2 2

˜ ’

I

Thus,

V (t, x (k) (t)) dt/ρk ’ r 2

2

(17.24) 2

I

and

∇x V (t, x (k) ) · x (k) dt/ρk ’ r 2 .

2

(17.25)

I

Hence,

μ

Hμ (t, x (k) (t)) dt/ρk ’ ’ 1 r 2.

2

(17.26)

2

I

17.2. Proofs of the theorems 219

Note that

|x (k) (t)| ¤ C x (k)

˜ ˜ = C.

X

If

|x (k) (t)| ’ ∞,

then, by hypothesis 3,

Hμ (t, x (k) (t)) Hμ (t, x (k) (t)) (k) 2

¤ lim sup |x (t)| ¤ 0.

˜

lim sup

|x (k) (t)|2

ρk

2

If

|x (k) (t)| ¤ C,

then

Hμ (t, x (k) (t))

’ 0.

ρk2

Hence,

Hμ (t, x (k) (t)) dt/ρk ¤ 0.

2

lim sup

I

Thus, by (17.26),

μ

’ 1 r 2 ¤ 0.

2

If r = 0, this contradicts the fact that μ > 2. If r = 0, then w(k) ’ 0 uniformly

˜

(k) |2 = v (k) 2 ’ 1. Thus, there is a renamed

in I by Lemma 17.8. Moreover, T |v ˜ ˜

(k) ’ v in N with |v|2 = 1/T. Hence, x (k) ’ v uniformly in

subsequence such that v ˜ ˜ ˜ ˜ ˜

(k) | ’ ∞ uniformly in I. Thus, by hypothesis 4,

I. Consequently, |x

V (t, x (k) (t)) (k) 2

(k)

V (t, x (t)) dt/ρk ≥ lim inf |x (t)| dt > 0.

˜

2

lim inf

|x (k) (t)|2

I e

This contradicts (17.24). Hence, the ρk are bounded, and the proof is complete.

The proof of Theorem 17.2 is similar to that of Theorem 17.1 with the exception

of inequality (17.18) resulting from hypothesis 2. In its place we reason as follows:

If x ∈ M, we have, by hypothesis 2A,

2

±|x(t)|2 dt ’ C (|x|q + 1) dt

G(x) ≥ ™ ’2

x

|x|<m |x|>m

’ C(1 + m 2’q + m ’q )

2 2

|x|q dt

≥ ™ ’ 2± x

x

|x|>m

≥ ™ (1 ’ [2±T 2 /4π 2 ]) ’ C |x|q dt

2

x

|x|>m

q

≥ (1 ’ [±T 2 /2π 2 ]) x ’C

2

x dt

X X

I

220 17. Second-Order Periodic Systems

q

≥ (1 ’ [±T 2 /2π 2 ]) x ’C

2

x

X X

q’2

= 1 ’ [±T 2 /2π 2 ] ’ C 2

x x X

X

by Lemma 17.8. Hence, we have

Lemma 17.9.

G(x) ≥ µ x X, ¤ ρ, x ∈ M,

2

(17.27) x X

for ρ > 0 suf¬ciently small, where µ < 1 ’ [±T 2 /2π 2 ].

The remainder of the proof is essentially the same.

In proving Theorem 17.3, we follow the proof of Theorem 17.1 until we reach

(17.26). Then we reason as follows. If

|x (k) (t)| ’ ∞,

then

Hμ (t, x (k) (t)) Hμ (t, x (k) (t)) (k) 2

≥ lim inf |x (t)| ≥ 0.

˜

lim inf

|x (k) (t)|2

ρk

2

If

|x (k) (t)| ¤ C,

then

Hμ (t, x (k) (t))