sequence {u k } ‚ E such that (2.9) holds.

10 2. Minimax Systems

De¬nition 2.9. A subset A of a Banach space E links a subset B of E if, for every

G ∈ C 1 (E, R) bounded on bounded sets and satisfying (2.18) there are a sequence

{u k } ‚ E and a constant a such that

b0 ¤ a < ∞

(2.19)

and (1.4) holds.

De¬nition 2.10. A subset A of a Banach space E links a subset B of E strongly if, for

every G ∈ C 1 (E, R) bounded on bounded sets and satisfying (2.18) and each positive,

nonincreasing, locally Lipschitz continuous function ψ(t) on [0, ∞) satisfying (2.8),

there is a sequence {u k } ‚ E such that (2.9) and (2.19) hold.

Theorem 2.11. If A links B [mm], then it links B strongly.

We note that Theorem 1.5 is a simple consequence of Theorem 2.4. In fact, we can

let K be the collection

K = {•(S) : • ∈ C(S, E), •(u) = u, u ∈ A}.

It is easily checked that K is a minimax system for A. Moreover, if A links B in the

old sense, then it links B [mm]. We can now apply Theorem 2.4.

In the following theorems, we allow a = a0 .

Theorem 2.12. Let K be a minimax system for a nonempty set A, and assume that A

links a subset B of E relative to K. Let G be a C 1 -functional satisfying (2.10), (2.11),

and

a0 := sup G ¤ b0 := inf G.

(2.20)

B

A

Assume that for any b > a, K ∈ K, and ¬‚ow σ (t) satisfying (2.12), (2.13) and such

that

σ (t)A © B = φ, t ∈ [0, T ],

(2.21)

˜

there is a K ∈ K such that (2.14) holds. Then the conclusions of Theorem 2.4 hold.

Moreover, if a0 = a, then there is a sequence satisfying (1.4) and

d(u k , B) ’ 0, k ’ ∞.

(2.22)

Theorem 2.13. Let K be a minimax system for a nonempty set A satisfying

K \A = φ, K ∈ K.

(2.23)

Let G ∈ C 1 satisfy (2.10) and (2.11). Assume that for any K ∈ K and ¬‚ow σ (t)

satisfying (2.12), (2.13), one has S(K ) ∈ K, where

S(u) = σ (d(u, A))u, u ∈ E.

(2.24)

Assume also that for each K ∈ K, there is a ρ > 0 such that G(u) is Lipschitz continu-

ous on

K ρ = {u ∈ K : d(u, A) ¤ ρ}.

(2.25)

Then the conclusions of Theorem 2.4 hold.

2.4. A variation 11

Theorem 2.14. Let K be a minimax system for a nonempty set A, and assume that there

is a subset B of E such that A links B relative to K. Assume that for any K ∈ K and ¬‚ow

σ (t) satisfying (2.12), (2.13), one has S(K ) ∈ K, where S(u) is given by (2.24). Let

G be a C 1 -functional satisfying (2.11) and (2.20). Assume also that for each K ∈ K

there is a ρ > 0 such that G(u) is Lipschitz continuous on K ρ given by (2.25). Then

the conclusions of Theorem 2.4 hold.

Corollary 2.15. Let K be a minimax system for a nonempty set A, and let G ∈ C 1

satisfy (2.10) and (2.11). Assume that for any b > a, K ∈ K, and ¬‚ow σ (t) satisfying

(2.12), (2.13), and (2.21) for

B = {u ∈ E\A : G(u) ≥ a0 },

(2.26)

˜ ˜

there is a K ∈ K satisfying K ‚ σ (T )E b . Then the conclusions of Theorem 2.4 hold.

2.4 A variation

Theorem 2.16. Let K be a minimax system for a nonempty subset A of E, and let G(u)

be a C 1 -functional on E. De¬ne

(2.27) b := sup inf G,

K ∈K K

and assume that b is ¬nite and satis¬es

b < b0 := inf G.

(2.28)

A

Then the conclusions of Theorem 2.4 hold with a replaced by b.

We also have the following. Let A = {0}. Then a minimax system for A can

be constructed as follows. K consists of the boundaries of all bounded, open sets ω

containing 0. That K is a minimax system for A follows from the fact that 0 ∈ •(ω)

and ‚•(ω) = •(‚ω) for all such sets and • ∈ (A). We have

Theorem 2.17. Let b be de¬ned by (2.27). Assume that G(0) < b < ∞. Then there is

a sequence satisfying

G(u k ) ’ b, G (u k )/ψ( u k ) ’ 0.

(2.29)

Theorem 2.18. Assume that b < ∞ and that there is an open set ω0 containing 0 such

that

G(0) = inf G.

‚ω0

Then there is a sequence satisfying (2.29).

We also have the counterpart of these theorems for the quantity a given by (2.6).

Theorem 2.19. Assume that ’∞ < a < G(0). Then the conclusions of Theorem 2.4

hold.

12 2. Minimax Systems

Theorem 2.20. Assume that a > ’∞ and that there is an open set ω0 containing 0

such that

G(0) = sup G.

‚ω0

Then the conclusions of Theorem 2.4 hold.

2.5 Weaker conditions

We now turn to the question as to what happens if some of the hypotheses of Theo-

rem 2.12 do not hold. We are particularly interested in what happens when (2.20) is

violated. In this case we let

B := {v ∈ B : G(v) < a0 }.

(2.30)

Note that

B = φ iff a0 ¤ b0 .

We assume that B = φ. Let ψ(t) be a positive, nonincreasing function on [0, ∞)

satisfying the hypotheses of Theorem 2.4 and such that

R+±

a0 ’ b0 < ψ(t) dt

(2.31)

±

for some ¬nite R ¤ d := d(B , A), where ± = d(0, B ). We assume d > 0. We have

Theorem 2.21. Let G be a C 1 -functional on E and A, B ‚ E be such that A links B

[mm] and

’∞ < b0 , a < ∞.

(2.32)

Assume that for any b > a, K ∈ K and ¬‚ow σ (t) satisfying (2.12), (2.13), (2.21), there

˜

is a K ∈ K satisfying (2.14). Under the hypotheses given above, for each δ > 0, there

is a u ∈ E such that

b0 ’ δ ¤ G(u) ¤ a + δ, G (u) < ψ(d(u, B )).

(2.33)

We can also consider a slightly different version of Theorem 2.21. We consider

the set

A := {u ∈ A : G(u) > b0 },