ˆ

By hypothesis, there is a ¬nite-dimensional subspace S0 = {0} of N containing p such

that F(v ’ z t (v)) ∈ S0 for all v ∈ ¯ © S0 . We note that •(u, t) maps ¯ © S0 — [0, T ]

ˆ

into S0 . For t in [0, T ], let •t (v) = •(v, t). Then

•t (v) = p, v ∈ ‚( © S0 ) = ‚ © S0 , 0 ¤ t ¤ T.

(15.33)

To see this, note that if v ∈ ‚ , then

v ’ p ¤ v ’ F σ (t)v + F σ (t)v ’ p .

ˆ ˆ

182 15. Weak Sandwich Pairs

Hence,

F σ (t)v ’ p > K T + δ ’ t K > 0,

ˆ v ∈‚ , 0¤t ¤T

(15.34)

since

t

F σ (t)v ’ v ¤ K

ˆ σ (s)v ds ¤ K t.

ˆ

0

Thus, (15.33) holds. Consequently, the Brouwer degree d(•t , © S0 , p) is de¬ned.

Since •t is continuous, we have

d(•T , © S0 , p) = d(•0 , © S0 , p) = d(I, © S0 , p) = 1.

(15.35)

Hence, there is a v ∈ such that F σ (T )v = p. Consequently, σ (T )v ∈ F ’1 ( p) = B.

ˆ ˆ

In view of (15.7), this implies

G(σ (T )v) ≥ b0 ,

ˆ

contradicting (15.30). Thus, (15.8) holds, and the proof is complete.

We can now give the proof of Theorem 15.2.

Proof. We take A = N, B = M, p = 0, and F = PN , the projection onto N. If

S is a ¬nite-dimensional subspace such that F S = {0}, we take S0 = F S. All of the

hypotheses of Theorem 15.4 are satis¬ed.

De¬nition 15.5. Let E, F be Banach spaces. We shall call a map J ∈ C(E, F) weak-

to-weak continuous if, for each sequence

u k ’ u weakly in E,

(15.36)

there exists a renamed subsequence such that

J (u k ) ’ J (u) weakly in F.

(15.37)

We have

Proposition 15.6. If A, B is a weak sandwich pair and J is a weak-to-weak continuous

diffeomorphism on the entire space having a derivative J (u) depending compactly on

u and satisfying

J (u)’1 ¤ C, u ∈ E,

(15.38)

then JA, JB is a weak sandwich pair.

Proof. Let G be a weak-to-weak continuously differentiable functional on E satisfying

’∞ < b0 := inf G ¤ a0 := sup G < ∞.

(15.39)

JB JA

Let

G 1 (u) = G(J u), u ∈ E.

15.2. Weak sandwich pairs 183

Then

(G 1 (u), h) = (G (J u), J (u)h).

If u k ’ u weakly, then there is a renamed subsequence such that

J (u k ) ’ J (u) weakly; J (u k ) ’ J (u).

Hence,

(G 1 (u k ), h) ’ (G (J u), J (u)h),

and G 1 is weak-to-weak continuously differentiable. Moreover,

’∞ < b0 := inf G = inf G(J u) = inf G 1

(15.40)

JB J u∈ J B B

¤ a0 := sup G = sup G(J u) = sup G 1 < ∞.

JA J u∈ J A A

Since A, B form a weak sandwich pair, there is a sequence {h k } ‚ E such that

G 1 (h k ) ’ c, b0 ¤ c ¤ a0 , G 1 (h k ) ’ 0.

(15.41)

If we set u k = J h k , this becomes

G(u k ) ’ c, b0 ¤ c ¤ a0 , G (u k )J (h k ) ’ 0.

(15.42)

In view of (15.38), this implies G (u k ) ’ 0. Thus, J A, J B is a sandwich pair.

Proposition 15.7. Let N be a closed subspace of a Hilbert space E with complement

M = M •{v 0 }, where v 0 is an element in E having unit norm, and let δ be any positive

number. Let •(t) ∈ C 1 (R) be such that

0 ¤ •(t) ¤ 1, •(0) = 1,

and

•(t) = 0, |t| ≥ 1.

Let

(15.43) F(v + w + sv 0 ) = v + [s + δ ’ δ•( w 2 /δ 2 )]v 0 , v ∈ N, w ∈ M, s ∈ R.

Then A = N = N • {v 0 }, B = F ’1 (δv 0 ) forms a weak sandwich pair.

Proof. De¬ne

J (v + w + sv 0 ) = v + w + [s ’ δ + δ•( w 2 /δ 2 )]v 0 , v ∈ N, w ∈ M, s ∈ R.

Then J is a diffeomorphism on E satisfying the hypotheses of Proposition 15.6. More-

over, A = J N and B = J [M + δv 0 ]. Since N and M + δv 0 form a weak sandwich

pair by Theorem 15.2, we see that A, B also form a weak sandwich pair (Proposi-

tion 15.6).

184 15. Weak Sandwich Pairs

15.3 Applications

Let A, B be positive, self-adjoint operators on L 2 ( ) with compact resolvents, where

‚ Rn . Let F(x, v, w) be a Carath´ odory function on — R2 such that

e

f (x, v, w) = ‚ F/‚v, g(x, v, w) = ‚ F/‚w

(15.44)

are also Carath´ odory functions satisfying

e

| f (x, v, w)| + |g(x, v, w)| ¤ C0 (|v| + |w| + 1), v, w ∈ R,

(15.45)

and

f (x, t y, tz)/t ’ ±+ (x)v + ’ ±’ (x)v ’ + β+ (x)w+ ’ β’ (x)w’ ,

(15.46)

g(x, t y, tz)/t ’ γ+ (x)v + ’ γ’ (x)v ’ + δ+ (x)w+ ’ δ’ (x)w’

(15.47)