Processing math: 59%
Research article Special Issues

Analysis of Weibull progressively first-failure censored data with beta-binomial removals

  • Received: 06 June 2024 Revised: 28 July 2024 Accepted: 31 July 2024 Published: 14 August 2024
  • MSC : 62F10, 62F15, 62N01, 62N02, 62N05

  • This study examined the estimations of Weibull distribution using progressively first-failure censored data, under the assumption that removals follow the beta-binomial distribution. Classical and Bayesian approaches for estimating unknown model parameters have been established. The estimations included scale and shape parameters, reliability and failure rate metrics as well as beta-binomial parameters. Estimations were considered from both point and interval viewpoints. The Bayes estimates were developed by using the squared error loss and generating samples for the posterior distribution through the Markov Chain Monte Carlo technique. Two interval estimation approaches are considered: approximate confidence intervals based on asymptotic normality of likelihood estimates and Bayes credible intervals. To investigate the performance of classical and Bayesian estimations, a simulation study was considered by various kinds of experimental settings. Furthermore, two examples related to real datasets were thoroughly investigated to verify the practical importance of the suggested methodologies.

    Citation: Refah Alotaibi, Mazen Nassar, Zareen A. Khan, Ahmed Elshahhat. Analysis of Weibull progressively first-failure censored data with beta-binomial removals[J]. AIMS Mathematics, 2024, 9(9): 24109-24142. doi: 10.3934/math.20241172

    Related Papers:

    [1] Nina Huo, Bing Li, Yongkun Li . Global exponential stability and existence of almost periodic solutions in distribution for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays. AIMS Mathematics, 2022, 7(3): 3653-3679. doi: 10.3934/math.2022202
    [2] Yongkun Li, Xiaoli Huang, Xiaohui Wang . Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. AIMS Mathematics, 2022, 7(4): 4861-4886. doi: 10.3934/math.2022271
    [3] Ardak Kashkynbayev, Moldir Koptileuova, Alfarabi Issakhanov, Jinde Cao . Almost periodic solutions of fuzzy shunting inhibitory CNNs with delays. AIMS Mathematics, 2022, 7(7): 11813-11828. doi: 10.3934/math.2022659
    [4] Yuwei Cao, Bing Li . Existence and global exponential stability of compact almost automorphic solutions for Clifford-valued high-order Hopfield neutral neural networks with D operator. AIMS Mathematics, 2022, 7(4): 6182-6203. doi: 10.3934/math.2022344
    [5] Jin Gao, Lihua Dai . Weighted pseudo almost periodic solutions of octonion-valued neural networks with mixed time-varying delays and leakage delays. AIMS Mathematics, 2023, 8(6): 14867-14893. doi: 10.3934/math.2023760
    [6] Hedi Yang . Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator. AIMS Mathematics, 2021, 6(2): 1865-1879. doi: 10.3934/math.2021113
    [7] Yanshou Dong, Junfang Zhao, Xu Miao, Ming Kang . Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. AIMS Mathematics, 2023, 8(9): 21828-21855. doi: 10.3934/math.20231113
    [8] Zhangir Nuriyev, Alfarabi Issakhanov, Jürgen Kurths, Ardak Kashkynbayev . Finite-time synchronization for fuzzy shunting inhibitory cellular neural networks. AIMS Mathematics, 2024, 9(5): 12751-12777. doi: 10.3934/math.2024623
    [9] Abdulaziz M. Alanazi, R. Sriraman, R. Gurusamy, S. Athithan, P. Vignesh, Zaid Bassfar, Adel R. Alharbi, Amer Aljaedi . System decomposition method-based global stability criteria for T-S fuzzy Clifford-valued delayed neural networks with impulses and leakage term. AIMS Mathematics, 2023, 8(7): 15166-15188. doi: 10.3934/math.2023774
    [10] Xiaofang Meng, Yongkun Li . Pseudo almost periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays and leakage delays on time scales. AIMS Mathematics, 2021, 6(9): 10070-10091. doi: 10.3934/math.2021585
  • This study examined the estimations of Weibull distribution using progressively first-failure censored data, under the assumption that removals follow the beta-binomial distribution. Classical and Bayesian approaches for estimating unknown model parameters have been established. The estimations included scale and shape parameters, reliability and failure rate metrics as well as beta-binomial parameters. Estimations were considered from both point and interval viewpoints. The Bayes estimates were developed by using the squared error loss and generating samples for the posterior distribution through the Markov Chain Monte Carlo technique. Two interval estimation approaches are considered: approximate confidence intervals based on asymptotic normality of likelihood estimates and Bayes credible intervals. To investigate the performance of classical and Bayesian estimations, a simulation study was considered by various kinds of experimental settings. Furthermore, two examples related to real datasets were thoroughly investigated to verify the practical importance of the suggested methodologies.



    As stated in [1], a nervous system in the real world, synaptic transmission is a noisy process caused by random fluctuations in neurotransmitter release and other probabilistic factors. Therefore, it is necessary to consider stochastic neural networks (NNs) because random inputs may change the dynamics of the (NN) [2,3,4,5].

    SICNNs, which were proposed in [6], have attracted the interest of many scholars since their introduction due to their special roles in psychophysics, robotics, adaptive pattern recognition, vision, and image processing. In the above applications, their dynamics play an important role. Thereupon, their various dynamics have been extensively studied (see [7,8,9,10,11,12,13] and references therein). However, there is limited research on the dynamics of stochastic SICNNs. Therefore, it is necessary to further study the dynamics of such NNs.

    On the one hand, research on the dynamics of NNs that take values from a non commutative algebra, such as quaternion-valued NNs [14,15,16], octonion-valued NNs [17,18,19,20], and Clifford-valued NNs [21,22,23], has gained the interest of many researchers because such neural networks can include typical real-valued NNs as their special cases, and they have superior multi-dimensional signal processing and data storage capabilities compared to real-valued NNs. It is worth mentioning that in recent years, many authors have conducted extensive research on various dynamics of Clifford-valued NNs, such as the existence, multiplicity and stability of equilibrium points, and the existence, multiplicity and stability of almost periodic solutions as well as the synchronization problems [22,23,24,25,26,27,28,29,30]. However, most of the existing results for the dynamics of Clifford-valued NNs has been obtained through decomposition methods [24,25,26,27]. However, the results obtained by decomposition methods are generally not convenient for direct application, and there is little research on Clifford-valued NNs using non decomposition methods [28,29,30]. Therefore, further exploration of using non decomposition methods to study the dynamics of Clifford-valued NNs has important theoretical significance and application value.

    On the other hand, Bohr's almost periodicity is a special case of Stepanov's almost periodicity, but there is little research on the Stepanov periodic oscillations of NNs [19,31,32,33], especially the results of Stepanov's almost periodic solutions of stochastic SICNNs with discrete and infinitely distributed delays have not been published yet.

    Motivated by the discussion above, our purpose of this article is to establish the existence and global exponential stability of Stepanov almost periodic solutions in the distribution sense for a stochastic Clifford-valued SICNN with mixed delays via non decomposition methods.

    The subsequent sections of this article are organized as follows. Section 2 introduces some concepts, notations, and basic lemmas and gives a model description. Section 3 discusses the existence and stability of Stepanov almost periodic solutions in the distribution sense of the NN under consideration. An example is provided in Section 4. Finally, Section 5 provides a brief conclusion.

    Let A={ϑPxϑeϑ,xϑR} be a real Clifford-algebra with N generators e=e0=1, and eh,h=1,2,,N, where P={,0,1,2,,ϑ,,12N}, e2i=1,i=1,2,,r,e2i=1,i=r+1,r+2,,m,eiej+ejei=0,ij and i,j=1,2,,N. For x=ϑPxϑeϑA, we indicate x=maxϑP{|xϑ|},xc=ϑxϑeϑ,x0=xxc, and for x=(x11,x12,,x1n,x21,x22,,x2n,,xmn)TAm×n, we denote x0=max{xij,1im,1jn}. The derivative of x(t)=ϑPxϑ(t)eϑ is defined by ˙x(t)=ϑP˙xϑ(t)eϑ and the integral of x(t)=ϑPxϑ(t)eϑ over the interval [a,b] is defined by bax(t)dt=ϑP(baxϑ(t)dt)eϑ.

    Let (Y,ρ) be a separable metric space and P(Y) the collection of all probability measures defined on Borel σ-algebra of Y. Denote by Cb(Y) the set of continuous functions f:YR with g:=supxY{|g(x)|}<.

    For gCb(Y), μ,νP(Y), let us define

    gL=supxy|g(x)g(y)|ρ(x,y),gBL=max{g,gL},
    ρBL(μ,ν):=supgBL1|Ygd(μν)|.

    According to [34], (Y,ρBL(,)) is a Polish space.

    Definition 2.1. [35] A continuous function g:RY is called almost periodic if for every ε>0, there is an (ε)>0 such that each interval with length has a point τ meeting

    ρ(g(t+τ),g(t))<ε,foralltR.

    We indicate by AP(R,Y) the set of all such functions.

    Let (X,) signify a separable Banach space. Denote by μ(X):=PX1 and E(X) the distribution and the expectation of X:(Ω,F,P)X, respectively.

    Let Lp(Ω,X) indicate the family of all X-valued random variables satisfying E(Xp)=ΩXpdP<.

    Definition 2.2. [21] A process Z:RLp(Ω,X) is called Lp-continuous if for any t0R,

    limtt0EZ(t)Z(t0)p=0.

    It is Lp-bounded if suptREZ(t)p<.

    For 1<p<, we denote by Lploc(R,X) the space of all functions from R to X which are locally p-integrable. For gLploc(R,X), we consider the following Stepanov norm:

    gSp=suptR(t+1tg(s)pds)1p.

    Definition 2.3. [35] A function gLploc(R,X) is called p-th Stepanov almost periodic if for any ε>0, it is possible to find a number >0 such that every interval with length has a number τ such that

    g(t+τ)g(t)Sp<ε.

    Definition 2.4. [9] A stochastic process ZLploc(R,Lp(Ω,X)) is said to be Sp-bounded if

    ZSps:=suptR(t+1tEZ(s)pds)1p<.

    Definition 2.5. [9] A stochastic process ZLloc(R,Lp(Ω,H)) is called Stepanov almost periodic in p-th mean if for any ε>0, it is possible to find a number >0 such that every interval with length has a number τ such that

    Z(t+τ)Z(t)Sps<ε.

    Definition 2.6. [9] A stochastic process Z:RLp(Ω,X)) is said to be p-th Stepanov almost periodic in the distribution sense if for each ε>0, it is possible to find a number >0 such that any interval with length has a number τ such that

    supaR(a+1adpBL(P[Z(t+τ)]1,P[Z(t)]1)dt)1p<ε.

    Lemma 2.1. [36] (Burkholder-Davis-Gundy inequality) If fL2(J,R), p>2, B(t) is Brownian motion, then

    E[suptJ|tt0f(s)dB(s)|p]CpE[Tt0|f(s)|2ds]p2,

    where cp=(pp+12(p1)p1)p2.

    The model that we consider in this paper is the following stochastic Clifford-valued SICNN with mixed delays:

    dxij(t)=[aij(t)xij(t)+CklNh1(i,j)Cklij(t)f(xkl(tτkl(t)))xij(t)+CklNh2(i,j)Bklij(t)0Kij(u)g(xkl(tu))duxij(t)+Lij(t)]dt+CklNh3(i,j)Eklij(t)δij(xij(tσij(t)))dωij(t), (2.1)

    where i=1,2,,m,j=1,2,,n, Cij(t) represents the cell at the (i,j) position, the h1-neighborhood Nh1(i,j) of Cij is given as:

    Nh1(i,j)={Ckl:max(|ki|,|lj|)h1,1km,1ln},

    Nh2(i,j),Nh3(i,j) are similarly defined, xij denotes the activity of the cell Cij, Lij(t):RA corresponds to the external input to Cij, the function aij(t):RA represents the decay rate of the cell activity, Cklij(t):RA,Bklij(t):RA and Eklij(t):RA signify the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the activity functions f():AA, and g():AA are continuous functions representing the output or firing rate of the cell Ckl, and τkl(t),σij(t):RR+ are the transmission delay, the kernel Kij(t):RR is an integrable function, ωij(t) represents the Brownian motion defined on a complete probability space, δij():AA is a Borel measurable function.

    Let (Ω, F, {Ft}t0, P) be a complete probability space in which {Ft}t0 is a natural filtration meeting the usual conditions. Denote by BF0([θ,0],An) the family of bounded, F0-measurable and An-valued random variables from [θ,0]An. The initial values of system (2.1) are depicted as

    xi(s)=ϕi(s),s[θ,0],

    where ϕiBF0([θ,0],A),θ=max1i,jn{suptRτij(t),suptRσij(t)}.

    For convenience, we introduce the following notations:

    a_0=minijΛa_0ij=minijΛinftRa0ij(t),ˉa0=maxijΛˉa0ij=maxijΛsuptRa0ij(t),Cklij+=suptRCklij(t),¯ac=maxijΛˉacij=maxijΛsuptRacij(t),Bklij+=suptRBklij(t),Eklij+=suptREklij(t),K+ij=suptRKij(t),τ+kl=suptRτkl(t),˙τ+kl=suptR˙τkl(t),σ+ij=suptRσij(t),˙σ+ij=suptR˙σij(t),ML=maxijΛL+ij=maxijΛsuptRLij(t),θ=maxijΛ{τ+ij,σ+ij},Λ={11,12,,1n,,mn}.

    Throughout this paper, we make the following assumptions:

    (A1) For ijΛ, f,g,δijC(A,A) satisfy the Lipschitz condition, and f,g are bounded, that is, there exist constants Lf>0,Lg>0,Lδij>0,Mf>0,Mg>0 such that for all x,yA,

    ||f(x)f(y)||Lf||xy||,||g(x)g(y)||Lg||xy||,||δij(x)δij(y)||Lδij||xy||,||f(x)||Mf,||g(x)||Mg;

    furthermore, f(0)=g(0)=δij(0)=0.

    (A2) For ijΛ, a0ijAP(R,R+),acijAP(R,A),τij,σijAP(R,R+)C1(R,R) satisfying 1˙τ+ij,1˙σ+ij>0, Cklij,Bklij,EklijAP(R,A), L=(L11,L12,,Lmn)Lploc(R,Lp(Ω,Am×n)) is almost periodic in the sense of Stepanov.

    (A3) For p>2,1p+1q=1,

    0<r1:=8p4maxijΛ{(pqa_0ij)pqqpa_0ij[(ˉacij)p+(CklNh1(i,j)(Cklij+)q)pq(2κLf+Mf)p+(CklNh2(i,j)(Bklij+)q)pq((2κLg+Mg)0|Kij(u)|du)p]+Cp(p22a_0ij)p22qpa_0ij(CklNh3(i,j)(Eklij+)q)pq(Lδij)p}<1,

    and for p=2,

    0<r2:=16maxijΛ{1(a_0ij)2[(ˉacij)2+CklNh1(i,j)(Cklij+)2(2κLf+Mf)2+CklNh2(i,j)(Bklij+)2×((2κLg+Mg)0|Kij(u)|du)2]+12a_0ijCklNh3(i,j)(Eklij+)2(Lδij)2}<1.

    (A4) For 1p+1q=1,

    0<qpa_0ρ1:=16p1qpa_0maxijΛ{(pqa_0ij)pq[(ˉacij)p+(CklNh1(i,j)(Cklij+)q)pq[2p1(Lf)p×CklNh1(i,j)epqa_0ijτkl+(2κ)p1˙τ+kl+(Mf)p]+(CklNh2(i,j)(Bklij+)q)pq[(2κLg×0|Kij(u)|du)p+(Mg0|Kij(u)|du)p]]+2p1Cp(p22a_0ij)p22×(CklNh3(i,j)(Eklij+)q)pq(Lδij)pepqa_0ijσ+ij1˙σ+ij}<1,(p>2),
    0<ρ2a_0:=32a_0maxijΛ{(1a_0ij)CklNh1(i,j)(Cklij+)2[(Lf)2CklNh1(i,j)ea_0ijτkl+(2κ)21˙τ+kl+(Mf)22]+CklNh3(i,j)(Eklij+)2(Lδij)2e2a_0ijσ+ij1˙σ+ij+12a_0ijCklNh2(i,j)(Bklij+)2(4κ2L2g+M2g)×(0|Kij(u)|du)2+(ˉacij)22a_0ij}<1,(p=2).

    (A5) The kernel Kij is almost periodic and there exist constants M>0 and u>0 such that |Kij(t)|Meut for all tR.

    Let X indicate the space of all Lp-bounded and Lp-uniformly continuous stochastic processes from R to Lp(Ω,Am×n), then with the norm ϕX=suptR{Eϕ(t)p0}1p, where ϕ=(ϕ11,ϕ12,,ϕmn)X, it is a Banach space.

    Set ϕ0=(ϕ011,ϕ012,,ϕ0mn)T, where ϕ0ij(t)=tetsa0ij(u)duLij(s)ds,tR,ijΛ. Then, ϕ0 is well defined under assumption (A2). Consequently, we can take a constant κ such that κϕ0X.

    Definition 3.1. [37] An Ft-progressively measurable stochastic process x(t)=(x11(t),x12(t),,xmn(t))T is called a solution of system (2.1), if x(t) solves the following integral equation:

    xij(t)=xij(t0)ett0a0ij(u)du+tt0etsa0ij(u)du[acij(s)xij(s)+CklNh1(i,j)Cklij(s)×f(xkl(sτkl(s)))xij(s)+CklNh2(i,j)Bklij(s)0Kij(u)g(x(su))duxij(s)+Lij(s)]ds+tt0etsa0ij(u)duCklNh3(i,j)Eklij(s)δij(xij(sσij(s)))dwij(s). (3.1)

    In (3.1), let t0, then one gets

    xij(t)=tetsa0ij(u)du[acij(s)xij(s)+CklNh1(i,j)Cklij(s)f(xkl(sτkl(s)))xij(s)+CklNh2(i,j)Bklij(s)0Kij(u)g(x(su))duxij(s)+Lij(s)]ds+tetsa0ij(u)du×CklNh3(i,j)Eklij(s)δij(xij(sσij(s)))dwij(s),tt0,ijΛ. (3.2)

    It is easy to see that if x(t) solves (3.2), then it also solves (2.1).

    Theorem 3.1. Assume that (A1)(A4) hold. Then the system (2.1) has a unique Lp-bounded and Lp-uniformly continuous solution in X={ϕX:ϕϕ0Xκ}, where κ is a constant satisfying κϕ0X.

    Proof. Define an operator ϕ:XX as follows:

    (Ψϕ)(t)=((Ψ11ϕ)(t),(Ψ12ϕ)(t),,(Ψmnϕ)(t))T,

    where (ϕ11,ϕ12,,ϕmn)TX, tR and

    (Ψijϕ)(t)=tetsa0ij(u)du[acij(s)ϕij(s)+CklNh1(i,j)Cklij(s)f(ϕkl(sτkl(s)))ϕij(s)+CklNh2(i,j)Bklij(s)0Kij(u)g(ϕkl(su))duϕij(s)+Lij(s)]ds+tetsa0ij(u)duCklNh3(i,j)Eklij(s)δij(ϕij(sσij(s)))dωij(s),ijΛ. (3.3)

    First of all, let us show that EΨϕ(t)ϕ0(t)p0κ for all ϕX.

    Noticing that for any ϕX, it holds

    ϕXϕ0X+ϕϕ0X2κ.

    Then, we deduce that

    EΨϕ(t)ϕ0(t)p04p1maxijΛ{Etetsa0ij(u)duacij(s)ϕij(s)p}+4p1maxijΛ{Etetsa0ij(u)du×CklNh1(i,j)Cklij(s)f(ϕkl(sτkl(s)))ϕij(s)dsp}+4p1maxijΛ{Etetsa0ij(u)du×CklNh2(i,j)Bklij(s)0Kij(u)g(ϕkl(su))duϕij(s)dsp}+4p1maxijΛ{Etetsa0ij(u)duCklNh3(i,j)Eklij(s)δij(ϕij(sσij(s)))dωij(s)p}:=F1+F2+F3+F4. (3.4)

    By the Hölder inequality, we have

    F24p1maxijΛ{E[teqptsa0ij(u)duds]pq[tepqtsa0ij(u)du×(CklNh1(i,j)Cklij(s)f(ϕkl(sτkl(s)))ϕij(s))pds]}4p1maxijΛ{(pqa_0ij)pqE[tepqtsa0ij(u)du(CklNh1(i,j)(Cklij(s))q)pq×ijΛ(2κLf)pϕij(s)pds]}4p1maxijΛ{(pqa_0ij)pqqpa_0ij(CklNh1(i,j)(Cklij+)q)pq(2κLf)p}ϕpX. (3.5)

    Similarly, one has

    F14p1maxijΛ{(pqa_0ij)pqqpa_0ij(ˉacij)p}ϕpX, (3.6)
    F34p1maxijΛ{(pqa_0ij)pqqpa_0ij(CklNh2(i,j)(Bklij+)q)pq(2κLg0|Kij(u)|du)p}ϕpX. (3.7)

    By the Burkolder-Davis-Gundy inequality and the Hölder inequality, when p>2, we infer that

    F44p1CpmaxijΛ{E[tetsa0ij(u)duCklNh3(i,j)Eklij(s)δij(ϕij(sσij(s)))2ds]p2}4p1CpmaxijΛ{E[e2tsa0ij(u)duCklNh3(i,j)Eklijδij(ϕij(sσij(s)))2ds]p2}4p1CpmaxijΛ{E[t(e2tsa0ij(u)du)pp2×1pds]p2p×p2×E[t(e2tsa0ij(u)du)1q×p2(CklNh3(i,j)Eklij(s)δijϕij(sσij(s))2)p2ds]}4p1CpmaxijΛ{(p22a_0ij)p22qpa_0ijECklNh3(i,j)Eklij(s)δij(ϕij(sσij(s)))p}4p1CpmaxijΛ{(p22a_0ij)p22qpa_0ij(CklNh3(i,j)(Eklij+)q)pq(Lδij)p}ϕpX. (3.8)

    When p=2, by the Itˆo isometry, it follows that

    F44maxijΛ{E[te2tsa0ij(u)duCklNh3(i,j)Eklij(s)δij(ϕij(sσij(s)))2Ads]}4maxijΛ{12a_0ijCklNh3(i,j)(Eklij+)2(Lδij)2}ϕ2X. (3.9)

    Putting (3.5)–(3.9) into (3.4), we obtain that

    Ψϕϕ0pX4p1maxijΛ{(pqa_0ij)pqqpa_0ij[(ˉacij)p+(CklNh1(i,j)(Cklij+)q)pq(2κLf)p+(CklNh2(i,j)(Bklij+)q)pq(2κLg0|Kij(u)|du)p]+Cp(p22a_0ij)p22qpa_0ij(CklNh3(i,j)(Eklij+)q)pq(Lδij)p}ϕpXκp,(p>2), (3.10)

    and

    Ψϕϕ02X4maxijΛ{1(aij)2[(ˉacij)2+CklNh1(i,j)(Cklij+)2(2κLf)2+CklNh2(i,j)(Bklij+)2(2κLg×0|Kij(u)|du)2]+12a_0ijCklNh3(i,j)(Eklij+)2(Lδij)2}ϕ2Xκ2,(p=2). (3.11)

    It follows from (3.10), (3.11) and (A3) that Ψϕϕ0Xκ.

    Then, using the same method as that in the proof of Theorem 3.2 in [21], we can show that Ψϕ is Lp-uniformly continuous. Therefore, we have Ψ(X)X.

    Last, we will show that Ψ is a contraction mapping. Indeed, for any ψ,φX, when p>2, we have

    E(Φφ)(t)(Φψ)(t)p04p1maxijΛ{Etetsa0ij(u)du(acij(s)φij(s)+acij(s)ψij(s))dsp}+4p1maxijΛ{Etetsa0ij(u)duCklNh1(i,j)Cklij(s)[f(φkl(sτkl(s)))φij(s)f(ψkl(sτkl(s)))ψij(s)]dsp}+4p1maxijΛ{Etetsa0ij(u)duCklNh2(i,j)Bklij(s)×[0Kij(u)g(φkl(su))duφij(s)0Kij(u)g(ψkl(su))duψij(u)]dsp}+4p1maxijΛ{Etetsa0ij(u)duCklNh3(i,j)Eklij(s)[δij(φij(sσij(s)))δij(ψij(sσij(s)))]dωij(s)p}4p1maxijΛ{(pqa_0ij)pqqpa_0ij[(ˉacij)p+(CklNh1(i,j)(Cklij+)q)pq(2κLf+Mf)p+(CklNh2(i,j)(Bklij+)q)pq((2κLg+Mg)0|Kij(u)|du)p]+Cp(p22a_0ij)p22qpa_0ij×(CklNh3(i,j)(Eklij+)q)pq(Lδij)p}φψpX. (3.12)

    Similarly, for p=2, we can get

    E(Φφ)(t)(Φψ)(t)204maxijΛ{1(a_0ij)2[(ˉacij)2+CklNh1(i,j)(Cklij+)2(2κLf+Mf)2+CklNh2(i,j)(Bklij+)2×((2κLg+Mg)0|Kij(u)|du)2]+12a_0ijCklNh3(i,j)(Eklij+)2(Lδij)2}φψ2X. (3.13)

    From (3.12) and (3.13) it follows that

    (Φφ)(t)(Φψ)(t)Xpr1φψX,(p>2),(Φφ)(t)(Φψ)(t)Xr2φψX,(p=2).

    Hence, by virtue of (A3) , \Psi is a contraction mapping. So, \Psi has a unique fixed point x in \mathbb{X}^* , i.e., (2.1) has a unique solution x in \mathbb{X}^* .

    Theorem 3.2. Assume that (A1) (A5) hold. Then the system (2.1) has a unique p -th Stepanov-like almost periodic solution in the distribution sense in \mathbb{X}^* = \{\phi\in\mathbb{X}:\|\phi-\phi^0\|_{\mathbb{X}}\leq \kappa\} , where \kappa is a constant satisfying \kappa\geq \|\phi^0\|_{\mathbb{X}} .

    Proof. From Theorem 3.1, we know that (2.1) has a unique solution x in \mathbb{X}^* . Now, let us show that x is Stepanov-like almost periodic in distribution. Since x\in \mathbb{X}^* , it is \mathcal{L}^p -uniformly continuous and satisfies \|x\|\leq 2\kappa . So, for any \varepsilon > 0 , there exists \delta\in (0, \varepsilon) , when |h| < \delta , we have \sup_{t\in\mathbb{R}} E\| x(t+h)-x(t) \|_0^p < \varepsilon . Hence, we derive that

    \begin{align} \sup\limits_{\xi\in\mathbb{R}} \int_\xi^{\xi+1} E\| x(t+h)-x(t) \|_0^p dt < \varepsilon. \end{align} (3.14)

    For the \delta above, according to (A2) , we have, for ij\in\Lambda ,

    \begin{align*} & |a_{ij}^0(t+\tau)-a_{ij}^0(t)| < \delta,\; \; \|a_{ij}^c(t+\tau)-a_{ij}^c(t)\|_{\flat}^p < \delta, \; \; \|C_{ij}^{kl}(t+\tau)-C_{ij}^{kl}(t)\|_{\flat}^p < \delta,\; \; \\ &|\tau_{ij}(t+\tau)-\tau_{ij}(t)| < \delta,\; \; \|B_{ij}^{kl}(t+\tau)-B_{ij}^{kl}(t)\|_{\flat}^p < \delta, \; \; \|E_{ij}^{kl}(t+\tau)-E_{ij}^{kl}(t)\|_{\flat}^p < \delta,\; \; \\ & |\sigma_{ij}(t+\tau)-\sigma_{ij}(t)| < \delta,\; \; \sup\limits_{\xi\in\mathbb{R}}\int_\xi^{\xi+1} \| L_{ij}(t+\tau)-L_{ij}(t) \|_{\flat}^p dt < \delta. \end{align*}

    As |\tau_{ij}(t+\tau)-\tau_{ij}(t)| < \delta , by (3.14), there holds

    \begin{align*} \sup\limits_{\xi\in\mathbb{R}}\int_\xi^{\xi+1} E\| x(s-\tau_{ij}(s+\tau))-x(s-\tau_{ij}(s)) \|_{0}^p ds < \varepsilon. \end{align*}

    Based on (3.2), we can infer that

    \begin{align*} x_{ij}(t+\tau) = & \int_{-\infty}^{t}e^{-\int_{s}^ta_{ij}^0(u+\tau)du}\bigg[ -a_{ij}^c(s+\tau)x_{ij}(s+\tau) +\sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s+\tau) \notag \\ & \times f(x_{kl}(s+\tau-\tau_{kl}(s+\tau))) x_{ij}(s+\tau)+ \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s+\tau)\int_{0}^{\infty}K_{ij}(u) \notag\\ & \times g(x_{kl}(s+\tau-u))dux_{ij}(s+\tau) +L_{ij}(s+\tau) \bigg]ds +\int_{-\infty}^te^{-\int_{s}^ta_{ij}(u+\tau)du} \notag\\ & \times \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s+\tau)\delta_{ij}(x_{ij}(s+\tau-\sigma_{ij}(s+\tau)))d[\omega_{ij}(s+\tau) -\omega_{ij}(\tau) ], \end{align*}

    in which ij\in\Lambda, \omega_{ij}(s+\tau)-\omega_{ij}(\tau) is a Brownian motion having the same distribution as \omega_{ij}(s) .

    Let us consider the process

    \begin{align} x_{ij}(t+\tau) = & \int_{-\infty}^{t}e^{-\int_{s}^ta_{ij}^0(u+\tau)du}\bigg[ -a_{ij}^c(s+\tau)x_{ij}(s+\tau) +\sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s+\tau) \\ & \times f(x_{kl}(s+\tau-\tau_{kl}(s+\tau))) x_{ij}(s+\tau)+ \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s+\tau)\int_{0}^{\infty}K_{ij}(u) \\ & \times g(x_{kl}(s+\tau-u))dux_{ij}(s+\tau) +L_{ij}(s+\tau) \bigg]ds +\int_{-\infty}^te^{-\int_{s}^ta_{ij}(u+\tau)du} \\ & \times \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s+\tau)\delta_{ij}(x_{ij}(s+\tau-\sigma_{ij}(s+\tau)))d\omega_{ij}(s). \end{align} (3.15)

    From (3.2) and (3.15), we deduce that

    \begin{align} & \int_\xi^{\xi+1} E\| x(t+\tau)-x(t) \|_0^p dt \\ \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_1}(i,j)} C_{ij}^{kl}(s+\tau) \\ & \times \bigg[f(x_{kl}(s+\tau-\tau_{kl}(s+\tau)))x_{ij}(s+\tau)-f(x_{kl}(s-\tau_{kl}(s)))x_{ij}(s+\tau)\bigg] ds\bigg\|_{\flat}^p dt \bigg\} \\ &+ 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_1}(i,j)} \big(C_{ij}^{kl}(s+\tau) - C_{ij}^{kl}(s) \big) \\ & \times f(x_{kl}(s-\tau_{kl}(s)))x_{ij}(s+\tau) ds\bigg\|_{\mathscr{A} }^p dt \bigg\} + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \\ &\sum\limits_{C_{kl}\in N_{h_1}(i,j)} C_{ij}^{kl}(s) \times f(x_{kl}(s-\tau_{kl}(s)))\big( x_{ij}(s+\tau) - x_{ij}(s) \big)ds\bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t \big(e^{-\int_s^t a_{ij}^0(u+\tau)du} - e^{-\int_s^t a_{ij}^0(u)du} \big) \sum\limits_{C_{kl}\in N_{h_1}(i,j)} C_{ij}^{kl}(t) \\ & \times f(x_{kl}(s-\tau_{kl}(s)))x_{ij}(s) ds\bigg\|_{\flat}^p dt \bigg\} + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \\ &\sum\limits_{C_{kl}\in N_{h_2}(i,j)} B_{ij}^{kl}(s+\tau) \bigg[\int_0^{\infty}K_{ij}(u) g(x_{kl}(s+\tau-u)) du x_{ij}(s+\tau) - \int_0^{\infty}K_{ij}(u) \\ & \times g(x_{kl}(s-u)) du x_{ij}(s+\tau)\bigg] ds\bigg\|_{\flat}^p dt \bigg\} + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \\ & \sum\limits_{C_{kl}\in N_{h_2}(i,j)} \big(B_{ij}^{kl}(s+\tau) - B_{ij}^{kl}(s) \big) \int_0^{\infty}K_{ij}(u) g(x_{kl}(s-u)) du x_{ij}(s+\tau) ds\bigg\|_{\flat}^p dt \bigg\} \\ &+ 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_2}(i,j)} B_{ij}^{kl}(s) \int_0^{\infty}K_{ij}(u) g(x_{kl}(s-u)) du \\ & \times \big( x_{ij}(s+\tau) - x_{ij}(s) \big)ds\bigg\|_{\flat}^p dt \bigg\} + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t \big(e^{-\int_s^t a_{ij}^0(u+\tau)du} \\ & - e^{-\int_s^t a_{ij}^0(u)du} \big) \sum\limits_{C_{kl}\in N_{h_2}(i,j)} B_{ij}^{kl}(s) \int_0^{\infty}K_{ij}(u) g(x_{kl}(s-u)) du x_{ij}(s) ds\bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} (L_{ij}(s+\tau) - L_{ij}(s)) ds \bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t \big(e^{-\int_s^t a_{ij}^0(u+\tau)du} - e^{-\int_s^t a_{ij}^0(u)du} \big) L_{ij}(s) ds \bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)} E_{ij}^{kl}(s+\tau) \\ & \times \bigg[ \delta_{ij}(x_{ij}(s+\tau-\sigma_{ij}(s+\tau)))- \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) \bigg] d\omega_{ij}(s) \bigg\|_{\mathscr{A} }^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)} \big(E_{ij}^{kl}(s+\tau) - E_{ij}^{kl}(s) \big) \\ & \times \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) d\omega_{ij}(s) \bigg\|_{\flat}^p dt \bigg\} + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t \big(e^{-\int_s^t a_{ij}^0(u+\tau)du} \\ & - e^{-\int_s^t a_{ij}^0(u)du} \big) \sum\limits_{C_{kl}\in N_{h_3}(i,j)} E_{ij}^{kl}(t) \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) d\omega_{ij}(s) \bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}^0(u+\tau)du} \big( a_{ij}^c(s)-a_{ij}^c(s+\tau) \big)x_{ij}(s+\tau) ds \bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t e^{-\int_s^t a_{ij}(u+\tau)du} a_{ij}^c(s)\big( x_{ij}(s)-x_{ij}(s+\tau) \big) ds \bigg\|_{\flat}^p dt \bigg\} \\ & + 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg\| \int_{-\infty}^t \big(e^{-\int_s^t a_{ij}(u+\tau)du} - e^{-\int_s^t a_{ij}(u)du} \big) (-a_{ij}^c(s))x_{ij}(s) ds \bigg\|_{\flat}^p dt \bigg\} \\ : = & \sum\limits_{i = 1}^{16} H_i. \end{align} (3.16)

    Employing the Hölder inequality, we can obtain

    \begin{align*} H_1 \leq & 32^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_f )^p \sum\limits_{C_{kl}\in N_{h_1}(i,j)} \int_\xi^{\xi+1} \bigg[ \int_{-\infty}^t e^{-\frac{p}{q}(t-s)\underline{a}_{ij}^0} \notag \\ & \times E \big\| \big [x(s+\tau-\tau_{kl}(s+\tau) ) - x(s-\tau_{kl}(s+\tau) ) \big] x(t+\tau) \big\|_0^p ds \notag\\ & + \int_{-\infty}^t e^{-\frac{p}{q}(t-s)\underline{a}_{ij}^0} E \big\| \big[ x(s-\tau_{kl}(s+\tau) ) - x(s-\tau_{kl}(s) ) \big] x(t+\tau) \big\|_0^p ds \bigg] dt \bigg\}. \end{align*}

    By a change of variables and Fubini's theorem, we infer that

    \begin{align} H_1 \leq & 32^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_f )^p \\ & \times \sum\limits_{C_{kl}\in N_{h_1}(i,j)} \int_\xi^{\xi+1} \bigg[ \int_{-\infty}^{t-\tau_{kl}(t+\tau)} \frac{1 }{1-{\dot{\tau}}_{kl}(s+\tau) } e^{-\frac{p}{q}}\underline{a}_{ij}^0(t-u-\tau_{kl}(s+\tau) ) \\ & \times E \big\| \big [x(u+\tau) - x(u) \big] x(t+\tau) \big\|_0^p du + \frac{q{\varepsilon}^p (2\kappa)^p } {p\underline{a}_{ij}^0 } \bigg] dt \bigg\} \\ \leq & 32^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_f )^p \sum\limits_{C_{kl}\in N_{h_1}(i,j)}\frac{e^{\frac{p}{q}}\underline{a}_{ij}^0 {\tau_{kl}}^+ (2\kappa)^p } {1-{\dot{\tau}}_{kl}^+ } \\ & \times \int_{-\infty}^{\xi} e^{-\frac{p}{q}(\xi-s)\underline{a}_{ij}^0} \bigg( \int_{s}^{s+1} E \big\| x(t+\tau) - x(t) \big\|_0^p dt \bigg) ds \bigg\} + \bigtriangleup_{H_1} \varepsilon , \end{align} (3.17)

    where

    \begin{align*} \bigtriangleup_{H_1} = 32^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \frac{q (2\kappa)^p } {p\underline{a}_{ij}^0 } \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_f )^p {\varepsilon}^{p-1} \bigg\} . \end{align*}

    Similarly, when p > 2 , one can obtain

    \begin{align} H_{11} \leq & 16^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg[ \int_{-\infty}^t e^{-2\int_s^t a_{ij}^0(u+\tau)du} E \bigg\| \sum\limits_{C_{kl}\in N_{h_2}(i,j)} E_{ij}^{kl}(s+\tau) \\ & \times \big( \delta_{ij}(x_{ij}(s+\tau-\sigma_{ij}(s+\tau)))- \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) \big) \bigg\|_{\flat}^2 ds \bigg]^{\frac{p}{2}} dt \bigg\} \\ \leq & 32^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \big(\frac{p-2}{2\underline{a}_{ij}^0} \big)^{\frac{p-2} {2} } \big( \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({E_{ij}^{kl}}^+ )^q \big)^{\frac{p}{q}} ( L_{ij}^{\delta})^p \frac{e^{\frac{p}{q}\underline{a}_{ij}^0 \sigma_{ij}^+ } } {1-\dot{\sigma}_{ij}^+ } \int_{-\infty}^{\xi} e^{-\frac{p}{q}(\xi-s)\underline{a}_{ij}^0 } \\ & \times \bigg( \int_s^{s+1} E \big\| x(t+\tau) - x(t) \big\|_0^p dt \bigg) ds\bigg\} +\bigtriangleup_{H_{11}^1 } \varepsilon, \end{align} (3.18)

    where

    \begin{align*} \bigtriangleup_{H_{11}^1} = 32^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \big(\frac{p-2}{2\underline{a}_{ij}^0} \big)^{\frac{p-2} {2} } \big( \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+ )^q \big)^{\frac{p}{q}} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}( L_{ij}^{\delta})^p \frac{q}{p\underline{a}_{ij}^0} {\varepsilon}^{p-1} \bigg\} , \end{align*}
    \begin{align} H_{12} \leq & 16^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg[ \int_{-\infty}^t e^{-2\int_s^t a_{ij}^0(u+\tau)du} E \bigg\| \sum\limits_{C_{kl}\in N_{h_3}(i,j)} \big( E_{ij}^{kl}(s+\tau) - E_{ij}^{kl}(s) \big) \\ & \times \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) \bigg\|_{\mathscr{A} }^2 ds \bigg]^{\frac{p}{2}} dt \bigg\} \\ \leq & 16^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \big(\frac{p}{q\underline{a}_{ij}^0} \big)^{\frac{p} {q} } \frac{q}{p\underline{a}_{ij}^0} ( L_{ij}^{\delta})^p \frac{e^{\frac{p}{q}\underline{a}_{ij}^0 \sigma_{ij}^+ } } {1-\dot{\sigma}_{ij}^+ } (2\kappa)^p \bigg\} \varepsilon^{\frac{p}{q}} : = \bigtriangleup_{H_{12}^1} \varepsilon , \end{align} (3.19)

    and when p = 2 , we have

    \begin{align} H_{11} \leq & 16 \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg[ \int_{-\infty}^t e^{-2\int_s^t a_{ij}^0(u+\tau)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^2 \\ & \times \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ( L_{ij}^{\delta} )^2 \| x_{ij}(s+\tau-\sigma_{ij}(s+\tau) ) - x_{ij}(s-\sigma_{ij}(s) ) \|_{\flat}^2 ds \bigg] dt \bigg\} \\ \leq & 32 \max\limits_{ij\in\Lambda} \bigg\{ \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^2 ( L_{ij}^{\delta} )^2 \frac{e^{2\underline{a}_{ij}^0 \sigma_{ij}^+} }{1-\dot{\sigma}_{ij}^+ } \int_{-\infty}^{\xi} e^{-2(\xi-s)\underline{a}_{ij}^0 } \\ & \times \bigg( \int_{s}^{s+1} E \| x(t+\tau) - x(t) \|_0^2 dt \bigg) ds \bigg\} + \bigtriangleup_{H_{11}^2} \varepsilon, \end{align} (3.20)

    where

    \begin{align*} \bigtriangleup_{H_{11}^2} = \frac{32}{\underline{a}_{ij}^0} \max\limits_{ij\in\Lambda} \bigg\{ \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^2 \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ( L_{ij}^{\delta} )^2 {\varepsilon} \bigg\} , \end{align*}
    \begin{align} H_{12} \leq & 16 \max\limits_{ij\in\Lambda} \bigg\{ \frac{1}{(\underline{a}_{ij}^0)^2} ( L_{ij}^{\delta})^2 \frac{e^{\underline{a}_{ij}^0 \sigma_{ij}^+ } } {1-\dot{\sigma}_{ij}^+ } 4\kappa^2 \bigg\} \varepsilon : = \bigtriangleup_{H_{12}^2} \varepsilon. \end{align} (3.21)

    In the same way, we can get

    \begin{align} H_2 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ ( \frac{mnp}{q\underline{a}_{ij}^0 })^{\frac{p}{q}} \frac{q}{p \underline{a}_{ij}^0 } (2\kappa M_f )^p \bigg\} \varepsilon^{\frac{p}{q} } : = \bigtriangleup_{H_{2}} \varepsilon, \end{align} (3.22)
    \begin{align} H_3 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ ( \frac{p}{q\underline{a}_{ij}^0 })^{\frac{p}{q}} \bigg ( \sum\limits_{C_{kl}\in N_{h_1}(i,j) } \big( {C_{ij}^{kl}}^+ \big)^q \bigg)^{\frac{p}{q}} (M_f )^p \int_{-\infty}^\xi e^{-\frac{p}{q} (\xi-s) \underline{a}_{ij}^0} \\ & \times \bigg(\int_s^{s+1} E \| x(t+\tau) - x(t) \|_0^p dt \bigg) ds \bigg\}, \end{align} (3.23)
    \begin{align} H_5 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0 } \bigg)^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_2}(i,j) } \big( {B_{ij}^{kl}}^+ \big)^q \bigg)^{\frac{p}{q}} \bigg( 2\kappa L_g\int_0^{\infty} |K_{ij}(u)| du \bigg)^p \int_{-\infty}^\xi e^{-\frac{p}{q} (\xi-s) \underline{a}_{ij}^0} \\ & \times \bigg(\int_s^{s+1} E \| x(t+\tau) - x(t) \|_0^p dt \bigg) ds\bigg\}, \end{align} (3.24)
    \begin{align} H_6 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ ( \frac{mnp}{q\underline{a}_{ij}^0 })^{\frac{p}{q}} \frac{q}{p \underline{a}_{ij}^0 } \bigg( (2\kappa M_g )\int_0^{\infty}|K_{ij}(u)|du \bigg)^p \bigg\} \varepsilon^{\frac{p}{q} } : = \bigtriangleup_{H_{6}} \varepsilon, \end{align} (3.25)
    \begin{align} H_7 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ ( \frac{p}{q\underline{a}_{ij}^0 })^{\frac{p}{q}} \bigg( \sum\limits_{C_{kl}\in N_{h_2}(i,j) } \big( {B_{ij}^{kl}}^+ \big)^q \bigg)^{\frac{p}{q}} \big( M_g\int_0^\infty |K_{ij}(u)|du \big)^p \int_{-\infty}^\xi e^{-\frac{p}{q} (\xi-s) \underline{a}_{ij}^0} \\ & \times \bigg(\int_s^{s+1} E \| x(t+\tau) - x(t) \|_0^p dt \bigg) ds \bigg\}, \end{align} (3.26)
    \begin{align} H_9 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \big( \frac{p}{q \underline{a}_{ij}^0}\big)^{\frac{p}{q}} \frac{q}{p \underline{a}_{ij}^0} \bigg\} \varepsilon^{p } : = \bigtriangleup_{H_{9}} \varepsilon, \end{align} (3.27)
    \begin{align} H_{14} \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \big( \frac{p}{q \underline{a}_{ij}^0}\big)^{\frac{p}{q}} \frac{q}{p \underline{a}_{ij}^0} (2\kappa)^p \bigg\} \varepsilon^{p } : = \bigtriangleup_{H_{14}} \varepsilon, \end{align} (3.28)
    \begin{align} H_{15} \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ ( \frac{p}{q\underline{a}_{ij}^0 })^{\frac{p}{q}} (\bar{a}_{ij}^c )^p \int_{-\infty}^\xi e^{-\frac{p}{q} (\xi-s) \underline{a}_{ij}^0} \bigg(\int_s^{s+1} E \| x(t+\tau) - x(t) \|_0^p dt \bigg) ds \bigg\}. \end{align} (3.29)

    Noting that

    \begin{align} &\bigg[ \int_{-\infty}^t \big| e^{-\int_s^t a_{ij}^0(u+\tau)du } - e^{-\int_s^t a_{ij}^0(u)du} \big|^{\frac{q}{p}} ds \bigg]^{\frac{p}{q}} \\ \leq & \bigg[ \int_{-\infty}^t e^{-\frac{q}{p}\underline{a}_{ij}^0 (t-s) } \bigg( \int_s^t | a_{ij}^0(u+\tau) -a_{ij}^0(u) | du \bigg)^{\frac{q}{p}} ds \bigg]^{\frac{p}{q}} \leq \bigg( \Gamma\big(\frac{q+p}{p}\big) \bigg)^{\frac{p}{q}} \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p+q}{q}} \varepsilon . \end{align} (3.30)

    We can gain

    \begin{align} H_4 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \Gamma\big(\frac{q+p}{p} \big) \bigg)^{\frac{p}{q}} \Gamma\big( \frac{q+p}{p} \big) \big(\frac{1}{\underline{a}_{ij}^0}\big)^{\frac{2(p+q)}{q}} \\ & \times \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (2\kappa M_f )^{p} \bigg\} \varepsilon^{\frac{p}{q}+1} : = \bigtriangleup_{H_{4}} \varepsilon, \end{align} (3.31)
    \begin{align} H_8 \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \Gamma\big(\frac{q+p}{p} \big) \bigg)^{\frac{p}{q}} \Gamma\big( \frac{q+p}{p} \big) \big(\frac{1}{\underline{a}_{ij}^0}\big)^{\frac{2(p+q)}{q}} \\ & \times \bigg( \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (2\kappa M_g\int_0^{\infty} |K_{ij}(u)|du )^p \bigg\} \varepsilon^{\frac{p}{q}+1} : = \bigtriangleup_{H_{8}} \varepsilon, \end{align} (3.32)
    \begin{align} H_{10} \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \Gamma\big(\frac{q+p}{p} \big) \bigg)^{\frac{p}{q}} \Gamma\big( \frac{q+p}{p} \big) \big(\frac{1}{\underline{a}_{ij}^0}\big)^{\frac{2(p+q)}{q}} (M_L)^p \bigg\} \varepsilon^{\frac{p}{q}+1} : = \bigtriangleup_{H_{10}} \varepsilon, \end{align} (3.33)
    \begin{align} H_{16} \leq & 16^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \Gamma\big(\frac{q+p}{p} \big) \bigg)^{\frac{p}{q}} \Gamma\big( \frac{q+p}{p} \big) \big(\frac{1}{\underline{a}_{ij}^0}\big)^{\frac{2(p+q)}{q}} (2\kappa \bar{a}_{ij}^c)^p \bigg\} \varepsilon^{\frac{p}{q}+1} : = \bigtriangleup_{H_{16}} \varepsilon, \end{align} (3.34)

    when p > 2 , we have

    \begin{align} H_{13} \leq & 16^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \int_\xi^{\xi+1} E \bigg[ \int_{-\infty}^t \bigg( e^{-\int_s^t a_{ij}^0(u+\tau)du } - e^{-\int_s^t a_{ij}^0(u)du} \bigg)^2 \\ & \times \sum\limits_{C_{kl}\in N_{h_3}(i,j)} E_{ij}^{kl}(s) \| \delta_{ij}(x_{ij}(s-\sigma_{ij}(s))) \|_{\flat}^2 ds \bigg]^{\frac{p}{2}} dt \bigg\} \\ \leq & 16^{p-1} C_p \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \Gamma\big(\frac{p}{p-2} \big) \bigg)^{\frac{p-2}{2}} \big( \frac{p-2}{2\underline{a}_{ij}^0} \big)^{\frac{p}{2}} \\ & \times \frac{q}{p\underline{a}_{ij}^0} \bigg( \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_{ij}^{\delta})^p \frac{e^{\frac{p}{q}\underline{a}_{ij}^0 \sigma_{ij}^+} } {1-{\dot{\sigma}}_{ij}^+} (2\kappa)^{p} \bigg\} \varepsilon : = \bigtriangleup_{H_{13}^1} \varepsilon, \end{align} (3.35)

    for p = 2 , we get

    \begin{align} H_{13} \leq & 16 \max\limits_{ij\in\Lambda} \bigg\{ \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^2 (L_{ij}^{\delta})^2 \frac{e^{2\underline{a}_{ij}^0 \sigma_{ij}^+} } {1-{\dot{\sigma}}_{ij}^+} \frac{ \Gamma(3) }{8(\underline{a}_{ij}^0)^3} (2\kappa)^{2} \bigg\} \varepsilon : = \bigtriangleup_{H_{13}^2} \varepsilon . \end{align} (3.36)

    Substituting (3.17)–(3.36) into (3.16), we have the following two cases:

    Case 1. When p > 2 , we have

    \begin{align*} & \int_\xi^{\xi+1} E\| x(t+\tau) -x(t) \|_0^p dt \\ \leq & H^1 \varepsilon + \rho^1 \int_{-\infty}^\xi e^{-(\xi-s)\frac{p}{q}\underline{a}^0}\bigg( \int_s^{s+1} E\| x(t+\tau)-x(t) \|_0^p dt \bigg) ds\\ \leq & H^1 \varepsilon + \rho^1\sup\limits_{s\in \mathbb{R}}\bigg( \int_s^{s+1} E\| x(t+\tau)-x(t) \|_0^p dt \bigg) \int_{-\infty}^\xi e^{-(\xi-s)\frac{p}{q}\underline{a}^0} ds\\ \leq & H^1 \varepsilon + \rho^1\frac{q}{p\underline{a}^0}\sup\limits_{s\in \mathbb{R}}\bigg( \int_s^{s+1} E\| x(t+\tau)-x(t) \|_0^p dt \bigg), \end{align*}

    where \rho^1 is the same as that in (A3) and H^1 = \bigtriangleup_{H_1} + \bigtriangleup_{H_2} +\bigtriangleup_{H_4} +\bigtriangleup_{H_6} +\bigtriangleup_{H_8} +\bigtriangleup_{H_9} +\bigtriangleup_{H_{10}} +\bigtriangleup_{H_{14}} +\bigtriangleup_{H_{16}} +\bigtriangleup_{H_{11}^1} +\bigtriangleup_{H_{12}^1} +\bigtriangleup_{H_{13}^1}.

    By (A4) , we know \rho^1 < \frac{p\underline{a}^0}{q} . Hence, we derive that

    \begin{align} \sup\limits_{\xi\in \mathbb{R}}\int_\xi^{\xi+1} E\| x(t+\tau)-x(t) \|_0^p dt \; < \; \frac{p\underline{a}^0 H^1}{p\underline{a}^0 - \rho^1 q} \varepsilon. \end{align} (3.37)

    Case 2. When p = 2 , we can obtain

    \begin{align*} & \int_\xi^{\xi+1} E\| x(t+\tau)-x(t) \|_0^2 dt \\ \leq & H^2 \varepsilon + \rho^2 \int_{-\infty}^\xi e^{-(\xi-s)\frac{p}{q}\underline{a}^0 }\bigg( \int_s^{s+1} E\| x(t+\tau)-x(t) \|_0^2 dt \bigg) ds, \end{align*}

    where \rho^2 is defined in (A4) and

    \begin{align*} H^2 = & 32 \max\limits_{ij\in\Lambda} \bigg\{ \bigg(\frac{2\kappa} {\underline{a}_{ij}^0 } \bigg)^{2} \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^2 (L_f )^2 \bigg\} {\varepsilon} + 16 \max\limits_{ij\in\Lambda} \bigg\{ \frac{m^2n^2}{(\underline{a}_{ij}^0)^2 } \bigg[ (2\kappa M_f )^2 +\bigg(2\kappa M_g \\ & \times \int_0^{\infty}|K_{ij}(u)|du \bigg)^2 \bigg]+ (\Gamma(2))^2 \bigg(\frac{1}{\underline{a}_{ij}^0}\bigg)^{4} \bigg[ \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^2 (2\kappa M_f)^2 + \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{ij}^{kl}}^+)^2 \\ & \times \bigg(2\kappa M_g \int_0^{\infty} |K_{ij}(u)|du \bigg)^2 +(M_L)^2 + (2\kappa \bar{a}_{ij}^c)^2\bigg] + \frac{1}{(\underline{a}_{ij}^0 )^2} \bigg[ \sum\limits_{C_{kl}\in N_{h_2}(i,j) }\big( {B_{ij}^{kl}}^+ \big)^2 \\ & \times \bigg(2\kappa M_g \int_0^{\infty} |K_{ij}(u)| du \bigg)^2 + (2\kappa)^2+1 \bigg] \bigg\} \varepsilon +\bigtriangleup_{H_{11}^2} +\bigtriangleup_{H_{12}^2} +\bigtriangleup_{H_{13}^2}. \end{align*}

    Similar to the previous case, by (A4) , we know \rho^2 < \underline{a}^0 and hence, we can get that

    \begin{align} \int_\xi^{\xi+1} E\| x(t+\tau)-x(t) \|_0^2 dt \; < \; \frac{\underline{a}^0 H^2}{\underline{a}^0 - \rho^2} \varepsilon. \end{align} (3.38)

    Noting that

    \begin{align*} &d_{BL}(P\circ[x(t+\tau)]^{-1},P\circ[x(t)]^{-1})\bigg)\\ \leq &\sup\limits_{\|f\|_{BL}\leq 1}\bigg|\int_{\mathscr{A}^{m\times n}}fd(P\circ[x(t+\tau)]^{-1}-P\circ[x(t)]^{-1})\bigg|\\ \leq&\sup\limits_{\|f\|_{BL}\leq 1}\bigg|\int_{\Omega}f(x(t+\tau))-f(x(t))\bigg|dP\\ \leq&\int_{\Omega}\|x(t+\tau)-x(t)\|_0dP\\ \leq&(E\|x(t+\tau)-x(t)\|_0^p)^{\frac{1}{p}}. \end{align*}

    Hence, we have

    \begin{align} &\sup\limits_{\xi\in\mathbb{R}}\bigg(\int_\xi^{\xi+1}d_{BL}^p(P\circ[x(t+\tau)]^{-1},P\circ[x(t)]^{-1})dt\bigg)^{\frac{1}{p}}\\ \leq &\bigg(\sup\limits_{\xi\in\mathbb{R}}\int_\xi^{\xi+1}E\|x(t+\tau)-x(t)\|_0^pd t\bigg)^{\frac{1}{p}}. \end{align} (3.39)

    Combining (3.37)–(3.39), we can conclude that x(t) is p -th Stepanov almost periodic in the distribution sense. The proof is complete.

    Similar to the proof of Theorem 3.7 in [21], one can easily show that.

    Theorem 3.3. Suppose that (A1) (A5) are fulfilled and let x be the Stepanov almost periodic solution in the distribution sense of system (2.1) with initial value \varphi . Then there exist constants \lambda > 0 and M > 0 such that for an arbitrary solution y with initial value \psi satisfies

    \begin{align*} E\| y(t)-x(t) \|_0^p \leq M \| \varphi -\psi \|_1 e^{-\frac{p}{q}\lambda t}, \; \; t > 0, \end{align*}

    where \| \varphi -\psi \|_1 = \sup_{a\in[-\theta, 0]} E\| \varphi(s) -\psi(s) \|_0^p , i.e., the solution x is globally exponentially stable.

    The purpose of this section is to demonstrate the effectiveness of the results obtained in this paper through a numerical example.

    In neural network (2.1), choose f(x) = \frac{1}{25} \sin x^0 e_0+\frac{1}{30} \sin (x^2+x^0) e_1+\frac{1}{35}\sin^2x^{12} e_2+\frac{1}{40}\tanh^2x^2 e_{12}, g(x) = \frac{1}{40} \sin x^0 e_0+\frac{1}{35} \sin (x^2+x^1) e_1+\frac{1}{30}\sin^2x^{12} e_2+\frac{1}{25}\tanh^2x^1 e_{12} , K_{ij}(t) = \frac{1}{e^{t}} and

    \begin{align*} \begin{bmatrix} a_{11}(t) \\ a_{12}(t) \\ a_{21}(t) \\ a_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} (7+2\text{sin}t) \; e_0 +\text{cos}t \; e_1 +\text{cos}\sqrt5t \; e_2+\text{sin}\sqrt3t \; e_{12} \\ (6-\text{cos}t) e_0+\text{sin}\sqrt3t e_2+\text{cos}\sqrt3t e_{12} \\ (8+\text{cos}\sqrt3t) \; e_0+\text{sin}\sqrt5t \; e_1+\text{sin}t \; e_{12} \\ (10-2\text{sin}\sqrt5t) \; e_0+\text{cos}\sqrt3t \; e_1+\text{cos}t \; e_2+\text{tanh}t \; e_{12} \\ \end{bmatrix}, \end{align*}
    \begin{align*} \begin{bmatrix} C_{11}(t) & C_{12}(t) \\ C_{21}(t) & C_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} B_{11}(t) & B_{12}(t) \\ B_{21}(t) & B_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} E_{11}(t) & E_{12}(t) \\ E_{21}(t) & E_{22}(t) \\ \end{bmatrix} \\ = \begin{bmatrix} 0.02+0.01\text{cost} & 0.03+0.2\sin \sqrt3 \text{t} \\ 0.05+0.07\sin \sqrt5 \text{t} & 0.02+0.05\cos \sqrt3 \text{t}\\ \end{bmatrix}e_0, \end{align*}
    \begin{align*} \begin{bmatrix} L_{11}(t) \\ L_{12}(t) \\ L_{21}(t) \\ L_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} (0.2|\cos|t) \; e_0+(0.2\text{cos}\sqrt3t) \; e_1 + 0.3\text{sin}(\sqrt{3} t) \; e_2 + (0.08\sin t+0.04 e^{-t}) \; e_{12}\\ (0.3\sin(\sqrt{2}t)+e^{-t}) \; e_0+(0.1\cos \sqrt{5} \text{t}+0.04e^{-t}) \; e_1 +0.2e^{-t} \; e_2 + 0.2\text{sin}t \; e_{12} \\ 0.02\sin t\; e_0 + 0.05\text{sin}\sqrt5t \; e_1 + (0.03\cos t+0.01e^{-t}) \; e_2 + 0.02\text{cos}\sqrt{3}t \; e_{12} \\ 0.08\text{sin}t \; e_0+ (0.04\text{cos}\sqrt5t +0.04e^{-t} ) \; e_1 +0.03\text{sin}\sqrt5t \; e_{12} \\ \end{bmatrix}, \end{align*}
    \begin{align*} \begin{bmatrix} \tau_{11}(t) & \tau_{12}(t) \\ \tau_{21}(t) & \tau_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} \sigma_{11}(t) & \sigma_{12}(t) \\ \sigma_{21}(t) & \sigma_{22}(t) \\ \end{bmatrix} = \begin{bmatrix} 0.03+0.009\text{sin0.6t} & 0.05+0.05\text{cos1.2t}\\ 0.02-0.008\text{sin1.1t} & 0.09+0.04\text{sin1.7t} \\ \end{bmatrix}, \end{align*}
    \begin{align*} \begin{bmatrix} \delta_{11}(x) \\ \delta_{12}(x) \\ \delta_{21}(x) \\ \delta_{22}(x) \\ \end{bmatrix} = \begin{bmatrix} \frac{1}{15} \sin \sqrt{3}x^0 e_0+\frac{1}{20} \sin x^2 e_1 +\frac{1}{30}\tanh^2x^1 e_{12} \\ 0.04 \sin x^0 e_1+0.03\sin^2x^{12} e_2+0.05\sin^2x^2 e_{12}\\ 0.02 \tanh x^2 e_0+0.06 \sin x^1 e_1+0.015\sin^2x^{0} e_2 \\ \frac{1}{20} \sin x^1 e_0+\frac{1}{15} \tanh x^2 e_1+\frac{1}{25}\sin x^{12} e_2+\frac{1}{40}\sin x^0 e_{12} \\ \end{bmatrix}, \end{align*}

    and let h_1 = h_2 = 1, h_3 = 0, m = n = 2 . Then we get

    \begin{align*} & L_f = L_g = M_g = M_f = 0.04,\; \underline{a}^0 = 5,\; \bar{a}^c = 1,\; M_L = 1, \; M = u = 1,\; L_{11}^{\delta} = \frac{1}{15} ,\; L_{12}^{\delta} = 0.05,\; \\ & L_{21}^{\delta} = 0.06 ,\; L_{22}^{\delta} = \frac{1}{15},\; \tau_{11}^{+} = \sigma_{11}^+ = 0.039 ,\; \tau_{12}^{+} = \sigma_{12}^+ = 0.1,\; \tau_{21}^{+} = \sigma_{21}^+ = 0.028 ,\; \tau_{22}^{+} = \sigma_{22}^+\\ & = 0.13,\; {\dot{\tau}}_{11}^{+} = {\dot{\sigma}}_{11}^+ = 0.0054 ,\; {\dot{\tau}}_{12}^{+} = {\dot{\sigma}}_{12}^+ = 0.006,\; {\dot{\tau}}_{21}^{+} = {\dot{\sigma}}_{21}^+ = 0.0088 ,\; {\dot{\tau}}_{22}^{+} = {\dot{\sigma}}_{22}^+ = 0.068,\\ & \sum\limits_{C_{kl}\in N_1(1,1)}{C_{11}^{kl}}^+ = \sum\limits_{C_{kl}\in N_1(1,1)}{B_{11}^{kl}}^+ = \sum\limits_{C_{kl}\in N_1(1,2)}{C_{12}^{kl}}^+ = \sum\limits_{C_{kl}\in N_1(1,2)}{B_{12}^{kl}}^+ = \sum\limits_{C_{kl}\in N_1(2,1)}{C_{21}^{kl}}^+ \\ & = \sum\limits_{C_{kl}\in N_1(2,1)}{B_{21}^{kl}}^+ = \sum\limits_{C_{kl}\in N_1(2,2)}{C_{22}^{kl}}^+ \; = \; \sum\limits_{C_{kl}\in N_1(2,2)}{B_{22}^{kl}}^+ = 0.45, \; \sum\limits_{C_{kl}\in N_0(1,1)}{E_{11}^{kl}}^+ = 0.03\\ & \sum\limits_{C_{kl}\in N_0(1,2)}{E_{12}^{kl}}^+ = 0.23,\; \sum\limits_{C_{kl}\in N_0(2,1)}{E_{21}^{kl}}^+ = 0.12,\; \sum\limits_{C_{kl}\in N_0(2,2)}{E_{22}^{kl}}^+ = \; 0.07. \end{align*}

    Take \kappa = 1 , p = \frac{21}{10}, q = \frac{21}{11} , then we have

    \begin{align*} r^1 < 0.6812 < 1, \; \; \; \frac{q}{p\underline{a}^0}\rho^1 < 0.6259 < 1. \end{align*}

    And when p = 2 , we have

    \begin{align*} r^2 < 0.6408 < 1, \; \; \frac{\rho^2}{\underline{a}^0} < 0.6786 < 1. \end{align*}

    Thus, all assumptions in Theorems 3.2 and 3.3 are fulfilled. So we can conclude that the system (2.1) has a unique S^p -almost periodic solution in the distribution sense which is globally exponentially stable.

    The results are also verified by the numerical simulations in Figures 14.

    Figure 1.  Global exponential stability of states x_{11}^0, x_{11}^1, x_{11}^2 and x_{11}^{12} of (2.1).
    Figure 2.  Global exponential stability of states x_{12}^0,x_{12}^1,x_{12}^2 and x_{12}^{12} of (2.1).
    Figure 3.  Global exponential stability of states x_{21}^0,x_{21}^1,x_{21}^2 and x_{21}^{12} of (2.1).
    Figure 4.  Global exponential stability of states x_{22}^0, x_{22}^1, x_{22}^2 and x_{22}^{12} of (2.1).

    From these figures, we can observe that when the four primitive components of each solution of this system take different initial values, they eventually tend to stabilize. It can be seen that these solutions that meet the above conditions do exist and are exponentially stable.

    In this article, we establish the existence and global exponential stability of Stepanov almost periodic solutions in the distribution sense for a class of stochastic Clifford-valued SICNNs with mixed delays. Even when network (2.1) degenerates into a real-valued NN, the results of this paper are new. In fact, uncertainty, namely fuzziness, is also a problem that needs to be considered in real system modeling. However, we consider only the disturbance of random factors and do not consider the issue of fuzziness. In a NN, considering the effects of both random perturbations and fuzziness is our future direction of effort.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the National Natural Science Foundation of China under Grant No. 12261098.

    The authors declare that they have no conflicts of interest.



    [1] U. Balasooriya, Failure-censored reliability sampling plans for the exponential distribution, J. Stat. Comput. Simul., 52 (1995), 337–349. https://doi.org/10.1080/00949659508811684 doi: 10.1080/00949659508811684
    [2] J. W. Wu, W. L. Hung, C. H. Tsai, Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan, Statistics, 37 (2003), 517–525. https://doi.org/10.1080/02331880310001598864 doi: 10.1080/02331880310001598864
    [3] J. W. Wu, W. L. Hung, C. Y. Chen, Approximate MLE of the scale parameter of the truncated Rayleigh distribution under the first failure-censored data, J. Inf. Optim. Sci., 25 (2004), 221–235. https://doi.org/10.1080/02522667.2004.10699604 doi: 10.1080/02522667.2004.10699604
    [4] S. J. Wu, C. Kuş, On estimation based on progressive first-failure-censored sampling, Comput. Statist. Data Anal., 53 (2009), 3659–3670. https://doi.org/10.1016/j.csda.2009.03.010 doi: 10.1016/j.csda.2009.03.010
    [5] M. Dube, H. Krishna, R. Garg, Generalized inverted exponential distribution under progressive first-failure censoring, J. Stat. Comput. Simul., 86 (2016), 1095–1114. https://doi.org/10.1080/00949655.2015.1052440 doi: 10.1080/00949655.2015.1052440
    [6] S. Saini, A. Chaturvedi, R. Garg, Estimation of stress-strength reliability for generalized Maxwell failure distribution under progressive first failure censoring, J. Stat. Comput. Simul., 91 (2021), 1366–1393. https://doi.org/10.1080/00949655.2020.1856846 doi: 10.1080/00949655.2020.1856846
    [7] M. Nassar, R. Alotaibi, A. Elshahhat, Statistical analysis of alpha power exponential parameters using progressive first-failure censoring with applications, Axioms, 11 (2022), 553. https://doi.org/10.3390/axioms11100553 doi: 10.3390/axioms11100553
    [8] S. K. Ashour, A. A. El-Sheikh, A. Elshahhat, Inferences and optimal censoring schemes for progressively first-failure censored Nadarajah-Haghighi distribution, Sankhya A, 84 (2022), 885–923. https://doi.org/10.1007/s13171-019-00175-2 doi: 10.1007/s13171-019-00175-2
    [9] M. S. Eliwa, E. A. Ahmed, Reliability analysis of constant partially accelerated life tests under progressive first failure type-Ⅱ censored data from Lomax model: EM and MCMC algorithms, AIMS Mathematics, 8 (2023), 29–60. https://doi.org/10.3934/math.2023002 doi: 10.3934/math.2023002
    [10] N. Alsadat, M. Abu-Moussa, A. Sharawy, On the study of the recurrence relations and characterizations based on progressive first-failure censoring, AIMS Mathematics, 9 (2024), 481–494. https://doi.org/10.3934/math.2024026 doi: 10.3934/math.2024026
    [11] H. K. Yuen, S. K. Tse, Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removals, J. Stat. Comput. Simul., 55 (1996), 57–71. https://doi.org/10.1080/00949659608811749 doi: 10.1080/00949659608811749
    [12] S. R. Huang, S. J. Wu, Estimation of Pareto distribution under progressive first-failure censoring with random removals, J. Chin. Statist. Assoc., 49 (2011), 82—97.
    [13] S. K. Ashour, A. A. El-Sheikh, A. Elshahhat, Inferences for Weibull lifetime model under progressively first-failure censored data with binomial random removals, Statist. Optim. Inf. Comput., 9 (2020), 47–60. https://doi.org/10.19139/soic-2310-5070-611 doi: 10.19139/soic-2310-5070-611
    [14] A. Elshahhat, V. K. Sharma, H. S. Mohammed, Statistical analysis of progressively first-failure-censored data via beta-binomial removals, AIMS Mathematics, 8 (2023), 22419–22446. https://doi.org/10.3934/math.20231144 doi: 10.3934/math.20231144
    [15] S. K. Singh, U. Singh, V. K. Sharma, Expected total test time and Bayesian estimation for generalized Lindley distribution under progressively Type-Ⅱ censored sample where removals follow the beta-binomial probability law, Appl. Math. Comput., 222 (2013), 402–419. https://doi.org/10.1016/j.amc.2013.07.058 doi: 10.1016/j.amc.2013.07.058
    [16] I. Usta, H. Gezer, Parameter estimation in Weibull distribution on progressively Type-Ⅱ censored sample with beta-binomial removals, Econom. Bus., 10 (2016), 505–515.
    [17] A. Kaushik, U. Singh, S. K. Singh, Bayesian inference for the parameters of Weibull distribution under progressive Type-Ⅰ interval censored data with beta-binomial removals, Comm. Statist. Simulation Comput., 46 (2017), 3140–3158. https://doi.org/10.1080/03610918.2015.1076469 doi: 10.1080/03610918.2015.1076469
    [18] P. K. Vishwakarma, A. Kaushik, A. Pandey, U. Singh, S. K. Singh, Bayesian estimation for inverse Weibull distribution under progressive Type-Ⅱ censored data with beta-binomial removals, Aust. J. Stat., 47 (2018), 77–94. http://dx.doi.org/10.17713/ajs.v47i1.578 doi: 10.17713/ajs.v47i1.578
    [19] P. K. Sangal, A. Sinha, Classical estimation in exponential power distribution under Type-Ⅰ progressive hybrid censoring with beta-binomial removals, Int. J. Agricult. Stat. Sci., 17 (2021), 1973–1988.
    [20] X. Jia, D. Wang, P. Jiang, B. Guo, Inference on the reliability of Weibull distribution with multiply Type-Ⅰ censored data, Reliab. Eng. Syst. Safety, 150 (2016), 171–181. https://doi.org/10.1016/j.ress.2016.01.025 doi: 10.1016/j.ress.2016.01.025
    [21] M. Nassar, M. Abo-Kasem, C. Zhang, S. Dey, Analysis of Weibull distribution under adaptive type-Ⅱ progressive hybrid censoring scheme, J. Indian. Soc. Probab. Stat., 19 (2018), 25–65. https://doi.org/10.1007/s41096-018-0032-5 doi: 10.1007/s41096-018-0032-5
    [22] E. Ramos, P. L. Ramos, F. Louzada, Posterior properties of the Weibull distribution for censored dat, Stat. Probab. Lett., 166 (2020), 108873. https://doi.org/10.1016/j.spl.2020.108873 doi: 10.1016/j.spl.2020.108873
    [23] T. Zhu, Statistical inference of Weibull distribution based on generalized progressively hybrid censored data, J. Comput. Appl. Math., 371 (2020), 112705. https://doi.org/10.1016/j.cam.2019.112705 doi: 10.1016/j.cam.2019.112705
    [24] J. K. Starling, C. Mastrangelo, Y. Choe, Improving Weibull distribution estimation for generalized Type Ⅰ censored data using modified SMOTE, Reliab. Eng. Syst. Safety, 211 (2021), 107505. https://doi.org/10.1016/j.ress.2021.107505 doi: 10.1016/j.ress.2021.107505
    [25] J. Ren, W. Gui, Statistical analysis of adaptive type-Ⅱ progressively censored competing risks for Weibull models, Appl. Math. Model., 98 (2021), 323–342. https://doi.org/10.1016/j.apm.2021.05.008 doi: 10.1016/j.apm.2021.05.008
    [26] M. Nassar, A. Elshahhat, Estimation procedures and optimal censoring schemes for an improved adaptive progressively type-Ⅱ censored Weibull distribution, J. Appl. Stat., 51 (2024), 1664–1688. https://doi.org/10.1080/02664763.2023.2230536 doi: 10.1080/02664763.2023.2230536
    [27] A. Xu, B. Wang, D. Zhu, J. Pang, X. Lian, Bayesian reliability assessment of permanent magnet brake under small sample size, IEEE Trans. Reliab., 2024, 1–11. https://doi.org/10.1109/TR.2024.3381072
    [28] M. Plummer, N. Best, K. Cowles, K. Vines, CODA: Convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7–11.
    [29] A. Henningsen, O. Toomet, maxLik: A package for maximum likelihood estimation in R, Comput. Stat., 26 (2011), 443–458. https://doi.org/10.1007/s00180-010-0217-1 doi: 10.1007/s00180-010-0217-1
    [30] E. T. Lee, J. W. Wang, Statistical methods for survival data analysis, John Wiley & Sons, Inc., 2003.
    [31] D. K. Bhaumik, K. Kapur, R. D. Gibbons, Testing parameters of a gamma distribution for small samples, Technometrics, 51 (2009), 326–334. https://doi.org/10.1198/tech.2009.07038 doi: 10.1198/tech.2009.07038
    [32] A. Elshahhat, B. R. Elemary, Analysis for Xgamma parameters of life under Type-Ⅱ adaptive progressively hybrid censoring with applications in engineering and chemistry, Symmetry, 13 (2021), 2112. https://doi.org/10.3390/sym13112112 doi: 10.3390/sym13112112
    [33] R. Alotaibi, A. Elshahhat, H. Rezk, M. Nassar, Inferences for alpha power exponential distribution using adaptive progressively type-Ⅱ hybrid censored data with applications, Symmetry, 14 (2022), 651. https://doi.org/10.3390/sym14040651 doi: 10.3390/sym14040651
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(954) PDF downloads(82) Cited by(1)

Figures and Tables

Figures(7)  /  Tables(18)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog