0387986995Basic TheoryC
.pdf408 XIII. SINGULARITIES OF THE SECOND KIND
Observation XIII-2-3. Set
(XIII.2.9) g1(x,u") = u, uj +R1(x,u') (j = 1,2,... ,n).
Then, by virtue of Assumption III, (XIII.2.7), and (XIII.2.8), for sufficiently small positive numbers No 1 and 5, there exists a positive constant c, independent of j, such that for any positive integer h, the estimates
(XIII.2.10) |
lR3(x,u)I ( clui |
+bhlxl-(n+1) |
(j = 1,2,... n) |
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and |
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(XIII.2.11) |
IR,(x,u) - R, (x,1U') I < clu"- u l |
(j = 1,2,... ,n) |
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hold whenever (x, u") and (x, u"') are in the domain |
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(XIII.2.12) |
lxl > No, Iargxl <oo, luil <6 (j =1,2,... ,n). |
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Here, by, is a positive constant depending on h. Furthermore, by virtue of Assump- tion II, it can be assumed without loss of generality that the constant c satisfies the condition
(XIII.2.13) |
c< r+1 |
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H |
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where H is a positive constant to be specified later. |
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From (XIII.2.5) and (XIII.2.9), it follows that |
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(XIII.2.14) |
d |
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ds = X'" [µuj + R, (x, u)] |
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Change system (XII/I.2.14) to a system of integral equations |
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(XIII.2.15) |
u, = Jt trR, (t, u) exp Ir - |
1(tr+l |
_ xr+1)1 |
dt |
(j = 1, 2,... , n), |
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where the paths of integration L3 start from x. The paths of integration must be chosen carefully so that uniformly convergent successive approximations can be defined in such a way that the limit is a solution uf(x) of (XIII.2.15) which satisfies the conditions (i) u , (x) (j = 1, 2, ... , n) are holomorphic in D(N,'y) and (ii) u., (x) ^ _ - 0 (j = 1, 2, ... , n) as x - oo in D(N, ry) for suitable positive numbers
N and ry.
Hereafter in this section, we explain how to choose paths of integration on the right-hand side of (XIII.2.15).
Observation XIII-2-4. Since for each j, the domain Dj (N, y, q,) contains the real half-line defined by x > N in its interior, their common part D(N, ry) is given by the inequalities
(XIII.2.16) |
l xI > N, -t < argx < e', |
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XIII. SINGULARITIES OF THE SECOND KIND |
in )(N). This implies that - 2 + 2 < wj + O(l:) < 2 - 2 The function O(C) is defined in a way similar to the previous case.
(c) For j E C3, from (XIII.2.38.3), it follows that
-17r+ I |
< wj - (r+ 1)e- 'y'+1r 17r< |
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lry'-7r< 7- 1>' |
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Let |
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o = N''+1 exp [i{(r+1)t'+.i'_ 2rr1 . |
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In the case when (r + 1)(t + e') >_ 2a - ry', the inequality |
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(XIII.2.41) |
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arglo ? argf2, |
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holds, whereas in the case when (r + 1)(t + e') < 2a - ', it holds that
(XIII.2.42) |
arg o < arg t;2 |
In the case when (XIII.2.41) holds, the lines Lj( are defined by (XIII.2.39) with
-(r+1)e- 2y'+a |
(r+1)e'+ 2ry'-7r |
in D(N). This implies that - 2 + 2 < wj + B({) < 2 - 2 . Since to is on arc
(A-3), O(l;) is defined as given above. In the case when (XIII.2.42) holds, the lines
Ljf are defined by (XIII.2.39) with
(r+1)e'+ try'-7r <9({) -(r+l)e- Zry'+7r
in D(N). This implies that - 2 + |
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Y. |
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< wj + 9({) < 2 - 2 |
In this case, o is |
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not on arc (A-3), but 8(e) is still defined in a way similar to the previous cases.
For all of the cases considered above, we prove the following lemma.
Lemma XIII-2-6. There exists a positive constant H independent of h such that the inequalities
(XIII.2.43) |
I Isi-11 I exp[ - r+ 1 a11 Ids) <Hjl:I-h l exp 1r l ] |
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(j = 1,2,... n) |
hold in a domain D(Nh) for any positive number h, where Nh is a sufficiently large positive constant depending only on h.
3. PROOF OF THEOREM XIII-1-2 |
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Consider a function Y(r) = (D2 + r2)h/2exp(Mjr) |
xp+ M, ) da, where |
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M2 = 1'U2I `os(+ + OW). Note that b > 0. Hence, for r > 0, (XIII.2.47) implies that Y(r) < M . On the other hand, Y(r) satisfies the differential equation dY =
M + h |
Y -1. Since |
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< 1 , it follows that |
dY > |
lit, Ib |
Y-1 |
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D2 + r2 } |
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D2+72 |
2D |
dr |
2(r + 1) |
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if M > h(r + 1) . |
Here, a use was made of the inequality M < D. Since Y(0) < |
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1i2 1b |
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- |
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2(r i 1) (r < 0). Therefore, (XIII.2.43) follows. |
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(r + 1) we obtain Y(r) < |
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M2 |
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1A2 ib |
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1µ, b |
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2(r + 1) |
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This completes the proof of Lemma XIII-2-6. In this case, again, H = |
µb , |
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where µ =min{µ2 }. 0
2
We fix the paths of integration on the right-hand sides of (XIII.2.15) as explained above. The constant H in (XIII.2.13) is chosen to be equal to the constant H given in Lemma XIII-2-6. Note that H is independent of h.
XIII-3. Proof of Theorem XIII-1-2
Let us construct a solution u2 = 02 (x) of (XIII.2.15) so that
(XIII.3.1) |
O2 (x) 0 |
(j = 1,2,... n) |
as x oo in (XIII.1.15).
Observation XIII-3-1. As in the previous section, denote by Do(Nh) the domain in the x-plane which corresponds to the domain D(Nh) in the -pllane. The domain
Do(Nh) depends on h. Set o i = Nh+1 exp {i(r + 1)(e + |
and denote by |
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xoh) the corresponding point in Do(Nh ). Also, setting h' = |
r + 1 |
(h = 1,2,3.... ), |
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consider the domain Do(Nh,). As mentioned in the previous section, the constant
H in (XIII.2.13) is chosen to be equal to the constant H in Lemma XIII-2-6.
Choose a positive constant 8 so that r H 1 { cb + Nh2 <_ S. Furthermore, by
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virtue of (XIII.2.13), Nh, can be chosen so large that Nh, > No and b < d (cf.
(XIII.2.12)). Fix Nh, in this way. Then, by a method similar to that in §XII-3 for each j (j = 1, 2, ... , n), successive approximations can be defined to construct a solution u2 = O,(x) of (XIII.2.15) which is holomorphic in Do(Nh-) and 101(x)l <
(j = 1,2,... ,n), where = xT+'. In this way, the existence of a bounded solution u2 = 02 (x) (j = 1, 2, ... , n) of system (XIII.2.15) is proved.