Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 916316, 21 pages
doi:10.1155/2009/916316
Research Article
Uniform Attractor for the Partly Dissipative
Nonautonomous Lattice Systems
Xiaojun Li and Haishen Lv
Department of Applied Mathematics, Hohai University, Nanjing, Jiangsu 210098, China
Correspondence should be addressed to Xiaojun Li, [email protected]
Received 25 March 2009; Accepted 17 June 2009
Recommended by Toka Diagana
The existence of uniform attractor in l
2
× l
2
is proved for the partly dissipative nonautonomous
lattice systems with a new class of external terms belonging to L
2
loc
R, l
2
, which are locally
asymptotic smallness and translation bounded but not translation compact in L
2
loc
R, l
2
.Itis
also showed that the family of processes corresponding to nonautonomous lattice systems with
external terms belonging to weak topological space possesses uniform attractor, which is identified
α
i
v
i
k
i
t
,i∈ Z,t>τ, 1.1
˙v
i
δ
i
v
i
− β
i
u
i
g
i
t
,i∈ Z,t>τ, 1.2
with initial conditions
u
i
i
t
i∈Z
,gt
g
i
t
i∈Z
belong to certain metric space, which will be given i n the following.
2 Advances in Difference Equations
Lattice dynamical systems occur in a wide variety of applications, where the
spatial structure has a discrete character, for example, chemical reaction theory, electrical
engineering, material science, laser, cellular neural networks with applications to image
processing and pattern recognition; see 1–4. Thus, a great interest in the study of infinite
lattice systems has been raising. Lattice differential equations can be considered as a spatial
or temporal discrete analogue of corresponding partial differential equations on unbounded
domains. It is well known that the long-time behavior of solutions of partial differential
equations on unbounded domains raises some difficulty, such as well-posedness and lack of
compactness of Sobolev embeddings for obtaining existence of global attractors. Authors in
5–7 consider the autonomous partial equations on unbounded domain in weighted spaces,
using the decaying of weights at infinity to get the compactness of solution semigroup. In
8–10, asymptotic compactness of the solutions is used to obtain existence of global compact
attractors for autonomous system on unbounded domain. Authors in 11 consider them
in locally uniform space. For non-autonomous partial differential equations on bounded
domain, many studies on the existence of uniform attractor have been done, for example
12–14.
For lattice dynamical systems, standard theory of ordinary differential equations can
be applied to get the well-posedness of it. “Tail ends” estimate method is usually used to get
asymptotic compactness of autonomous infinite-dimensional lattice, and by this the existence
of global compact attractor is obtained; see 15–17. Authors in 18, 19 also prove that the
function in L
2
loc
R, l
2
. We also show that locally asymptotic functions are translation bounded
in L
2
loc
R, l
2
, but not translation compact tr.c. in L
2
loc
R, l
2
. Since the locally asymptotic
smallness functions are not necessary to be translation compact in C
b
R, l
2
, compared with
20, the conditions on external terms of 1.1–1.3 can be relaxed in this paper.
This paper is organized as follows. In Section 2, we give some preliminaries and
present our main result. In Section 3, the existence of a family of processes for 1.1–
1.3 is obtained. We also show that the family of processes possesses a uniformly w.r.t
H
w
k
0
, 2.1
with the inner product ·, · and norm ·given by
u, v
i∈Z
u
i
v
i
,
u
2
u, u
i∈Z
u
2
i
. 2.2
For l
2
×l
2
u
1
,u
2
l
2
v
1
,v
2
l
2
i∈Z
u
1
i
u
2
2
.
2.3
Denote by L
2
loc
R, l
2
the space of function φs,s∈ R with values in l
2
that locally 2-power
integrable in the Bochner sense, that is,
t
2
t
1
φs
2
l
2
ds < ∞, ∀
t
1
,t
2
sup
t∈R
t1
t
φs
2
l
2
ds < ∞. 2.5
Denote by L
2,w
loc
R, l
2
the space L
2
loc
R, l
2
endow with the local weak convergence topology.
For each sequence u u
i
2u
i
− u
i−1
,i∈ Z.
2.6
Then
A BB
∗
B
∗
B,
B
∗
u, v
u, Bv
, ∀u, v ∈ l
2
.
2.7
4 Advances in Difference Equations
For convenience, initial value problem 1.1–1.3 can be written as
˙u ν
Au
τ
v
τ
v
i,τ
i∈Z
,τ∈ R, 2.10
where u u
i
i∈Z
,vv
i
i∈Z
,νAuν
i
Au
i
i∈Z
,fu, Bufu
i
, Bu
i
f
i
u
i
0,
Bu
i
0
0,f
i
u
i
,
Bu
i
u
i
≥ 0. 2.11
H
2
There exists a positive-value continuous function Q : R
sup
i∈Z
max
u
i
,Bu
i
∈−r,r
f
i,Bu
i
u
i
,
Bu
i
≤ Q
r
i
, : i ∈ Z
}
,ν
0
max
{
ν
i
, : i ∈ Z
}
< ∞,
0 <λ
0
min
{
λ
i
, : i ∈ Z
}
,λ
0
max
{
λ
i
, : i ∈ Z
}
< ∞,
0 <α
, : i ∈ Z
< ∞,
0 <δ
0
min
{
δ
i
, : i ∈ Z
}
,δ
0
max
{
δ
i
, : i ∈ Z
}
< ∞.
2.13
Let the external term ht,gt belong to L
2
b
R, l
2
, it follows from the standard theory
of ordinary differential equations that there exists a unique local solution u, v ∈ Cτ, t
0
,l
×Hg
0
, the set contains all translations of k
0
s,g
0
s in L
2
b
R, l
2
× L
2
b
R, l
2
. Take
the Σ
w
H
w
k
0
×H
w
g
0
the closure of Σ in L
2,w
loc
Σ Σ H
k
0
×H
g
0
,T
h
Σ
w
Σ
w
H
w
k
0
×H
w
u
i
,v
i
i∈Z
t ∈ Cτ,∞,l
2
× l
2
. Thus, there exists a family of processes {U
k,g
t, τ} from
l
2
×l
2
to l
2
×l
2
. In order to obtain the uniform attractor of the family of processes, we suppose
the external term is locally asymptotic smallness see Definition 4.5.LetE be a Banach space
which the processes acting in, for a given symbol space Ξ, the uniform w.r.t. σ ∈ Ξ ω-limit
set ω
τ,Ξ
B of B ⊂ E is defined by
ω
τ,Ξ
and H
1
–H
3
hold. Then the process {U
k
0
,g
0
} corresponding to problems 2.8–2.10 with external
term k
0
s, g
0
s possesses compact uniform w.r.t.τ ∈ R attractor A
0
in l
2
× l
2
which coincides
with uniform (w.r.t. ks, gs ∈H
w
k
0
×H
w
g
0
w
g
0
ω
0,A
H
w
k
0
×H
w
g
0
B
0
k,g∈H
w
k
0
×H
w
g
0
w
g
0
attracts the bounded set in l
2
× l
2
.
We also consider finite-dimensional approximation to the infinite-dimensional
systems 1.2-1.3 on finite lattices. For every positive integer n>0, let Z
n
Z ∩{−n ≤ i ≤ n},
consider the following ordinary equations with initial data in R
2n1
× R
2n1
:
˙u
i
ν
i
Au
i
λ
i
u
i
i
u
i
g
i
t
,i∈ Z
n
,t>τ,
u
τ
u
i
τ
|
i
|
≤n
u
i,τ
n
0
in
2n1
× R
2n1
, and these uniform attractors are upper semicontinuous
when n →∞. More precisely, we have the following theorem.
Theorem B. Assume that k
0
s, g
0
s ∈ L
2
b
R, l
2
× L
2
b
R, l
2
and H
1
–H
3
hold. Then for
every positive integer n, systems 2.17 possess compact uniform attractor A
n
0
A
n
0
, A
0
sup
a∈A
n
0
inf
b∈A
0
a − b
l
2
×l
2
. 2.19
3. Processes and Uniform Absorbing Set
In this section, we show that the process can be defined and there exists a bounded uniform
absorbing set for the family of processes.
Lemma 3.1. Assume that k
0
, g
0
∈ L
2
2
l
2
×l
2
≤
u, v
τ
2
l
2
×l
2
e
−
γ
0
/η
0
t−τ
g
0
s
2
L
2
b
R,l
2
1
η
0
γ
0
,
3.1
where η
0
min{α
0
,β
0
β
i
ν
i
|
Bu
i
|
2
i∈Z
λ
i
β
i
u
2
i
i∈Z
β
i
α
i
u
i
i∈Z
α
i
δ
i
u
2
i
−
i∈Z
β
i
α
i
u
i
v
i
i∈Z
α
i
v
i
g
i
i∈Z
β
i
λ
i
k
2
i
t
,
i∈Z
α
i
v
i
g
i
t
≤
1
2
i∈Z
α
i
u
2
i
α
i
v
2
i
i∈Z
λ
i
β
i
u
2
i
α
i
δ
i
v
2
i
≤
η
0
d
dt
u, v
t
2
l
2
×l
2
γ
0
u, vt
2
l
2
×l
2
≤
β
0
w
k
0
×H
w
g
0
,from12, Proposition V.4.2., we have
k
t
2
L
2
b
R,l
2
≤
k
0
t
2
L
2
b
R,l
2
. 3.7
From 3.6-3.7, applying Gronwall’s inequality of generalization see 12, Lemma II.1.3,
we get 3.1. The proof is completed.
It follows from Lemma 3.1 that the solution u, v of problem 2.8–2.10 is defined
for all t ≥ τ. Therefore, there exists a family processes acting in the space l
2
× l
2
: {U
k,g
} :
U
k,g
t, τu
τ
,v
τ
ut,vt, U
k,g
t, τ : l
2
× l
2
→ l
k,g
s, τ
U
k,g
t, τ
, ∀t ≥ s ≥ τ, τ ∈ R,
U
k,g
τ,τ
Id is the identity operator,τ∈ R.
3.8
Furthermore, the following translation identity holds:
U
k,g
t h, τ h
|
ut,vt
l
2
×l
2
≤ C
u,v
,
U
k,g
t, τ
u
τ
,v
τ
u
|
u
·
,v
·
∈K
k,g
. 3.11
Lemma 3.1 also shows that the family of processes possesses a uniform absorbing set
in l
2
× l
2
.
8 Advances in Difference Equations
Lemma 3.2. Assume that k
0
, g
0
∈ L
2
b
w
g
0
, that is, for any bounded set B ⊂ l
2
× l
2
, there exists t
0
t
0
τ,B ≥ τ,
k,g∈H
w
k
0
×H
w
g
0
U
k,g
t, τ
B ⊂ B
0
t−τ
1
η
0
β
0
λ
0
k
0
s
2
L
2
b
R,l
2
α
0
δ
β
0
λ
0
k
0
s
2
L
2
b
R,l
2
α
0
δ
0
g
0
s
2
X
τ, X
1
η
0
β
0
λ
0
k
0
s
2
L
2
b
R,l
2
α
0
δ
2
× l
2
|u, vt
2
l
2
×l
2
≤ 2X
2
}. The proof is completed.
4. Uniform Attractor
In this section, we establish the existence of uniform attractor for the non-autonomous lattice
systems 2.8–2.10.LetE be a Banach space, and let Ξ be a subset of some Banach space.
Definition 4.1. {U
σ
t, τ},σ∈ Ξ is said to be E×Ξ,E weakly continuous, if for any t ≥ τ, τ ∈
R, the mapping u, σ →{U
σ
t, τu is weakly continuous from E × Ξ to E.
A family of processes U
σ
t, τ, σ ∈ Ξ is said to be uniformly w.r.t.σ ∈ Ξ ω-limit
compact if for any τ ∈ R and bounded set B ⊂ E,theset
σ∈Ξ
s≥t
U
Ξ
0
and A
Ξ
satisfying
A
Ξ
0
A
Ξ
ω
0,Ξ
B
0
σ∈Ξ
K
σ
0
. 4.1
Furthermore, K
σ
0 is nonempty for all σ ∈ Ξ.
Let E be a Banach space and p ≥ 1, denote the space L
p
1
,t
2
denotes the restriction of the
set Σ to the segment t
1
, t
2
.
Proposition 4.4. A function σs is tr.c. in L
p
loc
R, E if and only if
i for any h ∈ R the set {
th
t
σsds | t ∈ R} is precompact in E;
ii there exists a function αs, αs → 0
s → 0
such that
t1
t
σs − σs l
ds < . 4.3
Denote by L
2
las
R, l
2
the set of all locally asymptotic smallness functions in L
2
loc
R, l
2
.
It is easy to see that L
2
las
R, l
2
⊂ L
2
b
R, l
2
. The next examples show that there exist functions
in L
2
b
R, l
2
but not in L
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0,i≤ 0,
⎧
⎪
⎪
⎪
⎪
⎪
2i − 4i
√
2i
t − 2i −
1
2i
, 2i
1
2i
≤ t ≤ 2i
3
4i
,
0, otherwise.
i ≥ 1.
4.4
10 Advances in Difference Equations
For every t ∈ 2i − 1/4i, 2i 3/4i, i ≥ 1,
t1
t
i∈Z
φ
i
2ids
2i3/4i
2i1/2i
√
2i − 4i
√
2i
s − 2i −
1
2i
2
ds
≤ 2i ×
1
4i
2i ×
1
2i
2i ×
1
4i
2 < ∞.
4.5
Thus,
sup
t∈R
|
≥N
φ
i
s
2
ds ≥
2i1/2i
2i
2ids 1. 4.7
Therefore, φt
/
∈L
2
las
R, l
2
.
Example 4.7. ϕtϕ
i
t
i∈Z
,
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
√
2k 4
2k
2
√
2k
2
,
√
2k − 4
2k
2
√
2k
t − 2k −
j
2k
−
1
2k
2
, 2k
j
2k
1
2k
2
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
√
2i
2i
2
√
2i
t − 2i
, 2i −
1
1
2i
2
≤ t ≤ 2i
2
2i
2
,
0, otherwise.
4.9
Here, Z
denote the positive integer set.
For every positive integer N>1, i ≥ N,andfort ∈ 2i − 1/2i
2
, 2i 2/2i
2
,
t1
t
|i|≥N
ϕ
2i1/
2i
2
2i
2ids
2i2/
2i
2
2i1/
2i
2
×
√
2i −
2i
2
√
2i
s − 2i −
,
4.10
which implies that
sup
t∈R
t1
t
|i|≥N
ϕ
i
s
2
ds ≤
3
2N
. 4.11
Therefore, ϕtϕ
i
t
i∈Z
∈ L
2
ϕ
1
s 2k − ϕ
1
s 2k l
2
≥ 1.
4.12
From Proposition 4.4, ϕtϕ
i
t
i∈Z
is not translation compact in L
2
loc
R, l
2
.
Remark 4.8. Example 4.7 shows that a locally asymptotic function is not necessary translation
compact in C
b
R, l
2
.
In the following, we give some properties of locally asymptotic smallness function.
12 Advances in Difference Equations
Lemma 4.9. L
. 4.13
Then, for any >0, there exists positive integer N
1
such that for every n ≥ N
1
,
sup
t∈R
t1
t
ψ
n
s − ψs
2
l
2
<. 4.14
Since ψ
n
∈ L
2
las
R, l
2
, there exist N
|
i
|
≥N
2
ψ
2
i
s
ds
≤ 2
⎛
⎝
sup
t∈R
t1
t
|
i
|
≥N
2
ψ
⎞
⎠
< 4.
4.16
Therefore, ψs ∈ L
2
las
R, l
2
. This completes the proof.
Lemma 4.10. Every translation compact function ws in L
2
loc
R, l
2
is locally asymptotic smallness.
Proof. Since ws is tr.c. in L
2
loc
R, l
2
,wegetthat{ws t | t ∈ R} is precompact in L
2
loc
R, l
2
.
By Proposition 4.3,wegetthat{ws t | t ∈ R}
0,1
is precompact in L0, 1; l
2
l
2
<, t∈ R. 4.17
For the given above, w
j
s ∈ L0, 1; l
2
implies that there exists positive integer N such
that
1
0
|i|≥N
w
j
s
2
ds < . 4.18
Advances in Difference Equations 13
Therefore,
ji
s
2
ds 2
1
0
|i|≥N
w
ji
s
2
ds
≤ 4,
4.19
which implies ws is locally asymptotic smallness. This completes the proof.
We now establish the uniform estimates on the tails of solutions of 2.8–2.10 as
n →∞.
Lemma 4.11. Assume that H
1
–H
g
0
satisfies
|
i
|
≥N
|
u
i
t
|
2
|
v
i
t
|
2
<. 4.20
Proof. Choose a smooth function θ such that 0 ≤ θs ≤ 1fors ∈ R
,and
θ
˙u, βφ
νAu, βφ
λu, βφ
f
u, Bu
,βφ
αv, βφ
k
t
,βφ
˙v, αψ
≥
η
0
2
d
dt
i∈Z
u
2
i
v
2
i
θ
|
i
|
N
, 4.23
14 Advances in Difference Equations
where η
i
θ
|
i 1
|
N
u
i1
− θ
|
i
|
N
u
i
≥
i∈Z
ν
i
β
i
Bu
|
N
− θ
|
i
|
N
u
i1
≥
i∈Z
ν
i
β
i
Bu
i
2
θ
i∈Z
θ
|
i
|
N
u
2
i
,
αv, βφ
βu, αψ
,
δv, αψ
≥ δ
0
α
0
i∈Z
θ
|
N
u
i
≤
1
2
λ
0
β
0
i∈Z
θ
|
i
|
N
u
2
i
1
2
β
0
2
δ
0
α
0
i∈Z
θ
|
i
|
N
v
2
i
1
2
α
0
2
δ
0
α
0
i∈Z
θ
|
N
γ
0
η
0
i∈Z
u
2
i
v
2
i
θ
|
i
|
N
≤
β
0
2
λ
|
i
|
N
g
2
i
t
4ν
0
β
0
M
0
X
2
N
, ∀t ≥ t
0
.
4.26
Thus,
i∈Z
u
v
2
i
τ
e
−
γ
0
/η
0
t−τ
η
0
γ
0
·
4ν
0
β
0
M
0
X
2
N
k
2
i
s
ds
t
τ
α
0
2
δ
0
α
0
e
−
γ
0
/η
0
t−s
t−s
i∈Z
θ
|
i
|
N
k
2
i
s
ds ≤
t
t−1
e
−
γ
0
/η
0
t−s
i∈Z
θ
|
i
|
N
k
2
i
s
ds ···
≤ e
−γ
0
/η
0
1
0
e
−
γ
0
/η
0
0
/η
0
s
i∈Z
θ
|
i
|
N
k
2
i
s t − 2
ds ···
≤
1 e
−γ
0
/η
0
e
−2γ
γ
0
/η
0
sup
t∈R
1
0
i∈Z
θ
|
i
|
N
k
2
i
s t
ds.
4.28
Similarly,
t
/η
0
sup
t∈R
1
0
i∈Z
θ
|
i
|
N
g
2
i
s t
ds. 4.29
Since k
0
t,g
0
t is locally asymptotic smallness, from 4.27–4.29 we get that for any >0,
if uτ,vτ
2
4ν
0
β
0
M
0
X
2
N
β
0
2
λ
0
β
0
1
1 − e
γ
0
/η
0
sup
t∈R
1
0
0
sup
t∈R
1
0
i∈Z
θ
|
i
|
N
g
2
0i
s t
ds
<, ∀t ≥ T.
4.30
The proof is completed.
Lemma 4.12. Assume that H
1
–H
k, g weaklyinL
2
loc
R, l
2
× l
2
, then for any t ≥ τ, τ ∈ R,
U
k
n
,g
n
u
n0
,v
n0
U
k,g
u
0
,v
0
n0
} is bou-
nded in l
2
× l
2
,byLemma 3.2,wegetthat
{
u
n
,v
n
t
}
is uniformly bounded in l
2
× l
2
. 4.32
Therefore, for all t ≥ τ, τ ∈ R,
u
n
,v
n
t
2
, it follow from 4.32 that
˙u
n
, ˙v
n
t
˙u
w
, ˙v
w
t
weak starin L
∞
R, l
2
× l
2
, as n −→ ∞. 4.34
In the following, we show that u
w
n
i
ψ
t
dt
t
τ
λ
i
u
ni
ψ
t
dt
t
τ
f
i
u
ni
,
ψ
t
dt, t ≥ τ,
t
τ
˙v
ni
ψ
t
dt
t
τ
δ
i
v
ni
ψ
t
dt −
t
loc
R, l
2
× l
2
.Letn →∞in 4.35,by4.34 we get
that u
w
,v
w
t is the solution of 2.8 and 2.9 with the initial data u
0
,v
0
. By the unique
solvability of problem 2.8–2.10,wegetthatu
w
,v
w
tu, vt. This completes the
proof.
Proof of Theorem A. From Lemmas 3.2, 4.11 and 4.12,andTheorem 4.2,wegettheresults.
5. Upper Semicontinuity of Attractors
In this section, we present the approximation to the uniform attractor A
H
w
k
0
×H
w
i
α
i
v
i
k
i
t
,i∈ Z
n
,t>τ,
˙v
i
δ
i
v
i
− β
i
u
i
g
i
t
,v
τ
v
i
τ
|
i
|
≤n
v
i,τ
|i|≤n
,τ∈ R, 5.2
and the periodic boundary conditions
u
n1
,v
n1
2
possess a unique solution u, vu
i
,v
i
|i|≤n
∈
Cτ, ∞,R
2n1
× R
2n1
, which continuously depends on initial data. Therefore, we can
associate a family of processes {U
n
k,g
t, τ}
H
w
k
0
×H
w
g
0
which satisfy similar properties
3.8–3.9. Similar to Lemma 3.2, we have the following result.
Lemma 5.1. Assume that k
0
0
×H
w
g
0
, that is, for any bounded set B
n
⊂ R
2n1
×R
2n1
, there exists t
0
t
0
τ,B
n
≥
τ,fort≥ t
0
,
k,g∈H
w
k
0
×H
w
g
w
k
0
×H
w
g
0
ω-limit compact. Similar to Lemma 4.12,ifu
n
m0
,v
n
m0
→ u
n
0
,v
n
0
in
l
2
× l
2
, k
m
,g
m
k, g weakly in L
2
0
,v
n
0
weakly in l
2
× l
2
,m−→ ∞. 5.5
Lemma 5.2. Assume that k
0
s, g
0
s ∈ L
2
b
R, l
2
×L
2
b
R, l
2
and H
1
–H
3
hold. Then the process
{U
w
k
0
×H
w
g
0
for the family of processes {U
n
k,g
t,τ}, k, g ∈H
w
k
0
×
H
w
g
0
, that is,
A
n
0
A
n
H
w
k
0
0
K
n,k,g
0
, 5.6
where B
1
is the uniform w.r.t. k, g ∈H
w
k
0
×H
w
g
0
absorbing set in R
2n1
×R
2n1
, and K
k,g
is kernel of the process {U
k,g
t, τ}. The uniform attractor uniformly w.r.t. k, g ∈H
w
k
0
w
g
0
and a bounded complete solution u
n
·,v
n
· ∈ CR, R
2n1
× R
2n1
such
that
u
n
t
,v
n
t
U
n
k
n
,g
n
0
,
u
n
t
,v
n
t
∈A
n
0
, ∀t ∈ R,n 1, 2,
5.7
Since k
n
,g
n
∈H
w
k
0
×H
w
n
k, g
weakly in L
2
loc
R, l
2
× l
2
, as n −→ ∞. 5.8
From Lemma 5.1,wegetthat
u
n
,v
n
t
l
2
×l
2
≤ C
1
n
s
≤
˙u
n
|
t − s
|
≤ C
2
|
t − s
|
,
v
n
t
− v
n
s
exists a subsequence of {u
n
,v
n
t} still denoted by {u
n
,v
n
t} and u
t
,v
t
∈ l
2
× l
2
such
that
u
n
,v
n
t
u
t
,v
n
t} and u
1
,v
1
t
such that {u
n
1
,v
n
1
t} converges to u
1
,v
1
t ∈ CI
1
,l
2
× l
2
. Using Ascoli’s theorem again,
we get, by induction, that there is a subsequence {u
n
j1
,v
n
j1
,v
j
t to I
j1
. Finally, taking a diagonal subsequence in the usual way, we
find that there exist a subsequence {u
n
n
,v
n
n
t} of {u
n
,v
n
t} and u, vt ∈ CR, l
2
× l
2
such that for any compact interval I ⊂ R
u
n
n
,v
n
n
t
∀t ∈ R. 5.14
Next, we show that ut,vt is the solution of 2.8–2.10. It follows from 5.10 that
˙u
n
t
, ˙v
n
t
˙u
t
, ˙v
t
weak star in L
∞
R, l
2
× l
2
n
i
− λ
i
u
n
i
− f
i
u
n
i
,
Bu
n
i
− α
i
v
n
i
k
n
i
2
, we have
I
˙u
n
i
t
ψ
t
dt −
I
ν
i
Au
n
i
ψ
t
dt −
dt
−
I
α
i
v
n
i
ψ
t
dt
I
k
n
i
t
ψ
t
dt, t ∈ R,
I
˙v
t
dt
I
g
n
i
t
ψ
t
dt, t ∈ R.
5.17
Letting n →∞,by5.8, 5.13, 5.15 and 5.17 we find that u, v satisfies
˙u
i
t
−ν
i
Au
i
−δ
i
v
i
β
i
u
i
g
i
t
, ∀t ∈ I, i ∈ Z.
5.18
Since I is arbitrary, we note that 5.18 are valid for all t ∈ R.From5.14 we find that u, v
is a bounded complete solution of 2.8–2.10. Therefore, u0,v0 ∈A
0
.By5.13 we get
that
u
n
n
0
,v
n
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