# Mild Solutions of Neutral Semilinear Stochastic Functional Dynamic Systems with Local Non-Lipschitz Coefficients

Authors: Feng Jiang
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics and engineering. In this paper, we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional dynamic systems under local non-Lipschitz conditions on the coefficients by means of the stopping time technique. We especially generalize and improve the results that appeared in Govinadan (2005), Bao and Hou (2010), and Jiang and Shen (2011).


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## Body

## 1. Introduction

Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics, biology engineering, and so forth [1]. Recently, for the case where the coefficients satisfy the global Lipschitz condition and the linear growth condition, many results are known [1–3]. However, the global Lipschitz condition, even the local Lipschitz condition, is seemed to be considerably strong when one discusses variable applications in real world. Reference [4] discussed the existence of mild solution to neutral semilinear stochastic functional systems with non-Lipschitz coefficients. Reference [5] discussed the existence and uniqueness of solutions to neutral stochastic functional systems with infinite delay under the local Lipschitz conditions in the Euclidean space.We focus on neutral semilinear stochastic functional dynamic systems for the case where the coefficients do not necessarily satisfy the global Lipschitz condition. Thus we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional systems with the condition, which was investigated by [6–8] as the Carathéodory-type conditions for strong solutions. Motivated by the above papers, in this paper, we will extend the existence and uniqueness of mild solutions of (2) under the non-Lipschitz conditions to the local non-Lipschitz conditions.The rest of this paper is organized as follows. In Section2, we introduce some preliminaries. In Section 3 we prove the existence and uniqueness of the mild solution.


## 2. Preliminaries

Throughout this paper, let(Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a normal filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all P-null sets). Moreover, let X,Y be two real separable Hilbert spaces, and we denote by 〈·,·〉X,〈·,·〉Y their inner products and by ∥·∥X,∥·∥Y their vector norms, respectively. We denote that L(Y,X) denotes the space of all bounded linear operators from Y into X, equipped with the usual operator norm ∥·∥. In this paper, we always use the same symbol ∥·∥ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let τ>0 and C=C([-τ,0];X) denote the family of all continuous X-valued functions η from [-τ,0] to X with norm ∥η∥C=supt∈[-τ,0]∥η(t)∥X. Here let Cℱ0b([-τ,0],X) be the family of all almost surely bounded, ℱ0-measurable, continuous random variables from [-τ,0] to X.Let{w(t):t≥0} denote a Y-valued Wiener process defined on the probability space (Ω,ℱ,{ℱt}t≥0,P) with covariance operator Q; that is, E〈w(t),x〉YE〈w(s),y〉X=(t∧s)〈Qx,y〉Y, for all x,y∈Y, where Q is a positive, self-adjoint, trace class operator on Y. In particular, we denote w(t) a Y-valued Q-Wiener process with respect to {ℱt}t≥0. To define stochastic integrals with respect to the Q-Wiener process w(t), we introduce the subspace Y0=Q1/2Y of Y which is endowed with the inner product 〈u,v〉Y0=〈Q-1/2u,Q-1/2v〉Y and is a Hilbert space. We assume that there exists a complete orthonormal system {ei} in Y, a bounded sequence of nonnegative real numbers λi such that Qei=λiei,i=1,2,…, and a sequence {βi}i≥1 of independent Brownian motions such that
(1)〈w(t),e〉=∑i=1∞‍λi〈ei,e〉βi(t),e∈Y,
and ℱt=ℱtw, where ℱtw is the σ-algebra generated by {w(s):0≤s≤t}. Let L20=L2(Y0,X) be the space of all Hilbert-Schmidt operators from Y0 to X. It turns out to be a separable Hilbert space equipped with the norm ∥u∥L202=tr((uQ1/2)(uQ1/2)*) for any u∈L20. Obviously, for any bounded operators u∈L20 this norm reduces to ∥u∥L202=tr(uQu*).Suppose that{S(t),t≥0} is an analytic semigroup with its infinitesimal generator A; for the literature related to semigroup theory, we suggest Pazy [9]. We suppose 0∈ϱ(A), the resolvent set of -A. For any α∈[0,1], it is possible to define the fractional power (-A)α which is a closed linear operator with its domain 𝒟((-A)α).Consider the following neutral semilinear stochastic functional systems:(2)d[x(t)-u(t,xt)]=[Ax(t)+f(t,xt)]dt+g(t,xt)dw(t),t≥0,x0(·)=φ∈Cℱ0b([-τ,0],X),
where xt={x(t+θ):-τ≤θ≤0} can be regarded as a C([-τ,0];X))-valued stochastic process; u:R+×C([-τ,0],X)→X,f:R+×C([-τ,0],X)→X,g:R+×C([-τ,0],X)→L(Y,X) are all Borel measurable; A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t),t≥0, in X.Definition 1.
A process{x(t),t∈[0,T]},0≤T<∞, is called a mild solution of (2) if(i)
x(t) is adapted to ℱt,t≥0 with ∫0T‍∥x(t)∥X2dt<∞ a.s.;(ii)
x(t)∈X has càdlàg paths on t∈[0,T] a.s.: and, for each t∈[0,T],x(t) satisfies the integral equation,
(3)x(t)=S(t)[φ(0)-u(0,φ)]+u(t,xt)+∫0t‍AS(t-s)u(s,xs)ds+∫0t‍S(t-s)f(s,xs)ds+∫0t‍S(t-s)g(s,xs)dw(s)for any x0(·)=φ∈Cℱ0b([-τ,0],X).To guarantee the existence and uniqueness of a mild solution to (2), the following much weaker conditions, instead of non-Lipschitz condition, are described.(H1)
A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t),t≥0, in X, and S(t) is uniformly bounded, for some constant 0<a∈R+,∥S(t)∥≤e-at,t≥0.(H2)
(a) There exists a functionH(t,r):R+×R+→R+ such that H(t,r) is locally integrable in t≥0 for any fixed r≥0 and is continuous monotone nondecreasing and concave in r for any fixed t∈[0,T]. Moreover, for any fixed t∈[0,T] and ξ∈X, the following inequality is satisfied:
(4)∥f(t,ξ)∥X2+∥g(t,ξ)∥L202≤H(t,∥ξ∥C2),t∈[0,T].
(b) For any constant K>0, the differential dynamic system
(5)dudt=KH(t,u),t∈[0,T],
has a global solution for any initial value u0.(H3)
(a) There exists a functionG(t,r):R+×R+→R+ such that G(t,r) is locally integrable in t≥0 for any fixed r≥0 and is continuous monotone nondecreasing and concave in r for any fixed t∈[0,T]. G(t,0)=0 for any fixed t∈[0,T]. Moreover, for any fixed t∈[0,T] and ξ,η∈X, the following inequality is satisfied:
(6)∥f(t,ξ)-f(t,η)∥X2+∥g(t,ξ)-g(t,η)∥L202≤G(t,∥ξ-η∥C2),t∈[0,T].
(b) For any constant K¯>0, if a nonnegative function z(t) satisfies that
(7)z(t)≤K¯∫0t‍G(s,z(s))ds,t∈[0,T],
then z(t)=0 holds for any t∈[0,T].(H4)
There exist a numberα∈[0,1] and a positive K0 such that, for any ξ,η∈X and t≥0,u(t,x)∈𝒟((-A)α) and
(8)∥(-A)αu(t,ξ)-(-A)αu(t,η)∥X≤K0∥ξ-η∥C.
Moreover, we assume that u(t,0)=0.Remark 2.
LetG(t,u)=L(t)G¯(u),t∈[0,T], where L(t)≥0 is locally integrable and G¯(u) is a concave nondecreasing function from R+ to R+ such that G¯(0)=0,G¯(u)>0 for u>0, and ∫0+‍(1/G-(u))du=∞. Then by comparison theorem of differential dynamic systems we know that assumption (H3-b) holds.Now let us give some concrete examples of the functionG¯(·). Let ζ>0 and let δ∈(0,1) be sufficiently small. Define
(9)G¯1(u)=ζu,u≥0,G¯2(u)={ulog(u-1),0≤u≤δ,δlog(δ-1)+G-2′(δ-)(u-δ),u>δ,
where G-2′ denotes the derivative of function G¯2. They are all concave nondecreasing functions satisfying ∫0+‍(1/G-i(u))du=∞(i=1,2). In particular, we see that Lipschitz condition in [3] and non-Lipschitz conditions in [4] are special cases of our proposed condition.To show our main results, we need the following lemma.Lemma 3 (see [9]).
If (H1) holds and0∈ϱ(A), then, for any β∈(0,1]:(i)
for eachx∈𝒟((-A)β),
(10)S(t)(-A)βx=(-A)βS(t)x;(ii)
there exist positive constantsMβ>0 and a∈R+ such that
(11)∥(-A)βS(t)∥≤Mβt-βe-at,t>0.


## 3. Existence and Uniqueness of the Mild Solution

In this section, we will establish the existence and uniqueness of the mild solution. When the coefficientsf and g satisfy the global Lipschitz condition and the linear growth condition, [7, 8] discussed the existence and uniqueness of the mild solution to stochastic differential dynamic systems. In [10], the existence and uniqueness of the mild solution of (2) under the non-Lipschitz condition are given as follows.Theorem 4.
If(H1)–(H4) hold for some α∈(1/2,1], then there exists a unique mild solution to (2), provided that
(12)γ:=4K02M1-α2a-2αΓ(2α-1)1-K0∥(-A)-α∥+K0∥(-A)-α∥<1,
where M1-α is defined in Lemma 3.Remark 5.
IfG(t,u)=K1u for some constant K1, then the condition (H3) implies a global Lipschitz condition, which is studied in [3]. In Remark 2, if L(t)=1, it is studied in [4]. Therefore, some of the results [3, 4] are improved and generalized.Now, we will replace (H3) by the following local non-Lipschitz condition.(H′3)
(a) For any integerN>0, there exists a function GN(t,r):R+×R+→R+ such that GN(t,r) is locally integrable in t≥0 for any fixed r≥0 and is continuous monotone nondecreasing and concave in r with GN(t,0)=0. Moreover, for any fixed t∈[0,T] and ξ,η∈X with ∥ξ∥C,∥η∥C≤N, the following inequality is satisfied:
(13)∥f(t,ξ)-f(t,η)∥X2+∥g(t,ξ)-g(t,η)∥L202≤GN(t,∥ξ-η∥C2),t∈[0,T].(b)
For any constantK¯>0, if a nonnegative function z(t) satisfies that
(14)z(t)≤K¯∫0t‍GN(s,z(s))ds,t∈[0,T],then z(t)=0 holds for any t∈[0,T].Remark 6.
Equation (13) is a generalization of the local Lipschitz condition. (H′3) is the local non-Lipschitz condition of this type for the end of wider applications.Theorem 7.
If(H1), (H2),(H′3),and (H4) hold for some α∈(1/2,1], then there exists a unique mild solution to (2), provided that
(15)3K02M1-α2a-2αΓ(2α-1)1-K0∥(-A)-α∥+K0∥(-A)-α∥<1,
where M1-α is defined in Lemma 3.Proof.
For anyN≥1, define the truncation functions fN and gN as follows:
(16)fN(t,xt)={f(t,xt),∥xt∥C≤N,f(t,Nxt∥xt∥C),∥xt∥C>N,gN(t,xt)={g(t,xt),∥xt∥C≤N,g(t,Nxt∥xt∥C),∥xt∥C≤N,
and then the functions fN and gN satisfy (H2) and
(17)∥fN(t,ξ)-fN(t,η)∥X2+∥gN(t,ξ)-gN(t,η)∥L202≤GN(t,∥ξ-η∥C2),ξ,η∈X,t∈[0,T].
By Theorem4, there exist the unique mild solutions xN(t) and xN+1(t), respectively, to the following stochastic systems:
(18)xN(t)-u(t,(xN)t)=S(t)[φ(0)-u(0,φ)]+∫0t‍AS(t-s)u(s,(xN)s)ds+∫0t‍S(t-s)fN(s,(xN)s)ds+∫0t‍S(t-s)gN(s,(xN)s)dw(s),xN+1(t)-u(t,(xN+1)t)=S(t)[φ(0)-u(0,φ)]+∫0t‍AS(t-s)u(s,(xN+1)s)ds+∫0t‍S(t-s)fN+1(s,(xN+1)s)ds+∫0t‍S(t-s)×gN+1(s,(xN+1)s)dw(s).
Define the stopping timeτN=T∧inf{t∈[0,T]:∥(xN)t∥C≥N}. Recall that, for a,b∈X, ε∈(0,1), ∥a-b∥X2≤1/(1-ε)∥a∥X2+1/ε∥b∥X2. Hence, for some ε,
(19)Esup0≤s≤t∧τN∥xN+1(s)-xN(s)∥X2≤1εEsup0≤s≤t∧τN∥u(s,(xN+1)s)-u(s,(xN)s)∥X2+31-εEsup0≤s≤t∧τN∥∫0s‍AS(s-r)×(u(r,(xN+1)r))-u(r,(xN)r)dr∫0s‍∥X2+31-εEsup0≤s≤t∧τN∥∫0s‍S(s-r)×((fN+1(r,(xN+1)r)-fN(r,(xN)r))dr∫0s‍∥X2+31-εEsup0≤s≤t∧τN∥∫0s‍S(s-r)×(gN+1(r,(xN+1)r)-gN(r,(xN)r))dw(r)∫0s‍∥X2=:∑i=14‍Ji.
From (H4), we have(20)J1≤1εEsup0≤s≤t∧τN∥u(s,(xN+1)s)-u(s,(xN)s)∥X2≤∥(-A)-α∥2K021εEsup0≤s≤t∧τN∥(xN+1)s-(xN)s∥C2≤∥(-A)-α∥2K021εEsup0≤s≤t∧τN∥(xN+1)(s)-(xN)(s)∥X2.
Applying the Hölder inequality, (H4), and Lemma3, we have
(21)J2≤31-εEsup0≤s≤t∧τN(∫0s∥(-A)1-αS(s-r)(-A)α‍×u(s,(xN+1)s)-u(s,(xN)s)(-A)1-α∥Xdr∫0s‍)2≤31-εEsup0≤s≤t∧τN(∫0s‍M1-αe-a(s-r)×(s-r)α-1∥(-A)α∥Xdr∫0s‍)2≤31-εK02M1-α2a-2αΓ(2α-1)×E(sup0≤s≤t∧τN∥(xN+1)(s)-(xN)(s)∥X2).
For0≤t≤τN, we see that fN+1(s,(xN)s)=fN(s,(xN)s)=f(s,(xN)s) and gN+1(s,(xN)s)=gN(s,(xN)s)=g(s,(xN)s). Thus, by (H′3) and the Jensen inequality, we obtain
(22)J3≤31-εT∫0tGN+1‍(r,E(sup0≤u≤r∥(xN+1)(u)-(xN)(u)∥X2sup0≤u≤r))dr.
By (H′3), Liu [1, Theorem 1.2.6, page 14] and the Jensen inequality, there exists a positive constant C5 such that
(23)J4≤31-εC5×∫0tGN+1(r,E(sup0≤u≤r∥(xN+1)(u)-(xN)(u)∥2))dr.
From (19)–(23), choosing ε=K0∥(-A)-α∥ we have
(24)Esup0≤s≤t∧τN∥xN+1(s)-xN(s)∥X2≤[31-K0∥(-A)-α∥K02M1-α2a-2αΓ(2α-1)∥(-A)-α∥K0+31-K0∥(-A)-α∥K02M1-α2a-2αΓ(2α-1)]×E(sup0≤s≤t∧τN∥(xN+1)(s)-(xN)(s)∥X2)+[31-K0∥(-A)-α∥T+3C51-K0∥(-A)-α∥]×∫0t‍GN+1(r,E(sup0≤u≤r∥(xN+1)(u)-(xN)2(u)∥2sup0≤u≤r))dr.
By (22) and the Gronwall inequality, there exists a constant K¯ such that
(25)Esup0≤s≤t∥xN+1(s∧τN)-xN(s∧τN)∥X2≤K¯∫0t‍GN+1(r∧τN,E(sup0≤u≤r∥(xN+1)(u∧τN)-(xN)(u∧τN)∥2sup0≤u≤r∥(xN+1)(u∧τN)))dr.
By (H′3-b), we have
(26)E(sup0≤s≤T∧τN∥xN+1(s)-xN(s)∥X2)=0,0≤t≤τN,which means thats for 0≤t≤τN, we obtain
(27)xN+1(s)=xN(s)a.e.
For eachω∈Ω, there exists an N0=N0(ω)>0 such that 0<T≤τN0. Define x(t) by x(t)=xN0(t),t∈[0,T]. Since x(t∧τN)=xN(t∧τN), then we have
(28)x(t∧τN)-u(t,xt∧τN)=S(t)[φ(0)-u(0,φ)]+∫0t∧τN‍AS(t-s)u(s,xs)ds+∫0t∧τN‍S(t-s)fN(s,xs)ds+∫0t∧τN‍S(t-s)gN(s,xs)dw(s)=S(t)[φ(0)-u(0,φ)]+∫0t∧τN‍AS(t-s)u(s,xs)ds+∫0t∧τN‍S(t-s)f(s,xs)ds+∫0t∧τN‍S(t-s)g(s,xs)dw(s).
LettingN→∞,
(29)x(t)-u(t,xt)=S(t)[φ(0)-u(0,φ)]+∫0t‍AS(t-s)u(s,xs)ds+∫0t‍S(t-s)f(s,xs)ds+∫0t‍S(t-s)g(s,xs)dw(s).
Hence we have thatx(t) is a mild solution to (2). The proof is complete.


## 4. Concluding Remarks

In the paper, by means of the stopping time technique, the conditions to assure the existence and uniqueness of mild solutions of neutral semilinear stochastic functional dynamic systems are given under the local non-Lipschitz condition, which is a generalization of the local Lipschitz condition. The paper generalizes the results in [8, 10].



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