Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 64, pp. 1–22.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.64

UPPER SEMICONTINUITY OF UNIFORM ATTRACTORS FOR SINGULAR

PERTURBED SECOND ORDER NONAUTONOMOUS DELAY LATTICE

SYSTEMS

YAO ZHOU, HONGLIANG LIU

Abstract. In this article, we consider the upper semicontinuity of the uniform attractors for
the singular perturbed second order nonautonomous delay lattice systems driven by the almost

periodic forces as the coefficient of second order derivative term tends to zero under the Hausdorff

semidistance. First we prove the existence of uniform attractors for the second order and the
corresponding first order nonautonomous delay lattice systems. Then we establish some prior

uniform estimations of solutions. Finally we study the upper semicontinuity of the uniform

attractors as the coefficient of second order derivative term tends to zero which showing the
relationship between the uniform attractors for second order and the corresponding first order

nonautonomous delay lattice systems.

1. Introduction

Let k ∈ N and

ℓ2 =
{
u = (um)m∈Zk : m = (m1, . . . ,mk) ∈ Zk, um ∈ R,

∑
m∈Zk

u2m < +∞
}
,

be a Hilbert space endowed with the inner product and norm:

(u, v) =
∑
m∈Zk

umvm, ∥u∥2 = (u, u), u = (um)m∈Zk , v = (vm)m∈Zk ∈ ℓ2.

Let Cb(R, ℓ2) denote the space of continuous bounded functions from R into ℓ2 and g0 = (g0,m)m∈Zk :

R → ℓ2 be an almost periodic function in the Bohr sense,

H(g0) = {g0(·+ r) : r ∈ R}
Cb(R,ℓ2)

(the closure in Cb(R, ℓ2)).
In this article, we consider the family of second-order nonautonomous delay lattice systems

with singular perturbation

ϵüm + u̇m + γ(Au̇)m + (Au)m + λmum + fm(uj |j ∈ Imq)

+hm(um(t− ϑ)) = gm(t), t ≥ τ, g(·) = (gm(·))m∈Zk ∈ H(g0), ϵ > 0,

um,τ (θ) = um(τ + θ) = u0,mτ (θ), τ ∈ R,

u̇m,τ (θ) = u̇m(τ + θ) = u1,mτ (θ), θ ∈ [−ϑ, 0], m ∈ Zk,

(1.1)

where m ∈ Zk, λm > 0, ϵ, ϑ > 0, γ ≥ 0, um, gm(t), fm(uj(t)|j ∈ Imq), hm(um(t − ϑ)) ∈ R,
u = (um)m∈Zk , A is a linear coupled operator, Imq = {j ∈ Zk : ∥j−m∥ = max1≤l≤k |jl−ml| ≤ q},
q ∈ N, fm(uj |j ∈ Imq) indicates that the state at the m-th lattice point can be related to the
states at its surrounding (2q + 1)k − 1 lattice points (the relationship may be nonlinear).

2020 Mathematics Subject Classification. 37C70, 34K31, 34L26, 37K60.
Key words and phrases. Singular perturbed second order delay lattice system;

uniform attractor; upper semicontinuity; continuous process; almost periodic function.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 22, 2025. Published June 26, 2025.

1



2 Y. ZHOU, H. LIU EJDE-2025/64

When ϵ = 0, (1.1) becomes the following family of first order nonautonomous delay lattice
systems

u̇m + γ(Au̇)m + (Au)m + λmum + fm(uj |j ∈ Imq) + hm(um(t− ϑ))

= gm(t), t ≥ τ, g = (gm)m∈Zk ∈ H(g0),

um,τ (θ) = um(τ + θ) = u0,mτ (θ), τ ∈ R, θ ∈ [−ϑ, 0], m ∈ Zk.
(1.2)

The lattice systems (1.1)-(1.2) can be used as mathematical models for the various coupled oscil-
lator systems (such as the system of coupled pendulum motions) and the dynamic network systems
with infinite nodes etc.. The attractors of various different types of lattice systems (consisting of
infinite dimensional ordinary differential equations) have been studied by many publications in the
last more than 20 years from the work of Bate et al in 2001 [2], including the existence and related
properties of the global attractor, pullback attractor, uniform attractor and random attractor, see
[1, 5, 7, 8, 10, 12, 13, 14, 15, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28] and the references therein.

The relationship between the attractors of first-order and second-order lattice systems is of
interesting topic. For the case of autonomous and nonautonomous lattice systems (1.1)-(1.2)
without delay and the coupled term of first order derivatives (that is, ϑ ≡ 0 and γ = 0), the
relationship between the global attractors and uniform attractors of (1.1) and (1.2) as ϵ → 0+

have been studied in [13, 25], respectively. In the case of ϑ ̸= 0 and γ ̸= 0, the phase space of
(1.1) and (1.2) are Banach spaces C([−ϑ, 0], ℓ2 × ℓ2) and C([−ϑ, 0], ℓ2) consisting of continuous
functions from a closed interval [−ϑ, 0] into the spaces ℓ2 × ℓ2 and ℓ2, respectively, which are
different from the Hilbert phase spaces in [13, 25]. As we know, there is no results about the
relationship between the uniform attractors of (1.1) and (1.2) until now.

Based on the works of [13, 25], here we consider the upper-semicontinuity of the uniform
attractors for the singular perturbated second order nonautonomous delay lattice system (1.1)
as ϵ → 0+, which gives the relationship between the uniform attractors of (1.1) and (1.2). Since
the Banach phase spaces cannot be decomposed into a direct sum of a finite dimensional space and
an infinite dimensional space with a small norm, so in proving the key prior uniformly bounded
estimations of solutions of systems, the asymptotic compactness of the solutions processes and the
convergence of the solutions sequences et al., we have to use new techniques different from those

in [13, 25]. Notice that the uniform attractors AH(g
0
)

ϵ of (1.1) and AH(g
0
)

0 of (1.2) are included
in different spaces C([−ϑ, 0], ℓ2 × ℓ2) and C([−ϑ, 0], ℓ2), respectively, to compare the relationship
between them, we take them in the bigger space C([−ϑ, 0], ℓ2× ℓ2). For our purpose, we construct
a compact set BH(g

0
)

0 ⊂ C([−ϑ, 0], ℓ2 × ℓ2) such that AH(g
0
)

0 is naturally embedded into BH(g
0
)

0 as
the first component. It is worth mentioning that because of the lack of the structure of operator
(I + γA)−1, the equivalent first order lattice equations of (1.2) may be not a locally coupled
lattice system. Generally, the proof of the asymptotic compactness of the solutions process for a
non-locally coupled lattice system is difficult. Fortunately, the linear boundedness of (I + γA)−1

here is just enough to solve this challenging problem.
In section 2, we present some spaces, some assumptions and the vector forms of (1.1) and (1.2).

In section 3, we prove the existence of uniform attractors AH(g0)
ϵ of (1.1) and AH(g0)

0 of (1.2). In
section 4, we establish some prior uniform estimations for the solutions of (1.1). In section 5, we

consider the upper semicontinuity of AH(g0)
ϵ as ϵ→ 0+.

2. Mathematical setting

Firstly, we present some concepts related with the uniform attractor for a family of processes.
Let X be a Banach space with norm ∥ · ∥X , B(X) be the union of all bounded sets of X and Σ be
a parameter set.

Definition 2.1. A two-parameters family of mappings {U(t, τ) : X → X, t ≥ τ ∈ R} is said to be
a continuous process on X, if (i) U(t, s)U(s, τ) = U(t, τ), for all t ≥ s ≥ τ ; (ii) U(τ, τ) = I (unit
operator), for all τ ∈ R; (iii) for all t ≥ τ ∈ R, U(t, τ) is continuous on X. {Uσ(t, τ)}t≥τ,σ∈Σ is
called a family of continuous processes in X with parameter σ ∈ Σ, if for each σ ∈ Σ, {Uσ(t, τ)}t≥τ
is a continuous process in X, where Σ is called a symbol space and σ ∈ Σ is a symbol.



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 3

Definition 2.2. A subset D0 of X is said to be uniformly (with respect to (w.r.t.) σ ∈ Σ)
absorbing for a family of processes {Uσ(t, τ)}t≥τ,σ∈Σ, if for any τ ∈ R and each bounded set
B ∈ B(X), there exists tτ,B ≥ 0 such that

⋃
σ∈Σ U

σ(t, τ)B ⊆ D0 for all t ≥ τ + tτ,B .

Definition 2.3. A closed set AΣ ⊆ X is said to be a uniform (w.r.t. σ ∈ Σ) attractor for a family
of continuous processes {Uσ(t, τ)}t≥τ,σ∈Σ, if

(i) limt→+∞ supσ∈Σ dh(U
σ(t, τ)B,AΣ) = 0 for any τ ∈ R and any bounded set B ∈ B(X),

where “dh(·, ·)” is the Hausdorff semidistance between two subsets of X;
(ii) AΣ is the minimal set (for inclusion relation) among those sets satisfying (i).

Definition 2.4. A family of processes {Uσ(t, τ)}t≥τ,σ∈Σ is said to be asymptotically compact
in X if for any τ ∈ R, B ∈ B(X), each sequence {tn} ⊂ [0,+∞) with tn → +∞ as n → ∞,
each sequence {un} ⊂ B and each sequence {σn} ⊂ Σ, the sequence {Uσn(tn + τ, τ)un} has a
convergent subsequence in X.

We make the following assumptions on the quantities in (1.1)- (1.2):

(A1) A is a linear operator on ℓ2 with decomposition A =
∑k
j=1Aj , Aj = B∗

jBj = BjB
∗
j , where

the operators Bj are defined by

(Bju)m =

l=m0∑
l=−m0

dj,lumjl
, |dj,l| ≤ a0 (constant), u = (um)m∈Zk ∈ ℓ2,

mjl = (m1, . . . ,mj−1,mj + l,mj+1, . . . ,mk) ∈ Zk, (Bju, v) = (u,B∗
j v) for u, v ∈ ℓ2,

j = 1, . . . , k.
(A2) ∀m ∈ Zk, 0 < λ0 ≤ λm ≤ λ0 <∞, where λ0, λ

0 are two positive constants.
(A3) g0 = (g0,m)m∈Zk , g′0 = (g′0,m)m∈Zk : R → ℓ2 are both almost periodic functions in the

Bohr sense.
(A4) For all m ∈ Zk, fm(·) ∈ C1(R(2q+1)k,R), fm(uj = 0|j ∈ Imq) = 0 and there exist

ρ ∈ C(R+,R+), b = (bm)m∈Zk ∈ ℓ2, such that

sup
m∈Zk

max
uj∈[−r,r],j∈Imq

|f ′m,j(uj |j ∈ Imq)| ≤ ρ(r),

fm(uj |j ∈ Imq)um ≥ Gm(uj |j ∈ Imq) ≥ −b2m,

where Gm(uj |j ∈ Imq) =
∫ um

0
fm(r, uj |j ∈ Imq \ {m})dr, f ′m,j(uj |j ∈ Imq) =

∂fm
∂uj

(uj |j ∈
Imq) and fm(r, uj |j ∈ Imq \ {m}) is the function fm(uj |j ∈ Imq) in which um is replaced
by r.

(A5) For all m ∈ Zk, hm ∈ C1(R,R), hm(0) = 0 and hm (s) is Lipschitz continuous in s:

|hm (s1)− hm (s2) | ≤ Lh|s1 − s2|, Lh ≥ 0, ∀s1, s2 ∈ R, m ∈ Zk,
where

0 ≤ Lh ≤

{
λ0

2 e
− 1

2 λ̃ϑ, ϵ = 0,

1
2

√
λ0ε0
ϵ e−

ε0
8ϵ ϑ, ϵ > 0,

λ̃ =

{
λ0

2 , γ = 0,

min{ 2
γ ,

λ0

2 }, γ > 0,

ε0 =

{
ϵλ0

1+3ϵλ0
, γ = 0,

min
{
ϵ
γ ,

ϵλ0

1+3ϵλ0

}
, γ > 0.

(A6) For all ϵ > 0, there exists a constant δϵ ≥ 0 such that∣∣∂Gm
∂uj

(uj |j ∈ Imq)
∣∣ = |G′

m,j(uj |j ∈ Imq)| ≤ δϵ|um|, m ̸= j, m ∈ Zk,

where

0 ≤ δϵ ≤ min
{ ε0λ0
4ϵ(2λ20(2q)

2k + 1)
,

1

4(2q)2k
}
.



4 Y. ZHOU, H. LIU EJDE-2025/64

Now we define some spaces. For each ϵ > 0 and u, v ∈ ℓ2, we define the inner products

(u, v)δλ = δ

k∑
j=1

(Bju,Bjv) + (λu, v)

= δ

k∑
j=1

(Bju,Bjv) +
∑
m∈Zk

λmumvm, δ = 1− ε0
ϵ
γ ∈ [0, 1]

and

(u, v)δλϵ = ϵ−1(u, v)δλ = ϵ−1δ

k∑
j=1

(Bju,Bjv) + ϵ−1
∑
m∈Zk

λmumvm.

By (A1) and (A2), the three norms ∥ · ∥, ∥ · ∥δλ, ∥ · ∥δλϵ are equivalent to each other. Let

ℓ2δλϵ =
(
ℓ2, (·, ·)δλϵ

)
, E = ℓ2 × ℓ2, H = ℓ2δλϵ × ℓ2,

then E, H are Hilbert spaces with the norms:

∥(u, v)T∥2E = ∥u∥2 + ∥v∥2, ∀(u, v)T ∈ E,

∥(u, v)T∥2H = ∥u∥2δλϵ + ∥v∥2 = ϵ−1δ∥Bu∥2 + ϵ−1
∑
m∈Zk

λmu
2
m + ∥v∥2, ∀(u, v)T ∈ H.

For the positive delay number ϑ > 0, write the Banach spaces ℓ2ϑ = C([−ϑ, 0], ℓ2), Eϑ =
C([−ϑ, 0], E) and Hϑ = C([−ϑ, 0], H) with norms, respectively, as follows:

∥u(·)∥2ℓ2ϑ = sup
−ϑ≤θ≤0

∥u(θ)∥2, ∀u(·) ∈ ℓ2ϑ,

∥(u(·), v(·))T∥2Eϑ
= ∥u(·)∥2ℓ2ϑ + ∥v(·)∥2ℓ2ϑ , ∀(u(·), v(·))T ∈ Eϑ,

∥(u(·), v(·))T∥2Hϑ
= sup

−ϑ≤θ≤0
∥u(θ)∥2δλϵ + ∥v(·)∥2ℓ2ϑ , ∀(u(·), v(·))

T ∈ Hϑ.

By (A3) and the Bochner-Amerio criteria, the sets {g0(·+ r)}r∈R, {g′0(·+ r)}r∈R are both precom-

pact in Cb(R, ℓ2) [4]. Thus H(g0) and H(g′0) = {g′0(·+ r) : r ∈ R}
Cb(R,ℓ2)

are compact in Cb(R, ℓ2).
We set

T (r) : g → T (r)g = g(·+ r), ∀g ∈ H(g0), r ∈ R,
then {T (r)}r∈R is a translation group acting onH(g0), (r, g) → T (r)g is continuous and T (r)H(g0) =
H(g0) for all r ∈ R.

Finally, we present the equivalent vector forms of systems (1.1)-(1.2). By (A1) and the Lax-
Milgram theorem, the operator (I + γA)−1 exists and it is linear bounded from ℓ2 into ℓ2: ∥(I +
γA)−1∥L(ℓ2,ℓ2) ≤ 1.

For θ ∈ [−ϑ, 0], t ∈ R, we write u = (um)m∈Z, λu = (λmum)m∈Z, f(u) = (fm(uj |j ∈ Imq))m∈Z,
h(u(t − ϑ)) = (hm(um(t − ϑ)))m∈Z, g(t) = (gm(t))m∈Z, ut(θ) = u(t + θ) = (um(t + θ))m∈Z,
u̇t(θ) = u̇(t+ θ) = (u̇m(t+ θ))m∈Z. Then (1.2) can be written as the following family of first order
delay lattice systems with initial condition:

u̇ = F0(ut(θ), t), t ≥ τ, θ ∈ [−ϑ, 0], g ∈ H(g0),

uτ (θ) = u(τ + θ) = u0,τ (θ), τ ∈ R, θ ∈ [−ϑ, 0],
(2.1)

where

F0(ut(θ), t) = (I + γA)−1[−Au(t)− λu(t)− f(u(t))− h (u(t− ϑ)) + g(t)], (2.2)

(I + γA)u̇+Au+ λu+ f(u) + h (u(t− ϑ)) = g(t), t ≥ τ, g ∈ H(g0), (2.3)

and (1.1) can be written as the following family of second order delay lattice systems with initial
conditions:

ϵü+ u̇+ γAu̇+Au+ λu+ f(u) + h (u(t− ϑ)) = g(t), t ≥ τ, g ∈ H(g0),

uτ (θ) = u(τ + θ) = u0,τ (θ), u̇τ (θ) = u̇(τ + θ) = u1,τ (θ), τ ∈ R, θ ∈ [−ϑ, 0].
(2.4)



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 5

For a fixed g ∈ H(g0) and ϵ > 0, let uϵ(t) be the solution of (2.4), set

vϵ = u̇ϵ +
ε0
ϵ
uϵ, uϵ,t(θ) = uϵ(t+ θ), vϵ,t(θ) = vϵ(t+ θ), θ ∈ [−ϑ, 0], t ∈ R, (2.5)

where ε0 is defined in (A5). Then problem (2.4) is equivalent to the following vector forms:

ψ̇ϵ(t) +Hϵψϵ(t) = Fϵ(ψϵ,t(θ), t), t ≥ τ, θ ∈ [−ϑ, 0], g ∈ H(g0), τ ∈ R,

ψϵ(τ)(θ) = ψϵ(τ + θ) = (uϵ(τ + θ), vϵ(τ + θ))T = (uϵ,τ (θ), vϵ,τ (θ))
T,

(2.6)

where ψϵ(t) = (uϵ(t), vϵ(t))
T, ψϵ,t = (uϵ,t, vϵ,t)

T,

Hϵψϵ(t) = Hϵψϵ,t(0)

=

(
ε0
ϵ uϵ(t)− vϵ(t)

1
ϵ [λuϵ(t) + (1− 1

ϵγε0)Auϵ(t)−
1
ϵ ε0(1− ε0)uϵ(t) + (1− ε0)vϵ(t) + γAvϵ(t)]

)
,

Fϵ(ψϵ,t(θ), t) =

(
0

− 1
ϵ f(uϵ,t(0))−

1
ϵh (uϵ,t(−ϑ)) +

1
ϵ g(t)

)
.

3. Existence of uniform attractors

We first consider the existence of uniform attractors for the family of continuous processes
defined by the solutions of (2.1) and (2.6) on the spaces ℓ2ϑ and Hϑ, respectively. Then based on
the transformation (2.5), we obtain the existence of a uniform attractor for the family of continuous
processes of solutions φϵ,t(·) = (uϵ,t(·), u̇ϵ,t(·))T of (2.4) in Eϑ.

Theorem 3.1. For the initial value problem (2.1), if (A1)–(A5) hold, then for each g ∈ H(g0),
τ ∈ R, and uτ (·) ∈ ℓ2ϑ, (2.1) has a unique solution ut(·) = u(t, τ, uτ (·)) ∈ ℓ2ϑ existing on t ∈
[τ,+∞), ut(·) is continuous in uτ (·) and u(·) = u(·, τ, uτ (θ)) ∈ C([τ−ϑ,+∞), ℓ2)∩C1([τ,+∞), ℓ2),
θ ∈ [−ϑ, 0]. Moreover, the solution maps:

Ug0 (t, τ) : ℓ
2
ϑ ∋ uτ (·) → ut(·) = u(t, τ, uτ (·)) ∈ ℓ2ϑ, t ≥ τ, τ ∈ R, (3.1)

generates a continuous process {Ug0 (t, τ)}t≥τ on ℓ2ϑ and the family of continuous processes

{Ug0 (t, τ)}t≥τ,g∈H(g0) possesses a unique compact uniform attractor AH(g
0
)

0 :

AH(g
0
)

0 = ∪g∈H(g
0
)Ag

0,t = ∪g∈H(g
0
)Ag

0,0 ⊂ ℓ2ϑ, ∀t ∈ R, (3.2)

where

Ag
0,t =

{
ut : ut(·) = u(t+ ·) : [−ϑ, 0] → ℓ2 is the global solution of (2.1),

∥ut(·)∥ℓ2ϑ ≤ r0, ∀t ∈ R
} (3.3)

with the invariance in the sense that Ug0 (t, τ)A
g
0,τ = Ag

0,t for t ≥ τ , τ ∈ R and r0 = 2
√

∥g0∥2

λ̃λ0
+ ∥b∥2

λ̃
.

Proof. (i) By (A3), for any g ∈ H(g
0
), g is almost periodic on R and H(g) = H(g

0
). By (A1)–

(A5) and the linear boundedness of (I + γA)−1 from ℓ2 into ℓ2, it follows that for τ , t ∈ R,
T > 0, θ ∈ [−ϑ, 0], ut(θ) = u(t + θ), F0(ut(·), t) is continuous from ℓ2ϑ × [τ, τ + T ] into ℓ2 and
locally Lipschitz in ut(·). Therefore, for any uτ (·) ∈ ℓ2ϑ, (2.1) has a unique (locally) solution
u(·) = u(·, τ, uτ (θ)) ∈ C([τ − ϑ, T0,max), ℓ

2) ∩ C1([τ, T0,max), ℓ
2), θ ∈ [−ϑ, 0], T0,max > τ and

ut(·) = u(t, τ, uτ (·)) is continuous in uτ (·) for t ∈ [τ, T0,max) [9, 21]. u(t) satisfies the initial value
and integral equation:

u(τ)(θ) = u(τ, τ, uτ (θ)) = uτ (θ),

u(t) = uτ (0) +

∫ t

τ

F0(us(θ), s)ds,

for θ ∈ [−ϑ, 0] and t ∈ [τ, T0,max). Taking the inner product of u(t) = u(t, τ, uτ (·)) ∈ ℓ2 (t ≥ τ)
with (2.3) and by (A1)–(A5), we have

d

dt
(∥u(t)∥2 + γ

k∑
j=1

∥Bju(t)∥2) + λ̃(∥u(t)∥2 + γ

k∑
j=1

∥Bju(t)∥2) +
λ0
2
∥u(t)∥2 (3.4)



6 Y. ZHOU, H. LIU EJDE-2025/64

≤ 2L2
h

λ0
∥u(t− ϑ)∥2 + 2∥g0∥2

λ0
+ 2∥b∥2, t ≥ τ, (3.5)

where ∥b∥2 =
∑
m∈Z b

2
m <∞, and

∥g∥2 = sup
t∈R

∑
m∈Zk

g2m(t) ≤ ∥g0∥2 = sup
t∈R

∑
m∈Zk

g20,m(t) <∞.

Applying Gronwall’s inequality on [τ, t] (t ≥ τ) to (3.5), we obtain

∥u(t)∥2 ≤ (∥u(τ)∥2 + γ

k∑
j=1

∥Bju(τ)∥2)e−λ̃(t−τ)

+
2L2

h

λ0λ̃
eλ̃ϑ∥uτ (·)∥2ℓ2ϑe

−λ̃(t−τ) +
r20
2
, t ≥ τ.

(3.6)

Thus, for θ ∈ [−ϑ, 0], set t+ θ instead of t, it holds from (3.6) that for t ≥ τ ,

∥u(t+ θ)∥2 ≤
(
∥u(τ)∥2 + γ

k∑
j=1

∥Bju(τ)∥2
)
eλ̃ϑe−λ̃(t−τ)

+
2L2

h

λ0λ̃
eλ̃ϑ∥uτ (·)∥2ℓ2ϑe

λ̃ϑe−λ̃(t−τ) +
r20
2
, t+ θ ≥ τ,

∥u(t+ θ)∥2 ≤ ∥uτ (·)∥2ℓ2ϑ , t+ θ ≤ τ.

(3.7)

So T0,max = +∞, and the solutions maps (3.1) generate a continuous process {Ug0 (t, τ)}t≥τ on ℓ2ϑ.
By the definition of {T (r)}r∈R, we have

Ug0 (t+ r, τ + r) = U
T (r)g
0 (t, τ), ∀g ∈ H(g

0
), t ≥ τ, τ, r ∈ R.

By (3.7),

∥ut(·)∥2ℓ2ϑ ≤
(
∥uτ (·)∥2ℓ2ϑ + γ

k∑
j=1

∥Bju(τ)∥2 +
2L2

h

λ0λ̃
eλ̃ϑ∥uτ (·)∥2ℓ2ϑ

)
eλ̃ϑe−λ̃(t−τ) +

r20
2

.
= b̃2(t, τ, ∥uτ (·)∥2ℓ2ϑ), t ≥ τ,

(3.8)

where b̃2(t, τ, uτ (·)) is continuous in t. Let τ ∈ R, g(1), g(2) ∈ H(g0), u
(1)
τ (·), u(2)τ (·) ∈ ℓ2ϑ, t ≥ τ ,

u(j)(t) = u(t, τ, g(j), u
(j)
τ (·)), j = 1, 2,

κ(t, τ, g(1), g(2), u(1)τ (·), u(2)τ (·)) = u(t, τ, g(1), u(1)τ (·))− u(t, τ, g(2), u(2)τ (·)),
then

(I + γA)κ̇+Aκ+ λκ+ f(u(1)(t))− f(u(2)(t))

+ h(u(1)(t− ϑ))− h(u(2)(t− ϑ))

= g(1)(t)− g(2)(t), t ≥ τ,

κτ (θ) = u(1)τ (θ)− u(2)τ (θ), τ ∈ R, θ ∈ [−ϑ, 0].

(3.9)

By (3.8),

∥f(u(1)(t))− f(u(2)(t))∥2 ≤ (2q + 1)2kρ2(∥b̃(t, τ, ∥u(1)τ (·)∥2ℓ2ϑ)∥+ ∥b̃(t, τ, ∥u(2)τ (·)∥2ℓ2ϑ)∥)∥κ(t)∥
2.

Taking the inner product of κ(t) ∈ ℓ2 (t ≥ τ) with (3.9) and applying Gronwall’s inequality, we
have

∥κ(t)∥2 ≤
(
∥κ(τ)∥2 + γ

k∑
j=1

∥Bjκ(τ)∥2
)
e−

∫ t
τ
C̃0(l,τ)dl

+
2L2

h

λ0
eλ̃ϑ∥κτ (·)∥2ℓ2ϑ

∫ τ

τ−ϑ
e−

∫ t
r
C̃0(l,τ)dldr

+
4

λ0
∥g(1) − g(2)∥2

∫ t

τ

e−
∫ t
r
C̃0(l,τ)dldr, t ≥ τ,

(3.10)



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 7

where

C̃0(t, τ) = λ̃− 4

λ0
(2q + 1)2kρ2(∥b̃(t, τ, ∥u(1)τ (·)∥2ℓ2ϑ)∥+ ∥b̃(t, τ, ∥u(2)τ (·)∥2ℓ2ϑ)∥).

Setting t+ θ instead of t, where θ ∈ [−ϑ, 0], it holds from (3.10) that

∥κ(t+ θ)∥2 ≤ C̃1(t, τ)
(
∥κτ (·)∥2ℓ2ϑ + ∥g(1) − g(2)∥2

)
, t ≥ τ, (3.11)

where

C̃1(t, τ) = max
{
C̃10(t, τ),

4

λ0
eλ̃ϑ

∫ t

τ

e−
∫ t
r
C̃0(l,τ)dldr

}
,

C̃10(t, τ) = (1 + γk(2m0 + 1)2a20)e
λ̃ϑe−

∫ t
τ
C̃0(l,τ)dl +

2L2
h

λ0
e2λ̃ϑ

∫ τ

τ−ϑ
e−

∫ t
r
C̃0(l,τ)dldr,

and (3.11) implies that {Ug0 (t, τ)}t≥τ,g∈H(g0) is a family of continuous processes from ℓ2ϑ ×H(g0)

into ℓ2ϑ.
(ii) Let B0 = {u(·) ∈ ℓ2ϑ : ∥u(·)∥ℓ2ϑ ≤ r0} ⊂ ℓ2ϑ (independent of (τ, g) ∈ R×H(g0)), then by

(3.8) , B0 is a uniformly bounded closed absorbing ball of {Ug0 (t, τ)}t≥τ,g∈H(g0) and there exists
TB0 ≥ 0 such that for any g ∈ H(g0), τ ∈ R, t− τ ≥ TB0 , U

g
0 (t, τ)B0 ⊆ B0. Additionally, by (2.2)

and (3.8), it holds that there exist positive constants b0, b1 (independent of (g, t, τ) ) such that
for any t ≥ τ , τ ∈ R, g ∈ H(g

0
),

∥Ug0 (t, τ)B0∥ℓ2ϑ = sup
ut∈Ug

0 (t,τ)B0

∥ut(·)∥ℓ2ϑ

= sup
ut∈U0(t,τ)B0

sup
θ∈[−ϑ,0]

∥u(t+ θ)∥ ≤ b0,

sup
ut∈U0(t,τ)B0

sup
θ∈[−ϑ,0]

∥F0(ut(θ), t)∥ ≤ b1.

(3.12)

(iii) Choose a smooth increasing function ϱ ∈ C1(R+, [0, 1]) that satisfies

ϱ(s) = 0, 0 ≤ s ≤ 1, 0 ≤ ϱ(s) ≤ 1, 1 ≤ s ≤ 2,

ϱ(s) = 1, s ≥ 2, ϱ′(s)| ≤ C0, s ∈ R+, C0 > 0.

Let g ∈ H(g
0
), τ ∈ R, uτ (·) ∈ B0,

u(t) = Ug0 (t, τ)uτ (·) = u(t, τ, uτ (·)) = (um(t, τ, uτ (·)))m∈Zk ∈ ℓ2, t ≥ τ.

By(3.12), ∥u(t)∥ ≤ b0, ∥u(t − ϑ)∥ ≤ b0, ∥u̇(t)∥ ≤ b1 for t ≥ τ . Let K be a positive integer,

xm = ϱ( ||m||
K )um, x = (xm)m∈Zk . Taking the inner product of (2.3) with x in ℓ2, we have

d

dt

∑
m∈Zk

ϱ(
∥m∥
K

)(u2m(t) + γ

k∑
j=1

(Bju(t))
2
m)

+ λ̃
∑
m∈Zk

ϱ(
| |m||
K

)(u2m(t) + γ

k∑
j=1

(Bju(t))
2
m) +

λ0
2

∑
m∈Zk

ϱ(
||m||
K

)u2m(t)

≤ 2L2
h

λ0

∑
m∈Zk

ϱ(
||m||
K

)u2m(t− ϑ) +
c1
K

+
2

λ0

∑
∥m∥≥2K

g2m(t) + 2
∑

∥m∥≥2K

b2m, t ≥ τ,

(3.13)

where c1 =
kC0m0a

2
0(2m0+1)2

2 (b21 + 3b20). Applying Gronwall’s inequality on [τ, t] to (3.13), we have∑
m∈Zk

ϱ(
∥m∥
K

)u2m(t) ≤
(
r20 + γka20(2m0 + 1)2r20 +

1

λ̃

2L2
h

λ0
eλ̃ϑr20

)
e−λ̃(t−τ)

+
1

λ̃

( 2

λ0
sup
r∈R

∑
∥m∥≥K

g2m(r) +
c1
K

+ 2
∑

∥m∥≥K

b2m

)
, t ≥ τ.

(3.14)



8 Y. ZHOU, H. LIU EJDE-2025/64

Hance, set t+ θ instead of t in (3.14), θ ∈ [−ϑ, 0], we have∑
m∈Zk

ϱ(
∥m∥
K

)u2m(t+ θ) ≤ r1e
λ̃ϑe−λ̃(t−τ) +

1

λ̃

( 2

λ0
sup
r∈R

∑
∥m∥≥K

g2m(r) +
c1
K

+ 2
∑

∥m∥≥K

b2m

)
,

where r1 = r20 + γka20(2m0 + 1)2r20 +
1
λ̃

2L2
h

λ0
eλ̃ϑr20. By (A3) and (A4), the compactness of H(g0) in

Cb(R, ℓ2), it follows that for any η > 0, there exists K0(η, g0, r0) ∈ N (independent of g ∈ H(g0))
and T0(η, r0) ≥ TB0 > 0 such that∑

m∈Zk

ϱ(
∥m∥
K

)u2m(t+ θ)

≤ r1e
λ̃ϑe−λ̃(t−τ) + sup

t∈R
sup

g∈H(g0 )

2

λ̃λ0

∑
∥m∥≥K

g2m(t) +
c1

λ̃K
+

2

λ̃

∑
∥m∥≥K

b2m

≤ η2

4
, ∀K ≥ K0(η, g0, r0), t ≥ τ + T0(η, r0), τ ∈ R.

Thus,

sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

|(Ug0 (t, τ)uτ )m(θ)|2 = sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

u2m(t+ θ)

≤ η2

4
, ∀uτ ∈ B0, t ≥ τ + T0(η, r0).

(iv) For each fixed τ ∈ R, any sequence {tn}+∞
n=1 ⊂ [ϑ,+∞) with tn → +∞ as n → ∞, any

sequence {un}+∞
n=1 ⊂ B0 and any sequence {gn}+∞

n=1 ⊂ H(g
0
), we use Arezla-Ascoli theorem to

prove that the sequence {u(gn)tn+τ = Ugn0 (tn + τ, τ)un}+∞
n=1 has a convergent subsequence in ℓ2ϑ. By

(3.12), it follows that {u(gn)tn+τ}
+∞
n=1 is uniformly bounded in ℓ2ϑ:

sup
1≤n<+∞

∥u(gn)tn+τ∥ℓ2ϑ = sup
1≤n<+∞

sup
−ϑ≤θ≤0

∥u(gn)(tn + τ + θ)∥ ≤ b0. (3.15)

Taking θ1, θ2 ∈ [−ϑ, 0] with θ1 ≤ θ2, where tn + θ1 ≥ 0, by (3.12), we have

∥u(gn)tn+τ (θ1)− u
(gn)
tn+τ (θ2)∥ = ∥

∫ tn+τ+θ2

tn+τ+θ1

F0(u
(gn)
tn+τ (θ), gn(s))ds∥ ≤ b1|θ2 − θ1|, ∀n,

which implies the equicontinuity of {u(gn)tn+τ}
+∞
n=1 in ℓ2ϑ. For any η > 0, by (iii) and tn → +∞ as

n→ ∞, there exists K0,η ∈ N such that for n ≥ K0,η, it follows that tn ≥ T0(η,B0) and

sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

|(Ugn0 (tn + τ, τ)un)m(θ)|2

= sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

|u(gn)tn+τ,m(θ)|2 ≤ η2

4
.

By (3.15), the set

Γ
(gn)
0,tn+τ

(θ) =
{
û
(gn)
tn+τ,m(θ) = (u

(gn)
tn+τ,m(θ))∥m∥≤2K0(η,g0,r0) ∈ R(4K0(η,g0,r0)+1)k

}
is precompact in R(4K0(η,g0,r0)+1)k and Γ

(gn)
0,tn+τ

(θ) can be covered by finite closed balls with radius η2
centered at the points in Γ

(gn)
0,tn+τ

(θ) ⊂ R(4K0(η,g0,r0)+1)k. It follows that for any η > 0, {u(gn)tn+τ}
+∞
n=1

is precompact in ℓ2. So {u(gn)tn+τ}
+∞
n=1 has a convergent subsequence in ℓ2ϑ. Since B0 is absorbing in

ℓ2ϑ, {U
g
0 (t, τ)}t≥τ,g∈H(g

0
) is uniformly (w.r.t. g ∈ H(g

0
)) asymptotically compact in ℓ2ϑ.

(v) According to [4, 16] and (i)–(iv), {Ug0 (t, τ)}t≥τ,g∈H(g0) possesses a unique compact uniform

attractor A
H(g

0
)

0 satisfying (3.2)–(3.3). The proof is complete. □



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 9

Remark 3.2. In view of [3, 4, 27], for any g ∈ H(g0), t ∈ R, {Ag
0,t}t∈R is the pullback attractor

of {Ug0 (t, τ)}t≥τ and

Ag
0,t = {ut|{ut(·), t ∈ R} is a complete bounded trajectory of {Ug0 (t, τ)}t≥τ}

= ∩r≥0∪s≥rUg0 (t, t− s)B0 ⊂ B0 ⊂ ℓ2ϑ.

that is, for all g ∈ H(g0), t ∈ R, Ag
0,t is compact in ℓ2ϑ; for all t ≥ τ ∈ R, Ug0 (t, τ)A

g
0,τ = Ag0,t;

for all B ⊂ B(ℓ2ϑ), lims→+∞ dh(U
g
0 (t, t − s)B,Ag0,t) = 0; moreover, if ut ∈ Ag

0,t, then there exists

uτ ∈ Ag
0,τ such that Ug0 (t, τ)uτ = ut for t ≥ τ ∈ R and ∥ut(·)∥ℓ2ϑ ≤ Cu (constant) for all t ∈ R.

We consider the system (2.6), for ϵ > 0 and δϵ in (A6), and can see that

β = 1− 2δϵ(2q)
2k ≥ 1

2
, δ1 =

ε0
2

− δϵϵ(2λ
2
0(2q)

2k + 1)

λ0
∈ [

ε0
4
,
ε0
2
], µ =

δ1
ϵ
.

Theorem 3.3. For the initial value problem (2.6) and ϵ > 0, if (A1)–(A6) hold, then for any
g ∈ H(g

0
), τ ∈ R and ψϵ,τ (·) = (uϵ,τ (·), vϵ,τ (·))T ∈ Hϑ, (2.6) has a unique solution

ψϵ,t(·) = ψϵ(t, τ, ψϵ,τ (·)) = (uϵ(t, τ, ψϵ,τ (·)), vϵ(t, τ, ψϵ,τ (·)))T ∈ H, t ≥ τ,

ψϵ,t(·) is continuous in ψϵ,τ (·) and
ψϵ(·) = ψϵ(·, τ, ψϵ,τ (θ)) ∈ C([τ − ϑ,+∞), H) ∩ C1([τ,+∞), H), θ ∈ [−ϑ, 0].

The solutions maps Ugϵ (t, τ) : Hϑ → Hϑ, ψϵ,τ (·) → ψϵ,t(·) = ψϵ(t, τ, ψϵ,τ (·)), t ≥ τ , generate
a continuous process {Ugϵ (t, τ)}t≥τ on Hϑ, and {Ugϵ (t, τ)}t≥τ,g∈H(g

0
) possesses a unique compact

uniform attractor KH(g
0
)

ϵ ⊂ Hϑ defined by

KH(g
0
)

ϵ = ∪g∈H(g0 )
Kgϵ,t = ∪g∈H(g0 )

Kgϵ,0 ⊂ Hϑ, ∀t ∈ R,
where

Kgϵ,t =
{
ψϵ,t(·) = ψϵ(t+ ·) : [−ϑ, 0] → H is the global solution of (2.6),

∥ψϵ,t(·)∥Hϑ
≤ rϵt ∈ R

}
.

with Ugϵ (t, τ)Kgϵ,τ = Kgϵ,t for t ≥ τ , g ∈ H(g0), and rϵ = 2
√

1
µϵβ ∥g0∥2 +

λ0

µϵ∥b∥2.

Proof. (i) By (A1)–(A6), it holds that for any τ ∈ R, g ∈ H(g
0
) and any ψϵ,τ (·) ∈ Hϑ, the system

(2.6) has a unique local solution ψϵ,t(·) = ψϵ(t, τ, ψϵ,τ (·)) = (uϵ(t, τ, ψϵ,τ (·), vϵ(t, τ, ψϵ,τ (·))T for t ∈
[τ, T1,max), ψϵ,t(·) is continuous in ψϵ,τ (·) and ψϵ(·) = ψϵ(·, τ, ψϵ,τ (θ)) ∈ C([τ − ϑ, T1,max), H) ∩
C1([τ, T1,max), H), θ ∈ [−ϑ, 0], where ψϵ(t) satisfies the following initial value and integral equation

ψϵ(τ)(θ) = ψϵ,τ (θ), θ ∈ [−ϑ, 0],

ψϵ(t) = ψϵ,τ (0) +

∫ t

τ

(Fϵ(ψϵ,s(θ), g(s))−Hϵψϵ,s(0))ds, t ∈ [τ, T1,max).
(3.16)

Taking the inner product of ψϵ(t) with (2.6) (t ≥ T1,max) in H, by (A1)–A6), we have

d

dt
[∥ψϵ(t)∥2H +

2

ϵ

∑
m∈Zk

(Gm(uϵ,j(t, τ)|j ∈ Imq) + b2m)] +
ε0λ0
2ϵ2

∥uϵ(t)∥2

+ µ[∥ψϵ(t, τ)∥2H +
2

ϵ

∑
m∈Zk

(Gm(uϵ,j(t, τ)|j ∈ Imq) + b2m)]

≤ 2L2
h

ϵβ
∥uϵ(t− ϑ)∥2 + 2

ϵβ
∥g0∥2 +

2λ0
ϵ

∥b∥2, t ≥ τ.

(3.17)

Applying Gronwall’s inequality to (3.17) on [τ, t] (t ≥ τ), we obtain

∥ψϵ(t)∥2H ≤
(
∥ψϵ(τ)∥2H +

2(2q + 1)kρ(∥uϵ(τ)∥)∥uϵ(τ)∥2

ϵ
+

2∥b∥2

ϵ

)
eµϑe−µ(t−τ)

+
4L2

h

ε0
eµϑ∥uϵ,τ (·)∥2ℓ2ϑe

−µ(t−τ) +
2

µϵβ
∥g0∥2 +

2λ0
µϵ

∥b∥2, t ≥ τ.

(3.18)



10 Y. ZHOU, H. LIU EJDE-2025/64

Set t+ θ instead of t in (3.18), where θ ∈ [−ϑ, 0], it holds that for t ≥ τ ,

∥ψϵ(t+ θ)∥2H ≤
(
∥ψϵ(τ)∥2H +

2(2q + 1)kρ(∥uϵ(τ)∥)∥uϵ(τ)∥2

ϵ
+

2∥b∥2

ϵ

)
eµϑe−µ(t−τ)

+
4L2

h

ε0
eµϑ∥uϵ,τ (·)∥2ℓ2ϑe

−µ(t−τ) +
r2ϵ
2
, t+ θ ≥ τ,

(3.19)

and for t + θ ≤ τ , ∥ψϵ(t + θ)∥2H ≤ ∥ψϵ,τ (·)∥2Hϑ
. Thus, T1,max = +∞, the solution ψϵ(·) ∈

C([τ −ϑ,+∞), H)∩C1((τ,+∞), H), θ ∈ [−ϑ, 0] and the solutions map Ugϵ (t, τ) (t ≥ τ) generates
a continuous process {Ugϵ (t, τ)}t≥τ on Hϑ. Moreover,

∥ψϵ,t(·)∥2Hϑ
= sup
θ∈[−ϑ,0]

∥ψϵ(t+ θ)∥2H

= sup
θ∈[−ϑ,0]

(δ
ϵ

k∑
j=1

∥Bjuϵ(t+ θ)∥2 + 1

ϵ
∥uϵ(t+ θ)∥2λ + ∥vϵ(t+ θ)∥2

)
≤

(
∥ψϵ,τ (·)∥2Hϑ

+
2(2q + 1)kρ(∥uϵ(τ)∥)∥uϵ(τ)∥2

ϵ
+

2∥b∥2

ϵ

)
eµϑe−µ(t−τ)

+
4L2

h

ε0
eµϑ∥uϵ,τ (·)∥2ℓ2ϑe

−µ(t−τ) +
r2ϵ
2

.
= b̃2ϵ(t, τ, ∥ψϵ,τ (·)∥2Hϑ

), t ≥ τ,

(3.20)

where b̃2ϵ(t, τ, ∥ψϵ,τ (·)∥2Hϑ
) is continuous in t. Let τ ∈ R, g(1), g(2) ∈ H(g

0
), ψ

(1)
ϵ,τ (·), ψ(2)

ϵ,τ (·) ∈ Hϑ,

ψ
(j)
ϵ (t) = ψϵ(t, τ, g

(j), ψ
(j)
ϵ,τ (·)), j = 1, 2,

κϵ(t) = κϵ(t, τ, g
(1), g(2), ψ(1)

ϵ,τ (·), ψ(2)
ϵ,τ (·)) = ψ(1)

ϵ (t)− ψ(2)
ϵ (t), t ≥ τ,

then
κ̇ϵ(t) +Hϵκϵ(t) = Fϵ(ψ

(1)
ϵ,t (θ), g

(1)(t))− Fϵ(ψ
(1)
ϵ,t (θ), g

(1)(t)), t ≥ τ,

κϵ(τ)(θ) = ψ(1)
ϵ,τ (θ)− ψ(2)

ϵ,τ (θ), θ ∈ [−ϑ, 0], τ ∈ R.
(3.21)

Taking the inner product of κϵ(t) ∈ H (t ≥ τ) with (3.21), we have

d

dt
∥κϵ(t)∥2H + C̃ϵ(t, τ)∥κϵ(t)∥2H +

ε0λ0
2ϵ2

∥u(1)ϵ (t)− u(2)ϵ (t)∥2

≤ 2L2
h

ϵ
∥u(1)ϵ (t− ϑ))− u(2)ϵ (t− ϑ)∥2 + 4

ϵ
∥g(1) − g(2)∥2, t ≥ τ,

(3.22)

where

C̃ϵ(t, τ) =
ε0
2ϵ

− 4

λ0
(2q + 1)2kρ2(

√
ϵ

λ0
(b̃ϵ(t, τ, ∥ψ(1)

ϵ,τ (·)∥2Hϑ
) + b̃ϵ(t, τ, ∥ψ(1)

ϵ,τ (·)∥2Hϑ
))).

Applying Gronwall’s inequality to (3.22) on [τ, t] (t ≥ τ), we obtain

∥κϵ(t)∥2 ≤ ∥κϵ(τ)∥2He−
∫ t
τ
C̃ϵ(l,τ)dl

+
4L2

h

ε0
eµϑ∥u(1)ϵ,τ (·)− u(2)ϵ,τ (·)∥2ℓ2ϑ

∫ τ

τ−ϑ
e−

∫ t
r
C̃ϵ(l,τ)dldr

+
4

ϵ
∥g(1) − g(2)∥2

∫ t

τ

e−
∫ t
r
C̃ϵ(l,τ)dldr, t ≥ τ.

Thus

∥κϵ(t+ θ)∥2 ≤ C̃ϵ,1(t, τ)
(
∥κϵ,τ (·)∥2Hϑ

+ ∥g(1) − g(2)∥2
)
, t ≥ τ, (3.23)

where

C̃ϵ,1(t, τ) = max
{
eµϑ +

4ϵL2
h

λ0ε0
e2µϑ

∫ τ

τ−ϑ
e−

∫ t
r
C̃ϵ(l,τ)dldr,

4

λ0
eµϑ

∫ t

τ

e−
∫ t
r
C̃ϵ(l,τ)dldr

}
,

and (3.23) implies that {Ugϵ (t, τ)}t≥τ,g∈H(g0) is continuous from Hϑ ×H(g0) into Hϑ.
(ii) From (3.20), the family {Ugϵ (t, τ)}t≥τ,g∈H(g0 )

has a (g, τ)-uniformly bounded closed absorb-

ing set Bϵ = {ψ ∈ Hϑ : ∥ψ∥Hϑ
≤ rϵ} ⊂ Hϑ and for any g ∈ H(g0), τ ∈ R, there exists TBϵ ≥ 0

(independent of (g, τ)) such that ∪g∈H(g0 )
Ugϵ (t, τ)Bϵ ⊆ Bϵ for t ≥ τ + TBϵ

. Moreover, by (3.20)



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 11

again, there exists a positive constants b0,ϵ, b1,ϵ (independent of (g, t, τ)) such that for t ≥ τ ,
g ∈ H(g0),

∥Ugϵ (t, τ)Bϵ∥Hϑ
= sup
ψϵ,t∈Ug

ϵ (t,τ)Bϵ

∥ψϵ,t(·)∥Hϑ

= sup
ψt∈Ug

ϵ (t,τ)Bϵ

sup
θ∈[−ϑ,0]

∥ψϵ(t+ θ)∥Hϑ
≤ b0,ϵ,

sup
ψϵ,t∈Ug

ϵ (t,τ)Bϵ

sup
θ∈[−ϑ,0]

∥Fϵ(ψϵ,t(θ), t)−Hϵψϵ(t)∥ ≤ b1,ϵ.

(3.24)

(iii) Fix g ∈ H(g
0
), τ ∈ R, ψϵ,τ (·) ∈ Bϵ, and let

ψϵ(t) = Ugϵ (t, τ)ψϵ,τ (·) = ψϵ(t, τ, ψϵ,τ (·))
= (uϵ,m(t, τ, ψϵ,τ (·)), vϵ,m(t, τ, ψϵ,τ (·)))Tm∈Z ∈ Hϑ, t ≥ τ,

be the solution of equation (2.6). Let K ∈ N, wϵ,m = ϱ( |m|
K )uϵ,m, zϵ,m = ϱ( |m|

K )vϵ,m, ỹϵ =

(wϵ, zϵ)
T = ((wϵ,m)m∈Zk , (zϵ,m)m∈Zk)T. Taking inner product (·, ·)H of (2.6) with ỹϵ, we have

d

dt

[ ∑
m∈Zk

ϱ(
∥m∥
K

)|ψϵ,m|2H +
2

ϵ

∑
m∈Zk

ϱ(
∥m∥
K

)(Gm(uϵ,j |j ∈ Imq) + b2m)
]

+
ε0λ0
2ϵ2

∑
m∈Zk

ϱ(
||m||
K

)u2ϵ,m(t)

+ µ
[ ∑
m∈Zk

ϱ(
∥m∥
K

)|ψϵ,m|2H +
2

ϵ

∑
m∈Zk

ϱ(
∥m∥
K

)(Gm(uϵ,j |j ∈ Imq) + b2m)
]

≤ 2L2
h

ϵβ

∑
m∈Zk

ϱ(
||m||
K

)u2ϵ,m(t− ϑ) +
δ3b

2
0,ϵ

ϵK

+
2

ϵβ

∑
∥m∥≥K

g2m(t) +
2λ0
ϵ

∑
∥m∥≥K

b2m, t ≥ τ,

(3.25)

where c2 = C0m0a
2
0(2m0+1)2, δ2 = 2+2ϵλ0+

ϵ
λ0
, δ3 = 2(λ0δc2δ2k+γc2k+δc2δ2k)+δϵδ2(2q)

2kC0q.

Applying Gronwall’s inequality on [τ, t] to (3.25), we obtain∑
m∈Zk

ϱ(
∥m∥
K

)|ψϵ,m(t)|2H

≤ (r2ϵ +
2

ϵ
(
ϵ(2q + 1)k

λ0
ρ(rϵ

√
ϵ

λ0
)r2ϵ + ∥b∥2) + 4L2

h

ε0
eµϑ∥uϵ,τ (·)∥2ℓ2ϑ)e

−µ(t−τ)

+
4δ3b

2
0,ϵ

ε0K
+

16

ε0
sup
r∈R

∑
∥m∥>K

g2m(r) +
8λ0
ε0

∑
∥m∥>K

b2m, t ≥ τ.

(3.26)

Hance, set t+ θ instead of t in (3.26), where θ ∈ [−ϑ, 0], we have∑
∥m∥>2K

|ψϵ,m(t+ θ)|2H

≤
(
r2ϵ +

2(2q + 1)k

λ0
ρ(rϵ

√
ϵ

λ0
)r2ϵ +

2

ϵ
∥b∥2 + 4L2

h

ε0
eµϑr2ϵ

)
eµϑe−µ(t−τ)

+
4δ3
ε0K

b20,ϵ +
16

ε0
sup
r∈R

∑
∥m∥>K

g2m(r) +
8λ0
ε0

∑
∥m∥>K

b2m, t ≥ τ.

(3.27)

Thus, by (3.27), for any η > 0, there exist Tϵ(η, rϵ) ≥ TBϵ > 0 and Kϵ(η, g0, rϵ) ∈ N (independent
of g) such that for t ≥ τ + Tϵ(η, rϵ), τ ∈ R,

sup
g∈H(g

0
)

sup
ψϵ,τ∈Bϵ

sup
θ∈[−ϑ,0]

∑
∥m∥>2Kϵ(η,g0,rϵ)

|(Ugϵ (t, τ)ψϵ,τ )m(θ)|2H ≤ η2

4
.



12 Y. ZHOU, H. LIU EJDE-2025/64

(iv) For any fixed τ ∈ R, any sequence {tn}+∞
n=1 ⊂ [ϑ,+∞) with tn → +∞ as n → ∞,

any sequence {ψϵ,n}+∞
n=1 ∈ Bϵ and any sequence {gn}+∞

n=1 ∈ H(g
0
), we show that the sequence

{ψ(gn)
ϵ,tn+τ = Ugnϵ (tn + τ, τ)ψϵ,n}+∞

n=1 has a convergent subsequence in Hϑ. By (3.24), it follows that

{ψ(gn)
ϵ,tn+τ}

+∞
n=1 is uniformly bounded in Hϑ. Taking θ1, θ2 ∈ [−ϑ, 0] with θ1 ≤ θ2, where tn+θ1 ≥ 0,

by (3.16) and (3.24), we have

∥ψ(gn)
ϵ,tn+τ (θ1)− ψ

(gn)
ϵ,tn+τ (θ2)∥ = ∥

∫ tn+τ+θ2

tn+τ+θ1

(Fϵ(ψ
(gn)
ϵ,tn+τ (θ), s)−Hϵψ

(gn)
ϵ,tn+τ (0))ds∥

≤ b1,ϵ|θ2 − θ1|,

which implies that {ψ(gn)
ϵ,tn+τ}

+∞
n=1 is equicontinuous in Hϑ. For any η > 0, by (iii) and tn → +∞

as n→ ∞, there exists Kϵ,η ∈ N such that for n ≥ Kϵ,η, tn ≥ τ + Tϵ(η, rϵ) and

sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

|(Ugnϵ (tn + τ, τ)ψϵ,n)m(θ)|2

= sup
θ∈[−ϑ,0]

∑
∥m∥>2K0(η,g0,r0)

|ψ(gn)
ϵ,tn+τ (θ)|

2
H ≤ η2

4
.

It follows that {ψ(gn)
ϵ,tn+τ (θ)}

+∞
n=1 is precompact in H. By Arezla-Ascoli theorem, {ψ(gn)

ϵ,tn+τ}
+∞
n=1 has

a convergent subsequence in Hϑ, that is, {Ugϵ (t, τ)}t≥τ,g∈H(g0 )
is uniformly (w.r.t. g ∈ H(g0))

asymptotically compact in Bϵ ⊂ Hϑ.
(v) It is the results from (i)–(iv) and [4, 16, 27]. The proof is complete. □

By the transformation (2.5), if ψϵ(t) = ψϵ,t(·) = (uϵ(t), vϵ(t))
T ∈ Hϑ, where vϵ = u̇ϵ+

ε0
ϵ uϵ, is a

solution of (2.6), then φϵ(t) = φϵ,t(·) = (uϵ,t(·), u̇ϵ,t(·))T is the a solution of the following system
(3.28) in Eϑ = C([−ϑ, 0], E):

φ̇ϵ(t) = F̃ (φϵ,t(θ), g(t)), t ≥ τ, g ∈ H(g0), τ ∈ R,

φϵ,τ (θ) = (uϵ,τ (θ), u̇ϵ,τ (θ))
T = (uϵ,τ (τ + θ), u̇ϵ(τ + θ))T, θ ∈ [−ϑ, 0],

(3.28)

where

F̃ (φϵ,t(θ), g(t)) =

(
u̇ϵ(t)

− 1
ϵ u̇ϵ(t)−

1
ϵγAu̇ϵ(t)−

1
ϵAuϵ(t)−

1
ϵλuϵ(t)

)
+

(
0

− 1
ϵ f(uϵ(t))−

1
ϵh (uϵ(t− ϑ)) + 1

ϵ g(t)

)
and ∥φϵ(t)∥2E = ∥uϵ(t)∥2 + ∥u̇ϵ(t)∥2 ≤ δ2∥ψϵ(t)∥2H .

From Theorem 3.3 and (2.5), we have the following result.

Theorem 3.4. For the system (3.28) and ϵ > 0, if (A1)–(A6) hold, then for any g ∈ H(g
0
),

τ ∈ R and φϵ,τ (·) = (uϵ,τ (·), u̇ϵ,τ (·))T ∈ Hϑ, (3.28) has a unique solution

φϵ,t(·) = φϵ(t, τ, φϵ,τ (·)) = (uϵ(t, τ, φϵ,τ (·)), u̇ϵ(t, τ, φϵ,τ (·)))T ∈ Eϑ, t ≥ τ,

φϵ,t(·) is continuous in φϵ,τ (·) and

φϵ(·) = φϵ(·, τ, φϵ,τ (θ)) ∈ C([τ − ϑ,+∞), E) ∩ C1([τ,+∞), E), θ ∈ [−ϑ, 0].

The solution maps

V gϵ (t, τ) : Eϑ → Eϑ, φϵ,τ (·) → φϵ,t(·) = φϵ(t, τ, φϵ,τ (·)), t ≥ τ,

generates a continuous process {V gϵ (t, τ)}t≥τ on Eϑ, V
g
ϵ (t, τ) = D−1

ϵ Ugϵ (t, τ)Dϵ, where
Dϵ : (a, b)

T → (a, b+ ε0
ϵ a)

T is a reversible operator from E into E.

{V gϵ (t, τ)}t≥τ,g∈H(g
0
) possesses a unique compact uniform attractor A H(g

0
)

ϵ ⊂ Hϑ given by

AH(g0 )
ϵ = ∪g∈H(g0 )

Ag
ϵ,t = ∪g∈H(g0 )

Ag
ϵ,0 = D−1

ϵ KH(g0 )
ϵ ⊂ Hϑ, ∀t ∈ R, (3.29)



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 13

where

Ag
ϵ,t =

{
{φϵ,t φϵ,t(·) = φϵ(t+ ·) : [−ϑ, 0] → E is the global solution of (3.28),

∥φϵ,t(·)∥Eϑ
≤ r̃ϵ for t ∈ R

}
.

with V gϵ (t, τ)Ag
ϵ,τ = Ag

ϵ,t for t ≥ τ , g ∈ H(g
0
) and r̃ϵ =

2√
ϵµ

√
δ2∥g0∥2 + λ0δ2∥b∥2.

4. Prior uniform estimations of solutions

To investigate the upper semicontinuity of uniform attractors AH(g
0
)

ϵ and the relationship be-

tween AH(g
0
)

ϵ and AH(g
0
)

0 as ϵ→ 0+, in this section, we establish some prior uniform estimations
for the solutions of (3.28) with respect to finite ϵ. Let the conditions (A1)–(A6) hold and ϵ > 0
be a given positive constant.

Lemma 4.1. For each ϵ ∈ (0, ϵ], g ∈ H(g
0
), t ∈ R and a constant q1 ≥ 0, let s ≥ 0,

φϵ(t) = φϵ,t(·) = φϵ(t, t− s, φϵ,t−s(·))
= (uϵ,t(·), u̇ϵ,t(·))T = V gϵ (t, t− s)φϵ,t−s(·) ∈ Eϑ,

be a solution of (3.28) with the initial value φϵ,t−s(·) ∈ Eϑ satisfying

ϵ∥u̇ϵ,t−s(·)∥2ℓ2ϑ + ∥uϵ,t−s(·)∥2ℓ2ϑ = sup
−ϑ≤θ≤0

(
ϵ∥u̇ϵ(t− s+ θ∥2 + ∥uϵ (t− s+ θ) ∥2

)
≤ q1, s ≥ 0.

(4.1)

Then there exist positive constants M1 = M1(ϵ), µ̄ = µ̄(ϵ), C1(q1, ϵ), K̃1(ϵ, q1), K̃2(ϵ, q1), M2 =
M2(ϵ, q1), M3 =M3(ϵ, q1) > 0 (independent of (g, t, ϵ)) and C2(q1, ϵ), C3(q1, ϵ) > 0 (depending on
ϵ) such that for any s ≥ 0, t ∈ R,

ϵ∥u̇ϵ,t(·)∥2ℓ2ϑ + ∥uϵ,t(·)∥2ℓ2ϑ = sup
−ϑ≤θ≤0

(
ϵ∥u̇ϵ (t+ θ) ∥2 + ∥uϵ (t+ θ) ∥2

)
≤M1 + C1(q1, ϵ)e

−µ̄s,∫ t+1

t

∥u̇ϵ(r)∥2dr ≤ K̃1(ϵ, q1),

∫ t

t−ϑ
∥u̇ϵ(r)∥2dr ≤ K̃2(ϵ, q1),

ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ = sup
−ϑ≤θ≤0

(
ϵ∥üϵ (t+ θ) ∥2 + ∥u̇ϵ (t+ θ) ∥2

)
≤M2 + C2(q1, ϵ)e

−µ̄s,
(4.2)

ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ + ∥uϵ,t(·)∥2ℓ2ϑ ≤M3 + C3(q1, ϵ)e
−µ̄s. (4.3)

Proof. For g ∈ H(g
0
), t ∈ R, let vϵ = u̇ϵ +

ε0
ϵ uϵ, where φϵ,t(·) = (uϵ,t(·), u̇ϵ,t(·))T is the solution

of problem (3.28) with initial data φϵ,t−s(·) ∈ Eϑ satisfying (4.1), then ψϵ(t) = ψϵ,t(·) = ψϵ(t, t−
s,Dϵφϵ,t−s(·)) = (uϵ(t), vϵ(t))

T ∈ Hϑ (s ≥ 0) is a solution of problem (2.6). It follows from (3.20)
that

sup
θ∈[−ϑ,0]

(λ0
ϵ
∥uϵ(t+ θ)∥2 + ∥vϵ(t+ θ)∥2

)
≤

(
∥ψϵ,t−s(·)∥2Hϑ

+
2(2q + 1)kρ(∥uϵ(t− s)∥)∥uϵ(t− s)∥2

ϵ
+

2∥b∥2

ϵ

)
eµϑe−µs

+
4L2

h

ε0
eµϑ∥uϵ,t−s(·)∥2ℓ2ϑe

µϑe−µs +
2

µϵβ
∥g0∥2 +

2λ0
µϵ

∥b∥2, s ≥ 0.

(4.4)

Multiplying both sides of (4.4) by ϵ, we have

sup
θ∈[−ϑ,0]

(
λ0∥uϵ(t+ θ)∥2 + ϵ∥vϵ(t+ θ)∥2

)
≤

(
ϵ∥ψϵ,t−s(·)∥2Hϑ

+ 2(2q + 1)kρ(∥uϵ(t− s)∥)∥uϵ(t− s)∥2 + 2∥b∥2
)
eµϑe−µs

+
4L2

hϵ

ε0
eµϑ∥uϵ,t−s(·)∥2ℓ2ϑe

µϑe−µs +
2

µβ
∥g0∥2 +

2λ0
µ

∥b∥2, s ≥ 0.

(4.5)



14 Y. ZHOU, H. LIU EJDE-2025/64

Since (1 + 3ϵλ0)
2 ≥ 12ϵλ0,and

ε20
ϵ ≤ λ0

4 ≤ λ0

2 , we have

sup
θ∈[−ϑ,0]

(
λ0∥uϵ(t+ θ)∥2 + ϵ∥vϵ(t+ θ)∥2

)
≥ λ0

2
∥uϵ,t(·)∥2ℓ2ϑ +

ϵ

2
∥u̇ϵ,t(·)∥2ℓ2ϑ ,

ϵ∥vϵ∥2 ≤ 2ϵ∥u̇ϵ∥2 +
λ0
2
∥uϵ∥2,

ϵ∥ψϵ,t−s(·)∥2Hϑ
≤ (a20(2m0 + 1)2k + λ0 + 2 +

λ0
2
)q1.

By (A5)–(A6), we have

ε0
4ϵ

≥

{
λ0

4(1+3ϵλ0)
, γ = 0,

min{ 1
4γ ,

λ0

4(1+3ϵλ0)
}, γ > 0,

.
= µ̄ > 0,

µ ≤ ε0
4ϵ

≤ µ =
δ1
ϵ

≤ ε0
2ϵ

≤ λ0
2
, eµϑ ≤ e

λ0
2 ϑ, e−µs ≤ e−µ̄s, ∀ϵ ∈ (0, ϵ], s ≥ 0,

ϵ

ε0
≤ max

{
γ,

1 + 3ϵλ0
λ0

}
= µ̌,

1

µ
≤ 4ϵ

ε0
≤ 4µ̌,

1

µβ
≤ 8µ̌.

(4.6)

By (4.5) and (4.6), we have

λ0
2
∥uϵ,t(·)∥2ℓ2ϑ +

ϵ

2
∥u̇ϵ,t(·)∥2ℓ2ϑ ≤ C̃1(q1, ϵ)e

−µ̄s +M0(ϵ), s ≥ 0,

where

C̃1(q1, ϵ) =

(
(a20(2m0 + 1)2k + λ0 + 2 +

λ0
2

+ 2(2q + 1)kρ(
√
q1)

)
e

λ0
2 ϑ

+ 4L2
he

λ0
2 ϑµ̌)q1 + 2∥b∥2e

λ0
2 ϑ,

M0(ϵ) = 16µ̌∥g0∥2 + 8µ̌λ0∥b∥2.

Hance

∥uϵ,t(·)∥2ℓ2ϑ + ϵ∥u̇ϵ,t(·)∥2ℓ2ϑ ≤ C1(q1, ϵ)e
−µ̄s +M1, ∀s ≥ 0, t ∈ R, (4.7)

where

C1(q1) =
2C̃1(q1, ϵ)

min{λ0, 1}
, M1 =

2M0(ϵ)

min{λ0, 1}
.

In particular,

∥uϵ,t(·)∥2ℓ2ϑ + ϵ∥u̇ϵ,t(·)∥2ℓ2ϑ ≤ C1(q1, ϵ) +M1, ∀t ∈ R, ϵ ∈ (0, ϵ]. (4.8)

Then

∥ −Auϵ(t)− λuϵ(t)− f(uϵ(t))− h (uϵ(t− ϑ)) + g(t)∥2

≤ 5[a40(2m0 + 1)4 + (λ0)2 + (2q + 1)2kρ2(
√
C1(q1, ϵ) +M1)](C1(q1, ϵ) +M1)

+ 5L2
h(C1(q1, ϵ) +M1) + 4∥g0∥2

.
= K̃3(ϵ, q1), s ≥ 0.

Taking the inner product of (2.4) with u̇ϵ, we have

ϵ
d

dt
∥u̇ϵ(t)∥2 + ∥u̇ϵ(t)∥2 ≤ K̃3(ϵ, q1), t ∈ R, s ≥ 0. (4.9)

Integrating both sides of (4.9) over [t, t+ 1] and [t− ϑ, t], respectively, we have

ϵ(∥u̇ϵ(t+ 1)∥2 − ∥u̇ϵ(t)∥2) +
∫ t+1

t

∥u̇ϵ(r)∥2dr ≤ K̃3(ϵ, q1), t ∈ R,

and

ϵ(∥u̇ϵ(t)∥2 − ∥u̇ϵ(t− ϑ)∥2) +
∫ t

t−ϑ
∥u̇ϵ(r)∥2dr ≤ ϑK̃3(ϵ, q1), t ∈ R.



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 15

Then for t ∈ R, ϵ ∈ (0, ϵ], we have∫ t+1

t

∥u̇ϵ(r)∥2dr ≤ K̃3(ϵ, q1) + ϵ∥u̇ϵ(t)∥2

≤ K̃3(ϵ, q1) + C1(q1) +M1
.
= K̃1(ϵ, q1),

(4.10)

and ∫ t

t−ϑ
∥u̇ϵ(r)∥2dr ≤ ϑK̃3(ϵ, q1) + ϵ∥u̇ϵ(t− ϑ)∥2

≤ ϑK̃3(ϵ, q1) + C1(q1) +M1
.
= K̃2(ϵ, q1).

(4.11)

(ii) Set ∥g′∥2 = supt∈R
∑
m∈Z g

′2
m(t) < ∞, ζϵ(t) = u̇ϵ(t). We differentiate equation (2.4), with

respect to t, to obtain

ϵζ̈ϵ + ζ̇ϵ + γAζ̇ϵ +Aζ + λζϵ +
( ∑
j∈Imq

f ′m,j(uϵ,j |j ∈ Imq)ζϵ,j

)
m∈Zk

+ (h′m (uϵ,m(t− ϑ)) ζϵ,m(t− ϑ))m∈Zk = g′(t),

(4.12)

where

ζ̇ϵ(t− s) = üϵ(t− s)

=
1

ϵ
(g(t− s)− h (uϵ (t− s− ϑ))− f(uϵ(t− s))− λuϵ(t− s)

−Auϵ(t− s)− γAu̇ϵ(t− s)− u̇ϵ(t− s)),

ζϵ,t−s(θ) = u̇ϵ,t−s(θ), s ≥ 0, t ∈ R, θ ∈ [−ϑ, 0].

Then

sup
−ϑ≤θ≤0

∥ζϵ,t−s(θ)∥2 + ϵ∥ζ̇(t− s)∥2

≤ q1
ϵ

(
2 + 7[∥g∥2 + L2

h + (2q + 1)2kρ2(
√
q1) + (λ0)2 + a40(2m0 + 1)4k

)
+
q1
ϵ

(
γ2ka40(2m0 + 1)4 + 1

) .
= q2(q1, ϵ).

Let

ṽϵ = ζ̇ϵ +
ε0
ϵ
ζϵ, ψ̃ϵ = (ζϵ, ṽϵ)

T .

Then problem (4.12) can be written as

·
ψ̃ϵ +Hϵψ̃ϵ = F̃ϵ(ψ̃ϵ, g

′, t), t ∈ R, s ≥ 0, (4.13)

where

Hϵψ̃ϵ =

(
ε0
ϵ ζϵ − ṽϵ

1
ϵλζϵ +

1
ϵ (1−

1
ϵγε0)Aζϵ −

1
ϵ2 ε0(1− ε0)ζϵ +

1
ϵ (1− ε0)ṽϵ +

1
ϵγAṽϵ

)
,

F̃ϵ(ψ̃ϵ, g
′, t) =

(
0

− 1
ϵ (
∑
j∈Imq

f ′m,j(uϵ,j |j ∈ Imq)ζϵ,j)m∈Zk

)
+

(
0

1
ϵ [(h

′
m (uϵ,m(t− ϑ)) ζϵ,m(t− ϑ))m∈Zk + g′(t)]

)
.

By computations,

2(Hϵψ̃ϵ, ψϵ)H ≥ ε0
ϵ
∥ψ̃ϵ∥2H +

1

ϵ
∥ṽϵ∥2. (4.14)

Taking the inner product of (4.13) with ψ̃ϵ in H, by (4.14), we get that for t ∈ R, s ≥ 0,

d

dt
∥ψ̃ϵ(t)∥2H +

ε0
2ϵ

∥ψ̃ϵ(t)∥2H

≤ 3

ϵ

(
(2q + 1)2kρ2(

√
C1(q1) +M1)∥u̇ϵ(t)∥2 + L2

h∥u̇ϵ(t− ϑ)∥2 + ∥g′0∥2
)
.

(4.15)



16 Y. ZHOU, H. LIU EJDE-2025/64

By (4.10) and (4.11), for t ∈ R, we have∫ t+1

t

∥u̇ϵ (r − ϑ) ∥2dr ≤
∫ t+1

t

∥u̇ϵ (r) ∥2dr +
∫ t

t−ϑ
∥u̇ϵ (r) ∥2dr

≤ K̃1(ϵ, q1) + K̃2(ϵ, q1), ϵ ∈ (0, ϵ],

and ∫ t+1

t

3

ϵ
((2q + 1)2kρ2(

√
C1(q1) +M1)∥u̇ϵ(r)∥2 + L2

h∥u̇ϵ(r − ϑ)∥2 + ∥g′0∥2)dr

≤ 3

ϵ
((2q + 1)2kρ2(

√
C1(q1) +M1)K̃1(ϵ, q1) + L2

h(K̃1(ϵ, q1) + K̃2(ϵ, q1)) + ∥g′0∥2)

.
=

1

ϵ
K̃4(ϵ, q1), ϵ ∈ (0, ϵ].

By applying Gronwall’s inequality on [t− s, t] (s ≥ 0) to (4.15), we have

∥ψ̃ϵ(t)∥2H ≤ ∥ψ̃ϵ(t− s)∥2He−µ̄s +
1

ϵ
K̃4(ϵ, q1)(1 + 2µ̌), t ∈ R, s ≥ 0.

where

ϵ∥ψ̃ϵ,t−s(·)∥2Hϑ
≤ (a20(2m0 + 1)2k + λ0)∥ζϵ,t−s(·)∥2ℓ2ϑ + 2ϵ∥ζ̇ϵ,t−s(·)∥2ℓ2ϑ +

λ0
2
∥ζϵ,t−s(·)∥2ℓ2ϑ

≤ (a20(2m0 + 1)2k + λ0 + 2 +
λ0
2
)q2(q1, ϵ).

Therefore,

λ0
2
∥ζϵ,t(·)∥2ℓ2ϑ +

ϵ

2
∥ζ̇ϵ,t(·)∥2ℓ2ϑ

≤ sup
θ∈[−ϑ,0]

ϵ∥ψ̃ϵ(t+ θ)∥2H

≤ (a20(2m0 + 1)2k + λ0 + 2 +
λ0
2
)q2(q1, ϵ)e

−µ̄s + K̃4(ϵ, q1)(1 + 2µ̌), s ≥ 0.

Then
ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ ≤M2 + C2(q1, ϵ)e

−µ̄s, t ∈ R, s ≥ 0, (4.16)

where

M2 =
2K̃4(ϵ, q1)(1 + 2µ̌)

min{λ0, 1}
, C2(q1, ϵ) =

4 + 2a20(2m0 + 1)2k + 2λ0 + λ0
min{λ0, 1}

q2(q1, ϵ).

Combining (4.7) and (4.16), we conclude (4.3). The proof is complete. □

Lemma 4.2. For any g ∈ H(g
0
), t ∈ R, s ≥ 0, ϵ ∈ (0, ϵ̄], let

φϵ,t(·) = φϵ(t, t− s, φϵ,t−s(·)) = (uϵ,t(·), u̇ϵ,t(·))T = V gϵ (t, t− s)φϵ,t−s(·) ∈ Eϑ,

be the solution of problem (3.28) with the initial value φϵ,t−s(·) ∈ Ag
ϵ,t−s ⊆ B̃ϵ, where B̃ϵ = {ψ ∈

Hϑ : ∥ψ∥Hϑ
≤ r̃ϵ} ⊂ Hϑ. Then

(i) there exists a positive constant M4 > 0 (independent of (g, ϵ)) such that

ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ + ∥uϵ,t(·)∥2ℓ2ϑ ≤ 2M4, ∀t ∈ R, ϵ ∈ (0, ϵ̄]. (4.17)

(ii) For any η > 0, there exists a I2(η) = I2(η, g0) ∈ N (independent of (g, ϵ)) such that

sup
−ϑ≤θ≤0

∑
∥m∥>2I2(η)

|uϵ,t,m(θ)|2 ≤ η2, ∀t ∈ R, ϵ ∈ (0, ϵ̄].

Proof. (i) Since V gϵ (t, t− s)Ag
ϵ,t−s = Ag

ϵ,t and φϵ,t−s(·) ∈ Ag
ϵ,t−s ⊆ B̃ϵ, we have

φϵ,t(·) = φϵ(t+ ·) = (uϵ,t(·), u̇ϵ,t(·))T = V gϵ (t, t− s)φϵ,t−s(·) ∈ Ag
ϵ,t ⊆ B̃ϵ,

and

ψϵ,t(·) = (uϵ,t(·), u̇ϵ,t(·) +
ε0
ϵ
uϵ,t(·))T = Ugϵ (t, t− s)ψϵ,t−s(·)



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 17

= DϵV
g
ϵ (t, t− s)φϵ,t−s(·) ∈ Kgϵ,t ⊆ Bϵ, ∀t ∈ R, s ≥ 0,

is the solution of (2.6). Again,

ϵ∥u̇ϵ,t−s(·)∥2ℓ2ϑ + ∥uϵ,t−s(·)∥2ℓ2ϑ

≤ 2ϵ

min{λ0, 1}

(
∥uϵ,t−s(·)∥2δλϵ + ∥u̇ϵ,t−s(·) +

ε0
ϵ
uϵ,t−s(·)∥2ℓ2ϑ

)
≤ 2ϵ

min{λ0, 1}

( 4

µϵβ
∥g0∥2 +

4λ0
µϵ

∥b∥2
)

≤ 32µ̌

min{λ0, 1}
(
2∥g0∥2 + λ0∥b∥2

) .
= q4(ϵ) = q4 (independent of (g, ϵ)).

Therefore, by Lemma 4.1, there exist positive constants M4 = M4(ϵ̄), µ̄ = µ̄(ϵ̄) (independent of
(g, ϵ)) and a finite positive constant C4(q4, ϵ) (depending on ϵ) such that for any t ∈ R,

ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ + ∥uϵ,t(·)∥2ℓ2ϑ ≤M4 + C4(q4, ϵ)e
−µ̄s, ∀s ≥ 0, ϵ ∈ (0, ϵ̄].

So, for each fixed t ∈ R and ϵ ∈ (0, ϵ̄], there must exists a large number τϵ > 0 (depending on ϵ)
such that C4(q4, ϵ)e

−µ̄s ≤M4 for all s ≥ τϵ, thus

ϵ∥üϵ,t(·)∥2ℓ2ϑ + ∥u̇ϵ,t(·)∥2ℓ2ϑ + ∥uϵ,t(·)∥2ℓ2ϑ ≤ 2M4, ∀t ∈ R, ϵ ∈ (0, ϵ̄], (4.18)

which implies that for any solution φϵ,t(·) of (3.28) in Ag
ϵ,t(·), (4.17) holds.

(ii) Similar to the proof of (3.27), it follows from (4.18) that there exists positive constants
q5(M4, ϵ̄) > 0, q6(M4, ϵ̄) > 0 (independent of ϵ) such that for K ∈ N, t ∈ R, s ≥ 0, θ ∈ [−ϑ, 0],∑

∥m∥>2K

|ψϵ,m(t+ θ)|2H =
∑

∥m∥>2K

(
1

ϵ
δ(Buϵ)

2
m(t+ θ) +

1

ϵ
λmu

2
ϵ,m(t+ θ) + v2ϵ,m(t+ θ))

≤
∑
m∈Zk

ϱ(
|m|
K

)|ψϵ,m(t+ θ)|2H

≤ q5(M4, ϵ̄)

ϵ
e−µ̄s +

q6(M4, ϵ̄)

ε0K
+

16

ε0
sup
r∈R

∑
∥m∥>K

g2m(r) +
8λ0
ε0

∑
|m|>K

b2m.

Thus, ∑
∥m∥>2K

(
λ0
2
u2

ϵ,m(t+ θ) +
ϵ

2
u̇2ϵ,m(t+ θ))

≤
∑

∥m∥>2K

(λmu
2
ϵ,m(t+ θ) + ϵv2ϵ,m(t+ θ))

≤ ϵ
∑

∥m∥>2K

|ψϵ,m(t+ θ)|2H

≤ q5(M4, ϵ̄)e
−µ̄s +

µ̌q6(M4, ϵ̄)

K
+ 16µ̌ sup

r∈R

∑
∥m∥≥K

g2m(r) + 8λ0µ̌
∑

∥m∥≥K

b2m

and ∑
∥m∥>2K

u2ϵ,m(t+ θ) ≤ 2

λ0
q5(M4, ϵ̄)e

−µ̄s +
λ0µ̌q6(M4, ϵ̄)

K
+

32

λ0
µ̌ sup
r∈R

∑
∥m∥≥K

g2m(r)

+ 16µ̌
∑

∥m∥≥K

b2m.
(4.19)

It follows that ∀η > 0, there exists I2(η) = I2(η,M4, g0, ϵ̄) ∈ N and T2(η) = T2(η, ,M4, ϵ̄) =

max
{
0, 1

µ̄ ln 16q5(M4,ϵ̄)
λ0η2

}
(independent of (g, ϵ)) such that for any t ∈ R, K ≥ I2(η), s ≥ T2(η), we



18 Y. ZHOU, H. LIU EJDE-2025/64

have

2

λ0
q5(M4, ϵ̄)e

−µ̄s ≤ η2

2
,
λ0µ̌q6(M4, ϵ̄)

K
+

32

λ0
µ̌ sup
r∈R

∑
∥m∥≥K

g2m(r) + 16µ̌
∑

∥m∥≥K

b2m ≤ η2

2
,

∑
∥m∥>2K

u2ϵ,m(t+ θ) ≤ η2, t ∈ R, K ≥ I2(η), s ≥ T2(η).

In particular,

sup
t∈R

sup
−ϑ≤θ≤0

∑
∥m∥>2I2(η)

|uϵ,m(t+ θ)|2 ≤ η2, ∀ϵ ∈ (0, ϵ̄].

The proof is complete. □

Lemma 4.3. For each φ̃ϵ(·) = (uϵ(·), w̃ϵ(·))T = ((uϵ,m(·))m∈Zk , (w̃ϵ,m(·))m∈Zk)T ∈ A
H(g0 )
ϵ it holds

that

∥φ̃ϵ(·)∥2Eϑ
= ∥uϵ(·)∥2ℓ2ϑ + ∥w̃ϵ(·)∥2ℓ2ϑ ≤ 2M4, ∀ϵ ∈ (0, ϵ̄],

and for any η > 0 there exists I3(η) ∈ N (independent of (g, ϵ)) such that

sup
−ϑ≤θ≤0

∑
∥m∥>2I3(η)

|uϵ,m(θ)|2 = sup
−ϑ≤θ≤0

∑
∥m∥>2I3(η)

|uϵ,m(θ)|2 ≤ η2, ∀ϵ ∈ (0, ϵ̄].

Proof. From Theorem 3.4, it follows that

AH(g
0
)

ϵ = ∪g∈H(g
0
)Ag

ϵ,0 ⊆ B̃ϵ ⊂ Hϑ.

Thus, for any fixed φ̃ϵ(·) = (uϵ(·), w̃ϵ(·))T = ((uϵ,m(·))m∈Zk , (w̃ϵ,m(·))m∈Zk)T ∈ A
H(g

0
)

ϵ , there must
exists a g ∈ H(g0) such that φ̃ϵ(·) ∈ Ag

ϵ,0. According to Lemma 4.2, the statements in Lemma 4.3
follow. □

5. Upper semicontinuity of uniform attractors

Now, we consider the upper semicontinuity of the uniform attractor AH(g
0
)

ϵ ⊂ Eϑ ⊆ ℓ2ϑ× ℓ2ϑ for
the second order delay lattice system (3.28) as ϵ→ 0+. When ϵ = 0, (2.4) is the first order delay

lattice system (2.1) with a uniform attractor AH(g0 )
0 ⊂ ℓ2ϑ. Notice that AH(g0 )

ϵ and AH(g0 )
0 are

in different spaces, to compare the relationship between them, we should take them in the same

bigger space ℓ2ϑ×ℓ2ϑ. For this purpose, basing on the structure of AH(g
0
)

ϵ and AH(g
0
)

0 , we introduce
the following set in ℓ2ϑ × ℓ2ϑ:

Bg0,t =
{(

ut
ωt

)
: ut(·) ∈ Ag

0,t and ωt(θ) = (I + γA)−1[−Au(t)− λu(t)− f(u(t))]

+ (I + γA)−1[−h(u(t− ϑ)) + g(t)], θ ∈ [−ϑ, 0]
}

⊂ Eϑ, t ∈ R, g ∈ H(g
0
),

where Ag
0,t is embedded into Bg0,t as the first component, that is, Π1Bg0,t = Ag

0,t, where Π1 :

(ut(·), ωt(·)) ∈ ℓ2ϑ × ℓ2ϑ → ut(·) ∈ ℓ2ϑ is the projector from ℓ2ϑ × ℓ2ϑ to ℓ2ϑ. Since (I + γA)−1g(·) ∈
C1
b (R, ℓ2) and (I + γA)−1[−Au− λu− f(u)− h (u(t− ϑ))] is continuous in u, so for fixed t ∈ R,

that Bg0,t is compact in ℓ2ϑ × ℓ2ϑ. Set

BH(g0 )
0 = ∪g∈H(g0 )

Bg0,t ⊂ Eϑ.

ThenAH(g
0
)

0 is naturally embedded into BH(g
0
)

0 as the first component, that is, Π1B
H(g

0
)

0 = AH(g
0
)

0 .
In the following, we show the upper semicontinuity:

lim
ϵ→0+

dh(A
H(g

0
)

ϵ ,BH(g
0
)

0 ) = 0.



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 19

Lemma 5.1. Let ϵ̄ > 0 be a given constant, conditions (A1)–(A6) hold and {ϵn}+∞
n=1 ⊂ (0, ϵ̄]

be an arbitrary sequence of positive numbers with ϵn → 0 as n → +∞. Taking φ(n)(·) =

(u(n)(·), ω(n)(·))T ∈ A
H(g

0
)

ϵn , then there exists a subsequence {ni} of {n} such that

φ(ni)(·) = (u(ni)(·), ω(ni)(·))T → (ū(·), v̄(·))T = φ̄(·) ∈ BH(g0 )
0 (ni → +∞) strongly in Eϑ.

Proof. By (3.29), A
H(g

0
)

ϵn = ∪g∈H(g
0
)A

g
ϵn,0

= ∪g∈H(g
0
)A

g
ϵn,t ⊂ Eϑ, for all t ∈ R, then for any

n ∈ N and any φ
(n)
0 (·) = (u

(n)
0 (·), ω(n)

0 (·))T ∈ A
H(g0 )
ϵn , there exists g(n) ∈ H(g0) such that φ

(n)
0 (·) =

(u
(n)
0 (·), ω(n)

0 (·))T ∈ Ag
(n)

ϵn,0
. Let

φ
(n)
t (·) = φ(n)(t+ ·) = φ(n)(t, 0, φ

(n)
0 (·))

= (u
(n)
t (·), u̇(n)t (·))T = (uϵn,t(·), u̇ϵn,t(·))T

= V g
(n)

ϵn (t, 0)φ
(n)
0 (·) ∈ Eϑ

be the solution of problem (3.28) with value φ
(n)
0 (·) ∈ Ag

(n)

ϵn,0
at t = 0; that is, φ

(n)
t (·) satisfies

ϵnü
(n) + u̇(n) + γAu̇(n) +Au(n) + λu(n) + f(u(n)) + h

(
u(n)(t− ϑ)

)
= g(n)(t),

u
(n)
t (·)|t=0 = u

(n)
0 (θ), u̇

(n)
t (·)|t=0 = ω

(n)
0 (θ), θ ∈ [−ϑ, 0].

By Theorem 3.4, we have

φ
(n)
t (·) = φ(n)(t+ ·) = (u

(n)
t (·), u̇(n)t (·))T ∈ Ag(n)

ϵn,t ⊆ B̃ϵn ⊂ Eϑ, ∀t ∈ R. (5.1)

By the compactness of H(g0) in Cb(R, ℓ2), there exists a subsequence of {g(n)(·)}+∞
n=1 (still denoted

by {g(n)(·)}+∞
n=1) such that

g(n)(·) → ḡ(·) ∈ H(g0) (n→ +∞) strongly in Cb(R, ℓ2).

In what follows, we prove that there exists a subsequence {ni} of {n} such that

φ
(ni)
t (·) = (u(ni)(t+ ·), u̇(ni)(t+ ·))T → (ū(t+ ·), ˙̄u(t+ ·))T ∈ Bḡ0,t (ni → +∞) in Eϑ

for t ∈ R, by using Arezla-Ascoli theorem and diagonal sequence method.

From (5.1) and Lemma 4.2(i), {φ(n)
t (θ) = φ

(n)
ϵn (t+ θ)}+∞

n=1 is uniformly bounded in ℓ2 × ℓ2 with
respect to θ ∈ [−ϑ, 0] and t ∈ R:

sup
t∈R

sup
1≤n<+∞

∥φ(n)
t (·)∥2Eϑ

= sup
t∈R

sup
1≤n<+∞

sup
−ϑ≤θ≤0

(∥u̇ϵn (t+ θ) ∥2 + ∥uϵn (t+ θ) ∥2) ≤ 2M4.

In particlualr,

sup
t∈R

sup
1≤n<+∞

(∥u̇(n) (t) ∥2 + ∥u(n) (t) ∥2) ≤ 2M4. (5.2)

Let Ji = [−i, i], i ∈ Z+, be a sequence of closed interval of R such that Ji ⊂ Ji+1,
⋃
i∈Z+

Ji = R.
Taking t1, t2 ∈ Ji, by mean value theorem and (5.2), we have

∥u(n)(t1)− u(n)(t2)∥ ≤
√
2M4|t1 − t2|,

which implies the equicontinuity of {u(n)(·)}+∞
n=1 ⊂ C1(R, ℓ2) in C(Ji, ℓ

2). Since E is a Hilbert

space, by (5.2), there exists a subsequence of {(u(n)(t), u̇(n)(t))T} (denoted still by {(u(n)(t), u̇(n)(t))T})
and (ū(t), ũ(t))T ∈ E such that

(u(n)(t), u̇(n)(t))T → (ū(t), ũ(t))T (n→ +∞) weakly in ℓ2 × ℓ2, ∀t ∈ R,

sup
t∈R

∥(ū(t), ũ(t))T∥2ℓ2×ℓ2 ≤ 2M4.

By Lemma 4.3, for any η > 0, there exists I4(η) ∈ N (independent of ϵn and n) such that for

u(n)(t) = (u
(n)
m (t))m∈Z,

sup
t∈R

∑
∥m∥>I4(η)

∥u(n)m (t)∥2 ≤ η2.



20 Y. ZHOU, H. LIU EJDE-2025/64

It obtain from the characteristics of a precompact set in ℓ2 that {u(n)(t)}∞n=1 is precompact in ℓ2,
i.e. for any fixed t ∈ R, {u(n)(t)}∞n=1 has a subsequence u(ni)(t) strongly convergent to ū(t) in ℓ2.

By Arezla-Ascoli theorem, {u(n)(·)} has a subsequence {u(n,1)(·)}+∞
n=1 such that

u(n,1)(·) → ū(·) (n→ +∞) strongly in C(J1, ℓ
2)

and for any k ∈ N, {u(n,i)(·)} has a subsequence {u(n,i+1)(·)} such that

u(n,i+1)(·) → ū(·) (n→ +∞) strongly in C(Ji+1, ℓ
2).

Taking the diagonal sequence of {u(n,i)(·)}, we obtain a subsequence {u(i,i)(·)} = {u(ii)(·)}, where
ii → +∞ as i→ +∞ and the corresponding subsequence ϵ(ii) → 0 as i→ +∞, such that for any
compact subset J ⊆ Ji ⊂ R,

u(ii)(·) → ū(·) (i→ +∞) strongly in C(J, ℓ2),

u(ii)(· − ϑ) → ū(· − ϑ) (i→ +∞) strongly in C(J, ℓ2),

u̇(ii)(·) → ˙̄u(·) (i→ +∞) weak star in L∞(J, ℓ2).

(5.3)

By (2.4),

u̇(ii)(t) = (I + γA)−1[−ϵ(kk)ü
(ii)(t)−Au(ii)(t)− λu(ii)(t)− f(u(ii)(t))]

+ (I + γA)−1[−h
(
u(ii)(t− ϑ)

)
+ g(ii)(t)], t ∈ R.

(5.4)

By (4.17) and 0 < ϵii → 0+ (i→ +∞), we have

sup
t∈R

√
ϵii∥ü(ii)(t)∥ ≤

√
2M4 <∞, lim

i→+∞
sup
t∈R

(ϵii∥ü(ii)(t)∥) = 0. (5.5)

By (5.4), (5.5), the continuity of f and h, the bounded linearity of A and (I + γA)−1, for any
compact subset J ⊂ R, t ∈ R, it follows that

u̇(ii)(t) → (I + γA)−1[−Aū(t)− λū(t)− f(ū(t))− h (ū(t− ϑ)) + ḡ(t)] (5.6)

as i→ +∞. By the uniqueness of the limit, it follows that

ũ(·) = ˙̄u(·) = (I + γA)−1[−Aū(·)− λū(·)− f(ū(·))− h (ū (· − ϑ)) + ḡ(·)] ∈ Cb(R, ℓ2). (5.7)

Thus, ū(t), t ∈ R, is a global bounded solution for the system (2.1) defined on R. By (5.2), we
have

sup
t∈R

sup
−ϑ≤θ≤0

(∥ ˙̄u (t+ θ) ∥2 + ∥ū (t+ θ) ∥2) ≤ 2M4.

By the structure of Aḡ
0,t and Bḡ0,t, (ū(t+ ·), ˙̄u(t+ ·))T ∈ Bḡ0,t for any t ∈ R. From (5.6), (5.7), we

have

(u(ii)(t+ ·), u̇(ii)(t+ ·))T → (ū(t+ ·), ˙̄u(t+ ·))T (i→ +∞)) in Eϑ for t ∈ R.

Specially,

φ
(ii)
0 (·) = (u

(ii)
0 (·), ω(ii)

0 (·))T = (u
(ii)
0 (·), u̇(ii)0 (·))T

→ (ū0(·), ˙̄u0(·))T = φ0,0(·) ∈ Bḡ0,0 ⊂ BH(g
0
)

0 (i→ +∞) in Eϑ.

The proof is complete. □

According to Lemma 5.1 and the contradiction, we obtain the following upper semicontinuity

of AH(g0 )
ϵ .

Theorem 5.2. Let conditions (A1)–(A6) hold. Then

lim
ϵ→0+

dh(A
H(g

0
)

ϵ ,BH(g
0
)

0 ) = 0 and limϵ→0+dh(Π1A
H(g

0
)

ϵ ,AH(g
0
)

0 ) = 0.



EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 21

Proof. If limϵ→0+ dh(A
H(g

0
)

ϵ ,BH(g
0
)

0 ) ̸= 0, then there exist η0 > 0 and {ϵn}+∞
n=1 ⊂ (0, 1] with ϵn → 0

as n→ +∞, and (u(n)(·), ω(n)(·))T ∈ AH(g0 )
ϵn such that

dh((u
(n)(·), ω(n)(·))T,BH(g

0
)

0 ) ≥ η0, n ∈ Z+. (5.8)

From Lemma 5.1, we obtain that {(u(n)(·), ω(n)(·))T}n∈Z+
has a subsequence converging to a point

in BH(g0 )
0 , which contradicts with (5.8). □

Acknowledgments. The authors would like to thank the anonymous referees for carefully read-
ing of the manuscript and for the useful suggestions and comments. This work is supported by
Natural Science Foundation of Hunan Province under Grant No. 2024JJ5320.

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Yao Zhou
School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Email address: yaozhou202401@163.com

Hongliang Liu (corresponding author)

School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Email address: math lhliang@163.com