implies G(u k ) ’ 0, showing that a = 0. Thus, A does not link B.

6.4 Notes and remarks

The results of this chapter are from [138]. For related material cf. [137] and [156].

Chapter 7

Sandwich Pairs

7.1 Introduction

In this chapter we discuss the situation in which one cannot ¬nd linking sets that

separate a functional G, i.e., satisfy

a0 := sup G ¤ b0 := inf G.

(7.1)

B

A

Are there weaker conditions that will imply the existence of a PS sequence

G(u k ) ’ a, G (u k ) ’ 0?

(7.2)

Our answer is yes, and we ¬nd pairs of subsets such that the absence of (7.1) produces

a PS sequence. We have

De¬nition 7.1. We shall say that a pair of subsets A, B of a Banach space E forms a

sandwich if, for any G ∈ C 1 (E, R), the inequality

’∞ < b0 := inf G ¤ a0 := sup G < ∞

(7.3)

B A

implies that there is a sequence satisfying

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

(7.4)

Unlike linking, the order of a sandwich pair is immaterial; i.e., if the pair A, B

forms a sandwich, so does B, A. Moreover, we allow sets forming a sandwich pair to

intersect. One sandwich pair has been studied in Chapter 3. In fact, Theorem 3.17 tells

us that if M, N are closed subspaces of a Hilbert space E, and M = N ⊥ , then M, N

form a sandwich pair if one of them is ¬nite-dimensional.

Until recently, only complementing subspaces have been considered. The purpose

of the present chapter is to show that other sets can qualify as well. In¬nite-dimensional

sandwich pairs will be considered in Chapter 15.

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_7,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

58 7. Sandwich Pairs

7.2 Criteria

In this section we present suf¬cient conditions for sets to qualify as sandwich pairs.

We have

Proposition 7.2. If A, B is a sandwich pair and J is a diffeomorphism on the entire

space having a derivative J satisfying

J (u)’1 ¤ C, u ∈ E,

(7.5)

then JA, JB is a sandwich pair.

Proof. Suppose G ∈ C 1 satis¬es

’∞ < b0 := inf G ¤ a0 := sup G < ∞.

(7.6)

JB JA

Let

G 1 (u) = G(J u), u ∈ E.

Then

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

(7.7)

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 sandwich pair, there is a sequence {h k } ‚ E such that

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

(7.8)

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

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

(7.9)

In view of (7.5), this implies G (u k ) ’ 0. Thus, J A, J B is a sandwich pair.

Proposition 7.3. 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

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

Assume that one of the subspaces M, N is ¬nite-dimensional. Then A = N = N •

{v 0 }, B = F ’1 (δv 0 ) form a sandwich pair.

7.2. Criteria 59

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 with its inverse having a derivative satisfying (7.5).

Moreover, J A = N and J B = M +δv 0 . Hence, J A, J B form a sandwich pair as long

as one of them is ¬nite-dimensional (Theorem 3.17). We now apply Proposition 7.2.

Theorem 7.4. Let N be a ¬nite dimensional subspace of a Banach space E. Let F be a

Lipschitz continuous map of E onto N such that F = I on N and

F(g) ’ F(h) ¤ K g ’ h , g, h ∈ E.

(7.11)

Let p be any point of N. Then A = N, B = F ’1 ( p) form a sandwich pair.

Proof. Let G be a C 1 -functional on E satisfying (7.3), where A, B are the subsets of

E speci¬ed in the theorem. If the theorem is not true, then there is a δ > 0 such that

G (u) ≥ 3δ

(7.12)

whenever

b0 ’ 3δ ¤ G(u) ¤ a0 + 3δ.

(7.13)

Since G ∈ C 1 (E, R), there is a locally Lipschitz continuous mapping Y (u) of

ˆ

E = {u ∈ E : G (u) = 0} into E such that

ˆ

Y (u) ¤ 1, u ∈ E,

and

(G (u), Y (u)) ≥ 2δ

whenever u satis¬es (7.13) (for the construction of such a map, cf., e.g., [112]). Let

= {u ∈ E : b0 ’ 2δ ¤ G(u) ¤ a0 + 2δ},

Q0

= {u ∈ E : b0 ’ δ ¤ G(u) ¤ a0 + δ},

Q1

= E\Q 0 ,

Q2

·(u) = ρ(u, Q 2 )/[ρ(u, Q 1 ) + ρ(u, Q 2 )].