Existence and Uniqueness of Mild Solutions to Some Neutral Stochastic Partial Functional Integrodifferential Equations with Non-Lipschitz Coefficients

Abstract

This paper presents the result on existence, uniqueness of mild solutions to neutral stochastic partial functional integrodifferential equations under the Carathéodory-type conditions on the coefficients. The results are obtained by using the method of successive approximation. An example is provided to illustrate the results of this work.

1. Introduction

In this paper, our objective is to study the existence of mild solution for the following neutral stochastic partial functional integrodifferential equation in a real separable Hilbert space: 𝑑𝑢(𝑡)−𝐻𝑡,𝑢_𝑡=𝐴𝑢(𝑡)−𝐻𝑡,𝑢_𝑡𝑑𝑡+^𝑡₀𝐵(𝑡−𝑠)𝑢(𝑠)−𝐻𝑠,𝑢_𝑠𝑑𝑠+𝐹𝑡,𝑢_𝑡𝑑𝑡+𝐺𝑡,𝑢_𝑡(𝑡)[],𝑢𝑑𝑤,for𝑡∈0,𝑇₀(⋅)=𝜑∈𝐶^𝑏_ℱ0([−𝑟,0];ℍ),𝑟>0, (1.1) where 𝑢_𝑡(𝜃)=𝑢(𝑡+𝜃)for𝜃∈[−𝑟,0]. The mappings 𝐻∶ℝ₊×𝐶_𝑟→ℍ, 𝐹∶ℝ₊×𝐶_𝑟→ℍ, and 𝐺∶ℝ₊×𝐶_𝑟→ℒ(𝕂,ℍ) are Borel measurable.

Neutral stochastic partial functional differential equations have atracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on.

Qualitative properties such as existence, uniqueness, and stability for various stochastic differential and integrodifferential systems have been extensively studied by many researchers; see, for instance, [1–6] and the references therein. The problem of the existence and uniqueness of solution for neutral stochastic partial functional differential equations in the case where the coefficients do not satisfy the global Lipschitz condition was investigated by Taniguchi [7], Turo [8], Cao et al. [9], and recently by Jiang and Shen [10].

Stimulated by the above works, we consider the existence and uniqueness of mild solutions to (1.1) under some carathéodory-type conditions to the Hilbert space with the Lipschitz condition in [11] and the non-Lipschitz condition in [12] being regarded as special cases.

Our main results concerning (1.1) rely essentially on techniques using strongly continuous family of operators{𝑅(𝑡),𝑡≥0}, defined on the Hilbert spaceℍand called the resolvent (precise definition will be given below).

The contents of the paper are as follows. In Section 2, we summarize several important working tools on the Wiener process and deterministic integrodifferential equations that will be used to develop our results. In Section 3, we study the existence of mild solutions for the neutral system (1.1) using the theory of resolvent operators and by means of successive approximation (the Picard iteration). In Section 4, we provide an example to illustrate our main approach.

2. Wiener Process and Deterministic Integrodifferential Equations

2.1. Wiener Process

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 ℍ and 𝕂be two real separable Hilbert spaces; we denote by⟨⋅,⋅⟩_ℍ, ⟨⋅,⋅⟩_𝕂 their inner products and by ‖⋅‖_ℍ, ‖⋅‖_𝕂 their vectors norms, respectively. We denote by ℒ(ℍ,𝕂) the space of all bounded linear operator from ℍ into 𝕂, 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 𝑟>0 and 𝐶_𝑟=𝐶([−𝑟,0];ℍ) denote the family of all continuous ℍ-valued functions 𝜉 from [−𝑟,0] to ℍ with norm ‖𝜉‖_𝐶=sup_{𝑡∈[−𝑟,0]}‖𝜉(𝑡)‖_ℍ. Let 𝐶^𝑏_ℱ0([−𝑟,0];ℍ) be the family of all almost surely bounded, ℱ₀-measurable,𝐶([−𝑟,0];ℍ)-valued random variables.

Let {𝑤(𝑡)∶𝑡≥0} denote a 𝕂-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⟩_𝕂⟨𝑤(𝑠),𝑦⟩_𝕂=(𝑡∧𝑠)⟨𝑄𝑥,𝑦⟩_𝕂, for all 𝑥,𝑦∈𝕂, where 𝑄 is a positive, self-adjoint, trace class operator on 𝕂. In particular, we denote 𝑤(𝑡) a 𝕂-valued𝑄-Wiener process with respect to {ℱ_𝑡}_𝑡≥0. To define stochastic integrals with respect to the 𝑄-Wiener process𝑤(𝑡), we introduce the subspace 𝕂₀=𝑄^1/2𝕂of𝕂 endowed with the inner product ⟨𝑢,𝑣⟩_𝕂0=⟨𝑄^−1/2𝑢,𝑄^−1/2𝑣⟩_𝕂 as a Hilbert space. We assume that there exists a complete orthonormal system {𝑒_𝑖} in 𝕂, a bounded sequence of nonnegative real numbers 𝜆_𝑖 such that 𝑄𝑒_𝑖=𝜆_𝑖𝑒_𝑖, 𝑖=1,2,…, and a sequence {𝛽_𝑖(𝑡)}_𝑖>1 of independent standard Brownian motions such that𝑤(𝑡)=^+∞_𝑖=1𝜆_𝑖𝛽_𝑖(𝑡)𝑒_𝑖,𝑡≥0, (2.1) and ℱ_𝑡=ℱ^𝑤_𝑡, where ℱ^𝑤_𝑡 is the 𝜎-algebra generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}. Let ℒ⁰₂=ℒ₂(𝕂₀,ℍ) be the space of all Hilbert-Schmidt operators from 𝕂₀toℍ. It turns out to be a separable Hilbert space equipped with the norm ‖𝑣‖_ℒ⁰₂=tr((𝑣𝑄^1/2)(𝑣𝑄^1/2)^∗) for any 𝑣∈ℒ⁰₂. Obviously, for any bounded operator 𝑣∈ℒ⁰₂, this norm reduces to ‖𝑣‖²_ℒ02=tr(𝑣𝑄𝑣^∗).

2.2. Partial Integrodifferential Equations

In this section, we recall some fundamental results needed to establish our results. Regarding the theory of resolvent operators, we refer the reader to [13, 14]. Throughout the paper, 𝑋 is a Banach space 𝐴and𝐵(𝑡)are closed linear operators on 𝑋. 𝑌 represents the Banach space𝐷(𝐴)equipped with the graph norm defined by 𝑦_𝑌∶=𝐴𝑦+𝑦for𝑦∈𝑌. (2.2) The notations𝐶([0,+∞);𝑌), 𝐵(𝑌,𝑋)stand for the space of all continuous functions from[0,+∞)into𝑌, the set of all bounded linear operators from𝑌into𝑋, respectively. We consider the following Cauchy problem: 𝑣^′(𝑡)=𝐴𝑣(𝑡)+^𝑡₀𝐵()𝑣(𝑠)𝑣(0)𝑡−𝑠𝑑𝑠,for𝑡≥0,=𝑣₀∈𝑋. (2.3)

Definition 2.1 (see [13]). A resolvent operator for (2.3) is a bounded linear operator-valued function 𝑅(𝑡)∈ℒ(𝑋) for 𝑡≥0, having the following properties. (i)𝑅(0)=𝐼 and |𝑅(𝑡)|≤𝑁𝑒^𝛽𝑡 for some constants 𝑁 and 𝛽. (ii)For each 𝑥∈𝑋, 𝑅(𝑡)𝑥 is strongly continuous for 𝑡≥0. (iii)𝑅(𝑡)∈ℒ(𝑌) for 𝑡≥0. For 𝑥∈𝑌, 𝑅(⋅)𝑥∈𝐶¹([0,+∞);𝑋)∩𝐶([0,+∞);𝑌) and 𝑅^′(𝑡)𝑥=𝐴𝑅(𝑡)𝑥+^𝑡₀𝐵()𝑅(𝑠)(𝑡)𝑡−𝑠𝑥𝑑𝑠=𝑅𝐴𝑥+^𝑡₀𝑅(𝑡−𝑠)𝐵(𝑠)𝑥𝑑𝑠,for𝑡≥0. (2.4)In what follows we suppose the following assumptions.(H1) A is the infinitesimal generator of a strongly continuous semigroup on 𝑋. (H2) For all𝑡≥0, 𝐵(𝑡) is closed linear operator from 𝐷(𝐴)to 𝑋 and 𝐵(𝑡)∈𝐵(𝑌,𝑋). For any𝑦∈𝑌, the map𝑡→𝐵(𝑡)𝑦is bounded and differentiable, and the derivative𝑡→𝐵^′(𝑡)𝑦 is bounded uniformly continuous on ℝ⁺.

The resolvent operator plays an important role to study the existence of solutions and to give a variation of constants formula for nonlinear systems. We need to know when the linear system 2.1 has a resolvent operator. For more details on resolvent operators, we refer to [13, 14]. The following theorem gives a satisfactory answer to this problem, and it will be used in this work to develop our main results.

Theorem 2.2 (see [13]). Assume that (H1)-(H2) hold. Then there exists a unique resolvent operator of the Cauchy problem (2.3).

In the following, we give some results for the existence of solutions for the following integrodifferential equation: 𝑣^′(𝑡)=𝐴𝑣(𝑡)+^𝑡₀𝐵()𝑣(𝑠)(𝑡)𝑣(0)𝑡−𝑠𝑑𝑠+𝑞,for𝑡≥0=𝑣₀∈𝑋, (2.5) where𝑞∶[0,+∞[→𝑋 is a continuous function.

Definition 2.3 (see [13]). A continuous function 𝑣∶[0,+∞)→𝑋 is said to be a strict solution of (2.5) if(i)𝑣∈𝐶¹([0,+∞);𝑋)∩𝐶([0,+∞);𝑌), (ii)𝑣 satisfies (2.5), for 𝑡≥0.

Remark 2.4. From this definition, we deduce that 𝑣(𝑡)∈𝐷(𝐴), the function𝐵(𝑡−𝑠)𝑣(𝑠)is integrable, for all𝑡≥0, and𝑠∈[0,t].

Theorem 2.5 (see [13]). Assume that (H1)-(H2) hold. If v is a strict solution of (2.5), then 𝑣(𝑡)=𝑅(𝑡)𝑣₀+^𝑡₀𝑅(𝑡−𝑠)𝑞(𝑠)𝑑𝑠,for𝑡≥0. (2.6)

Accordingly, we make the following definition.

Definition 2.6 (see [13]). For 𝑣₀∈𝑋, a function 𝑣∶[0,+∞)→𝑋 is called a mild solution of (2.5) if 𝑣 satisfies (2.6).

The next theorem provides sufficient conditions for the regularity of solutions of (2.5).

Theorem 2.7 (see [13]). Let𝑞∈𝐶¹([0,+∞);𝑋) and𝑣 be defined by (2.6). If 𝑣₀∈𝐷(𝐴), then𝑣 is a strict solution of (2.5).

3. Existence of Mild Solutions of (1.1)

Definition 3.1. A process {𝑢(𝑡),𝑡∈[0,𝑇]}, 0≤𝑇<+∞, is called a mild solution of (1.1) if(i)𝑢(𝑡) is ℱ_𝑡-adapted,𝑡≥0with∫^𝑇₀‖𝑢‖²_ℍ𝑑𝑡<+∞ a.s.;(ii)𝑢(𝑡)∈ℍhas continuous paths on 𝑡∈[0,𝑇] a.s., and, for each 𝑡∈[0,𝑇], 𝑢(𝑡) satisfies the integral equation 𝑢(𝑡)−𝐻𝑡,𝑢_𝑡=𝑅(𝑡)𝜑(0)−𝐻0,𝜑+^𝑡₀𝑅(𝑡−𝑠)𝐹𝑠,𝑢_𝑠+𝑑𝑠_𝑡0𝑅(𝑡−𝑠)𝐺𝑠,𝑢_𝑠𝑑𝑤(𝑠), (3.1) for any𝑢₀(⋅)=𝜑∈𝐶^𝑏_ℱ0([−𝑟,0];ℍ).

In the rest of this paper we replace 𝑋 by ℍ in (H1) and (H2).

To guarantee the existence and uniqueness of a mild solution to (1.1), the following much weaker conditions, instead of the non-Lipschitz condition, are used.(H3)(a) There exists a function 𝐿(𝑡,𝑦)∶ℝ₊×ℝ₊→ℝ₊ such that 𝐿(𝑡,𝑦) is locally integrable in 𝑡≥0 and is continuous monotone nondecreasing and concave in 𝑦 for any fixed𝑡∈[0,𝑇]. Moreover, for any fixed 𝑡∈[0,𝑇] and 𝜁∈ℍ, the following inequality is satisfied: 𝐹(𝑡,𝜁)²_ℍ+𝐺(𝑡,𝜁)²_ℒ02≤𝐿𝑡,𝜁²_𝐶,𝑡∈[0,𝑇]. (3.2)(b) For any constant𝐾^∗>0, the differential equation 𝑑𝑥𝑑𝑡=𝐾^∗𝐿(𝑡,𝑥),𝑡∈[0,𝑇], (3.3) has a global solution for any initial value𝑥₀.(H4)(a) There exists a function 𝜆(𝑡,𝑦)∶ℝ₊×ℝ₊→ℝ₊ such that 𝜆(𝑡,𝑦) is locally integrable in 𝑡≥0 for any fixed 𝑦≥0 and is continuous monotone nondecreasing and concave in 𝑦 for any fixed 𝑡∈[0,𝑇]. 𝐺(𝑡,0)=0 for any fixed𝑡∈[0,𝑇]. Moreover, for any fixed 𝑡∈[0,𝑇] and 𝜁,𝜂∈ℍ, the following inequality is satisfied: 𝐹(𝑡,𝜁)−𝐹(𝑡,𝜂)²_ℍ+𝐺(𝑡,𝜁)−𝐺(𝑡,𝜂)²_ℒ02≤𝜆𝑡,𝜁−𝜂²_𝐶,𝑡∈[0,𝑇]. (3.4)(b) For any constant𝐾>0, if a nonnegative function 𝑧(𝑡) satisfies that 𝑧(𝑡)≤𝐾^𝑡₀𝜆(𝑠,𝑧(𝑠))𝑑𝑠,𝑡∈[0,𝑇], (3.5) then 𝑧(𝑡)=0 holds for any 𝑡∈[0,𝑇].(H5) The mapping 𝐻(𝑡,𝑥) satisfies that there exists a positive𝐾 such that, for any 𝜁,𝜂∈ℍ and 𝑡≥0, 𝐻(𝑡,𝜁)−𝐻𝑡,𝜂_ℍ≤𝐾𝜁−𝜂_𝐶. (3.6)

Remark 3.2. Let𝜆(𝑡,𝑢)=𝛽(𝑡)𝜆(𝑢), 𝑡∈[0,𝑇], where𝛽(𝑡)≥0 is locally integrable and𝜆(𝑢)is a concave nondecreasing function fromℝ₊toℝ₊such that𝜆(0)=0,𝜆(𝑢)>0 for 𝑢>0and∫₀₊(1/𝜆(𝑢))𝑑𝑢=∞. Then, by the comparison theorem of differential equations we know that assumption (H4-(b)) holds.
Now let us give some concrete examples of the function𝜆(⋅). Let𝜉>0and let𝛿∈(0,1)be sufficient small. Define 𝜆₁(𝑢)=𝜉𝑢,𝑢≥0,𝜆₂(𝑢)=𝑢log𝑢⁻¹𝛿,0≤𝑢≤𝛿,𝛿log⁻¹+𝜆^′₂(𝛿−)(𝑢−𝛿),𝑢>𝛿, (3.7) where𝜆^′₂denotes the derivative of function𝜆₂. They are all concave nondecreasing functions satisfying∫₀₊(1/𝜆_𝑖(𝑢))𝑑𝑢=∞(𝑖=1,2).

In the following, we establish the existence and uniqueness theorem of the mild solution.

Theorem 3.3. If (H1)–(H5) hold, then there exists a unique mild solution to (1.1), provided that 𝐾<1. (3.8)

Proof. To obtain the existence of the solution to (1.1), we consider the Picard iteration which is defined by 𝑢⁰(𝑡)(𝑡)𝜑(0)[],𝑢=𝑅,𝑡∈0,𝑇₀(𝑡)=𝜑(𝑡),𝑡∈[−𝑟,0], (3.9) and𝑢^𝑛for𝑛≥1is defined by 𝑢^𝑛(𝑡)(𝑡)[],𝑢=𝜑for𝑡∈−𝑟,0_𝑛(𝑡)−𝐻𝑡,𝑢^𝑛_𝑡=𝑅(𝑡)𝜑(0)−𝐻0,𝜑+^𝑡₀𝑅(𝑡−𝑠)𝐹𝑠,𝑢_𝑠^𝑛−1+𝑑𝑠_𝑡0𝑅(𝑡−𝑠)𝐺𝑠,𝑢_𝑠^𝑛−1𝑑𝑤(𝑠),𝑡∈[0,𝑇]. (3.10)(3.11) The proof is divided into the following three steps.Step 1. We claim that the sequence{𝑢^𝑛(𝑡),𝑛≥0} is bounded. From (3.11), for0≤𝑡≤𝑇, 𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛(𝑠)−𝐻(𝑠,𝑢^𝑛_𝑠(𝑠))²_ℍ≤3𝐸sup_{0≤𝑠≤𝑡}𝑅(𝑡)[𝜑(0)−𝐻(0,𝜑)]²_ℍ+3𝐸sup_{0≤𝑠≤𝑡}^𝑡₀𝑅(𝑡−𝑠)𝐹(𝑠,𝑢_𝑠^𝑛−1)𝑑𝑠²_ℍ+3𝐸sup_{0≤𝑠≤𝑡}^𝑡₀𝑅(𝑡−𝑠)𝐺(𝑠,𝑢_𝑠^𝑛−1)𝑑𝑤(𝑠)²_ℍ=∶3𝐼₁+𝐼₂+𝐼₃. (3.12) By (H5), 𝐼₁≤𝑀(1+𝐾)²𝐸𝜑²_𝐶, (3.13) where 𝑀=sup_{0≤𝑡≤𝑇}|𝑅(𝑡)|².
Note that 𝐸sup_{0≤𝑠≤𝑡}𝑢_𝑠^𝑛−12_𝐶≤𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛−1(𝑠)²_ℍ+𝐸𝜑²_𝐶. (3.14) By (H3) and the Jensen inequality, we obtain 𝐼₂≤𝑀𝑇^𝑡₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛−1(𝑙)²_ℍ+𝜑²_𝐶𝑑𝜏. (3.15) By (H3), Liu and Hu [15, Theorem 1.2.5, page 14], and the Jensen inequality, there exists a positive constant𝐶₁such that 𝐼₃≤𝐶₁^𝑡₀𝐸𝐺𝜏,𝑢^𝑛−1(𝜏)²_ℍ𝑑𝜏≤𝐶₁^𝑡₀𝐸𝐿𝜏,𝑢^𝑛−1(𝜏)²_ℍ𝑑𝜏≤𝐶₁^𝑡₀𝐿𝜏,𝐸𝑢^𝑛−1(𝜏)²_ℍ𝑑𝜏. (3.16) Since𝐿 is continuous monotone nondecreasing in𝑦, we obtain 𝐼₃≤𝐶₁^𝑡₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛−1(𝑙)²_ℍ+𝜑²_𝐶𝑑𝜏. (3.17) Recall that, for𝑎,𝑏∈𝑋, 𝜀∈(0,1), ‖𝑎−𝑏‖²_𝑋≤1/(1−𝜀)‖𝑎‖²_𝑋+(1/𝜀)‖𝑏‖²_𝑋. Hence, substituting (3.13)−(3.16) into (3.12) yields 𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛(𝑠)²_ℍ+𝜑²_𝐶≤11−𝐾𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛(𝑠)−𝐻𝑠,𝑢^𝑛_𝑠²_ℍ+1𝐾𝐸sup_{0≤𝑠≤𝑡}𝐻𝑠,𝑢^𝑛_𝑠²_ℍ+𝐸𝜑²_𝐶≤3𝑀(1+𝐾)²𝐸𝜑1−𝐾^2𝐶+𝐸𝜑²_𝐶+𝐾𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛(𝑠)²_ℍ+𝜑²_𝐶+3𝑀𝑇+𝐶₁1−𝐾^𝑡₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛−1(𝑙)²_ℍ+𝜑²_𝐶≤()𝑑𝜏3𝑀1+𝐾²(1−𝐾)²𝐸𝜑²_𝐶+1𝐸𝜑1−𝐾^2𝐶+3𝑀𝑇+𝐶₁(1−𝐾)^2𝑡₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛−1(𝑙)²_ℍ+𝜑²_𝐶𝑑𝜏. (3.18) Assumption (H3-(b)) indicates that there is a solution𝑥_𝑡that satisfies 𝑥_𝑡=𝐶₂𝐸𝜑²_𝐶+𝐶₃^𝑡₀𝐿𝑟,𝑥_𝑟𝑑𝑟, (3.19) where 𝐶₂=3𝑀(1+𝐾)²/(1−𝐾)²+1/(1−𝐾), 𝐶₃=3(𝑀𝑇+𝐶₁)/(1−𝐾)².
Since𝐸‖𝜑‖²_𝐶<∞, from (3.17), we have 𝐸(sup_{0≤𝑠≤𝑡}‖𝑢^𝑛_𝑠‖²_𝐶)≤𝑥_𝑡≤𝑥_𝑇<∞, which shows the boundedness of the {𝑢^𝑛(𝑡),𝑛≥0}.Step 2. We claim that {𝑢^𝑛(𝑡),𝑛≥0}is a Cauchy sequence. For all 𝑛,𝑚≥0and 𝑡∈[0,𝑇], from (3.11), (H4), and Step 1, we have 𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛+1(𝑠)−𝐻(𝑠,𝑢_𝑠^𝑛+1(𝑠))−𝑢^𝑚+1(𝑠)+𝐻(𝑠,𝑢_𝑠^𝑚+1(𝑠))²_ℍ≤2𝐸sup_{0≤𝑠≤𝑡}^𝑠₀𝑅(𝑡−𝑠)[𝐹(𝑠,𝑢_𝜏^𝑛+1)−𝐹(𝑠,𝑢_𝜏^𝑚+1)]𝑑𝜏²_ℍ+2𝐸sup_{0≤𝑠≤𝑡}^𝑡₀𝑅(𝑡−𝑠)[𝐺(𝜏,𝑢^𝑛_𝑠)−𝐺(𝜏,𝑢^𝑚_𝑠)]𝑑𝑤(𝑠)²_ℍ≤2𝑀+𝐶₄^𝑠₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛(𝑙)−𝑢^𝑚(𝑙)²_ℍ𝑑𝜏, (3.20) where 𝐶₄is a generic constant used by Liu and Hu [15,Theorem 1.2.6, page 14]. Therefore, 𝐸sup_{0≤𝑠≤𝑡}𝑢²⁺¹(𝑠)−𝑢^𝑚+1(𝑠)²_ℍ≤11−𝐾𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛+1(𝑠)−𝐻𝑠,𝑢_𝑠^𝑛+1(𝑠)−𝑢^𝑚+1(𝑠)+𝐻𝑠,𝑢_𝑠^𝑚+1(𝑠)²_ℍ+1𝐾𝐸sup_{0≤𝑠≤𝑡}𝐻𝑠,𝑢_𝑠^𝑛+1−𝐻𝑠,𝑢_𝑠^𝑚+12_ℍ≤2𝑀1+𝐶₄1−𝐾^𝑠₀𝐿𝜏,𝐸sup_{0≤𝑙≤𝜏}𝑢^𝑛(𝑙)−𝑢^𝑚(𝑙)²_ℍ𝑑𝜏+𝐾𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛+1(𝑠)−𝑢^𝑚+1(𝑠)²_ℍ. (3.21) Let 𝑧(𝑡)=lim_{𝑛,𝑚→+∞}sup𝐸sup_{0≤𝑠≤𝑡}𝑢^𝑛(𝑠)−𝑢^𝑚(𝑠)²_ℍ. (3.22) By assumption (H4-(b)) and the Fatou lemma, we have 𝑧(𝑡)≤𝐶₅^𝑡₀𝜆(𝑠,𝑧(𝑠))𝑑𝑠, (3.23) where𝐶₅=2𝑀(1+𝐶₄)/(1−𝐾)². By assumption (H4-(b)) we obtain𝑧(𝑡)=0. This shows that {𝑢^𝑛(𝑡),𝑛≥0}is Cauchy.Step 3. We claim the existence and uniqueness of the solution to (1.1). The Borel-Cantelli lemma shows that, as 𝑛→∞, 𝑢^𝑛(𝑡)→𝑢(𝑡) holds uniformly for0≤𝑡≤𝑇. Hence, taking limits on both sides of (3.11), we obtain that𝑢(𝑡) is a solution to (1.1). This shows the existence. The uniqueness of the solution could be obtained by the same procedure as Step 2. The proof is complete.

Remark 3.4. If𝜆(𝑡,𝑢)=𝐾₁𝑢for some constant𝐾₁, then condition (H4) implies a global Lipschitz condition.

4. Application

We conclude this work with an example of the form𝜕𝑥()−𝜕𝑡𝑡,𝜉⁰_−𝑟ℎ((=𝜕𝑡,𝑥𝑡+𝜃,𝜉))𝑑𝜃²𝜕𝜉²𝑥(𝑡,𝜉)−⁰_−𝑟ℎ((+𝑡,𝑥𝑡+𝜃,𝜉))𝑑𝜃_𝑡0𝑏()𝜕𝑡−𝑠²𝜕𝜉²𝑥(𝑠,𝜉)−⁰_−𝑟ℎ((+𝑡,𝑥𝑡+𝜃,𝜉))𝑑𝜃𝑑𝑠_0−𝑟𝑓(((((𝑡)[],𝑥()−𝑡,𝑥𝑡+𝜃,𝜉))𝑑𝜃+𝑔𝑡,𝑥𝑡+𝜃,𝜉))𝑑𝑤,for𝑡≥0,𝜉∈0,𝜋𝑡,0⁰_−𝑟ℎ((𝑥()−𝑡,𝑥𝑡+𝜃,0))𝑑𝜃=0,for𝑡≥0,𝑡,𝜋⁰_−𝑟ℎ((𝑥()𝑡,𝑥𝑡+𝜃,𝜋))𝑑𝜃=0,for𝑡≥0,𝜃,𝜉=𝑥₀(𝜃,𝜉),for𝜃∈[−𝑟,0],𝜉∈[0,𝜋], (4.1) where𝑤(𝑡) denotes an ℝ-valued Brownian motion, ℎ,𝑓,𝑔∶ℝ⁺×ℝ→ℝare continuous functions,𝑏∶ℝ⁺→ℝis continuous, and 𝑥₀∶[−𝑟,0]×[0,𝜋]→ℝis a given continuous function such that 𝑥₀(⋅)∈𝐿²([0,𝜋]) is ℱ₀-measurable and satisfies 𝐸‖𝑥₀‖²<∞.

Let ℍ=𝐿²([0,𝜋])with the norm‖⋅‖ and𝑒_𝑛∶=2/𝜋sin(𝑛𝑥), (𝑛=1,2,3,…) denote the completed orthonormal basis inℍ. Let𝑤(𝑡)∶=∑^∞_𝑛=1𝜆_𝑛𝛽_𝑛(𝑡)𝑒_𝑛(𝜆_𝑛>0), where𝛽_𝑛(𝑡)are one-dimensional standard Brownian motion mutually independent on a usual complete probability space(Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ).

Define𝐴∶ℍ→ℍby𝐴=𝜕²/𝜕𝑧², with domain𝐷(𝐴)=𝐻²([0,𝜋])∩𝐻¹₀([0,𝜋]).

Then𝐴ℎ=−∑^∞_𝑛=1𝑛²⟨ℎ,𝑒_𝑛⟩𝑒_𝑛, ℎ∈𝐷(𝐴), where𝑒_𝑛, 𝑛=1,2,3,…, is also the orthonormal set of eigenvectors of𝐴. It is wellknown that 𝐴 is the infinitesimal generator of a strongly continuous semigroup onℍ; thus,(𝐻1) is true.

Let𝐵∶𝐷(𝐴)⊂ℍ→ℍbe the operator defined by 𝐵(𝑡)(𝑧)=𝑏(𝑡)𝐴𝑧for𝑡≥0and 𝑧∈𝐷(𝐴).

We suppose that (i)there exists a positive constant 𝐿_ℎ, 0<√𝜋𝐿_ℎ𝑟<1, such that ℎ𝑡,𝜁₁−ℎ𝑡,𝜁₂≤𝐿_ℎ𝜁₁−𝜁₂,for𝑡≥0,𝜁₁,𝜁₂∈ℝ; (4.2)(ii)there exists a constant 𝐿_𝑓, 0<√𝜋𝐿_𝑓𝑟<1, such that 𝑓𝑡,𝜁₁−𝑓𝑡,𝜁₂²≤𝐿_𝑓𝜆𝑡,𝜁₁−𝜁₂²_𝐶,for𝑡≥0,𝜁₁,𝜁₂∈ℝ; (4.3)(iii)for 𝑡≥0and𝜁₁,𝜁₂∈ℝ, |𝑔(𝑡,𝜁₁)−𝑔(𝑡,𝜁₂)|²≤𝜆(𝑡,‖𝜁₁−𝜁₂‖²_𝐶).

Let 𝐶=𝐶([−𝑟,0];ℍ) and define the operators 𝐻,𝐹,𝐺∶ℝ⁺×𝐶→ℍ by𝐻𝑡,𝜙(𝜉)=⁰_−𝑟ℎ(𝜃𝜉)[]𝐹(𝜉)=𝑡,𝜙)(𝑑𝜃,for𝜉∈0,𝜋,𝜙∈𝐶,𝑡,𝜙⁰_−𝑟𝑓(𝜃𝜉)[]𝐺(𝜉)(𝜃𝜉.𝑡,𝜙)(𝑑𝜃,for𝜉∈0,𝜋,𝜙∈𝐶,𝑡,𝜙=𝑔𝑡,𝜙)() (4.4) If we put𝑢(𝑡)()[],𝜑(𝜃𝜉)=𝑥𝑡,𝜉,for𝑡≥0,𝜉∈0,𝜋)(=𝑥₀(𝜃,𝜉),for𝜃∈[−𝑟,0],𝜉∈[0,𝜋]. (4.5) then (4.1) takes the following abstract form: 𝑑𝑢(𝑡)−𝐻𝑡,𝑢_𝑡=𝐴𝑢(𝑡)−𝐻𝑡,𝑢_𝑡𝑑𝑡+^𝑡₀𝐵(𝑡−𝑠)𝑢(𝑠)−𝐻𝑠,𝑢_𝑠𝑑𝑠+𝐹𝑡,𝑢_𝑡𝑑𝑡+𝐺𝑡,𝑢_𝑡(𝑡)[]𝑢𝑑𝑤,for𝑡∈0,𝑇₀=𝜑. (4.6) Moreover, if 𝑏 is bounded and 𝐶¹ function such that 𝑏^′ is bounded and uniformly continuous, then (H1) and (H2) are satisfied, and hence, by Theorem 2.2, (2.3) has a resolvent operator (𝑅(𝑡))_𝑡≥0 on ℍ. Thus, all the assumptions of Theorem 3.3 are fulfilled. Therefore, the existence of a unique mild solution of (4.1) follows.