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Research article

Commutators of multilinear θ-type generalized fractional integrals on non-homogeneous metric measure spaces

  • Received: 23 November 2021 Revised: 28 February 2022 Accepted: 02 March 2022 Published: 15 March 2022
  • MSC : 42B35, 47B37, 28A35

  • Let Iα,m be the multilinear θ-type generalized fractional integrals and bσ be the vector with each bσi~RBMO(μ). The boundedness for Iα,m and the iterated multi-commutators Iα,m,bσ on Lebesgue spaces over non-homogeneous spaces are showed in this paper.

    Citation: Xiangxing Tao, Jiahui Wang. Commutators of multilinear θ-type generalized fractional integrals on non-homogeneous metric measure spaces[J]. AIMS Mathematics, 2022, 7(6): 9627-9647. doi: 10.3934/math.2022535

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  • Let Iα,m be the multilinear θ-type generalized fractional integrals and bσ be the vector with each bσi~RBMO(μ). The boundedness for Iα,m and the iterated multi-commutators Iα,m,bσ on Lebesgue spaces over non-homogeneous spaces are showed in this paper.



    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}t, P ) be a complete probability space in which \{\mathscr{F}_t\}_{t\geqslant 0} is a natural filtration meeting the usual conditions. Denote by B_{\mathscr{F}_0}([-\theta, 0], \mathscr{A}^n) the family of bounded, \mathscr{F}_0 -measurable and \mathscr{A}^n -valued random variables from [-\theta, 0]\rightarrow \mathscr{A}^n . The initial values of system (2.1) are depicted as

    x_i(s) = \phi_i(s),\; \; \; s\in[-\theta,0],

    where \phi_i\in B_{\mathscr{F}_0} ([-\theta, 0], \mathscr{A}), \theta = \max\limits_{1\leq i, j\leq n}\{ \sup\limits_{t\in \mathbb{R}}\tau_{ij}(t), \sup\limits_{t\in \mathbb{R}}\sigma_{ij}(t)\} .

    For convenience, we introduce the following notations:

    \begin{align*} &\underline{a}^0 = \min\limits_{ij\in\Lambda} \underline{a}_{ij}^0 = \min\limits_{ij\in\Lambda}\inf\limits_{t\in\mathbb{R}} a_{ij}^0(t),\; \; \bar{a}^0 = \max\limits_{ij\in\Lambda} \bar{a}_{ij}^0 = \max\limits_{ij\in\Lambda}\sup\limits_{t\in\mathbb{R}} a_{ij}^0(t),\; \; {C_{ij}^{kl}}^+ = \sup\limits_{t\in\mathbb{R}}\|C_{ij}^{kl}(t) \|_{\flat},\\ &\bar{a^c} = \max\limits_{ij\in\Lambda} \bar{a}_{ij}^c = \max\limits_{ij\in\Lambda}\sup\limits_{t\in\mathbb{R}} \|a_{ij}^c(t)\|_{\flat}, \; \; {B_{ij}^{kl}}^+ = \sup\limits_{t\in\mathbb{R}} \|B_{ij}^{kl}(t)\|_{\flat},\; \; {E_{ij}^{kl}}^+ = \sup\limits_{t\in\mathbb{R}}\|E_{ij}^{kl}(t)\|_{\flat}, \\ & K_{ij}^+ = \sup\limits_{t\in\mathbb{R}}K_{ij}(t),\; \; \tau_{kl}^+ = \sup\limits_{t\in\mathbb{R}}\tau_{kl}(t),\; \; \dot{\tau}_{kl}^+ = \sup\limits_{t\in\mathbb{R}}\dot{\tau}_{kl}(t), \sigma_{ij}^+ = \sup\limits_{t\in \mathbb{R}}\sigma_{ij}(t),\; \; \dot{\sigma}_{ij}^+ = \sup\limits_{t\in\mathbb{R}}\dot{\sigma}_{ij}(t), \\ & M_L = \max\limits_{ij\in\Lambda} L_{ij}^+ = \max\limits_{ij\in\Lambda}\sup\limits_{t\in\mathbb{R}}\|L_{ij}(t)\|_{\flat},\; \; \theta = \max\limits_{ij\in\Lambda}\{\tau_{ij}^+,\sigma_{ij}^+\},\; \; \Lambda = \{11,12,\cdots,1n,\cdots,mn\}. \end{align*}

    Throughout this paper, we make the following assumptions:

    (A1) For ij\in\Lambda , f, g, \delta_{ij}\in C(\mathscr{A}, \mathscr{A}) satisfy the Lipschitz condition, and f, g are bounded, that is, there exist constants L_f > 0, L_g > 0, L_{ij}^{\delta} > 0, M_f > 0, M_g > 0 such that for all x, y\in \mathscr{A} ,

    \begin{align*} ||f(x)-f(y)||_{\flat}\leq L_f||x-y||_{\flat},\; \; ||g(x)-g(y)||_{\flat}\leq L_g||x-y||_{\flat},\\ ||\delta_{ij}(x)-\delta_{ij}(y)||_{\flat}\leq L_{ij}^{\delta}||x-y||_{\flat} , ||f(x)||_{\flat}\leq M_f ,\; \; ||g(x)||_{\flat}\leq M_g; \end{align*}

    furthermore, f(0) = g(0) = \delta_{ij}(0) = 0 .

    (A2) For ij\in\Lambda , {a}_{ij}^0\in AP(\mathbb{R}, \mathbb{R}^+), {a}_{ij}^c\in AP(\mathbb{R}, \mathscr{A}), \tau_{ij}, \sigma_{ij}\in AP(\mathbb{R}, \mathbb{R}^+)\cap C^1(\mathbb{R}, \mathbb{R}) satisfying 1-\dot{\tau}_{ij}^+, 1-\dot{\sigma}_{ij}^+ > 0 , C_{ij}^{kl}, B_{ij}^{kl}, E_{ij}^{kl}\in AP(\mathbb{R}, \mathscr{A}) , L = (L_{11}, L_{12}, \cdots, L_{mn})\in\mathscr{L}_{loc}^p(\mathbb{R}, L^p(\Omega, \mathscr{A}^{m\times n})) is almost periodic in the sense of Stepanov.

    (A3) For p > 2, \frac{1}{p}+\frac{1}{q} = 1 ,

    \begin{align*} 0 < r^1: = & \frac{8^p}{4}\max\limits_{ij\in\Lambda}\bigg\{ \bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}}\frac{q}{p\underline{a}_{ij}^0}\bigg[ (\bar{a}_{ij}^c)^p + \bigg(\sum\limits_{C_{kl}\in N_{h_1}(i,j)}({C_{ij}^{kl}}^+)^q\bigg)^{\frac{p}{q}}(2\kappa L_f+M_f)^p \\ & + \bigg(\sum\limits_{C_{kl}\in N_{h_2}(i,j)}({B_{ij}^{kl}}^+)^q\bigg)^{\frac{p}{q}} \bigg( (2\kappa L_g+M_g) \int_0^{\infty}|K_{ij}(u)|du\bigg)^{p}\bigg] \\ & +C_p\bigg(\frac{p-2}{2\underline{a}_{ij}^0}\bigg)^{\frac{p-2}{2}}\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\bigg\} < 1, \end{align*}

    and for p = 2 ,

    \begin{align*} 0 < r^2: = & 16\max\limits_{ij\in\Lambda}\bigg\{\frac{1}{(\underline{a}_{ij}^0)^2} \bigg[ (\bar{a}_{ij}^c)^2 + \sum\limits_{C_{kl}\in N_{h_1}(i,j)}({C_{ij}^{kl}}^+)^2 (2\kappa L_f+M_f)^2+ \sum\limits_{C_{kl}\in N_{h_2}(i,j)}({B_{ij}^{kl}}^+)^2 \\ & \times\bigg((2\kappa L_g+M_g)\int_0^{\infty}|K_{ij}(u)|du\bigg)^2\bigg] + \frac{1}{2\underline{a}_{ij}^0} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}({E_{ij}^{kl}}^+)^2 (L_{ij}^{\delta})^2\bigg\} < 1. \end{align*}

    (A4) For \frac{1}{p}+\frac{1}{q} = 1 ,

    \begin{align*} 0 < \frac{q}{p\underline{a}^0}\rho^1 : = & 16^{p-1} \frac{q}{p\underline{a}^0} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0 }\bigg)^{\frac{p}{q}} \bigg[ (\bar{a}_{ij}^c )^p + \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} \bigg[ 2^{p-1} (L_f )^p \notag\\ & \times \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}^+ } +(M_f )^p \bigg] +\bigg( \sum\limits_{C_{kl}\in N_{h_2}(i,j) } \big( {B_{ij}^{kl}}^+ \big)^q \bigg)^{\frac{p}{q}} \bigg[ \bigg( 2\kappa L_g \\ & \times \int_0^{\infty} |K_{ij}(u)|du \bigg)^{p} + \bigg( M_g \int_0^\infty |K_{ij}(u)|du \bigg)^p \bigg] \bigg] + 2^{p-1} C_p \big(\frac{p-2}{2\underline{a}_{ij}^0} \big)^{\frac{p-2} {2} } \\ & \times \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}^+ } \bigg\} < 1, \,\, (p > 2), \end{align*}
    \begin{align*} 0 < \frac{\rho^2}{\underline{a}^0}: = & \frac{32}{\underline{a}^0} \max\limits_{ij\in\Lambda} \bigg\{ \bigg(\frac{1}{\underline{a}_{ij}^0}\bigg) \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^2 \bigg[ (L_f )^2 \sum\limits_{C_{kl}\in N_{h_1}(i,j)}\frac{e^{\underline{a}_{ij}^0 {\tau_{kl}}^+} (2\kappa)^2 } {1-{\dot{\tau}}_{kl}^+ } +\frac{(M_f)^2}{2} \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{1}{2\underline{a}_{ij}^0 } \sum\limits_{C_{kl}\in N_{h_2}(i,j) } \big( {B_{ij}^{kl}}^+ \big)^2 (4\kappa^2 L_g^2+M_g^2) \\ & \times \bigg( \int_0^{\infty} |K_{ij}(u)| du \bigg)^2 +\frac{(\bar{a}_{ij}^c)^2}{2\underline{a}_{ij}^0} \bigg\} < 1,\,\, (p = 2). \end{align*}

    (A5) The kernel K_{ij} is almost periodic and there exist constants M > 0 and u > 0 such that |K_{ij}(t)|\leq Me^{-ut} for all t\in \mathbb{R} .

    Let \mathbb{X} indicate the space of all \mathcal{L}^p -bounded and \mathcal{L}^p -uniformly continuous stochastic processes from \mathbb{R} to \mathcal{L}^p(\Omega, \mathscr{A}^{m\times n}) , then with the norm \|\phi\|_{\mathbb{X}} = \sup\limits_{t\in\mathbb{R}}\big\{E\|\phi(t)\|_0^p\big\}^{\frac{1}{p}}, where \phi = (\phi_{11}, \phi_{12}, \ldots, \phi_{mn})\in \mathbb{X} , it is a Banach space.

    Set \phi^0 = (\phi_{11}^0, \phi_{12}^0, \ldots, \phi_{mn}^0)^T , where \phi_{ij}^0(t) = \int_{-\infty}^te^{-\int_s^ta_{ij}^0(u)du}L_{ij}(s)ds, t\in\mathbb{R}, ij\in\Lambda . Then, \phi^0 is well defined under assumption (A2) . Consequently, we can take a constant \kappa such that \kappa\geq \|\phi^0\|_{\mathbb{X}} .

    Definition 3.1. [37] An \mathscr{F}_t -progressively measurable stochastic process x(t) = (x_{11}(t), x_{12}(t), \ldots, x_{mn}(t))^T is called a solution of system (2.1), if x(t) solves the following integral equation:

    \begin{align} x_{ij}(t) = & x_{ij}(t_0)e^{-\int_{t_0}^ta_{ij}^0(u)du}+\int_{t_0}^{t}e^{-\int_{s}^ta_{ij}^0(u)du}\bigg[ -a_{ij}^c(s)x_{ij}(s)+\sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s) \\ & \times f(x_{kl}(s-\tau_{kl}(s)))x_{ij}(s)+ \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s)\int_{0}^{\infty}K_{ij}(u)g(x(s-u))dux_{ij}(s) \\ &+L_{ij}(s) \bigg]ds+\int_{t_0}^te^{-\int_{s}^ta_{ij}^0(u)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(x_{ij}(s-\sigma_{ij}(s)))dw_{ij}(s). \end{align} (3.1)

    In (3.1), let t_0\rightarrow -\infty , then one gets

    \begin{align} x_{ij}(t) = & \int_{-\infty}^{t}e^{-\int_{s}^ta_{ij}^0(u)du}\bigg[ -a_{ij}^c(s)x_{ij}(s)+\sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s)f(x_{kl}(s-\tau_{kl}(s)))x_{ij}(s) \\ & + \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s)\int_{0}^{\infty}K_{ij}(u)g(x(s-u))dux_{ij}(s)+L_{ij}(s) \bigg]ds+\int_{-\infty}^te^{-\int_{s}^ta_{ij}^0(u)du} \\ & \times \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(x_{ij}(s-\sigma_{ij}(s)))dw_{ij}(s), t\geq t_0,ij\in\Lambda. \end{align} (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 \mathcal{L}^p -bounded and \mathcal{L}^p -uniformly continuous solution 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. Define an operator \phi:\mathbb{X}^*\to \mathbb{X} as follows:

    (\Psi\phi)(t) = ( (\Psi_{11}\phi)(t),(\Psi_{12}\phi)(t),\ldots,(\Psi_{mn}\phi)(t) )^T,

    where (\phi_{11}, \phi_{12}, \ldots, \phi_{mn})^T\in\mathbb{X} , t\in\mathbb{R} and

    \begin{align} (\Psi_{ij}\phi)(t) = &\int_{-\infty}^{t}e^{-\int_{s}^ta_{ij}^0(u)du}\bigg[ -a_{ij}^c(s)\phi_{ij}(s)+ \sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s)f(\phi_{kl}(s-\tau_{kl}(s)))\phi_{ij}(s) \\ & + \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s)\int_{0}^{\infty}K_{ij}(u)g(\phi_{kl}(s-u))du\phi_{ij}(s)+L_{ij}(s) \bigg]ds\\ &+\int_{-\infty}^te^{-\int_{s}^ta_{ij}^0(u)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))d\omega_{ij}(s), ij\in\Lambda. \end{align} (3.3)

    First of all, let us show that E\|\Psi \phi(t)-\phi^0(t)\|_{0}^p\leq \kappa for all \phi\in \mathbb{X}^* .

    Noticing that for any \phi\in\mathbb{X}^* , it holds

    \|\phi\|_{\mathbb{X}}\leq \|\phi^0 \|_{\mathbb{X}}+\|\phi-\phi^0 \|_{\mathbb{X}}\leq 2\kappa.

    Then, we deduce that

    \begin{align} & E\|\Psi \phi(t)-\phi^0(t)\|_{0}^p \\ \leq & 4^{p-1}\max\limits_{ij\in\Lambda} \bigg\{ E\bigg\|\int_{-\infty}^{t} -e^{-\int_{s}^ta_{ij}^0(u)du} a_{ij}^c(s)\phi_{ij}(s) \bigg\|_{\flat}^p \bigg\} +4^{p-1}\max\limits_{ij\in\Lambda}\bigg\{E\bigg\|\int_{-\infty}^{t} e^{-\int_{s}^ta_{ij}^0(u)du} \\ &\times \sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s) f(\phi_{kl}(s-\tau_{kl}(s)))\phi_{ij}(s)ds\bigg\|_{\flat}^p\bigg\} + 4^{p-1}\max\limits_{ij\in\Lambda} \bigg\{E\bigg\|\int_{-\infty}^{t} e^{-\int_{s}^ta_{ij}^0(u)du} \\ &\times \sum\limits_{C_{kl}\in N_{h_2}(i,j)}B_{ij}^{kl}(s)\int_{0}^{\infty}K_{ij}(u)g(\phi_{kl}(s-u))du\phi_{ij}(s)ds\bigg\|_{\flat}^p\bigg\}\\ & + 4^{p-1}\max\limits_{ij\in\Lambda} \bigg\{E\bigg\|\int_{-\infty}^{t} e^{-\int_{s}^ta_{ij}^0(u)du} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))d\omega_{ij}(s)\bigg\|_{\flat}^p\bigg\}\\ : = &F_1+F_2+F_3+F_4 . \end{align} (3.4)

    By the Hölder inequality, we have

    \begin{align} F_2\leq & 4^{p-1}\max\limits_{ij\in\Lambda}\bigg\{E\bigg\|\bigg[ \int_{-\infty}^te^{-\frac{q}{p}\int_s^ta_{ij}^0(u)du}ds \bigg]^{\frac{p}{q}} \bigg[\int_{-\infty}^te^{-\frac{p}{q}\int_s^ta_{ij}^0(u)du} \\ & \times\bigg(\sum\limits_{C_{kl}\in N_{h_1}(i,j)}C_{ij}^{kl}(s)f(\phi_{kl}(s-\tau_{kl}(s)))\phi_{ij}(s)\bigg)^pds\bigg]\bigg\|_{\flat}\bigg\} \\ \leq & 4^{p-1} \max\limits_{ij\in\Lambda}\bigg\{ \bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}} E\bigg[ \int_{-\infty}^t e^{-\frac{p}{q}\int_{s}^ta_{ij}^0(u)du}\bigg(\sum\limits_{C_{kl}\in N_{h_1}(i,j)}(\|C_{ij}^{kl}(s)\|_{\flat})^q\bigg)^{\frac{p}{q}} \\ &\times\sum\limits_{ij\in\Lambda}(2\kappa L_f)^p\|\phi_{ij}(s)\|_{\flat}^pds \bigg] \bigg\} \\ \leq & 4^{p-1}\max\limits_{ij\in\Lambda}\bigg\{ \bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}} \frac{q}{p\underline{a}_{ij}^0} \bigg(\sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg) ^{\frac{p}{q}} (2\kappa L_f )^p \bigg\}\|\phi\|_{\mathbb{X}}^{p}. \end{align} (3.5)

    Similarly, one has

    \begin{align} F_1 \leq & 4^{p-1}\max\limits_{ij\in\Lambda}\bigg\{\bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}} \frac{ q}{p\underline{a}_{ij}^0} (\bar{a}_{ij}^c)^p \bigg\}\|\phi\|_{\mathbb{X}}^{p}, \end{align} (3.6)
    \begin{align} F_3 \leq & 4^{p-1}\max\limits_{ij\in\Lambda}\bigg\{ \bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}} \frac{ q}{p\underline{a}_{ij}^0} \bigg (\sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{ij}^{kl}}^+)^q \bigg ) ^{\frac{p}{q}} \bigg( 2\kappa L_g \int_0^\infty |K_{ij}(u)|du \bigg )^{p} \bigg \} \|\phi\|_{\mathbb{X}}^{p}. \end{align} (3.7)

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

    \begin{align} F_4\leq&4^{p-1}C_p\max\limits_{ij\in\Lambda} \bigg\{E\bigg[\int_{-\infty}^{t}\bigg\| e^{-\int_{s}^ta_{ij}^0(u)du}\sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))\bigg\|_\flat^2ds\bigg]^{\frac{p}{2}}\bigg\}\\ \leq & 4^{p-1} C_p \max\limits_{ij\in\Lambda}\bigg\{ E\bigg[ e^{-2\int_s^ta_{ij}^0(u)du} \bigg\|\sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))\bigg\|_{\flat}^2ds \bigg]^{\frac{p}{2}} \bigg\} \\ \leq & 4^{p-1} C_p \max\limits_{ij\in\Lambda}\bigg \{ E\bigg [ \int_{-\infty}^t (e^{-2\int_s^ta_{ij}^0(u)du})^{\frac{p}{p-2}\times\frac{1}{p}}ds \bigg ]^{\frac{p-2}{p}\times\frac{p}{2}} \\ &\times E\bigg[ \int_{-\infty}^t (e^{-2\int_s^ta_{ij}^0(u)du})^{\frac{1}{q}\times\frac{p}{2}} \bigg( \bigg\|\sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}\phi_{ij}(s-\sigma_{ij}(s))\bigg\|_{\flat}^2\bigg)^{\frac{p}{2}}ds \bigg] \bigg\} \\ \leq& 4^{p-1} C_p \max\limits_{ij\in\Lambda}\bigg\{ \bigg( \frac{p-2}{2\underline{a}_{ij}^0} \bigg)^{\frac{p-2}{2}} \frac{q}{p\underline{a}_{ij}^0} E\bigg\|\sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))\bigg\|_{\flat}^p\bigg\} \\ \leq & 4^{p-1} C_p \max\limits_{ij\in\Lambda}\bigg\{ \bigg( \frac{p-2}{2\underline{a}_{ij}^0} \bigg)^{\frac{p-2}{2}} \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 \bigg\} \|\phi\|_{\mathbb{X}}^p. \end{align} (3.8)

    When p = 2 , by the It \rm \hat{o} isometry, it follows that

    \begin{align} F_4\leq & 4 \max\limits_{ij\in\Lambda}\bigg\{ E\bigg[ \int_{-\infty}^t e^{-2\int_s^ta_{ij}^0(u)du} \bigg\| \sum\limits_{C_{kl}\in N_{h_3}(i,j)}E_{ij}^{kl}(s)\delta_{ij}(\phi_{ij}(s-\sigma_{ij}(s)))\bigg\|_{\mathscr{A} }^2ds \bigg] \bigg\} \\ \leq & 4 \max\limits_{ij\in\Lambda}\bigg\{ \frac{1}{2\underline{a}_{ij}^0} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}({E_{ij}^{kl}}^+)^2 (L_{ij}^{\delta})^2 \bigg\} \|\phi\|_{\mathbb{X}}^2. \end{align} (3.9)

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

    \begin{align} \|\Psi\phi-\phi^0\|_{\mathbb{X}}^p \leq & 4^{p-1} \max\limits_{ij\in\Lambda}\bigg\{ \bigg(\frac{p}{q\underline{a}_{ij}^0}\bigg)^{\frac{p}{q}} \frac{q}{p\underline{a}_{ij}^0} \bigg[ (\bar{a}_{ij}^c)^p +\bigg(\sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (2\kappa L_f)^p \\ & + \bigg(\sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{ij}^{kl}}^+)^q \bigg) ^{\frac{p}{q}} \bigg(2\kappa L_g\int_0^\infty |K_{ij}(u)|du\bigg)^{p} \bigg] \\ & + C_p \bigg( \frac{p-2}{2\underline{a}_{ij}^0} \bigg)^{\frac{p-2}{2}} \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 \bigg\}\|\phi\|_{\mathbb{X}}^{p} \leq \kappa^{p}, \,\,(p > 2), \end{align} (3.10)

    and

    \begin{align} \|\Psi\phi-\phi^0\|_{\mathbb{X}}^2 \leq & 4 \max\limits_{ij\in\Lambda}\bigg\{ \frac{1}{(a_{ij}^-)^2} \bigg[ (\bar{a}_{ij}^c)^2 + \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{ij}^{kl}}^+)^2 (2\kappa L_f)^2 + \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{ij}^{kl}}^+)^2 \bigg(2\kappa L_g \\ & \times\int_0^\infty |K_{ij}(u)|du\bigg)^2 \bigg] + \frac{1}{2\underline{a}_{ij}^0} \sum\limits_{C_{kl}\in N_{h_3}(i,j)}({E_{ij}^{kl}}^+)^2 (L_{ij}^{\delta})^2 \bigg\}\|\phi\|_{\mathbb{X}}^2 \leq \kappa^2, \,\, (p = 2). \end{align} (3.11)

    It follows from (3.10), (3.11) and (A3) that \|\Psi\phi-\phi^0\|_{\mathbb{X}} \leq \kappa .

    Then, using the same method as that in the proof of Theorem 3.2 in [21], we can show that \Psi\phi is \mathcal{L}^p -uniformly continuous. Therefore, we have \Psi({\mathbb{X}}^*) \subset \mathbb{X}^* .

    Last, we will show that \Psi is a contraction mapping. Indeed, for any \psi, \varphi \in \mathbb{X}^* , when p > 2 , we have

    \begin{align} & E\|(\Phi \varphi)(t)-(\Phi \psi)(t)\|_{0}^{p} \\ \leq & 4^{p-1} \max _{ij\in\Lambda} \bigg\{ E \bigg\| \int_{-\infty}^{t} e^{-\int_{s}^{t} a_{ij}^0(u) d u} (-a_{ij}^c(s)\varphi_{ij}(s)+a_{ij}^c(s)\psi_{ij}(s)) d s \bigg\|_{\flat}^{p} \bigg\} \\ &+4^{p-1} \max _{ij\in\Lambda} \bigg\{ E \bigg\| \int_{-\infty}^{t} e^{-\int_{s}^{t} a_{ij}^0(u) d u} \sum\limits_{C_{kl}\in N_{h_1}(i,j)} C_{i j}^{kl}(s) \bigg[f\big(\varphi_{kl}(s-\tau_{kl}(s)) \big)\varphi_{ij}(s) \\ & -f\big( \psi_{kl}(s-\tau_{kl}(s)) \big)\psi_{ij}(s) \bigg] d s \bigg\|_{\flat}^{p} \bigg\} +4^{p-1} \max _{ij\in\Lambda} \bigg\{E \bigg\| \int_{-\infty}^{t} e^{-\int_{s}^{t} a_{ij}^0(u) d u} \sum\limits_{C_{kl}\in N_{h_2}(i,j)} B_{i j}^{kl}(s) \\ & \times \bigg[ \int_0^{\infty} K_{ij}(u)g\big(\varphi_{kl}(s-u)\big) du\varphi_{ij}(s) -\int_0^{\infty} K_{ij}(u)g\big(\psi_{kl}(s-u)\big) du\psi_{ij}(u) \bigg] d s \bigg\|_{\flat}^{p}\bigg\} \\ & + 4^{p-1} \max _{ij\in\Lambda} \bigg\{E \bigg\| \int_{-\infty}^{t} e^{-\int_{s}^{t} a_{ij}^0(u) d u} \sum\limits_{C_{kl}\in N_{h_3}(i,j)} E_{i j}^{kl}(s)\bigg[\delta_{ij}\big(\varphi_{ij}(s-\sigma_{i j}(s))\big) \\ & -\delta_{ij}\big(\psi_{ij}(s-\sigma_{i j}(s))\big)\bigg] d \omega_{ij}(s) \bigg\|_{\flat}^{p}\bigg\} \\ \leq & 4^{p-1} \max\limits_{ij\in\Lambda} \bigg\{ \bigg( \frac{p}{q\underline{a}_{ij}^0} \bigg)^{\frac{p}{q}} \frac{q}{p\underline{a}_{ij}^0} \bigg[ (\bar{a}_{ij}^c)^p + \bigg( \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{i j}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (2\kappa L_f+M_f)^p \\ & + \bigg( \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{i j}^{kl}}^+)^q \bigg)^{\frac{p}{q}} \bigg( (2\kappa L_g+M_g)\int_0^{\infty}|K_{ij}(u)|du \bigg)^p \bigg] + C_p \bigg( \frac{p-2}{2\underline{a}_{ij}^0} \bigg)^{\frac{p-2}{2}} \frac{q}{p\underline{a}_{ij}^0} \\ & \times \bigg(\sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^q \bigg)^{\frac{p}{q}} (L_{ij}^{\delta})^p \bigg\} \| \varphi-\psi \|_{\mathbb{X}}^p. \end{align} (3.12)

    Similarly, for p = 2 , we can get

    \begin{align} & E\|(\Phi \varphi)(t)-(\Phi \psi)(t)\|_{0}^{2} \\ \leq & 4 \max\limits_{ij\in\Lambda} \bigg\{ \frac{1}{(\underline{a}_{ij}^0)^2} \bigg[ (\bar{a}_{ij}^c)^2 + \sum\limits_{C_{kl}\in N_{h_1}(i,j)} ({C_{i j}^{kl}}^+)^2 (2\kappa L_f+M_f)^2 + \sum\limits_{C_{kl}\in N_{h_2}(i,j)} ({B_{i j}^{kl}}^+)^2 \\ & \times \bigg( (2\kappa L_g+M_g)\int_0^{\infty}|K_{ij}(u)|du \bigg)^2 \bigg] + \frac{1}{2\underline{a}_{ij}^0} \sum\limits_{C_{kl}\in N_{h_3}(i,j)} ({E_{ij}^{kl}}^+)^2 (L_{ij}^{\delta})^2 \bigg\} \| \varphi-\psi \|_{\mathbb{X}}^2. \end{align} (3.13)

    From (3.12) and (3.13) it follows that

    \begin{align*} \|(\Phi \varphi)(t)-(\Phi \psi)(t)\|_{\mathbb{X}} \leq \sqrt[p]{r^1}\| \varphi- \psi \|_{\mathbb{X}}, (p > 2),\\ \|(\Phi \varphi)(t)-(\Phi \psi)(t)\|_{\mathbb{X}} \leq \sqrt{r^2}\| \varphi- \psi \|_{\mathbb{X}}, (p = 2). \end{align*}

    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.



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