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On the Integral Manifold for a System of Differential Equations with Piecewise Constant Argument
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
201, 75]90 Ž1996.
0242
On the Integral Manifold for a System of Differential Equations with Piecewise Constant Argument Garyfalos Papaschinopoulos School of Engineering, Democritus Uni¨ ersity of Thrace, 67100 Xanthi, Greece Submitted by Jack K. Hale Received October 14, 1994
The existence of an integral manifold for a system of differential equations with piecewise constant argument is studied. Q 1996 Academic Press, Inc.
INTRODUCTION We consider the system of differential equations with piecewise constant argument of the form x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , y Ž t . . y9 Ž t . s A Ž t . y Ž t . q C Ž t . y Ž w t x . q g Ž t , x Ž t . , y Ž t . . ,
Ž 1.
where t g R s Žy`, `., w?x denotes the greatest integer function, B Ž t . is an r = r continuous matrix for t g R, AŽ t ., C Ž t . are k = k continuous matrices for t g R, and f : R = R r = R k ¬ R r , g: R = R r = R k ¬ R k are continuous functions on R = R r = R k , small Lipschitzian in x and y. A pair of functions Ž x Ž t ., y Ž t .., x: Žy`, `. ¬ R r , y: Žy`, `. ¬ R k , is a solution of Ž1. if the following conditions are satisfied: Ži. The functions x Ž t ., y Ž t . are continuous on Žy`, `.. Žii. The derivatives x9Ž t ., y9Ž t . exist on Žy`, `. except possibly at the points t s n, n g Z s . . . , y1, 0, 1, . . . 4 where one-sided derivatives exist. Žiii. Ž x Ž t ., y Ž t .. satisfies Ž1. on every interval w n, n q 1., n g Z. 75 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
76
GARYFALOS PAPASCHINOPOULOS
A function ¨ : R = R r ¬ R k determines an integral manifold for Ž1. if the following statements are true: Ži. ¨ is continuous and bounded on R = R r. Žii. For any solution x Ž t . of the differential equation x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , ¨ Ž t , x Ž t . . . we have that Ž x Ž t ., ¨ Ž t, x Ž t ... is a solution of Ž1.. Moreover if Ž x Ž t ., y Ž t .. is a solution of Ž1. with sup< y Ž t .<, t g R4 - ` it follows that y Ž t . s ¨ Ž t, x Ž t .., t g R. In what follows we denote by < ? < any norm. We say that the linear differential equation z9 Ž t . s A Ž t . z Ž t . ,
t g R,
Ž 2.
where AŽ t . is a k = k continuous matrix for t g R, has an exponential dichotomy on R if there exist a projection P Ž P 2 s P . and constants K G 1, a ) 0 such that < Z Ž t . PZy1 Ž s . < F KeyaŽ tys. , < Z Ž t . Ž Ik y P . Zy1 Ž s . < F KeyaŽ syt . ,
tGs s G t,
Ž 3.
where ZŽ t . is the fundamental matrix solution of Ž2. such that ZŽ0. s Ik , Ik is the k = k indentity matrix, and t, s g R. In this paper under some conditions on the matrices AŽ t ., B Ž t ., C Ž t . and the functions f Ž t, x, y ., g Ž t, x, y . and assuming that Ž2. has an exponential dichotomy Ž3. we prove that there exists an integral manifold ¨ Ž t, x . for Ž1. Žsee Proposition 1 below.. We note that results concerning integral manifolds for equations of the form Ž1., where B Ž t . s 0, C Ž t . s 0, t g R, are included in the papers w1, 5, 7x. We also note that differential equations with piecewise constant argument ŽEPCA. describe hybrid dynamical systems Ža combination of continuous and discrete. and so combine properties of both differential and difference equations. These equations may also have applications in certain biomedical models w2x. Results concerning asymptotic stability and oscillatory behavior of the solutions of differential equations with piecewise constant argument are included in w3, 13, 14x and the references cited therein. In addition w14x contains results on the relationship between equations with piecewise constant argument and impulsive equations.
77
INTEGRAL MANIFOLD
Moreover, the structural stability, the asymptotic behavior of the solutions of a class of differential equations with piecewise constant argument, has been studied in w10]12x.
MAIN RESULTS We prove now our main results. We need three lemmas. The first lemma concerns existence and uniqueness of the solutions of a differential equation with piecewise constant argument. LEMMA 1. such that
Suppose that B Ž s ., s g R, is an r = r continuous matrix on R < B < s sup < B Ž s . < , s g R 4 s d ,
0-d-1
Ž 4.
and h: R = R r ¬ R r is a continuous function on R = R r such that for all s g R, x 1 , x 2 g R r < h Ž s, x 1 . y h Ž s, x 2 . < F 2 p < x 1 y x 2 < ,
Ž 5.
where p is a constant satisfying 0-p-
1yd 4
.
Ž 6.
Then if t g R, x g R r the initial ¨ alue problem x9 Ž s . s B Ž s . x Ž w s x . q h Ž s, x Ž s . . ,
xŽ t. s x
Ž 7.
has a unique solution x Ž s . satisfying the integral equation x Ž s. s LŽ s. x Ž w sx. q
s
Hw s xh Ž u, x Ž u . . du,
s g R,
Ž 8.
where the matrix LŽ s . is defined as L Ž s . s Ir q
s
Hw s xB Ž u . du
Ž 9.
and it is in¨ ertible for all s g R, Ir is the r = r identity matrix. Moreo¨ er if m s w s x, n s w t x, then x m s x Žw s x. satisfies the problem x mq 1 s Dm x m q Hm ,
x n s Ly1 Ž t . x y
ž
t
Hn h Ž u, x Ž u . . du
/
, Ž 10 .
78
GARYFALOS PAPASCHINOPOULOS
where Dm s Ir q
mq1
Hm
Hm s
B Ž u . du,
mq1
Hm
h Ž u, x Ž u . . du.
Ž 11 .
Proof. Using Ž4. and Ž9. Žresp. Ž11.. it is obvious that LŽ s . Žresp. Dm . is invertible for all s g R Žresp. m g Z . and < LŽ s. < F 1 q d ,
< Ly1 Ž s . < F Ž 1 y d . y1 ,
s g R,
Ž 12 .
< Dm < F 1 q d ,
< Dmy1 < F Ž 1 y d . y1 ,
m g Z.
Ž 13 .
For s g R and m s w s x we define LŽ s. s LŽ s. , We prove now that satisfies Ž8. and Ž10. continuous functions w n, n q 1x4 and T n be
s g m, m q 1 . ,
L Ž m q 1 . s Dm .
the problem Ž7. has a unique solution x Ž s . which for s g w n, n q 1x. Let F n be the space of the x n : w n, n q 1x ¬ R r with < x n < s sup< x n Ž s .<, s g the operator defined on F n as
T n x n Ž s . s L Ž s . Ly1 Ž t . x y
ž
q
s
Hn h Ž u, x
n
t
Hn h Ž u, x
n
Ž u . . du
/
Ž u . . du,
Ž 14 .
where n F s F n q 1. It is obvious that T n is in F n. We show now that T n is a contraction on F n. If x 1n, x 2n g F n from Ž5., Ž12., Ž13., and Ž14. we take < T n x 1n Ž s . y T n x 2n Ž s . < F
4p 1yd
< x 1n y x 2n < ,
s g w n, n q 1 x ,
where < x 1n y x 2n < s sup< x 1n Ž s . y x 2n Ž s .<, s g w n, n q 1x4 , which implies that T n is a contraction on F n, since Ž6. holds. Therefore there exists a unique x n g F n such that T n x nŽ s. s x nŽ s. ,
s g w n, n q 1 x .
Ž 15 .
Then from Ž14. and Ž15. it is obvious that x Ž s . s x n Ž s ., s g w n, n q 1x, is the unique solution of Ž7. satisfying Ž8. and Ž10. for s g w n, n q 1x. Working inductively backward we can define the solution x Ž s . of Ž7. in every interval w m, m q 1x, m g . . . , n y 2, n y 14 , by the following procedure:
79
INTEGRAL MANIFOLD
Let F m be the space of the continuous functions x m : w m, m q 1x ¬ R r with < x m < s sup< x m Ž s .<, s g w m, m q 1x4 and T m y be the operator defined on F m as m y1 mq1 Tm x mq1 y y x Ž s . s L Ž s . Dm
ž
q
s
Hmh Ž u, x
m
mq1
Hm
h Ž u, x m Ž u . . du
/
Ž u . . du,
where s g w m, m q 1x and x mq 1 Ž s . is the solution of Ž7. for t g w m q 1, m m q 2x. Arguing as above we can prove that T m y is a contraction on F m m and so T y has a unique fixed point x m g F . Then it is obvious that the function x Ž s . s x m Ž s ., s g w m, m q 1x, m g . . . , n y 2, n y 14 , is the unique solution of the problem Ž7. on the space Žy`, n x which satisfies Ž8. and Ž10.. Working now inductively forward we define the solution x Ž s . of Ž7. in every interval w m, m q 1x, m g n q 1, n q 2, . . . 4 , by using the operator Tm q, m my1 Tm q qx Ž s . s L Ž s . x m
s
Hmh Ž u, x
m
s g w m, m q 1 x ,
Ž u . . du,
where x m g F m , F m has been defined previously, and x my1 Ž s . is the solution of Ž7. for s g w m y 1, m x. Using Ž5. and Ž6. we can easily prove m that T m and let x m g F m be the unique fixed q is a contraction on F m point of T q. Then the function x Ž s . s x m Ž s ., s g w m, m q 1x, m g n q 1, n q 2, . . . 4 , is the unique solution of the problem Ž7. on the space w n q 1, `. which satisfies Ž8. and Ž10.. Thus the proof of the lemma is complete. LEMMA 2. Let B Ž s . be an r = r continuous matrix such that Ž4. holds and h i : R = R r ¬ R r , i s 1, 2, be continuous functions on R = R r such that for all s g R, x 1 , x 2 g R r < h i Ž s, x 1 . y h i Ž s, x 2 . < F 2 p < x 1 y x 2 < ,
i s 1, 2
Ž 16 .
and < h1 Ž s, x 1 . y h 2 Ž s, x 2 . < F 2 p < x 1 y x 2 < q pE,
Ž 17 .
where E and p are constants which satisfy E G 0,
0-p-
1yd 2 eŽ1 q d .
.
Ž 18 .
80
GARYFALOS PAPASCHINOPOULOS
Suppose that b is a positi¨ e constant. Then if x i Ž s ., s g R, is a solution of the problem x9 Ž s . s B Ž s . x Ž w s x . q h i Ž s, x Ž s . . ,
xiŽ t . s x i,
Ž 19 .
where t g R, x i g R r , i s 1, 2, the following relation is satisfied, < x 1 Ž s . y x 2 Ž s . < F N1 < x 1 y x 2 < e b < myn< q p Ž N2 e b < myn < q e . E,
sgR,
Ž 20 .
where m s w s x, n s w t x, N1 s t 2 Ž1 q d . e1q dt , N2 s N1Ž1 q 2 pe .Ž1 q by1ty1 ., d s Ž e b y 1.y1 , t s Ž1 y d y 2 peŽ1 q d ..y1 . Proof. Relations Ž4., Ž16., and Ž18. imply that all the conditions of Lemma 1 are satisfied for Ž19.. Then for i s 1, 2 it holds that x i Ž s. s LŽ s. x i Ž w sx. q
s
s g R.
Ž 21 .
Ž u . < du.
Ž 22 .
i
Hw s xh Ž u, x Ž u . . du, i
Therefore from Ž12., Ž17., and Ž21. we obtain for s g R 2 < < x 1 Ž s . y x 2 Ž s . < F Ž 1 q d . < x 1m y x m
s
Hm < x Ž u . y x
q pE q 2 p
1
2
Applying Gronwall’s lemma w4, p. 19x to Ž22. for s g w m, m q 1x and since from Ž18., 2 p - 1 it follows that 2 < < x 1 Ž s . y x 2 Ž s . < F Ž 1 q d . e < x 1m y x m q peE,
s g R.
Ž 23 .
From Lemma 1 the function x mi , i s 1, 2, satisfies i i i x mq 1 s Dm x m q Hm ,
x ni s Ly1 Ž t . x i y
ž
t
i
Hn h Ž u, x Ž u . . du i
/
where Hmi s
mq1
Hm
h i Ž u, x i Ž u . . du,
i s 1, 2.
Ž 24 . ,
81
INTEGRAL MANIFOLD
Now let s G t, from which m G n. Then from Ž24. we obtain my1
x mi s x ni q
Ý Ž Hki q Ž Dk y Ir . xki . ,
i s 1, 2, m G n.
Ž 25 .
ksn
Using Ž4., Ž11., Ž17., Ž23., and Ž25. we get 2 < < x 1m y x m F < x 1n y x n2 < q p Ž 1 q 2 pe . E Ž m y n .
my1
q Ž 2 ep Ž 1 q d . q d .
Ý
ksn
< xk1 y xk2 < ,
mGn.
Ž 26 .
2 < On setting sm s eyb Ž myn. < x 1m y x m , from Ž26. and the relation Ž m y n. yb Ž myn. y1 ?e - b , m G n, we have
sm F sn q Ž 1 q 2 pe . pby1 E my1
q Ž 2 ep Ž 1 q d . q d .
Ý
ksn
eyb Ž my k .sk ,
mGn.
Ž 27 .
When the discrete Gronwall’s lemma w6, p. 21x is applied to Ž27. and Ž18. is used it follows that 2 < < x 1m y x m F Ž < x 1n y x n2 < q Ž 1 q 2 pe . pby1 E . e d e b Ž myn. ,
m G n. Ž 28 .
Suppose now that s F t and so m F n. Then, ny1
x mi s x ni q
Ý Ž Ž Ir y Dk . xki y Hki . ,
i s 1, 2, n G m.
Ž 29 .
ksm
From relations Ž4., Ž11., Ž17., Ž18., Ž23., and Ž29. we obtain 2 < < x 1m y x m F t < x 1n y x n2 < q t p Ž 1 q 2 pe . E Ž n y m .
ny1
q t Ž 2 ep Ž 1 q d . q d .
Ý
ksmq1
< xk1 y xk2 < ,
mFn y 1.
Ž 30 . 2 < Moreover, if nm s eyb Ž nym. < x 1m y x m , then from Ž30. and the relation Ž n y m. eyb Ž nym. - by1, n G m, we get
nm F tnn q t p Ž 1 q 2 pe . by1 E ny1
q t Ž 2 ep Ž 1 q d . q d .
Ý
ksmq1
eyb Ž kym.nk ,
Ž 31 .
82
GARYFALOS PAPASCHINOPOULOS
where m F n y 1. Applying Lemma 1 w9, p. 458x to Ž31. and using Ž18. we have 2 < < x 1m y x m F Ž < x 1n y x n2 < q Ž 1 q 2 pe . pby1 E . t e dt e b Ž nym. ,
m F n y 1.
Ž 32 . Furthermore relations Ž12., Ž17., Ž24. imply that < x 1n y x n2 < F Ž 1 y d . y1 Ž < x 1 y x 2 < q pE . q 2 pŽ1 y d .
y1
t
Hn < x Ž u . y x 1
2
Ž u . < du.
Ž 33 .
Then from Ž18., Ž23., and Ž33. we take < x 1n y x n2 < F t < x 1 y x 2 < q Ž 1 q 2 ep . t pE.
Ž 34 .
Finally, relations Ž23., Ž28., Ž32., and Ž34. imply that Ž20. is satisfied. This completes the proof of the lemma. For the remainder of this paper we need the following definition: The difference equation
cnq 1 s Tn cn ,
n g Z,
Ž 35 .
Tn is a k = k invertible matrix for n g Z, has an exponential dichotomy on Z if there exist a projection P Ž P 2 s P . and constants K G 1, a ) 0 such that < G Ž n, m . < F Keya < nym < ,
n, m g Z,
Ž 36 .
where G Ž n, m . s
½
Cn P Cmy1 , yCn Ž Ik y P .
nGm Cmy1 ,
m ) n,
Cn is the fundamental matrix solution of Ž35. such that C0 s Ik , Ik is the k = k identity matrix. LEMMA 3. Suppose that B Ž t . is an r = r matrix continuous on R which satisfies Ž4., AŽ t . is a k = k continuous matrix on R, such that < A Ž t . < F M,
t g R,
Ž 37 .
where M is a constant, M ) 1, and Eq. Ž2. has exponential dichotomy Ž3..
83
INTEGRAL MANIFOLD
Let C Ž t . be a k = k continuous matrix on R such that for all t g R
MeyM 2K
min
½
1
fŽ a.
,
ey2 b
fŽe.
5
,
Ž 38 .
where K, a are defined in Ž3., e is a constant, 0 - e - a , 2 b s a y e , and f : R ¬ R is a function, f Ž t . s Ž e t q 1.rŽ e t y 1., t g R. Consider also that f : R = R r = R k ¬ R r is a continuous function on R = R r = R k such that for all t g R, x 1 , x 2 g R r , y 1 , y 2 g R k < f Ž t , x 1 , y1 . y f Ž t , x 2 , y 2 . < F p Ž < x 1 y x 2 < q < y1 y y 2 <. ,
Ž 39 .
where p is a constant, 0-p-
1yd
,
2 eŽ1 q d .
Ž 40 .
d is defined in Ž4.. Finally, let g: R = R r = R k ¬ R k be a continuous function on R = R r = R k such that for all t g R, x g R r , y g R k < g Ž t, x , y. < F m,
Ž 41 .
m is a positi¨ e constant, and for all t g R, x 1 , x 2 g R r , y 1 , y 2 g R k g Ž t , x 1 , y1 . y g Ž t , x 2 , y 2 . F q Ž < x 1 y x 2 < q < y1 y y 2 <. ,
Ž 42 .
where q is a constant which satisfies the inequality q-Ns
M 2 Ž e b y 1 . eyM
Ž N3 q 1 . Ž M Ž e b y 1 . q 3 L Ž M q d . e Mq2 b .
,
Ž 43 .
N3 s peŽ1 q t dŽ1 q d .Ž1 q pe . e bq t d ., L s 2 Ke e f Ž e ., d is defined in Ž20.. Then for e¨ ery x 0 g R r there exists a unique solution Ž x Ž t ., y Ž t .. of Ž1. such that x Ž 0. s x 0 ,
sup < y Ž t . < , t g R 4 - `.
Ž 44 .
Proof. We consider the difference equation wnq 1 s Ž Tn q Sn . wn s R n wn ,
n g Z,
where Tn s Znq1 Zy1 n ,
Sn s Znq1
nq1
Hn
Zy1 Ž u . C Ž u . du,
Ž 45 .
84
GARYFALOS PAPASCHINOPOULOS
ZŽ t . is defined in Ž3.. From Ž37. and Gronwall’s lemma w4, p. 19x we take < Z Ž t . Zy1 Ž s . < F e M < tys < ,
t , s g R.
Ž 46 .
Therefore using Ž38., Ž46. we have that R n is an invertible matrix for n g Z. Since Ž2. has an exponential dichotomy Ž3., it is obvious that Eq. Ž35. with Tn defined in Ž45. has an exponential dichotomy Ž36.. Then from Ž36., Ž38., Ž46. we can apply the roughness of exponential dichotomies for difference equations w8, p. 276x to Ž45. and so there exists a projection Q Ž Q 2 s Q . such that < G Ž n, m . < F Ley2 b < nym < ,
n, m g Z,
Ž 47 .
where G Ž n, m . s
½
Wn QWmy1 ,
nGm
yWn Ž Ik y Q .
Wmy1 ,
m ) n,
Wn is the fundamental matrix solution of Ž45. such that W0 s Ik . We consider the space C of the continuous functions y: R ¬ R k such that < y < s sup eyb < n < < y Ž t .<, t g R4 - `, n s w t x. Let y g C and x 0 g R r. From Ž39. for all t g R, x 1 , x 2 g R r it follows that f Ž t , x1 , y Ž t . . y f Ž t , x2 , y Ž t . . F p< x1 y x2 <.
Ž 48 .
Since Ž40. and Ž48. hold, from Lemma 1 the problem x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , y Ž t . . ,
x Ž 0. s x 0
Ž 49 .
has a unique solution x Ž t . s x Ž t, 0, x 0 .. Let H be the operator defined on C as H y Ž t . s Z Ž t . Zy1 n q
ž
q ZŽ t . bn s
`
Ý
ksy`
jk s Zkq1
t y1
Hn Z
t y1
Hn Z
/
Ž u . C Ž u . du bn
Ž u . g Ž u, x Ž u . , y Ž u . . du,
Ž 50 .
G Ž n, k q 1 . jk ,
Ž 51 . kq1
Hk
y1
Z
Ž u . g Ž u, x Ž u . , y Ž u . . du,
where ZŽ t . is defined in Ž3. and x Ž u. s x Ž u, 0, x 0 ..
n s wtx,
85
INTEGRAL MANIFOLD
Using relations Ž38., Ž41., Ž46., Ž47., Ž50., and Ž51. we take < H y Ž t . < F ml ,
t g R,
Ž 52 .
where l s My2 e M Ž M q Le M Ž M q d . f Ž2 b .. It is easy to show that bn satisfies the difference equation bnq 1 s R n bn q j n ,
n g Z.
Then the function H y is continuous at t s n g Z and so it is obvious that H y is continuous on R. Therefore H is in C . We prove now that H is a contraction on C . Let y 1 , y 2 g C and x i Ž t . s x i Ž t, 0, x 0 . be the solution of the problem xXi Ž t . s B Ž t . x i Ž w t x . q f Ž t , x i Ž t . , yi Ž t . . ,
x i Ž 0 . s x 0 , i s 1, 2.
Relation Ž39. implies that eyb < n< f Ž t , x 1 Ž t . , y 1 Ž t . . y f Ž t , x 2 Ž t . , y 2 Ž t . . F peyb < n< x 1 Ž t . y x 2 Ž t . q p y 1 y y 2 ,
n s wtx,
Ž 53 .
where < y 1 y y 2 < s sup eyb < n < < y 1Ž t . y y 2 Ž t .<, t g R4 . Using Ž53. and arguing as in the proof of Lemma 2 we take eyb < n < x 1 Ž t . y x 2 Ž t . F N3 < y 1 y y 2 < ,
t g R.
Ž 54 .
From the relations Ž38., Ž42., Ž46., Ž47., Ž50., Ž51., and Ž54. we get eyb < n < H y 1 Ž t . y H y 2 Ž t . F Ž M q d . My1 e M ey b < n< < bn1 y bn2 < q qMy1 e M Ž N3 q 1 . < y 1 y y 2 < , ey b < n< < bn1 y bn2 < F bni s
3 Lq Ž N3 q 1 . e Mq 2 b M Ž e b y 1.
`
Ý
ksy`
jki s Zkq1
tgR, Ž 55 .
< y1 y y 2 < ,
Ž 56 .
G Ž n, k q 1 . jki , kq1
Hk
Zy1 Ž u . g Ž u, x i Ž u . , yi Ž u . . du.
Therefore relations Ž43., Ž55., and Ž56. imply that H is a contraction on C and so there exists a unique y g C such that H y s y.
Ž 57 .
86
GARYFALOS PAPASCHINOPOULOS
From Ž50., Ž51., and Ž57. it follows that Ž x Ž t ., y Ž t .., x Ž t . satisfies the problem Ž49., is a solution of Ž1. which satisfies Ž44.. Suppose that there exists another solution Ž x Ž t ., y Ž t .. of Ž1. such that sup < y Ž t . < , t g R 4 - `.
x Ž 0. s x 0 , Then y Ž t . satisfies
y Ž t . s Z Ž t . Zy1 n q
ž
q ZŽ t .
t y1
Hn Z
t y1
Hn Z
/
Ž u . C Ž u . du yn
Ž u . g Ž u, x Ž u . , y Ž u . . du.
Ž 58 .
From the continuity of the function y Ž t . it follows that yn satisfies ynq 1 s R n yn q j n ,
j n s Znq1
nq1
Hn
Zy1 Ž u . g Ž u, x Ž u . , y Ž u . . du,
ngZ.
Then since Ž41., Ž46., and Ž47. hold, from Lemma 2.7 w8, p. 272x we obtain yn s
`
Ý
ksy`
G Ž n, k q 1 . jk .
Ž 59 .
Therefore from Ž58. and Ž59. it follows that y is a fixed point of H . Then x Ž t . s x Ž t ., y Ž t . s y Ž t .. This completes the proof of the lemma. In the following proposition we find conditions so that the integral manifold for Ž1. exists. PROPOSITION 1. Suppose that all the conditions of Lemma 3 are satisfied. Moreo¨ er consider that the constant q defined in Ž42. satisfies the inequality q - min
½
1 2g
,
MeyM
v Ž M q d . q 2 Ž N2 q e . p q 1
5
,
Ž 60 .
g s N1 e M M y 2 Ž M q Le M q 2 b Ž M q d . f Ž b .., v s Le M M y 1 ŽŽ1 q 2 pe . f Ž2 b . q 2 pN2 e 2 bf Ž b .., and N1 , N2 are defined in Ž20., b , d , f Ž?. in Ž38., and M, p, L in Ž37., Ž39., Ž43., respecti¨ ely. Then there exists a function ¨ : R = R r ¬ R k which determines an integral manifold for Ž1. and for all t g R, x g R r , x 1 , x 2 g R r the function ¨ Ž t, x . satisfies the relations ¨ Ž t , x . F ml ,
¨ Ž t , x1 . y ¨ Ž t , x2 . F u < x1 y x2 <,
where m Ž resp. l. is defined in Ž41. Ž resp. Ž52.. and u s g qrŽ1 y g q ..
Ž 61 .
87
INTEGRAL MANIFOLD
Proof. Let C be the space of the functions ¨ : R = R r ¬ R k continuous and bounded on R = R r such that the relations Ž61. hold. From Ž60. it is obvious that u - 1. Then if ¨ g C relations Ž39. and Ž61. imply that the relation f Ž s, x 1 , ¨ Ž s, x 1 . . y f Ž s, x 2 , ¨ Ž s, x 2 . . F 2 p < x 1 y x 2 <
Ž 62 .
holds for all s g R, x 1 , x 2 g R r. Therefore since Ž4., Ž40., and Ž62. hold, from Lemma 1 the equation x9 Ž s . s B Ž s . x Ž w s x . q f Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ,
sgR
Ž 63 .
has a unique solution x Ž s . s x Ž s, t, x , ¨ . such that x Ž t . s x , t g R, x g R r. If n s w t x we define the operator K on C as K ¨ Ž t , x . s Z Ž t . Zy1 n q
ž
q ZŽ t . b Ž n, ¨ , x . s
`
t y1
Hn Z
t y1
Hn Z
Ž s . C Ž s . ds b Ž n, ¨ , x .
/
Ž s . g Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ds,
G Ž n, m q 1 . zm ,
Ý msy`
zm s Zmq1
Ž 64 .
Ž 65 . mq1
Hm
Zy1 Ž s . g Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ds,
where ZŽ t . Žresp. GŽ n, m.. is defined in Ž3. Žresp. Ž47.. and x Ž s . s x Ž s, t, x , ¨ .. From Ž38., Ž41., Ž46., Ž47., Ž64., and Ž65. it holds that K ¨ Ž t , x . F ml ,
t g R, x g R r .
Ž 66 .
Consider now x 1 , x 2 g R r and let x 1Ž s . s x 1Ž s, t, x 1 , ¨ ., x 2 Ž s . s x 2 Ž s, t, x 2 , ¨ . be the solutions of Ž63. such that x 1Ž t . s x 1 , x 2 Ž t . s x 2 . Then since Ž4., Ž40., and Ž62. hold, by setting h1Ž s, x . s h 2 Ž s, x . s f Ž s, x , ¨ Ž s, x .. and E s 0 in Lemma 2 we take x 1 Ž s . y x 2 Ž s . F N1 < x 1 y x 2 < e b < myn< ,
s g R,
Ž 67 .
where m s w s x. Moreover from Ž42. and Ž61. we get for all s g R g Ž s, x 1 Ž s . , ¨ Ž s, x 1 Ž s . . . y g Ž s, x 2 Ž s . , ¨ Ž s, x 2 Ž s . . . F q Ž 1 q u . x1Ž s . y x 2 Ž s . .
Ž 68 .
88
GARYFALOS PAPASCHINOPOULOS
Hence using Ž38., Ž46., Ž64., Ž67., Ž68. we have for t g R, K ¨ Ž t , x 1 . y K ¨ Ž t , x 2 . F Ž M q d . My1 e M b Ž n, ¨ , x 1 . y b Ž n, ¨ , x 2 . q Ž 1 q u . qN1 My1 e M < x 1 y x 2 < .
Ž 69 .
Moreover relations Ž46., Ž47., Ž65., Ž67., Ž68. imply that b Ž n, ¨ , x 1 . y b Ž n, ¨ , x 2 . F Ž 1 q u . LN1 My1 qe Mq2 bf Ž b . < x 1 y x 2 < .
Ž 70 . Hence from Ž69. and Ž70. it follows that K¨ Ž t , x1 . y K¨ Ž t , x 2 . F u < x1 y x 2 <.
Ž 71 .
Arguing as in Lemma 3 we can prove that K ¨x Ž t . s K ¨ Ž t, x . is continuous on R for every x g R r. Furthermore for all t, t 0 g R, x , x 0 g R r it is obvious that K¨ Ž t , x . y K¨ Ž t0 , x 0 . F K¨ Ž t , x . y K¨ Ž t , x 0 . q K¨ Ž t , x 0 . y K¨ Ž t0 , x 0 . .
Ž 72 .
Then it is obvious that relations Ž71. and Ž72. imply that KŽ t, x . is continuous on R = R r and so from Ž66. and Ž71. K is in C . We prove now that K is a contraction on C . Let ¨ 1 , ¨ 2 g C . Then using Ž39. and the relation u - 1, for all s g R, x 1 , x 2 g R r it follows that f Ž s, x 1 , ¨ i Ž s, x 1 . . y f Ž s, x 2 , ¨ i Ž s, x 2 . . F 2 p < x 1 y x 2 < ,
i s 1, 2.
Ž 73 . Moreover from Ž61. for all s g R, x 1 , x 2 g R r it holds that ¨ 1 Ž s, x 1 . y ¨ 2 Ž s, x 2 . F u < x 1 y x 2 < q < ¨ 1 y ¨ 2 < ,
Ž 74 .
where < ¨ 1 y ¨ 2 < s sup< ¨ 1Ž s, x . y ¨ 2 Ž s, x .<, s g R, x g R r 4 . Hence from Ž39. and the relation u - 1 we get f Ž s, x 1 , ¨ 1 Ž s, x 1 . . y f Ž s, x 2 , ¨ 2 Ž s, x 2 . . F 2 p < x 1 y x 2 < q p < ¨ 1 y ¨ 2 < .
Ž 75 . Let x i Ž s . s x i Ž s, t, x , ¨ i ., i s 1, 2, be the solution of Ž19. with h i Ž s, x . s f Ž s, x , ¨ i Ž s, x .., i s 1, 2, such that x 1Ž t . s x 2 Ž t . s x . Since Ž4., Ž40., Ž73.,
89
INTEGRAL MANIFOLD
and Ž75. hold, from Lemma 2 we have x 1 Ž s . y x 2 Ž s . F Ž N2 e b < myn < q e . p < ¨ 1 y ¨ 2 < ,
s g R.
Ž 76 .
Furthermore, using Ž42., Ž74., and u - 1 we obtain g Ž s, x 1 Ž s . , ¨ 1 Ž s, x 1 Ž s . . . y g Ž s, x 2 Ž s . , ¨ 2 Ž s, x 2 Ž s . . . F 2 q x1Ž s . y x 2 Ž s . q q < ¨ 1 y ¨ 2 < .
Ž 77 .
Then for all t g R and x g R r relations Ž38., Ž46., Ž47., Ž64., Ž65., Ž76., Ž77. imply that K ¨ 1 Ž t , x . y K ¨ 2 Ž t , x . F Ž M q d . My1 e M b Ž n, ¨ 1 , x . y b Ž n, ¨ 2 , x . q qe M My1 Ž 1 q 2 Ž N2 q e . p . < ¨ 1 y ¨ 2 < , Ž 78 . b Ž n, ¨ 1 , x . y b Ž n, ¨ 2 , x . F q v < ¨ 1 y ¨ 2 < .
Ž 79 .
Therefore from Ž60., Ž78., and Ž79. it is obvious that K is a contraction on C and so there exists a unique ¨ g C such that K¨ s ¨ .
Ž 80 .
So using Ž64., Ž65., and Ž80. we can show that Ž x Ž t ., ¨ Ž t, x Ž t .. , x Ž t . satisfies Ž63., is a solution of Ž1.. Suppose now that Ž x Ž t ., y Ž t .. is a solution of Ž1. such that sup< y Ž t .<, t g R4 - `. Then using Lemma 3 we have that y Ž t . s ¨ Ž t, x Ž t .., t g R. Therefore ¨ Ž t, x . determines an integral manifold for Ž1.. This completes the proof of the proposition.
ACKNOWLEDGMENT I thank the referee for his valuable suggestions.
REFERENCES 1. N. N. Bogolyubov and Y. A. Mitropolski, ‘‘Asymptotic Methods in the Theory of Nonlinear Oscillations,’’ 2nd ed. wEnglish transl.x, Gordon & Breach, New York, 1962. 2. S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential}continuous dynamics, in ‘‘Nonlinear Phenomena in Mathematical Sciences’’ ŽV. Lakshmikantham, Ed.., pp. 179]187, Academic Press, New York, 1982. 3. K. L. Cooke and J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 Ž1984., 265]297. 4. W. A. Coppel, ‘‘Stability and Asymptotic Behavior of Differential Equations,’’ Heath, Boston, 1965.
90
GARYFALOS PAPASCHINOPOULOS
5. J. K. Hale, ‘‘Ordinary Differential Equations,’’ Wiley, New York, 1969. 6. K. S. Miller, ‘‘Linear Difference Equations,’’ Benjamin, New York, 1968. 7. K. J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl. 51 Ž1975., 243]255. 8. K. J. Palmer, Exponential dichotomies, the Shadowing}Lemma and transversal homoclinic points, Dynam. Report. Ser. Dynam. Syst. Appl. 1 Ž1988., 265]306. 9. G. Papaschinopoulos, On the summable manifold, Math. Japon. 33, No. 3 Ž1988., 457]468. 10. G. Papaschinopoulos, On asymptotic behavior of the solutions of a class of perturbed differential equations with piecewise constant argument and variable coefficients, J. Math. Anal. Appl. 185 Ž1994., 490]500. 11. G. Papaschinopoulos, Some results concerning a class of differential equations with piecewise constant argument, Math. Nachr. 166 Ž1994., 193]206. 12. G. Papaschinopoulos, Exponential dichotomy, topological equivalence and structural stability for differential equations with piecewise constant argument, Analysis, 14 Ž1994., 239]247. 13. J. Wiener and K. L. Cooke, Oscillations in systems of differential equations with piecewise constant argument, J. Math. Anal. Appl. 137 Ž1989., 221]239. 14. J. Wiener, ‘‘Generalized Solutions of Functional Differential Equations,’’ World Scientific, Singapore, 1993.
201, 75]90 Ž1996.
0242
On the Integral Manifold for a System of Differential Equations with Piecewise Constant Argument Garyfalos Papaschinopoulos School of Engineering, Democritus Uni¨ ersity of Thrace, 67100 Xanthi, Greece Submitted by Jack K. Hale Received October 14, 1994
The existence of an integral manifold for a system of differential equations with piecewise constant argument is studied. Q 1996 Academic Press, Inc.
INTRODUCTION We consider the system of differential equations with piecewise constant argument of the form x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , y Ž t . . y9 Ž t . s A Ž t . y Ž t . q C Ž t . y Ž w t x . q g Ž t , x Ž t . , y Ž t . . ,
Ž 1.
where t g R s Žy`, `., w?x denotes the greatest integer function, B Ž t . is an r = r continuous matrix for t g R, AŽ t ., C Ž t . are k = k continuous matrices for t g R, and f : R = R r = R k ¬ R r , g: R = R r = R k ¬ R k are continuous functions on R = R r = R k , small Lipschitzian in x and y. A pair of functions Ž x Ž t ., y Ž t .., x: Žy`, `. ¬ R r , y: Žy`, `. ¬ R k , is a solution of Ž1. if the following conditions are satisfied: Ži. The functions x Ž t ., y Ž t . are continuous on Žy`, `.. Žii. The derivatives x9Ž t ., y9Ž t . exist on Žy`, `. except possibly at the points t s n, n g Z s . . . , y1, 0, 1, . . . 4 where one-sided derivatives exist. Žiii. Ž x Ž t ., y Ž t .. satisfies Ž1. on every interval w n, n q 1., n g Z. 75 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
76
GARYFALOS PAPASCHINOPOULOS
A function ¨ : R = R r ¬ R k determines an integral manifold for Ž1. if the following statements are true: Ži. ¨ is continuous and bounded on R = R r. Žii. For any solution x Ž t . of the differential equation x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , ¨ Ž t , x Ž t . . . we have that Ž x Ž t ., ¨ Ž t, x Ž t ... is a solution of Ž1.. Moreover if Ž x Ž t ., y Ž t .. is a solution of Ž1. with sup< y Ž t .<, t g R4 - ` it follows that y Ž t . s ¨ Ž t, x Ž t .., t g R. In what follows we denote by < ? < any norm. We say that the linear differential equation z9 Ž t . s A Ž t . z Ž t . ,
t g R,
Ž 2.
where AŽ t . is a k = k continuous matrix for t g R, has an exponential dichotomy on R if there exist a projection P Ž P 2 s P . and constants K G 1, a ) 0 such that < Z Ž t . PZy1 Ž s . < F KeyaŽ tys. , < Z Ž t . Ž Ik y P . Zy1 Ž s . < F KeyaŽ syt . ,
tGs s G t,
Ž 3.
where ZŽ t . is the fundamental matrix solution of Ž2. such that ZŽ0. s Ik , Ik is the k = k indentity matrix, and t, s g R. In this paper under some conditions on the matrices AŽ t ., B Ž t ., C Ž t . and the functions f Ž t, x, y ., g Ž t, x, y . and assuming that Ž2. has an exponential dichotomy Ž3. we prove that there exists an integral manifold ¨ Ž t, x . for Ž1. Žsee Proposition 1 below.. We note that results concerning integral manifolds for equations of the form Ž1., where B Ž t . s 0, C Ž t . s 0, t g R, are included in the papers w1, 5, 7x. We also note that differential equations with piecewise constant argument ŽEPCA. describe hybrid dynamical systems Ža combination of continuous and discrete. and so combine properties of both differential and difference equations. These equations may also have applications in certain biomedical models w2x. Results concerning asymptotic stability and oscillatory behavior of the solutions of differential equations with piecewise constant argument are included in w3, 13, 14x and the references cited therein. In addition w14x contains results on the relationship between equations with piecewise constant argument and impulsive equations.
77
INTEGRAL MANIFOLD
Moreover, the structural stability, the asymptotic behavior of the solutions of a class of differential equations with piecewise constant argument, has been studied in w10]12x.
MAIN RESULTS We prove now our main results. We need three lemmas. The first lemma concerns existence and uniqueness of the solutions of a differential equation with piecewise constant argument. LEMMA 1. such that
Suppose that B Ž s ., s g R, is an r = r continuous matrix on R < B < s sup < B Ž s . < , s g R 4 s d ,
0-d-1
Ž 4.
and h: R = R r ¬ R r is a continuous function on R = R r such that for all s g R, x 1 , x 2 g R r < h Ž s, x 1 . y h Ž s, x 2 . < F 2 p < x 1 y x 2 < ,
Ž 5.
where p is a constant satisfying 0-p-
1yd 4
.
Ž 6.
Then if t g R, x g R r the initial ¨ alue problem x9 Ž s . s B Ž s . x Ž w s x . q h Ž s, x Ž s . . ,
xŽ t. s x
Ž 7.
has a unique solution x Ž s . satisfying the integral equation x Ž s. s LŽ s. x Ž w sx. q
s
Hw s xh Ž u, x Ž u . . du,
s g R,
Ž 8.
where the matrix LŽ s . is defined as L Ž s . s Ir q
s
Hw s xB Ž u . du
Ž 9.
and it is in¨ ertible for all s g R, Ir is the r = r identity matrix. Moreo¨ er if m s w s x, n s w t x, then x m s x Žw s x. satisfies the problem x mq 1 s Dm x m q Hm ,
x n s Ly1 Ž t . x y
ž
t
Hn h Ž u, x Ž u . . du
/
, Ž 10 .
78
GARYFALOS PAPASCHINOPOULOS
where Dm s Ir q
mq1
Hm
Hm s
B Ž u . du,
mq1
Hm
h Ž u, x Ž u . . du.
Ž 11 .
Proof. Using Ž4. and Ž9. Žresp. Ž11.. it is obvious that LŽ s . Žresp. Dm . is invertible for all s g R Žresp. m g Z . and < LŽ s. < F 1 q d ,
< Ly1 Ž s . < F Ž 1 y d . y1 ,
s g R,
Ž 12 .
< Dm < F 1 q d ,
< Dmy1 < F Ž 1 y d . y1 ,
m g Z.
Ž 13 .
For s g R and m s w s x we define LŽ s. s LŽ s. , We prove now that satisfies Ž8. and Ž10. continuous functions w n, n q 1x4 and T n be
s g m, m q 1 . ,
L Ž m q 1 . s Dm .
the problem Ž7. has a unique solution x Ž s . which for s g w n, n q 1x. Let F n be the space of the x n : w n, n q 1x ¬ R r with < x n < s sup< x n Ž s .<, s g the operator defined on F n as
T n x n Ž s . s L Ž s . Ly1 Ž t . x y
ž
q
s
Hn h Ž u, x
n
t
Hn h Ž u, x
n
Ž u . . du
/
Ž u . . du,
Ž 14 .
where n F s F n q 1. It is obvious that T n is in F n. We show now that T n is a contraction on F n. If x 1n, x 2n g F n from Ž5., Ž12., Ž13., and Ž14. we take < T n x 1n Ž s . y T n x 2n Ž s . < F
4p 1yd
< x 1n y x 2n < ,
s g w n, n q 1 x ,
where < x 1n y x 2n < s sup< x 1n Ž s . y x 2n Ž s .<, s g w n, n q 1x4 , which implies that T n is a contraction on F n, since Ž6. holds. Therefore there exists a unique x n g F n such that T n x nŽ s. s x nŽ s. ,
s g w n, n q 1 x .
Ž 15 .
Then from Ž14. and Ž15. it is obvious that x Ž s . s x n Ž s ., s g w n, n q 1x, is the unique solution of Ž7. satisfying Ž8. and Ž10. for s g w n, n q 1x. Working inductively backward we can define the solution x Ž s . of Ž7. in every interval w m, m q 1x, m g . . . , n y 2, n y 14 , by the following procedure:
79
INTEGRAL MANIFOLD
Let F m be the space of the continuous functions x m : w m, m q 1x ¬ R r with < x m < s sup< x m Ž s .<, s g w m, m q 1x4 and T m y be the operator defined on F m as m y1 mq1 Tm x mq1 y y x Ž s . s L Ž s . Dm
ž
q
s
Hmh Ž u, x
m
mq1
Hm
h Ž u, x m Ž u . . du
/
Ž u . . du,
where s g w m, m q 1x and x mq 1 Ž s . is the solution of Ž7. for t g w m q 1, m m q 2x. Arguing as above we can prove that T m y is a contraction on F m m and so T y has a unique fixed point x m g F . Then it is obvious that the function x Ž s . s x m Ž s ., s g w m, m q 1x, m g . . . , n y 2, n y 14 , is the unique solution of the problem Ž7. on the space Žy`, n x which satisfies Ž8. and Ž10.. Working now inductively forward we define the solution x Ž s . of Ž7. in every interval w m, m q 1x, m g n q 1, n q 2, . . . 4 , by using the operator Tm q, m my1 Tm q qx Ž s . s L Ž s . x m
s
Hmh Ž u, x
m
s g w m, m q 1 x ,
Ž u . . du,
where x m g F m , F m has been defined previously, and x my1 Ž s . is the solution of Ž7. for s g w m y 1, m x. Using Ž5. and Ž6. we can easily prove m that T m and let x m g F m be the unique fixed q is a contraction on F m point of T q. Then the function x Ž s . s x m Ž s ., s g w m, m q 1x, m g n q 1, n q 2, . . . 4 , is the unique solution of the problem Ž7. on the space w n q 1, `. which satisfies Ž8. and Ž10.. Thus the proof of the lemma is complete. LEMMA 2. Let B Ž s . be an r = r continuous matrix such that Ž4. holds and h i : R = R r ¬ R r , i s 1, 2, be continuous functions on R = R r such that for all s g R, x 1 , x 2 g R r < h i Ž s, x 1 . y h i Ž s, x 2 . < F 2 p < x 1 y x 2 < ,
i s 1, 2
Ž 16 .
and < h1 Ž s, x 1 . y h 2 Ž s, x 2 . < F 2 p < x 1 y x 2 < q pE,
Ž 17 .
where E and p are constants which satisfy E G 0,
0-p-
1yd 2 eŽ1 q d .
.
Ž 18 .
80
GARYFALOS PAPASCHINOPOULOS
Suppose that b is a positi¨ e constant. Then if x i Ž s ., s g R, is a solution of the problem x9 Ž s . s B Ž s . x Ž w s x . q h i Ž s, x Ž s . . ,
xiŽ t . s x i,
Ž 19 .
where t g R, x i g R r , i s 1, 2, the following relation is satisfied, < x 1 Ž s . y x 2 Ž s . < F N1 < x 1 y x 2 < e b < myn< q p Ž N2 e b < myn < q e . E,
sgR,
Ž 20 .
where m s w s x, n s w t x, N1 s t 2 Ž1 q d . e1q dt , N2 s N1Ž1 q 2 pe .Ž1 q by1ty1 ., d s Ž e b y 1.y1 , t s Ž1 y d y 2 peŽ1 q d ..y1 . Proof. Relations Ž4., Ž16., and Ž18. imply that all the conditions of Lemma 1 are satisfied for Ž19.. Then for i s 1, 2 it holds that x i Ž s. s LŽ s. x i Ž w sx. q
s
s g R.
Ž 21 .
Ž u . < du.
Ž 22 .
i
Hw s xh Ž u, x Ž u . . du, i
Therefore from Ž12., Ž17., and Ž21. we obtain for s g R 2 < < x 1 Ž s . y x 2 Ž s . < F Ž 1 q d . < x 1m y x m
s
Hm < x Ž u . y x
q pE q 2 p
1
2
Applying Gronwall’s lemma w4, p. 19x to Ž22. for s g w m, m q 1x and since from Ž18., 2 p - 1 it follows that 2 < < x 1 Ž s . y x 2 Ž s . < F Ž 1 q d . e < x 1m y x m q peE,
s g R.
Ž 23 .
From Lemma 1 the function x mi , i s 1, 2, satisfies i i i x mq 1 s Dm x m q Hm ,
x ni s Ly1 Ž t . x i y
ž
t
i
Hn h Ž u, x Ž u . . du i
/
where Hmi s
mq1
Hm
h i Ž u, x i Ž u . . du,
i s 1, 2.
Ž 24 . ,
81
INTEGRAL MANIFOLD
Now let s G t, from which m G n. Then from Ž24. we obtain my1
x mi s x ni q
Ý Ž Hki q Ž Dk y Ir . xki . ,
i s 1, 2, m G n.
Ž 25 .
ksn
Using Ž4., Ž11., Ž17., Ž23., and Ž25. we get 2 < < x 1m y x m F < x 1n y x n2 < q p Ž 1 q 2 pe . E Ž m y n .
my1
q Ž 2 ep Ž 1 q d . q d .
Ý
ksn
< xk1 y xk2 < ,
mGn.
Ž 26 .
2 < On setting sm s eyb Ž myn. < x 1m y x m , from Ž26. and the relation Ž m y n. yb Ž myn. y1 ?e - b , m G n, we have
sm F sn q Ž 1 q 2 pe . pby1 E my1
q Ž 2 ep Ž 1 q d . q d .
Ý
ksn
eyb Ž my k .sk ,
mGn.
Ž 27 .
When the discrete Gronwall’s lemma w6, p. 21x is applied to Ž27. and Ž18. is used it follows that 2 < < x 1m y x m F Ž < x 1n y x n2 < q Ž 1 q 2 pe . pby1 E . e d e b Ž myn. ,
m G n. Ž 28 .
Suppose now that s F t and so m F n. Then, ny1
x mi s x ni q
Ý Ž Ž Ir y Dk . xki y Hki . ,
i s 1, 2, n G m.
Ž 29 .
ksm
From relations Ž4., Ž11., Ž17., Ž18., Ž23., and Ž29. we obtain 2 < < x 1m y x m F t < x 1n y x n2 < q t p Ž 1 q 2 pe . E Ž n y m .
ny1
q t Ž 2 ep Ž 1 q d . q d .
Ý
ksmq1
< xk1 y xk2 < ,
mFn y 1.
Ž 30 . 2 < Moreover, if nm s eyb Ž nym. < x 1m y x m , then from Ž30. and the relation Ž n y m. eyb Ž nym. - by1, n G m, we get
nm F tnn q t p Ž 1 q 2 pe . by1 E ny1
q t Ž 2 ep Ž 1 q d . q d .
Ý
ksmq1
eyb Ž kym.nk ,
Ž 31 .
82
GARYFALOS PAPASCHINOPOULOS
where m F n y 1. Applying Lemma 1 w9, p. 458x to Ž31. and using Ž18. we have 2 < < x 1m y x m F Ž < x 1n y x n2 < q Ž 1 q 2 pe . pby1 E . t e dt e b Ž nym. ,
m F n y 1.
Ž 32 . Furthermore relations Ž12., Ž17., Ž24. imply that < x 1n y x n2 < F Ž 1 y d . y1 Ž < x 1 y x 2 < q pE . q 2 pŽ1 y d .
y1
t
Hn < x Ž u . y x 1
2
Ž u . < du.
Ž 33 .
Then from Ž18., Ž23., and Ž33. we take < x 1n y x n2 < F t < x 1 y x 2 < q Ž 1 q 2 ep . t pE.
Ž 34 .
Finally, relations Ž23., Ž28., Ž32., and Ž34. imply that Ž20. is satisfied. This completes the proof of the lemma. For the remainder of this paper we need the following definition: The difference equation
cnq 1 s Tn cn ,
n g Z,
Ž 35 .
Tn is a k = k invertible matrix for n g Z, has an exponential dichotomy on Z if there exist a projection P Ž P 2 s P . and constants K G 1, a ) 0 such that < G Ž n, m . < F Keya < nym < ,
n, m g Z,
Ž 36 .
where G Ž n, m . s
½
Cn P Cmy1 , yCn Ž Ik y P .
nGm Cmy1 ,
m ) n,
Cn is the fundamental matrix solution of Ž35. such that C0 s Ik , Ik is the k = k identity matrix. LEMMA 3. Suppose that B Ž t . is an r = r matrix continuous on R which satisfies Ž4., AŽ t . is a k = k continuous matrix on R, such that < A Ž t . < F M,
t g R,
Ž 37 .
where M is a constant, M ) 1, and Eq. Ž2. has exponential dichotomy Ž3..
83
INTEGRAL MANIFOLD
Let C Ž t . be a k = k continuous matrix on R such that for all t g R
MeyM 2K
min
½
1
fŽ a.
,
ey2 b
fŽe.
5
,
Ž 38 .
where K, a are defined in Ž3., e is a constant, 0 - e - a , 2 b s a y e , and f : R ¬ R is a function, f Ž t . s Ž e t q 1.rŽ e t y 1., t g R. Consider also that f : R = R r = R k ¬ R r is a continuous function on R = R r = R k such that for all t g R, x 1 , x 2 g R r , y 1 , y 2 g R k < f Ž t , x 1 , y1 . y f Ž t , x 2 , y 2 . < F p Ž < x 1 y x 2 < q < y1 y y 2 <. ,
Ž 39 .
where p is a constant, 0-p-
1yd
,
2 eŽ1 q d .
Ž 40 .
d is defined in Ž4.. Finally, let g: R = R r = R k ¬ R k be a continuous function on R = R r = R k such that for all t g R, x g R r , y g R k < g Ž t, x , y. < F m,
Ž 41 .
m is a positi¨ e constant, and for all t g R, x 1 , x 2 g R r , y 1 , y 2 g R k g Ž t , x 1 , y1 . y g Ž t , x 2 , y 2 . F q Ž < x 1 y x 2 < q < y1 y y 2 <. ,
Ž 42 .
where q is a constant which satisfies the inequality q-Ns
M 2 Ž e b y 1 . eyM
Ž N3 q 1 . Ž M Ž e b y 1 . q 3 L Ž M q d . e Mq2 b .
,
Ž 43 .
N3 s peŽ1 q t dŽ1 q d .Ž1 q pe . e bq t d ., L s 2 Ke e f Ž e ., d is defined in Ž20.. Then for e¨ ery x 0 g R r there exists a unique solution Ž x Ž t ., y Ž t .. of Ž1. such that x Ž 0. s x 0 ,
sup < y Ž t . < , t g R 4 - `.
Ž 44 .
Proof. We consider the difference equation wnq 1 s Ž Tn q Sn . wn s R n wn ,
n g Z,
where Tn s Znq1 Zy1 n ,
Sn s Znq1
nq1
Hn
Zy1 Ž u . C Ž u . du,
Ž 45 .
84
GARYFALOS PAPASCHINOPOULOS
ZŽ t . is defined in Ž3.. From Ž37. and Gronwall’s lemma w4, p. 19x we take < Z Ž t . Zy1 Ž s . < F e M < tys < ,
t , s g R.
Ž 46 .
Therefore using Ž38., Ž46. we have that R n is an invertible matrix for n g Z. Since Ž2. has an exponential dichotomy Ž3., it is obvious that Eq. Ž35. with Tn defined in Ž45. has an exponential dichotomy Ž36.. Then from Ž36., Ž38., Ž46. we can apply the roughness of exponential dichotomies for difference equations w8, p. 276x to Ž45. and so there exists a projection Q Ž Q 2 s Q . such that < G Ž n, m . < F Ley2 b < nym < ,
n, m g Z,
Ž 47 .
where G Ž n, m . s
½
Wn QWmy1 ,
nGm
yWn Ž Ik y Q .
Wmy1 ,
m ) n,
Wn is the fundamental matrix solution of Ž45. such that W0 s Ik . We consider the space C of the continuous functions y: R ¬ R k such that < y < s sup eyb < n < < y Ž t .<, t g R4 - `, n s w t x. Let y g C and x 0 g R r. From Ž39. for all t g R, x 1 , x 2 g R r it follows that f Ž t , x1 , y Ž t . . y f Ž t , x2 , y Ž t . . F p< x1 y x2 <.
Ž 48 .
Since Ž40. and Ž48. hold, from Lemma 1 the problem x9 Ž t . s B Ž t . x Ž w t x . q f Ž t , x Ž t . , y Ž t . . ,
x Ž 0. s x 0
Ž 49 .
has a unique solution x Ž t . s x Ž t, 0, x 0 .. Let H be the operator defined on C as H y Ž t . s Z Ž t . Zy1 n q
ž
q ZŽ t . bn s
`
Ý
ksy`
jk s Zkq1
t y1
Hn Z
t y1
Hn Z
/
Ž u . C Ž u . du bn
Ž u . g Ž u, x Ž u . , y Ž u . . du,
Ž 50 .
G Ž n, k q 1 . jk ,
Ž 51 . kq1
Hk
y1
Z
Ž u . g Ž u, x Ž u . , y Ž u . . du,
where ZŽ t . is defined in Ž3. and x Ž u. s x Ž u, 0, x 0 ..
n s wtx,
85
INTEGRAL MANIFOLD
Using relations Ž38., Ž41., Ž46., Ž47., Ž50., and Ž51. we take < H y Ž t . < F ml ,
t g R,
Ž 52 .
where l s My2 e M Ž M q Le M Ž M q d . f Ž2 b .. It is easy to show that bn satisfies the difference equation bnq 1 s R n bn q j n ,
n g Z.
Then the function H y is continuous at t s n g Z and so it is obvious that H y is continuous on R. Therefore H is in C . We prove now that H is a contraction on C . Let y 1 , y 2 g C and x i Ž t . s x i Ž t, 0, x 0 . be the solution of the problem xXi Ž t . s B Ž t . x i Ž w t x . q f Ž t , x i Ž t . , yi Ž t . . ,
x i Ž 0 . s x 0 , i s 1, 2.
Relation Ž39. implies that eyb < n< f Ž t , x 1 Ž t . , y 1 Ž t . . y f Ž t , x 2 Ž t . , y 2 Ž t . . F peyb < n< x 1 Ž t . y x 2 Ž t . q p y 1 y y 2 ,
n s wtx,
Ž 53 .
where < y 1 y y 2 < s sup eyb < n < < y 1Ž t . y y 2 Ž t .<, t g R4 . Using Ž53. and arguing as in the proof of Lemma 2 we take eyb < n < x 1 Ž t . y x 2 Ž t . F N3 < y 1 y y 2 < ,
t g R.
Ž 54 .
From the relations Ž38., Ž42., Ž46., Ž47., Ž50., Ž51., and Ž54. we get eyb < n < H y 1 Ž t . y H y 2 Ž t . F Ž M q d . My1 e M ey b < n< < bn1 y bn2 < q qMy1 e M Ž N3 q 1 . < y 1 y y 2 < , ey b < n< < bn1 y bn2 < F bni s
3 Lq Ž N3 q 1 . e Mq 2 b M Ž e b y 1.
`
Ý
ksy`
jki s Zkq1
tgR, Ž 55 .
< y1 y y 2 < ,
Ž 56 .
G Ž n, k q 1 . jki , kq1
Hk
Zy1 Ž u . g Ž u, x i Ž u . , yi Ž u . . du.
Therefore relations Ž43., Ž55., and Ž56. imply that H is a contraction on C and so there exists a unique y g C such that H y s y.
Ž 57 .
86
GARYFALOS PAPASCHINOPOULOS
From Ž50., Ž51., and Ž57. it follows that Ž x Ž t ., y Ž t .., x Ž t . satisfies the problem Ž49., is a solution of Ž1. which satisfies Ž44.. Suppose that there exists another solution Ž x Ž t ., y Ž t .. of Ž1. such that sup < y Ž t . < , t g R 4 - `.
x Ž 0. s x 0 , Then y Ž t . satisfies
y Ž t . s Z Ž t . Zy1 n q
ž
q ZŽ t .
t y1
Hn Z
t y1
Hn Z
/
Ž u . C Ž u . du yn
Ž u . g Ž u, x Ž u . , y Ž u . . du.
Ž 58 .
From the continuity of the function y Ž t . it follows that yn satisfies ynq 1 s R n yn q j n ,
j n s Znq1
nq1
Hn
Zy1 Ž u . g Ž u, x Ž u . , y Ž u . . du,
ngZ.
Then since Ž41., Ž46., and Ž47. hold, from Lemma 2.7 w8, p. 272x we obtain yn s
`
Ý
ksy`
G Ž n, k q 1 . jk .
Ž 59 .
Therefore from Ž58. and Ž59. it follows that y is a fixed point of H . Then x Ž t . s x Ž t ., y Ž t . s y Ž t .. This completes the proof of the lemma. In the following proposition we find conditions so that the integral manifold for Ž1. exists. PROPOSITION 1. Suppose that all the conditions of Lemma 3 are satisfied. Moreo¨ er consider that the constant q defined in Ž42. satisfies the inequality q - min
½
1 2g
,
MeyM
v Ž M q d . q 2 Ž N2 q e . p q 1
5
,
Ž 60 .
g s N1 e M M y 2 Ž M q Le M q 2 b Ž M q d . f Ž b .., v s Le M M y 1 ŽŽ1 q 2 pe . f Ž2 b . q 2 pN2 e 2 bf Ž b .., and N1 , N2 are defined in Ž20., b , d , f Ž?. in Ž38., and M, p, L in Ž37., Ž39., Ž43., respecti¨ ely. Then there exists a function ¨ : R = R r ¬ R k which determines an integral manifold for Ž1. and for all t g R, x g R r , x 1 , x 2 g R r the function ¨ Ž t, x . satisfies the relations ¨ Ž t , x . F ml ,
¨ Ž t , x1 . y ¨ Ž t , x2 . F u < x1 y x2 <,
where m Ž resp. l. is defined in Ž41. Ž resp. Ž52.. and u s g qrŽ1 y g q ..
Ž 61 .
87
INTEGRAL MANIFOLD
Proof. Let C be the space of the functions ¨ : R = R r ¬ R k continuous and bounded on R = R r such that the relations Ž61. hold. From Ž60. it is obvious that u - 1. Then if ¨ g C relations Ž39. and Ž61. imply that the relation f Ž s, x 1 , ¨ Ž s, x 1 . . y f Ž s, x 2 , ¨ Ž s, x 2 . . F 2 p < x 1 y x 2 <
Ž 62 .
holds for all s g R, x 1 , x 2 g R r. Therefore since Ž4., Ž40., and Ž62. hold, from Lemma 1 the equation x9 Ž s . s B Ž s . x Ž w s x . q f Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ,
sgR
Ž 63 .
has a unique solution x Ž s . s x Ž s, t, x , ¨ . such that x Ž t . s x , t g R, x g R r. If n s w t x we define the operator K on C as K ¨ Ž t , x . s Z Ž t . Zy1 n q
ž
q ZŽ t . b Ž n, ¨ , x . s
`
t y1
Hn Z
t y1
Hn Z
Ž s . C Ž s . ds b Ž n, ¨ , x .
/
Ž s . g Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ds,
G Ž n, m q 1 . zm ,
Ý msy`
zm s Zmq1
Ž 64 .
Ž 65 . mq1
Hm
Zy1 Ž s . g Ž s, x Ž s . , ¨ Ž s, x Ž s . . . ds,
where ZŽ t . Žresp. GŽ n, m.. is defined in Ž3. Žresp. Ž47.. and x Ž s . s x Ž s, t, x , ¨ .. From Ž38., Ž41., Ž46., Ž47., Ž64., and Ž65. it holds that K ¨ Ž t , x . F ml ,
t g R, x g R r .
Ž 66 .
Consider now x 1 , x 2 g R r and let x 1Ž s . s x 1Ž s, t, x 1 , ¨ ., x 2 Ž s . s x 2 Ž s, t, x 2 , ¨ . be the solutions of Ž63. such that x 1Ž t . s x 1 , x 2 Ž t . s x 2 . Then since Ž4., Ž40., and Ž62. hold, by setting h1Ž s, x . s h 2 Ž s, x . s f Ž s, x , ¨ Ž s, x .. and E s 0 in Lemma 2 we take x 1 Ž s . y x 2 Ž s . F N1 < x 1 y x 2 < e b < myn< ,
s g R,
Ž 67 .
where m s w s x. Moreover from Ž42. and Ž61. we get for all s g R g Ž s, x 1 Ž s . , ¨ Ž s, x 1 Ž s . . . y g Ž s, x 2 Ž s . , ¨ Ž s, x 2 Ž s . . . F q Ž 1 q u . x1Ž s . y x 2 Ž s . .
Ž 68 .
88
GARYFALOS PAPASCHINOPOULOS
Hence using Ž38., Ž46., Ž64., Ž67., Ž68. we have for t g R, K ¨ Ž t , x 1 . y K ¨ Ž t , x 2 . F Ž M q d . My1 e M b Ž n, ¨ , x 1 . y b Ž n, ¨ , x 2 . q Ž 1 q u . qN1 My1 e M < x 1 y x 2 < .
Ž 69 .
Moreover relations Ž46., Ž47., Ž65., Ž67., Ž68. imply that b Ž n, ¨ , x 1 . y b Ž n, ¨ , x 2 . F Ž 1 q u . LN1 My1 qe Mq2 bf Ž b . < x 1 y x 2 < .
Ž 70 . Hence from Ž69. and Ž70. it follows that K¨ Ž t , x1 . y K¨ Ž t , x 2 . F u < x1 y x 2 <.
Ž 71 .
Arguing as in Lemma 3 we can prove that K ¨x Ž t . s K ¨ Ž t, x . is continuous on R for every x g R r. Furthermore for all t, t 0 g R, x , x 0 g R r it is obvious that K¨ Ž t , x . y K¨ Ž t0 , x 0 . F K¨ Ž t , x . y K¨ Ž t , x 0 . q K¨ Ž t , x 0 . y K¨ Ž t0 , x 0 . .
Ž 72 .
Then it is obvious that relations Ž71. and Ž72. imply that KŽ t, x . is continuous on R = R r and so from Ž66. and Ž71. K is in C . We prove now that K is a contraction on C . Let ¨ 1 , ¨ 2 g C . Then using Ž39. and the relation u - 1, for all s g R, x 1 , x 2 g R r it follows that f Ž s, x 1 , ¨ i Ž s, x 1 . . y f Ž s, x 2 , ¨ i Ž s, x 2 . . F 2 p < x 1 y x 2 < ,
i s 1, 2.
Ž 73 . Moreover from Ž61. for all s g R, x 1 , x 2 g R r it holds that ¨ 1 Ž s, x 1 . y ¨ 2 Ž s, x 2 . F u < x 1 y x 2 < q < ¨ 1 y ¨ 2 < ,
Ž 74 .
where < ¨ 1 y ¨ 2 < s sup< ¨ 1Ž s, x . y ¨ 2 Ž s, x .<, s g R, x g R r 4 . Hence from Ž39. and the relation u - 1 we get f Ž s, x 1 , ¨ 1 Ž s, x 1 . . y f Ž s, x 2 , ¨ 2 Ž s, x 2 . . F 2 p < x 1 y x 2 < q p < ¨ 1 y ¨ 2 < .
Ž 75 . Let x i Ž s . s x i Ž s, t, x , ¨ i ., i s 1, 2, be the solution of Ž19. with h i Ž s, x . s f Ž s, x , ¨ i Ž s, x .., i s 1, 2, such that x 1Ž t . s x 2 Ž t . s x . Since Ž4., Ž40., Ž73.,
89
INTEGRAL MANIFOLD
and Ž75. hold, from Lemma 2 we have x 1 Ž s . y x 2 Ž s . F Ž N2 e b < myn < q e . p < ¨ 1 y ¨ 2 < ,
s g R.
Ž 76 .
Furthermore, using Ž42., Ž74., and u - 1 we obtain g Ž s, x 1 Ž s . , ¨ 1 Ž s, x 1 Ž s . . . y g Ž s, x 2 Ž s . , ¨ 2 Ž s, x 2 Ž s . . . F 2 q x1Ž s . y x 2 Ž s . q q < ¨ 1 y ¨ 2 < .
Ž 77 .
Then for all t g R and x g R r relations Ž38., Ž46., Ž47., Ž64., Ž65., Ž76., Ž77. imply that K ¨ 1 Ž t , x . y K ¨ 2 Ž t , x . F Ž M q d . My1 e M b Ž n, ¨ 1 , x . y b Ž n, ¨ 2 , x . q qe M My1 Ž 1 q 2 Ž N2 q e . p . < ¨ 1 y ¨ 2 < , Ž 78 . b Ž n, ¨ 1 , x . y b Ž n, ¨ 2 , x . F q v < ¨ 1 y ¨ 2 < .
Ž 79 .
Therefore from Ž60., Ž78., and Ž79. it is obvious that K is a contraction on C and so there exists a unique ¨ g C such that K¨ s ¨ .
Ž 80 .
So using Ž64., Ž65., and Ž80. we can show that Ž x Ž t ., ¨ Ž t, x Ž t .. , x Ž t . satisfies Ž63., is a solution of Ž1.. Suppose now that Ž x Ž t ., y Ž t .. is a solution of Ž1. such that sup< y Ž t .<, t g R4 - `. Then using Lemma 3 we have that y Ž t . s ¨ Ž t, x Ž t .., t g R. Therefore ¨ Ž t, x . determines an integral manifold for Ž1.. This completes the proof of the proposition.
ACKNOWLEDGMENT I thank the referee for his valuable suggestions.
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