Result.Math. 48 (200S) 310-32S 1422-6383/0S/040310-16 DorIO.1007/s0002S-00S-0203-z © Birkhiiuser Verlag, Basel, 200S
Results in Mathematics
POSITIVE PERIODIC SOLUTIONS OF SYSTEMS OF FIRST ORDER ORDINARY
DIFFERENTIAL EQUATIONS
Donal O'Regan and Haiyan Wang*
Abstract Consider the n-dimensional nonautonomous system x(t) = A(t)G(x(t)) - B(t)F(x(t - r(t))) Let U
=
(Ul, ... ,Un ),
fJ = limllull_o~, f~ = limllull_oo~, i =
max.=l, ... ,nUJ} and Foo ther Fo
= max'=l, ... ,nU~}.
1, ... ,n, F
=
(f', ... ,r), Fo
=
Under some quite general conditions, we prove that ei-
= 0 and F = 00, or Fo = 00 and F = 0, guarantee the existence of positive periodic solutions 00
00
for the system for all A > O. Furthermore, we show t.hat Fo
= F = 0, or Fo = F = 00 guarantee the 00
00
multiplicity of positive periodic solutions for the system for sufficiently large, or small A, respectively. We also establish the nonexistence of the system when either Fo and F 00
> 0,
or Fo and F 00
< 00 for
sufficiently large, or small A, respectively. We shall use fixed point theorems in a cone.
Keywords: positive periodic solutions, existence, fixed point theorem.
1
Introduction
The existence of periodic solutions of the equation of the form
x'(t)
=
a(t)g(x(t» - >"b(t)f(x(t -
ret»~).
(Ll)
and its generalizations have attracted much attention in the literature. See, e.g., Chow [1], Hadeler and Tomiuk [5], Kuang [8, 9J, Kuang and Smith [lOJ, Tang and Kuang [12J. The equation of the form (1.1) has 2000 Mathematics Subject Classification. Primary 34K13 • Corresponding author
O'Regan and Wang
311
been proposed as models for a variety of population dynamics and physiological processes such as production of blood cells, respiration, and cardiac arrhythmias, See, for example, the above references, and [4, 11, 16]. Motivated by multiple-species ecological models, it is natural to explore nonautonomous n-dimensional systems. Nonautonomous systems are more realistic since real-world models often require us to incorporate temporal inhomogeneity in the models. One of the methods of incorporating temporal nonuniformity of the environments in models is to assume that the parameters are periodic with the same period of the time variable. In this paper, we shall study the existence of positive w-periodic solutions for the nonautonomous n-dimensional system (1.2)
x(t) = A(t)G(x(t)) - >.B(t)F(x(t - T(t))),
where A(t) G(x)
= diag[al (t), a2(t), ... ,an(t)], B(t) = diag[bl (t), b2(t), . .. ,bn(t)], F(x) = [fl (x), F(x), ... ,r(xW,
= [gl(X),g2(X), ... ,gn(x)]T and>' >
°is a positive parameter.
In(1.2), we assume that (HI) ai, bi E C(IR, [0,00)) are w-periodic functions such that T
= 1, ... , n.
E C(IR, JR.) is an w-periodic function.
(H2) fi: JR.+ (H3)
J;;' ai(t)dt > 0, J;;' bi(t)dt > 0, i
fi(U)
-->
[0,00) is continuous, gi : JR.+
-->
°
[I, L], < 1 < L < 00 is continuous, i
= 1, ...
,n.
°
> for Ilull > 0, i = 1, ... , n.
In order to state our theorems, we introduce some notation. Let
fi = lim fi(u) o
Iiuli-+o
Fo =. max
t.=l, ... ,n
Ilull '
un,
i I 00=
l'
1m
Iiuli-+oo
Ji(u) nn' 1 -II-II ,uE,,+, l = , ... ,n U
t.=l, ... ,n
A solution u(t) = (UI(t), ... , un(t)) is positive if, for each i = 1, ... , n, Ui(t) ~
at least one component which is positive on lR.. Our main results are:
(1.3)
Foo = . max u~}·
°for all
t E IR and there is
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Theorem 1.1 Assume (H1)-(H2) hold. (a). If Fo
= 0 and F = 00, then for all >. > 0 (1.2) has a positive w-periodicsolution.
(b). If Fo
= 00 and F = 0, then for all >. > 0 (1.2) has a positive w-periodic solution.
00
00
Theorem 1.2 Assume (H1)-(H3) hold. (a). If Fo
= 0 or F = 0, then there exists a >'0 > 0 such that (1.2) has a positive w-periodic solution for 00
>. > >'0' (b). If Fo
= 00 or F = 00, then there exists a >'0 > (} such that (1.2) has a positive w-periodic solution for 00
o < >. < >'0. (c). If Fo = F 00 = 0, then there exists a >'0 > 0 such that (1.2) has two positive w-periodic solutions for
>. > >'0' (d). If Fo = F 00 =
00,
then there exists a >'0 > 0 such that (1.2) has two positive w-periodic solutions for
o < A < AO. (e). If Fo <
00
and F 00 < 00, then there exists a >'0 > 0 such that for all 0 < >. < >'0 (1.2) has no positive
w-periodic solution. (f). If Fo > 0 and F 00 > 0, then there exists a >'0 > 0 such that for all >. > >'0 (1.2) has no positive w-periodic solution.
For n = 1, the existence, multiplicity and nonexistence of positive w-periodic solution of (1.2) with a parameter>' was discussed in Wang [14]. Jiang, Wei and Zhang [6] obtained some existence results for the case when when gi == 1, i
= 1, ... , n.
In a recent paper, Wang, Kuang and Fen [15] proved multiplicity and
nonexistence results for a similar equation when gi == 1, i
= 1, ... , n.
This paper is organized in the following ways. In Section 2, we transform (1.2) into a system of integral equations, and then to a fixed point problem of an equivalent operator in a cone. FUrther, we establish two inequalities which allow us to estimate the operator. In Section 3, we apply the fixed point index to show the existence, multiplicity and nonexistence of positive w-periodic solutions of (1.2) based on the inequalities.
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O'Regan and Wang
2
Preliminaries
In this section, we recall some concepts and conclusions on the fixed point index in a cone in [2, 3, 7J. Let E be a Banach space and K be a closed, nonempty subset of E. K is said to be a cone if (i) au + {3v E K for all u, v E K and all a, {3 ;:: 0 and (ii) u, -u E K imply u = O. Assume 0 is a bounded open subset in E with the boundary 80, and let T : K n n -> K is completely continuous such that Tx the fixed point index itT, K n 0, K) is defined. If itT, K n 0, K)
#
#x
for x E 00 n K, then
0, then T has a fixed point in K n O.
The following well-known result on the fixed point index is crucial in our arguments. Lemma 2.1 ([2,3,7]). Let E be a Banach space and K a cone in E. Forr > 0, define Kr = {u E K : IIxll < r}. Assume that T : K r
(i) If IITxl1 (ii) If IITxl1
;:: Ilxll
->
K is completely continuous such that Tx ~ x for x
E
oKr = {u
E
K : Ilxll = r}.
for x E 8Kro then itT, K" K) = O.
5: Ilxll for x E 8Kro then itT, K" K) = 1.
In order to apply Lemma 2.1 to (1.2), let X be the Banach space defined by
X with a norm
lIuli
n
=
L
sup
;=1 tE[O,wj
= {u(t) E C(lR, lRn) : u(t + w) = u(t), t E lR}
lu;(t)l, for u = (Ul' ... , un) EX. For u E X or lRf., Ilull denotes the norm of u
in X or lRf., respectively. Define K =
{u = (Ul, ... ,un) EX: Ui(t)
;:: O'f(1 - ~D sup IUi(t)l, i = 1, ... , n, t E [O,wJ,} 1 - O'i tE[O,wj
where O'i = e- J; a,(t)dt, i = 1, ... , n. It is clear K is cone in X. For r
> 0, define Or
=
{u E K: Ilull < r}.
It is clear that 80r =
{u E K: Ilull = r}. Let T,\ : K
->
X
be a map with components (Tl, ... , 11'):
T,{u(t)
=
r
Ait
Hw
Gi(t, s)bi(s)fi(u(s - r(s)))ds, i
where
e- It' a,(8)g'(u(8))d8
Gi(t, s)
= 1 _ e- I; a,(8)g'(u(8))d8
=
1, .. . ,n,
(2.1)
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314
Note that aL 1 -1 • L ~Gi(t,s)~-I--I' t~s~t+w,i=I, ... ,n.
-ui
-O"i
c K and TA : K
Lemma 2.2 Assume (Hl)-(H2) hold. Then TA(K)
--->
K is continuous and completely
continuous.
PROOF
In view of the definition of K, for u E K, we have, i
(T{u)(t + w)
t+ t+w
= AI =A
2w
I
t+w
t
I t+
w
Gi(t + W, s)bi(s)Ji(u(S - r(s)))ds
Gi(t + w, 0 + w)bi(O + w)fi(u(O + w - r(O + w)))dO Gi(t, s)bi(s)fi(u(S - r(s)))ds
=
A
=
(l1 u )(t).
t
= 1, ... , n,
It is easy to see that ftt+w bi(s)fi(u(s-r(s)))ds is a constant because ofthe periodicity of b;(t)J'(u(t-r(t))). Notice that, for u E K and t E [D,w], i = 1, ... , n, aL
l1u(t) 2: ~A 1-
=
a;
ar LA
1 - a;
It+w bi(s)J'(u(s - r(s)))ds t
r bi(s)fi(U(S - r(s)))ds
10
a!) =aI,(1 ' - L' 1 - ai
2: ar(1 -
~D
1 - ai
Thus TA(K)
1
l
IA
1 - ai
0
w
.
b;(s)f'(u(s - r(s)))ds
sup IT{u(t)l. tE[O,w]
c K and it is easy to show that TA : K
-+
K is continuous and completely continuous.
0
Lemma 2.3 Assume that (Hl)-(H2) hold. Then u E K is a positive periodic solution of (1.2) if and only if it is a fixed point of TA in K.
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O'Regan and Wang
PROOF If u
= (UJ,"" Un) u;(t)
E
K and T>.u = u, then, for i = 1, ... ,n,
=
d It+w diY' Gi(t, s)bi(s)t(u(s - T(s»)ds)
=
AGi(t, t + w)bi(t + w)fi(u(t + w - T(t + w» - AGi(t, t)bi(t)fi(U(t - T(t»)
t
+ ai(t)gi(u(t»T{u(t) =
A[Gi(t, t + w) - Gi(t, t)]bi(t)t(u(t - T(t»)
=
ai(t)l(u(t»Ui(t) - Abi(t)fi(u(t - T(t»).
+ ai(t)gi(u(t»T{u(t)
Thus u is a positive w-periodic solution of (1.2). On the other hand, if u = (UJ,"" un) is a positive w-periodic function of (1.2), then Abi(t)fi(u(t - T(t») = a,(t)gi(u(t»Ui(t) - ui(t) and
T{u(t)
=
I t+ I t+ A
t
w
=
t
W
Gi(t,s)bi(s)fi(U(S - T(s»)ds Gi(t, s)(ai(s)gi(u(s»Ui(S) - u;(s»ds
I t+w Gi(t, s)ai(s)gi(u(s»Ui(S)ds - It+w Gi(t, s)u:(s)ds t+w = I Gi(t, s)ai(s)gi(u(s»u,(s)ds - Gi(t, s)u,(s)11+ t + I w Gi(t, s)ai(s)gi(u(s»Ui(S)ds =
t
t
w
t
-
t
= Ui(t).
Thus, T>.u = u, Furthermore, in view of the proof of Lemma 2.2, we also have Ui(t) 2': for t E [D,w]. That is, u is a fixed point of T>. in K. Define
r
= mini=J, ... ,n{l~tf
C7tJ~~fl) SUPtE[O,w) Ui(t)
0
Jaw b,(s)ds}mini=l, ... ,n{C7tJ~~fl)} > Dand we have the following lemma.
Lemma 2.4 Assume that (H1)-(H2) hold. For any TJ
> D and u = (uJ,"" un) E K, if there exists a
component fi of F such that fi(u(t» 2': ~j=J Uj(t)TJ for t E [D,w], then
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316
PROOF
Since u
E
K and ji(U(t)) 2: E7=1 Uj(t)T} for t
For each i = 1, .. , n, let ji(t) : IR+
->
E
[O,wj, we have
IR+ be the function given by
p(t) = max{fi(u) : u E IR~ and Ilull :::; t}.
Lemma 2.5 ([13]) Assume (H2) holds. Then
jJ = fJ and jfx, = ffx" i = 1, ... , n.
Lemma 2.6 Assume (H1)-(H2) hold and let r > o. If there exits an c > 0 such that p(r):::; cr, i = 1, ... ,n, then
n where C- = Ei=1
PROOF
R1 J.w0 bi(s)ds.
From the definition of T
A,
for u
:::;
n
E
anT! we have
r
1
~ 1 - 0"1 A Jo b;(s)ji(r)ds
1 l : :; L ---=---r 1 n
i=1
0";
0
w
b;(s)dsAcllull
= ACcllull·
o The following two lemmas are weak forms of Lemmas 2.4 and 2.6.
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O'Regan and Wang
Lemma 2.7 Assume (HI )-(H3) hold. If u E ann r > 0, then IIT>.ull ~ where
uL
Amr i=~~,)I-'uf
r
Jo bi(s)ds}
mr = min{fi(u): u E lR+ and ur::; Ilull ::; r, i = 1, ... ,n.} > 0, and u = rnini=1, ... ,n{0'\~;;I)}.
PROOF Note r
=
Ilull
mr for t'E
fi(U(t)) ~
= L:~=1 sup[O,w] IUi(t)1
[O,w], i
= 1, ... ,n.
~ L:~=1 inf[o,w] IUi(t)1 ~ u L:~=1 sup [O,w] IUi(t)1
= ur.
Thus
A slight modification of the proof in Lemma 2.4 yields the result.
o Lemma 2.8 Assume (Hl)-(H3) hold. If u E ann r > 0, then
where
Mr
=
max{fi(u) : u E lR+ and Ilull ::; r, i = 1, ... , n} > 0 and
C is the positive constant defined in
Lemma 2.6 PROOF Since Ji(u(t)) ::; guarantees the result.
3
for t E [O,w], i
= 1, ... , n,
a slight modification of the proof in Lemma 2.6
0
Proof of Theorem 1.1
PROOF i
Mr
Part (a). F o
= 1, ... , n.
= 0 implies
that fj
Therefore, we can choose r1
= 0, i = 1, ... , n.
It follows from Lemma 2.5 that jj
= 0,
> 0 so that ji(rd ::; Cr1, i = 1, ... , n, where the constant c > 0
satisfies
AcC < 1, and
C is the positive constant defined in Lemma 2.6.
We have by Lemma 2.6 that
IIT>.ull ::; AcC11u11 < Ilull Now, since F 00 such that
= 00,
for
u E
anr,.
there exists a component fi of F such that f;'"
= 00.
Therefore, there is an
if > 0
318
for
u=
O'Regan and Wang
(Ul' ... , un) E lR+ and
Let r2 = max{2rl'
Ilull ~ iI , where 1/ > 0 is chosen so that
~iI}, where a = mini=I •...• n{'Ttl~~f:)}. If u = (Ul' ... ,un) E anr" then n
min '" Ui(t) ~ allull = a
-
i=l
ar2 > iI, -
which implies that n
fi(U(t)) ~ 1/ ~ Ui(t) for t E [O,w]. i=l
It follows from Lemma 2.4 that
By Lemma 2.1,
It follows from the additivity of the fixed point index that
Thus, i(T>.,n r ,
\
nr"K)
i- 0, which implies T>.
of the fixed point index. The fixed point u E nr , Part (b). If Fa =
00,
has a fixed point u E n r , \
there exists a component
n
r1
u = (Ul' ... , Un) E lR+
and
lIull
~ rl,
Ji such that fj =
00.
where 1/ > 0 is chosen so that
n
fi(u(t)) ~ 1/ ~ Ui(t), for t E [0, w]. i=1
n
r,
by the existence property
is the desired positive solution of (1.2).
that
for
\
Therefore, there is an rl > 0 such
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O'Regan and Wang
Lemma 2.4 implies that
We now determine i
nr,.
Foe
= 0 implies that f':x, = 0, i = 1, ... ,n.
= 1, ... , n. Therefore there is an r2 > 2rl
It follows from Lemma 2.5 that
j':x, = 0,
such that
where the constant c > 0 satisfies
>..cC < 1, and
C is the positive constant defined in Lemma 2.6.
Thus, we have by Lemma 2.6 that
By Lemma 2.1, i(T)"nr"K) = 0 and i(T)"nr"K) = 1.
It follows from the additivity of the fixed point index that i(T)" point in
4
nr2 \ Or"
nr,
which is the desired positive solution of (1.2).
\ Or" K) = 1. Thus, T), has a fixed
0
Proof of Theorem 1.2
PROOF Part (a). Fix a number rl > O. Lemma 2.7 implies that there exists a >"0 > 0 such that IIT),ull> Ilull, for .u E an,,,>,, > >"0. If Fa
= 0, then f~ = 0, i = 1, ... , n.
It follows from Lemma 2.5 that
jJ = 0, i = 1, ... , n. Therefore, we can choose 0
< r2 < rl so that
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320
where the constant c > 0 satisfies AcC and
C is the positive constant defined in Lemma 2.6.
If F 00
= 0, then 1/:0 = 0, i = 1, ... , n.
an
> 2Tl such that
T3
where the constant c
We have by Lemma 2.6 that
It follows from Lemma 2.5 that
j/:o = 0, i = 1, ... , n. Therefore there is
> 0 satisfies AcC
and
< 1,
C is the positive constant defined in Lemma 2.6. IIT),ull
< 1, Thus, we have by Lemma 2.6 that
:s; AcCliul1 < Ilull
for
u E
ao r3 ·
It follows from Lemma 2.1 that
i(T)"Or"K) Thus i(T)" Or, \
nr" K)
OT3 \ nT, according to Fo
= 0,
= -1 and i(T)"Or3 \
nr" K)
= 0 or F 00 = 0, respectively.
Part (b). Fix a number
Tl >
=1
and i(T)"Or3,K)
< Ilull,
for u E
ao
T"
0
where
T}
0 such that
Therefore, there is a positive number T2
> 0 is chosen so that ArT} > 1.
>'0>
< A < Ao.
such that
:s; T2,
nr• or
Consequently, (1.2) has a positive solution for A > AO'
= 00, there exists a component Ii of F such that IJ = 00.
for u = (Ul, ... , un) E lR'i- and Ilull
= 1.
= 1. Hence, T), has a fixed point in Or, \
O. Lemma 2.8 implies that there exists a IIT),ull
If Fo
i(T)"Or.,K)
< Tl
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O'Regan and Wang
Then n
fi(U(t)) ~ TJ
L Ui(t), i=l
for u = (U1' ... , Un) E anr>! t E [0, w]. Lemma 2.4 implies that
If F 00 =
for
u=
Let
1'3
00,
there exists a component fi of F such that f!x, =
(U1' ... , un)
=
E
lR+. and
Ilull
~
00.
Therefore, there is an
iI > 0 such that
iI , where TJ > 0 is chosen so that
= mini=1, ... ,n{ut1~;!:)}' If u = (U1' ... ,Un) E an r., then
max{2T1'~}, where a
n
min "Ui(t) ~ allull = aT3 ~ o
iI,
i=l
which implies that n
fi(u(t)) ~ TJ
L Ui(t) for t E [O,w]. i=l
It follows from Lemma 2.4 that
It follows from Lemma 2.1 that
or
nr • \ nr ,
according to Fo =
00
or F 00 =
00,
respectively. Consequently, (1.2) has a positive solution for
o < A < AO. Part (c). Fix two numbers 0 < 1'3 < 1'4. Lemma 2.7 implies that there exists a AO for A > AO,
IIT>.ull > Ilull,
for
u E an r "
(i = 3,4).
> 0 such that we have,
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322
Since Fo = 0 and F 00 = 0, it follows from the proof of Theorem 1.2 (a) that we can choose 0
< rl < r3/2
and rz > 2r4 such that
IITAul1 < Ilull, for u E anr" (i = 1,2). It follows from Lemma 2.1 that
and
i(TA,nr.,K) =0, i(TA,nr.,K) =0 and hence, i(TA,nr • \nr"K) = -1 and i(TA,nr , \nr.,K) = 1. Thus, TA has two fixed points Ul(t) and
U2(t) such that Ul(t) E
nr •
\
n
and uz(t) E nr ,
r}
\
nr. , which are the desired distinct positive periodic
solutions of (1.2) for A > Ao satisfying
Part (d). Fix two numbers 0 < r3
< r4. Lemma 2.8 implies that there exists a Ao > 0 such that we have,
for 0 < A < AO,
IITAul1 < lIull, for u E anr Since Fo and r2
;!
(i = 3,4).
= 00 and F = 00, it follows from the proof of Theorem 00
1.2 (b) that we can choose 0 < rl < r3/2
> 2r4 such that
It follows from Lemma 2.1 that
and
i(TA,nr.,K) = 1, i(TA,nr.,K) = 1 and hence, i(T A, nr •
\
nr "
uz(t) such that Ul(t) E
K)
= 1 and i(TA, n r , \ nr • , K) = -1.
nr • \ nr }
and U2(t) E
nr , \ nr •
,
Thus, T A has two fixed points Ul (t) and
which are the desired distinct positive periodic
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O'Regan and Wang
solutions of (1.2) for 0
Part (e). Since Fo
< A < AO satisfying
<
00
and F 00
< 00, then
f~
< 00 and
f~
< 00,
i = 1, ... , n. It is easy to show (see
[13]) that there exists an e > 0 such that fi(U) :s:; ellull for u
R+, i =
E
1, ... , n.
Assume v(t) is a positive solution of (1.2) . We will show that this leads to a contradiction for 0 < A < AO, where
AO =
n
1
1
rw
2:i=1 R
Jo bi(s)dse
.
In fact, for 0 < A < AO, since TAV(t) = v(t) for t E [O,w], we find IIvll
IIT>.vll n
"" max T{v(t) LO
t; n
:s:;
1 1 - O"j
{w
10
bi(s)dSAellvll
< Ilvll, which is a contradiction. Part (f). Since Fo > 0 and F 00 > 0, there exist two components fi and fi of F such that f& > 0 and
fix, > O. It is easy to show
(see [13]) that there exist positive numbers
1},
TI such that (4.2)
and
(4.3) here
0"
u!'(l-u!)
= mini=I, ... ,n{~} . Assume v(t) = (VI, ... , v n) is a positive solution of (1.2). We will show that
this leads to a contradiction for A > AO =
f,;.
In fact, if Ilvll :s:; TI, (4.2) implies that n
fi(v(t)) ~ 1}
L Vi(t) , i=l
for t E [O,wj.
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O'Regan and Wang
On the other hand, if
Ilvll > 1'1, then n
min ' " Vi(t) ~ o
CTllvll > CTT1,
i=l
which, together with (4.3), implies that n
fj(v(t)) ~
1] Lv;(t),
for t E [O,wl·
i=l
Since
T>. v(t) = v(t) for t E [0, wj, it follows from Lemma Ilvll
which is a contradiction.
2.4 that, for A > AO,
IIT>.vll
0
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(Polish) Mat. Stos. 6 (1976), 23-40 . Department of Mathematics National University of Ireland Galway, Ireland Email address:
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Department of Mathematical Sciences and Applied Computing Arizona State University Phoenix, AZ 85069-7100, U.S .A. E-mail address:
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