Proof. It is easy to show that such a collection K is indeed a minimax system. Indeed,

if K ∈ K and • ∈ (A), then

ψ(t)u = t•(u) + (1 ’ t)u

satis¬es the stipulations above, and consequently •(K ) ∈ K. Since each of the mem-

bers of K is compact, it follows that every C 1 -functional is Lipschitz continuous on

some set K ρ de¬ned by (2.25) for ρ > 0 suf¬ciently small. Moreover, if σ (t) ∈

C(R+ — E, E) is such that σ (0) = I, one has S(K ) ∈ K, where

S(u) = σ (d(u, A))u, u ∈ E.

This follows from the fact that S(t)u = σ (td(u, A))u is in ∈ C([0, 1] — E, E) and

satis¬es

S(0)u = u, u ∈ E,

and

S(t)u = u, t ∈ [0, 1], u ∈ A.

Consequently, S(K ) = S(1)K ∈ K by hypothesis. It now follows that the conclusion

of Theorem 2.14 holds.

Note. It is not required to have a satisfy

a ¤ inf G.

B

If it does, one obtains additional information as described in [74].

3.5. Examples of linking sets 21

3.5 Examples of linking sets

We now discuss various subsets of a Banach space E with respect to linking. First we

have

Proposition 3.9. [122] Let A, B be two closed, bounded subsets of E such that E\A

is path connected. If A links B [hm], then B links A [hm].

The next proposition gives a very useful method of checking the linking of two

sets.

Proposition 3.10. [122] Let F be a continuous map from E to Rn , and let Q ‚ E

be such that F0 = F| Q is a homeomorphism of Q onto the closure of a bounded open

’1

subset of Rn . If p ∈ , then F0 (‚ ) links F ’1 ( p) [hm].

Proposition 3.11. [122] If H is a homeomorphism of E onto itself and A links B [hm],

then H A links H B [hm].

The following examples were given in [122].

Example 1. Let B be an open set in E, and let A consist of two points e1 , e2 with

/¯

e1 ∈ B and e2 ∈ B. Then A links ‚ B [hm]. ‚ B links A [hm] as well if ‚ B is bounded.

Example 2. Let M, N be closed subspaces such that dim N < ∞ and E = M • N.

Let

B R = {u ∈ E : u < R}

(3.9)

and take A = ‚ B R © N, B = M. Then A links B [hm].

Example 3. We take M, N as in Example 2. Let w0 = 0 be an element of M, and take

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

A

= ‚ Bδ © M, 0 < δ < R.

B

Then A and B link each other [hm].

Example 4. Take M, N as before and let v 0 = 0 be an element of N. We write

N = {v 0 } • N . We take

= {v ∈ N : v ¤ R} ∪ {sv 0 + v : v ∈ N , s ≥ 0, sv 0 + v = R},

A

= {w ∈ M : w ≥ δ} ∪ {sv 0 + w : w ∈ M, s ≥ 0, sv 0 + w = δ},

B

where 0 < δ < R. Then A links B [hm].

Example 5. This is the same as Example 4 with A replaced by A = ‚ B R © N.

Example 6. Let M, N be as in Example 2. Take A = ‚ Bδ © N, and let v 0 be any

element in ‚ B1 © N. Take B to be the set of all u of the form

u = w + sv 0 , w ∈ M,

22 3. Examples of Minimax Systems

satisfying any of the following:

(a) w ¤ R, s = 0

(b) w ¤ R, s = 2R0

(c) w = R, 0 ¤ s ¤ 2R0 ,

where 0 < δ < min(R, R0 ). Then A and B link each other [hm].

Example 7. Let M, N be as in Example 2. Let v 0 be in ‚ B1 © N and write N =

{v 0 } • N . Let

¯

A = ‚ Bδ © N, Q = Bδ © N,

and

B = {w ∈ M : w ¤ R} ∪ {w + sv 0 : w ∈ M, s ≥ 0, w + sv 0 = R},

where 0 < δ < R. Then A and B link each other [hm].

Example 8. Let M, N be closed subspaces of E, one of which is ¬nite-dimensional,

and such that

E = M • N.

If

B R := {u ∈ E : u < R},

then M © ‚ B R links N [hm] for each R > 0.

Example 9. Let M, N be closed subspaces of E such that

E = M • N,

with one of them being ¬nite-dimensional. Let w0 be an element of M\{0}, and let

0 < δ < r < R. Take

= {v ∈ N : δ ¤ v ¤ R} ∪ {sw0 + v : v ∈ N, s ≥ 0, sw0 + v = δ}

A

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

= ‚ Br © M, 0 < δ < r < R.

B

Then A and B link each other [hm].

Example 10. Let M, N be closed subspaces of E such that

E = M • N,

with one of them being ¬nite-dimensional. Let w0 be an element of M\{0}, and let

0 < r < R,

= {w ∈ M : w = R},

A

= {v ∈ N : v ≥ r } ∪ {u = v + sw0 : v ∈ N, s ≥ 0, u = r }.

B

Then A links B [hm].

3.5. Examples of linking sets 23

Example 11. Let M, N be as in Example 2. Take A = ‚ Bδ © N, and let v 0 be any

element in ‚ B1 © N. Take B to be the set of all u of the form

u = w + sv 0 , w ∈ M,

satisfying any of the following:

(a) s = 0

(b) s = 2R0

where 0 < δ < R0 . Then A links B [hm].

Example 12. Let M, N be as in Example 2. Take A = ‚ Bδ © N, and let v 0 be any

element in ‚ B1 © N. Take B to be the set of all u of the form

u = w + sv 0 , w ∈ M,

satisfying any of the following:

(a) w ¤ R, s = 0,

(b) w = R, s > 0,

where 0 < δ < ∞. Then A links B [hm].

Example 13. Let M be a closed subspace of a Hilbert space E with complement

N • {v 0 }, where v 0 is an element in E having unit norm, and let δ be any positive