’ 0.

ρk

2

Hence,

Hμ (t, x (k) (t)) dt/ρk ¤ 0.

2

lim sup

I

In view of (17.39), this implies that

μ

’ 1 r 2 ¤ 0,

2

contrary to hypothesis 3. Hence, the ρk are bounded, and the proof is complete.

The proof of Theorem 17.6 is similar to that of Theorem 17.5 with the exception

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

17.3. Nonconstant solutions 225

If x ∈ M, we have, by hypothesis 2A,

≥ ™ ’2 ±|x(t)|2 dt ’ C (|x|q + 1) dt

2

G(x) x

|x|<m |x|>m

’ C(1 + m 2’q + m ’q )

≥ ™ ’ 2± x |x|q dt

2 2

x

|x|>m

≥ ™ (1 ’ [2±T 2 /4π 2 ]) ’ C |x|q dt

2

x

|x|>m

q

(1 ’ [±T 2 /2π 2 ]) x 2

≥ ’C x dt

X X

I

q

(1 ’ [±T 2 /2π 2 ]) x 2

≥ ’C x

X X

q’2

= 1 ’ [±T 2 /2π 2 ] ’ C 2

x x X

X

by Lemma 17.8. Hence, we have

Lemma 17.10.

G(x) ≥ µ x X, ¤ ρ, x ∈ M,

2

(17.40) x X

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

The remainder of the proof is essentially the same, but in this case c > µ > 0,

obviating the need to consider the situation when c = 0.

In proving Theorem 17.7, we follow the proof of Theorem 17.5. In particular, it

follows that r > 0. Moreover, (17.21) and (17.23) imply that

H (t, x (k)(t)) dt ’ ’c.

(17.41)

I

On the other hand, by hypothesis 1A, we have

0 ← [x (k) ]· V (t, x (k) (t)) dt/ρk

˜ ’2

2 2

I

’2

≥ [x (k) ]· (|x (k) (t)|2 + ρk ) dt

2

˜ ’ 2C ˜

I

’ r 2 ’ 2C |x(t)|2 dt.

˜

I

Hence, x(t) ≡ 0. Let 0 ‚ I be the set on which x(t) = 0. The measure of

˜ ˜ is

0

(k) (t)| ’ ∞ as k ’ ∞ for t ∈

positive. Moreover, |x 0 . Thus,

H (t, x (k)(t)) dt ¤ H (t, x (k)(t)) dt + W (t) dt ’ ’∞

I\

I 0 0

by hypothesis 4A. But this contradicts (17.41). Hence, the ρk are bounded, and the

proof is complete.

226 17. Second-Order Periodic Systems

17.4 Notes and remarks

The periodic, nonautonomous problem

x(t) = ∇x V (t, x(t))

¨

(17.42)

has an extensive history in the case of singular systems (cf., e.g., Ambrosetti“Coti

Zelati [2]). The ¬rst to consider it for potentials satisfying (17.3) were Berger and the

author [21] in 1977. We proved the existence of solutions to (17.42) under the condition

that

V (t, x) ’ ∞ as |x| ’ ∞

uniformly for a.e. t ∈ I. Subsequently, Willem [159], Mawhin [93], Mawhin“Willem

[95], Tang [151], [152], Tang“Wu [154], [153], Wu“Tang [160] and others proved

existence under various conditions (cf. the references given in these publications).

The periodic problem (17.1) was studied by Mawhin“Willem [96],[95], Long [88],

Tang“Wu [155] and others (cf. the refernces quoted in them). Tang“Wu [155] proved

the existence of solutions of problem (17.1) under the following hypotheses:

(I) V (t, x) ’ ∞ as |x| ’ ∞

uniformly for a.e. t ∈ I,

(II) ∃ a ∈ C(R+ , R+ ), b ∈ L 1 (0, T, R+ )

such that

|V (t, x)| + |∇V (t, x)| ¤ a(|x|)b(t) ∀x ∈ Rn and a.e. t ∈ [0, T ],

and

(III) ∃ 0 < μ < 2, M >0

such that

∇V (t, x) · x ¤ μV (t, x) ∀ |x| ≥ M and a.e. t ∈ [0, T ].

Rabinowitz [103] proved existence under stronger hypotheses. In particular, he

assumed

(I ) ∃ constants a1 , a2 > 0, μ0 > 1,

such that

V (t, x) ≥ a1 |x|μ0 + a2 ∀ x ∈ Rn and a.e. t ∈ [0, T ]

in place of (I), and

(III ) ∃ 0 < μ < 2, M > 0

such that

0 < ∇V (t, x) · x ¤ μV (t, x) ∀ |x| ≥ M and a.e. t ∈ [0, T ]

in place of (III). Mawhin“Willem [96] proved existence for the case of convex

potentials, while Long [88] studied the problem for even potentials. They assumed

that V (t, x) is subquadratic in the sense that

∃ a3 < (2π/T )2 and a4

17.4. Notes and remarks 227

such that

|V (t, x)| ¤ a3 |x|2 + a4 ∀x ∈ Rn and a.e. t ∈ [0, T ].

Mawhin“Willem [95] also studied the problem for a bounded nonlinearity. Tang“Wu

[155] also proved the existence of solutions if one replaces (I) with

T

V (t, x) dt ’ ∞ as |x| ’ ∞