5.5 Appendix
5.5.2 Appendix II: The Proof of Lemma 5.16
This appendix gives the proof of Lemma 5.17. The proof consists of several claims which will be proved respectively.
Claim 1: If y∈ [y5y1, then y· s will exit W on the boundary of the regions R or P . Proof. This directly follows by Lemma 5.9
Claim 2: If y∈ [y3y4, then y· s will immediately enter the region Q.
Proof. Let y(0) ∈ [y3y4. For small ε > 0, we have u(0) > u∗ and 0 < w < w∗, which yields to z0(0) < 0 and z(0+) < 0. As a result, it will enter Q immediately.
Claim 3: If y∈ [y2y3, then y· s will immediately enter the region S.
Proof. To begin with, we show that g(w)− p(u) > 0 provide y ∈ [y2y3. Indeed, by Mean Value Theorem, we have
g(w)− p(u)
ε = g0(w0)w− p0(u0)(u− K)
= g0(w0)[−ψ(λ2) sin θ sin φ− ψ(λ3) cos φ]
−p0(u0)[− cos θ sin φ − sin θ sin φ − cos φ] + O(ε).
(5.67) for some u0, w0and (u0, w0) approaches to (K, 0) as ε approaches to 0. Since v = 0, we have
− cos θ sin φ = λ3
λ1cos φ + λ2
λ1sin θ sin φ + O(ε).
Substituting it into equation (5.67), we have The coefficient of sin θ sin φ can be shown to be positive by showing that
−g0(0)ψ(λ2)− p0(K)λ2− λ1
The coefficient of cos φ can be shown to be positive by showing that λ3ψ(λ3)
Let y(0) ∈ [y2y3, we have we have u(0) > u∗, v(0) = 0 and g(w(0))− p(u(0)) > 0 for ε > 0 small enough, which yields to, we have v0(0) > 0 and v(0+) > 0. As a result, it will enter the region S immediately.
Claim 4: If y∈ [y1y2, then y· s will exit W on the boundary of the regions R, P or S.
Proof. Let y(0)∈ [y2y3, we have u(0) > u∗, v(0) < 0, w(0) > 0, z(0) > 0. Since z0= cz + l(w)q(u), we have z0(s) > 0 until u(s) = u∗. If there exist s0> 0 such that u(s0) = u∗, it will tough the boundary P1, P12 or R1 and enters P , R or S as discussing in the last section. On the other hand, if u(s) > u∗ for all time s > 0, we have w is increasing to infinity. There exist s1 > 0 such that w(s) > w∗ for all time s > s1. As a result, g(w(s))− p(u(s)) > 0 for all time s > s1. If there is s2 > s1 such that v(s2) > 0, it enters the region S. If v(s) ≤ 0 for all s > s1, we have v and u are bounded above.
However, it contradicts to dv0 = cv + h(u)[g(w)− p(u)].
Claim 5: If y∈ [y4y5, then y· s will exit W on the boundary of the regions R.
Proof. We know from Lemma 5.5 that y0· s will enter the region R, which is an open set. Since [y4y5
is very close to y0, the assertion directly follows from the continuous dependence of the solution on the initial data.
Claim 6 The set F (Σ) is not simply connected.
Proof. This proof follows Dunbar’s idea in the Appendix II of [9] with a slight modification. Let Λ =: P∪ Q ∪ R ∪ S. For a fixed w, the projection of the sets of Λ on the u-v-z space is shown in Figure 14 for w > w∗, Figure 13 for w = w∗, and Figure 12 for w < w∗. The coordinate v of each figure is then ”compressed” in the subspace v = 0 respectively by a strong deformation retraction, as shown in Figure 15. Synthesizing the three cases of Figure 15 in u-w-z space yields to Figure ??, where the deformation retraction of ∂Λ is the boundary of the two wedges
{u > u∗, g(w)− p(u) < 0, z > 0} and {0 < u < u∗, g(w)− p(u) > 0, z < 0}.
The deformation retraction of F (∂Σ) must lie in the boundary of the two wedges. The Claim 1 to 5 imply that the boundary ∂Σ will be mapped to a closed curve visiting R, P , S, and Q in turns at least once. It follows that the deformation retraction of F (∂Σ) surrounds (u = u∗, w = w∗, z = 0), and cannot be homotopic to a point in W− since W−dose not contain (u = u∗, w = w∗, z = 0). Thus F (Σ) is not simply connected.
u v
z
u∗ p−1(g(w)) S¯
Q¯
R¯
2
Figure 12: The graph of Λ in the case w < w∗
u v
z
u∗
S¯ P¯
R¯
Q¯
Figure 13: The graph of Λ in the case w = w∗
u
Figure 16: The deformation retrace of Λ in the u-w-z space
68
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