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Asymptotic behaviour of a neural field lattice model with delays

  • Received: 01 March 2020 Revised: 01 April 2020
  • Primary: 34A33, 34D05; Secondary: 35B41, 92B20

  • The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space 2ρ under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.

    Citation: Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays[J]. Electronic Research Archive, 2020, 28(2): 1037-1048. doi: 10.3934/era.2020056

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  • The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space 2ρ under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.



    Neural field models are often represented as evolution equations generated as continuum limits of computational models of neural fields theory. They are tissue level models that describe the spatio-temporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons. See Coombes et al. [4] and the literature therein. A particularly influential model is that proposed by S. Amari in [1] (see also Chapter 3 of Coombes et al. [4] by Amari):

    tu(t,x)=u(t,x)+ΩK(xy)H(u(t,y)θ)dy,xΩR,

    where θ>0 is a given threshold and H : R R is the Heaviside function.

    The continuum neural models may lose their validity in capturing detailed dynamics at discrete sites when the discrete structures of neural systems become dominant. Lattice models, e.g., [2,6,9,10], can used to describe dynamics at each site of the neural field. Han & Kloeden [7] introduced and investigated the following lattice version of the Amari model:

    ddtui(t)=fi(ui(t))+jZdki,jH(uj(t)θ)+gi(t),iZd.

    Delays are often included in neural field models to account for the transmission time of signals between neurons. In addition, to facilitate the analysis, the Heaviside function can be replaced by a simplifying sigmoidal function such as

    σε(x)=11+ex/ε,xR,0<ε<1.

    In this paper we consider the autonomous neural field lattice system with delays

    ddtui(t)=fi(ui(t))+jZdki,jσε(uj(tτj)θ)+gi,iZd. (1)

    Throughout this paper we assume that the delays τj>0 are uniformly bounded, i.e., satisfy

    Assumption 1. There exists a constant h(0,) that 0τih for all iZd.

    and that the interconnection matrix (ki,j)i,jZd satisfies

    Assumption 2. ki,j0 for all i,jZd and there exists a constant κ>0 such that jZdki,j κ for each iZd.

    The main goal of this paper is to investigate asymptotic behaviour of solutions to the neural lattice system with delays (1), in particular, the attractor for the semidynamical system generated by its solutions. The initial conditions for such delay systems have the form

    ui(s)=ψi(s),s[h,0],iZd, (2)

    for appropriate functions ψi.

    We follow Han & Kloeden [7] and consider a weighted space of bi-infinite real valued sequences with vectorial indices i=(i1,,id)Zd.

    In particular, given a positive sequence of weights (ρi)iZd, we consider the separable Hilbert space

    2ρ:={u=(ui)iZd:iZdρiu2i<}

    with the inner product

    u,v:=iZdρiuiviforu=(ui)iZd,v=(vi)iZd2ρ

    and norm

    uρ:=iZdρiu2i.

    We assume that the ρi satisfy the following assumption.

    Assumption 3. ρi>0 for all iZd and ρΣ:=iZdρi<.

    The appropriate function space for the solutions of the lattice system with delays (1) is the Banach space C([h,0],2ρ) of continuous functions by v : [h,0] 2ρ with the norm

    vC([h,0],2ρ)=maxs[h,0]v(s)ρ.

    For a solution u(t)=(ui(t))iZd2ρ of (1) we denote by ut the segment of the solution in C([h,0],2ρ) defined by ut(s)=u(t+s) for each s[h,0]. The corresponding initial condition (2) must then satisfy (ψi())iZd C([h,0],2ρ).

    For any u=(ui)iZd2ρ, we define the operator f by

    f(u):=(fi(ui))iZd.

    To ensure that the f(u) take values in 2ρ for every u2ρ and has necessary dissipative properties, we make the following standing assumptions on the fi throughout the rest of the paper.

    Assumption 4. The functions fi:RR are continuously differential with weighted equi-locally bounded derivatives, i.e., there exists a non-decreasing function L()C(R+,R+) such that

    supiZdmaxs[r,r]|fi(s)|L(ρir),rR+,iZd;

    Assumption 5. fi(0)=0 for all iZd;

    Assumption 6. There exist constants α>0 and βi with β=(βi)iZd2ρ such that

    sfi(s)α|s|2+β2i,sR,iZd.

    It was shown in [7] that Assumption 4 implies that fi is locally Lipschitz with

    |fi(x)fi(y)|L(ρi(|x|+|y|))|xy|,iZd,x,yR.

    Since

    ρi|ui|ρΣ(iZdρiu2i)1/2=ρΣuρ,

    it follows

    |fi(ui)fi(vi)|L(ρi(|ui|+|vi|))|uivi|L(ρΣ(uρ+vρ))|uivi|

    for every u=(ui)iZd and v=(vi)iZd. The following lemma from [7] states the Lipschitz and dissipative properties of the operator f.

    Lemma 2.1. Assume that Assumptions 4–6 hold. Then f:2ρ2ρ is locally Lipschitz and satisfies the dissipativity condition

    f(u),uαu2ρ+β2ρ.

    For any v C([h,0],2ρ) we define the operator Kτ by Kτ(v)=(Kτ,i(v))iZd by

    Kτ,i(v)=jZdki,jσε(vj(τj)θ),iZd.

    Lemma 2.2. The operator Kτ maps C([h,0],2ρ) to 2ρ.

    Proof. The function σε takes values in the unit interval [0, 1], so

    |Kτ,i(v)|jZdki,jκ,iZd,vC([h,0],2ρ).

    Then

    Kτ(v)2ρ=iZdρi|Kτ,i(v)|2κ2ρΣ<.

    Remark 1. The function σε is differentiable with a uniformly bounded derivative

    ddxσε(x)1εfor allxR.

    Hence it is globally Lipschitz with the Lipschitz constant Lσ=1ε.

    Finally, we suppose that the constant forcing term g:=(gi)iZd satisfies the following assumption.

    Assumption 7. g 2ρ.

    The lattice differential equation (1) can be rewritten as an infinitely dimensional ordinary differential equation on 2ρ,

    ddtu(t)=Gτ(t,ut):=f(u)+Kτ(ut)+g, (3)

    where Gτ(t,ut):=(Gτ,i(t,ut))iZd.

    In this section we study the existence and uniqueness of solutions of the differential equation (3). To this end, we will need the following auxiliary Lemma 3.1.

    Let ZdN :={i=(i1,i2,,id)Zd:|i1|,|i2|,,|id|N} and define

    KNτ,i(v):=jZdNki,jσε(vj(τj)θ),iZd.

    Lemma 3.1. The mapping vKNτ,i(v) is continuous from C([h,0],2ρ) to R for every iZd.

    Proof. Let vnv0 in C([h,0],2ρ). Since Assumption 1: 0τjh for each jZd, we see that (vn(τj))jZd (v0(τj))jZd in 2ρ. Thus for every ε>0 there exist an M(ε)>0 such that

    jZdρj|vnj(τj)v0j(τj)|2<ε2,nM(ε).

    Considering only the jZdN appearing in the sum defining KNτ,i, we obtain

    |vnj(τj)v0j(τj)|<ε/ρN,nM(ε),jZdN,

    where ρN := minjZdNρj.

    The mapping xσε(xθ) is continuous for all xR. Since there are a finite number of terms in the sum in the definition of KNτ,i, it follows from the elementary inequality

    |(a1+b1)(a2+b2)||a1a2|+|b1b2|,a1,a2,b1,b2R1

    that the mapping vKNτ,i(v) is continuous.

    Theorem 3.2. Suppose that Assumptions 1–7 hold. Then for each r>0 there exists a(r)>0 such that for every ψ C([h,0],2ρ) satisfying ψC([h,0],2ρ) r, the lattice delay equation (3) has at least one solution defined on [0,a(r)]. Moreover, the solution u() C1([0,a],2ρ).

    Proof. Step 1. First, we claim that Gτ(t,ut) is well defined and bounded.

    It is easy to see that Gτ(t,ut) is well defined since f(u), Kτ(ut) and g are all well defined. As for the boundedness, we denote that

    |Gτ,i(ut)||fi(ui(t))|+|Kτ,i(ut)|+|gi|. (4)

    Since fi is locally Lipschitz and satisfies fi(0) = 0 by Assumption 4-5, we see that

    |fi(ui(t))|L(ρi|ui(t)|)|ui(t)|L(ρΣu(t)ρ)|ui(t)|.

    Then we obtain

    (iZdρi|fi(ui(t))|2)12L(ρΣu(t)ρ)u(t)ρ. (5)

    For the second term with delay, we have |Kτ,i(ut)|κ by Assumption 2, which gives

    (iZdρi|Kτ,i(ut)|2)12ρΣκ, (6)

    where we have used Assumption 3.

    Finally, for the last term g, Assumption 7 gives

    gρ<. (7)

    Using (5), (6) and (4) in (4) we conclude that Gτ is well defined and bounded.

    Step 2. Next, we claim that the maps Gτ,i:C([h,0],2ρ) R are continuous for all iZd.

    We consider {unt}nN C([h,0],2ρ) and u0t C([h,0],2ρ) such that unt u0t in C([h,0],2ρ). Then

    |Gτ,i(unt)Gτ,i(u0t)||fi(uni(t))fi(u0i(t))|+|Kτ,i(unt)Kτ,i(u0t)|. (8)

    By the local Lipschitz continuity of fi,

    |fi(uni(t))fi(u0i(t))|L(ρΣ(unt(0)ρ+u0t(0)ρ))|uni(t)u0i(t)|, (9)

    which shows that this term converges to zero.

    Next for the second term on the right-hand side

    |Kτ,i(unt)Kτ,i(u0t)|=|jZdki,jσε(unj(tτj)θ)jZdki,jσε(u0j(tτj)θ)|=jZdki,j|σε(unj(tτj)θ)σε(u0j(tτj)θ)|
    jZdNki,j|σε(unj(tτj)θ)σε(u0j(tτj)θ)|+jZdZdNki,j|σε(unj(tτj)θ)σε(u0j(tτj)θ)|LσjZdNki,j|unj(tτj)u0j(tτj)|+2jZdZdNki,j. (10)

    On one hand, since jZdki,j κ, for every ε>0, there exists a N1(ε) such that jZdZdNki,j 4ε when NN1(ε): we assume that the N in (10) is such an N. On the other hand, jZdNki,j|unj(tτj)u0j(tτj)| < ε when n>M(ε) by Lemma 3.1 for all N. Thus Kτ,i is continuous.

    Using (9) and (10) in (8), we complete the proof of the claim.

    Having the two steps above, by Theorem 10 in Caraballo et al. [3], for each r>0 there exists a(r)>0 such that if ψC([h,0],2ρ) and ψC([h,0],2ρ)r, then the problem (3) has at least one solution defined on [0,a(r)].

    Step 3. Finally, we claim the following inequality holds:

    |i|Kρi|Gτ,i(ut)|2C(ρΣu(t)ρ)(maxs[h,0]|i|Kρiu2i(t+s)+bK),

    where bK0+ as K, and C()>0 is a continuous non-decreasing function.

    The proof is as follows.

    |i|Kρi|Gτ,i(ut)|23|i|Kρi|fi(ui(t))|2+3|i|Kρi|jZdki,jσε(uj(tτj)θ)|2+3|i|Kρi|gi|23|i|KρiL2(ρi|ui(t)|)|ui(t)|2+3|i|Kρiκ2+3|i|Kρi|gi|23L2(ρΣu(t)ρ)|i|Kρi|ui(t)|2+3|i|Kρiκ2+3|i|Kρi|gi|2C(ρΣu(t)ρ)(maxs[h,0]|i|Kρiu2i(t+s)+bK).

    By Corollary 13 in [3], we also conclude that the solution u() C1([0,a],2ρ).

    Here we will establish some estimates of the solutions, which imply that the solutions are bounded uniformly with respect to bounded sets of initial conditions and all positive values of time.

    Proposition 1. Suppose that Assumptions 1–7 hold. Then every solution u() of (3) with u0 = ψ C([h,0],2ρ) verifies

    ut2C([h,0],2ρ)R1eαtψ2C([h,0],2ρ)+R2, (11)

    where Rj > 0, j=1,2, are constants depending on the parameters of the problem.

    Proof. We multiply the ith component of (1) by ρiui(t) and sum over i to obtain

    12ddtu2ρ=iZdρiuifi(ui)+iZd(ρiuijZdki,jσε(uj(tτj)θ))+iZdρiuigi. (12)

    By Assumption 6 and ρi>0 we have

    ρifi(ui(t))ui(t)αρiu2i(t)+ρiβ2i,

    so

    iZdρifi(ui(t))ui(t)αu(t)2ρ+β2ρ.

    Since function σε takes values in the unit interval, using Young's inequality we obtain

    |ρiuijZdki,jσε(uj(tτj)θ)||ρiuijZdki,j|α4ρiu2i+1αρi(jZdki,j)2α4ρiu2i+1αρiκ2,

    so

    iZd|ρiuijZdki,jσε(uj(tτj)θ)|α4u(t)2ρ+1αρΣκ2.

    The last term on the right hand of (12) satisfies

    iZdρigiui(t)α4iZdρiu2i(t)+1αiZdρig2iα4u(t)2ρ+1αg2ρ.

    In summary, collecting the inequalities above, we obtain

    12ddtu(t)2ρ12αu(t)2ρ+β2ρ+1α(ρΣκ2+g2ρ).

    Integrating both sides of this differential inequality yields

    u(t)2ρu(0)2ρeαt+2α(β2ρ+1α(ρΣκ2+g2ρ))(1eαt). (13)

    Let θ[h,0]. Replacing t by t+θ in (13) and using

    u(t+θ)ρ=ψ(t+θ)ρψC([h,0],2ρ),t+θ<0,

    we obtain

    u(t+θ)2ρψ2ρeα(t+θ)+2α(β2ρ+1α(ρΣκ2+g2ρ))(1eα(t+θ)).

    Finally, using that θ[h,0] and neglecting the negative terms yields

    ut2C([h,0],2ρ)eαheαtψ2C([h,0],2ρ)+2α(β2ρ+1α(ρΣκ2+g2ρ))=R1eαtψ2C([h,0],2ρ)+R2, (14)

    where

    R1:=eαh,R2:=2α(β2ρ+1α(ρΣκ2+g2ρ)).

    Having the existence of the solution of problem (3), moreover, we now establish the uniqueness of the solution with the additional assumption that

    Assumption 8. There exists a constant ˜κ>0 such jZdk2i,jρj˜κ for each iZd.

    Lemma 3.3. Suppose that Assumptions 1–8 hold. Then the solution u of problem (3) is unique.

    Proof. Assumption 8 implies that the operator Kτ:C([h,0],2ρ)2ρ is Lipschitz. In fact,

    iZdρi|Kτ,i(ut)Kτ,i(vt)|2iZdρi|jZdki,j(σε(uj(tτj)θ)σε(vj(tτj)θ))|2iZdρi(jZdki,j|σε(uj(tτj)θ)σε(vj(tτj)θ)|)2iZdρi(LσjZdki,j|uj(tτj)vj(tτj)|)2iZdρiL2σ(jZdki,jρjρj|uj(tτj)vj(tτj)|)2ρΣ˜κL2σutvt2C([h,0],2ρ).

    Hence, suppose that we have two different solutions u, v of problem (3) with the same initial condition u(s)=v(s)=ψ(s), s[h,0].

    Set w=uv, we obtain that

    12ddtw2ρL(ρΣ(uρ+vρ))w2ρ+wρC1ρΣ˜κLσwtC([h,0],2ρ)L(2ρΣ(R1ψ2C([h,0],2ρ)+R2))w2ρ+wρC1ρΣ˜κLσwtC([h,0],2ρ)Cwt2C([h,0],2ρ),

    where C=max{L(2ρΣ(R1ψ2C([h,0],2ρ)+R2)),C1ρΣ˜κLσ}.

    Integrating from 0 to t then gives

    w(t)2ρ2Ct0wτ2C([h,0],2ρ)dτ+w(0)2ρ.

    Let θ[h,0]. Replacing t by t+θ in the inequality above and using w(t+θ)ρ=0 when t+θ<0. We obtain

    w(t+θ)2ρ2Ct+θ0wτ2C([h,0],2ρ)dτ+w(0)2ρ.

    Then take the supremum on θ,

    wt2C([h,0],2ρ)2Ct0wτ2C([h,0],2ρ)dτ+w(0)2ρ.

    By Gronwall's inequality, we have

    wt2C([h,0],2ρ)2Cte2Ctw(0)2ρ+w(0)2ρ. (15)

    Since w(0)=0, we obtain that w0.

    The proof of the next corollary follows easily using (15).

    Corollary 1. The map (t,ψ)ut is continuous.

    Proposition 1 implies that every local solution of (1) can be extended globally, which, with the uniqueness of the solution, will allow us to define a semigroup in terms of the solution mapping and to conclude that it has a bounded absorbing set.

    When Assumptions 1–8 hold, Theorem 3.2 and Lemma 3.3 ensure the local existence and uniqueness of solutions of the delayed lattice system (3), while Proposition 1 shows that the solutions are, in fact, globally defined.

    We can thus define a semigroup of operators S : R+×C([h,0],2ρ) C([h,0],2ρ) by

    S(t,ψ)=ut,

    where ut is the unique solution to (3) with u0=ψ. The semigroup map S is continuous in its variables by Corollary 1.

    It also follows from inequality (11) that the semigroup has a bounded absorbing set.

    Corollary 2. The bounded set defined by

    B0:={ψC([h,0],2ρ):ψC([h,0],2ρ)R0},

    with R0:=1+R2, is absorbing for the semigroup S.

    Our aim is to study the asymptotic behaviour of solution of problem (1). In particular, we will show the existence of a global attractor. For this we will apply the following well-known results about the existence of global attractors, see [8] and [5].

    Theorem 4.1. Let xS(t,x) be continuous for any t0. Assume that S is asymptotically compact and possesses a bounded absorbing set B0. Then there exists a global compact attractor A, which is the minimal closed set attracting any bounded set. If, moreover, the space X is connected and the map tS(t,x) is continuous for any xX, then the set A is connected.

    To show the asymptotic compactness of the semigroup, we need to estimate the tails of solutions of (3), i.e., their higher dimensional components, see [2].

    Lemma 4.2. Suppose that Assumptions 1–8 hold and let B be a bounded set of C([h,0],2ρ). Then, for any ε>0 there exist T(ε,B) and M(ε,B) such that

    maxs[h,0]|i|>2M(ε,B)ρi|ui(t+s)|2<ε,tT,

    for any initial condition ψB and the corresponding solution u() of (3) with u0 = ψ.

    Proof. Define a smooth function ξ satisfying

    ξ(s){=0,       0s1,[0,1],   1s2,=1,       s2.

    Let M be a fixed (and large) integer to be specified later, and set

    vi(t)=ξM(|i|)ui(t)  with  ξM(|i|)=ξ(|i|M),   iZd,

    where || denotes the Euclidean norm. We multiply the ith component of (1) by ρivi, then summing over i Zd, and since u() C1([0,],2ρ), we have

    12ddtiZdρiξM(|i|)|ui(t)|2=iZdρiξM(|i|)ui(t)dui(t)dt=iZdρiξM(|i|)ui(t)fi(ui(t))+iZdρiξM(|i|)ui(t)gi+iZdρiξM(|i|)ui(t)jZdki,jσε(uj(tτj)θ). (16)

    First, by Assumption 6,

    ρiξM(|i|)ui(t)fi(ui)αρiξM(|i|)u2i(t)+ρiξM(|i|)β2i. (17)

    Then, since function σε takes values in the unit interval, using Young's inequality,

    |ρiξM(|i|)ui(t)jZdki,jσε(uj(tτj)θ)||ρiξM(|i|)ui(t)κ|α4ρiξM(|i|)u2i(t)+κ2αρiξM(|i|). (18)

    And using Young's inequality again,

    iZdρiξM(|i|)giui(t)=|i|MρiξM(|i|)giui(t)α4iZdρiξM(|i|)u2i(t)+1α|i|Mρig2i. (19)

    Inserting the estimations (17), (18) and (19) into (16), then

    12ddtiZdρiξM(|i|)u2i(t)12αiZdρiξM(|i|)u2i(t)+iZdρiξM(|i|)β2i+κ2αiZdρiξM(|i|)+1α|i|Mρig2i. (20)

    We now estimate each term on the right hand side of the above inequality. Note that

    iZdρiξM(|i|)β2i=|i|MρiξM(|i|)β2i|i|Mρiβ2i.

    Since β = (βi)iZd 2ρ, then for every ε>0 there exists I1(ε)>0 such that

    iZdρiξM(|i|)β2i|i|Mρiβ2i<16ε  whenMI1(ε). (21)

    Similarly, since ρΣ = iZdρi < , then for every ε>0, there exists I2(ε)>0 such that

    iZdρiξM(|i|)=|i|MρiξM(|i|)|i|Mρi<α6κ2εwhenMI2(ε). (22)

    In addition, since g = (gi)iZd 2ρ by Assumption 7, for every ε>0 there exists I3(ε)>0 such that

    |i|Mρig2iα6ε,  tR,whenMI3(ε). (23)

    Finally, for any ε>0, choosing I(ε):=max{I1(ε),I2(ε),I3(ε)}, inserting the estimations (21), (22) and (23) into (20) results in

    ddtiZdρiξM(|i|)u2i(t)αiZdρiξM(|i|)u2i(t)+ε,  MI(ε).

    It follows immediately from Gronwall's lemma that

    iZdρiξM(|i|)u2i(t)eαtiZdρiξM(|i|)u2i(0)+εα.

    In a similar way as in Proposition 1 we have

    maxs[h,0]iZdρiξM(|i|)u2i(t+s)eαheαtmaxs[h,0]iZdρiξM(|i|)ψ2i(s)+εα.

    Thus, there exist T(ε,B) and M(ε,B) such that

    maxs[h,0]|i|>2Mρi|ui(t+s)|2εiftT.

    In order to apply Theorem 4.1, we need to prove that S generated by the delay lattice system (3) is asymptotically compact.

    Lemma 4.3. Suppose that Assumptions 1–8 hold. Then the semigroup S is asymptotically compact.

    Proof. We consider ξn := untn = S(tn,ψn), where ψn B, a bounded set in C([h,0],2ρ). From (14) there is a C>0 such that

    untn(s)C,s[h,0],nN.

    For fixed s[h,0] we can find a subsequence (which we still denote by un) such that

    un(tn+s)ζ(s)  in  2ρ.

    In fact, the weak convergence here is strong, which follows from Lemma 4.2. Indeed, there exists N1>0, when nN1, we have tn>T (where T is the constant in Lemma 4.2). Moreover, for any μ>0 there exist K2(μ) and N2(μ) such that

    |i|>K2ρi|uni(tn+s)|2<μ,|i|>K2ρi|ζi(s)|2<μ,|i|K2ρi|uni(tn+s)ζi(s)|2<μ

    if nmax{N1,N2(μ)}. Hence

    un(tn+s)ζ(s)2ρ|i|K2ρi|uni(tn+s)ζi(s)|2+|i|>K2ρi|uni(tn+s)ζi(s)|2|i|K2ρi|uni(tn+s)ζi(s)|2+2|i|>K2ρi|uni(tn+s)|2+2|i|>K2ρi|ζi(s)|2<5μ.

    Thus, {un(tn+s)} is precompact in 2ρ for any s[h,0]. Since Gτ is a bounded map, Proposition 1 and the integral representation of solutions imply that

    un(tn+s)un(tn+t)ρtsGτ(untn+τ)ρdτK(ts)ifhs<t0.

    Then, the Ascoli-Arzelà theorem implies that ξn is relatively compact in C([h,0],2ρ).

    Remark 2. If Assumption 8 guarranteeing uniqueness of solutions does not hold, then the lattice model (1) generates a set-valued semi-dynamical system, which can be shown to have a global attractor using essentially the same Lemmas as above.



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