(3.28)

w∈M

and

m 1 := sup G(v) = ∞.

(3.29)

v∈N

Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞)

such that (2.8) holds. Then there are a constant c ∈ R and a sequence {u k } ‚ E such

that

G(u k ) ’ c, m0 ¤ c ¤ m1, G (u k )/ψ( Pu k ) ’ 0,

(3.30)

where P is the (orthogonal) projection onto M.

Proof. We may assume dim N < ∞; otherwise, we can consider ’G in place of G.

Let A be the set ‚ B R © N, and take B = M, where R > 0 is arbitrary. Then A links B

[hm] by Example 2 above. We now apply Theorem 2.21. Note that a0 ¤ m 1 , m 0 = b0 ,

and (2.31) holds for R suf¬ciently large. We also note that

a R ¤ sup G ¤ m 1

B R ©N

by taking (s)u = (1 ’ s)u, u ∈ E. Hence, by Theorem 2.21, for each δ > 0, there

is a u ∈ E such that

m 0 ’ δ ¤ G(u) ¤ m 1 + δ, G (u) < ψ(d(u, B )).

Since this is true for each δ > 0, we obtain the desired conclusion.

3.7. A sandwich theorem 27

An immediate consequence is

Corollary 3.18. Under the hypotheses of Theorem 3.17, there are a constant c ∈ R

and a sequence {u k } ‚ E such that

G(u k ) ’ c, m 0 ¤ c ¤ m 1, (1 + Pu k )G (u k ) ’ 0,

(3.31)

where P is the (orthogonal) projection onto M.

The following is a consequence of Theorem 2.23.

Theorem 3.19. Under the hypotheses of Theorem 3.17, for any sequence {Rk } ‚ R+

such that Rk ’ ∞, there are a constant c ∈ R and a sequence {u k } ‚ E such that

m1 ’ m0

G(u k ) ’ c, m 0 ¤ c ¤ m 1, (Rk + u k ) G (u k ) ¤ .

(3.32)

ln(4/3)

Proof. We may assume dim N < ∞. Let Ak be the set ‚ B Rk © N, and take Bk = M.

Then, for each k, Ak links Bk [hm] by Example 2 above. We now apply Theorem 2.23.

Note that ±k = Rk and ak0 ¤ m 1 , m 0 = bk0 . Take

m1 ’ m0

ψk (t) = .

[2Rk + t] ln(4/3)

Since Rk + d(u, Bk ) ≥ u , we see that (3.32) holds for each k.

The following is another consequence of Theorem 2.23.

Theorem 3.20. Let N be a closed subspace of a Hilbert space E, and let M = N ⊥ .

Assume that at least one of the subspaces M, N is ¬nite-dimensional. Let G be a

C 1 -functional on E such that

m 0 := sup inf G(v + w) = ’∞

(3.33)

v∈N w∈M

and

m 1 := inf sup G(v + w) = ∞.

(3.34)

w∈M v∈N

Then, for any µ > 0 and any sequence {Rk } ‚ R+ such that Rk ’ ∞, there are a

constant c ∈ R and a sequence {u k } ‚ E such that

G(u k ) ’ c, m 0 ’ µ ¤ c ¤ m 1 + µ,

(3.35)

m 1 ’ m 0 + 3µ

(Rk + u k ) G (u k ) ¤ .

ln(5/4)

Proof. We may assume dim N < ∞. Let µ > 0 be given. Then there is a u µ = v µ + wµ

such that

m 0 ’ µ < inf G(v µ + w), sup G(v + wµ ) < m 1 + µ.

w∈M v∈N

28 3. Examples of Minimax Systems

Note that ±k = Rk and ak0 ¤ m 1 , m 0 = bk0 . Take

m 1 ’ m 0 + 3µ

ψk (t) = .

[3Rk + t] ln(5/4)

Note that

2Rk

m 1 ’ m 0 + 2µ < ψk (t) dt.

Rk

We now apply Theorem 2.23. Thus, there is a sequence such that

G(u k + u µ ) ’ c, m 0 ’ µ ¤ c ¤ m 1 + µ,

and

m 1 ’ m 0 + 3µ

[3Rk + d(u k , Bk )] G (u k + u µ ) ¤ .

ln(5/4)

Since Rk + d(u, Bk ) ≥ u , we have

2Rk + d(u k , Bk ) ≥ Rk + u k ≥ u µ + u k ≥ u µ + u k

when Rk ≥ u µ . Let h k = u k + u µ . Then we have

G(h k ) ’ c, m 0 ’ µ ¤ c ¤ m 1 + µ,

m 1 ’ m 0 + 3µ

(Rk + h k ) G (h k ) ¤ .

ln(5/4)

Since µ was arbitrary, we see that (3.35) holds.

Here are some consequences.

Theorem 3.21. Let G be a C 1 -functional on E such that

a0 = sup G < ∞.

(3.36)

E

If ψ satis¬es the hypotheses of Theorem 2.4, then there is a sequence {u k } ‚ E such

that

G(u k ) ’ a0 , G (u k )/ψ( u k ) ’ 0.

(3.37)

The same holds if

b0 = inf G > ’∞,

(3.38)

E

with

G(u k ) ’ b0 , G (u k )/ψ( u k ) ’ 0.

(3.39)