ˆ

Otherwise there would be a sequence {v k } ‚ B such that

|v k ’ u|w ’ 0 (G (v k ), h(u)) ¤ δ.

(15.18) and

180 15. Weak Sandwich Pairs

ˆ

Since B is bounded in E, v k ’ u weakly in E and (10.5) implies that

(G (v k ), h(u)) ’ (G (u), h(u)) ≥ 2δ

(15.19)

˜ ˆ

in view of (15.14). This contradicts (15.17). Let B be the set B with the inherited

˜ ˜

topology of E. It is a metric space, and W (u) © B is an open set in this space. Thus,

˜ ˜ ˜

{W (u) © B}, u ∈ B, is an open covering of the paracompact space B (cf., e.g., [77]).

Consequently, there is a locally ¬nite re¬nement {W„ } of this cover. For each „, there

is an element u „ such that W„ ‚ W (u „ ). Let {ψ„ } be a partition of unity subordinate

to this covering. Each ψ„ is locally Lipschitz continuous with respect to the norm |u|w

and consequently with respect to the norm of E. Let

˜

Y (u) = ψ„ (u)h(u „ ), u ∈ B.

(15.20)

Then Y (u) is locally Lipschitz continuous with respect to both norms. Moreover,

Y (u) ¤ ψ„ (u) h(u „ ) ¤ 1

(15.21)

and

ˆ

(G (u), Y (u)) = ψ„ (u)(G (u), h(u „ )) ≥ δ, u ∈ B.

(15.22)

ˆ

For u ∈ ¯ © E, let σ (t)u be the solution of

σ (t) = ’Y (σ (t)), t ≥ 0, σ (0) = u.

(15.23)

ˆ

Note that σ (t)u will exist as long as σ (t)u is in B. Moreover, it is continuous in (u, t)

with respect to both topologies.

ˆ ˆ

Next, we note that if u ∈ ¯ © E, we cannot have σ (t)u ∈ B and G(σ (t)u) > b0 ’δ

for 0 ¤ t ¤ T : for by (15.23), (15.22),

d G(σ (t)u)/dt = (G (σ ), σ ) = ’(G (σ ), Y (σ )) ¤ ’δ

(15.24)

ˆ ˆ

as long as σ (t)u ∈ B. Hence, if σ (t)u ∈ B for 0 ¤ t ¤ T , we would have

G(σ (T )u) ’ G(u) ¤ ’δT = ’(a0 ’ b0 + 4δ).

(15.25)

Thus, we would have G(σ (T )u) < b0 ’ 4δ. On the other hand, if σ (s)u exists for

ˆ

0 ¤ s < T, then σ (t)u ∈ B. To see this, note that

t

u ’ σ (t)u = z t (u) := Y (σ (s)u)ds.

(15.26)

0

By (15.21),

z t (u) ¤ t.

(15.27)

Consequently,

σ (t)u ¤ u + t < R.

(15.28)

15.2. Weak sandwich pairs 181

ˆ ˆ

Thus, σ (t)u ∈ B. We can now conclude that for each u ∈ ¯ © E, there is a t ≥ 0 such

that σ (s)u exists for 0 ¤ s ¤ t and G(σ (t)u) ¤ b0 ’ δ. Let

ˆ

u ∈ ¯ © E.

Tu := inf{t ≥ 0 : G(σ (t)u) ¤ b0 ’ δ},

(15.29)

Then σ (t)u exists for 0 ¤ t ¤ Tu and Tu < T . Moreover, Tu is continuous in u. De¬ne

σ (t)u, 0 ¤ t ¤ Tu ,

σ (t)u =

ˆ

σ (Tu )u, Tu ¤ t ¤ T,

ˆ ˆ

for u ∈ ¯ © E. For u ∈ ¯ \ E, de¬ne σ (t)u = u, 0 ¤ t ¤ T . Then σ (t)u is continuous

ˆ ˆ

in (u, t), and

u ∈ ¯.

G(σ (T )u) ¤ b0 ’ δ,

ˆ

(15.30)

Let

v ∈ ¯,

•(v, t) = F σ (t)v,

ˆ 0 ¤ t ¤ T.

(15.31)

Then • is a continuous map of ¯ — [0, T ] to N. Let

¯

K = {(u, t) : u = σ (t)v, v ∈ Q, t ∈ [0, T ]}.

ˆ

˜

Then K is a compact subset of E — R. To see this, let (u k , tk ) be any sequence in K .

¯

Then u k = σ (tk )v k , where v k ∈ Q. Since Q is bounded, there is a subsequence such

¯

that v k ’ v 0 weakly in E and tk ’ t0 in [0, T ]. Since Q is convex and bounded, v 0 is

¯ ˜

in Q and |v k ’ v 0 |w ’ 0. Since σ (t) is continuous in E — R, we have

ˆ

u k = σ (tk )v k

ˆ σ (t0 )v 0 = u 0 ∈ K .

ˆ

ˆ ˜

Each u 0 ∈ B has a neighborhood W (u 0 ) in E and a ¬nite-dimensional subspace S(u 0 )

ˆ

such that Y (u) ‚ S(u 0 ) for u ∈ W (u 0 )© B. Since σ (t)u is continuous in (u, t), for each

ˆ

˜

(u 0 , t0 ) ∈ K , there are a neighborhood W (u 0 , t0 ) ‚ E — R and a ¬nite-dimensional

subspace S(u 0 , t0 ) ‚ E such that z t (u) ‚ S(u 0 , t0 ) for (u, t) ∈ W (u 0 , t0 ), where

ˆ

ˆ

t

Y (σ (s)u)ds,

ˆ u ∈ E,

0

z t (u) := u ’ σ (t)u =

ˆ ˆ

(15.32)

ˆ

u ∈ E.

0,

Since K is compact, there are a ¬nite number of points (u j , t j ) ‚ K such that K ‚

W = ∪W (u j , t j ). Let S be a ¬nite-dimensional subspace of E containing p and all