and we note that A = φ iff a0 ¤ b0 . We assume that ψ satis¬es the hypotheses of

Theorem 2.4 and

R+β

a0 ’ b0 < ψ(t) dt

(2.35)

β

holds for some ¬nite R ¤ d := d(A , B), where β = d(0, A ). Assume that A = φ.

We have

2.6. Some consequences 13

Theorem 2.22. Assume that for any b < b0 , K ∈ K, and ¬‚ow σ (t) satisfying

G(σ (t)v) > b, v ∈ K , t > 0,

(2.36)

G(σ (t)v) > b0 , v ∈ B, t > 0,

(2.37)

and

σ (t)B © A = φ, t ∈ [0, T ],

(2.38)

˜

there is a K ∈ K such that

˜ σ (t)B ∪ σ (T )[E b ∪ K ],

K‚

(2.39)

t ∈[0,T ]

where

E ± = {u ∈ E : G(u) ≥ ±}.

(2.40)

If B links A [mm] and

’∞ < b0 , a<∞

(2.41)

hold, then, for each δ > 0, there is a u ∈ E such that

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

(2.42)

2.6 Some consequences

We now discuss some methods that follow from Theorems 2.21 and 2.22. Let {Ak , Bk }

be a sequence of pairs of subsets of E. Let Kk be a minimax system for Ak . For

G ∈ C 1 (E, R), let

ak0 = sup G, bk0 = inf G,

(2.43)

Bk

Ak

and

ak = inf sup G.

(2.44)

K ∈Kk K

We assume ak < ∞ for each k. We de¬ne

Bk := {v ∈ Bk : G(v) < ak0 },

(2.45)

Ak := {u ∈ Ak : G(u) > bk0 }

(2.46)

dk := d(Ak , Bk ), dk := d(Ak , Bk ).

(2.47)

14 2. Minimax Systems

We have

Theorem 2.23. Assume that Ak links Bk [mm] for each k, that

dk ’ ∞ as k ’ ∞,

(2.48)

and for each k, there is a positive, nonincreasing function ψk (t) on [0, ∞) satisfying

the hypotheses of Theorem 2.4 and such that

Rk +±k

ak0 ’ bk0 < ψk (t)dt,

(2.49)

±k

where ±k = d(0, Bk ) and Rk ¤ dk . Then, under the hypotheses of Theorem 2.21, there

is a sequence {u k } ‚ E such that

bk0 ’ (1/k) ¤ G(u k ) ¤ ak + (1/k)

(2.50)

and

G (u k ) ¤ ψk (d(u k , Bk )).

(2.51)

Theorem 2.24. Assume that Bk links Ak [mm] for each k, that

dk ’ ∞ as k ’ ∞,

(2.52)

and that, for each k, there is a positive, nonincreasing function ψk (t) on [0, ∞) satis-

fying the hypotheses of Theorem 2.4 and such that

Rk +βk

ak0 ’ bk0 < ψk (t)dt,

(2.53)

βk

where βk = d(0, Ak ) and Rk ¤ dk . Then, under the hypotheses of Theorem 2.22, there

is a sequence {u k } ‚ E such that

bk0 ’ (1/k) ¤ G(u k ) ¤ ak + (1/k)

(2.54)

and

G (u k ) ¤ ψk (d(u k , Ak )).

(2.55)

We combine the proofs of Theorems 2.23 and 2.24.

Proof. For each k, take Rk equal to dk or dk , as the case may be. We may assume

that bk0 < ak0 for each k. Otherwise the conclusions of the theorems follow from

Theorem 2.4. We can now apply Theorems 2.21 and 2.22 for each k to conclude that

there is a u k ∈ E such that

bk0 ’ (1/k) ¤ G(u k ) ¤ ak + (1/k),

and either

G (u k ) < ψk (d(u k , Bk ))

or

G (u k ) < ψk (d(u k , Ak )),

as the case may be.

2.7. Notes and remarks 15

2.7 Notes and remarks

The concept of a minimax system is related to that of [74] (cf. Section 3.4). The ¬rst

situation that allowed a0 = a appeared in [75]. The ¬rst to consider the case ψ(t) =

1/(1 + |t|) was [39]. The ¬rst to consider the general case of arbitrary nonincreasing

ψ was [107]. The theorems of Section 2.4 are due to [107]. The theorems of Sections

2.2, 2.3, 2.5, and 2.6 are due to [134].

Chapter 3

Examples of Minimax Systems

3.1 Introduction

In this chapter we present some linking methods which are special cases of linking

with respect to minimax systems. We consider three useful methods and show that

they provide minimax systems. We then consider examples and applications.

3.2 A method using homeomorphisms

One linking method can be described as follows. Let E be a Banach space and let

be the set of all continuous maps = (t) from E — [0, 1] to E such that

1. (0) = I , the identity map.

2. For each t ∈ [0, 1), (t) is a homeomorphism of E onto E and ’1 (t) ∈ C(E —

[0, 1), E).

3. (1)E is a single point in E and (t)A converges uniformly to (1)E as t ’ 1 for

each bounded set A ‚ E.

4. For each t0 ∈ [0, 1) and each bounded set A ‚ E,

’1

sup { (t)u + (t)u } < ∞.