Abstract
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 :where is the set of integer, for any , and the sequence {ti} is such that the derived sequence is equipotentially almost periodic. Moreover, 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 Ii are appropriate functions, is given by xt(s) = x(t + s), for any s ∈ [ − θ, 0], and is an measurable random variable such that E‖ξ‖2 < ∞. Let θ > 0 be a given constant and let
be continuous everywhere except for a finite number of points s at which ϕ(s−) and ϕ(s+) exist and satisfy , endowed with the normsuch that ϕ(s, ⋅) is -measurable for each s ∈ [ − θ, 0] and .
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 and denote two real separable Hilbert spaces. We denote by the set of all linear bounded operators from into , 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 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 with Hurst parameter H ∈ (0, 1) is a Gaussian process with continuous sample paths such that E[βH(t)] = 0 andfor all s, t ≥ 0. It is known that fBm {βH(t), t ≥ 0} with admits the following Wiener integral representation:where W is a standard Brownian motion and the kernel KH(t, s) is given bywhere cH > 0 is a constant satisfying . For any function σ ∈ L2(0, T), the Wiener integral of σ with respect to βH is defined byfor any T > 0, where . A -valued, -adapted fBm BH with Hurst index H can be defined bywhere are independent fBms with the same Hurst parameter , which is a complete orthonormal basis in that is a bounded sequence of non-negative real numbers satisfying Qen = λnen, and Q is non-negative self-adjoint trace class operator with .
Let denote the space of all such that is a Hilbert–Schmidt operator. The norm is defined by . Generally, σ is called a Q-Hilbert–Schmidt operator from to .
Definition 2.1Letsuch thatthen the stochastic integral ofσwith respect to fBmBHis defined by
RemarkIfis a bounded sequence of non-negative real numbers such that the nuclear operatorQsatisfiesQen = λen, assuming that there exists a positive constantKσsuch thatuniformly in [0, T], then it is obvious thatis uniformly convergent fort ∈ [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
is said to be continuous if
for all
, and it is said to be bounded if there exists
N> 0, such that
E‖
X(
t)‖
2≤
Nfor all
. For convenience, we list the following concepts and notations:
• is Banach space when it is equipped with norm .
• Let be the set consisting of all real sequences such that , and , ,and represent the left and right limits of x(t) at the point , respectively.
• Let be the space consisting of all stochastically bounded functions such that b(⋅) is stochastically continuous at t for any , and for all , .
• Let be the space of all piecewise stochastic process such that
• for any , b(⋅, ϕ) is stochastically continuous at point t for any and for all ;
and b(t, ⋅) is stochastically continuous at , for .
• For k < i, t − tk = t − ti + ti − tk ≥ t − ti + (i − k)α, if , and ti < t ≤ ti+1 (see [27]).
Definition 2.2([28]). The family of the sequencewill be called equipotentially almost periodic if for anyɛ > 0; there exists a relatively dense setofand an integersuch that the inequalityholds for eachand.
Definition 2.3
{
b(
t),
t≥ 0}
is said to be square-mean piecewise almost periodic if the following conditions are fulfilled:a) For anyɛ > 0, there exists a positive numberδ = δ(ɛ) such that if the pointst′ andt″ belong to the same interval of continuity and |t′ − t″| < δ, thenE‖b(t′) − b(t″)‖2 < ɛ.
b) For anyɛ > 0, there existsl(ɛ) > 0, such that every interval of lengthl(ɛ) contains a numberτwith the property
Definition 2.4(compare with [28]). A sequenceis called square-mean almost periodic if for anyɛ > 0, there exists a natural numberN = N(ɛ) such that, for each, there is at least one integerpin the segment [k, k + N], for which inequalityholds for all.
Definition 2.5The functionis said to be square-mean piecewise almost periodic inuniformly inφ ∈ Λ, whereis compact if for anyɛ > 0, there exitsl(ɛ, Λ) > 0 such that any interval of lengthl(ɛ, Λ) contains at least a numberτfor whichfor eachx ∈ Λ,, satisfying |t − ti| > ɛ. The collection of all such processes is denoted by.
Lemma 2.1Let the functionbe square-mean piecewise almost periodic inuniformly for, whereis compact. Iffis a Lipschitz function in the following sense,for all,, and a constantM2 > 0, then for any.
ProofNoting that is square-mean almost periodic, we can conclude that ϕt = {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 satisfyingNoting that 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 satisfyingfor any , with |t − ti| > ɛ. Using the elementary inequality |a + b|2 ≤ 2(|a|2 + |b|2) and condition (3), we havefor all . Combining (4) and (5), one can show thatwhich implies that f(t, ϕt) 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
for some
γ> 0,
M> 0. Moreover,
R(
λ,
A) is almost periodic, where
λ∈
ρ(
A).
(H2) Let . Moreover, there exists a positive constant Mb such that
for any
.
(H3) Let and let be a square-mean piecewise almost periodic sequence satisfying
for some positive constant
MI.
Recall the notion of a mild solution for Eq. 1.
Definition 3.1An-progressive processis called a mild solution of the system(1)onif it satisfies the corresponding stochastic integral equationfor allt ≥ t0and for each.
(
H1) − (
H3)
be satisfied. Then,(1)has a unique square-mean piecewise almost periodic mild solution wheneverConsider the following equation:
with
t≥
t0. It is easy to verify that the above equation is equivalent to
(7). Define the operator
on
by
for all
. To prove the theorem, it is sufficient to show that the next statements hold:
I) is square-mean piecewise almost periodic.
II) admits a unique fixed point.
Proof of Statement (I)This will be done in two steps.
Step 1We claim that .Let . By the uniform continuity of S(t), we can see that, for any ɛ > 0, there exists a number δ > 0 between 0 and such thatfor all t′, t″ ∈ (ti, ti+1), t′ < t″ as 0 < t″ − t′ < δ, where . It follows from the inequality |a + b + c|3 ≤ 3(a2 + b2 + c2) thatfor all t′, t″ ∈ (ti, ti+1), t′ < t″. By the assumptions (H1), (H2), and (H3), we have thatandfor all t′, t″ ∈ (ti, ti+1), t′ < t″. Moreover, we also have thatandfor all t′, t″ ∈ (ti, ti+1), t′ < t″. Combining these with Hölder’s inequality and (9), we get thatandfor all t′, t″ ∈ (ti, ti+1), t′ < t″ provided |t″ − t′| < δ. Similarly, by the assumptions (H1) and (H3) and (9), one can see thatfor all t′, t″ ∈ (ti, ti+1), t′ < t″ provided |t″ − t′| < δ. Thus, we have shown that the estimateholds for all t′, t″ ∈ (ti, ti+1), t′ < t″ provided |t″ − t′| < δ, which means .
Step 2We prove the almost periodicity of .For Φ1(t), let ti < t < ti+1; by (H1), (H2), and Hölder’s inequality, we have thatwhere . By Lemma 2.1 and (H2), we find that for any ɛ > 0 and , there exists a real number l(ɛ, Λ) > 0 such that every interval of length l(ɛ, Λ) contains at least a constant τ andfor each x ∈ Λ, |t − ti| > ɛ, since , where is compact.For s ∈ [tj + η, tj+1 − η], , j ≤ i, t − s ≥ t − ti + ti − (tj+1 − η) ≥ t − ti + α(i + j − 1) + η, we haveFor , by the mean value theorem of integral, we get thatSimilarly, we can show thatwhere C1, C2 are two positive constants. Thus, we have introduced the next estimate: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 estimateholds for every interval of length l(ɛ) containing a number τ. By using (H1) and the computation of fBm, we haveFurthermore, by Hölder’s inequality, we havewhere . In the same way as that of handling Φ1(t), one can introduce the estimatewhere N2 is a positive constant, and hence Φ2(t) is piecewise square-mean almost periodic.For , let βi = Ii(x(ti)). For ti < t ≤ ti+1, |t − ti| > ɛ, , by (2), one has ti+q+1 > t + τ > ti+q. From (H3), it follows that βi is a square-mean almost periodic sequence, for any ɛ > 0; there exists such a natural number N = N(ɛ) that, for an arbitrary , there is at least one integer p > 0 in the segment [k, k + N] such that the inequalityholds for all . We getwhich implies that . Thus, we have proved that and is square-mean piecewise almost periodic.
Proof of Statement (II)Given and assuming that x(t), y(t) ∈ B are both almost periodic solutions of (1) and x(t) ≠ y(t), then we haveFrom (H1), (H2), (H3) and the Cauchy-Schwarz inequality, we have thatandIt follows thatfor each , which implies thatThis means that 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:
Lemma 4.1([30]). Let a nonnegative piecewise continuous functiont↦v(t) satisfy the inequalityfort ≥ t0, whereC ≥ 0, u(σ) > 0, αi ≥ 0,, andσi,are the first kind discontinuity points of the functionv. Then, the following estimate holds:
Assume that (H1) − (H3) hold. The almost periodic solutions to(15)are asymptotically stable in the square-mean sense if
ProofLet x(t) and x*(t) be two square-mean piecewise almost periodic mild solutions of (15); we then have thatfor all t ≥ 0. By using Cauchy–Schwartz’s inequality, Fubini’s theorem, and assumptions (H1) − (H3), we deduce thatfor t ≥ 0. Multiplying both sides of the above inequality by eγt, we getfor t ≥ 0, which implies thatfor t ≥ 0. Combining this with Lemma 4.1, we get thatfor t ≥ 0. So,for t ≥ 0. Thus, we get the desired estimateand 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:where r is a constant and BH(t) is a fractional Brownian motion. Denote X = L2(Ω, L2([0, π])) and define A: D(A) ⊆ X → X given by with the following domain:
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. TakeandThus, one hasand
Let α = 1. Then, (17) has a square-mean piecewise almost periodic mild solution, provided that by Theorem 3.1, and moreover the solution of (17) is asymptotically stable in the square-mean sense provided that 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 in a Hilbert space. An example is presented to illustrate our theoretical results. Fractional Brownian motion BH with admits different Wiener integral representation from fractional Brownian motion with . It is difficult to get the square-mean piecewise almost periodic mild solutions of ISDEs driven by fractional Brownian motion with in a Hilbert space properly due to estimation without moment.
Statements
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.
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.
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.
References
1.
TsokosCPPadgettWJ. Random Integral Equations with Applications to Life Sciences and Engineering. New York: Academic Press (1974).
2.
ArthiGParkJHJungHY. Existence and Exponential Stability for Neutral Stochastic Integrodifferential Equations with Impulses Driven by a Fractional Brownian Motion. Commun Nonlinear Sci Numer Simulation (2016) 32:145–57. 10.1016/j.cnsns.2015.08.014
3.
CaraballoT. Existence of Mild Solutions to Stochastic Delay Evolution Equations with a Fractional Brownian Motion and Impulses. Stoch Anal Appl (2015) 33:244–58. 10.1080/07362994.2014.981641
4.
ChengLHuLRenY. Perturbed Impulsive Neutral Stochastic Functional Differential Equations. Qual Theor Dyn. Syst. (2021) 20:27. 10.1007/s12346-021-00469-7
5.
ShiJLiCSunJ. Stability of Impulsive Stochastic Differential Delay Systems and its Application to Impulsive Stochastic Neural Networks. Nonlinear Anal (2011) 74:3099–111. 10.1016/j.na.2011.01.026
6.
LiuXLiuJXieW. Existence and Uniqueness Results for Impulsive Hybrid Stochastic Delay Systems. Appl Nonlinear Anal (2010) 17:37–53. 10.1016/j.nahs.2009.11.004
7.
SakthivelRLuoJ. Asymptotic Stability of Nonlinear Impulsive Stochastic Differential Equations. Stat Probab Lett (2009) 79:1219–23. 10.1016/j.spl.2009.01.011
8.
WuXYanLZhangWChenL. Exponential Stability of Impulsive Stochastic Delay Differential Systems. Discrete Dyn Nat Soc (2012) 2012:1–15. 10.1155/2012/296136
9.
ChenHXiaoS. Existence and Exponential Stability for Impulsive Stochastic Partial Functional Differential Equations. J Math Phys (2017) 58:032701. 10.1063/1.4976727
10.
MaSKangY. Periodic Averaging Method for Impulsive Stochastic Differential Equations with Lévy Noise. Appl Maths Lett (2019) 93:91–7. 10.1016/j.aml.2019.01.040
11.
GuoYZhuQWangF. Stability Analysis of Impulsive Stochastic Functional Differential Equations. Commun Nonlinear Sci Numer Simulation (2020) 82:105013. 10.1016/j.cnsns.2019.105013
12.
LiM. Multi-fractional Generalized Cauchy Process and its Application to Teletraffic. Physica A: Stat Mech its Appl (2020) 550:123982. 10.1016/j.physa.2019.123982
13.
LiM. Generalized Fractional Gaussian Noise and its Application to Traffic Modeling. Physica A (2020) 579:12638. 10.1016/j.physa.2021.126138
14.
LiM. Modified Multifractional Gaussian Noise and its Application. Phys Scr (2021) 96:125002. 10.1088/1402-4896/ac1cf6
15.
ZachevskyIZeeviYY. Single-Image Superresolution of Natural Stochastic Textures Based on Fractional Brownian Motion. IEEE Trans Image Process (2014) 23:2096–108. 10.1109/TIP.2014.2312284
16.
KsendalB. Fractional Brownian Motion in Finance. London: Springer (2008).
17.
HeydariMHMahmoudiMRShakibaAAvazzadehZ. Chebyshev Cardinal Wavelets and Their Application in Solving Nonlinear Stochastic Differential Equations with Fractional Brownian Motion. Commun Nonlinear Sci Numer Simulation (2018) 64:98–121. 10.1016/j.cnsns.2018.04.018
18.
HeydariMHAvazzadehZMahmoudiMR. Chebyshev Cardinal Wavelets for Nonlinear Stochastic Differential Equations Driven with Variable-Order Fractional Brownian Motion. Chaos, Solitons & Fractals (2019) 124:105–24. 10.1016/j.chaos.2019.04.040
19.
LiZXuLZhouJ. Viability for Coupled SDEs Driven by Fractional Brownian Motion. Appl Maths Optimization (2021) 84:55C98. 10.1007/s00245-021-09761-z
20.
CuiJYanL. Controllability of Neutral Stochastic Evolution Equations Driven by Fractional Brownian Motion. Acta Mathematica Scientia (2017) 37:108–18. 10.1016/s0252-9602(16)30119-9
21.
BiaginiFHuYØksendalBZhangT. Stochastic Calculus for Fractional Brownian Motion and Applications. New York: Springer (2008).
22.
HuY. Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. Mem AMS (2005) 175:825. 10.1090/memo/0825
23.
MishuraY. Stochastic Calculus for Fractional Brownian Motion and Related. New York: Springer (2008).
24.
NurlartD. The Malliavin Calculus and Related Topics. New York: Springer (2006).
25.
StamovGT. Almost Periodic Solutions of Impulsive Differential Equations. New York: Springer (2012).
26.
LiuJZhangC. Existence and Stability of Almost Periodic Solutions to Impulsive Stochastic Differential Equations. CUBO (2013) 15:77–96. 10.4067/s0719-06462013000100005
27.
HenríquezHRDe AndradeBRabeloM. Existence of Almost Periodic Solutions for a Class of Abstract Impulsive Differential Equations. ISRN Math Anal (2011) 2011:1–21. 10.5402/2011/632687
28.
SamoilenkoAMPerestyukNA. Impulsive Differential Equations. Singapore: World Scientific (1995).
29.
YoshizawaT. Stability Theory and Existence of Periodic Sotlutions and Almost Periodic Solutions. New York: Springer (1975).
30.
WangYLiuZ. Almost Periodic Solutions for Stochastic Differential Equations with Lévy Noise. Nonlinearity (2012) 25:2803–21. 10.1088/0951-7715/25/10/2803
Summary
Keywords
fractional Brownian motion, square-mean piecewise almost periodic solution, impulsive systems, stochastic functional differential equation, stability
Citation
Gao L and Sun X (2021) Almost Periodic Solutions to Impulsive Stochastic Delay Differential Equations Driven by Fractional Brownian Motion With < H < 1. Front. Phys. 9:783125. doi: 10.3389/fphy.2021.783125
Received
25 September 2021
Accepted
14 October 2021
Published
22 November 2021
Volume
9 - 2021
Edited by
Ming Li, Zhejiang University, China
Reviewed by
Mohammad Hossein Heydari, Shiraz University of Technology, Iran
Yong Ren, Anhui Normal University, China
Yaozhong Hu, University of Alberta, Canada
Updates
Copyright
© 2021 Gao and Sun.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xichao Sun, sunxichao626@126.com
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
Disclaimer
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.