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Research article

Modified prairie dog optimization algorithm for global optimization and constrained engineering problems


  • Received: 03 August 2023 Revised: 23 September 2023 Accepted: 06 October 2023 Published: 13 October 2023
  • The prairie dog optimization (PDO) algorithm is a metaheuristic optimization algorithm that simulates the daily behavior of prairie dogs. The prairie dog groups have a unique mode of information exchange. They divide into several small groups to search for food based on special signals and build caves around the food sources. When encountering natural enemies, they emit different sound signals to remind their companions of the dangers. According to this unique information exchange mode, we propose a randomized audio signal factor to simulate the specific sounds of prairie dogs when encountering different foods or natural enemies. This strategy restores the prairie dog habitat and improves the algorithm's merit-seeking ability. In the initial stage of the algorithm, chaotic tent mapping is also added to initialize the population of prairie dogs and increase population diversity, even use lens opposition-based learning strategy to enhance the algorithm's global exploration ability. To verify the optimization performance of the modified prairie dog optimization algorithm, we applied it to 23 benchmark test functions, IEEE CEC2014 test functions, and six engineering design problems for testing. The experimental results illustrated that the modified prairie dog optimization algorithm has good optimization performance.

    Citation: Huangjing Yu, Yuhao Wang, Heming Jia, Laith Abualigah. Modified prairie dog optimization algorithm for global optimization and constrained engineering problems[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19086-19132. doi: 10.3934/mbe.2023844

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  • The prairie dog optimization (PDO) algorithm is a metaheuristic optimization algorithm that simulates the daily behavior of prairie dogs. The prairie dog groups have a unique mode of information exchange. They divide into several small groups to search for food based on special signals and build caves around the food sources. When encountering natural enemies, they emit different sound signals to remind their companions of the dangers. According to this unique information exchange mode, we propose a randomized audio signal factor to simulate the specific sounds of prairie dogs when encountering different foods or natural enemies. This strategy restores the prairie dog habitat and improves the algorithm's merit-seeking ability. In the initial stage of the algorithm, chaotic tent mapping is also added to initialize the population of prairie dogs and increase population diversity, even use lens opposition-based learning strategy to enhance the algorithm's global exploration ability. To verify the optimization performance of the modified prairie dog optimization algorithm, we applied it to 23 benchmark test functions, IEEE CEC2014 test functions, and six engineering design problems for testing. The experimental results illustrated that the modified prairie dog optimization algorithm has good optimization performance.



    Periodicity is such an important phenomenon; people have studied it for a long time, and many researchers have investigated the properties about asymptotically almost automorphic, almost periodic, asymptotically almost periodic and S-asymptotically ω-periodic solutions of various determinate differential systems (see, e.g., [1,2,3,4]). The differential equations of fractional order have received great attention in recent years (see, e.g., [1,5,6,7,8]). Cuevas and de Souza[9] considered the S-asymptotically ω-periodic solution of the semilinear integro-differential equation of fractional order given by

    {x(t)=t0(ts)α2Γ(α1)Ax(s)ds+f(t,x(t)),x(0)=c0.

    Moreover, in [10], Cuevas and de Souza considered the S-asymptotically ω-periodic solutions of the following form:

    {x(t)=t0(ts)α2Γ(α1)Ax(s)ds+f(t,xt),x(0)=ψ0B,

    where B is some abstract phase space. Shu et al. [11] considered the existence of the S-asymptotically ω-periodic solutions of the following Caputo fractional differential equations with infinite delay

    {Dαt(x(t)+F(t,xt))+A(x(t))=G(t,xt),t0x(0)=ψ0B,

    where 0<α<1. Recently, stochastic perturbations have been taken into consideration in the mathematical systems [12]; specially, Liu and Sun [13] made an initial contribution to the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution and got the existence results of the almost automorphic in distribution solutions for a kind of stochastic differential equations with Lévy noise. Additionally, Li [14] considered the weighted pseudo almost automorphic solutions for nonautonomous stochastic partial differential equations driven by Lévy noise. Ma et al. [15] investigated the existence of almost periodic solutions for a class of fractional impulsive neutral stochastic differential equations with infinite delay.

    Inspired by the work mentioned above, we investigate the existence of the mild solutions and the S-asymptotically ω-periodic solutions in distribution in an abstract space for a class of stochastic fractional functional differential equations driven by Lévy noise of the form

    {dD(t,xt)=t0(ts)α2Γ(α1)AD(s,xs)dsdt+f(t,xt)dt+g(t,xt)dw(t)+|u|U<1F(t,x(t),u)˜N(dt,du)+|u|U1G(t,x(t),u)N(dt,du),x0=ϕCbF0([τ,0],X), (1.1)

    where 1<α<2, D(t,φ)=φ(0)+h(t,φ), the operator A:D(A)XX is a linear densely defined operator of sectorial type on a Banach space X, and f,g,F and G are functions subject to some additional conditions. The convolution integral in (1.1) is known as the Riemann-Liouville fractional integral [16]. We introduce the concept of a Poisson square-mean S-asymptotically ω-periodic solution for (1.1) in order to correspond to the effect of the Lévy noise. We make an initial consideration of the S-asymptotically ω-periodic solution in distribution in an abstract space C for (1.1).

    In Section 2, we review and introduce some concepts about the square mean S-asymptotically ω-periodic process and some of their basic properties. We show the existence and uniqueness of the mild solution and the S-asymptotically ω-periodic solution in distribution to (1.1) in Sections 3 and 4, respectively. An example and the conclusions of this paper are given in the last two sections.

    Let (Ω,F,P) be a complete probability space equipped with some filtration {Ft}t0 which satisfies the usual conditions, and (H,||) and (U,||) are real separable Hilbert spaces. L(U,H) denote the space of all bounded linear operators from U to H which with the usual operator norm L(U,H) is a Banach space. L2(P,H) is the space of all H-valued random variables X such that E|X|2=Ω|X|2dP<. For XL2(P,H), let X:=(Ω|X|2dP)1/2; it is well known that (L2(P,H),) is a Hilbert space. We denote M2([0,T],H) for the collection of stochastic processes x(t):[0,T]L2(P,H) such that ET0|x(s)|ds<. Let τ>0 and we introduce the space.

    C:=C([τ,0];H) denotes the family of all right-continuous functions with left hand limits φ from [τ,0] to H. If x(t)M2([τ,T],H) with right-continuous functions and with left-hand limits; then, xt(θ)=x(t+θ)C for each θ[τ,0]. The space C is assumed to be equipped with the norm φC=supτθ0|φ(θ)|H. CbF0([τ,0];H) denotes the family of all almost surely bounded F0-measurable, C-valued random variables. {xt}tR is regarded as a C-valued stochastic process. In the following discussion, we always consider the Lévy processes that are U-valued.

    Let L be a Lévy process on U; we write ΔL(t)=L(t)L(t) for all t0. We define a counting Poisson random measure N on (U{0}) through

    N(t,O)={0st:ΔL(s)(ω)O}=Σ0stχO(ΔL(s)(ω))

    for any Borel set O in (U{0}), where χO is the indicator function. We write ν()=E(N(1,)) and call it the intensity measure associated with L. We say that a Borel set O in (U{0}) is bounded below if 0ˉO, where ˉO is a closure of O. If O is bounded below, then N(t,O)< almost surely for all t0 and (N(t,O),t0) is a Poisson process with the intensity ν(O). So, N is called a Poisson random measure. For each t0 and O bounded below, the associated compensated Poisson random measure ˜N is defined by ˜N(t,O)=N(t,O)tν(O) (see [17,18]).

    Proposition 2.1. (see [17]) (Lévy-Itô decomposition). If L is a U-valued Lévy process, then there exist aU, a U-valued Wiener process w with the covariance operator Q, the so-called Q-wiener process and an independent Poisson random measure N on R+×(U{0}) such that, for each t0,

    L(t)=at+w(t)+|u|U<1u˜N(t,du)+|u|U1uN(t,du), (2.1)

    where the Poisson random measure N has the intensity measure ν which satisfies U(|y|2U1)ν(dy)< and ˜N is the compensated Poisson random measure of N.

    For more properties of the Lévy process and Q-Wiener processes, we refer the readers to [19] and [20]. We assume the covariance operator Q of w is of the trace class, i.e., TrQ< and the Lévy process L is defined on the filtered probability space (Ω,F,P,(Ft)tR+) in this paper. We also denote b:=|y|U1ν(dx) throughout the paper.

    In [13], if for any sR, limtsx(t)x(s)=0, then the stochastic process x:RL2(P,H) is said to be L2-continuous; if x=suptRx(t)<, the stochastic process x:RL2(P,H) is said to be L2-bounded. Denote by Cb(R;L2(P,H)) the Banach space of all L2-bounded and L2-continuous mappings from R to L2(P,H) endowed with the norm .

    Definition 2.1. 1) An L2-continuous stochastic process x:R+L2(P,H) is said to be square-mean S-asymptotically ω-periodic if there exists ω>0 such that limtx(t+ω)x(t)=0. The collection of all S-asymptotically ω-periodic stochastic processes x:R+L2(P,H) is denoted by SAPω(L2(P,H)).

    2) A function g:R+×CL(U,L2(P,H)), (t,φ)g(t,φ) is said to be square-mean S-asymptotically ω-periodic in t for each φC if g is continuous in the following sense:

    E(g(t,ϕ)g(t,φ))Q1/22L(U,L2(P,H))0as(t,φ)(t,ϕ)

    and

    limtE(g(t+ω,ϕ)g(t,ϕ))Q1/22L(U,L2(P,H))=0

    for each ϕC.

    3) A function F:R+×C×UL2(P,H), (t,ϕ,u)F(t,ϕ,u) with UF(t,ϕ,u)2ν(du)< is said to be Poisson square-mean S-asymptotically ω-periodic in t for each ϕC if F is continuous in the following sense:

    UF(t,ϕ,u)F(t,φ,u)2ν(du)0as(t,φ)(t,ϕ),

    and that

    limtUF(t+ω,ϕ,u)F(t,ϕ,u)2ν(du)=0

    for each ϕC.

    Remark 2.1. Any square-mean S-asymptotically ω-periodic process x(t) is L2-bounded and, by [3], SAPω(L2(P,H)) is a Banach space when it is equipped with the norm

    x:=suptR+x(t)=suptR+(E|x(t)|2)12.

    For the sequel, we introduce some definitions about square-mean S-asymptotically ω-periodic functions with parameters.

    Definition 2.2. 1) A function f:R+×CL2(P,H) is said to be uniformly square-mean S-asymptotically ω-periodic in t on bounded sets if for every bounded set K of C, we have limtf(t+ω,ϕ)f(t,ϕ)=0 uniformly on ϕC.

    2) A function g:R+×CL(U,L2(P,H)) is said to be uniformly square-mean S-asymptotically ω-periodic on bounded sets if for every bounded set K of C, we have

    limtE(g(t+ω,ϕ)g(t,ϕ))Q1/22L(U,L2(P,H))=0

    uniformly on ϕK.

    3) A function F:R+×C×UL2(P,H) with UF(t,ϕ,u)2ν(du)< is said to be uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets if for every bounded set K of C,

    limtUF(t+ω,ϕ,u)F(t,ϕ,u)2ν(du)=0

    uniformly on ϕK.

    Lemma 2.1. Let f:R+×CL2(P,H),(t,ϕ)f(t,ϕ) be uniformly square-mean S-asymptotically ω-periodic in t on bounded sets of C, and assume that f satisfies the Lipschitz condition in the sense f(t,ϕ)f(t,φ)2Lϕφ2C for all ϕ,φC and tR, where L is independent of t. Then for any square-mean S-asymptotically ω-periodic process Y:RL2(P,H), the stochastic process F:RL2(P,H) given by F(t):=f(t,Yt) is square-mean S-asymptotically ω-periodic.

    Proof. Since Y(t)SAPω(L2(P,H)), the range of Y(t) is a bounded set in L2(P,H), which means that {Yt}tR+ is also a bounded set in C. Then, limtF(t+ω,Yt+ω)F(t,Yt+ω)=0. For any ϵ>0, T(ϵ) such that F(t+ω,Yt+ω)F(t,Yt+ω)<ϵ/2 and Yt+ωYt<ϵ2L. We get

    F(t+ω)F(t)F(t+ω,Yt+ω)F(t,Yt)=F(t+ω,Yt+ω)F(t,Yt+ω)+F(t+ω,Yt+ω)F(t,Yt)ϵ.

    Lemma 2.2. Let F:R+×C×UL2(P,H) be uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets of C and F satisfy the Lipschitz condition in the sense

    UF(t,ϕ,u)F(t,φ,u)2ν(du)Lϕφ2C

    for all ϕ,φC and tR, where L is independent of t. Then, for any square-mean S-asymptotically ω-periodic process Y(t):RL2(P,H), the stochastic process ˜F:R×UL2(P,H) given by ˜F(t,u):=F(t,Yt,u) is Poisson square-mean S-asymptotically ω-periodic.

    Proof. Since F is uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets of C and Y(t)SAPω(L2(P,H), the range R(Y) of Y(t) is a bounded set in L2(P,H), namely, {Yt}tR is bounded in C; we get limtUF(t+ω,ϕ,u)F(t,ϕ,u)2ν(du)=0 uniformly for ϕ{Yt}tR+. For any ϵ>0, there is a T(ϵ)>0 such that for any tT(ϵ), the inequalities UF(t+ω,ϕ,u)F(t,ϕ,u)2ν(du)ϵ/4,ϕ{Yt}tR+ and Yt+ωYt2<ϵ2L hold. Note that

    ˜F(t+ω,u)˜F(t,u)=F(t+ω,Yt+ω,u)F(t+ω,Yt,u)+F(t+ω,Yt,u)F(t,Yt,u);

    so, for the above ϵ, when tT(ϵ),

    U˜F(t+ω,u)˜F(t,u)2ν(du)2UF(t+ω,Yt+ω,u)F(t,Yt+ω,u)2ν(du)+2UF(t,Yt+ω,u)F(t,Yt,u)2ν(du)ϵ/2+2LYt+ωYt2ϵ.

    We deduce that

    limtU˜F(t+ω,u)˜F(t,u)2ν(du)=0,

    which means that ˜F(t,u) is Poisson square-mean S-asymptotically ω-periodic.

    Let P(C) be the space of Borel probability measures on C; for P1,P2P(C), denote the metric dL as follows:

    dL(P1,P2)=supfL|Cf(σ)P1(dσ)Cf(σ)P2(dσ)|,

    where

    L={f:CR:|f(ϕ)f(φ)|ϕφCand|f()|1}.

    Definition 2.3. A stochastic process xt:RC is said to be S-asymptotically ω-periodic in distribution if the law μ(t) of xt is a P(C)-valued S-asymptotically ω-periodic mapping, i.e., there is a positive number ω such that

    limtdL(μ(t+ω),μ(t))=0.

    Lemma 2.3. Any square-mean S-asymptotically ω-periodic solution of (1.1) is necessarily S-asymptotically ω-periodic in distribution.

    Proof. Let x(t)SPAω(L2(P,H)) be a solution of (1.1), then, ω>0,

    limtx(t+ω)x(t)=0. (2.2)

    We need to show that the law μ(t) of xt satisfies

    limtdL(μ(t+ω),μ(t))=0,

    which is equivalent to show that, ϵ>0, T>0,

    supfL|Cf(σ)μ(t+ω)(dσ)Cf(σ)μ(t)(dσ)|ϵ,tT.

    For any fL,

    |E(f(xxωt))E(f(xϕt))|E(2xxωtxϕtC).

    According to (2.2), there is a Tϵ>0 satisfying Exxωtxϕt2Cϵ2. For the arbitrary fL, we get

    supfL|Cf(σ)μ(t+ω)(dσ)Cf(σ)μ(t)(dσ)|ϵ,tT.

    The proof is completed.

    The definitions and the properties about sectorial operators have been studied well in the past decades; for details, see [21,22].

    Definition 2.4. Let X be an Banach space; A:D(A)XX is a closed linear operator. A is said to be a sectorial operator of the type μ and angle θ if there exist 0<θ<π/2,M>0 and μR such that the resolvent ρ(A) of A exists outside of the sector μ+Sθ={μ+λ:λC,|arg(λ)|<θ} and (λA)1M|λμ| when λ does not belong to μ+Sθ.

    Definition 2.5. (see [23]) Let A be a closed and linear operator with domain the D(A) defined on a Banach space X. We call A the generator of a solution operator if there exist μR and a strongly continuous function Sα:R+L(X,X) such that {λα:Re(λ)>μ}ρ(A) and λα1(λαA)1x=0eλtSα(t)dt, where Re(λ)>μ, xX. In this case, Sα() is called the solution operator generated by A.

    If A is a sectorial operator of the type μ with 1<θ<π(1α2), then A is the generator of a solution operator given by Sα(t)=12πiγeλtλα1(λαA)1dλ, where γ is a suitable path lying outside of the sector μ+Sθ. Cuesta and Mendizabal [24] showed that, if A is a sectorial operator of the type μ<0, for some M>0 and 0<θ<π(1π2), there is C>0 such that

    Sα(t)CM1+|μ|tα,t0. (2.3)

    Definition 3.1. An Ft-progressively measurable stochastic process {x(t)}tR is called a mild solution of (1.1) if it satisfies the corresponding stochastic integral equation:

    x(t)=Sα(t)(ϕ(0)+h(0,ϕ))h(t,xt)+t0Sα(ts)f(s,xs)ds+t0Sα(ts)g(s,xs)dw(s)+t0|u|<1Sα(ts)F(s,x(s),u)˜N(ds,du)+t0|u|1Sα(ts)G(s,x(s),u)N(ds,du),x0=ϕ()CbF0([τ,0];H) (3.1)

    for all t0.

    In the following discussion, we impose the following conditions.

    (H1) A is a sectorial operator of the type μ<0 and angle θ with 0θπ(1α/2).

    (H2) h(t,0)=0 and, for all φ,ψC there exists a constant k0(0,1) such that |h(t,φ)h(t,ψ)|k0φψC.

    (H3) f:R+×CL2(P,H), g:R+×CL(U,L2(P,H)), f(t,0)=0, g(t,0)=0, F:R+×C×UL2(P,H), G:R+×C×UL2(P,H), F(t,0,u)=0 and G(t,0,u)=0. For all tR+,

    f(t,ϕ)f(t,φ)2Lϕφ2C,E(g(t,ϕ)g(t,φ))Q1/22L(U,L2(P,H))Lϕφ2C,|u|U<1F(t,ϕ,u)F(t,φ,u)2ν(du)Lϕφ2C,|u|U1G(t,ϕ,u)G(t,φ,u)2ν(du)Lϕφ2C,

    where the constant L>0 is independent of t.

    Theorem 3.1. If (H1)–(H2) hold, then the Cauchy problem (1.1) has a unique mild solution.

    Proof. Define x00=ϕ, and x0(t)=Sα(t)(ϕ(0)+f(0,ϕ)) for t0.

    Set xn0=ϕ; for n=1,2,..., T(0,), we define the sequence of successive approximations to (1.1) as follows:

    xn(t)+h(t,xnt)=Sα(t)(ϕ(0)+h(0,ϕ))+t0Sα(ts)f(s,xn1s))ds+t0Sα(ts)g(s,xn1s)dw(s)+t0|u|<1Sα(ts)F(s,xn1s,u)˜N(ds,du)+t0|u|1Sα(ts)G(s,xn1s,u)N(ds,du), (3.2)

    for t[0,T]. Obviously, x0()Cb(R,L2(P,H)) and x0()2c, where c=2(CM)2(1+k20)ϕ2C is a positive constant.

    Esup0st|xn(t)+h(t,xnt)|2=5Esup0st|Sα(t)(ϕ(0)+h(0,ϕ))|2+5Esup0st|s0Sα(sr)f(r,xn1r)dr|2+5Esup0st|s0Sα(sr)g(r,xn1r)dw(r)|2+5Esup0st|s0|u|<1Sα(sr)F(r,xn1r,u)˜N(dr,du)|2+5Esup0st|s0|u|1Sα(sr)G(r,xn1r,u)N(dr,du)|2=5i=1Ii.

    Obviously, I15c, and

    I2=5Esup0st|s0Sα(sr)f(r,xn1r)dr|25Esup0sts0Sα(sr)drs0Sα(sr)|f(r,xn1r)|2dr5CM|μ|1/απαsin(π/α)LEt0Sα(tr)xn1r2Cdr5(CM)2|μ|1/απαsin(π/α)LEt0xn1r2Cdr.

    It follows from Itô's isometry that

    I3=5Esup0st|s0Sα(sr)g(r,xn1r)dw(r)|5(CM)2Et0g(r,xn1r)Q1/22L(U,L2(P,H))dr5(CM)2LEt0xn1r2Cdr.

    By using the properties of integrals for Poisson random measures, we get

    I4=Esup0st|s0|u|<1Sα(tr)F(r,xn1r,u)˜N(dr,du)|25(CM)2LEt0xn1r2Cdr,
    I5=5Esup0st|s0|u|1Sα(sr)G(r,xn1r,u)N(dr,du)|210(CM)2[t0E|u1|(11+|μ|(ts)α)2|G(r,xn1r,u)|2ν(du)ds+t011+|μ|(tr)α|u|1ν(du)dsEt0(|u|1|G(r,xn1r,u)|2ν(du))ds]10(CM)2L(1+b|μ|1/απαsin(π/α))Et0xn1r2Cdr.

    Note that

    |xn(t)|2=|xn(t)+h(t,xnt)h(t,xnt)|211k0(|xn(t)+h(t,xnt)|2+k0(1k0)xnt2C);

    by taking the expectation on both sides of the above inequality, we get

    Esup0st|xn(s)|211k0Esup0st|xn(s)+h(s,xns)|2+k0Esup0stxns2C.

    Combining the estimations for I1I5, we get

    Esup0st|xn(s)|211k05(CM)2[(1+k20)ϕ2C]+11k05(CM)2L×(|μ|1/απαsin(π/α)+2+2(1+b|μ|1/απαsin(π/α)))Et0xn1r2Cdr=c1+c2Esup0sts0xn1r2Cdr.

    Then, for any arbitrary positive integer ˜k, we have

    max1n˜kEsup0st|xn(s)|2c1+c2t0ϕ2Cdr+c2t0sup0rtmax1n˜kExn1(r)2Cdr.

    By the Gronwall inequality, we get

    max1n˜kEsup0st|xn(s)|2(c1+c2ϕ2T)ec3t.

    Due to the arbitrary ˜k, we have

    Esup0sT|xn(s)|2(c1+c2ϕ2CT)ec3T. (3.3)

    So, xn(t)M2([0,T],H).

    Obviously, we have

    E(sup0st|xn+1(s)xn(s)|2)11k0E(sup0st(|xn+1(s)xn(s)+h(s,xn+1s)h(s,xns)|2+k0E(sup0st|xn+1(s)xn(s)|2),

    namely,

    E(sup0st|xn+1(s)xn(s)|2)1(1k0)2Esup0st|xn+1(s)xn(s)+h(s,xn+1s)h(s,xns)|2,

    and

    Esup0st|[xn+1(s)xn(s)]+[h(t,xn+1s)h(t,xns)]|24Esup0st|s0Sα(sr)[f(r,xnr))f(r,xn1r))]dr|2+4Esup0st|s0Sα(sr)[g(r,xnr)g(r,xn1r)]dw(r)|2+4Esup0st|t0|u|<1Sα(sr)[F(r,xnr,u)F(r,xn1r,u)]˜N(dr,du)|2+4Esup0st|s0|u|1Sα(sr)[G(r,xnr,u)G(r,xn1r,u)]N(dr,du)|2.

    By the fact that

    Esup0st|s0Sα(sr)[f(r,xnr))f(r,xn1r))]dr|2LEsup0sts0Sα(sr)drs0Sα(sr)xnrxn1r2CdrL(CM)2|μ|1/απαsin(π/α)Et0xnrxn1r2Cdr,
    Esup0st|s0Sα(sr)[g(r,xnr)g(r,xn1r)]dw(r)|2Et0S2α(tr)[g(r,xnr)g(r,xn1r)]Q1/22L(U,L2(P,H))drL(CM)2Et0xnrxn1r2Cdr,
    Esup0stt0|u|<1Sα(sr)[F(r,xnr,u)F(r,xn1r,u)]˜N(dr,du)|2LEt0S2α(tr)xnrxn1r2CdrL(CM)2Et0xnrxn1r2Cdr

    and

    Esup0st|s0|u|1Sα(sr)[G(r,xnr,u)G(r,xn1r,u)]N(dr,du)|22CMt0Sα(tr)dr|u|1ν(du)Et0|u|1|G(r,xnr,u)G(r,xn1r,u)|2ν(du)dr+2(CM)2t0E|u|1|G(r,xnr,u)G(r,xn1r,u)|2ν(du)ds(2(CM)2|μ|1/απαsin(π/α)b+2(CM)2)LEt0xnrxn1r2Cdr,

    we get

    E(sup0st|xn+1(s)xn(s)|2)ct0E(sup0rs|xn(r)xn1(r)|2)dr,

    where c=4L(CM)2(1k0)2(|μ|1/απαsin(π/α)+4+2|μ|1/απαsin(π/α)b)L. Note that

    Esup0st|x1(s)x0(s)|2c0,

    where c0=5[k20c+Lc(1+2b)(CM|μ|1/απαsin(π/α))2+4Lc(CM)2|μ|12απ2αsin(π/2α)] is a positive number; by induction, we get

    E(sup0st|xn+1(s)xn(s)|2)c0(c6t)nn!. (3.4)

    Taking t=T in (3.4), we have

    E(sup0tT|xn+1(t)xn(t)|2)c0(c6T)nn!. (3.5)

    Hence

    P{sup0tT|xn+1(t)xn(t)|>12n}c0[c6T]nn!.

    Note that n=0c0[c6T]nn!<; by using the Borel-Cantelli lemma, we can get a stochastic process x(t) on [0,T] such that xn(t) uniformly converges to x(t) as n almost surely.

    It is easy to check that x(t) is a unique mild solution of (1.1). The proof of the theorem is complete.

    Lemma 4.1. If x(t)SAPω(L2(P,H) and T(ts)L(R,R) then Γ1(t)=t0T(ts)x(s)dsSAPω(L2(P,H)).

    The proof process is similar to that of Lemma 1 in [25], so we omit it.

    Lemma 4.2. If x(t)SAPω(L2(P,H)), then

    Γ2(t)=t0Sα(ts)x(s)dw(s)SAPω(L2(P,H)).

    Proof. It is obvious that Γ2(t) is L2-continuous since

    Γ2(t+ω)Γ2(t)2=t+ω0Sα(t+ωs)x(s)dw(s)t0Sα(ts)x(s)dw(s)2=2ω0Sα(t+ωs)x(s)dw(s)2+2t+ωωSα(t+ωs)x(s)dw(s)t0Sα(ts)x(s)dw(s)22ω0Sα(t+ωs)x(s)dw(s)2+2t0Sα(ts)(x(s+ω)x(s))dw(s)2.

    Since x(t)SAPω(L2(P,H)), for any ϵ>0, we can choose Tϵ>0 such that when t>Tϵ, x(t+ω)x(t)<ϵ. For the above ϵ, we have

    2t0Sα(ts)(x(s+ω)x(s))dw(s)24Tϵ0Sα(ts)2x(s+ω)x(s)2ds+4tTϵSα(ts)2x(s+ω)x(s)2ds.

    Note that

    2ω0Sα(t+ωs)x(s)dw(s)22(CM)21+|μ|2t2αω0x(s)2ds0,t,

    that

    Tϵ0Sα(ts)2x(s+ω)x(s)2ds4(CM)21+|μ|2(tTϵ)2αxTϵ0,t

    and that

    tTϵSα(ts)2x(s+ω)x(s)2dsϵ2(CM)2|μ|2/απ2αsin(π/2α);

    we get limtΓ2(t+ω)Γ2(t)=0. So Γ2(t)SAPω(L2(P,H)).

    The following lemma is made obvious by using Lemmas 2.1 and 2.2 and a similar discussion as that for Lemma 4.2.

    Lemma 4.3. If x(t)SAPω(L2(P,H)) and F:R+×C×UL2(P,H) is uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets of C, then

    Γ3(t)=t0|u|U<1Sα(ts)F(s,xs,u)˜N(du,ds)SAPω(L2(P,H))

    and

    Γ4(t)=t0|u|U1Sα(ts)G(s,xs,u)N(du,ds)SAPω(L2(P,H)).

    Theorem 4.1. Assume that (H1)–(H3) are satisfied and h,fandg are uniformly square-mean S-asymptotically ω-periodic on bounded sets of C. FandG are uniformly Poisson square-mean S-asymptotically ω-periodic on bounded sets of C. Then, (1.1) has a unique S-asymptotically ω-periodic solution in distribution if

    5k20+5(CM)2L(|μ|1/α  παsin(π/α))2(1+b)+20L(CM)2|μ|1/2α  π2αsin(π/2α)<1.

    Proof. Let us first show the existence of the square-mean S-asymptotically ω-periodic solution of (1.1); so, we consider the operator Φ acting on the Banach space SAPω(L2(P,H)) given by

    Φx(t)=Sα(t)(ϕ(0)+h(0,ϕ))h(t,xt)+t0Sα(ts)f(s,xs)ds+t0Sα(ts)g(s,xs)dw(s)+t0|u|<1Sα(ts)F(s,xs,u)˜N(ds,du)+t0|u|1Sα(ts)G(s,xs,u)N(ds,du). (4.1)

    From a previous assumption one can easily see that Φx(t) is well defined and L2-continuous. Moreover, from Lemma 2.1, Lemma 4.1, Lemma 4.2, and Lemma 4.3 we infer that Φ maps SAPω(L2(P,H)) into itself. Next, we prove that Φ is a strict contraction on SAPω(L2(P,H)). Indeed, for x,˜xSAPω(L2(P,H)), we get

    Φˉx(t)Φ˜x(t)25k20ˉxt˜xt2C+5(CM)2L(|μ|1/απαsin(π/α))2ˉx(t)˜x(t)2+20L(CM)2|μ|1/2απ2αsin(π/2α)ˉx(t)˜x(t)2+10L(CM)2(|μ|1/απαsin(π/α))2bˉx(t)˜x(t)2=[5k20+5(CM)2L(|μ|1/απαsin(π/α))2(1+b)+20L(CM)2|μ|1/2απ2αsin(π/2α)]×ˉx(t)˜x(t)2.

    Since 5k20+5(CM)2L(|μ|1/απαsin(π/α))2(1+b)+20L(CM)2|μ|1/2απ2αsin(π/2α)<1, it follows that Φ is a contraction mapping on SAPω(L2(P,H)). By the classical Banach fixed-point principle, there exists a unique xSAPω(L2(P,H)) such that Φx=x, which is the unique square-mean S-asymptotically ω-periodic solution of (1.1). By Lemma 2.3, we deduce that (1.1) has a unique S-asymptotically ω-periodic solution in distribution. The proof is now complete.

    In this section, an example is provided to illustrate the results obtained in previous sections. Let H=L2([0,π]) and w(t) be an H-valued Wiener process; given τ>0, we consider the following initial problem

    {d[x(t,ξ)sint8x(tτ,ξ)]=t0(ts)α2Γ(α1)2ξ2[x(s,ξ)sins8x(tτ,ξ)]dsdt+18(sinln(t+1)+cost)x(tτ,ξ)dt+18(sinln(t+1)+cost)x(tτ,ξ)dw(t)+|u|U<118(cost+ln(t+1)t)x(tτ,ξ)˜N(dt,du),x(t,0)=x(t,π)=0,x0(θ,ξ)=ϕ(θ,ξ)CbF0([τ,0],H),θ[τ,0],ξ[0,π]. (5.1)

    The operator A:HH by A=2x2 and D(A)={zH:zH,z(0)=z(π)=0} is the infinitesimal generator of a strongly continuously cosine family[26]. Based on the estimates on the norms of the operators of Theorems 3.3 and 3.4 in [27], the operators Sq(t) in the mild solution of (5.1) satisfy Sq(t)L(H,H)3, and h(t,φ)=sint8x(tτ,ξ); obviously, the function h satisfies (H2) and k0=18.

    f(t,ϕ)=18(sinln(t+1)+cost)ϕ(tτ),g(t,ϕ)=18(sinln(t+1)+cost)ϕ(tτ),
    F(t,ϕ,u)=18(cost+ln(t+1)t)ϕ(tτ)

    satisfy (H3), and L can be chosen as L=14. According to Theorem 3.1 in Section 3, the Cauchy problem (5.1) has a mild solution. Moreover, if

    5k20+5(CM)2L(|μ|1/απαsin(π/α))2(1+b)+20L(CM)2|μ|1/2απ2αsin(π/2α)564+45(|μ|1/απαsin(π/α))2(1+b)+90|μ|1/2απ2αsin(π/2α)<1,

    the Cauchy problem (5.1) has a unique S-asymptotically ω-periodic solution in distribution.

    In this work, inspired by the idea in [13], we established the concept of a Poisson square-mean S-asymptotically ω-periodic solution for (1.1) in order to correspond to the effect of Lévy noise. Furthermore, we made an initial consideration of the S-asymptotically ω-periodic solution in distribution in an abstract space C for (1.1). We established the existence and uniqueness of a mild solution of a class of stochastic fractional differential evolution equations with delay and Piosson jumps. First, the existence and the uniqueness of the mild solution for this type of equation are derived by means of the successive approximation under Lipschitz conditions. We also obtained sufficient conditions for the existence and the uniqueness of the S-asymptotically ω-periodic solution in distribution. To the best of our knowledge, this is the first attempt to discuss this property for these kinds of stochastic fractional functional differential equations.

    This work was supported by the Support Plan on Science and Technology for Youth Innovation of Universities in Shandong Province (NO. 2021KJ086).

    The authors declare that there is no conflict of interest.



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