyV(x)=|x|p. If the function V(x) satisfies liminf|x|→∞⁡V(x)=∞, existence of the global solution still holds.After completing the proof of the general theorem, we continue to examine it in both ways. On one hand, (5) and (6) are the two main results whose understanding can be enriched as the following corollary shows.Corollary 6.
Letx(t) be a positive stochastic process with properties (5) and (6). If 0<γ<p<∞ and 0<γ¯<q<∞, then(26)limsupt→∞⁡Extγ≤Kγ,limsupt→∞⁡1t∫0tExsγ¯ds≤Lγ¯,where Kγ and Lγ¯ are positive constants, which may be dependent on Kp, Lq, and γ,γ¯.Proof.
By the Lyapunov inequality, for anypth integrable random variable X, we have (27)EXγ1/γ≤EXp1/p,which gives the first result. By the Lyapunov inequality, the Hölder inequality, and (6),(28)limsupt→∞⁡1t∫0tExsγ¯ds≤limsupt→∞⁡1t∫0tExsqγ-/qds≤limsupt→∞⁡1t∫0tdsq-γ¯/q∫0tExsqdsγ-/q=limsupt→∞⁡1t∫0tExsqdsγ-/q≤Lqγ-/q. This completes the proof.On the other hand, condition (19) is inconvenient to be checked because it is unrelated to functions f and g explicitly. To make Theorem 4 more applicable, one natural alternative is to look for other simplified conditions on f and g. Applying (7) to V=|x|p leads to(29)LVt,x,Y=pxp-2xTft,x,Y+p2xp-2gt,x,Y2+pp-22xp-4xTgt,x,Y2≔I1+I2+I3,where Y∈Rn×L is defined in (11). By (19), if we can test(30)I1+I2+I3≤Φɛt,x,Y-axq+oxq,where q and a represent positive constants, q>p, o(|x|q) denotes some h(x)∈C(Rn) which satisfies |x|-qh(x)→0 as |x|→∞. By Lemma 2 we can easily decide that supx∈Rn[-b|x|q+o(|x|q)]<∞, which implies that Theorem 4 will hold.In the next section, we give some alternative conditions to guarantee Theorem4, which shows coefficients f and g how to determine existence of global solution to (4) and boundedness of this solution.


## 4. Main Results

To match (30), we will impose the following two groups of conditions on the functions f and g, which shows that the growth of both f and g is polynomial or controlled by polynomial speed.For anyx∈Rn,Y∈Rn×L, and t≥0,(A1)
xTf(t,x,Y)≤-σ|x|α+2+∑l=1L‍σ¯l|yl|α+2e-ɛq¯lt+o(|x|α+2),(A2)
|g(t,x,Y)|≤λ|x|β+1+∑l=1L‍λ¯l|yl|β+1e-ɛq¯lt+o(|x|β+1),and(B1)
|x|-1|f(t,x,Y)|≤γ|x|α+∑l=1L‍γ¯l|yl|αe-ɛq¯lt+o(|x|α),(B2)
|x|-1|g(t,x,Y)|≤λ|x|β+∑l=1L‍λ¯l|yl|βe-ɛq¯lt+o(|x|β),(B3)
-|x|-4|xTg(t,x,Y)|2≤-σ|x|2β+∑l=1L‍σ¯l|yl|2βe-ɛq¯lt+o(|x|2β),where all parameters are positive. Since e-ɛq¯lt is decreasing in ɛ, the above conditions will still hold when ɛ is replaced by any ɛ′∈(0,ɛ].For the purpose of simplicity, define some notations:(31)λ~=λ+∑l=1Lλ¯l;λ^=λ+∑l=1Lλ¯lql-1;(32)σ^=σ-∑l=1Lσ¯lql-1;γ^=γ+∑l=1Lγ¯lql-1.Theorem 7.
Let Assumption1 hold. Under conditions (A1) and (A2), if α≥2β and(33)2σ^>λ~λ^H2β-α,then, for any initial data x0∈Rn, there exists a unique global solution x(t,x0) to (4) which obeys (5) and (6), where p∈(0,p0), q∈(0,α+p0). H(x)=0 if x<0, and H(x)=1 if x≥0, and p0 is defined as(34)p0=1+2σ^λ~λ^H2β-α-1.In particular, p0=∞ if α>2β.Proof.
By (33) and (34), we have p0>2. For any p∈(2,p0), by conditions (A1) and (A2), we estimate I1~I3. By (A1) and the Young inequality, (35)I1=pxp-2xTft,x,Y≤pxp-2-σxα+2+∑l=1Lσ¯lylα+2e-ɛq¯lt+oxα+2≤-pσxα+p+p∑l=1Lσ¯le-ɛq¯ltp-2xα+p+α+2ylα+pα+p+oxα+p≤-pσxα+p+p∑l=1Lσ¯lp-2xα+p+α+2ylα+pe-ɛq¯ltα+p+oxα+p.
Letxi∈R and δi≥0. By the Cauchy inequality, we have(36)∑i=1nδixi2≤∑i=1nδi∑i=1nδixi2.
Recall the elementary inequality: for any0<ρ<1,  a,b∈R,(37)a+b2≤a2ρ+b21-ρ.Noting that p>2, by (A2), and combining the Young inequality and inequalities (36) and (37) we estimate(38)I2+I3=p2xp-2gt,x,Y2+pp-22xp-4xTgt,x,Y2≤pp-12xp-2gt,x,Y2≤pp-12xp-2λxβ+1+∑l=1Lλ¯lylβ+1e-ɛq¯lt+oxβ2≤pp-12ρxp-2λxβ+1+∑l=1Lλ¯lylβ+1e-ɛq¯lt2+pp-121-ρox2β+p≤pp-12ρxp-2λ~λx2β+2+∑l=1Lλ¯lyl2β+2e-2ɛq¯lt+ox2β+p≤pp-12ρλ~λx2β+p+∑l=1Lλ¯lp-2x2β+p+2β+2e-ɛq¯ltyl2β+p2β+p+ox2β+p. We therefore have(39)I1+I2+I3≤Φɛ+I,where (40)Φɛ=pα+2α+p∑l=1Lσ¯le-ɛq¯ltylα+p-ql-1xα+p+β+1pp-1λ~ρ2β+p∑l=1Lλ¯le-ɛq¯ltyl2β+p-ql-1x2β+p,I=-pσxα+p+pα+p∑l=1Lσ¯lp-2+α+2ql-1xα+p+pp-12ρλ~λ+∑l=1Lλ¯lp-2+2β+2ql-12β+px2β+p+oxα+p≤-pσxα+p+p∑l=1Lσ¯lql-1xα+p+pp-12ρλ~λ+∑l=1Lλ¯lql-1x2β+p+oxα+p≔-paρxα+p+oxα+p, in which we have used ql≤1(1≤l≤L). By (33) and (34), (41)aρ=1=σ-∑l=1Lσ¯ql-1-p-1λ~2λ+∑l=1Lλ¯lql-1H2β-α>σ^-p0-1λ~2λ^H2β-α=0. Choose ρ sufficiently near 1 such that a(ρ)>0, which shows that (39) has similar expression to (30). By Theorem 4 and Corollary 6, there almost surely exists a unique global solution to (4), and for any p∈(0,p0) and q∈(0,α+p0), this solution still holds properties (5) and (6), as required.Ifα≤2β, for any initial data x0∈R×n, we give the following lemma for existence of global solution to (4).Lemma 8.
Let Assumption1 hold. For positive constants p and K, letting V(x)=|x|p-pln|x|, if for any x∈R×n, Y∈R×nL and t≥0,(42)LVt,x,Y≤Φx,Y+K, where Φ(x,Y) defined by (16). Then, for any given x0∈R×n, there exists a unique global solution to (4).Proof.
Similar to the proof of Theorem4, for any initial data x0∈R×n, there almost surely exists a unique maximal local solution x(t)=x(t,x0)(-τ0≤t<σ) to (4). Similar to the stopping time (20), define the stopping time (here we still use σk) (43)σk=inf⁡-τ0≤t<σ:xt∉k-1,kfor  each  integerk. We still need to prove P(σk≤t)→0 as k→∞ for any t≥0. For any y>0, let Uy=yp-plny and Dk≔{x:k-1<|x|<k}. By the definition of σk, we have x(σk)∈∂Dk when σk<∞, which implies that (44)Vxσk≥Uk-1∧Uk≔uk. We therefore have (45)Pσk≤tuk≤Pσk≤tVxσk≤EVxt∧σk≤EVx0+E∫0tkLVxsds≤EVx0+E∫0tkΦxs,Ys+Kds≤const+Kt. Note that u(k)→∞ as k→∞. Letting k→∞, for any t≥0, we have (46)Pσk≤t⟶0,ask⟶∞, as required.Applying this lemma, we may obtain the following theorem.Theorem 9.
Let Assumption1 hold. Under conditions (B1), (B2), and (B3), if α≤2β and(47)2σ^>λ~λ^+2γ^Hα-2β,then, for any initial data x0∈R×n, there exists a unique global solution to (4) and this solution has properties (5) and (6) in which 0<p<p0 and 0<q<2β+p0 and p0 is defined as(48)p0=2-σ^-1λ~λ^+2γ^Hα-2β.Proof.
By (48), p0∈(0,2). Fix any p∈(0,p0); using (B1), (B2), and (B3), we estimate I1~I3 respectively. By the Young inequality and (B1), we have (49)I1=pxp-2xTft,x,Y≤pxpγxα+∑l=1Lγ¯lylαe-ɛq¯lt+oxα≤pγxα+p+p∑l=1Lγ¯le-ɛq¯ltpxα+p+αylα+pα+p+oxα+p≤pγxα+p+p∑l=1Lγ¯lpxα+p+αylα+pe-ɛq¯ltα+p+oxα+p. By (B2) and elementary inequalities (36) and (37), for any ρ∈(0,1), we have(50)I2=p2xp-2gt,x,Y2≤p2xpλxβ+∑l=1Lλ¯le-ɛq¯ltylβ+oxβ2≤p2ρxpλxβ+∑l=1Lλ¯le-ɛq¯ltylβ2+ox2β+p≤p2ρxpλ~λx2β+∑l=1Lλ¯le-2ɛq¯ltyl2β+ox2β+p≤p2ρλ~λx2β+p+∑l=1Lλ¯le-ɛq¯ltpx2β+p+2βyl2β+p2β+p+ox2β+p≤pλ~λ2ρx2β+p+pλ~2ρ∑l=1Lλ¯lpx2β+p+2βyl2β+pe-ɛq¯lt2β+p+ox2β+p. Noting that p<2, by the Young inequality and condition (B3), we have the following estimate: (51)I3=pp-22xp-4xTgt,x,Y2≤pp-22xpσx2β-∑l=1Lσ¯le-ɛq¯ltyl2β+ox2β≤pp-22σx2β+p-∑l=1Lσ¯lpx2β+p+2βyl2β+pe-ɛq¯lt2β+p+ox2β+p. We therefore have(52)I1+I2+I3≤Φɛ+I,where (53)Φɛ=αpα+p∑l=1Lγ¯le-ɛq¯ltylα+p-ql-1xα+p+pλ~βρ2β+p∑l=1Lλ¯le-ɛq¯ltyl2β+p-ql-1x2β+p-pp-2β2β+p∑l=1Lσ¯le-ɛq¯ltyl2β+p-ql-1x2β+p,I=pγxα+p+p∑l=1Lγ¯lp+αql-1α+pxα+p+pλ~λ2ρx2β+p+ox2β+p+pλ~2ρ∑l=1Lλ¯lp+2βql-12β+px2β+p+pp-22σ-∑l=1Lσ¯lp+2βql-12β+px2β+p≤pγ+p∑l=1Lγ¯lql-1xα+p+pλ~2ρλ+∑l=1Lλ¯lql-1x2β+p+pp-22σ-∑l=1Lσ¯lql-1x2β+p+ox2β+p≔-aρx2β+p+ox2β+p, where we have used ql≤1(1≤l≤L). By conditions (47) and (48) we have (54)2paρ=1=-2γ+∑l=1Lγ¯lql-1Hα-2β-λ~λ+∑l=1Lλ¯lql-1-p-2σ-∑l=1Lσ¯lql-1=-2γ^Hα-2β-λ~λ^-p-2σ^>-2γ^Hα-2β-λ~λ^-p0-2σ^=0. Choose ρ sufficiently near 1 such that a(ρ)>0, which shows that (52) has similar expression to (30). By the similar proof process to Theorem 7, we may obtain the desired result. This completes this proof.Theorem7 shows that the drift coefficient f makes a dominant role when α≥2β; in particular, g only needs to satisfy condition (A2) when α>2β, while in Theorem 9 the diffusion coefficient g is dominant. That is, we depend on the environmental noise to suppress the explosion of solutions and guarantee the boundedness. However, we obtained that the order of moment is lower than Theorem 7.


## 5. Examples

In this section, we investigate the moment boundedness of two 1-dimensional stochastic multipantograph equations.Example 1.
Consider(55)dxt=-4x5t+u1tx2tx30.25t+x4tx0.5tu2t+x2tdt+x2t+u1txtx0.25t+xtx0.5tu2t+xtdwtwith initial data x(0)=x0∈R, t≥0, where u1(t)=e-0.75ɛt, u2(t)=e-0.5ɛt, ɛ>0.
Definey1(t)=x(0.25t), y2(t)=x(0.5t), and (56)ft,x,y=-4x5+e-0.75ɛtx2y13+e-0.5ɛtx4y2+x2,gt,x,y=x2+e-0.75ɛtxy1+e-0.5ɛtxy2+x.Obviously, f and g satisfy the local Lipschitz condition. Then by the Young inequality we estimate (57)xft,x,y=-4x6+e-0.75ɛtx3y13+e-0.5ɛtx5y2+x3≤-4x6+e-0.75ɛtx6+y162+e-0.5ɛt5x6+y266+x3≤-83x6+12e-0.75ɛty16+16e-0.5ɛty26+x3 which shows that condition (A1) will hold if α=4,σ=8/3,σ¯1=1/2,σ¯2=1/6. Obviously, we have (58)gt,x,y=x2+e-0.75ɛtxy1+e-0.5ɛtxy2+x≤x2+12e-0.75ɛtx2+y12+12e-0.5ɛtx2+y22+x≤2x2+12e-0.75ɛty12+12e-0.5ɛty22+x,which shows that condition (A2) will hold if β=1, λ=2, λ¯1=1/2, and λ¯2=1/2.
Substituting the above parameters into (32) gives σ^=1/3>0. Noting that α>2β, applying Theorem 7, we have the following result.
For any initial datax0∈R, (55) admits a unique global solution x(t,x0) which has the properties (5) and (6) for any p,q∈R+.Example 2.
Consider(59)dxt=-x3t+u1tx2tx0.25t+u2txtx20.5t+x2tdt+20x3t+u1tx2tx0.25t+u2tx2tx0.5t+xtdwtwith initial data x(0)=x0∈R×,   t≥0, where u1(t),  u2(t),  y1, and  y2 are defined the same as in Example 1.
Define(60)ft,x,y=-x3+e-0.75ɛtx2y1+e-0.5ɛtxy22+x2,gt,x,y=20x3+e-0.75ɛtx2y1+e-0.5ɛtx2y2+x.Obviously, f and g satisfy the local Lipschitz condition. By the Young inequality, we compute(61)x-1ft,x,y=x-1-x3+e-0.75ɛtx2y1+e-0.5ɛtxy22+x2≤x2+e-0.75ɛtx2+y122+e-0.5ɛty22+x≤32x2+12e-0.75ɛty12+e-0.5ɛty22+x, which shows that condition (B1) will hold if α=2, γ=3/2, γ¯1=1/2, and γ¯2=1. Applying the Young inequality again, we have(62)x-1gt,x,y=x-120x3+e-0.75ɛtx2y1+e-0.5ɛtx2y2+x≤20x2+e-0.75ɛtx2+y122+e-0.5ɛtx2+y222+1≤21x2+12e-0.75ɛty12+12e-0.5ɛty22+1, which shows that condition (B2) will hold if β=2, λ=21, λ¯1=1/2, and λ¯2=1/2.
Consider the elementary inequality: for anyρ>1,  a,b∈R, (63)a+b2≥a2ρ-b2ρ-1, and recall inequality (37): for any ρ>1, 0<δ<1, we estimate (64)x-2gt,x,y2=20x2+e-0.75ɛtxy1+e-0.5ɛtxy2+12≥202x4ρ-e-0.75ɛtxy1+e-0.5ɛtxy2+12ρ-1≥202x4ρ-1ρ-1e-0.75ɛtxy1+e-0.5ɛtxy22δ+11-δ≥202x4ρ-2δρ-1e-0.75ɛtx2y12+e-0.5ɛtx2y22-1ρ-11-δ≥202x4ρ-1δρ-1e-0.75ɛtx4+y14+e-0.5ɛtx4+y24-1ρ-11-δ≥202ρ-2δρ-1x4-1δρ-1e-0.75ɛty14-1δρ-1e-0.5ɛty24-1ρ-11-δ. Therefore, condition (B3) is satisfied with β=2 and (65)σ=202ρ-2δρ-1,σ¯1=σ¯2=1δρ-1. Substituting the above parameters into (31) and (32) gives λ~=22, λ^=24, and (66)σ^=202ρ-8δρ-1. Let δ→1, maximizing σ^ by choosing ρ=20/(20-8)>1, yield (67)σ^=20-82. Noting that α<2β, then we have 2σ^=2(20-8)2>λ~λ^=528; that is, condition (47) is satisfied and P0=2-σ^-1λ~λ^=0.3.
Applying Theorem9, we have the following result.
For any initial datax0∈R×, (59) admits a unique global solution x(t,x0) which has properties (5) and (6) in which p∈(0,0.3) and q∈(0,4.3).


## 6. Further Remarks

This paper is devoted to the asymptotic properties of the stochastic multipantograph equations. We investigate the existence and uniqueness of the global solution and its moment boundedness. Besides obtaining a general theorem, we obtain two sufficient criteria which can be much more easily verifiable than the general theorem. Two examples demonstrate our results.Since (3) arises in the analysis of the dynamics of an overhead current collection system for an electric locomotive and applied to engineering and applied mathematics fields, the asymptotic behavior of stochastic multipantograph equation has meaningful interpretations (cf. [4, 26]). The main idea and the method developed in this paper have the potential to investigate some other unbounded delay stochastic systems, such as neural networks, infinite-delay Kolmogorov-type systems, and Volterra equations in mathematical biology.



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