Impulsive systems arise naturally in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time. Impulsive stochastic modeling has come to play an important role in many branches of science where more and more people have encountered impulsive stochastic differential equations. For example, a stochastic model for drug distribution in a biological system was described by Tsokos and Padgett [1] as a closed system with a simplified heart, one organ, or capillary bed, and recirculation of blood with a constant rate of flow, where the heart is considered as a mixing chamber of constant volume. Recently, there has been a significant development in impulsive stochastic differential equations (ISDEs). The existence and stability of ISDEs were investigated in [2–11] and the references therein.

On the other hand, in recent years, there has been considerable interest in studying fractional Brownian motions (fBms) due to their compact properties and applications in various scientific areas, including telecommunications [12, 13], turbulence [14], image processing [15], and finance [16]. Stochastic differential equations (SDEs) driven by fBms attract the interest of researchers [2, 3, 17–21]. Taking the time delay into account, the theory of stochastic differential equations has been generalized to stochastic functional differential equations; it makes the dynamics more complex and the system may lose stability and show almost periodicity. Arthi et al. [2] considered the existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by fractional Brownian motion (fBm), and Caraballo [3] studied the existence of mild solutions to stochastic delay evolution equations with fBm and impulses.

In this paper, we are concerned with the existence and stability of almost periodic mild solutions to the following impulsive stochastic functional differential system driven by fBm with Hurst index H ∈ ( 1 2 , 1 ) : d x t = A x t + b t , x t d t + σ t d B H t , t ≠ ± t i , i ∈ Z , △ x t i = x t i + − x t i − = I i x t i , i ∈ Z , x t 0 = ξ = ξ t : − θ ≤ t ≤ 0 , (1) where Z is the set of integer, for any i , k ∈ Z , and the sequence {t _i } is such that the derived sequence { t i k = t i + k − t i } is equipotentially almost periodic. Moreover, A : D ( A ) ⊂ H → H is a linear bounded operator, ρ(A) is the resolvent set of A, and for λ ∈ ρ(A), R(λ, A) is the resolvent of A. In addition, b, σ, and I _i are appropriate functions, x t ( ⋅ ) : [ − θ , 0 ] → H is given by x _t (s) = x(t + s), for any s ∈ [ − θ, 0], and ξ ∈ C θ is an F t 0 − measurable random variable such that E‖ξ‖² < ∞. Let θ > 0 be a given constant and let

C θ = ϕ : [ − θ , 0 ] × Ω → H , ϕ be continuous everywhere except for a finite number of points s at which ϕ(s ⁻) and ϕ(s ⁺) exist and satisfy ϕ ( s − ) = ϕ ( s ) , endowed with the norm ‖ ϕ ‖ C θ = sup − θ ≤ s ≤ 0 E ‖ ϕ s ‖ 2 1 2 , such that ϕ(s, ⋅) is F 0 -measurable for each s ∈ [ − θ, 0] and sup s ∈ [ − θ , 0 ] E ‖ ϕ ( s ) ‖ 2 < ∞ .

There are several difficulties with our problems. First, there is the delay for the impulsive stochastic differential equations. Second, about the stochastic differential equations driven by fractional Brownian motion, the classical stochastic integral failed for lack of the martingale property. Third, there is no strong solution for stochastic partial delay differential equations driven by fractional Brownian motion. The lifting space method, mild solutions, fixed point theorem, and semigroup theory will be used to overcome these difficulties.

The paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. Section 3 is devoted to stating the existence and uniqueness of the mild square-mean piecewise almost periodic solution to (1). In Section 4, we show the stability of the mild square-mean piecewise almost periodic solution. An example is provided to illustrate the effectiveness of the results.

Let ( H , ‖ ⋅ ‖ H , ( ⋅ , ⋅ ) H ) and ( K , ‖ ⋅ ‖ K , ( ⋅ , ⋅ ) K ) denote two real separable Hilbert spaces. We denote by L ( H , K ) the set of all linear bounded operators from H into K , equipped with the usual operator norm ‖ ⋅‖ and use |⋅| to denote the Euclidean norm of a vector. In this article, we use the symbol ‖⋅‖ to denote the norms of operators regardless of the spaces involved when no confusion possibly arises. Let ( Ω , F , { F t } t ≥ 0 , P ) be a filtered complete probability space satisfying the usual condition.

In this section, we study the existence of the square-mean piecewise almost periodic solution to (1). We first present some assumptions as follows:

(H1) Let the bounded linear operator A be an infinitesimal generator of an analytic semigroup {S(t), t ≥ 0} such that

Recall the notion of a mild solution for Eq. 1.

Definition 3.1

An F t -progressive process { x ( t ) } t ∈ R is called a mild solution of the system (1) on R if it satisfies the corresponding stochastic integral equation x t = S t x 0 + ∫ t 0 t S t − s b s , x s d s + ∫ t 0 t S t − s σ s d B H s + ∑ t 0 < t < t i S t − t i I i x t i , (7) for all t ≥ t ₀ and for each t 0 ∈ R .

Theorem 3.1

Let (H1) − (H3) be satisfied. Then, (1) has a unique square-mean piecewise almost periodic mild solution whenever Θ ≔ M 2 M b γ 2 + M 2 M I 1 − e − γ α 2 < 1 . (8) Consider the following equation: x t = ∫ − ∞ t S t − s b s , x s d s + ∫ − ∞ t S t − s σ s d B H s + ∑ t i < t S t − t i I i x t i , with t ≥ t ₀. It is easy to verify that the above equation is equivalent to (7). Define the operator L on A P T ( R , L 2 ( Ω , H ) ) by L x t : = ∫ − ∞ t S t − s b s , x s d s + ∫ − ∞ t S t − s σ s d B H s + ∑ t i < t S t − t i I i x t i ≡ Φ 1 t + Φ 2 t + Φ 3 t , for all t ∈ R . To prove the theorem, it is sufficient to show that the next statements hold:

I) L x ( t ) is square-mean piecewise almost periodic.

Proof of Statement (I)This will be done in two steps.

Step 1

We claim that L x ( t ) ∈ U P C .

Let i ∈ Z . By the uniform continuity of S(t), we can see that, for any ɛ > 0, there exists a number δ > 0 between 0 and min { ε b ̃ , ε σ ̃ 2 H } such that ‖ S t ″ − t ′ − I ‖ 2 ≤ min γ 2 ε b ̃ , γ 2 H ε H 2 H σ ̃ , 1 − e − γ α 2 ε I ̃ , (9) for all t′, t″ ∈ (t _i , t _i+1), t′ < t″ as 0 < t″ − t′ < δ, where b ̃ = 36 M 2 ‖ b ‖ ∞ 2 , σ ̃ = 36 H ( 2 H − 1 ) M 2 ‖ σ ‖ ∞ 2 , I ̃ = 9 M 2 ‖ I ‖ ∞ 2 . It follows from the inequality |a + b + c|³ ≤ 3(a ² + b ² + c ²) that E ‖ L x t ′ − L x t ″ ‖ 2 ≤ 3 E ‖ Φ 1 t ′ − Φ 1 t ″ ‖ 2 + 3 E ‖ Φ 2 t ′ − Φ 2 t ″ ‖ 2 + 3 E ‖ Φ 3 t ′ − Φ 3 t ″ ‖ 2 , for all t′, t″ ∈ (t _i , t _i+1), t′ < t″. By the assumptions (H1), (H2), and (H3), we have that E ‖ ∫ t ′ t ″ S t ″ − s b s , x s d s ‖ 2 ≤ M 2 ∫ t ′ t ″ e − γ t ″ − s d s ∫ t ′ t ″ e − γ t ″ − r E ‖ b s , x r ‖ 2 d r ≤ M 2 t ″ − t ′ 2 sup t ∈ R E ‖ b t , x t ‖ 2 , (10) and E ‖ ∫ t ′ t ″ S t ″ − s σ s d B H s ‖ 2 ≤ H 2 H − 1 M 2 ‖ σ ‖ ∞ 2 ∫ t ′ t ″ e − γ t ″ − s H d s 2 H ≤ M 2 t ″ − t ′ 2 H sup t ∈ R E ‖ b t , x t ‖ 2 , (11) for all t′, t″ ∈ (t _i , t _i+1), t′ < t″. Moreover, we also have that E ‖ ∫ − ∞ t ′ S t ′ − s − S t ″ − s b s , x s d s ‖ 2 = E ‖ ∫ − ∞ t ′ I − S t ″ − t ′ S t ′ − s b s , x s d s ‖ 2 ≤ M 2 ‖ I − S t ″ − t ′ ‖ 2 ∫ − ∞ t ′ e − γ t ′ − s d s ∫ − ∞ t ′ e − γ t ′ − r E ‖ b s , x r ‖ 2 d r ≤ M 2 ‖ I − S t ″ − t ′ ‖ 2 γ − 2 sup t ∈ R E ‖ b t , x t ‖ 2 , (12) and E ‖ ∫ − ∞ t ′ S t ′ − s − S t ″ − s σ s d B H s ‖ 2 ≤ H 2 H − 1 M 2 ‖ I − S t ″ − t ′ ‖ 2 ∫ − ∞ t ′ e − γ t ′ − s H ‖ σ s ‖ L 0 2 1 H d s 2 H ≤ H 2 H − 1 M 2 ‖ I − S t ″ − t ′ ‖ 2 ‖ σ ‖ ∞ 2 H γ 2 H , (13) for all t′, t″ ∈ (t _i , t _i+1), t′ < t″. Combining these with Hölder’s inequality and (9), we get that E ‖ Φ 1 t ′ − Φ 1 t ″ ‖ 2 ≤ 2 M 2 γ 2 γ 2 ε b ̃ ‖ b ‖ ∞ 2 + 2 δ 2 M 2 ‖ b ‖ ∞ 2 ≤ ε 9 , and E ‖ Φ 2 t ′ − Φ 2 t ″ ‖ 2 ≤ 2 H 2 H − 1 M 2 ( ‖ I − S t ″ − t ′ ‖ 2 ‖ σ ‖ ∞ 2 H γ 2 H + ‖ σ ‖ ∞ 2 t ″ − t ′ 2 H ) ≤ ε 9 , for all t′, t″ ∈ (t _i , t _i+1), t′ < t″ provided |t″ − t′| < δ. Similarly, by the assumptions (H1) and (H3) and (9), one can see that E ‖ Φ 3 t ′ − Φ 3 t ″ ‖ 2 ≤ E ‖ ∑ t i < t ′ S t ′ − t i I i x t i − ∑ t i < t ″ S t ″ − t i I i x t i ‖ 2 ≤ E ‖ ∑ t i < t ″ S t ′ − t i I i x t i − S t ″ − t i I i x t i ‖ 2 ≤ M 2 ‖ I − S t ″ − t ′ ‖ 2 ∑ t i < t ′ e − γ t ′ − t i ∑ t i < t ′ e − γ t ′ − t i E ‖ I i x t i ‖ 2 ≤ M 2 ‖ I − S t ″ − t ′ ‖ 2 ∑ t i < t ′ e − γ t ′ − t i 2 ‖ I ‖ ∞ 2 ≤ M 2 ‖ I − S t ″ − t ′ ‖ 2 1 1 − e − γ α 2 ‖ I ‖ ∞ 2 ≤ ε 9 , for all t′, t″ ∈ (t _i , t _i+1), t′ < t″ provided |t″ − t′| < δ. Thus, we have shown that the estimate E ‖ L x t ′ − L x t ″ ‖ 2 < ε , holds for all t′, t″ ∈ (t _i , t _i+1), t′ < t″ provided |t″ − t′| < δ, which means L x ( t ) ∈ U P C .

Step 2

We prove the almost periodicity of L x ( t ) .

For Φ₁(t), let t _i < t < t _i+1; by (H1), (H2), and Hölder’s inequality, we have that E Φ 1 t + τ − Φ 1 t 2 = E ∫ − ∞ t + τ S t + τ − s b s , x s d s − ∫ − ∞ t S t − s b s , x s d s 2 = E ∫ − ∞ t S t − s b s + τ , x s + τ − b s , x s d s 2 ≤ E ∫ − ∞ t M e − γ t − s b s + τ , x s + τ − b s , x s d s 2 ≤ M 2 γ ∑ j = − ∞ i − 1 ∫ t j + η t j + 1 − η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s + ∑ j = − ∞ i − 1 ∫ t j t j + η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s + ∑ j = − ∞ i − 1 ∫ t j + 1 − η t j + 1 e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s + ∫ t i t i + η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s , where η = min { ε , α 2 } . By Lemma 2.1 and (H2), we find that for any ɛ > 0 and i ∈ Z , there exists a real number l(ɛ, Λ) > 0 such that every interval of length l(ɛ, Λ) contains at least a constant τ and E b t + τ , x t + τ − b t , x t 2 < ε , ∀ t ∈ R , for each x ∈ Λ, |t − t _i | > ɛ, since b ∈ AP T ( R , L 2 ( Ω , H ) ) , where Λ ⊂ C θ is compact.

For s ∈ [t _j + η, t _j+1 − η], j ∈ Z , j ≤ i, t − s ≥ t − t _i + t _i − (t _j+1 − η) ≥ t − t _i + α(i + j − 1) + η, we have ∑ j = − ∞ i − 1 ∫ t j + η t j + 1 − η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s ≤ ε ∑ j = − ∞ i − 1 ∫ t j + η t j + 1 − η e − γ t − s d s ≤ ε γ ∑ j = − ∞ i − 1 e − γ t − t j + 1 + η ≤ ε γ ∑ j = − ∞ i − 1 e − γ α i − j + 1 ≤ ε γ 1 − e − γ α .

For s ∈ [ t j , t j + η ] , j ∈ Z , j ≤ i , by the mean value theorem of integral, we get that ∑ j = − ∞ i − 1 ∫ t j t j + η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s ≤ 2 sup s ∈ R E ‖ b s , x s ‖ 2 ∑ j = − ∞ i − 1 ∫ t j t j + η e − γ t − s d s ≤ 2 ‖ b ‖ ∞ 2 ε e γ η ∑ j = − ∞ i − 1 e − γ t − t j ≤ 2 ‖ b ‖ ∞ 2 ε e γ η e γ t − t i ∑ j = − ∞ i − 1 e − γ α i − j ≤ 2 ‖ b ‖ ∞ 2 e γ 2 1 − e − γ α ε . Similarly, we can show that ∑ j = − ∞ i − 1 ∫ t j + 1 − η t j + 1 e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s ≤ C 1 ε , ∫ t i t i + η e − γ t − s E ‖ b s + τ , x s + τ − b s , x s ‖ 2 d s ≤ C 2 ε , where C ₁, C ₂ are two positive constants. Thus, we have introduced the next estimate: E Φ 1 t + τ − Φ 1 t 2 < N 1 ε , where N ₁ is a positive constant, which implies that Φ₁(t) is square-mean piecewise almost periodic.

We now show that Φ₂(t) is square-mean piecewise almost periodic. Recall that t↦σ(t) is piecewise almost periodic if for each ɛ > 0 there exists a real number l(ɛ) > 0 such that the estimate σ s + τ − σ s 2 < ε , ∀ t ∈ R , | t − t i | > ε , i ∈ Z (14) holds for every interval of length l(ɛ) containing a number τ. By using (H1) and the computation of fBm, we have E Φ 2 t + τ − Φ 2 t 2 = E ∫ − ∞ t + τ S t + τ − s σ s d B H s − ∫ − ∞ t S t − s σ s d B H s 2 = E ∫ − ∞ t S t − s σ s + τ d B H s + τ − ∫ − ∞ t S t − s σ s d B H s 2 = E ∫ − ∞ t S t − s σ s + τ − σ s d B ̃ H s 2 = H 2 H − 1 ∑ n = 1 ∞ ∫ − ∞ t ∫ − ∞ t S t − u σ u + τ − σ u Q 1 2 e n × S t − v σ v + τ − σ v Q 1 2 e n ⋅ u − v 2 H − 2 d u d v ≤ H 2 H − 1 M 2 ∑ n = 1 ∞ ∫ − ∞ t ∫ − ∞ t e − γ t − u σ u + τ − σ u Q 1 2 e n × e − γ t − v σ v + τ − σ v Q 1 2 e n ⋅ u − v 2 H − 2 d u d v ≤ H 2 H − 1 M 2 ∫ − ∞ t e − γ t − s σ s + τ − σ s L 2 0 1 H d s 2 H . Furthermore, by Hölder’s inequality, we have E Φ 2 t + τ − Φ 2 t 2 ≤ H 2 H − 1 M 2 ∫ − ∞ t e − γ t − s 2 H − 1 d s 2 H − 1 2 H ∫ − ∞ t e − γ t − s σ s + τ − σ s L 2 0 2 d s ≤ H 2 H − 1 M 2 2 H − 1 γ 2 H − 1 2 H ∑ j = − ∞ i − 1 ∫ t j + η t j + 1 − η e − γ t − s σ s + τ − σ s L 2 0 2 d s + ∑ j = − ∞ i − 1 ∫ t j t j + η e − γ t − s σ s + τ − σ s L 2 0 2 d s + ∑ j = − ∞ i − 1 ∫ t j + 1 − η t j + 1 e − γ t − s σ s + τ − σ s L 2 0 2 d s + ∫ t i t i + η e − γ t − s σ s + τ − σ s L 2 0 2 d s , where η = min { ε , α 2 } . In the same way as that of handling Φ₁(t), one can introduce the estimate E Φ 2 t + τ − Φ 2 t 2 < N 2 ε , where N ₂ is a positive constant, and hence Φ₂(t) is piecewise square-mean almost periodic.