Mean Square Almost Periodic Solutions for Impulsive Stochastic Differential Equations with Delays

Abstract

We establish a result on existence and uniqueness on mean square almost periodic solutions for a class of impulsive stochastic differential equations with delays, which extends some earlier works reported in the literature.

1. Introduction

Impulsive effects widely exist in many evolution processes of real-life phenomena in which states are changed abruptly at certain moments of time, involving such areas as population dynamics and automatic control [1–3]. Because delay is ubiquitous in the dynamical system, impulsive differential equations with delays have received much interesting in recent years, intensively researched, some important results are obtained [4–9]. And almost periodic solutions for abstract impulsive differential equations and for impulsive neural networks with delay have been discussed by G. T. Stamov and I. M. Stamova [10], and Stamov and Alzabut [11].

However, besides delay and impulsive effects, stochastic effects likewise exist in real system. A lot of dynamic systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of components, changes in the interconnections of subsystems, sudden environment changes, and so on [12–14]. Moreover, differential descriptor systems also have abrupt changes [15, 16]. Recently, a large number of stability criteria of stochastic system with delays have been reported [17–19]. Almost periodic solutions to some functional integro-differential stochastic evolution equations and to some stochastic differential equations have been studied by Bezandry and Diagana [20], and Bezandry [21]. Huang and Yang investigated almost periodic solution for stochastic cellular neural networks with delays [22]. Because it is not easy to deal with the case of coexistence of impulsive, delay and stochastic effects in a dynamical system, there are few results about this problems [23–25]. To the best of our knowledge, there exists no result on the existence and uniqueness of mean square almost periodic solutions for impulsive stochastic differential equations with delays.

Motivated by the above discussions, the main aim of this paper is to study the mean square almost periodic solutions for impulsive stochastic differential equations with delays. By employing stochastic analysis, delay differential inequality technique and fixed points theorem, we obtain some criteria to ensure the existence and uniqueness of mean square almost periodic solutions.

The rest of this paper is organized as follows: in Section 2, we introduce a class of impulsive stochastic differential equations with delays, and the relating notations, definitions and lemmas which would be used later; in Section 3, a new sufficient condition is proposed to ensure the existence and uniqueness of mean square almost periodic solutions; in Section 4, an example is constructed to show the effectiveness of our results. Finally, a conclusion is given in Section 5.

2. Preliminaries

Let ℝ=(−∞,+∞), ℕ={1,2,3,…}, and ℬ={{𝑡_𝑘}∶𝑡₀=0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯,lim_𝑘→+∞𝑡_𝑘=+∞} be the set of all sequence unbounded and strictly increasing. For 𝑥∈ℝ^𝑛 and 𝐴∈ℝ^𝑛×𝑛, let ‖𝑥‖ be any vector norm, and denote the induced matrix norm and the matrix measure, respectively, by‖𝐴‖=sup_𝑥≠0‖𝐴𝑥‖‖𝑥‖,𝜇(𝐴)=lim_ℎ→0⁺‖𝐼+ℎ𝐴‖−1ℎ.(2.1) The norm and measure of vector and matrix are ‖𝑥‖=max_𝑖|𝑥_𝑖|, ‖𝐴‖=max_𝑖∑^𝑛𝑗=1|𝑎_𝑖𝑗|, 𝜇(𝐴)=max_𝑖{𝑎_𝑖𝑖+∑^{𝑛𝑗≠𝑖}|𝑎_𝑖𝑗|}.

Consider the following a class of Itô impulsive stochastic differential equations with delay𝑑𝑥(𝑡)=[𝐴Δ𝑥𝑥((𝑡𝑡))+=𝐵𝑥𝑓𝑡(_𝑘𝑡,𝑥(𝑡))+𝐶𝑔(𝑡,𝑥(𝑡−ℎ))+𝐼(𝑡)]𝑑𝑡+𝜎(𝑡,𝑥(𝑡))𝑑𝜔(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,−𝑥𝑡^−𝑘𝑥(=𝑡)𝐷=_𝑘𝑥𝜙(𝑡𝑡)^−𝑘,+−𝑉_𝑘ℎ≤𝑥𝑡𝑡≤^−𝑘0,+𝛽_𝑘,𝑡=𝑡_𝑘,𝑘∈ℕ,(2.2) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the solution process, 𝐴,𝐵,𝐶,𝐷_𝑘∈ℝ^𝑛×𝑛 are constant matrices, 𝑓(𝑡,𝑥)=(𝑓₁(𝑡,𝑥),…,𝑓_𝑛(𝑡,𝑥))^𝑇, 𝑔(𝑡,𝑥)=(𝑔₁(𝑡,𝑥),…,𝑔_𝑛(𝑡,𝑥))^𝑇, 𝐼(𝑡)=(𝐼₁(𝑡),…,𝐼_𝑛(𝑡))^𝑇, 𝜎(𝑡,𝑥)=(𝜎_𝑖𝑗(𝑡,𝑥))_𝑛×𝑛 is the diffusion coefficient matrix, 𝑉_𝑘(𝑥)=(𝑉_1𝑘(𝑥),…,𝑉_𝑛𝑘(𝑥))^𝑇 is impulsive function, ℎ>0 is delay; 𝑡_𝑘∈ℬ is impulsive time, 𝛽_𝑘=(𝛽_1𝑘,…,𝛽_𝑛𝑘)^𝑇 is a constant vector, 𝜔(𝑡)=(𝜔₁(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brown motion defined on a complete probability space (Ω,ℱ,ℙ) with a natural filtration {ℱ_𝑡}_𝑡≥0 generated by 𝜔(𝑡), and denote by ℱ the associated 𝜎-algebra generated by 𝜔(𝑡) with the probability measure ℙ. Moreover, the initial conditions 𝜙(𝑡)=(𝜙₁(𝑡),…,𝜙_𝑛(𝑡))^𝑇∈𝑃𝐶𝐵^𝑏ℱ₀([−ℎ,0],ℝ^𝑛)=^Δ𝑃𝐶𝐵^𝑏ℱ₀. Denote by 𝑃𝐶𝐵^𝑏ℱ₀ the family of all bounded ℱ₀-measurable, 𝑃𝐶([−ℎ,0],ℝ^𝑛)-valued random variable 𝜁, satisfying 𝐸‖𝜁‖²=𝐸(sup_{−ℎ≤𝜃≤0}‖𝜁(𝜃)‖²)<+∞, where 𝑃𝐶([−ℎ,0],ℝ^𝑛)={𝜁∶[−ℎ,0]→ℝ^𝑛 is continuous}. 𝐸 denotes the expectation of stochastic process.

Let (ℍ,‖⋅‖) be a Hilbert space and (Ω,ℱ,ℙ) be a complete probability space. Define 𝐿²(ℙ,ℍ) to be the space of all ℍ-value random variable 𝑌 such that𝐸‖𝑌‖²=_Ω‖𝑌‖²𝑑ℙ<∞.(2.3)

It is then routine to check that 𝐿²(ℙ,ℍ) is a Hilbert space when it is equipped with its natural norm ‖⋅‖₂ defined by‖𝑌‖₂=_Ω‖𝑌‖²𝑑ℙ^1/2<∞,(2.4) for each 𝑌∈𝐿²(ℙ,ℍ).

Definition 2.1 (see [25]). For any 𝜙∈𝑃𝐶𝐵^𝑏ℱ₀, a function 𝑥(𝑡)∶[−ℎ,+∞)→𝐿²(ℙ,ℍ) is said to be solution of system (2.2) on [−ℎ,+∞) satisfying initial value condition, if the following conditions hold:(i)𝑥(𝑡) is absolutely continuous on each interval (𝑡_𝑘,𝑡_𝑘+1)∈[0,+∞),𝑘∈ℕ;(ii) for any 𝑡_𝑘∈[0,+∞),𝑘∈ℕ, 𝑥(𝑡^+𝑘) and 𝑥(𝑡^−𝑘) exist and 𝑥(𝑡^+𝑘)=𝑥(𝑡_𝑘);(iii)𝑥(𝑡) satisfies (2.2) for almost everywhere in [−ℎ,+∞) and at impulsive points 𝑡=𝑡_𝑘 situated in [0,+∞),𝑘∈ℕ, may have discontinuity points of the first kind.
Obviously, the solution defined by definition 1 is piecewise continuous.

Definition 2.2 (see [26]). The set of sequences {𝑡^𝑗𝑘}, 𝑡^𝑗𝑘=𝑡_𝑘+𝑗−𝑡_𝑘,𝑘∈ℕ,𝑗∈ℕ,{𝑡_𝑘}∈ℬ is said to be uniformly almost periodic if for any 𝜀>0, there exists relatively dense set of 𝜀-almost periods common for any sequences.

Definition 2.3. A piecewise continuous function 𝑥(𝑡)∶[−ℎ,+∞)→𝐿²(ℙ,ℍ) with discontinuity points of first kind at 𝑡=𝑡_𝑘 is said to be mean square almost periodic, if(i) the set of sequence {𝑡^𝑗𝑘} is uniformly almost periodic;(ii) for any 𝜀>0, there exists 𝛿>0, such that if the points 𝑡′ and 𝑡^ belong to one and the same interval of continuity of 𝑥(𝑡) and satisfy the inequality |𝑡^−𝑡^|<𝛿, then 𝐸‖𝑥(𝑡^)−𝑥(𝑡^)‖²<𝜀;(iii) for any 𝜀>0, there exists a relatively dense set 𝑇 such that if 𝜏∈𝑇, then 𝐸‖𝑥(𝑡+𝜏)−𝑥(𝑡)‖²<𝜀 for all 𝑡∈[−ℎ,+∞) satisfying the condition |𝑡−𝑡_𝑘|>𝜀,𝑘∈ℕ.

The collection of all functions 𝑥(𝑡)∶[−ℎ,+∞)→𝐿²(ℙ,ℍ) with discontinuity points of the first kind at 𝑡=𝑡_𝑘 which are mean square almost periodic is denoted by 𝐴𝑃([−ℎ,+∞);𝐿²(ℙ,ℍ)), one can check that 𝐴𝑃([−ℎ,+∞);𝐿²(ℙ,ℍ)) is a Banach space when it is equipped with the norm:‖𝑥‖_∞=sup_𝑡∈ℝ𝐸‖𝑥(𝑡)‖²^1/2.(2.5)

Let (𝐵₁,‖⋅‖₁) and (𝐵₂,‖⋅‖₂) be Banach space and 𝐿²(ℙ,𝐵₁) and 𝐿²(ℙ,𝐵₂) be their corresponding 𝐿²-space, respectively.

Lemma 2.4 (see [20]). Let 𝑓∶ℝ×𝐿²(ℙ,𝐵₁)→𝐿²(ℙ,𝐵₂),(𝑡,𝑥)↦𝑓(𝑡,𝑥) be mean square almost periodic in 𝑡∈ℝ uniformly in 𝑥∈𝐾, where 𝐾⊂𝐿²(ℙ,𝐵₁) is compact. Suppose that there exists 𝐿_𝑓>0 such that 𝐸‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖²²≤𝐿_𝑓𝐸‖𝑥−𝑦‖²¹(2.6) for all 𝑥,𝑦∈𝐿²(ℙ,𝐵₁) and for each 𝑡∈ℝ. Then for any mean square almost periodic function 𝜓(𝑡)∶ℝ→𝐿²(ℙ,𝐵₁), 𝑓(𝑡,𝜓(𝑡)) is mean square almost periodic.

In this paper, we always assume that:(A1) det(𝐼+𝐷_𝑘)≠0 and the sequence {𝐷_𝑘},𝑘∈ℕ, is almost periodic, where 𝐼∈𝑅^𝑛×𝑛 is the identity matrix;(A2) the set of {𝑡^𝑗𝑘} is uniformly almost periodic and 𝜃=inf_𝑘{𝑡^1𝑘}>0.

Recall [2], consider the following linear system of system(2.2)Δ𝑥𝑥𝑡̇_𝑘(𝑡)=𝐴𝑥(𝑡),𝑡≠𝑡_𝑘,=𝐷_𝑘𝑥𝑡^−𝑘,𝑘∈ℕ,(2.7) that if 𝑈_𝑘(𝑡,𝑠) is the Cauchy matrix for the system𝑥̇(𝑡)=𝐴𝑥(𝑡),𝑡_𝑘−1≤𝑡<𝑡_𝑘,(2.8) then the Cauchy matrix for the system (2.7) is in the form𝑊(𝑡,𝑠)=⎧⎪⎪⎨⎪⎪⎩𝑈^𝑘𝑈(_𝑘+1𝑡,𝑠),𝑡_𝑘−1≤𝑠≤𝑡<𝑡_𝑘,𝑈_𝑘+1𝑡,𝑡_𝑘𝐼+𝐷_𝑘𝑈_𝑘𝑡_𝑘,𝑠,𝑡_𝑘−1≤𝑠<𝑡_𝑘≤𝑡<𝑡_𝑘+1,𝑡,𝑡_𝑘^𝑖+1_𝑗=𝑘𝐼+𝐷_𝑘𝑈_𝑗𝑡_𝑗,𝑡_𝑗+1𝐼+𝐷_𝑖𝑈_𝑖𝑡_𝑖,𝑠,𝑡_𝑖−1≤𝑠<𝑡_𝑖<𝑡_𝑘≤𝑡<𝑡_𝑘+1.(2.9)

As the special case of Lemma  1 in [10], we have the following lemma.

Lemma 2.5. Assume that (A1), (A2) and the following condition hold. For the Cauchy matrix 𝑊(𝑡,𝑠) of system (2.7), there exist positive constants 𝑀 and 𝜆 such that ‖𝑊(𝑡,𝑠)‖≤𝑀𝑒^{−𝜆(𝑡−𝑠)},𝑡≥𝑠,𝑡,𝑠∈ℝ.(2.10) Then for any 𝜀>0,𝑡≥𝑠,𝑡,𝑠∈ℝ,|𝑡−𝑡_𝑘|>𝜀,|𝑠−𝑡_𝑘|>𝜀,𝑘∈ℕ, there must be exist a relatively dense set T of 𝜀-almost periodic of the matrix 𝐴 and a positive constant Γ such that for 𝜏∈𝑇, it follows: ‖𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)‖≤𝜀Γ𝑒^{(−𝜆/2)(𝑡−𝑠)}.(2.11)

Lemma 2.6 (see [6]). Let 𝑊(𝑡,𝑠) be the Cauchy matrix of the linear system (2.7). Given a constant 𝜂≥‖𝐼+𝐷_𝑘‖ for all 𝑘∈ℕ, if 𝜂≥1 and 𝜃=inf_𝑘{𝑡^1𝑘}>0, then ‖𝑊(𝑡,𝑠)‖≤𝜂𝑒^{(𝜇(𝐴)+(ln𝜂/𝜃))(𝑡−𝑠)},𝑡≥𝑠.(2.12)

Introduce the following conditions:(A3) The functions 𝑓,𝑔∶ℝ×𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) are mean square almost periodic in 𝑡∈ℝ uniformly in 𝑥∈Θ, where Θ⊂𝐿²(ℙ,ℍ) is compact, and 𝑓(0,0)=𝑔(0,0)=0. Moreover, there exist 𝐿_𝑓,𝐿_𝑔>0 such that𝐸𝐸‖‖𝑓𝑔((𝑡𝑡,,𝑥𝑥))−−𝑓𝑔((𝑡𝑡,,𝑦𝑦))‖‖²₂≤𝐿_𝑓𝐸‖𝑥−𝑦‖²,≤𝐿_𝑔𝐸‖𝑥−𝑦‖²,(2.13)

for all stochastic processes 𝑥,𝑦∈𝐿²(ℙ,ℍ) and 𝑡∈ℝ.(A4) The function 𝜎∶ℝ×𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) is mean square almost periodic in 𝑡∈ℝ uniformly in 𝑥∈Θ′, where Θ′⊂𝐿²(ℙ,ℍ) is compact, and 𝜎(0,0)=0. Moreover, there exists 𝐿_𝜎>0 such that𝐸‖𝜎(𝑡,𝑥)−𝜎(𝑡,𝑦)‖²≤𝐿_𝜎𝐸‖𝑥−𝑦‖²,(2.14)

for all stochastic processes 𝑥,𝑦∈𝐿²(ℙ,ℍ) and 𝑡∈ℝ.(A5) The function 𝐼_𝑖(𝑡)∶ℝ→ℝ is almost periodic in the sense of Bohr, {𝛽_𝑘}_𝑘∈ℕ is almost periodic sequence and there exists a constant 𝛾₀>0, such thatmaxmax_𝑘||𝛽_𝑘||,sup_𝑡‖𝐼(𝑡)‖≤𝛾₀.(2.15)(A6) The sequence of functions 𝑉_𝑘(𝑥)∶𝐿²(ℙ,ℍ)→𝐿²(ℙ,ℍ) is mean square almost periodic uniformly with respect to 𝑥∈Θ^, where Θ^⊂𝐿²(ℙ,ℍ) is compact. Moreover, there exists 𝐿_𝑉>0 such that𝐸‖‖𝑉_𝑘(𝑥)−𝑉_𝑘(𝑦)‖‖²≤𝐿_𝑉𝐸‖𝑥−𝑦‖²(2.16) for all stochastic processes 𝑥,𝑦∈𝐿²(ℙ,ℍ).

Lemma 2.7 (see [26]). If conditions (A1)–(A6) are satisfied, then for each 𝜀>0, there exists 𝜀₁,0<𝜀₁<𝜀 and relatively dense sets 𝑇 of real numbers and 𝑄 of integral numbers, such that(i)𝐸‖𝑓(𝑡+𝜏,𝑦)−𝑓(𝑡,𝑦)‖²<𝜀, 𝐸‖𝑔(𝑡+𝜏,𝑦)−𝑔(𝑡,𝑦)‖²<𝜀,𝑡∈ℝ,𝜏∈𝑇, |𝑡−𝑡_𝑘|>𝜀,𝑘∈ℕ, 𝑦∈𝐿²(ℙ,ℍ);(ii)𝐸‖𝜎(𝑡+𝜏,𝑦)−𝜎(𝑡,𝑦)‖²<𝜀,𝑡∈ℝ,𝜏∈𝑇,|𝑡−𝑡_𝑘|>𝜀,𝑘∈ℕ,𝑦∈𝐿²(ℙ,ℍ);(iii)‖𝐼(𝑡+𝜏)−𝐼(𝑡)‖²<𝜀,𝑡∈ℝ,𝜏∈𝑇,|𝑡−𝑡_𝑘|>𝜀;(iv)𝐸‖𝑉_𝑘+𝑞(𝑦)−𝑉_𝑘(𝑦)‖²<𝜀,𝑞∈𝑄,𝑘∈ℕ;(v)‖𝛽_𝑘+𝑞−𝛽_𝑘‖²<𝜀,𝑞∈𝑄,𝑘∈ℕ;(vi)‖𝑡_𝑘+𝑞−𝜏‖²<𝜀₁,𝑞∈𝑄,𝜏∈𝑇,𝑘∈ℕ.

Lemma 2.8 (see [26]). Let condition (A2) holds. Then for each 𝑝>0, there exists a positive integer 𝑁 such that on each interval of length 𝑝, there are no more than 𝑁 elements of the sequence {𝑡_𝑘}, that is, 𝑖(𝑠,𝑡)≤𝑁(𝑡−𝑠)+𝑁,(2.17) where 𝑖(𝑠,𝑡) is the number of points 𝑡_𝑘 in the interval (𝑠,𝑡).

3. Main Results

Theorem 3.1. Assume that (A1)–(A6) hold, then there exists a unique mean square almost periodic solution of system (2.2) if the following conditions are satisfied: There exists a constant 𝜂≥1, such that ‖𝐼+𝐷_𝑘‖≤𝜂,𝑘∈ℕ and 𝜇(𝐴)+ln𝜂𝜃^Δ=−𝜆<0.(3.1) Furthermore, 𝜌=6𝜂²𝜆2²‖𝐵‖²𝐿^2𝑓+‖𝐶‖²𝐿^2𝑔+𝑁²(1−𝑒^−𝜆)²𝐿^2𝑉+𝐿^2𝜎2𝜆<1.(3.2)

Proof. Let 𝐷={𝜑(𝑡)∈𝐿²(ℙ,ℍ)∶𝜑(𝑡)=(𝜑₁(𝑡),…,𝜑_𝑛(𝑡))^𝑇}⊂𝐴𝑃([−ℎ,+∞);𝐿²(ℙ,ℍ)) satisfying the equality 𝐸‖𝜑‖²<𝐾, where 𝐾=2𝜂²𝛾²⁰((1/𝜆)+(𝑁/(1−𝑒^−𝜆)))²>0.
Set 𝑥(𝑡)=+𝑊(𝑡,0)𝜙₀+^𝑡0𝑊(𝑡,𝑠)[𝐵𝑓(𝑠,𝑥(𝑠))+𝐶𝑔(𝑠,𝑥(𝑠−ℎ))+𝐼(𝑠)]𝑑𝑠_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑥𝑡_𝑘+𝛽_𝑘+^𝑡0𝑊(𝑡,𝑠)𝜎(𝑠,𝑥(𝑠))𝑑𝜔(𝑠),𝑡≥0.(3.3) where 𝜙₀=𝑥(0), it is easy to see that 𝑥(𝑡) given by (3.3) is the solution of system (2.2) according to [2] and Lemma  2.2 in [27].
By Lemma 2.6 and the conditions of Theorem, we have ‖𝑊(𝑡,𝑠)‖≤𝜂𝑒^{−𝜆(𝑡−𝑠)},𝑡≥𝑠,𝑡,𝑠∈ℝ.(3.4)
For 𝑧(𝑡)∈𝐷, we define the operator 𝐿 in the following way (𝐿𝑧)(𝑡)=+^𝑡0𝑊(𝑡,𝑠)[𝐵𝑓(𝑠,𝑧(𝑠))+𝐶𝑔(𝑠,𝑧(𝑠−ℎ))+𝐼(𝑠)]𝑑𝑠_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑧𝑡_𝑘+𝛽_𝑘+^𝑡0𝑊(𝑡,𝑠)𝜎(𝑠,𝑧(𝑠))𝑑𝜔(𝑠).(3.5)
Define subset 𝐷^∗⊂𝐷, 𝐷^∗={𝑧∈𝐷∶𝐸‖𝑧−𝑧₀‖²≤𝜌𝐾/(1−𝜌)}, and 𝑧₀=∫^𝑡0𝑊(𝑡,𝑠)𝐼(𝑠)𝑑𝑠+∑_{0≤𝑡𝑘<𝑡}𝑊(𝑡,𝑡_𝑘)𝛽_𝑘.
We have 𝐸‖‖𝑧₀‖‖²≤2𝐸‖‖‖^𝑡0𝑊(𝑡,𝑠)𝐼(𝑠)𝑑𝑠‖‖‖²+2𝐸‖‖‖‖_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝛽_𝑘‖‖‖‖²≤2^𝑡0𝜂𝑒^{−𝜆(𝑡−𝑠)}sup_𝑠‖𝐼(𝑠)‖𝑑𝑠²+2⎡⎢⎢⎣_{0≤𝑡𝑘<𝑡}𝜂𝑒^{−𝜆(𝑡−𝑡𝑘)}max_𝑘||𝛽_𝑘||⎤⎥⎥⎦²≤2𝜂²𝛾²⁰1𝜆+1−𝑁𝑒^−𝜆²=𝐾.(3.6) Then for ∀𝑧∈𝐷^∗, from the definition of 𝐷^∗ and (3.6), since (𝑎+𝑏)²≤2𝑎²+2𝑏², we have 𝐸‖𝑧‖²=≤2𝐸‖‖𝑧𝜌−𝑧₀+𝑧₀‖‖²≤2𝐸‖‖𝑧−𝑧₀‖‖²+‖‖𝑧₀‖‖²𝐾1−𝜌+𝐾=2𝐾1−𝜌.(3.7)
For ∀𝑧∈𝐷^∗, we have ‖‖𝐿𝑧−𝑧₀‖‖=‖‖‖+^𝑡0𝑊(𝑡,𝑠)[𝐵𝑓(𝑠,𝑧(𝑠))+𝐶𝑔(𝑠,𝑧(𝑠−ℎ))]𝑑𝑠_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑧𝑡_𝑘+^𝑡0𝑊(𝑡,𝑠)𝜎(𝑠,𝑧(𝑠))𝑑𝜔(𝑠)‖‖‖‖.(3.8)
Since (𝑎+𝑏+𝑐)²≤3𝑎²+3𝑏²+3𝑐², it follows 𝐸‖‖𝐿𝑧−𝑧₀‖‖²≤3𝐸^𝑡0‖𝑊(𝑡,𝑠)‖‖𝐵𝑓(𝑠,𝑧(𝑠))+𝐶𝑔(𝑠,𝑧(𝑠−ℎ))‖𝑑𝑠²+3𝐸⎛⎜⎜⎝‖‖‖‖_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑧𝑡_𝑘‖‖‖‖⎞⎟⎟⎠²+3𝐸^𝑡0‖𝑊(𝑡,𝑠)𝜎(𝑠,𝑧(𝑠))𝑑𝜔(𝑠)‖².(3.9) For first term of the right-hand side, using (3.7), (A3) and Cauchy-Schwarz inequality, we have 𝐸^𝑡0‖𝑊(𝑡,𝑠)‖‖𝐵𝑓(𝑠,𝑧(𝑠))+𝐶𝑔(𝑠,𝑧(𝑠−ℎ))‖𝑑𝑠²≤𝜂²^𝑡0≤𝜂₂𝑒^{−𝜆(𝑡−𝑠)}𝑑𝑠⋅^𝑡0𝑒^{−𝜆(𝑡−𝑠)}⋅𝐸‖𝐵𝑓(𝑠,𝑧(𝑠))+𝐶𝑔(𝑠,𝑧(𝑠−ℎ))‖²𝑑𝑠^𝑡0≤𝜂₂𝑒^{−𝜆(𝑡−𝑠)}𝑑𝑠⋅^𝑡0𝑒^{−𝜆(𝑡−𝑠)}⋅2‖𝐵‖²𝐿^2𝑓𝐸‖𝑧(𝑠)‖²+2‖𝐶‖²𝐿^2𝑔𝐸‖𝑧(𝑠−ℎ)‖²𝑑𝑠⋅2𝐾1−𝜌𝜆2²‖𝐵‖²𝐿^2𝑓+‖𝐶‖²𝐿^2𝑔.(3.10) As to the second term, using (3.7), (A6) and Cauchy-Schwarz inequality, we can write 𝐸⎛⎜⎜⎝‖‖‖‖_{0≤𝑡𝑘<𝑡}𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑧𝑡_𝑘‖‖‖‖⎞⎟⎟⎠²≤𝜂²⎛⎜⎜⎝_{0≤𝑡𝑘<𝑡}𝑒^{−𝜆(𝑡−𝑡𝑘)}⎞⎟⎟⎠⋅⎛⎜⎜⎝_{0≤𝑡𝑘<𝑡}𝑒^{−𝜆(𝑡−𝑡𝑘)}𝐸‖‖𝑉_𝑘𝑧𝑡_𝑘‖‖²≤𝜂₂⎞⎟⎟⎠⎛⎜⎜⎝_{0≤𝑡𝑘<𝑡}𝑒^{−𝜆(𝑡−𝑡𝑘)}⎞⎟⎟⎠⋅⎛⎜⎜⎝_{0≤𝑡𝑘<𝑡}𝑒^{−𝜆(𝑡−𝑡𝑘)}𝐿^2𝑉𝐸‖‖𝑧𝑡_𝑘‖‖²≤𝜂₂⎞⎟⎟⎠⋅2𝐾1−𝜌𝐿^2𝑉⋅𝑁²(1−𝑒^−𝜆)².(3.11) As far as last term is concerned, using (3.7), (A4), and the Itô isometry theorem, we obtain 𝐸^𝑡0‖𝑊(𝑡,𝑠)𝜎(𝑠,𝑧(𝑠))𝑑𝜔(𝑠)‖²≤≤𝜂^𝑡0₂‖𝑊(𝑡,𝑠)‖²𝐸‖𝜎(𝑠,𝑧(𝑠))‖²𝑑𝑠^𝑡0𝑒^{−2𝜆(𝑡−𝑠)}𝐿^2𝜎𝐸‖𝑧(𝑠)‖²𝑑𝑠≤𝜂²⋅2𝐾1−𝜌⋅𝐿2𝜆^2𝜎.(3.12) Thus, by combining (3.9)–(3.12), it follows that 𝐸‖‖𝐿𝑧−𝑧₀‖‖²≤3𝜂²⋅2𝐾1−𝜌𝜆2²‖𝐵‖²𝐿^2𝑓+‖𝐶‖²𝐿^2𝑔+𝑁²(1−𝑒^−𝜆)²𝐿^2𝑉+𝐿^2𝜎2𝜆=𝜌𝐾1−𝜌.(3.13) By Lemmas 2.5 and 2.6, one can obtain ‖𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)‖≤𝜀Γ𝑒^{−(𝜆/2)(𝑡−𝑠)}.(3.14) Let 𝜏∈𝑇,𝑞∈𝑄, where the sets 𝑇 and 𝑄 are determined in Lemma 2.7, and we assume that 0<𝜀<1, then ‖𝐿=𝑧‖‖‖(𝑡+𝜏)−𝐿𝑧(𝑡)‖_𝑡0+[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝐵𝑓(𝑠+𝜏,𝑧(𝑠+𝜏))+𝐶𝑔(𝑠+𝜏,𝑧(𝑠+𝜏−ℎ))+𝐼(𝑠+𝜏)]𝑑𝑠_𝑡0+𝑊(𝑡,𝑠)−{[[𝐵𝐵𝑓𝑓((𝑠𝑠,+𝑧(𝜏𝑠,)𝑧)+(𝑠𝐶+𝑔𝜏()𝑠),+𝑧(𝐶𝑠𝑔−(𝑠ℎ+))+𝜏,𝐼𝑧((𝑠𝑠)+]𝑑𝜏𝑠}_{0≤𝑡𝑘<𝑡}−ℎ))+𝐼(𝑠+𝜏)]+_{0≤𝑡𝑘<𝑡}𝑊𝑡+𝜏,𝑡_𝑘+𝑞−𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞+𝛽_𝑘+𝑞+_𝑡0𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞−𝑉_𝑘𝑧𝑡_𝑘+𝛽_𝑘+𝑞−𝛽_𝑘+[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))]𝑑𝜔(𝑠)_𝑡0𝑊(𝑡,𝑠)[𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))−𝜎(𝑠,𝑧(𝑠))]𝑑𝜔(𝑠)‖‖‖.(3.15) Therefore, we have 𝐸‖𝐿𝑧(𝑡+𝜏)−𝐿𝑧(𝑡)‖²≤3𝐸+‖‖‖^𝑡0[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝐵𝑓(𝑠+𝜏,𝑧(𝑠+𝜏))+𝐶𝑔(𝑠+𝜏,𝑧(𝑠+𝜏−ℎ))+𝐼(𝑠+𝜏)]𝑑𝑠_𝑡0𝑊(𝑡,𝑠){−[𝐵[𝐵𝑓𝑓(𝑠(+𝑠,𝑧𝜏,(𝑧𝑠)()𝑠++𝐶𝜏𝑔))(+𝑠,𝑧𝐶(𝑔𝑠(+𝑠+−ℎ𝜏,))𝑧+(𝑠𝐼+(𝑠𝜏)]−𝑑ℎ𝑠}))‖‖‖+₂𝐼(𝑠+𝜏)]+3𝐸+‖‖‖‖_{0≤𝑡𝑘<𝑡0≤𝑡𝑘<𝑡}𝑊𝑡+𝜏,𝑡_𝑘+𝑞−𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞+𝛽_𝑘+𝑞𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞−𝑉_𝑘𝑧𝑡_𝑘+𝛽_𝑘+𝑞−𝛽_𝑘‖‖‖‖²+3𝐸+‖‖‖^𝑡0[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))]𝑑𝜔(𝑠)_𝑡0𝑊(𝑡,𝑠)[𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))−𝜎(𝑠,𝑧(𝑠))]𝑑𝜔(𝑠)‖‖‖².(3.16) We first evaluate the first term of the right hand side 𝐸‖‖‖+^𝑡0[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝐵𝑓(𝑠+𝜏,𝑧(𝑠+𝜏))+𝐶𝑔(𝑠+𝜏,𝑧(𝑠+𝜏−ℎ))+𝐼(𝑠+𝜏)]𝑑𝑠_𝑡0𝑊(𝑡,−𝑠)[𝐵{[𝑓𝐵(𝑓𝑠,(𝑧𝑠(+𝑠)𝜏),+𝑧(𝐶𝑠𝑔+(𝑠𝜏,)𝑧)(+𝑠𝐶+𝑔−(ℎ𝑠)+)+𝜏,𝐼𝑧((𝑠𝑠)]+𝑑𝜏𝑠}−‖‖‖ℎ₂))+𝐼(𝑠+𝜏)]≤2𝐸×^𝑡0‖‖𝑊𝐵𝑓(𝑡(𝑠++𝜏,𝜏𝑠,𝑧+(𝑠𝜏)+−𝜏𝑊))+(𝑡𝐶,𝑠𝑔)‖(𝑠+𝜏,𝑧(𝑠+𝜏−ℎ))+𝐼(𝑠+𝜏)‖₂+2𝐸_𝑡0𝑑𝑠+𝐶‖(𝑊𝑔(𝑠(𝑡+,𝑠𝜏)‖,𝑧‖(𝐵𝑠+(𝑓𝜏(−𝑠+ℎ)𝜏),−𝑧(𝑔𝑠(+𝑠,𝜏𝑧)()𝑠−−𝑓ℎ()𝑠)),𝑧(𝐼(𝑠()𝑠)+)𝜏)−𝐼(𝑠))‖‖‖₂≤𝑐₁𝑑𝑠𝜀,(3.17) where 𝑐₁=(96𝜂²/𝜆²)[‖𝐵‖²𝐿^2𝑓⋅((𝐾/(1−𝜌))+1)+‖𝐶‖²𝐿^2𝑔⋅((𝐾/(1−𝜌))+1)+𝛾²⁰+1].
For the second term, we can estimate that 𝐸‖‖‖‖+_{0≤𝑡𝑘<𝑡}𝑊_{0≤𝑡𝑘<𝑡}𝑡+𝜏,𝑡_𝑘+𝑞−𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞+𝛽_𝑘+𝑞𝑊𝑡,𝑡_𝑘𝑉_𝑘+𝑞𝑧𝑡_𝑘+𝑞−𝑉_𝑘𝑧𝑡_𝑘+𝛽_𝑘+𝑞−𝛽_𝑘‖‖‖‖²≤2𝐸‖‖‖‖_{0≤𝑡𝑘<𝑡}𝑊(𝑡+𝜏,𝑡_𝑘+𝑞)−𝑊(𝑡,𝑡_𝑘)𝑉_𝑘+𝑞(𝑧(𝑡_𝑘+𝑞))+𝛽_𝑘+𝑞‖‖‖‖²+2𝐸‖‖‖‖_{0≤𝑡𝑘<𝑡}𝑊(𝑡,𝑡_𝑘)[𝑉_𝑘+𝑞(𝑧(𝑡_𝑘+𝑞))−𝑉_𝑘(𝑧(𝑡_𝑘))+𝛽_𝑘+𝑞−𝛽_𝑘]‖‖‖‖²≤𝑐₂𝜀,(3.18) where 𝑐₂=(8𝜂²𝑁²/(1−𝑒^−𝜆))[𝐿^2𝑉⋅((𝐾/(1−𝜌))+1)+𝛾²⁰+1].
For the last term, using (A4) and Itô isometry identity, we have 𝐸+‖‖‖^𝑡0[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)][𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))]𝑑𝜔(𝑠)_𝑡0𝑊(𝑡,𝑠)[𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))−𝜎(𝑠,𝑧(𝑠)]𝑑𝜔(𝑠)‖‖‖²≤2𝐸‖‖‖^𝑡0[𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠),𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))]𝑑𝜔(𝑠)‖‖‖²+2𝐸‖‖‖^𝑡0𝑊(𝑡,𝑠)[𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))−𝜎(𝑠,𝑧(𝑠))]𝑑𝜔(𝑠)‖‖‖²≤+2𝐸2𝐸^𝑡0‖_𝑡0𝑊(𝑡+𝜏,𝑠+𝜏)−𝑊(𝑡,𝑠)‖²‖𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))‖²𝑑𝑠≤𝑐₃‖𝑊(𝑡,𝑠)‖²‖𝜎(𝑠+𝜏,𝑧(𝑠+𝜏))−𝜎(𝑠,𝑧(𝑠))‖²𝑑𝑠𝜀,(3.19) where 𝑐₃=(2/𝜆)[Γ²𝐿^2𝜎(𝐾/(1−𝜌))+1].
Combining (3.17), (3.18) and (3.19), it follows that𝐸‖𝐿𝑧(𝑡+𝜏)−𝐿𝑧(𝑡)‖²≤𝑐₀𝜀,(3.20) where 𝑐₀=3(𝑐₁+𝑐₂+𝑐₃).
So, 𝐿𝑧∈𝐷^∗, that is 𝐿 is self-mapping from 𝐷^∗ to 𝐷^∗ by (3.13) and (3.20).
Secondly, we will show 𝐿 is contracting operator in 𝐷^∗.
For ∀𝑥,𝑦∈𝐷^∗, ‖𝐿𝑥−𝐿𝑦‖=‖‖‖++^𝑡0𝐶𝑊[𝑔(𝑡(,𝑠𝑠,)𝑥𝐵(𝑠[𝑓−(ℎ𝑠),)𝑥−(𝑠𝑔))(𝑠−,𝑦𝑓((𝑠𝑠−,𝑦ℎ(𝑠)))])]𝑑𝑠_{0≤𝑡𝑘<𝑡}+𝑊𝑡,𝑡_𝑘𝑉_𝑘𝑥𝑡_𝑘−𝑉_𝑘𝑦𝑡_𝑘_𝑡0𝑊(𝑡,𝑠)[𝜎(𝑠,𝑥(𝑠))−𝜎(𝑠,𝑦(𝑠))]𝑑𝜔(𝑠)‖‖‖.(3.21) By a minor modification of the proof of (3.13), we can obtain 𝐸‖𝐿𝑥−𝐿𝑦‖²≤6𝜂²𝜆2²‖𝐵‖²𝐿^2𝑓+‖𝐶‖²𝐿^2𝑔+𝑁²(1−𝑒^−𝜆)²𝐿^2𝑉+𝐿^2𝜎2𝜆sup_𝑡𝐸‖𝑥(𝑡)−𝑦(𝑡)‖²=𝜌‖𝑥−𝑦‖^2∞,(3.22) and therefore, ‖𝐿𝑥−𝐿𝑦‖_∞≤𝜌‖𝑥−𝑦‖_∞, it follows that 𝐿 is contracting operator in 𝐷^∗, so there exists a unique mean square almost periodic solution of (2.2) by the fixed points theorem.

4. Example

Consider the following impulsive stochastic differential equation with delay𝑑𝑥_𝑖(𝑡)=𝑎_𝑖𝑥_𝑖(𝑡)+²_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+²_𝑗=1+0.5𝑥_𝑖𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡−0.1)+𝐼_𝑖(𝑡)𝑑𝑡Δ𝑥(𝑡)=𝑥𝑡_𝑘(𝑡)𝑑𝜔_𝑖(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,𝑥(𝑡)=𝜙(𝑡),−𝑥−ℎ𝑡^−𝑘≤=𝑡≤𝐷0_𝑘𝑥,𝑡^−𝑘+𝑉_𝑘𝑥𝑡^−𝑘+𝛽_𝑘,𝑡=𝑡_𝑘,𝑘∈ℕ,(4.1) where [𝑡_𝑘0.0=1s𝑘i,n𝑘𝑥₁∈ℕ,𝑓(𝑥(𝑡))=[sin𝑥₁(𝑡),sin𝑥₂(𝑡)]^𝑇,𝑔(𝑥(𝑡−0.1))=[cos𝑥₁(𝑡−0.1),cos𝑥₂(𝑡−0.1)]^𝑇,𝑉_𝑖𝑘=(𝑡),0.01cos𝑥₂(𝑡)]^𝑇,𝛽_𝑘=0.1,𝐼(𝑡)=[0.1,0.1]^𝑇,𝛾₀=0.1, for convenience, we can choose𝐴=⎡⎢⎢⎣−02−03⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣−00.1−00.1⎤⎥⎥⎦,𝐶=⎡⎢⎢⎣00.2−00.2⎤⎥⎥⎦,𝐷_𝑘=⎡⎢⎢⎣−00.5−00.5⎤⎥⎥⎦.(4.2) Then 𝜇(𝐴)=−2,‖𝐼+𝐷_𝑘‖=1/2,‖𝐵‖=0.1,‖𝐶‖=0.2,𝐿_𝑓=𝐿_𝑔=1,𝐿_𝑉=0.01,𝐿_𝜎=0.5. Choose 𝜃=inf_𝑘{𝑡^1𝑘}=0.01,𝜂=1,𝑁=6. By simple calculation, we have 𝜆=−(𝜇(𝐴)+(ln𝜂/𝜃))=2,𝜌≐0.8139<1,𝐾≐1.107,(𝜌𝐾/(1−𝜌))≐4.841.

Let 𝐷^∗={𝑧∈𝐷∶𝐸‖𝑧−𝑧₀‖²≤4.841}, so, by Theorem 3.1, system (4.1) has a unique mean square almost periodic solution in 𝐷^∗.

Remark 4.1. Since there exist no results for almost periodic solutions for impulsive stochastic differential equations with delays, one can easily see that all the results in [10, 11, 20–22, 28] and the references therein cannot be applicable to system (4.1). This implies that the results of this paper are essentially new.

5. Conclusion

In this paper, a class of Itô impulsive stochastic differential equations with delays has been investigated. We conquer the difficulty of coexistence of impulsive, delay and stochastic factors in a dynamic system, and give a result for the existence and uniqueness of mean square almost periodic solutions. The results in this paper extend some earlier works reported in the literature. Moreover, our results have important applications in almost periodic oscillatory stochastic delayed neural networks with impulsive control.

Acknowledgment

This work is supported by the National Science Foundation of China (no. 10771199).