Almost Periodic Solutions to Impulsive Stochastic Delay Differential Equations Driven by Fractional Brownian Motion With 12 < H < 1

In this article, we study the existence and uniqueness of square-mean piecewise almost periodic solutions to a class of impulsive stochastic functional differential equations driven by fractional Brownian motion. Moreover, the stability of the mild solution is obtained. To illustrate the results obtained in the paper, an impulsive stochastic functional differential equation driven by fractional Brownian motion is considered.

1 Introduction

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 $H\in (12,1)$:

$\left\{\begin{array}{cc}\hfill & dx\left(t\right)=\left[Ax\left(t\right)+b\left(t,x_t\right)\right]dt+\sigma \left(t\right)d{B}^H\left(t\right),t\ne \pm t_i,i\in \mathbb{Z},\hfill \\ \hfill & △x\left(t_i\right)=x\left(t_i^{+}\right)-x\left(t_i^{-}\right)=I_i\left(x\left(t_i\right)\right),i\in \mathbb{Z},\hfill \\ \hfill & x_{t_0}=\xi =\left\{\xi \left(t\right):-\theta \le t\le 0\right\},\hfill \end{array}\right.(1)$

where $\mathbb{Z}$ is the set of integer, for any $i,k\in \mathbb{Z}$, and the sequence {t_i} is such that the derived sequence $\{t_i^k=t_{i+k}-t_i\}$ is equipotentially almost periodic. Moreover, $A:D(A)\subset \mathbb{H}\to \mathbb{H}$ 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, $x_t(\cdot ):[-\theta ,0]\to \mathbb{H}$ is given by x_t(s) = x(t + s), for any s ∈ [ − θ, 0], and $\xi \in {\mathcal{C}}_{\theta }$ is an ${\mathcal{F}}_{t_0}-$measurable random variable such that E‖ξ‖² < ∞. Let θ > 0 be a given constant and let

${\mathcal{C}}_{\theta }=\left\{\varphi :[-\theta ,0]\times \mathrm{\Omega }\to \mathbb{H},\varphi \right.$ be continuous everywhere except for a finite number of points s at which ϕ(s⁻) and ϕ(s⁺) exist and satisfy $\left.\varphi (s^{-})=\varphi (s)\right\}$, endowed with the norm

$\Vert \varphi {\Vert }_{{\mathcal{C}}_{\theta }}=\underset{-\theta \le s\le 0}{\sup }{\left(E\Vert \varphi \left(s\right){\Vert }^2\right)}^{\frac{1}{2}},$

such that ϕ(s, ⋅) is ${\mathcal{F}}_0$-measurable for each s ∈ [ − θ, 0] and $\underset{s\in [-\theta ,0]}{\sup }E\Vert \varphi (s){\Vert }^2<\infty$.

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.

2 Preliminaries

Let $(\mathbb{H},\Vert \cdot {\Vert }_{\mathbb{H}},{(\cdot ,\cdot )}_{\mathbb{H}})$ and $(\mathbb{K},\Vert \cdot {\Vert }_{\mathbb{K}},{(\cdot ,\cdot )}_{\mathbb{K}})$ denote two real separable Hilbert spaces. We denote by $\mathbb{L}(\mathbb{H},\mathbb{K})$ the set of all linear bounded operators from $\mathbb{H}$ into $\mathbb{K}$, 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 $(\mathrm{\Omega },\mathcal{F},{\{{\mathcal{F}}_t\}}_{t\ge 0},P)$ be a filtered complete probability space satisfying the usual condition.

2.1 Fractional Brownian Motion

In this subsection, we briefly introduce some useful results about fBm and the corresponding stochastic integral taking values in a Hilbert space. For more details, refer to Hu [22], Mishura [23], Nualart [24], and references therein.

A real standard fractional Brownian motion $\{{\beta }^H(t),t\in \mathbb{R}\}$ with Hurst parameter H ∈ (0, 1) is a Gaussian process with continuous sample paths such that E[β^H(t)] = 0 and

$E\left[{\beta }^H\left(t\right){\beta }^H\left(s\right)\right]=\frac{1}{2}\left(|t{|}^{2H}+|s{|}^{2H}-|t-s{|}^{2H}\right),$

for all s, t ≥ 0. It is known that fBm {β^H(t), t ≥ 0} with $H>\frac{1}{2}$ admits the following Wiener integral representation:

${\beta }^H\left(t\right)={\int }_0^t{K}_H\left(t,s\right)dW\left(s\right),$

where W is a standard Brownian motion and the kernel K_H(t, s) is given by

$K_H\left(t,s\right)=c_H{\int }_s^t{\left(u-s\right)}^{H-\frac{3}{2}}{\left(\frac{u}{s}\right)}^{H-\frac{1}{2}}du,s<t,$

where c_H > 0 is a constant satisfying $E{({\beta }_1^H)}^2=1$. For any function σ ∈ L²(0, T), the Wiener integral of σ with respect to β^H is defined by

${\int }_0^T\sigma \left(s\right)d{\beta }^H\left(s\right)={\int }_0^T{K}_H^{*}\sigma \left(s\right)dW\left(s\right),$

for any T > 0, where $K_H^{*}\sigma (s)={\int }_s^T\frac{\partial K_H}{\partial r}(r,s)dr$. A $\mathbb{K}$-valued, ${\mathcal{F}}_t$-adapted fBm B^H with Hurst index H can be defined by

$B^H\left(t\right)=\sum _{n=1}^{\infty }\sqrt{{\lambda }_n}{e}_n{\beta }_n^H\left(t\right),$

where ${\beta }_n^H,n=\mathrm{1,2},\cdot \cdot \cdot$ are independent fBms with the same Hurst parameter $H\in (\frac{1}{2},1),\{e_n,n\in N\}$, which is a complete orthonormal basis in $\mathbb{K},\{{\lambda }_n,n\in N\}$ that is a bounded sequence of non-negative real numbers satisfying Qe_n = λ_ne_n, and Q is non-negative self-adjoint trace class operator with $TrQ=\sum _{n=1}^{\infty }{\lambda }_n<+\infty$.

Let ${\mathbb{L}}_2^0(\mathbb{H},\mathbb{K})$ denote the space of all $\sigma \in \mathbb{L}(\mathbb{H},\mathbb{K})$ such that $\sigma Q^{\frac{1}{2}}$ is a Hilbert–Schmidt operator. The norm is defined by $\Vert \sigma {\Vert }_{{\mathbb{L}}_2^0}^2=\sum _{n=1}^{\infty }\Vert \sqrt{{\lambda }_n}\sigma e_n{\Vert }^2$. Generally, σ is called a Q-Hilbert–Schmidt operator from $\mathbb{H}$ to $\mathbb{K}$.

Definition 2.1 Let $\sigma :[0,T]\to L_2^0(\mathbb{H},\mathbb{K})$ such that

$\sum _{n=1}^{\infty }\Vert K_H^{*}\left(\sigma e_n\right){\Vert }_{{\mathbb{L}}_2^0}^2<\infty ,$

then the stochastic integral of σ with respect to fBm B^H is defined by

${\int }_0^t\sigma \left(s\right)d{B}^H\left(s\right)≔\sum _{n=1}^{\infty }{\int }_0^t\sigma \left(s\right)Q^{\frac{1}{2}}{e}_{n}d{\beta }^H\left(s\right)=\sum _{n=1}^{\infty }{\int }_0^t\left(K_H^{*}\left(\sigma \left(s\right)Q^{\frac{1}{2}}{e}_n\right)\right)\left(s\right)dW\left(s\right).$

Remark If ${\{{\lambda }_n\}}_{n\in \mathbb{N}}$ is a bounded sequence of non-negative real numbers such that the nuclear operator Q satisfies Qe_n = λe_n, assuming that there exists a positive constant K_σ such that $\Vert \sigma {\Vert }_{{\mathbb{L}}_2^0}^2\le K_{\sigma }$ uniformly in [0, T], then it is obvious that $\sum _{n=1}^{\infty }\Vert \sigma Q^{\frac{1}{2}}{e}_n\left.\right){\Vert }^2$ is uniformly convergent for t ∈ [0, T].

2.2 Piecewise Almost Periodic Stochastic Processes

In this subsection, we recall some notations about the square-mean piecewise almost periodic stochastic process and introduce some lemmas. For further details, we refer to Takens and Teissier [25] and Liu [26].

Recall that a stochastic process $X:\mathbb{R}\to L^2(\mathrm{\Omega };\mathbb{H})$ is said to be continuous if

$\underset{t\to s}{\lim }E\Vert X\left(t\right)-X\left(s\right){\Vert }^2=0,$

for all $s\in \mathbb{R}$, and it is said to be bounded if there exists N > 0, such that E‖X(t)‖² ≤ N for all $t\in \mathbb{R}$. For convenience, we list the following concepts and notations:

• $L^2(\mathrm{\Omega },\mathbb{H})$ is Banach space when it is equipped with norm $\Vert \cdot {\Vert }_{L^2(\mathrm{\Omega },\mathbb{H})}$.

• Let $\mathcal{T}$ be the set consisting of all real sequences ${\{t_i\}}_{i\in \mathbb{Z}}$ such that $\alpha ={\inf }_{i\in \mathbb{Z}}(t_{i+1}-t_i)>0$, and $\underset{i\to \infty }{\lim }{t}_i=\infty ,\underset{i\to -\infty }{\lim }{t}_i=-\infty$, $x(t_i^{-})$,and $x(t_i^{+})$ represent the left and right limits of x(t) at the point $t_i,i\in \mathbb{Z}$, respectively.

• Let $PC(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))$ be the space consisting of all stochastically bounded functions $b:\mathbb{R}\to L^2(\mathrm{\Omega },\mathbb{H})$ such that b(⋅) is stochastically continuous at t for any $t\notin {\{t_i\}}_{i\in \mathbb{Z}}$, and $b(t_i)=b(t_i^{-})$ for all $i\in \mathbb{Z}$, ${\{t_i\}}_{i\in \mathbb{Z}}\in \mathcal{T}$.

• Let $PC(\mathbb{R}\times {\mathcal{C}}_{\theta },L^2(\mathrm{\Omega },\mathbb{H}))$ be the space of all piecewise stochastic process $b:\mathbb{R}\times {\mathcal{C}}_{\theta }\to L^2(\mathrm{\Omega },\mathbb{H})$ such that

• for any $\varphi \in {\mathcal{C}}_{\theta }$, b(⋅, ϕ) is stochastically continuous at point t for any $t\notin {\{t_i\}}_{i\in \mathbb{Z}}$ and $b(t_i,\varphi )=b(t_i^{-},\varphi )$ for all $i\in \mathbb{Z}$;

and b(t, ⋅) is stochastically continuous at $\varphi \in {\mathcal{C}}_{\theta }$, for $t\in \mathbb{R}$.

• For k < i, t − t_k = t − t_i + t_i − t_k ≥ t − t_i + (i − k)α, if ${\{t_i\}}_{i\in \mathbb{Z}}\in \mathcal{T}$, and t_i < t ≤ t_i+1 (see [27]).

Definition 2.2 ([28]). The family of the sequence $\{t_i^k=t_{i+k}-t_i\},i\in \mathbb{Z},k\in \mathbb{Z}$ will be called equipotentially almost periodic if for any ɛ > 0; there exists a relatively dense set ${\mathbb{Q}}_{\epsilon }$ of $\mathbb{R}$ and an integer $q\in \mathbb{Z}$ such that the inequality

$|t_{i+q}-t_i-\tau |<\epsilon ,(2)$

holds for each $\tau \in {\mathbb{Q}}_{\epsilon }$ and $i\in \mathbb{Z}$.

Definition 2.3 A function {b(t), t ≥ 0} is said to be square-mean piecewise almost periodic if the following conditions are fulfilled:

a) For any ɛ > 0, there exists a positive number δ = δ(ɛ) such that if the points t′ and t″ belong to the same interval of continuity and |t′ − t″| < δ, then E‖b(t′) − b(t″)‖² < ɛ.

b) For any ɛ > 0, there exists l(ɛ) > 0, such that every interval of length l(ɛ) contains a number τ with the property

$\underset{t\in \mathbb{R}}{\sup }E\Vert b\left(t+\tau \right)-b\left(t\right){\Vert }^2<\epsilon ,$

which satisfies the condition $|t-t_i|>\epsilon ,i\in \mathbb{Z}$.Let ${\mathcal{AP}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega };\mathbb{H}))$ denote the space of all square-mean piecewise almost periodic functions. Obviously ${\mathcal{AP}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega };\mathbb{H}))$ endowed with the supremum norm is a Banach space. Let $\mathcal{U}PC(\mathbb{R};L^2(\mathrm{\Omega };\mathbb{H}))$ be the space of all functions $b\in PC(\mathbb{R},L^2(\mathrm{\Omega };\mathbb{H}))$ such that b satisfies the condition (a) in Definition 2.3. It is easy to check that $\mathcal{U}PC(\mathbb{R};L^2(\mathrm{\Omega };\mathbb{H}))$ is a Banach space with the norm

$\Vert X{\Vert }_{\infty }=\underset{t\in \mathbb{R}}{\sup }{\left(E\Vert X\left(t\right){\Vert }^2\right)}^{\frac{1}{2}},$

for each $X\in \mathcal{U}PC(\mathbb{R};L^2(\mathrm{\Omega };\mathbb{H}))$.

Definition 2.4 (compare with [28]). A sequence $\{x_i\}:\mathbb{Z}\to L^2(\mathrm{\Omega },\mathbb{H})$ is called square-mean almost periodic if for any ɛ > 0, there exists a natural number N = N(ɛ) such that, for each $k\in \mathbb{Z}$, there is at least one integer p in the segment [k, k + N], for which inequality

$E\Vert x_{i+p}-x_i{\Vert }^2<\epsilon ,$

holds for all $i\in \mathbb{Z}$.

Definition 2.5 The function $b(t,\phi )\in PC(\mathbb{R}\times {\mathcal{C}}_{\theta },L^2(\mathrm{\Omega },\mathbb{H}))$ is said to be square-mean piecewise almost periodic in $t\in \mathbb{R}$ uniformly in φ ∈ Λ, where $\mathrm{\Lambda }\subseteq {\mathcal{C}}_{\theta }$ is compact if for any ɛ > 0, there exits l(ɛ, Λ) > 0 such that any interval of length l(ɛ, Λ) contains at least a number τ for which

$\underset{t\in \mathbb{R}}{\sup }E\Vert b\left(t+\tau ,x\right)-b\left(t,x\right){\Vert }^2<\epsilon ,$

for each x ∈ Λ, $t\in \mathbb{R}$, satisfying |t − t_i| > ɛ. The collection of all such processes is denoted by ${\mathcal{AP}}_{\mathcal{T}}(\mathbb{R}\times {\mathcal{C}}_{\theta },L^2(\mathrm{\Omega },\mathbb{H}))$.

Lemma 2.1 Let the function $f:\mathbb{R}\times {\mathcal{C}}_{\theta }\to L^2(\mathrm{\Omega };\mathbb{H})$ be square-mean piecewise almost periodic in $t\in \mathbb{R}$ uniformly for $y\in {\mathcal{C}}_{\theta }$, where $\mathrm{\Lambda }\subset {\mathcal{C}}_{\theta }$ is compact. If f is a Lipschitz function in the following sense,

$E\Vert f\left(t,y\right)-f\left(t,\tilde{y}\right)\Vert <M_2\left(\Vert y-\tilde{y}{\Vert }_{{\mathcal{C}}_{\theta }}\right),(3)$

for all $y,\tilde{y}\in {\mathcal{C}}_{\theta }$, $t\in \mathbb{R}$, and a constant M₂ > 0, then for any $\varphi (\cdot )\in {\mathcal{AP}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega };\mathbb{H})),f(\cdot ,{\varphi }_{\cdot })\in {\mathcal{AP}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega };\mathbb{H}))$.

Proof Noting that $\varphi (t):\mathbb{R}\to L^2(\mathrm{\Omega };\mathbb{H})$ is square-mean almost periodic, we can conclude that ϕ_t = {ϕ(t + s), − θ ≤ s ≤ 0, θ > 0} is square-mean almost periodic by Theorem 1.2.7 of [29]. Thus, for each ɛ > 0, there exists a constant l(ɛ) > 0 such that every interval with the length l(ɛ) contains a number τ satisfying

$E\Vert {\varphi }_{t+\tau }-{\varphi }_t{\Vert }_{{\mathcal{C}}_{\theta }}^2\le \frac{\epsilon }{4M^2},\forall t\in \mathbb{R}.(4)$

Noting that $f:\mathbb{R}\times {\mathcal{C}}_{\theta }\to L^2(\mathrm{\Omega };\mathbb{H})$ is square-mean piecewise almost periodic, we can see that for any ɛ > 0, there exits l(ɛ, Λ) > 0 such that each interval with length l(ɛ, Λ) contains at least a number τ satisfying

$E\Vert f\left(t+\tau ,{\varphi }_t\right)-f\left(t,{\varphi }_t\right){\Vert }^2\le \frac{\epsilon }{4},\forall t\in \mathbb{R},(5)$

for any $x\in \mathrm{\Lambda }(\subseteq {\mathcal{C}}_{\theta })$, $t\in \mathbb{R}$ with |t − t_i| > ɛ. Using the elementary inequality |a + b|² ≤ 2(|a|² + |b|²) and condition (3), we have

$\begin{array}{ll}\hfill & E\Vert f\left(t+\tau ,{\varphi }_{t+\tau }\right)-f\left(t,{\varphi }_t\right){\Vert }^2\hfill \\ \hfill & \le 2E\Vert f\left(t+\tau ,{\varphi }_{t+\tau }\right)-f\left(t,{\varphi }_{t+\tau }\right){\Vert }^2+2E\Vert f\left(t,{\varphi }_{t+\tau }\right)-f\left(t,{\varphi }_t\right){\Vert }^2\hfill \\ \hfill & \le 2E\Vert f\left(t+\tau ,{\varphi }_{t+\tau }\right)-f\left(t,{\varphi }_{t+\tau }\right){\Vert }^2+2M_2^{2}E\Vert {\varphi }_{t+\tau }-{\varphi }_t{\Vert }_{{\mathcal{C}}_{\theta }}^2,\hfill \end{array}$

for all $t\in \mathbb{R}$. Combining (4) and (5), one can show that

$\underset{t\in \mathbb{R}}{\sup }E\Vert f\left(t+\tau ,{\varphi }_{t+\tau }\right)-f\left(t,{\varphi }_t\right){\Vert }^2\le 2\cdot \frac{\epsilon }{4}+2M^2\cdot \frac{\epsilon }{4M^2}\le \epsilon ,$

which implies that f(t, ϕ_t) is square-mean piecewise almost periodic.

3 Existence of Square-Mean Piecewise Almost Periodic Solution

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

$\Vert S\left(t\right)\Vert \le M{e}^{-\gamma t},t\ge 0(6)$

for some γ > 0, M > 0. Moreover, R(λ, A) is almost periodic, where λ ∈ ρ(A).

(H2) Let $b\in {\mathcal{AP}}_{\mathcal{T}}(\mathbb{R}\times {\mathcal{C}}_{\theta },L^2(\mathrm{\Omega },\mathbb{H}))$. Moreover, there exists a positive constant M_b such that

$E\Vert b\left(t,x\right)-b\left(t,\tilde{x}\right){\Vert }^2\le M_b\Vert x-\tilde{x}{\Vert }_{{\mathcal{C}}_{\theta }}^2,$

for any $x,\tilde{x}\in {\mathcal{C}}_{\theta }$.

(H3) Let $\sigma \in {\mathcal{AP}}_{\mathcal{T}}(\mathbb{R}{L}_2^0(\mathrm{\Omega },L^2(\mathrm{\Omega },\mathbb{H})))$ and let $\{I_i\left(\right.x(t_i),i\in \mathbb{Z}\}$ be a square-mean piecewise almost periodic sequence satisfying

$E{‖I_i\left(x\right)-I_i\left(y\right)‖}^2\le M_{I}E{‖x-y‖}^2.$

for some positive constant M_I.

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

Definition 3.1 An ${\mathcal{F}}_t$-progressive process ${\{x(t)\}}_{t\in \mathbb{R}}$ is called a mild solution of the system (1) on $\mathbb{R}$ if it satisfies the corresponding stochastic integral equation

$\begin{array}{cc}\hfill x\left(t\right)=S\left(t\right)x_0+& {\int }_{t_0}^{t}S\left(t-s\right)b\left(s,x_s\right)ds\hfill \\ \hfill & +{\int }_{t_0}^{t}S\left(t-s\right)\sigma \left(s\right)d{B}^H\left(s\right)+\sum _{t_0<t<t_i}S\left(t-t_i\right)I_i\left(x\left(t_i\right)\right),\hfill \end{array}(7)$

for all t ≥ t₀ and for each $t_0\in \mathbb{R}$.

Theorem 3.1 Let (H1) − (H3) be satisfied. Then, (1) has a unique square-mean piecewise almost periodic mild solution whenever

$\mathrm{\Theta }≔\frac{M^2{M}_b}{{\gamma }^2}+\frac{M^2{M}_I}{{\left(1-e^{-\gamma \alpha }\right)}^2}<1.(8)$

Consider the following equation:

$x\left(t\right)={\int }_{-\infty }^{t}S\left(t-s\right)b\left(s,x_s\right)ds+{\int }_{-\infty }^{t}S\left(t-s\right)\sigma \left(s\right)d{B}^H\left(s\right)+\sum _{t_i<t}S\left(t-t_i\right)I_i\left(x\left(t_i\right)\right),$

with t ≥ t₀. It is easy to verify that the above equation is equivalent to (7). Define the operator $\mathcal{L}$ on $\mathcal{A}{\mathcal{P}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))$ by

$\begin{array}{ll}\hfill \mathcal{L}x\left(t\right):& ={\int }_{-\infty }^{t}S\left(t-s\right)b\left(s,x_s\right)ds\hfill \\ \hfill & +{\int }_{-\infty }^{t}S\left(t-s\right)\sigma \left(s\right)d{B}^H\left(s\right)+\sum _{t_i<t}S\left(t-t_i\right)I_i\left(x\left(t_i\right)\right)\hfill \\ \hfill & \equiv {\mathrm{\Phi }}_1\left(t\right)+{\mathrm{\Phi }}_2\left(t\right)+{\mathrm{\Phi }}_3\left(t\right),\hfill \end{array}$

for all $t\in \mathbb{R}$. To prove the theorem, it is sufficient to show that the next statements hold:

I) $\mathcal{L}x(t)$ is square-mean piecewise almost periodic.

II) $\mathcal{L}$ admits a unique fixed point.

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

Step 1 We claim that $\mathcal{L}x(t)\in \mathcal{U}PC$.

Let $i\in \mathbb{Z}$. By the uniform continuity of S(t), we can see that, for any ɛ > 0, there exists a number δ > 0 between 0 and $\min \{\sqrt{\frac{\epsilon }{\tilde{b}}},\sqrt[2H]{\frac{\epsilon }{\tilde{\sigma }}}\}$ such that

$\Vert S\left(t^{″}-t^{\prime }\right)-I{\Vert }^2\le \min \left\{\frac{{\gamma }^2\epsilon }{\tilde{b}},\frac{{\gamma }^{2H}\epsilon }{H^{2H}\tilde{\sigma }},\frac{{\left(1-e^{-\gamma \alpha }\right)}^2\epsilon }{\tilde{I}}\right\},(9)$

for all t′, t″ ∈ (t_i, t_i+1), t′ < t″ as 0 < t″ − t′ < δ, where $\tilde{b}=36M^2\Vert b{\Vert }_{\infty }^2,\tilde{\sigma }=36H(2H-1)M^2\Vert \sigma {\Vert }_{\infty }^2,\tilde{I}=9M^2\Vert I{\Vert }_{\infty }^2$. It follows from the inequality |a + b + c|³ ≤ 3(a² + b² + c²) that

$\begin{array}{ll}\hfill E\Vert \mathcal{L}x\left(t^{\prime }\right)-\mathcal{L}x\left(t^{″}\right){\Vert }^2\le 3& E\Vert {\mathrm{\Phi }}_1\left(t^{\prime }\right)-{\mathrm{\Phi }}_1\left(t^{″}\right){\Vert }^2\hfill \\ \hfill & +3E\Vert {\mathrm{\Phi }}_2\left(t^{\prime }\right)-{\mathrm{\Phi }}_2\left(t^{″}\right){\Vert }^2+3E\Vert {\mathrm{\Phi }}_3\left(t^{\prime }\right)-{\mathrm{\Phi }}_3\left(t^{″}\right){\Vert }^2,\hfill \end{array}$

for all t′, t″ ∈ (t_i, t_i+1), t′ < t″. By the assumptions (H1), (H2), and (H3), we have that

$\begin{array}{cc}E\Vert {\int }_{t^{\prime }}^{t^{″}}S\left(t^{″}-s\right)b\left(s,x_s\right)ds{\Vert }^2\hfill & \le M^2{\int }_{t^{\prime }}^{t^{″}}{e}^{-\gamma \left(t^{″}-s\right)}ds{\int }_{t^{\prime }}^{t^{″}}{e}^{-\gamma \left(t^{″}-r\right)}E\Vert b\left(s,x_r\right){\Vert }^{2}dr\hfill \\ \hfill & \le M^2{\left(t^{″}-t^{\prime }\right)}^2\underset{t\in \mathbb{R}}{\sup }E\Vert b\left(t,x_t\right){\Vert }^2,\hfill \end{array}(10)$

and

$\begin{array}{cc}\hfill E\Vert {\int }_{t^{\prime }}^{t^{″}}S\left(t^{″}-s\right)\sigma \left(s\right)d{B}^H\left(s\right){\Vert }^2& \le H\left(2H-1\right)M^2\Vert \sigma {\Vert }_{\infty }^2{\left({\int }_{t^{\prime }}^{t^{″}}{e}^{-\frac{\gamma \left(t^{″}-s\right)}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le M^2{\left(t^{″}-t^{\prime }\right)}^{2H}\underset{t\in \mathbb{R}}{\sup }E\Vert b\left(t,x_t\right){\Vert }^2,\hfill \end{array}(11)$

for all t′, t″ ∈ (t_i, t_i+1), t′ < t″. Moreover, we also have that

$\begin{array}{cc}\hfill E\Vert {\int }_{-\infty }^{t^{\prime }}& \left[S\left(t^{\prime }-s\right)-S\left(t^{″}-s\right)\right]b\left(s,x_s\right)ds{\Vert }^2\hfill \\ \hfill & =E\Vert {\int }_{-\infty }^{t^{\prime }}\left[I-S\left(t^{″}-t^{\prime }\right)\right]S\left(t^{\prime }-s\right)b\left(s,x_s\right)ds{\Vert }^2\hfill \\ \hfill & \le M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2{\int }_{-\infty }^{t^{\prime }}{e}^{-\gamma \left(t^{\prime }-s\right)}ds{\int }_{-\infty }^{t^{\prime }}{e}^{-\gamma \left(t^{\prime }-r\right)}E\Vert b\left(s,x_r\right){\Vert }^{2}dr\hfill \\ \hfill & \le M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2{\gamma }^{-2}\underset{t\in \mathbb{R}}{\sup }E\Vert b\left(t,x_t\right){\Vert }^2,\hfill \end{array}(12)$

and

$\begin{array}{cc}\hfill E\Vert {\int }_{-\infty }^{t^{\prime }}& \left[S\left(t^{\prime }-s\right)-S\left(t^{″}-s\right)\right]\sigma \left(s\right)d{B}^H\left(s\right){\Vert }^2\hfill \\ \hfill & \le H\left(2H-1\right)M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2{\left({\int }_{-\infty }^{t^{\prime }}{e}^{-\frac{\gamma \left(t^{\prime }-s\right)}{H}}\Vert \sigma \left(s\right){\Vert }_{L_0^2}^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le H\left(2H-1\right)M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2\Vert \sigma {\Vert }_{\infty }^2{\left(\frac{H}{\gamma }\right)}^{2H},\hfill \end{array}(13)$

for all t′, t″ ∈ (t_i, t_i+1), t′ < t″. Combining these with Hölder’s inequality and (9), we get that

$E\Vert {\mathrm{\Phi }}_1\left(t^{\prime }\right)-{\mathrm{\Phi }}_1\left(t^{″}\right){\Vert }^2\le \frac{2M^2}{{\gamma }^2}\frac{{\gamma }^2\epsilon }{\tilde{b}}\Vert b{\Vert }_{\infty }^2+2{\delta }^2{M}^2\Vert b{\Vert }_{\infty }^2\le \frac{\epsilon }{9},$

and

$\begin{array}{ll}\hfill E& \Vert {\mathrm{\Phi }}_2\left(t^{\prime }\right)-{\mathrm{\Phi }}_2\left(t^{″}\right){\Vert }^2\hfill \\ \hfill & \le 2H\left(2H-1\right)M^2\left(\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2\Vert \sigma {\Vert }_{\infty }^2{\left(\frac{H}{\gamma }\right)}^{2H}+\Vert \sigma {\Vert }_{\infty }^2{\left(t^{″}-t^{\prime }\right)}^{2H}\right)\le \frac{\epsilon }{9},\hfill \end{array}$

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

$\begin{array}{ll}\hfill E\Vert {\mathrm{\Phi }}_3\left(t^{\prime }\right)& -{\mathrm{\Phi }}_3\left(t^{″}\right){\Vert }^2\le E\Vert \sum _{t_i<t^{\prime }}S\left(t^{\prime }-t_i\right)I_i\left(x\left(t_i\right)\right)-\sum _{t_i<t^{″}}S\left(t^{″}-t_i\right)I_i\left(x\left(t_i\right)\right){\Vert }^2\hfill \\ \hfill & \le E\Vert \sum _{t_i<t^{″}}\left[S\left(t^{\prime }-t_i\right)I_i\left(x\left(t_i\right)\right)-S\left(t^{″}-t_i\right)I_i\left(x\left(t_i\right)\right)\right]{\Vert }^2\hfill \\ \hfill & \le M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2\left(\sum _{t_i<t^{\prime }}{e}^{-\gamma \left(t^{\prime }-t_i\right)}\right)\left(\sum _{t_i<t^{\prime }}{e}^{-\gamma \left(t^{\prime }-t_i\right)}E\Vert I_i\left(x\left(t_i\right)\right){\Vert }^2\right)\hfill \\ \hfill & \le M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2{\left(\sum _{t_i<t^{\prime }}{e}^{-\gamma \left(t^{\prime }-t_i\right)}\right)}^2\Vert I{\Vert }_{\infty }^2\hfill \\ \hfill & \le M^2\Vert I-S\left(t^{″}-t^{\prime }\right){\Vert }^2{\left(\frac{1}{1-e^{-\gamma \alpha }}\right)}^2\Vert I{\Vert }_{\infty }^2\le \frac{\epsilon }{9},\hfill \end{array}$

for all t′, t″ ∈ (t_i, t_i+1), t′ < t″ provided |t″ − t′| < δ. Thus, we have shown that the estimate

$E\Vert \mathcal{L}x\left(t^{\prime }\right)-\mathcal{L}x\left(t^{″}\right){\Vert }^2<\epsilon ,$

holds for all t′, t″ ∈ (t_i, t_i+1), t′ < t″ provided |t″ − t′| < δ, which means $\mathcal{L}x(t)\in \mathcal{U}PC$.

Step 2 We prove the almost periodicity of $\mathcal{L}x(t)$.

For Φ₁(t), let t_i < t < t_i+1; by (H1), (H2), and Hölder’s inequality, we have that

$\begin{array}{ll}\hfill & E{‖{\mathrm{\Phi }}_1\left(t+\tau \right)-{\mathrm{\Phi }}_1\left(t\right)‖}^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t+\tau }S\left(t+\tau -s\right)b\left(s,x_s\right)ds-{\int }_{-\infty }^{t}S\left(t-s\right)b\left(s,x_s\right)ds‖}^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t}S\left(t-s\right)\left[b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right)\right]ds‖}^2\hfill \\ \hfill & \le E{\left({\int }_{-\infty }^{t}M{e}^{-\gamma \left(t-s\right)}‖b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right)‖ds\right)}^2\hfill \\ \hfill & \le \frac{M^2}{\gamma }\left(\sum _{j=-\infty }^{i-1}{\int }_{t_j+\eta }^{t_{j+1}-\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\right.\hfill \\ \hfill & +\sum _{j=-\infty }^{i-1}{\int }_{t_j}^{t_j+\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\hfill \\ \hfill & +\sum _{j=-\infty }^{i-1}{\int }_{t_j+1-\eta }^{t_{j+1}}{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\hfill \\ \hfill & \left.+{\int }_{t_i}^{t_i+\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\right),\hfill \end{array}$

where $\eta =\min \{\epsilon ,\frac{\alpha }{2}\}$. By Lemma 2.1 and (H2), we find that for any ɛ > 0 and $i\in \mathbb{Z}$, 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

$E{‖b\left(t+\tau ,x_{t+\tau }\right)-b\left(t,x_t\right)‖}^2<\epsilon ,\forall t\in \mathbb{R},$

for each x ∈ Λ, |t − t_i| > ɛ, since $b\in {\mathcal{AP}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))$, where $\mathrm{\Lambda }\subset {\mathcal{C}}_{\theta }$ is compact.

For s ∈ [t_j + η, t_j+1 − η], $j\in \mathbb{Z}$, j ≤ i, t − s ≥ t − t_i + t_i − (t_j+1 − η) ≥ t − t_i + α(i + j − 1) + η, we have

$\begin{array}{ll}\hfill & \sum _{j=-\infty }^{i-1}{\int }_{t_j+\eta }^{t_{j+1}-\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\hfill \\ \hfill & \le \epsilon \sum _{j=-\infty }^{i-1}{\int }_{t_j+\eta }^{t_{j+1}-\eta }{e}^{-\gamma \left(t-s\right)}ds\le \frac{\epsilon }{\gamma }\sum _{j=-\infty }^{i-1}{e}^{-\gamma \left(t-t_{j+1}+\eta \right)}\hfill \\ \hfill & \le \frac{\epsilon }{\gamma }\sum _{j=-\infty }^{i-1}{e}^{-\gamma \alpha \left(i-j+1\right)}\le \frac{\epsilon }{\gamma \left(1-e^{-\gamma \alpha }\right)}.\hfill \end{array}$

For $s\in [t_j,t_j+\eta ],j\in \mathbb{Z},j\le i$, by the mean value theorem of integral, we get that

$\begin{array}{ll}\hfill \sum _{j=-\infty }^{i-1}& {\int }_{t_j}^{t_j+\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\hfill \\ \hfill & \le 2\underset{s\in \mathbb{R}}{\sup }E\Vert b\left(s,x_s\right){\Vert }^2\sum _{j=-\infty }^{i-1}{\int }_{t_j}^{t_j+\eta }{e}^{-\gamma \left(t-s\right)}ds\le 2\Vert b{\Vert }_{\infty }^2\epsilon e^{\gamma \eta }\sum _{j=-\infty }^{i-1}{e}^{-\gamma \left(t-t_j\right)}\hfill \\ \hfill & \le 2\Vert b{\Vert }_{\infty }^2\epsilon e^{\gamma \eta }{e}^{\gamma \left(t-t_i\right)}\sum _{j=-\infty }^{i-1}{e}^{-\gamma \alpha \left(i-j\right)}\le \frac{2\Vert b{\Vert }_{\infty }^2{e}^{\frac{\gamma }{2}}}{1-e^{-\gamma \alpha }}\epsilon .\hfill \end{array}$

Similarly, we can show that

$\begin{array}{l}\hfill \sum _{j=-\infty }^{i-1}{\int }_{t_j+1-\eta }^{t_{j+1}}{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\le C_1\epsilon ,\\ \hfill {\int }_{t_i}^{t_i+\eta }{e}^{-\gamma \left(t-s\right)}E\Vert b\left(s+\tau ,x_{s+\tau }\right)-b\left(s,x_s\right){\Vert }^{2}ds\le C_2\epsilon ,\end{array}$

where C₁, C₂ are two positive constants. Thus, we have introduced the next estimate:

$E{‖{\mathrm{\Phi }}_1\left(t+\tau \right)-{\mathrm{\Phi }}_1\left(t\right)‖}^2<N_1\epsilon ,$

where N₁ 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

${‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}^2<\epsilon ,\forall t\in \mathbb{R},|t-t_i|>\epsilon ,i\in \mathbb{Z}(14)$

holds for every interval of length l(ɛ) containing a number τ. By using (H1) and the computation of fBm, we have

$\begin{array}{ll}\hfill & E{‖{\mathrm{\Phi }}_2\left(t+\tau \right)-{\mathrm{\Phi }}_2\left(t\right)‖}^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t+\tau }S\left(t+\tau -s\right)\sigma \left(s\right)d{B}^H\left(s\right)-{\int }_{-\infty }^{t}S\left(t-s\right)\sigma \left(s\right)d{B}^H\left(s\right)‖}^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t}S\left(t-s\right)\sigma \left(s+\tau \right)d{B}^H\left(s+\tau \right)-{\int }_{-\infty }^{t}S\left(t-s\right)\sigma \left(s\right)d{B}^H\left(s\right)‖}^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t}S\left(t-s\right)\left[\sigma \left(s+\tau \right)-\sigma \left(s\right)\right]d{\tilde{B}}^H\left(s\right)‖}^2\hfill \\ \hfill & =H\left(2H-1\right)\sum _{n=1}^{\infty }{\int }_{-\infty }^t{\int }_{-\infty }^t‖S\left(t-u\right)\left[\sigma \left(u+\tau \right)-\sigma \left(u\right)\right]Q^{\frac{1}{2}}{e}_n‖\hfill \\ \hfill & \times ‖S\left(t-v\right)\left[\sigma \left(v+\tau \right)-\sigma \left(v\right)\right]Q^{\frac{1}{2}}{e}_n‖\cdot {\left|u-v\right|}^{2H-2}dudv\hfill \\ \hfill & \le H\left(2H-1\right)M^2\sum _{n=1}^{\infty }{\int }_{-\infty }^t{\int }_{-\infty }^t{e}^{-\gamma \left(t-u\right)}‖\left[\sigma \left(u+\tau \right)-\sigma \left(u\right)\right]Q^{\frac{1}{2}}{e}_n‖\hfill \\ \hfill & \times e^{-\gamma \left(t-v\right)}‖\left[\sigma \left(v+\tau \right)-\sigma \left(v\right)\right]Q^{\frac{1}{2}}{e}_n‖\cdot {\left|u-v\right|}^{2H-2}dudv\hfill \\ \hfill & \le H\left(2H-1\right)M^2{\left({\int }_{-\infty }^t{\left(e^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}\right)}^{\frac{1}{H}}ds\right)}^{2H}.\hfill \end{array}$

Furthermore, by Hölder’s inequality, we have

$\begin{array}{ll}\hfill & E{‖{\mathrm{\Phi }}_2\left(t+\tau \right)-{\mathrm{\Phi }}_2\left(t\right)‖}^2\hfill \\ \hfill & \le H\left(2H-1\right)M^2{\left({\int }_{-\infty }^t{e}^{\frac{-\gamma \left(t-s\right)}{2H-1}}ds\right)}^{\frac{2H-1}{2H}}\left({\int }_{-\infty }^t{e}^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}^{2}ds\right)\hfill \\ \hfill & \le H\left(2H-1\right)M^2{\left(\frac{2H-1}{\gamma }\right)}^{\frac{2H-1}{2H}}\left(\sum _{j=-\infty }^{i-1}{\int }_{t_j+\eta }^{t_{j+1}-\eta }{e}^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}^{2}ds\right.\hfill \\ \hfill & +\sum _{j=-\infty }^{i-1}{\int }_{t_j}^{t_j+\eta }{e}^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}^{2}ds\hfill \\ \hfill & +\sum _{j=-\infty }^{i-1}{\int }_{t_j+1-\eta }^{t_{j+1}}{e}^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}^{2}ds\hfill \\ \hfill & \left.+{\int }_{t_i}^{t_i+\eta }{e}^{-\gamma \left(t-s\right)}{‖\sigma \left(s+\tau \right)-\sigma \left(s\right)‖}_{L_2^0}^{2}ds\right),\hfill \end{array}$

where $\eta =\min \{\epsilon ,\frac{\alpha }{2}\}$. In the same way as that of handling Φ₁(t), one can introduce the estimate

$E{‖{\mathrm{\Phi }}_2\left(t+\tau \right)-{\mathrm{\Phi }}_2\left(t\right)‖}^2<N_2\epsilon ,$

where N₂ is a positive constant, and hence Φ₂(t) is piecewise square-mean almost periodic.

For ${\mathrm{\Phi }}_3(t)={\sum }_{t_i<t}S(t-t_i)I_i(x(t_i)),i\in \mathbb{Z}$, let β_i = I_i(x(t_i)). For t_i < t ≤ t_i+1, |t − t_i| > ɛ, $|t-t_{i+1}|>\epsilon ,i\in \mathbb{Z}$, by (2), one has t_i+q+1 > t + τ > t_i+q. From (H3), it follows that β_i is a square-mean almost periodic sequence, for any ɛ > 0; there exists such a natural number N = N(ɛ) that, for an arbitrary $k\in \mathbb{Z}$, there is at least one integer p > 0 in the segment [k, k + N] such that the inequality

$E\Vert {\beta }_{i+p}-{\beta }_i{\Vert }^2<\epsilon ,$

holds for all $i\in \mathbb{Z}$. We get

$\begin{array}{ll}\hfill & E{‖{\mathrm{\Phi }}_3\left(t+\tau \right)-{\mathrm{\Phi }}_3\left(t\right)‖}^2\hfill \\ \hfill & =E{‖\sum _{t_i<t+\tau }S\left(t+\tau -t_i\right){\beta }_i-\sum _{t_i<t}S\left(t-t_i\right){\beta }_i‖}^2\hfill \\ \hfill & \le E{‖S\left(t-t_i\right)\left({\beta }_{i+q}-{\beta }_i\right)‖}^2\hfill \\ \hfill & \le M^2\sum _{t_i<t}{e}^{-\gamma \left(t-t_i\right)}\sum _{t_i<t}{e}^{-\gamma \left(t-t_i\right)}E\Vert {\beta }_{i+q}-{\beta }_i{\Vert }^2\hfill \\ \hfill & \le \frac{M^2\epsilon }{{\left(1-e^{-\gamma \alpha }\right)}^2},\hfill \end{array}$

which implies that ${\mathrm{\Phi }}_3(t)\in \mathcal{A}{\mathcal{P}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))$. Thus, we have proved that $\mathcal{L}x(t)\in \mathcal{A}{\mathcal{P}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))$ and $\mathcal{L}x(t)$ is square-mean piecewise almost periodic.

Proof of Statement (II) Given $B=\{u\in \mathcal{A}{\mathcal{P}}_{\mathcal{T}}(\mathbb{R},L^2(\mathrm{\Omega },\mathbb{H}))\}$ and assuming that x(t), y(t) ∈ B are both almost periodic solutions of (1) and x(t) ≠ y(t), then we have

$\begin{array}{ll}\hfill & E\Vert \mathcal{L}x\left(t\right)-\mathcal{L}y\left(t\right){\Vert }^2\hfill \\ \hfill & =E{‖{\int }_{-\infty }^{t}S\left(t-s\right)\left[b\left(s,x_s\right)-b\left(s,y_s\right)\right]ds+\sum _{t_i<t}S\left(t-t_i\right)\left[I_i\left(x\left(t_i\right)\right)-I_i\left(y\left(t_i\right)\right)\right]‖}^2\hfill \\ \hfill & \le 2\left(E{‖{\int }_{-\infty }^{t}S\left(t-s\right)\left[b\left(s,x_s\right)-b\left(s,y_s\right)\right]ds‖}^2\right.\hfill \\ \hfill & \left.+E{‖\sum _{t_i<t}S\left(t-t_i\right)\left[I_i\left(x\left(t_i\right)\right)-I_i\left(y\left(t_i\right)\right)\right]‖}^2\right)\hfill \\ \hfill & \equiv 2\left(A_1+A_2\right).\hfill \end{array}$

From (H1), (H2), (H3) and the Cauchy-Schwarz inequality, we have that

$\begin{array}{ll}\hfill A_1& \le E{\left({\int }_{-\infty }^t{M}^2{e}^{-2\gamma \left(t-s\right)}‖b\left(s,x_s\right)-b\left(s,y_s\right)‖ds\right)}^2\hfill \\ \hfill & \le \left({\int }_{-\infty }^t{M}^2{e}^{-\gamma \left(t-s\right)}ds\right)\left({\int }_{-\infty }^t{e}^{-\gamma \left(t-s\right)}E{‖b\left(s,x_s\right)-b\left(s,y_s\right)‖}^{2}ds\right)\hfill \\ \hfill & \le \frac{M^2}{\gamma }{\int }_{-\infty }^t{e}^{-\gamma \left(t-s\right)}{M}_{b}E{‖x_s-y_s‖}_{{\mathcal{C}}_{\theta }}^{2}ds\hfill \end{array}$

$\le \frac{M^2}{{\gamma }^2}{M}_b\underset{r\in \mathbb{R}}{\sup }E{‖x\left(r\right)-y\left(r\right)‖}^2=\frac{M^2{M}_b}{{\gamma }^2}{‖x-y‖}_{\infty }^2,$

and

$\begin{array}{ll}\hfill A_2& \le E{‖\sum _{t_i<t}{M}^2{e}^{-2\gamma \left(t-t_i\right)}\left[I_i\left(x\left(t_i\right)\right)-I_i\left(y\left(t_i\right)\right)\right]‖}^2\hfill \\ \hfill & \le \left(\sum _{t_i<t}{M}^2{e}^{-\gamma \left(t-t_i\right)}\right)\left(\sum _{t_i<t}{e}^{-\gamma \left(t-t_i\right)}E{‖I_i\left(x\left(t_i\right)\right)-I_i\left(y\left(t_i\right)\right)‖}^2\right)\hfill \\ \hfill & \le \frac{M^2}{1-e^{-\gamma \alpha }}\left(\sum _{t_i<t}{e}^{-\gamma \left(t-t_i\right)}{M}_{I}E{‖\left(x\left(t_i\right)\right)-\left(y\left(t_i\right)\right)‖}^2\right)\hfill \\ \hfill & \le \frac{M^2{M}_I}{{\left(1-e^{-\gamma \alpha }\right)}^2}\underset{r\in \mathbb{R}}{\sup }E{‖x\left(r\right)-y\left(r\right)‖}^2=\frac{M^2{M}_I}{{\left(1-e^{-\gamma \alpha }\right)}^2}{‖x-y‖}_{\infty }^2.\hfill \end{array}$

It follows that

$E\Vert \mathcal{L}x\left(t\right)-\mathcal{L}y\left(t\right){\Vert }^2\le \mathrm{\Theta }{‖x-y‖}_{\infty }^2,$

for each $t\in \mathbb{R}$, which implies that

$\Vert \mathcal{L}x\left(t\right)-\mathcal{L}y\left(t\right){\Vert }_{\infty }\le \sqrt[2]{\mathrm{\Theta }}{‖x-y‖}_{\infty }.$

This means that $\mathcal{L}$ is a contraction when (8) holds and statement (II) follows.

4 Asymptotic Stability

In this section, we are interested in the asymptotical stability of the almost periodic mild solution to (1) with t₀ = 0. For convenience, we rewrite the equation as follows:

$\left\{\begin{array}{cc}\hfill & dx\left(t\right)=\left[Ax\left(t\right)+b\left(t,x_t\right)\right]dt+\sigma \left(t\right)d{B}^H\left(t\right),t\in \left[t_0,\infty \right),t\ne \pm t_{i}i\in \mathbb{Z},\hfill \\ \hfill & △x\left(t_i\right)=x\left(t_i^{+}\right)-x\left(t_i^{-}\right)=I_i\left(x\left(t_i\right)\right),i\in \mathbb{Z},\hfill \\ \hfill & x_{t_0}=\xi =\left\{\xi \left(t\right):-\theta \le t\le 0\right\}.\hfill \end{array}\right.(15)$

Lemma 4.1 ([30]). Let a nonnegative piecewise continuous function t↦v(t) satisfy the inequality

$v\left(t\right)\le C+{\int }_{t_0}^{t}u\left(\sigma \right)v\left(\sigma \right)d\sigma +\sum _{t_0<{\sigma }_i<t}{\alpha }_{i}v\left({\sigma }_i\right),$

for t ≥ t₀, where C ≥ 0, u(σ) > 0, α_i ≥ 0, $i\in \mathbb{Z}$, and σ_i, $i\in \mathbb{Z}$ are the first kind discontinuity points of the function v. Then, the following estimate holds:

$v\left(t\right)\le C\prod _{t_0<{\sigma }_i<t}\left(1+{\alpha }_i\right)e^{{\int }_{t_0}^{t}u\left(\sigma \right)d\sigma }.$

Theorem 4.1 Assume that (H1) − (H3) hold. The almost periodic solutions to (15) are asymptotically stable in the square-mean sense if

$\frac{1}{\alpha }\ln \left(1+\frac{3M^2{M}_I}{1-e^{-\gamma \alpha }}\right)-\gamma +\frac{3M^2{M}_b}{\gamma }<0,(16)$

ProofLet x(t) and x*(t) be two square-mean piecewise almost periodic mild solutions of (15); we then have that

$\begin{array}{ll}\hfill E{‖x\left(t\right)-x^{*}\left(t\right)‖}^2& =E‖S\left(t\right)\left[\xi -{\xi }^{*}\right]+{\int }_0^{t}S\left(t-s\right)\left[b\left(s,x_s\right)-b\left(s,x_s^{*}\right)\right]ds\hfill \\ \hfill & {+\sum _{0<t_i<t}S\left(t-t_i\right)\left[I_i\left(x\left(t_i\right)\right.-I_i\left(x^{*}\left(t_i\right)\right)\right]‖}^2,\hfill \end{array}$

for all t ≥ 0. By using Cauchy–Schwartz’s inequality, Fubini’s theorem, and assumptions (H1) − (H3), we deduce that

$\begin{array}{ll}\hfill E& {‖x\left(t\right)-x^{*}\left(t\right)‖}^2\le 3E{‖S\left(t\right)\left[\xi -{\xi }^{*}\right]‖}^2+3E{‖{\int }_0^{t}S\left(t-s\right)\left[b\left(s,x_s\right)-b\left(s,x_s^{*}\right)\right]ds‖}^2\hfill \\ \hfill & +3E{‖\sum _{0<t_i<t}S\left(t-t_i\right)\left[I_i\left(x\left(t_i\right)\right)\right.-I_i\left(x^{*}\left(t_i\right)\right)‖}^2\hfill \\ \hfill & \le 3M^2{e}^{-2\gamma t}{‖\xi -{\xi }^{*}‖}^2+3E{\left({\int }_0^t‖M{e}^{-\gamma \left(t-s\right)}\left[b\left(s,x_s\right)-b\left(s,x_s^{*}\right)\right]‖ds\right)}^2\hfill \\ \hfill & +3E{‖\sum _{0<t_i<t}M{e}^{-\gamma \left(t-t_i\right)}\left[I_i\left(x\left(t_i\right)\right)\right.-I_i\left(x^{*}\left(t_i\right)\right)‖}^2\hfill \\ \hfill & \le 3M^2{e}^{-\gamma t}{‖\xi -{\xi }^{*}‖}^2+3{\int }_0^t{M}^2{e}^{-\gamma \left(t-s\right)}ds{\int }_0^t{e}^{-\gamma \left(t-s\right)}E{‖b\left(s,x_s\right)-b\left(s,x_s^{*}\right)‖}^{2}ds\hfill \\ \hfill & +3\left(\sum _{0<t_i<t}{M}^2{e}^{-\gamma \left(t-t_i\right)}\right)\left(\sum _{0<t_i<t}{e}^{-\gamma \left(t-t_i\right)}E{‖I_i\left(x\left(t_i\right)\right)-I_i\left(x^{*}\left(t_i\right)\right)‖}^2\right)\hfill \\ \hfill & \le 3M^2{e}^{-\gamma t}{‖\xi -{\xi }^{*}‖}^2+\frac{3M^2}{\gamma }{M}_b{\int }_0^t{e}^{-\gamma \left(t-s\right)}{‖x_s-x_s^{*}‖}_{{\mathcal{C}}_{\theta }}^{2}ds\hfill \\ \hfill & +\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\left(\sum _{0<t_i<t}{e}^{-\gamma \left(t-t_i\right)}E{‖x\left(t_i\right)-x^{*}\left(t_i\right)‖}^2\right)\hfill \\ \hfill & \le 3M^2{e}^{-\gamma t}{‖\xi -{\xi }^{*}‖}^2+\frac{3M^2}{\gamma }{M}_b{\int }_0^t{e}^{-\gamma \left(t-s\right)}\left(\underset{0\le r\le s}{\sup }E{‖x\left(r\right)-x^{*}\left(r\right)‖}^2\right)ds\hfill \\ \hfill & +\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\left(\sum _{0<t_i<t}{e}^{-\gamma \left(t-t_i\right)}E{‖x\left(t_i\right)-x^{*}\left(t_i\right)‖}^2\right),\hfill \end{array}$

for t ≥ 0. Multiplying both sides of the above inequality by e^γt, we get

$e^{\gamma t}E{‖x\left(t\right)-x^{*}\left(t\right)‖}^2\le 3M^2{‖\xi -{\xi }^{*}‖}^2+\frac{3M^2}{\gamma }{M}_b{\int }_0^t{e}^{\gamma s}\left(\underset{0\le r\le s}{\sup }E{‖x\left(r\right)-x^{*}\left(r\right)‖}^2\right)ds$

$+\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\left(\sum _{0<t_i<t}{e}^{\gamma t_i}E{‖x\left(t_i\right)-x^{*}\left(t_i\right)‖}^2\right),$

for t ≥ 0, which implies that

$\begin{array}{ll}\hfill \underset{0\le s\le t}{\sup }{e}^{\gamma s}E{‖x\left(s\right)-x^{*}\left(s\right)‖}^2& \le 3M^2{‖\xi -{\xi }^{*}‖}^2+\frac{3M^2}{\gamma }{M}_b{\int }_0^t\underset{0\le r\le s}{\sup }{e}^{\gamma s}E{‖x\left(r\right)-x^{*}\left(r\right)‖}^{2}ds\hfill \\ \hfill & +\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\left(\sum _{0<t_i<t}\underset{0\le r\le t_i}{\sup }{e}^{\gamma r}E{‖x\left(r\right)-x^{*}\left(r\right)‖}^2\right),\hfill \end{array}$

for t ≥ 0. Combining this with Lemma 4.1, we get that

$\begin{array}{ll}\hfill \underset{0\le s\le t}{\sup }{e}^{\gamma s}E{‖x\left(s\right)-x^{*}\left(s\right)‖}^2& \le 3M^2{‖\xi -{\xi }^{*}‖}^2\prod _{0<t_i<t}\left(1+\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\right)e^{{\int }_0^t\frac{3M^2}{\gamma }{M}_{b}d\sigma }\hfill \\ \hfill & \le 3M^2{‖\xi -{\xi }^{*}‖}^2{\left(1+\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\right)}^{\frac{t}{\alpha }}{e}^{\frac{3M^2}{\gamma }{M}_{b}t},\hfill \end{array}$

for t ≥ 0. So,

$e^{\gamma t}E{‖x\left(t\right)-x^{*}\left(t\right)‖}^2\le 3M^2{‖\xi -{\xi }^{*}‖}^2{\left(1+\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\right)}^{\frac{t}{\alpha }}{e}^{\frac{3M^2}{\gamma }{M}_{b}t},$

for t ≥ 0. Thus, we get the desired estimate

$\begin{array}{ll}\hfill E{‖x\left(t\right)-x^{*}\left(t\right)‖}^2& \le 3M^2{‖\xi -{\xi }^{*}‖}^2{\left(1+\frac{3M^2}{1-e^{-\gamma \alpha }}{M}_I\right)}^{\frac{t}{\alpha }}{e}^{\frac{3M^2}{\gamma }{M}_{b}t}{e}^{-\gamma t}\hfill \\ \hfill & \le 3M^2{‖\xi -{\xi }^{*}‖}^2{e}^{\left[\frac{1}{\alpha }\ln \left(1+\frac{3M^2{M}_I}{1-e^{-\gamma \alpha }}\right)-\gamma +\frac{3M^2{M}_b}{\gamma }\right]t},\hfill \end{array}$

and the square-mean piecewise almost periodic solution of (15) is asymptotically stable in the square-mean sense because of (16). This completes the proof.

5 An Example

Consider the semilinear impulsive stochastic partial functional differential equations of the following form:

$\left\{\begin{array}{cc}\hfill & dv\left(t,x\right)=\left[\frac{{\partial }^2}{\partial x^2}v\left(t,x\right)+2a\sin \left(x\left(t-r\right)\right)\sin t\right]dt\hfill \\ \hfill & +\cos td{B}^H\left(t\right),t\ne \pm t_{i}i\in \mathbb{Z},\hfill \\ \hfill & △x\left(t_i\right)=x\left(t_i^{+}\right)-x\left(t_i^{-}\right)=2a\cos \left(x\left(t_i\right)\right),i\in \mathbb{Z},\hfill \\ \hfill & v\left(t,0\right)=v\left(t,\pi \right)=0,\hfill \\ \hfill & x_{t_0}=\xi =\left\{\xi \left(s\right):-\theta \le s\le 0\right\},\hfill \end{array}\right.(17)$

where r is a constant and B^H(t) is a fractional Brownian motion. Denote X = L²(Ω, L²([0, π])) and define A: D(A) ⊆ X → X given by $A=\frac{{\partial }^2}{\partial x^2}$ with the following domain:

$D\left(A\right)=\left\{v\left(\cdot \right)\in X:v^{″}\in X,v^{\prime }\in X\text{\hspace{0.17em}are\hspace{0.17em}absolutely\hspace{0.17em}continous\hspace{0.17em}on\hspace{0.17em}}\left[0,\pi \right]\right\}.$

It is well known that a strongly continuous semigroup {S(t)}_t≥0 generated by the operator A satisfies ‖S(t)‖ ≤ e^−t, for t ≥ 0. Take

$b\left(t,x_t\right)=2a\sin \left(x_t\right)\sin t,$

and

$I_i\left(x\left(t_i\right)\right)=2a\cos \left(x\left(t_i\right)\right).$

Thus, one has

$E\Vert b\left(t,x_t\right)-b\left(t,y_t\right){\Vert }^2\le 4a^2\Vert x_t-y_t{\Vert }_{{\mathcal{C}}_{\theta }}^2,$

and

$\Vert I_i\left(x\right)-I_i\left(y\right){\Vert }^2\le 4a^2\Vert x-y{\Vert }^2.$

Let α = 1. Then, (17) has a square-mean piecewise almost periodic mild solution, provided that $0<a^2<\frac{1}{16}$ by Theorem 3.1, and moreover the solution of (17) is asymptotically stable in the square-mean sense provided that $0<a^2<\frac{1}{36}$ by Theorem 4.1.

6 Conclusion

In this article, we have investigated the existence and asymptotic stability of square-mean piecewise almost periodic mild solutions for a class of impulsive stochastic delay differential equations driven by fractional Brownian motion with the Hurst parameter $H\in (\frac{1}{2},1)$ in a Hilbert space. An example is presented to illustrate our theoretical results. Fractional Brownian motion B^H with $H\in (0,\frac{1}{2})$ admits different Wiener integral representation from fractional Brownian motion with $H\in (\frac{1}{2},1)$. It is difficult to get the square-mean piecewise almost periodic mild solutions of ISDEs driven by fractional Brownian motion with $H\in (0,\frac{1}{2})$ in a Hilbert space properly due to estimation without moment.

Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author Contributions

LG and XS carried out the mathematical studies, participated in the sequence alignment, drafted the manuscript and participated in the design of the study and performed proof of results. All authors read and approved the submitted version.

Funding

This work was supported by the National Natural Science Foundation of China, No. 11971101; Natural Science Foundation of Anhui Province, No.1808085MA02; and Natural Science Foundation of Bengbu University, Nos. 2020ZR04zd and BBXY2020KYQD05.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors are indebted to Professor Litan Yan for his encouragement and helpful discussion. The authors are grateful to the referees and the associate editor for valuable comments and suggestions to improve this article.

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