2

∞ ∞ ∞

1 1 1

k ’2

¤ |±k | ¤ |k±k | = ™ π /3.

2 2 22

x 2 x

∞

2π 2π 2π

k=’∞ k=’∞ k=1

This proves the lemma for the case T = 2π. Otherwise, we let y(t) = x(T t/2π). Then

T 2π

= y 2, ™ = y 2,

™ =y ∞.

2 2

x x x ∞

2π T

17.2. Proofs of the theorems 217

Thus,

2

T T T

= ¤ ™ = ™

2 2 2 2

x y y x

2π 2π 2π

and

π πT T

=y ¤ ™ = ™ = ™ .

2 2 2

x y x x

∞ ∞

6 6 2π 12

Note that it follows that

¤C x X, x ∈ X.

x ∞

We de¬ne

G(x) = x

™ ’2 V (t, x(t)) dt, x ∈ X.

2

(17.17)

I

For each x ∈ X, write x = v + w, where v ∈ N, w ∈ M. For convenience, we shall

use the following equivalent norm for X:

2 2 2

=w

™ +v .

x X

If x ∈ M and

12 2

™ = ρ2 = m,

2

x

T

¤ m, and we have by hypothesis 2 that V (t, x) ¤

then Lemma 17.8 implies that x ∞

±. Hence,

2

≥ ™ ’2 ± dt

(17.18) G(x) x

|x|<m

≥ ρ 2 ’ 2±T ≥ 0.

We also note that hypothesis 1 implies

G(v) ¤ 0, v ∈ N.

(17.19)

Take

12 2

= ‚ Bρ © M, ρ2 = m,

A

T

=

B N,

where

Bσ = {x ∈ X : x < σ }.

X

By Example 2 of Section 3.4, A links B. Moreover, if R is suf¬ciently large,

sup[’G] ¤ 0 ¤ inf[’G].

(17.20)

B

A

218 17. Second-Order Periodic Systems

Hence, we may apply Theorem 3.4 to conclude that there is a sequence {x (k) } ‚ X

such that

G(x (k) ) = x (k) V (t, x (k) (t)) dt ’ c ≥ 0,

2

™ ’2

(17.21)

I

(G (x (k) ), z)/2 = (x (k) , z ) ’ ∇x V (t, x (k) ) · z(t) dt ’ 0,

™ ™ z ∈ X,

(17.22)

I

and

(G (x (k) ), x (k) )/2 = x (k) ∇x V (t, x (k) ) · x (k) dt ’ 0.

™ ’

2

(17.23)

I

If

ρk = x (k) ¤ C,

X

then there is a renamed subsequence such that x (k) converges to a limit x ∈ X weakly

in X and uniformly on I. From (17.22), we see that

(G (x), z)/2 = (x, z ) ’

™™ ∇x V (t, x(t)) · z(t) dt = 0, z ∈ X,

I

from which we conclude easily that x is a solution of (17.1).

If

ρk = x (k) X ’ ∞,

let x (k) = x (k) /ρk . Then x (k) X = 1. Let x (k) = w(k) + v (k) , where w(k) ∈ M and

˜ ˜ ˜ ˜ ˜ ˜

(k) ∈ N. There is a renamed subsequence such that [ x (k) ]· ’ r and x (k) ’ „,

v

˜ ˜ ˜

2 + „ 2 = 1. From (17.21) and (17.23), we obtain

where r

[x (k) ]· V (t, x (k) (t)) dt/ρk ’ 0

˜ ’2

2 2

I

and