Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 360871, 17 pages
doi:10.1155/2009/360871
Research Article
Nonlinear Discrete Periodic Boundary Value
Problems at Resonance
Ruyun Ma1 and Huili Ma2
1
2
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
College of Economics and Management, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma, ruyun ma@126.com
Received 25 June 2009; Revised 4 October 2009; Accepted 6 December 2009
Recommended by Kanishka Perera
Let T ∈ N be an integer with T > 2, and let T : {1, . . . , T }. We study the existence of solutions of
nonlinear discrete problems Δ2 ut − 1 λk atut gt, ut ht, t ∈ T, u0 uT , u1 uT 1, where a, h : T → R with a > 0, λk is the kth eigenvalue of the corresponding linear
eigenvalue problem.
Copyright q 2009 R. Ma and H. Ma. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Initialed by Lazer and Leach 1, much work has been devoted to the study of existence result
for nonlinear periodic boundary value problem
y x m2 yx g x, yx ex,
y0 y2π,
x ∈ 0, 2π,
y 0 y 2π,
1.1
where m ≥ 0 is an integer. Results from the paper have been extended to partial differential
equations by several authors. The reader is referred, for detail, to Landesman and Lazer 2,
Amann et al. 3, Brézis and Nirenberg 4, Fučı́k and Hess 5, and Iannacci and Nkashama
6 for some reference along this line. Concerning 1.1, results have been carried out by
many authors also. Let us mention articles by Mawhin and Ward 7, Conti et al. 8, Omari
and Zanolin 9, Ding and Zanolin 10, Capietto and Liu 11, Iannacci and Nkashama 12,
Chu et al. 13, and the references therein.
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However, relatively little is known about the discrete analog of 1.1 of the form
Δ2 ut − 1 λk atut gt, ut ht,
u0 uT ,
t ∈ T,
u1 uT 1,
1.2
where T : {1, . . . , T }, a, h : T → R with a > 0, gt, s : T × R → R is continuous in s. The
likely reason is that the spectrum theory of the corresponding linear problem
Δ2 ut − 1 λk atut 0,
u0 uT ,
t ∈ T,
u1 uT 1
1.3
was not established until 14. In 14, Wang and Shi showed that the linear eigenvalue
problem 1.3 has exactly T real eigenvalues
μ0 < μ1 ≤ μ2 < · · · < μT −2 ≤ μT −1 ,
when T is odd,
μ0 < μ1 ≤ μ2 < · · · ≤ μT −2 < μT −1 ,
when T is even.
1.4
Suppose that these above eigenvalues have N 1 different values λk , k 0, 1, . . . , N. Then
1.4 can be rewritten as
λ0 < λ1 < · · · < λN .
1.5
For each λk , we denote its eigenspace by Mk . If dim Mk 1, then we assume that Mk :
span{ψk } in which ψk is the eigenfunction of λk . If dim Mk 2, then we assume that Mk :
span{ψk , ϕk } in which ψk and ϕk are two linearly independent eigenfunctions of λk .
It is the purpose of this paper to prove the existence results for problem 1.2 when
there occurs resonance at the eigenvalue λk and the nonlinear function g may “touching” the
eigenvalue λk1 . To have the wit, we have what follows.
Theorem 1.1. Let a, h : T → R with a > 0, gt, s : T × R → R is continuous in s, and for some
r ∗ < 0 < R∗ ,
gt, x ≥ At,
∀x ≥ R∗ ,
gt, x ≤ Bt,
∀x ≤ r ∗ ,
1.6
where A, B : T → R are two given functions. Suppose for some 1 ≤ k ≤ N − 1,
dim Mk1 2.
1.7
Assume that for all ε > 0, there exist a constant R Rε > 0 and a function b : T → R such that
gt, u ≤ Γt εat|u| bt,
t ∈ T, |u| ≥ R,
1.8
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where Γ : T → R is a given function satisfying
0 ≤ Γt ≤ λk1 − λk ,
t ∈ T,
1.9
and for at least T/2 2 points in 1, T ,
Γt < λk1 − λk ,
1.10
where r denotes the integer part of the real number r.
Then 1.2 has at least one solution provided
T
htvt <
t1
g tvt vt>0
g− tvt,
1.11
vt<0
0, and
where v ∈ Mk , v /
g t lim inf gt, u,
u → ∞
g− t lim sup gt, u.
u → −∞
1.12
In 12, Iannacci and Nkashama proved the analogue of Theorem 1.1 for continuoustime nonlinear periodic boundary value problems 1.1. Our paper is motivated by Iannacci
and Nkashama 12. However, as we will see below, there are big differences between
the continuous case and the discrete case. The main tool we use is the Leray-Schauder
continuation theorem see Mawhin 15, Theorem IV.5.
Finally, we note that when at ≡ 1 in 1.2, the existence of odd solutions or even
solutions was investigated by R. Ma and H. Ma 16 under some parity conditions on the
nonlinearities. The existence of solutions of second-order discrete problem at resonance was
studied by Rodriguez in 17, in which the nonlinearity is required to be bounded. For other
results on discrete boundary value problems, see Kelley and Peterson 18, Agarwal and
O’Regan 19, Rachunkova and Tisdell 20, Yu and Guo 21, Atici and Cabada 22, Bai and
Xu 23. However, these papers do not address the problem under “asymptotic nonuniform
resonance” conditions.
2. Preliminaries
Let
{0, 1, . . . , T 1}.
T
2.1
−→ R | u0 uT , u1 uT 1 .
D : u : T
2.2
Let
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Then D is a Hilbert space under the inner product
u, v T
atutvt,
2.3
t1
and the corresponding norm is
1/2
T
atutut
.
u : u, u 2.4
t1
Thus,
ψk , ϕk 0 if dim Mk 2,
ψj , ψk 0, for j, k ∈ {0, 1, . . . , N}, j /
k,
ϕj , ϕk 0, for j, k ∈ {0, 1, . . . , N}, j / k.
2.5
In the rest of the paper, we always assume that
ψk 1, for k ∈ {0, 1, . . . , N},
ϕk 1 if dim Mk 2.
2.6
Define a linear operator L : D → D by
Lut −Δ2 ut − 1,
t ∈ T,
Lu0 : Lu1,
2.7
LuT 1 : LuT .
Lemma 2.1 see 16. Let u, w ∈ D. Then
T
T
wkΔ2 uk − 1 − ΔukΔwk.
k1
2.8
k1
Similar to 12, Lemma 3, we can prove the following.
Lemma 2.2 see 12. Suppose that
i there exist A, B : T → R and real numbers r < 0 < R, such that
gt, x ≥ At,
∀x ≥ R,
gt, x ≤ Bt,
∀x ≤ r,
2.9
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5
ii there exist α, β : T → 0, ∞ and a constant B0 > 0 such that
gt, u ≤ αt|u| βt,
t ∈ T, |u| ≥ B0 .
2.10
Then for each real number κ > 0, there is a decomposition
gt, x qκ t, x eκ t, x
2.11
0 ≤ xqκ t, x, t ∈ T, x ∈ R,
qκ t, u ≤ αt|u| βt κ, t ∈ T, |u| ≥ max{1, B0 },
2.12
of g satisfying
2.13
and there exists a function σκ : T → 0, ∞ depending on r, R, and g such that
|eκ t, x| ≤ σκ t,
t ∈ T, x ∈ R.
2.14
3. Existence of Periodic Solutions
In this section, we need to give some lemmas first, which have vital importance to prove
Theorem 1.1.
For convenience, we set
as dimMk 1.
ϕk : 0,
3.1
Thus, for any u ∈ D, we have the following Fourier expansion:
ut a0 N
ai ψi t bi ϕi t ,
t ∈ T.
3.2
i1
Let us write
t,
ut ut u0 t u
u⊥ t ut − u0 t,
3.3
where
ut a0 k−1
ai ϕi t bi ψi t ,
i1
u t ak ϕk t bk ψk t,
0
u
t N
ik1
ai ϕi t bi ψi t .
3.4
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Lemma 3.1. Suppose that for 1 ≤ k ≤ N − 1, λk1 is an eigenvalue of 1.3 of multiplicity 2. Let
Γ : T → R be a given function satisfying
0 ≤ Γt ≤ λk1 − λk ,
t ∈ T,
3.5
and for at least T/2 2 points in 1, T ,
Γt < λk1 − λk .
3.6
Then there exists a constant δ δΓ > 0 such that for all u ∈ D, one has
T 2
Δ2 ut − 1 λk atut Γtatut ut u0 t − u
t ≥ δu⊥ .
3.7
t1
Proof. For u ∈ D,
N
Δ2 ut − 1 −at
ai λi ψi t bi λi ϕi t .
3.8
i1
in D, we have
Taking into account the orthogonality of u, u0 , and u
T Δ2 ut − 1 λk atut Γtatut ut u0 t − u
t
t1
T T
2
Δ2 ut − 1 λk atut ut Γtat ut u0 t
t1
t1
T Δ2 u
t − 1 λk at
ut Γtat
ut −
ut
t1
T Δ2 u0 t − 1 λk atu0 t u0 t
3.9
t1
T T
2
−Δut2 λk atu2 t Γtat ut u0 t
t1
T t1
u2 t − Γtat
u2 t
ut2 − λk at
Δ
t1
T
T
T
≥ λk − λk−1 atu2 t Δ
u2 t.
ut2 − λk at Γtat
t1
t1
t1
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Set
Λu λk − λk−1 T
atu2 t.
3.10
t1
Then,
Λu ≥ δ1 u2 ,
3.11
where δ1 is a positive constant less than λk − λk−1 .
Let
ΛΓ u T
T
u2 t.
ut2 − λk at Γtat
Δ
t1
3.12
t1
We claim that ΛΓ u ≥ 0 with the equality holding only if u
A0 ψk1 B0 ϕk1 , where A0 , B0 ∈
R are constants.
In fact, we have from Lemma 2.1 that
ΛΓ u T
T
u2 t
ut2 − λk at Γtat
Δ
t1
−
t1
T
T
u
tΔ2 u
t − 1 − λk at Γtat
u2 t
t1
t1
T N
N
λi at ai ψi t bi ϕi t
ai ψi t bi ϕi t
t1 ik1
ik1
T
− λk at Γtat
t1
≥
N
ai ψi t bi ϕi t
2
ik1
N
T N
λj at aj ψj t bj ϕj t
ai ψi t bi ϕi t
t1 ik1
jk1
T
− λk1 at
t1
N
N
⎞
⎛
N
ai ψi t bi ϕi t ⎝
aj ψj t bj ϕj t ⎠
ik1
ai aj λj
N
N
jk1
N
N
T
T
atψi tψj t bi bj λj atψi tψj t
t1
ik1 jk1
−
N
ai aj λk1
ik1 jk1
ik1 jk1
T
t1
atψi tψj t
t1
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−
N
N
bi bj λk1
N
N
a2j λj − λk1 bj2 λj − λk1
jk1
atψi tψj t
t1
ik1 jk1
T
jk1
N jk1
a2j bj2
λj − λk1 ≥ 0.
3.13
Obviously, ΛΓ u 0 implies that ak2 · · · aN bk2 · · · bN 0, and accordingly
u
t A0 ψk1 t B0 ϕk1 t for some A0 , B0 ∈ R.
Next we prove that ΛΓ u 0 implies u
0. Suppose to the contrary that u
/ 0.
We note that u
has at most T/21 zeros in T. Otherwise, u
must have two consecutive
zeros in T, and subsequently, u
≡ 0 in 0, T 1 by 1.3. This is a contradiction.
Using 3.6 and the fact that u
has at most T/2 1 zeros in T, it follows that
ΛΓ u T
ut2
λk1 at − λk at − Γtat
t1
atλk1 − λk − Γt
ut2
3.14
t∈T,
ut /
0
> 0,
u 0. Hence, u
0.
which contradicts ΛΓ We claim that there is a constant δ2 δ2 Γ > 0 such that
u ≥ δ2 ΛΓ u2 .
3.15
Assume that the claim is not true. Then we can find a sequence {
un } ⊂ D and u
∈ D,
such that, by passing to a subsequence if necessary,
0 ≤ ΛΓ un ≤
1
,
n
−→ 0,
un − u
un 1,
3.16
n −→ ∞.
3.17
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From 3.17, it follows that
T
T
T
T
2
2
2
2
un t − Δ
ut un t 1 − u
n t − ut 1 − u
t Δ
t1
t1
t1
t1
≤
T T 2
2
un t 1 − u
un t − u
2 t 1 2 t
t1
t1
T
2 |
un t| |
un t 1 − u
t 1| |
ut 1| |
un t − u
t|
t1
−→ 0.
3.18
By 3.12, 3.16, and 3.17, we obtain, for n → ∞,
T
T
ut2 ,
un t2 −→
Δ
λk at Γtat
t1
3.19
t1
and hence
T
T
ut2 ,
ut2 ≤
Δ
λk at Γtat
t1
3.20
t1
that is,
ΛΓ u ≤ 0.
3.21
By the first part of the proof, u
0, so that, by 3.19, Tt1 Δ
un t2 → 0, a contradiction
with the second equality in 3.16.
2
u2 u2 the proof is complete.
Set δ min{δ1 , δ2 } > 0 and observing that u⊥ Lemma 3.2. Let Γ be as in Lemma 3.1 and let δ > 0 be associated with Γ by that lemma. Let ε > 0.
Let p : T → R be a function satisfying
0 ≤ pt ≤ Γt ε.
3.22
Then for all u ∈ D, one has
T 2
Δ2 ut − 1 λk atut ptatut ut u0 t − u
t ≥ δ − εu⊥ .
t1
3.23
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Proof. Using the computations in the proof of Lemma 3.1 and 3.22, we obtain
T Δ2 ut − 1 λk atut ptatut ut u0 t − u
t
t1
T T
2
Δ2 ut − 1 λk atut ut ptat ut u0 t
t1
t1
T Δ2 u
t − 1 λk at
ut ptat
ut −
ut
t1
T Δ2 u0 t − 1 λk atu0 t u0 t
t1
≥
T ut2
ut2 − λk at ptat Δ
3.24
t1
T −Δut2 λk atut2
t1
≥
T T
ut2 − εat
ut2
ut2 − λk at Γtat
Δ
t1
t1
T −Δut2 λk atut2
t1
2
≥ δu⊥ − ε
u 2 .
So that, using 3.7, 3.8, the relation u
t that
N
ik1 ai ψi tbi ϕi t, and Lemma
2.1, it follows
T 2
Δ2 ut − 1 λk atut ptatut ut u0 t − u
t ≥ δ − εu⊥ .
3.25
t1
Proof of Theorem 1.1. The proof is motivated by Iannacci and Nkashama 12.
Let δ > 0 be associated to the function Γ by Lemma 3.1. Then, by assumption 1.8,
there exist Rδ > 0 and b : T → R, such that
gt, u ≤
Γt δ
at|u| bt,
4
3.26
for all t ∈ T and all u ∈ R with |u| ≥ R. Hence, 1.2 is equivalent to
Δ2 ut − 1 λk atut q1 t, ut e1 t, ut ht,
u0 uT ,
u1 uT 1,
3.27
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where q1 and e1 satisfy 2.12 and 2.14 with κ 1. Moreover, by 2.13
q1 t, u ≤
δ
Γt at|u| bt 1,
4
t ∈ T, |u| > max{1, R}.
3.28
Let B > max{1, R}, so that
bt 1 δ
< at,
4
|u|
t ∈ T, |u| > B.
3.29
It follows from 3.28 and 3.29 that
−1
0 ≤ u q1 t, u ≤
δ
at,
Γt 2
t ∈ T, |u| ≥ B.
3.30
Define γ : T × R → R by
⎧
⎪
u−1 q1 t, u,
|u| ≥ B,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ −1 u
u
t,
B
q
B
1
−
Γtat,
0 ≤ u < B,
1
γt, u B
B
⎪
⎪
⎪
⎪
⎪
u ⎪
−1 u
⎪
⎪
1
Γtat, −B < u ≤ 0.
⎩B q1 t, −B
B
B
3.31
So we have
0 ≤ γt, u ≤
δ
Γt at,
2
t ∈ T, u ∈ R.
3.32
Define f : T × R → R
ft, u e1 t, u q1 t, u − γt, uu.
3.33
Then there exists ν : T → 0, ∞ such that
ft, u ≤ νt,
t ∈ T, u ∈ R.
3.34
Therefore, 1.2 is equivalent to
Δ2 ut − 1 λk atut γt, utut ft, ut ht,
u0 uT ,
u1 uT 1.
3.35
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To prove that 1.2 has at least one solution in D, it suffices, according to the LeraySchauder continuation method 15, to show that all of the possible solutions of the family of
equations
Δ2 ut − 1 λk atut 1 − η τatut ηγt, utut ηft, ut ηht,
u0 uT ,
t ∈ T,
u1 uT 1
3.36
in which η ∈ 0, 1, τ ∈ 0, λk1 − λk with τ < δ/4, τ fixed are bounded by a constant K0
which is independent of η and u.
Notice that, by 3.32, we have
δ
0 ≤ 1 − η τat ηγt, u ≤ Γt at,
2
t ∈ T, u ∈ R.
3.37
It is clear that for η 0, 3.36 has only the trivial solution. Now if u ∈ D is a solution
of 3.36 for some η ∈ 0, 1, using Lemma 3.2 and Cauchy’s inequality, we obtain
0
T ut u0 t − u
t
Δ2 ut − 1 λk atut 1 − η τat ηγt, ut ut
t1
T ut u0 t − u
t
ηft, ut − ηht
t1
2
δ ⊥
u u0 ν h,
≥
u − ζ u 2
3.38
where
ζ
√
T
!
mint∈T at
2
.
3.39
So we conclude that
0≥
2
δ ⊥
u − β u⊥ u0 ,
2
3.40
for some constant β > 0, depending only on a, ν and h but not on u or η. Taking α βδ−1 ,
we get
1/2
⊥
.
u ≤ α α2 2αu0 3.41
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We claim that there exists ρ > 0, independent of u and η, such that for all possible
solutions of 3.36
u < ρ.
3.42
Suppose on the contrary that the claim is false. Then there exists {ηn , un } ⊂ 0, 1 × D
with un ≥ n and for all n ∈ N,
Δ2 un t − 1 λk atun t 1 − ηn τatun t ηn gt, un t ηn ht,
un 0 un T ,
un 1 un T 1.
3.43
From 3.41, it can be shown that
0
un −→ ∞,
−1
⊥ 0 −→ 0,
un un 3.44
and accordingly, u⊥n u0n −1 is bounded in D.
Setting vn un /un , we have
Δ2 vn t − 1 λk atvn t τatvn t
gt, un t
ht
ηn τatvn t − ηn
,
ηn
un un vn 0 vn T ,
t ∈ T,
3.45
vn 1 vn T 1.
Define an operator A : D → D by
Awt : Δ2 wt − 1 λk atwt τatwt,
Aw0 : AwT ,
t ∈ T,
3.46
Aw1 : AwT 1.
Then A−1 : D → D is completely continuous since D is finite dimensional. Now, 3.45 is
equivalent to
#
" g·, un ·
h·
ηn τa·vn · − ηn
vn t A ηn
t,
un un −1
t ∈ T.
3.47
By 3.26, it follows that {g·, un ·/un } is bounded. Using 3.47, we may assume that
taking a subsequence and relabeling if necessary vn → v in D, · , ||v|| 1 and v0 vT , v1 vT 1.
On the other hand, using 3.41, we deduce immediately that
⊥
vn −→ 0,
n −→ ∞.
3.48
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Therefore,
vt ak ϕk t bk ψk t,
t ∈ T.
3.49
Rewrite vn vn0 vn⊥ , and let, taking a subsequence and relabeling if necessary,
vn0 −→ v∗ ,
3.50
in D.
Set
I {t ∈ T : v∗ t > 0},
I− {t ∈ T : v∗ t < 0}.
3.51
∅ or I− /
∅.
Since ut /
≡ 0 in T, I /
We claim that
lim un t ∞,
n→∞
lim un t −∞,
n→∞
∀t ∈ I ,
3.52
∀t ∈ I− .
3.53
We may assume that I /
∅, and only deal with the case t ∈ I . The other case can be
treated by similar method.
It follows from 3.50 that
0
vn − v∗ : max vn0 t − v∗ t | t ∈ T −→ 0,
∞
n −→ ∞,
3.54
which implies that for all n sufficiently large,
vn0 t ≥
1 ∗
v t > 0,
2
3.55
∀t ∈ I .
On the other hand, we have from 3.44, 3.55, and the fact ||un || ≥ ||u0n || that there exists
N > 0 such that for n > N and t ∈ I ,
un t u0n t
u⊥n t
un vn0 t
u⊥ t
n
un ≥
1
un vn0 t.
2
3.56
This together with 3.55 implies that for n ≥ N,
un t ≥
Therefore, 3.52 holds.
1
un v∗ t,
4
t ∈ T .
3.57
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Now let us come back to 3.43. Multiplying both sides of 3.43 by vn0 and summing
from 1 to T , we get that
T
2
ht − gt, un t vn0 t.
0 ≤ ηn−1 1 − ηn τ vn0 un 3.58
t1
Combining this with 3.52 and 3.53, it follows that
T
htv∗ t ≥
t1
g tv∗ t vt>0
g− tv∗ t.
3.59
vt<0
However, this contradicts 1.11.
Example 3.3. By 16, the eigenvalues and eigenfunctions of
Δ2 yt − 1 λyt 0,
y0 y7,
3.60
y1 y8
can be listed as follows:
λ0 0,
ϕ0 1,
λ1 2 − 2 cos
2π
,
7
ψ1 t sin
2πt
,
7
ϕ1 t cos
2πt
,
7
λ2 2 − 2 cos
4π
,
7
ψ2 t sin
4πt
,
7
ϕ2 t cos
4πt
,
7
λ3 2 − 2 cos
6π
,
7
ψ2 t sin
6πt
,
7
ϕ2 t cos
6πt
.
7
3.61
Let us consider the nonlinear discrete periodic boundary value problem
Δ2 yt − 1 λ1 yt g t, yt ht,
y0 y7,
3.62
y1 y8,
where
" # π
s
5 gt, s λ2 − λ1 · sin
·
s
,
t
7
2 1 s2
t, s ∈ T × R.
3.63
Obviously, g t ∞, g− t −∞, and dimM2 2. If we take that
" #
π
5 ,
t
Γt λ2 − λ1 · sin
7
2 3.64
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then
Γ1 λ2 − λ1 ;
Γ j < λ2 − λ1 ,
for j 2, . . . , 7.
3.65
Now, it is easy to verify that g satisfies all conditions of Theorem 1.1. Consequently, for any
7-periodic function h : Z → R, 3.62 has at least one solution.
Acknowledgments
This work was supported by the NSFC no. 10671158, the NSF of Gansu Province no.
3ZS051-A25-016, NWNU-KJCXGC-03-17, NWNU-KJCXGC-03-18, the Spring-Sun program
no. Z2004-1-62033, SRFDP no. 20060736001, and the SRF for ROCS, SEM 2006 311.
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