systems to study the matched asymptotic expansion for the singularly perturbed boundary .... method of asymptotic matching principle originated by Van Dyke.
SIAM J. AI, Vol. 49, No.
Ma. pp. 26-54, February 1989
(C) 1989 Society for Industrial and Applied Mathematics 002
SHADOWING LEMMA AND SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS* XIAO-BIAO LINt Abstract. A complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in I". The validity of such expansions is deduced from a generalized Shadowing Lemma, where the inner and outer approximations are treated like pseudo-orbits in the classical dynamical system theory.
Key words, shadowing lemma, Fredholm alternatives, exponential diehotomics, inner and outer expansions, matching principles AMS(MOS) subject classifications, 34E15, 34B15, 34C35
1. Introduction. In the past few years, several people have attempted to bring some of the methods of dynamical systems to bear on singularly perturbed boundary value problems, e.g., for centermanifolds see Fenichel [7] and Carr and Pego (unpublished manuscript), for the Lyapunov-Schmidt method see Hale and Sakamoto [12]. The early work of Hoppensteadt [13] also used the idea of dynamical systems. In this paper we shall use the method developed in the theory of dynamical systems to study the matched asymptotic expansion for the singularly perturbed
boundary value problem
e =f(x, t, e), (1.1)
a
""
d /. We -->
d+ is
qg’(’r) f(qi(’r), 6, 0)q(7")=0
(2.3)i and the adjoint equation
.i’(7")+ fx(qi(7"),
(2.4)i
ti,
are important in our investigations. Let Pl(r), z
of (2.3)0 and p(r),
(H2)
0)1//(7")--0
+ be any nontrivial bounded solution
z- be any nontrivial bounded solution of (2.3); then
Blx(qo(O), 0)" qg,(O) 0,
B(q(O), O)p(O) O.
It should be clear that ql(r), r, 1 0,
0-O, 0 < fl < 1, if x( t, e) and g( t, e) are defined as in Theorems 2.1 and 2.2 for [0, to and are periodic with period to for then x (t, e) is a formal approximation of (2.6) with residuals and jumps as O(e(m+l)). There exists a unique exact periodic solution Xexact(t, E) of period to in a small neighborhood ofx(t, e) provided that 0 < e 0 is sufficiently small and 0< [F’I = fill, c>0 and IQ()I -< fill-’. From Lemma 3.9, CO(P-(O, e), P(O, F’))-- 1,
2MK Ii{g}llz,
k
1.
-
< 1/2, therefore k= II{g/}il < o, and Yk= il{v,}ll < Let u,(t)= [t_ t], mo < < ml" then u(t) is continuous in its domain of definition and by adding (4.4) through k 1, u(t) u+(t) g + g. Observ that Since 2KM e
k=l v(t) + v(t)
is a closed operator; thus
d{u,} {} x {g}. We have the estimate
il{ui}llRMg
,
II{gJ}llz+tl{v,}tl
(2KMe-’)
To prove uniqueness, we show that the only bounded and continuous solution of (4.1), with {}x{g}=0, is {u}0. Assume the contrary; suppose at certain t, we have 0 v Qiu(li), and I(I Q)u(t,)l IO’u(t)l, as the case I(i Q)u( can be studied similarly. First, from Qu(ti) pi(ti)Qiu(ti) + P+(ti)Oiu(ti), applying i to the equation, we have
IOiu( ti)l IOiP+’( t)Oiu( ti)l pr(to be a projection with P1s(O). Assume that [Qr 0.
j=0
This relation is denoted by E{f(t, e)} and f is said to be in the domain of
#%(t), E is called the expansion operator, =o E, and shall be called an asymptotic sum of
=:o ite%(t). is immediately obvious from Taylor’s formula that if each (Oi/Oei)f(t, e) exists
and is a continuous function of x e, then f(t, e) is in the domain of E. The following lemma shows that E has a right inverse and is a generalization of a lemma of Borel and Ritt. The proof shall be omitted. LEMMA 5.1. There exists an (nonunique) asymptotic sum f(t, e), J, e for >each formal power series each Moreover, t). O, e) (c)/Oei)f( t, exists and #%( for is continuous in and e. Two functions, f(t, e) and g(t, e), both in the domain of E, are said to be asymptotically equivalent, denoted by f(t, e) g(t, e), if and only if E (f(t, e)) E (g(t, e)). For any formal power series .:oe%(t), E-l(j:oe.’q(t)) forms a nonempty
=o
42
XIAO-BIAO LIN
equivalence class and shall be denoted by
[.ioeJpj(t)],
or [f(t,e)] with
[E)o eJtPj( t)].
fE
Let F(u, u2,’’’, Uk, t, e) be C in all the variables. Assume that f(t,e)--gi(t, e) [.--o #j(t)], i= 1,..., k. Then
F(fl(t, e),
,fk(t, e), t, e) F(gl(t, e),
gk(t, e), t, e).
Therefore,
e@(t)
F
ep(t)
,.’.,
=0
[F(f(t,e),’’’,fk(t,e),t,e)]
,t,e
=0
is well defined.
.,
DEFINITION 5.2. Define F (j=o e Jqj(t),. j=o eJq(t), t, e) as a formal power the of which is expansion asymptotic series j=o eJg/j(t), F([E)=o e o (t)],. [j=o #q(t)], t, e). The relation shall be denoted by
,
F
(5.1)
j o e tpj(t),"
E=o e"q;(t), t, e
)
E=o eJd/j(t)
It is now clear that termwise summation and multiplication by scalar functions of formal power series, as well as multiplication of two formal power series, can be defined by using Definition 5.2. Moreover, if each oj(t)C we define We remark here that this definition is not merely d/dtYq=o#qj(t) defYj= o eJi(t). formal, i.e., there exists at least one function f(t,e)_[Yq=o#q(t)] such that
,
(d/dt)f(t, e) [E-_-o ej(t)]. LEMMA 5.3. Each term (t) of (5.1) can be computed recursively by the following equation"
E eJqj(t) =F
(5.2)
j=O
,
eJqj(t) \j=O
eJq(t), t,e +O(e "+’)
j=O
m k with mi >= m, 1 = 1,
’
SHADOWING LEMMA AND SINGULAR PERTURBATIONS
possesses an exponential dichotomy in [+, with the projections being Any solution of (5.8)j in Ee+(O,j) can be written as follows: 0 o + y(z) T(z, 0) P.,.(0)yj(0)
45
P(s) and P(s).
T(’r, s)P(s)Gj(’’ .)(s) ds
(5.12)j
+
Joo T(r’ s)P(s)Gj(’"
.)(s) ds.
We refer the verification of (5.12)j to Lemma 3.6(ii). From (5.12)j, we have o o Now substituting yjo(0 P,(0)yj(0)
+ P(0)yy(0)into (5.10)j, we have B,x(y(O), 0)P(0)y.(0) + Bl/(Yo(0), 0)P,(0)ys-(0) + H.(yl(0), ")=0. (5.13)j Notice that P(O)@Y{B,,,(y(O),O)=N by virtue of (H2). Thus, n(0)y(0) is uniquely solvable from (5.13)j. By substituting into (5.12)j, y(r) has been completely
,
0 determined. And yj(’) Ee+(O,j) by Lemma 3.6(ii). It is straightforward to verify that the formal power series j=o eJYY(’r) thus obtained is a formal solution for (5.5). We shall not render the details here. We remark that ys.(-) is determined by the growth condition at T--+oO rather than matching principles as commonly used. However, there is a matching of y.(’) with outer layer that can be proved as the consequence of our construction and that is useful in the sequel. Consider the inner expansion of the outer formal solution
(5.14) =0
=0
It is easy to show that x.(-) is a polynomial of degree ---j. We can now state the following result. THEOREM 5.5. The formal solution Yj=o e"y(-) of (5.5), with yo(’)=qo(") is uniquely computable from (5.8)j, (5.10)j, and (5.11)j reeursively. Moreover, we have y.(’) x(’) E/( % j). Proof. From (5.14) and the fact that j=o eJx)(t) formally satisfies (1.1) without boundary conditions, we easily derive that j=o eJx() formally satisfies
y’(’) =f(y(’), a + e-, e).
x(-) satisfies (5.8)o (in fact, x(r)=pl(a)=const.), and x.(-) satisfies (5.8)j 0 with (y,(r),’",yj_l(r))=(x(’),’",xj_,(r)). Assuming that y(r)-x(r)e 0l and the solution yj(r) is in the codimension one subspace is {y(7.)ly(O)+/-q’(O)}, which is complementary to Y’. From hypothesis (H3), uniquely solvable from (5.23). Once (5.23) is satisfied, from standard property of Fredholm operators, y(7.)7]i is uniquely determined by (5.20) and (5.21)j. See Lemma 3.7. e zj formally satisfy (5.17) is straightforward. The proof that eY(Z) and Similar to the boundary layers, y.(7.) is determined by the growth condition (5.21 rather than by matching principles. However, the match of yj(7.) with the outer expansion can be proved as a consequence of our construction. Consider the inner expansion of the outer formal solutions
r_
.=o
=o
eJxj,,(7.)
(5.24)
=0
j=o
E
e
j=o
We assume that each
Xj,2(7.)
E
eJx. i(i/l e,.j
ti+e 7"+
2=0 eJ
ti+e 7.+
j=o
E
ej
j=o
,," + X(t),.
(t) has been extended C to teE. However, (7.)}.j=o and {x,2(7.)}=o do not depend on the extension, xj,(7.) and Xj,2(7.) are, in
=o =o
e x,(7.)i or j=o eJxj,2(7.), fact, polynomials of degree = 1 is a constant such that [z[ < To- 1, =< =< r- 1, then [E m-lj=O ei’2[ =< To if e e 1 ao as e --> 0. We may notice that Lemma 3.2 is stated for semi-infinite intervals. However, we can extend (6.1) and (6.2) so that Lemma 3.2 applies. (3) The extension of f(t, x, e) to f(t, x, e) seems to be essential in the sequel. For < r-1, extend the definition of f(t, x, e) in a neighborhood of 1- 0 also depends on g. Suppose we can prove that there exists eo > 0 such that for all 0 < g =< o, C(g) >_- C > 0, and (6.9) is valid for all 0 < e Co, the desired results can be obtained by setting e g in (6.9). Here we refer back to the proof of Lemma 3.10 and make the folloWing observations. If Co> 0 is sufficiently small, and 0 - 0 uniformly with respect to g; and (iv) from (3.4) O(e)/Oe-Oa(O)/Oe O(e), uniformly with respect to g. Thus the dependence of g of the equation (6.7) does not matter. Let us now m-1 consider the restriction of (6.7) on [ -e/3-1 -"j=om-1 e j E/3-1 -E=o erj] If eo is small and 0 < e g Co, then j=o ej 0 is a constant. Our result in (5) also shows that O(P(-, ), P(r+,’e)) => r- 1. Let Q(r) Clel, C > 0 is a constant for all the common points t,/e, 1,’ bc the projections p(r-, e), parallel to P(r+, e), hcrc is one of the 3r-1 common points of the 3r subintervals, then [Q(r) O(1/e), 0< e 2. It should be clear no less than e/3- >_that if eo is sufficiently small, all the assumptions of Theorem 4.4 are satisfied, in
"-
e -.
r
r
,
SHADOWING LEMMA AND SINGULAR PERTURBATIONS
53
particular, Ii-11 o(1/e) and e--o(11-11-=), Therefore, we obtain a unique solution Xexact(ET’, E) in a neighborhood of the orbit of x(e’, e). It follows from the estimate in Theorem 4.4 that
(6.10)
[Xexact(ET" e)-x(er, e)]}= O(e(m+l)-l).
sup r[a/e,b/e]
We now consider Ix(r, e)- (-, e)i where (r, e) is the composite expansion in (2.5). For t[t_+e,t-e], by virtue of the fact yj--x,i--1 Eu+(%j) and y-x, Ea-(%j), Ix(r, e)-(r, e)[ O(e+). For [t_, t_ + e ] by viue of the fact =0
8
=0
j =o
j=0
=o
j=o
,
Ix(r, e)-g(r, e)l= O(e+)). This is similar for t[t-e t]. Therefore (6.) sup (IXexat(r, )--(, Recall that our approximations x(r, e) and g(r, e) depend on m, and should be denoted by x(r, e, m) and 2(r, e, m). For any m 0, we can always choose m > m such that (m + 1)-1 m + 1. It is easy to see that
,
,
Ix(t, m)-x(t, m,)l= o(+’)), I(t, We now apply (6.10) and (6.11) to x(t, e, m) and (t, e, m), and the desired estimates in Theorem 2.2 follow easily. The proof of Theorem 2.3 uses Theorem 4.5 and is analogous to those of Theorem 2.1 and 2.2. Details shall be omitted. Acknowledgments. I am grateful for many encouraging discussions with S.-N. Chow, J. Hale, F. Hoppensteadt, L. Kelly, and D. Yen. This work was done when I was visiting Michigan State University.
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