(17.6)

4. There is a subset e ‚ I of positive measure such that

V (t, x)

> 0, t ∈ e.

(17.7) lim inf

|x|2

|x|’∞

We have

Theorem 17.1. Under the above hypotheses, system (17.1) has a solution.

As a variant of Theorem 17.1, we have

Theorem 17.2. The conclusion in Theorem 17.1 is the same if we replace hypothesis

2 with

2A. There is a constant q > 2 such that

V (t, x) ¤ C(|x|q + 1), t ∈ I, x ∈ Rn ,

and there are constants m > 0, ± < 2π 2 /T 2 such that

V (t, x) ¤ ±|x|2 , |x| ¤ m, t ∈ I, x ∈ Rn .

We also have

Theorem 17.3. The conclusion of Theorem 17.1 holds if we replace hypothesis 3 with

3A. There is a constant μ < 2 such that

Hμ (t, x)

≥ ’W (t) ∈ L 1 (I ), |x| ≥ C, t ∈ I, x ∈ Rn ,

(17.8)

|x| 2

and

Hμ (t, x)

≥ 0.

(17.9) lim inf

|x|2

|x|’∞

And we have

Theorem 17.4. The conclusion of Theorem 17.1 holds if we replace hypothesis 1 with

1A.

0 ¤ V (t, x) ¤ C(|x|2 + 1), t ∈ I, x ∈ Rn .

and hypothesis 3 with

17.2. Proofs of the theorems 215

3B. The function given by

H (t, x) = 2V (t, x) ’ ∇x V (t, x) · x

(17.10)

satis¬es

H (t, x) ¤ W (t) ∈ L 1 (I ), |x| ≥ C, t ∈ I, x ∈ Rn ,

(17.11)

and

H (t, x) ’ ’∞, |x| ’ ∞, t ∈ I, x ∈ Rn .

(17.12)

Theorems 17.1“17.4 show the existence of solutions, which conceivably could be

constants. The following theorems provide the existence of non-constant solutions.

Theorem 17.5. If we replace hypothesis 4 in Theorem 17.1 with

4A. There are constants β > 2π 2 /T 2 and C such that

V (t, x) ≥ β|x|2 , |x| > C, t ∈ I, x ∈ Rn ,

then system (17.1) has a nonconstant solution.

As a variant of Theorem 17.5, we have

Theorem 17.6. The conclusion in Theorem 17.5 is the same if we replace hypothesis

2 with hypothesis 2A.

We also have

Theorem 17.7. The conclusion of Theorem 17.5 holds if we replace hypothesis 1 with

hypothesis 1A and hypothesis 3 with hypothesis 3B.

We shall prove Theorems 17.1“17.7 in the next section. We use the linking method

of Chapter 2.

17.2 Proofs of the theorems

We now give the proof of Theorem 17.1.

Proof. Let X be the set of vector functions x(t) described above. It is a Hilbert space

with norm satisfying

n

= .

2 2

x xj

X H1

j =1

We also write

n

2 2

= ,

x xj

j =1

where · is the L 2 (I ) norm.

216 17. Second-Order Periodic Systems

Let

N = {x(t) ∈ X : x j (t) ≡ constant, 1 ¤ j ¤ n},

and M = N ⊥ . The dimension of N is n, and X = M • N. The following is known

(cf., e.g., Proposition 1.3 of [95]).

Lemma 17.8. If x ∈ M, then

T

¤ ™

2 2

x x

∞

12

and

T

x¤ x.

™

2π

Proof. It suf¬ces to prove the lemma for continuously differential periodic functions.

First, consider the case T = 2π. Using Fourier series, we have

∞

x= ±k •k ,

(17.13)

k=’∞

where

±k = (x, •k ),

¯ k = 0, ±1, ±2, . . . ,

(17.14)

and

1

•k (x) = √ eikx , k = 0, ±1, ±2, . . .

(17.15)

2π

Thus,

∞

2

|±k |2 .

=

(17.16) x

k=’∞

If x ⊥ M, then ±0 = 0, and

∞

2

|k±k |2 = x 2

¤ ™ .

x

k=’∞