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

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).

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 Section 2, 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,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a normal filtration {ℱ_𝑡}_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=sup_{𝑡∈[−𝜏,0]}‖𝜂(𝑡)‖_𝑋. Here let 𝐶^𝑏_ℱ0([−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,{ℱ_𝑡}_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 {ℱ_𝑡}_𝑡≥0. To define stochastic integrals with respect to the 𝑄-Wiener process 𝑤(𝑡), we introduce the subspace 𝑌₀=𝑄^1/2𝑌 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 {𝑒_𝑖} in 𝑌, a bounded sequence of nonnegative real numbers 𝜆_𝑖 such that 𝑄𝑒_𝑖=𝜆_𝑖𝑒_𝑖,𝑖=1,2,…, and a sequence {𝛽_𝑖}_𝑖≥1 of independent Brownian motions such that ⟨𝑤(𝑡),𝑒⟩=^∞_𝑖=1𝜆_𝑖𝑒_𝑖,𝑒𝛽_𝑖(𝑡),𝑒∈𝑌, (1) and ℱ_𝑡=ℱ^𝑤_𝑡, where ℱ^𝑤_𝑡 is the 𝜎-algebra generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}. Let 𝐿⁰₂=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‖²_𝐿02=tr((𝑢𝑄^1/2)(𝑢𝑄^1/2)^∗) for any 𝑢∈𝐿⁰₂. Obviously, for any bounded operators 𝑢∈𝐿⁰₂ this norm reduces to ‖𝑢‖²_𝐿02=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: 𝑑𝑥(𝑡)−𝑢𝑡,𝑥_𝑡=𝐴𝑥(𝑡)+𝑓𝑡,𝑥_𝑡𝑑𝑡+𝑔𝑡,𝑥_𝑡(𝑡)𝑥𝑑𝑤,𝑡≥0,₀(⋅)=𝜑∈𝐶^𝑏_ℱ0([−𝜏,0],𝑋), (2) where 𝑥_𝑡={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 ℱ_𝑡,𝑡≥0 with ∫^𝑇₀‖𝑥(𝑡)‖²_𝑋𝑑𝑡<∞ 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, 𝑥(𝑡)=𝑆(𝑡)𝜑(0)−𝑢0,𝜑+𝑢𝑡,𝑥_𝑡+^𝑡₀𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑠𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑤(𝑠) (3)for any 𝑥₀(⋅)=𝜑∈𝐶^𝑏_ℱ0([−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 function 𝐻(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: 𝑓(𝑡,𝜉)²_𝑋+𝑔(𝑡,𝜉)²_𝐿02≤𝐻𝑡,𝜉²_𝐶,𝑡∈[0,𝑇]. (4) (b) For any constant 𝐾>0, the differential dynamic system 𝑑𝑢()[],𝑑𝑡=𝐾𝐻𝑡,𝑢,𝑡∈0,𝑇 (5) has a global solution for any initial value 𝑢₀.(H3) (a) There exists a function 𝐺(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: 𝑓(𝑡,𝜉)−𝑓𝑡,𝜂²_𝑋+𝑔(𝑡,𝜉)−𝑔𝑡,𝜂²_𝐿02≤𝐺𝑡,𝜉−𝜂²_𝐶,𝑡∈[0,𝑇]. (6) (b) For any constant 𝐾>0, if a nonnegative function 𝑧(𝑡) satisfies that 𝑧(𝑡)≤𝐾^𝑡₀𝐺(𝑠,𝑧(𝑠))𝑑𝑠,𝑡∈[0,𝑇], (7) then 𝑧(𝑡)=0 holds for any 𝑡∈[0,𝑇].(H4) There exist a number 𝛼∈[0,1] and a positive 𝐾₀ such that, for any 𝜉,𝜂∈𝑋 and 𝑡≥0,𝑢(𝑡,𝑥)∈𝒟((−𝐴)^𝛼) and (−𝐴)^𝛼𝑢(𝑡,𝜉)−(−𝐴)^𝛼𝑢𝑡,𝜂_𝑋≤𝐾₀𝜉−𝜂_𝐶. (8) Moreover, we assume that 𝑢(𝑡,0)=0.

Remark 2. Let 𝐺(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 function 𝐺(⋅). Let 𝜁>0 and let 𝛿∈(0,1) be sufficiently small. Define 𝐺₁(𝑢)=𝜁𝑢,𝑢≥0,𝐺₂(𝑢)=𝑢log𝑢⁻¹𝛿,0≤𝑢≤𝛿,𝛿log⁻¹+𝐺^′₂(𝛿−)(𝑢−𝛿),𝑢>𝛿, (9) where 𝐺^′₂ denotes the derivative of function 𝐺₂. 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 and 0∈𝜚(𝐴), then, for any 𝛽∈(0,1]∶(i)for each 𝑥∈𝒟((−𝐴)^𝛽), 𝑆(𝑡)(−𝐴)^𝛽𝑥=(−𝐴)^𝛽𝑆(𝑡)𝑥; (10)(ii)there exist positive constants 𝑀_𝛽>0 and 𝑎∈𝑅₊ such that (−𝐴)^𝛽𝑆(𝑡)≤𝑀_𝛽𝑡^−𝛽𝑒^−𝑎𝑡,𝑡>0. (11)

3. Existence and Uniqueness of the Mild Solution

In this section, we will establish the existence and uniqueness of the mild solution. When the coefficients 𝑓 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 𝛾∶=4𝐾²₀𝑀²_1−𝛼𝑎^−2𝛼Γ(2𝛼−1)1−𝐾₀(−𝐴)^−𝛼+𝐾₀(−𝐴)^−𝛼<1, (12) where 𝑀_1−𝛼 is defined in Lemma 3.

Remark 5. If 𝐺(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 integer 𝑁>0, there exists a function 𝐺_𝑁(𝑡,𝑟)∶𝑅₊×𝑅₊→𝑅₊ such that 𝐺_𝑁(𝑡,𝑟) is locally integrable in 𝑡≥0 for any fixed 𝑟≥0 and is continuous monotone nondecreasing and concave in 𝑟 with 𝐺_𝑁(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: 𝑓(𝑡,𝜉)−𝑓𝑡,𝜂²_𝑋+𝑔(𝑡,𝜉)−𝑔𝑡,𝜂²_𝐿02≤𝐺_𝑁𝑡,𝜉−𝜂²_𝐶,𝑡∈[0,𝑇]. (13)(b) For any constant 𝐾>0, if a nonnegative function 𝑧(𝑡) satisfies that 𝑧(𝑡)≤𝐾^𝑡₀𝐺_𝑁(𝑠,𝑧(𝑠))𝑑𝑠,𝑡∈[0,𝑇], (14)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 3𝐾²₀𝑀²_1−𝛼𝑎^−2𝛼Γ(2𝛼−1)1−𝐾₀(−𝐴)^−𝛼+𝐾₀(−𝐴)^−𝛼<1, (15) where 𝑀_1−𝛼 is defined in Lemma 3.

Proof. For any 𝑁≥1, define the truncation functions 𝑓_𝑁 and 𝑔_𝑁 as follows: 𝑓_𝑁𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡,𝑥_𝑡𝐶𝑓≤𝑁,𝑡,𝑁𝑥_𝑡𝑥_𝑡𝐶,𝑥_𝑡𝐶𝑔>𝑁,_𝑁𝑡,𝑥_𝑡=𝑔𝑡,𝑥_𝑡,𝑥_𝑡𝐶𝑔≤𝑁,𝑡,𝑁𝑥_𝑡𝑥_𝑡𝐶,𝑥_𝑡𝐶≤𝑁, (16) and then the functions 𝑓_𝑁 and 𝑔_𝑁 satisfy (H2) and 𝑓_𝑁(𝑡,𝜉)−𝑓_𝑁𝑡,𝜂²_𝑋+𝑔_𝑁(𝑡,𝜉)−𝑔_𝑁𝑡,𝜂²_𝐿02≤𝐺_𝑁𝑡,𝜉−𝜂²_𝐶,𝜉,𝜂∈𝑋,𝑡∈[0,𝑇]. (17)
By Theorem 4, there exist the unique mild solutions 𝑥_𝑁(𝑡) and 𝑥_𝑁+1(𝑡), respectively, to the following stochastic systems: 𝑥_𝑁(𝑡)−𝑢𝑡,𝑥_𝑁𝑡(𝑡𝜑(0)+=𝑆)−𝑢0,𝜑_𝑡0𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑁𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓_𝑁𝑠,𝑥_𝑁𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔_𝑁𝑠,𝑥_𝑁𝑠(𝑠),𝑥𝑑𝑤_𝑁+1(𝑡)−𝑢𝑡,𝑥_𝑁+1𝑡(𝑡𝜑(0)+=𝑆)−𝑢0,𝜑_𝑡0𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑁+1𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓_𝑁+1𝑠,𝑥_𝑁+1𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)×𝑔_𝑁+1𝑠,𝑥_𝑁+1𝑠𝑑𝑤(𝑠). (18)
Define the stopping time 𝜏_𝑁=𝑇∧inf{𝑡∈[0,𝑇]∶‖(𝑥_𝑁)_𝑡‖_𝐶≥𝑁}. Recall that, for 𝑎,𝑏∈𝑋, 𝜀∈(0,1), ‖𝑎−𝑏‖²_𝑋≤1/(1−𝜀)‖𝑎‖²_𝑋+1/𝜀‖𝑏‖²_𝑋. Hence, for some 𝜀, 𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋≤1𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑢𝑠,𝑥_𝑁+1𝑠−𝑢𝑠,𝑥_𝑁𝑠²_𝑋+31−𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}^𝑠₀()×𝑢𝑥𝐴𝑆𝑠−𝑟𝑟,_𝑁+1𝑟−𝑢𝑟,𝑥_𝑁𝑟𝑑𝑟²_𝑋+31−𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}^𝑠₀𝑆()×𝑓𝑠−𝑟_𝑁+1𝑟,𝑥_𝑁+1𝑟−𝑓_𝑁𝑟,𝑥_𝑁𝑟𝑑𝑟²_𝑋+31−𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}^𝑠₀𝑆()×𝑔𝑠−𝑟_𝑁+1𝑟,𝑥_𝑁+1𝑟−𝑔_𝑁𝑟,𝑥_𝑁𝑟𝑑𝑤(𝑟)²_𝑋=∶⁴_𝑖=1𝐽_𝑖. (19)
From (H4), we have 𝐽₁≤1𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑢𝑠,𝑥_𝑁+1𝑠−𝑢𝑠,𝑥_𝑁𝑠²_𝑋≤(−𝐴)^−𝛼2𝐾²₀1𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1𝑠−𝑥_𝑁𝑠²_𝐶≤(−𝐴)^−𝛼2𝐾²₀1𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋. (20)
Applying the Hölder inequality, (H4), and Lemma 3, we have 𝐽₂≤31−𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}^𝑠₀(−𝐴)^1−𝛼𝑆(𝑠−𝑟)(−𝐴)^𝛼×𝑢𝑠,𝑥_𝑁+1𝑠−𝑢𝑠,𝑥_𝑁𝑠𝑋𝑑𝑟²≤31−𝜀𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}^𝑠₀𝑀_1−𝛼𝑒^{−𝑎(𝑠−𝑟)}×(𝑠−𝑟)^𝛼−1(−𝐴)^𝛼_𝑋𝑑𝑟²≤3𝐾1−𝜀²⁰𝑀²_1−𝛼𝑎^−2𝛼Γ(2𝛼−1)×𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋. (21)
For 0≤𝑡≤𝜏_𝑁, we see that 𝑓_𝑁+1(𝑠,(𝑥_𝑁)_𝑠)=𝑓_𝑁(𝑠,(𝑥_𝑁)_𝑠)=𝑓(𝑠,(𝑥_𝑁)_𝑠) and 𝑔_𝑁+1(𝑠,(𝑥_𝑁)_𝑠)=𝑔_𝑁(𝑠,(𝑥_𝑁)_𝑠)=𝑔(𝑠,(𝑥_𝑁)_𝑠). Thus, by (H^′3) and the Jensen inequality, we obtain 𝐽₃≤3𝑇1−𝜀^𝑡₀𝐺_𝑁+1𝑟,𝐸sup_{0≤𝑢≤𝑟}𝑥_𝑁+1(𝑢)−𝑥_𝑁(𝑢)²_𝑋𝑑𝑟. (22)
By (H^′3), Liu [1, Theorem 1.2.6, page 14] and the Jensen inequality, there exists a positive constant 𝐶₅ such that 𝐽₄≤3𝐶1−𝜀⁵×^𝑡₀𝐺_𝑁+1𝑟,𝐸sup_{0≤𝑢≤𝑟}𝑥_𝑁+1(𝑢)−𝑥_𝑁(𝑢)²𝑑𝑟. (23)
From (19)–(23), choosing 𝜀=𝐾₀‖(−𝐴)^−𝛼‖ we have 𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋≤(−𝐴)^−𝛼𝐾₀+31−𝐾₀(−𝐴)^−𝛼𝐾²₀𝑀²_1−𝛼𝑎^−2𝛼Γ(2𝛼−1)×𝐸sup_{0≤𝑠≤𝑡∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋+31−𝐾₀(−𝐴)^−𝛼𝑇+3𝐶₅1−𝐾₀(−𝐴)^−𝛼×^𝑡₀𝐺_𝑁+1𝑟,𝐸sup_{0≤𝑢≤𝑟}𝑥_𝑁+1(𝑢)−𝑥_𝑁2(𝑢)²𝑑𝑟. (24)
By (22) and the Gronwall inequality, there exists a constant 𝐾 such that 𝐸sup_{0≤𝑠≤𝑡}𝑥_𝑁+1𝑠∧𝜏_𝑁−𝑥_𝑁𝑠∧𝜏_𝑁²_𝑋≤𝐾^𝑡₀𝐺_𝑁+1𝑟∧𝜏_𝑁,𝐸sup_{0≤𝑢≤𝑟}𝑥_𝑁+1𝑢∧𝜏_𝑁−𝑥_𝑁𝑢∧𝜏_𝑁²𝑑𝑟. (25)
By (H^′3-b), we have 𝐸sup_{0≤𝑠≤𝑇∧𝜏𝑁}𝑥_𝑁+1(𝑠)−𝑥_𝑁(𝑠)²_𝑋=0,0≤𝑡≤𝜏_𝑁, (26)which means thats for 0≤𝑡≤𝜏_𝑁, we obtain 𝑥_𝑁+1(𝑠)=𝑥_𝑁(𝑠)a.e. (27)
For each 𝜔∈Ω, there exists an 𝑁₀=𝑁₀(𝜔)>0 such that 0<𝑇≤𝜏_𝑁0. Define 𝑥(𝑡) by 𝑥(𝑡)=𝑥_𝑁0(𝑡),𝑡∈[0,𝑇]. Since 𝑥(𝑡∧𝜏_𝑁)=𝑥_𝑁(𝑡∧𝜏_𝑁), then we have 𝑥𝑡∧𝜏_𝑁−𝑢𝑡,𝑥_{𝑡∧𝜏𝑁}(𝑡𝜑(0)+=𝑆)−𝑢0,𝜑_{𝑡∧𝜏𝑁0}𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑠+𝑑𝑠_{𝑡∧𝜏𝑁0}𝑆(𝑡−𝑠)𝑓_𝑁𝑠,𝑥_𝑠+𝑑𝑠_{𝑡∧𝜏𝑁0}𝑆(𝑡−𝑠)𝑔_𝑁𝑠,𝑥_𝑠(𝑠)(𝑡𝜑(0)+𝑑𝑤=𝑆)−𝑢0,𝜑_{𝑡∧𝜏𝑁0}𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑠+𝑑𝑠_{𝑡∧𝜏𝑁0}𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+^{𝑡∧𝜏𝑁}₀𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑤(𝑠). (28)
Letting 𝑁→∞, 𝑥(𝑡)−𝑢𝑡,𝑥_𝑡(𝑡𝜑(0)+=𝑆)−𝑢0,𝜑_𝑡0𝐴𝑆(𝑡−𝑠)𝑢𝑠,𝑥_𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝑔𝑠,𝑥_𝑠𝑑𝑤(𝑠). (29)
Hence we have that 𝑥(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].

Acknowledgments

The work is supported by the Fundamental Research Funds for the Central Universities under Grant 2722013JC080, China Postdoctoral Science Foundation funded project under Grant 2012M511615, the Natural Science Foundation of Hubei Province of China and the National Science Foundation of China.