Volume 2010, Article ID 626054,18pages doi:10.1155/2010/626054
Research Article
Existence of Positive Solutions of
Nonlinear Second-Order Periodic Boundary
Value Problems
Ruyun Ma, Chenghua Gao, and Ruipeng Chen
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma,ruyun [email protected]
Received 31 August 2010; Revised 30 October 2010; Accepted 8 November 2010
Academic Editor: Irena Rach ˚unkov´a
Copyrightq2010 Ruyun Ma et al. 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.
This paper is devoted to study the existence of periodic solutions of the second-order equationx ft, x, wheref is a Carath´eodory function, by combining a new expression of Green’s function together with Dancer’s global bifurcation theorem. Our main results are sharp and improve the main results by Torres2003.
1. Introduction
Let us say that the following linear problem:
xatx0, t∈0, T, 1.1
x0 xT, x0 xT 1.2
isnonresonant when its unique solution is the trivial one. It is well known that1.1,1.2
is nonresonant then, provided that his a L1-function, the Fredholm’s alternative theorem
implies that the inhomogeneous problem
xatxht, t∈0, T,
always has a unique solution which, moreover, can be written as
xt T
0
Gt, shsds, 1.4
whereGt, sis the Green’s function related to1.1,1.2. In recent years, the conditions,
H Problem1.1, 1.2is nonresonant and the corresponding Green’s function Gt, sis positive on0, T×0, T;
H− Problem1.1, 1.2is nonresonant and the corresponding Green’s function Gt, sis negative on0, T×0, T,
have become the assumptions in the searching for positive solutions of singular second-order equations and systems; see for instance Chu and Torres2, Chu et al.3, Franco and Torres
4, Jiang et al. 5, and Torres6. Moreover, the positiveness of Green’s function implies that an antimaximum principle holds, which is a fundamental tool in the development of the monotone iterative technique; see Cabada et al.7and Torres and Zhang8.
The classical condition implyingHis anLp-criteria proved in Torres9 and based on an antimaximum principle given in8. For the sake of completeness, let us recall the following result.
For any 1≤α≤ ∞, letKαbe the best Sobolev constant in the inequality
Cu2
α≤ u˙ 22, u∈ H:H010, T 1.5
given explicitly bysee10
Kα inf u∈H\{0}
u˙ 22
u2
α ,
Kα ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
2π αT12/α
2 2α
1−2/α Γ1/α
Γ1/21/α 2
, 1≤α <∞,
4
T, α∞.
1.6
Throughout the paper, “a.e.” means “almost everywhere”. Givena∈L10, T, we writea0
ifa ≥ 0 for a.e.t ∈ 0, Tand it is positive in a set of positive measure. Similarly,a ≺ 0 if
−a0.
Theorem Asee9, Corollary 2.3. Assume thata∈Lp0, Tfor some1 ≤ p ≤ ∞witha 0
and moreover
ap< K 2p∗, T
1.7
with1/p1/p∗1. Then Condition (H) holds.
Theorem Bsee9, Theorem 2.2. Assume thata∈ Lp0, Tfor some1 ≤p ≤ ∞witha ≺0.
Then Condition (H−) holds.
To study the existence and multiplicity of positive solutions of the related nonlinear problem
xatxgt, x, t∈0, T,
x0 xT, x0 xT,
1.8
it is necessary to find the explicit expression ofGt, s.
Letϕbe the unique solution of the initial value problem
ϕatϕ0, ϕ0 1, ϕ0 0, 1.9
and letψbe the unique solution of the initial value problem
ψatψ0, ψ0 0, ψ0 1. 1.10
Let
D:ϕT ψT−2. 1.11
Atici and Guseinov 11 showed that the Green’s function Gt, s of 1.1, 1.2 can be explicitly given as
Gt, s ψT
D ϕtϕs− ϕT
D ψtψs
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ψT−1
D ϕtψs−
ϕT−1
D ϕsψt, 0≤s≤t≤T, ψT−1
D ϕsψt−
ϕT−1
D ϕtψs, 0≤t≤s≤T.
1.12
Torres9also studied the Green’s functionGt, sof1.1,1.2. Letube the unique solution of the initial value problem
uatu0, u0 0, u0 1, 1.13
and letvbe the unique solution of the initial value problem
vatv0, vT 0, vT −1. 1.14
Let
α: 1
Then the Green’s functionKof1.1,1.2given in9is in the form
Kt, s αut vt− 1 v0
⎧ ⎪ ⎨ ⎪ ⎩
vtus, 0≤s≤t≤T,
vsut, 0≤t≤s≤T.
1.16
However, there is a mistake in1.16.
It is the purpose of this paper to point out that the Green’s function in1.16, which is induced by the two linearly independent solutionsuandvof1.13and1.14, should be corrected to the form
Gt, s αus vs
v0 ut vt− 1
v0
⎧ ⎪ ⎨ ⎪ ⎩
vtus, 0≤s≤t≤T,
vsut, 0≤t≤s≤T.
1.17
This will be done in Section 2. Finally in Section 3, we study the existence of one-sign solutions of the nonlinear problem
xft, x, t∈0, T,
x0 xT, x0 xT.
1.18
The proofs of the main results are based on the properties of G and the Dancer’s global bifurcation theorem; see12.
2. Preliminaries
Denote
Λ−:{a∈Lp0, T:a≺0},
Λ :a∈Lp0, T:a0, a
p< K 2p∗
for some 1≤p≤ ∞.
2.1
Recall thatuis a unique solution of IVP1.13andvis a unique solution of IVP1.14.
Lemma 2.1. Leta∈Lp0, T. Then
uT v0. 2.2
Proof. Since the WronskianWu, vtis constant, it follows that
−uT uT vT uT vT
ut vt ut vt
u0 v0
u0 v0
−v0. 2.3
Lemma 2.2. LetGt, sbe the Green’s function of 1.1,1.2. Then
iG:0, T×0, T → Ris continuous;
iifor a givens∈0, T,G0, s GT, s;
iiifor a givens∈0, T,Gt0, s GtT, s;
ivfor a givens ∈0, T,Gt, sas a function oftis a solution of 1.1in the intervals
0, sands, T.
Lemma 2.3. Leta∈ Λ∪Λ−. Then the Green’s functionGt, sinduced byuandvis explicitly given by1.17, that is,
Gt, s us vs
v02v0−uTut vt−
1
v0
⎧ ⎨ ⎩
vtus, 0≤s≤t≤T,
vsut, 0≤t≤s≤T.
2.4
Remark 2.4. Notice that it is not necessary to assume that
v0/0. 2.5
In fact, ifa∈Λ, then from13, Remark in Page 3328, we have
λ1a≥π T
2
1− ap
K 2p∗
>0, 2.6
whereλ1ais the first eigenvalue of the antiperiodic boundary value problem
x λatx0, x0 −xT, x0 −xT. 2.7
Now, by the same method to prove8, Lemma 2.1, we may get that the solutionvof the IVP
1.14has at most one zero in0, T. SincevT 0, we must have thatv0/0.
Ifa∈Λ−, we claim thatvt>0 fort∈0, T. Suppose on the contrary that there exists
τ ∈0, Tsuch that
vτ 0, vt>0, fort∈τ, T. 2.8
Then
vt −atvt≥0, t∈τ, T, 2.9
which means that
vt≥T−t, t∈τ, T. 2.10
Proof ofLemma 2.3.In the proof of9, Proposition 2.0.1, the Green function was assumed to have the form
Kt, s αut βvt− 1 v0
⎧ ⎨ ⎩
vtus, 0≤s≤t≤T,
vsut, 0≤t≤s≤T.
2.11
However, for aboveKt, s, it is impossible to find constantsαandβ, such that
Kt0, s KtT, s, s∈0, T. 2.12
So, we have to assume that the Green’s function is of the form
Gt, s αsut βsvt− 1 v0
⎧ ⎨ ⎩
vtus, 0≤s≤t≤T,
vsut, 0≤t≤s≤T.
2.13
ByLemma 2.2ii, we have thatG0, s GT, sfors∈0, T. Thus
βsv0 G0, s GT, s αsuT, s∈0, T, 2.14
which together with2.2imply that
βs αs, s∈0, T. 2.15
From2.13and2.15, we have
Gtt, s αs
ut vt− 1 v0
⎧ ⎨ ⎩
vtus, 0≤s < t≤T,
vsut, 0≤t < s≤T, 2.16
and, fors∈0, T,
Gt0, s αs
1v0− vs
v0, GtT, s αs
uT−1 us
v0. 2.17
Applying this andLemma 2.2iii, it follows that
αs us vs
2v0−uTv0. 2.18
Denote
M: max
Finally, we state a result concerning the global structure of the set of positive solutions of parameterized nonlinear operator equations, which is essentially a consequence of Dancer
12, Theorem 2.
Suppose thatEis a real Banach space with norm·. LetKbe a cone inE. A nonlinear mappingA :0,∞×K → Eis said to be positiveifA0,∞×K ⊆K. It is said to beK -completely continuousifAis continuous and maps bounded subsets of0,∞×Kto precompact subset ofE. Finally, a positive linear operatorV onEis said to be a linear minorantforAif
Aλ, u≥λVxforλ, u∈0,∞×K. IfBis a continuous linear operator onE, denoterB
the spectrum radius ofB. Define
cKB {λ∈0,∞: ∃x∈K with x1, xλBx}. 2.20
Lemma 2.5see14, Lemma 2.1. Assume that
iKhas a nonempty interior andEK−K;
iiA:0,∞×K → EisK-completely continuous and positive,Aλ,0 0forλ∈R,
A0, u 0foru∈K, and
Aλ, u λBuFλ, u, 2.21
where B : E → Eis a strongly positive linear compact operator on E with rB > 0, and F :
0,∞×K → EsatisfiesFλ, u◦uasu → 0locally uniformly inλ.
Then there exists an unbounded connected subsetCof
DKA {λ, u∈0,∞×K:uAλ, u, u /0} ∪
rB−1,0 2.22
such thatrB−1,0∈ C.
Moreover, ifAhas a linear minorantV, and there exists a
μ, y∈0,∞×K 2.23
such thaty1 andμV y≥y, thenCcan be chosen in
DKA∩ 0, μ
×K. 2.24
3. Main Results
In this section, we consider the existence of positive solutions of nonlinear periodic boundary value problem
xft, x, t∈0, T,
x0 xT, x0 xT, 3.1
3.1.
a
∈
Λ
By Theorem A,a∈ΛimpliesGt, s>0 on0, T×0, T, and subsequentlyM > m >0. Let us define
P:x∈C0, T|xt≥0 on 0, T, min
t xt≥ m Mx∞
. 3.2
Lemma 3.1see9, Theorem 3.2. Let us assume that there exista∈Λand0< r < Rsuch that
ft, x atx≥0, ∀x∈ m Mr, M mR
, a.e. t∈0, T. 3.3
Then3.1has a positive solution provided one of the following conditions holds
i
ft, x atx≥ M
Tm2x, ∀x∈ m
Mr, r
, a.e. t∈0, T,
ft, x atx≤ 1
TMx, ∀x∈
R,M mR
, a.e. t∈0, T;
3.4
ii
ft, x atx≤ 1
TMx, ∀x∈ m
Mr, r
, a.e. t∈0, T,
ft, x atx≥ M
Tm2x, ∀x∈
R,M mR
, a.e. t∈0, T.
3.5
Let
γ∗t: ft,m/Mr atm/Mr
m/Mr , Γ
∗t: ft,M/mR atM/mR
M/mR , 3.6
ft, x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Γ∗tx, x≥ M
mR, ft, x atx, m
Mr ≤x≤ M
mR, γ∗tx, 0≤x≤ m
Mr.
3.7
Let
γt:min
ft, s ats s |s∈
mr M, r
, Γt:max
ft, s ats s |s∈
R,MR
m
,
γt:max
ft, s ats
s |s∈ mr
M, r
, Γt:min
ft, s ats
Theorem 3.2. Assume that
(A1) There exista∈Λ∩C0, Tand0< r < Rsuch that
ft, x atx >0, ∀x∈
m Mr,
M mR
, a.e. t∈0, T. 3.9
Then3.1has a positive solution provided one of the following conditions holds
iμ0γ<1< μ0Γ;
iiμ0Γ<1< μ0γ.
Hereμ0βdenotes the principal eigenvalue of
xatxμβtx, t∈0, T,
x0 xT, x0 xT. P
Remark 3.3. Let a ∈ Λ and β 0. Then μ0β > 0. Moreover, μ0β is simple and the
corresponding eigenfunctionψ0∈intP.
In fact,3.10is equivalent to
xt μ T
0
Gt, sβsxsds:μAxt. 3.10
SinceG > 0 on0, T×0, T, it follows thatAP⊂ intP. From Krein-Rutman theorem, see15, Theorem 19.3, we may get the desired results.
Remark 3.4. Theorem 3.2is a partial generalization ofLemma 3.1. It is enough to prove that the conditionionfinTheorem 3.2holds when the conditioniinLemma 3.1holds.
First, we claim that
iμ0M/Tm2<1;
iiμ01/TM>1.
To this end, let us denote byλ0the principal eigenvalue of the linear problem
uatuλu, u0 uT, u0 uT, 3.11
andϕthe corresponding eigenfunction withϕ∈int P. Then applying the facts thatG≥m
andG /≡m,
λ0 μ0
M Tm2
· M
Tm2, 3.12 ϕ∞≥ϕt
λ0 T
0
Gt, sϕsds
> λ0m m
MTϕ∞,
which together with3.12, imply that
μ0
M Tm2
<1. 3.14
By the same method, with obvious changes, we may show thatμ01/TM>1.
Now, we proveμ0γ<1< μ0Γ.
Define the operatorsS1, S2:C0, T → C0, Tby
S1ut M Tm2
T
0
Gt, susds,
S2ut T
0
Gt, sγsusds,
3.15
respectively.
Sinceγt≥M/Tm2, by15, Theorem 19.3, we getrS
2≥rS1, whererSi,i1,2, is the spectrum radius ofSi. Thus,μ0γ 1/rS2≤1/rS2 μ0M/Tm2<1.
Similarly,μ0Γ≥μ01/TM>1.
Remark 3.5. The conditionsμ0γ<1< μ0Γandμ0Γ<1< μ0γare optimal.
Let, 1, 2be positive constants with1< 2, and
1
81≤at≤ 1
82. 3.16
Let us consider the problem
uatu at u, u0 uT, u0 uT. 3.17
Obviously, forft, s ats at s, we have that
γt Γt at ,
μ0
γμ0
Γμ0at .
3.18
Forj1,2, the principal eigenvalueμ01/8jof
x
1 8j
xμ·
1
8j
·x, t∈0, T,
x0 xT, x0 xT
is
μ0
1
8j
18j
8 j
1. 3.20
Applying the fact that
μ0
1
81
≤μ0
γμ0
Γ≤μ0
1
82
, 3.21
thoughμ0Γis a little bit smaller than 1, the existence of positive solutions of3.17will not
be guaranteed in this case.
Proof ofTheorem 3.2. We only provei.iican be proved by a similar method.
To study the existence of positive solutions of3.1, let us consider the parameterized problem
xatxμft, x, t∈0, T,
x0 xT, x0 xT.
3.22
Notice that
ft, x γ∗txξt, x, ft, s Γ∗tsζt, s, 3.23
with
lim x→0
ξt, x
x 0, slim→∞ ζt, s
s 0, a.e. t∈0, T. 3.24
Thus,3.22can be rewritten as
xatxμγ∗txμξt, x, t∈0, T,
x0 xT, x0 xT. 3.25
Denote
Ex∈C10, T|x0 xT, x0 xT 3.26
equipped with the norm · max{x∞,x∞}. Let
FromLemma 2.5, there exists a continuumCof solutions of3.25joiningμ0γ∗,0
to infinity inΦ. Moreover,C\ {μ0γ∗,0} ⊂Φ.
Now, we divide the proof into two steps.
Step 1. We show thatC joiningμ0γ∗,0toμ0Γ∗,∞inΦ. So, C ∩{1} ×E/∅, and
accordingly,3.25has at least one positive solutionu. Suppose thatηk, yk∈Cwith
ηkyk −→ ∞. 3.28
We firstly show that{ηk}is bounded.
In fact, it follows from the definition offand Condition3.9that
ft, s
s ≥et, a.e. t∈0, T, s∈0,∞ 3.29
for somee∈L10, Twithet>0 a.e. on0, T.
We claim thatykhas to change its sign in0, Tifηk → ∞. In fact,
ykt atykηk
f t, yk
yk
yk 3.30
yields thatykt>0 askis large enough. However, this contradicts the boundary condition
yk0 ykT.
Therefore,{ηk}is bounded. Now,{ηk, yk}k∈Nsatisfy
ykatykηkΓ∗tykηkζ t, yk
, t∈0, T,
yk0 ykT, yk0 ykT.
3.31
Let
vk: yk yk
. 3.32
Then
vkatvkηkΓ∗tvkηk ζ t, yk
yk
vk, t∈0, T,
vk0 vkT, vk0 vkT.
Equation3.33is equivalent to
vkt ηk
T
0 Gt, s
Γ∗sv
ks
ζ s, yks
yks
vks
ds. 3.34
Set
wkt: Γ∗tvkt
ζ t, ykt
ykt vkt. 3.35
Sinceft, ykt Γ∗tyktζt, ykt, it follows from3.7and the factyk>0 on0, Tthat
ζ t, ykt
ykt
≤ |Γ∗t| f t, ykt
ykt
≤ |Γ∗t|γt|Γt||at|max
f t, ykt
ykt
:
m
Mr ≤ykt≤ M
mR
≤ |Γ∗t|γt|Γt||at|maxft, τ
τ : m
Mr ≤τ≤ M mR , 3.36 which implies
ζ t, ykt
ykt
≤σt, t∈0, T 3.37
for some functionσ ∈ L10, T, independent ofk. Thus, it follows from3.9and3.6that {wkt}∞k1is bounded uniformly inC0, T. It is easy to check that{ηk
T
0 Gt, swksds} ⊂ C10, T. This together with the fact that C10, T imbeded compactly intoC0, T implies
that, after taking a subsequence and relabeling if necessary,vk → v∗ inC0, Tfor some v∗∈C0, Tandηk → η∗for someη∗∈0,∞, and using Lebesgue dominated convergence theorem, we get
v∗t η∗ T
0
Gt, sΓ∗sv∗sds. 3.38
This implies thatv∗∈W2,10, Tand
v∗atv∗η∗Γ∗tv∗, t∈0, T,
and subsequently,
η∗μ0Γ∗. 3.40
Therefore,Cjoinsμ0γ∗,0toμ0Γ∗,∞inΦ.
Step 2. We show thatuis actually a solution of3.1. To this end, we only prove that
xatxft, x, t∈0, T,
x0 xT, x0 xT
3.41
has no positive solutionywithy∞< r ory∞> MR/m.
In fact, suppose on the contrary thatyis a positive solution of3.41withy∞ < r.
Then we have from3.7,3.8and the definition offthat
f t, yt
yt ≥γt, t∈0, T. 3.42
Since
yt atyt 1·f t, yt
yt yt, y0 yT, y
0 yT, 3.43
wt atwt μ0
γ·γtwt, w0 wT, w0 wT, 3.44
wherewis the corresponding eigenfunction ofμ0γwithw > 0. Multiplying both sides of
equation in 3.43byw and multiplying both sides of equation in3.44 byy, integrating from 0 toTand subtracting, we get
T
0
f t, yt yt −μ0
γ·γt ytwtdt0, 3.45
which together with3.42implies that
μ0
γ≥1. 3.46
However, this contradicts the assumption thatμ0γ<1.
Next, suppose on the contrary that y is a positive solution of 3.41 with y∞ > MR/m. Then we have from3.7and3.8and the definition offthat
f t, yt
Since
yt atyt 1·f t, yt
yt yt, y0 yT, y
0 yT, 3.48
zt atzt μ0
Γ·Γtzt, z0 zT, z0 zT, 3.49
wherezis the corresponding eigenfunction ofμ0Γwithz >0. Multiplying both sides of the
equation in3.48byzand multiplying both sides of the equation in3.49byy, integrating from 0 toTand subtracting, we get
T
0
f t, yt yt −μ0
Γ·Γt ytztdt0, 3.50
which together with3.47implies that
μ0
Γ≤1. 3.51
However, this contradicts the assumption thatμ0Γ>1.
Let
bt: ft,−m/Mr−atm/Mr
−m/Mr , Bt:
ft,−M/mR−atM/mR
−M/mR ,
! ft, x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Btx, x≤ −M mR,
ft, x atx, −M
mR≤x≤ − m Mr,
btx, x≥ −m Mr.
3.52
Let
bt:min
ft, sats
s |s∈
−r,−mr M
, Bt:max
ft, sats
s |s∈
−MR m ,−R
,
bt:max
ft, sats
s |s∈
−r,−mr M
, Bt:min
ft, sats
s |s∈
−MR m ,−R
.
3.53
Theorem 3.6. Assume that
(H1) There exista∈Λ∩C0, Tand0< r < Rsuch that
ft, x atx <0, ∀x∈
−M mR,−
m Mr
, a.e. t∈0, T. 3.54
Then3.1has a negative solution provided one of the following conditions holds
iμ0b<1< μ0B;
iiμ0B<1< μ0b.
3.2.
a
∈
Λ
−By Theorem B,a∈Λ−impliesGt, s<0 on0, T×0, T, and subsequentlym < M <0. Let us define
P−:x∈C0, T|xt≥0 on0, T, min
t xt≥ M
mx∞
. 3.55
Letμ0βdenote the principal eigenvalue of
xatxμβtx, t∈0, T,
x0 xT, x0 xT. 3.56
Then, it is easy to see from Krein-Rutman theorem thatμ0β>0 provided thata∈Λ−and β≺0. Moreover,μ0βis simple and the corresponding eigenfunctionψ0 ∈intP−.
Let
qt:min
ft, s ats
s |s∈
Mr m , r
, Qt:max
ft, s ats
s |s∈
R,mR M
,
qt:max
ft, s ats s |s∈
Mr
m , r
, Qt:min
ft, s ats s |s∈
R,mR
M
.
3.57
Applying the knowledge of the sign of Green’s function whena∈Λ−and the similar argument to proveTheorem 3.2with obvious changes, we may prove the following.
Theorem 3.7. Let us assume that there exista∈Λ−∩C0, Tand0< r < Rsuch that
ft, x atx <0, ∀x∈
M mr,
m MR
Then3.1has a negative solution provided one of the following conditions holds
iμ0q<1< μ0Q;
iiμ0Q<1< μ0q.
Let
pt:min
ft, sats
s |s∈
−r,−Mr m
, Pt:max
ft, sats
s |s∈
−mR M,−R
,
pt:max
ft, sats s |s∈
−r,−Mr m
, Pt:min
ft, sats s |s∈
−mR
M,−R
.
3.59
Theorem 3.8. Let us assume that there exista∈Λ−∩C0, Tand0< r < Rsuch that
ft, x atx >0, ∀x∈
−m MR,−
M mr
, a.e. t∈0, T. 3.60
Then3.1has a negative solution provided one of the following conditions holds
iμ0p<1< μ0P;
iiμ0P<1< μ0p.
Remark 3.9. Very recently, Zhang 16 studied conditions onaso that the operator Lax xatxadmits the maximum principle or the antimaximum principle with respect to the periodic boundary condition. By exploiting Green’s functions, eigenvalues, rotation numbers, and their estimates, he gave several optimal conditions. The Green’s function in Zhang16
and the one in2.4are same. In fact, in the nonresonance case, Problem1.1,1.2has a unique Green’s function. In the resonance case, Problem1.1,1.2has no Green’s function any more.
Remark 3.10. It is worth remarking that Cabada and Cid 17, and Cabada et al.18have improved theLp-criteria in Torres9to the case thatamay change its sign, and established the similar results for periodic one-dimensionalp-Laplacian problems.
Acknowledgments
The authors are very grateful to the anonymous referees for their valuable suggestions. This paper is supported by the NSFCno. 11061030, NWNU-KJCXGC-03-17, the Fundamental Research Funds for the Gansu Universities.
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