0 ¤ •(t) ¤ 1, •(0) = 1,

and

•(t) = 0, |t| ≥ 1.

Let

2

/δ 2 )]v 0 ,

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

Assume that one of the subspaces M, N is ¬nite-dimensional. Take

A = [M • {v 0 }] © ‚ B R

and

B = {v + r v 0 : v ∈ N, r = δ•( v /δ 2 )}.

2

Then A links B [hm] provided 0 < δ < R.

‚ E, then ‚

Proposition 3.12. [122] If K is any subset of a bounded open set

links K [hm].

24 3. Examples of Minimax Systems

3.6 Various geometries

We now apply the theorems of the preceding sections to various geometries in Banach

space. As before, we assume that G ∈ C 1 (E, R) and that ψ satis¬es the hypotheses of

Theorem 2.4.

Theorem 3.13. Assume that there is a δ > 0 such that

G(0) ¤ ± ¤ G(u), u ∈ ‚ Bδ ,

(3.10)

and that there are a R0 < ∞ and a •0 ∈ ‚ B1 such that

G(R•0 ) ¤ γ , R > R0 .

(3.11)

Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a

sequence {u k } ‚ E such that

G(u k ) ’ c, ± ¤ c ¤ γ, G (u k )/ψ( u k ) ’ 0.

(3.12)

Proof. We take A = {0, R•0 }, B = ‚ Bδ . Then A = {R•0 }. Note that a given by

(2.6) is ¬nite for each R since

a R ¤ max G(r •0 ).

0¤r¤R

We apply Theorem 2.24. We note that in each case

aR ¤ γ , R > 0,

(3.13)

since the mapping

(s)u = (1 ’ s)u

(3.14)

(which is in ) satis¬es

G( (s)u) ¤ γ , 0 ¤ s ¤ 1, u ∈ A.

(3.15)

˜

This implies (3.13). We replace ψ(t) with ψ(t) = ψ(t + δ), which also satis¬es the

hypotheses of Theorem 2.4. By Theorem 2.24, we can ¬nd a sequence satisfying

˜

± ’ (1/k) ¤ G(u k ) ¤ γ + (1/k), G (u k )/ψ(d(u k , B)) ’ 0.

(3.16)

This implies (3.12), since u ¤ d(u, B) + δ.

Theorem 3.14. Let M, N be closed subspaces of E such that

E = M • N, M = E, N = E,

(3.17)

with

dim M < ∞ or dim N < ∞.

(3.18)

3.6. Various geometries 25

Let G ∈ C 1 (E, R) be such that

G(v) ¤ γ , v ∈ ‚ B R © N, R > R0 ,

(3.19)

and

G(w) ≥ ±, w ∈ M.

(3.20)

Then, for each function ψ(t) satisfying the hypotheses of Theorem 2.4, there is a

sequence {u k } ‚ E such that

G(u k ) ’ c, ± ¤ c ¤ γ, G (u k )/ψ(d(u k , M)) ’ 0.

(3.21)

Proof. This time we take A and B as in Example 2 above. Thus, A links B [hm].

Again, a R given by (2.6) is ¬nite for each R since

a R ¤ max G(u).

¯

u∈ B R ©N

Again we see that we can apply Theorem 2.24 to conclude that the desired sequence

exists.

Theorem 3.15. Let M, N be as in Theorem 3.14, and let G ∈ C 1 (E, R) satisfy

G(v) ¤ ±, v ∈ N,

(3.22)

G(w) ≥ ±, w ∈ ‚ Bδ © M,

(3.23)

G(sw0 + v) ¤ γ , s ≥ 0, v ∈ N, sw0 + v = R > R0 ,

(3.24)

for some w0 ∈ ‚ B1 © M, where 0 < δ < R0 . Then, for each function ψ(t) satisfying

the hypotheses of Theorem 2.4, there is a sequence {u k } ‚ E such that (3.12) holds.

Proof. Here we take A, B as in Example 3 above. Thus, A and B link each other [hm].

Here

A = {sw0 + v : s ≥ 0, v ∈ N, sw0 + v = R}.

Again, for each R, the quantity a given by (2.6) is ¬nite since

a R ¤ max G,

Q

where

Q = {sw0 + v : s ≥ 0, v ∈ N, sw0 + v ¤ R}.

We now apply Theorem 2.24 to conclude that the desired sequence exists.

Theorem 3.16. Let M, N be as in Theorem 3.14, and let v 0 ∈ ‚ B1 © N. Take N =

{v 0 } • N . Let G ∈ C 1 (E, R) be such that

G(v) ¤ γ , v ∈ ‚ B R © N, R > R0 ,

(3.25)

G(w) ≥ ±, w ∈ M, w ≥ δ,

(3.26)

G(sv 0 + w) ≥ ±, s ≥ 0, w ∈ M, sv 0 + w = δ,

(3.27)

where 0 < δ < R0 . Then, for each function ψ(t) satisfying the hypotheses of Theo-

rem 2.4, there is a sequence {u k } ‚ E such that (3.12) holds.

26 3. Examples of Minimax Systems

Proof. We take A, B as in Example 5 above. Thus, A links B [hm]. As before, we

note that a R < ∞ for each R. Hence, (3.12) holds by Theorem 2.24.

3.7 A sandwich theorem

We now discuss a very useful theorem that allows one to consider functionals that are

bounded from below on one subspace and bounded from above on another with no

correlation between the bounds. This provides such functionals with the same advan-

tages as those that are semibounded. One drawback is the requirement that one of the

subspaces be ¬nite-dimensional. This condition will be removed in Chapter 15 if we

assume that the functional satis¬es more than the mere continuity of its derivative.

Theorem 3.17. 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